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<!DOCTYPE html> <html lang="en"> <head> <meta content="text/html; charset=utf-8" http-equiv="content-type"/> <title>Complex interpolation of weighted Sobolev spaces with boundary conditions</title>  <meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport"/> <link href="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/css/bootstrap.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/ar5iv-fonts.0.7.9.min.css" rel="stylesheet" type="text/css"/> <link href="/static/browse/0.3.4/css/latexml_styles.css" rel="stylesheet" type="text/css"/> <script src="https://cdn.jsdelivr.net/npm/bootstrap@5.3.0/dist/js/bootstrap.bundle.min.js"></script> <script src="https://cdnjs.cloudflare.com/ajax/libs/html2canvas/1.3.3/html2canvas.min.js"></script> <script src="/static/browse/0.3.4/js/addons_new.js"></script> <script src="/static/browse/0.3.4/js/feedbackOverlay.js"></script> <meta content="" lang="en" name="keywords"/> <base href="/html/2503.14636v1/"/></head> <body> <nav class="ltx_page_navbar"> <nav class="ltx_TOC"> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.SS1" title="In 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.1 </span>Notation</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.SS2" title="In 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.2 </span>Weighted function spaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.SS3" title="In 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.3 </span>Properties of Banach spaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.SS4" title="In 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.4 </span>Embeddings</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.SS5" title="In 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2.5 </span>The complex interpolation method</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Trace spaces of Besov and Triebel-Lizorkin spaces</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.SS1" title="In 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.1 </span>The zeroth-order trace operator</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.SS2" title="In 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3.2 </span>The higher-order trace operator</span></a></li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Trace spaces of Bessel potential and Sobolev spaces</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS1" title="In 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.1 </span>Trace spaces of weighted spaces on the half-space</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS2" title="In 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2 </span>Density results</span></a> <ol class="ltx_toclist ltx_toclist_subsection"> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS2.SSS1" title="In 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2.1 </span>Bessel potential spaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsubsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS2.SSS2" title="In 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4.2.2 </span>Sobolev spaces</span></a></li> </ol> </li> </ol> </li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Trace theorem for boundary operators</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"> <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Complex interpolation</span></a> <ol class="ltx_toclist ltx_toclist_section"> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS1" title="In 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.1 </span>Interpolation results for weighted spaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS2" title="In 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.2 </span>Main results about complex interpolation of weighted spaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_subsection"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS3" title="In 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6.3 </span>The proofs of Theorems <span class="ltx_text ltx_ref_tag">6.4</span> and <span class="ltx_text ltx_ref_tag">6.5</span></span></a></li> </ol> </li> </ol></nav> </nav> <div class="ltx_page_main"> <div class="ltx_page_content"> <article class="ltx_document ltx_authors_1line" lang="en"> <h1 class="ltx_title ltx_title_document">Complex interpolation of weighted Sobolev spaces with boundary conditions</h1> <div class="ltx_authors"> <span class="ltx_creator ltx_role_author"> <span class="ltx_personname">Floris B. Roodenburg </span><span class="ltx_author_notes"> <span class="ltx_contact ltx_role_address">Delft Institute of Applied Mathematics <br class="ltx_break"/>Delft University of Technology <br class="ltx_break"/>P.O. Box 5031 <br class="ltx_break"/>2600 GA Delft <br class="ltx_break"/>The Netherlands </span> <span class="ltx_contact ltx_role_email"><a href="mailto:f.b.roodenburg@tudelft.nl">f.b.roodenburg@tudelft.nl</a> </span></span></span> </div> <div class="ltx_abstract"> <h6 class="ltx_title ltx_title_abstract">Abstract.</h6> <p class="ltx_p" id="id1.1"><span class="ltx_text" id="id1.1.1">We characterise the complex interpolation spaces of weighted vector-valued Sobolev spaces with and without boundary conditions on the half-space and on smooth bounded domains. The weights we consider are power weights that measure the distance to the boundary and do not necessarily belong to the class of Muckenhoupt <math alttext="A_{p}" class="ltx_Math" display="inline" id="id1.1.1.m1.1"><semantics id="id1.1.1.m1.1a"><msub id="id1.1.1.m1.1.1" xref="id1.1.1.m1.1.1.cmml"><mi id="id1.1.1.m1.1.1.2" xref="id1.1.1.m1.1.1.2.cmml">A</mi><mi id="id1.1.1.m1.1.1.3" xref="id1.1.1.m1.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="id1.1.1.m1.1b"><apply id="id1.1.1.m1.1.1.cmml" xref="id1.1.1.m1.1.1"><csymbol cd="ambiguous" id="id1.1.1.m1.1.1.1.cmml" xref="id1.1.1.m1.1.1">subscript</csymbol><ci id="id1.1.1.m1.1.1.2.cmml" xref="id1.1.1.m1.1.1.2">𝐴</ci><ci id="id1.1.1.m1.1.1.3.cmml" xref="id1.1.1.m1.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.1.1.m1.1c">A_{p}</annotation><annotation encoding="application/x-llamapun" id="id1.1.1.m1.1d">italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> weights. First, we determine the higher-order trace spaces for weighted vector-valued Besov, Triebel-Lizorkin, Bessel potential and Sobolev spaces. This allows us to derive a trace theorem for boundary operators and to interpolate spaces with boundary conditions. Furthermore, we derive density results for weighted Sobolev spaces with boundary conditions.</span></p> </div> <div class="ltx_keywords"> <h6 class="ltx_title ltx_title_keywords">Key words and phrases: </h6> <span class="ltx_text" id="id3.id1">Complex interpolation, weighted function spaces, vector-valued function spaces, Besov spaces, Triebel-Lizorkin spaces, Bessel potential spaces, Sobolev spaces, traces, boundary operators</span> </div> <div class="ltx_classification"> <h6 class="ltx_title ltx_title_classification">2020 Mathematics Subject Classification: </h6> <span class="ltx_text" id="id4.id1">Primary: 46B70, 46E35; Secondary: 46E40</span> </div> <div class="ltx_acknowledgements"> <span class="ltx_text" id="id5.id1">The author is supported by the VICI grant VI.C.212.027 of the Dutch Research Council (NWO). The author thanks Emiel Lorist and Mark Veraar for valuable comments and suggestions</span> </div> <nav class="ltx_TOC ltx_list_toc ltx_toc_toc"><h6 class="ltx_title ltx_title_contents">Contents</h6> <ol class="ltx_toclist"> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">1 </span>Introduction</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">2 </span>Preliminaries</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">3 </span>Trace spaces of Besov and Triebel-Lizorkin spaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">4 </span>Trace spaces of Bessel potential and Sobolev spaces</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">5 </span>Trace theorem for boundary operators</span></a></li> <li class="ltx_tocentry ltx_tocentry_section"><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6" title="In Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_title"><span class="ltx_tag ltx_tag_ref">6 </span>Complex interpolation</span></a></li> </ol></nav> <section class="ltx_section" id="S1"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">1. </span>Introduction</h2> <div class="ltx_para" id="S1.p1"> <p class="ltx_p" id="S1.p1.3">This paper aims to characterise complex interpolation spaces of weighted Sobolev spaces on a domain <math alttext="\mathcal{O}" class="ltx_Math" display="inline" id="S1.p1.1.m1.1"><semantics id="S1.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p1.1.m1.1.1" xref="S1.p1.1.m1.1.1.cmml">𝒪</mi><annotation-xml encoding="MathML-Content" id="S1.p1.1.m1.1b"><ci id="S1.p1.1.m1.1.1.cmml" xref="S1.p1.1.m1.1.1">𝒪</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.1.m1.1c">\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.1.m1.1d">caligraphic_O</annotation></semantics></math> with power weights <math alttext="w^{\partial\mathcal{O}}_{\gamma}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}" class="ltx_math_unparsed" display="inline" id="S1.p1.2.m2.3"><semantics id="S1.p1.2.m2.3a"><mrow id="S1.p1.2.m2.3b"><msubsup id="S1.p1.2.m2.3.4"><mi id="S1.p1.2.m2.3.4.2.2">w</mi><mi id="S1.p1.2.m2.3.4.3">γ</mi><mrow id="S1.p1.2.m2.3.4.2.3"><mo id="S1.p1.2.m2.3.4.2.3.1" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.2.m2.3.4.2.3.2">𝒪</mi></mrow></msubsup><mrow id="S1.p1.2.m2.3.5"><mo id="S1.p1.2.m2.3.5.1" stretchy="false">(</mo><mi id="S1.p1.2.m2.1.1">x</mi><mo id="S1.p1.2.m2.3.5.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S1.p1.2.m2.3.6" rspace="0.278em">:=</mo><mi id="S1.p1.2.m2.2.2">dist</mi><msup id="S1.p1.2.m2.3.7"><mrow id="S1.p1.2.m2.3.7.2"><mo id="S1.p1.2.m2.3.7.2.1" stretchy="false">(</mo><mi id="S1.p1.2.m2.3.3">x</mi><mo id="S1.p1.2.m2.3.7.2.2">,</mo><mo id="S1.p1.2.m2.3.7.2.3" lspace="0em" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S1.p1.2.m2.3.7.2.4">𝒪</mi><mo id="S1.p1.2.m2.3.7.2.5" stretchy="false">)</mo></mrow><mi id="S1.p1.2.m2.3.7.3">γ</mi></msup></mrow><annotation encoding="application/x-tex" id="S1.p1.2.m2.3c">w^{\partial\mathcal{O}}_{\gamma}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S1.p1.2.m2.3d">italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_x ) := roman_dist ( italic_x , ∂ caligraphic_O ) start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S1.p1.3.m3.1"><semantics id="S1.p1.3.m3.1a"><mrow id="S1.p1.3.m3.1.1" xref="S1.p1.3.m3.1.1.cmml"><mi id="S1.p1.3.m3.1.1.2" xref="S1.p1.3.m3.1.1.2.cmml">γ</mi><mo id="S1.p1.3.m3.1.1.1" xref="S1.p1.3.m3.1.1.1.cmml">></mo><mrow id="S1.p1.3.m3.1.1.3" xref="S1.p1.3.m3.1.1.3.cmml"><mo id="S1.p1.3.m3.1.1.3a" xref="S1.p1.3.m3.1.1.3.cmml">−</mo><mn id="S1.p1.3.m3.1.1.3.2" xref="S1.p1.3.m3.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p1.3.m3.1b"><apply id="S1.p1.3.m3.1.1.cmml" xref="S1.p1.3.m3.1.1"><gt id="S1.p1.3.m3.1.1.1.cmml" xref="S1.p1.3.m3.1.1.1"></gt><ci id="S1.p1.3.m3.1.1.2.cmml" xref="S1.p1.3.m3.1.1.2">𝛾</ci><apply id="S1.p1.3.m3.1.1.3.cmml" xref="S1.p1.3.m3.1.1.3"><minus id="S1.p1.3.m3.1.1.3.1.cmml" xref="S1.p1.3.m3.1.1.3"></minus><cn id="S1.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S1.p1.3.m3.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p1.3.m3.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S1.p1.3.m3.1d">italic_γ > - 1</annotation></semantics></math> measuring the distance to the boundary. The motivation for this stems from the study of partial differential equations (PDEs) and is twofold.</p> </div> <div class="ltx_para" id="S1.p2"> <p class="ltx_p" id="S1.p2.6">First, it is well known that complex interpolation serves as an essential tool for studying differential operators and evolution equations, see, e.g., the monographs <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib6" title="">6</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib55" title="">55</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib61" title="">61</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib72" title="">72</a>]</cite>. For example, if a sectorial operator <math alttext="A" class="ltx_Math" display="inline" id="S1.p2.1.m1.1"><semantics id="S1.p2.1.m1.1a"><mi id="S1.p2.1.m1.1.1" xref="S1.p2.1.m1.1.1.cmml">A</mi><annotation-xml encoding="MathML-Content" id="S1.p2.1.m1.1b"><ci id="S1.p2.1.m1.1.1.cmml" xref="S1.p2.1.m1.1.1">𝐴</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.1.m1.1c">A</annotation><annotation encoding="application/x-llamapun" id="S1.p2.1.m1.1d">italic_A</annotation></semantics></math> on a Banach space <math alttext="X" class="ltx_Math" display="inline" id="S1.p2.2.m2.1"><semantics id="S1.p2.2.m2.1a"><mi id="S1.p2.2.m2.1.1" xref="S1.p2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.p2.2.m2.1b"><ci id="S1.p2.2.m2.1.1.cmml" xref="S1.p2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.p2.2.m2.1d">italic_X</annotation></semantics></math> has bounded imaginary powers, then the domain of <math alttext="A^{\theta}" class="ltx_Math" display="inline" id="S1.p2.3.m3.1"><semantics id="S1.p2.3.m3.1a"><msup id="S1.p2.3.m3.1.1" xref="S1.p2.3.m3.1.1.cmml"><mi id="S1.p2.3.m3.1.1.2" xref="S1.p2.3.m3.1.1.2.cmml">A</mi><mi id="S1.p2.3.m3.1.1.3" xref="S1.p2.3.m3.1.1.3.cmml">θ</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p2.3.m3.1b"><apply id="S1.p2.3.m3.1.1.cmml" xref="S1.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p2.3.m3.1.1.1.cmml" xref="S1.p2.3.m3.1.1">superscript</csymbol><ci id="S1.p2.3.m3.1.1.2.cmml" xref="S1.p2.3.m3.1.1.2">𝐴</ci><ci id="S1.p2.3.m3.1.1.3.cmml" xref="S1.p2.3.m3.1.1.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.3.m3.1c">A^{\theta}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.3.m3.1d">italic_A start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="\theta\in(0,1)" class="ltx_Math" display="inline" id="S1.p2.4.m4.2"><semantics id="S1.p2.4.m4.2a"><mrow id="S1.p2.4.m4.2.3" xref="S1.p2.4.m4.2.3.cmml"><mi id="S1.p2.4.m4.2.3.2" xref="S1.p2.4.m4.2.3.2.cmml">θ</mi><mo id="S1.p2.4.m4.2.3.1" xref="S1.p2.4.m4.2.3.1.cmml">∈</mo><mrow id="S1.p2.4.m4.2.3.3.2" xref="S1.p2.4.m4.2.3.3.1.cmml"><mo id="S1.p2.4.m4.2.3.3.2.1" stretchy="false" xref="S1.p2.4.m4.2.3.3.1.cmml">(</mo><mn id="S1.p2.4.m4.1.1" xref="S1.p2.4.m4.1.1.cmml">0</mn><mo id="S1.p2.4.m4.2.3.3.2.2" xref="S1.p2.4.m4.2.3.3.1.cmml">,</mo><mn id="S1.p2.4.m4.2.2" xref="S1.p2.4.m4.2.2.cmml">1</mn><mo id="S1.p2.4.m4.2.3.3.2.3" stretchy="false" xref="S1.p2.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.4.m4.2b"><apply id="S1.p2.4.m4.2.3.cmml" xref="S1.p2.4.m4.2.3"><in id="S1.p2.4.m4.2.3.1.cmml" xref="S1.p2.4.m4.2.3.1"></in><ci id="S1.p2.4.m4.2.3.2.cmml" xref="S1.p2.4.m4.2.3.2">𝜃</ci><interval closure="open" id="S1.p2.4.m4.2.3.3.1.cmml" xref="S1.p2.4.m4.2.3.3.2"><cn id="S1.p2.4.m4.1.1.cmml" type="integer" xref="S1.p2.4.m4.1.1">0</cn><cn id="S1.p2.4.m4.2.2.cmml" type="integer" xref="S1.p2.4.m4.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.4.m4.2c">\theta\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.4.m4.2d">italic_θ ∈ ( 0 , 1 )</annotation></semantics></math> is the complex interpolation space <math alttext="[X,D(A)]_{\theta}" class="ltx_Math" display="inline" id="S1.p2.5.m5.3"><semantics id="S1.p2.5.m5.3a"><msub id="S1.p2.5.m5.3.3" xref="S1.p2.5.m5.3.3.cmml"><mrow id="S1.p2.5.m5.3.3.1.1" xref="S1.p2.5.m5.3.3.1.2.cmml"><mo id="S1.p2.5.m5.3.3.1.1.2" stretchy="false" xref="S1.p2.5.m5.3.3.1.2.cmml">[</mo><mi id="S1.p2.5.m5.2.2" xref="S1.p2.5.m5.2.2.cmml">X</mi><mo id="S1.p2.5.m5.3.3.1.1.3" xref="S1.p2.5.m5.3.3.1.2.cmml">,</mo><mrow id="S1.p2.5.m5.3.3.1.1.1" xref="S1.p2.5.m5.3.3.1.1.1.cmml"><mi id="S1.p2.5.m5.3.3.1.1.1.2" xref="S1.p2.5.m5.3.3.1.1.1.2.cmml">D</mi><mo id="S1.p2.5.m5.3.3.1.1.1.1" xref="S1.p2.5.m5.3.3.1.1.1.1.cmml">⁢</mo><mrow id="S1.p2.5.m5.3.3.1.1.1.3.2" xref="S1.p2.5.m5.3.3.1.1.1.cmml"><mo id="S1.p2.5.m5.3.3.1.1.1.3.2.1" stretchy="false" xref="S1.p2.5.m5.3.3.1.1.1.cmml">(</mo><mi id="S1.p2.5.m5.1.1" xref="S1.p2.5.m5.1.1.cmml">A</mi><mo id="S1.p2.5.m5.3.3.1.1.1.3.2.2" stretchy="false" xref="S1.p2.5.m5.3.3.1.1.1.cmml">)</mo></mrow></mrow><mo id="S1.p2.5.m5.3.3.1.1.4" stretchy="false" xref="S1.p2.5.m5.3.3.1.2.cmml">]</mo></mrow><mi id="S1.p2.5.m5.3.3.3" xref="S1.p2.5.m5.3.3.3.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p2.5.m5.3b"><apply id="S1.p2.5.m5.3.3.cmml" xref="S1.p2.5.m5.3.3"><csymbol cd="ambiguous" id="S1.p2.5.m5.3.3.2.cmml" xref="S1.p2.5.m5.3.3">subscript</csymbol><interval closure="closed" id="S1.p2.5.m5.3.3.1.2.cmml" xref="S1.p2.5.m5.3.3.1.1"><ci id="S1.p2.5.m5.2.2.cmml" xref="S1.p2.5.m5.2.2">𝑋</ci><apply id="S1.p2.5.m5.3.3.1.1.1.cmml" xref="S1.p2.5.m5.3.3.1.1.1"><times id="S1.p2.5.m5.3.3.1.1.1.1.cmml" xref="S1.p2.5.m5.3.3.1.1.1.1"></times><ci id="S1.p2.5.m5.3.3.1.1.1.2.cmml" xref="S1.p2.5.m5.3.3.1.1.1.2">𝐷</ci><ci id="S1.p2.5.m5.1.1.cmml" xref="S1.p2.5.m5.1.1">𝐴</ci></apply></interval><ci id="S1.p2.5.m5.3.3.3.cmml" xref="S1.p2.5.m5.3.3.3">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.5.m5.3c">[X,D(A)]_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S1.p2.5.m5.3d">[ italic_X , italic_D ( italic_A ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib64" title="">64</a>]</cite>. Typically, <math alttext="D(A)" class="ltx_Math" display="inline" id="S1.p2.6.m6.1"><semantics id="S1.p2.6.m6.1a"><mrow id="S1.p2.6.m6.1.2" xref="S1.p2.6.m6.1.2.cmml"><mi id="S1.p2.6.m6.1.2.2" xref="S1.p2.6.m6.1.2.2.cmml">D</mi><mo id="S1.p2.6.m6.1.2.1" xref="S1.p2.6.m6.1.2.1.cmml">⁢</mo><mrow id="S1.p2.6.m6.1.2.3.2" xref="S1.p2.6.m6.1.2.cmml"><mo id="S1.p2.6.m6.1.2.3.2.1" stretchy="false" xref="S1.p2.6.m6.1.2.cmml">(</mo><mi id="S1.p2.6.m6.1.1" xref="S1.p2.6.m6.1.1.cmml">A</mi><mo id="S1.p2.6.m6.1.2.3.2.2" stretchy="false" xref="S1.p2.6.m6.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p2.6.m6.1b"><apply id="S1.p2.6.m6.1.2.cmml" xref="S1.p2.6.m6.1.2"><times id="S1.p2.6.m6.1.2.1.cmml" xref="S1.p2.6.m6.1.2.1"></times><ci id="S1.p2.6.m6.1.2.2.cmml" xref="S1.p2.6.m6.1.2.2">𝐷</ci><ci id="S1.p2.6.m6.1.1.cmml" xref="S1.p2.6.m6.1.1">𝐴</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p2.6.m6.1c">D(A)</annotation><annotation encoding="application/x-llamapun" id="S1.p2.6.m6.1d">italic_D ( italic_A )</annotation></semantics></math> is a Sobolev space on a spatial domain that incorporates the boundary conditions of the boundary value problem. Thus, with the aid of complex interpolation, we can identify domains of fractional powers, which in turn play an important role for perturbation techniques in the theory of maximal regularity, see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib14" title="">14</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib36" title="">36</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib44" title="">44</a>]</cite>. Complex interpolation of spaces with boundary conditions was studied by Grisvard and Seeley in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib24" title="">24</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib64" title="">64</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib65" title="">65</a>]</cite> and, due to the numerous applications, the topic is now widely addressed in the literature, see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib8" title="">8</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib22" title="">22</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib72" title="">72</a>]</cite> and the references therein.</p> </div> <div class="ltx_para" id="S1.p3"> <p class="ltx_p" id="S1.p3.11">Secondly, solutions to PDEs on a domain <math alttext="\mathcal{O}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S1.p3.1.m1.1"><semantics id="S1.p3.1.m1.1a"><mrow id="S1.p3.1.m1.1.1" xref="S1.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p3.1.m1.1.1.2" xref="S1.p3.1.m1.1.1.2.cmml">𝒪</mi><mo id="S1.p3.1.m1.1.1.1" xref="S1.p3.1.m1.1.1.1.cmml">⊆</mo><msup id="S1.p3.1.m1.1.1.3" xref="S1.p3.1.m1.1.1.3.cmml"><mi id="S1.p3.1.m1.1.1.3.2" xref="S1.p3.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S1.p3.1.m1.1.1.3.3" xref="S1.p3.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.1.m1.1b"><apply id="S1.p3.1.m1.1.1.cmml" xref="S1.p3.1.m1.1.1"><subset id="S1.p3.1.m1.1.1.1.cmml" xref="S1.p3.1.m1.1.1.1"></subset><ci id="S1.p3.1.m1.1.1.2.cmml" xref="S1.p3.1.m1.1.1.2">𝒪</ci><apply id="S1.p3.1.m1.1.1.3.cmml" xref="S1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p3.1.m1.1.1.3.1.cmml" xref="S1.p3.1.m1.1.1.3">superscript</csymbol><ci id="S1.p3.1.m1.1.1.3.2.cmml" xref="S1.p3.1.m1.1.1.3.2">ℝ</ci><ci id="S1.p3.1.m1.1.1.3.3.cmml" xref="S1.p3.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.1.m1.1c">\mathcal{O}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.1.m1.1d">caligraphic_O ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> may exhibit blow-up behaviour near the boundary of <math alttext="\mathcal{O}" class="ltx_Math" display="inline" id="S1.p3.2.m2.1"><semantics id="S1.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p3.2.m2.1.1" xref="S1.p3.2.m2.1.1.cmml">𝒪</mi><annotation-xml encoding="MathML-Content" id="S1.p3.2.m2.1b"><ci id="S1.p3.2.m2.1.1.cmml" xref="S1.p3.2.m2.1.1">𝒪</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.2.m2.1c">\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.2.m2.1d">caligraphic_O</annotation></semantics></math>. Using weighted spaces with weights of the form <math alttext="w_{\gamma}^{\partial\mathcal{O}}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}" class="ltx_math_unparsed" display="inline" id="S1.p3.3.m3.3"><semantics id="S1.p3.3.m3.3a"><mrow id="S1.p3.3.m3.3b"><msubsup id="S1.p3.3.m3.3.4"><mi id="S1.p3.3.m3.3.4.2.2">w</mi><mi id="S1.p3.3.m3.3.4.2.3">γ</mi><mrow id="S1.p3.3.m3.3.4.3"><mo id="S1.p3.3.m3.3.4.3.1" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.3.m3.3.4.3.2">𝒪</mi></mrow></msubsup><mrow id="S1.p3.3.m3.3.5"><mo id="S1.p3.3.m3.3.5.1" stretchy="false">(</mo><mi id="S1.p3.3.m3.1.1">x</mi><mo id="S1.p3.3.m3.3.5.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S1.p3.3.m3.3.6" rspace="0.278em">:=</mo><mi id="S1.p3.3.m3.2.2">dist</mi><msup id="S1.p3.3.m3.3.7"><mrow id="S1.p3.3.m3.3.7.2"><mo id="S1.p3.3.m3.3.7.2.1" stretchy="false">(</mo><mi id="S1.p3.3.m3.3.3">x</mi><mo id="S1.p3.3.m3.3.7.2.2">,</mo><mo id="S1.p3.3.m3.3.7.2.3" lspace="0em" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S1.p3.3.m3.3.7.2.4">𝒪</mi><mo id="S1.p3.3.m3.3.7.2.5" stretchy="false">)</mo></mrow><mi id="S1.p3.3.m3.3.7.3">γ</mi></msup></mrow><annotation encoding="application/x-tex" id="S1.p3.3.m3.3c">w_{\gamma}^{\partial\mathcal{O}}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.3.m3.3d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT ( italic_x ) := roman_dist ( italic_x , ∂ caligraphic_O ) start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> for some suitable value of <math alttext="\gamma" class="ltx_Math" display="inline" id="S1.p3.4.m4.1"><semantics id="S1.p3.4.m4.1a"><mi id="S1.p3.4.m4.1.1" xref="S1.p3.4.m4.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.4.m4.1b"><ci id="S1.p3.4.m4.1.1.cmml" xref="S1.p3.4.m4.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.4.m4.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S1.p3.4.m4.1d">italic_γ</annotation></semantics></math>, one can nevertheless study problems which may be ill-posed in unweighted spaces. Moreover, with weighted spaces certain compatibility conditions needed in the unweighted case can be avoided. Many authors have employed weighted spaces to solve (stochastic) PDEs, see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib15" title="">15</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib37" title="">37</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib38" title="">38</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib39" title="">39</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib40" title="">40</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib41" title="">41</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib56" title="">56</a>]</cite>. In most of these works, homogeneous weighted Sobolev spaces are used which are known to form a complex interpolation scale, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib54" title="">54</a>, Proposition 2.4]</cite>. Recently, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>]</cite> an alternative approach to solving PDEs on the half-space is presented via the <math alttext="H^{\infty}" class="ltx_Math" display="inline" id="S1.p3.5.m5.1"><semantics id="S1.p3.5.m5.1a"><msup id="S1.p3.5.m5.1.1" xref="S1.p3.5.m5.1.1.cmml"><mi id="S1.p3.5.m5.1.1.2" xref="S1.p3.5.m5.1.1.2.cmml">H</mi><mi id="S1.p3.5.m5.1.1.3" mathvariant="normal" xref="S1.p3.5.m5.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p3.5.m5.1b"><apply id="S1.p3.5.m5.1.1.cmml" xref="S1.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S1.p3.5.m5.1.1.1.cmml" xref="S1.p3.5.m5.1.1">superscript</csymbol><ci id="S1.p3.5.m5.1.1.2.cmml" xref="S1.p3.5.m5.1.1.2">𝐻</ci><infinity id="S1.p3.5.m5.1.1.3.cmml" xref="S1.p3.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.5.m5.1c">H^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.5.m5.1d">italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>-calculus for the Laplacian on inhomogeneous Sobolev spaces with weights outside the Muckenhoupt <math alttext="A_{p}" class="ltx_Math" display="inline" id="S1.p3.6.m6.1"><semantics id="S1.p3.6.m6.1a"><msub id="S1.p3.6.m6.1.1" xref="S1.p3.6.m6.1.1.cmml"><mi id="S1.p3.6.m6.1.1.2" xref="S1.p3.6.m6.1.1.2.cmml">A</mi><mi id="S1.p3.6.m6.1.1.3" xref="S1.p3.6.m6.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p3.6.m6.1b"><apply id="S1.p3.6.m6.1.1.cmml" xref="S1.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S1.p3.6.m6.1.1.1.cmml" xref="S1.p3.6.m6.1.1">subscript</csymbol><ci id="S1.p3.6.m6.1.1.2.cmml" xref="S1.p3.6.m6.1.1.2">𝐴</ci><ci id="S1.p3.6.m6.1.1.3.cmml" xref="S1.p3.6.m6.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.6.m6.1c">A_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.6.m6.1d">italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> class. To further investigate this <math alttext="H^{\infty}" class="ltx_Math" display="inline" id="S1.p3.7.m7.1"><semantics id="S1.p3.7.m7.1a"><msup id="S1.p3.7.m7.1.1" xref="S1.p3.7.m7.1.1.cmml"><mi id="S1.p3.7.m7.1.1.2" xref="S1.p3.7.m7.1.1.2.cmml">H</mi><mi id="S1.p3.7.m7.1.1.3" mathvariant="normal" xref="S1.p3.7.m7.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p3.7.m7.1b"><apply id="S1.p3.7.m7.1.1.cmml" xref="S1.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S1.p3.7.m7.1.1.1.cmml" xref="S1.p3.7.m7.1.1">superscript</csymbol><ci id="S1.p3.7.m7.1.1.2.cmml" xref="S1.p3.7.m7.1.1.2">𝐻</ci><infinity id="S1.p3.7.m7.1.1.3.cmml" xref="S1.p3.7.m7.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.7.m7.1c">H^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.7.m7.1d">italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>-calculus and its consequences, it is crucial to have access to complex interpolation for these weighted Sobolev spaces. In particular, in the upcoming work <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib48" title="">48</a>]</cite>, domains of fractional powers and perturbation techniques will be used to obtain the <math alttext="H^{\infty}" class="ltx_Math" display="inline" id="S1.p3.8.m8.1"><semantics id="S1.p3.8.m8.1a"><msup id="S1.p3.8.m8.1.1" xref="S1.p3.8.m8.1.1.cmml"><mi id="S1.p3.8.m8.1.1.2" xref="S1.p3.8.m8.1.1.2.cmml">H</mi><mi id="S1.p3.8.m8.1.1.3" mathvariant="normal" xref="S1.p3.8.m8.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S1.p3.8.m8.1b"><apply id="S1.p3.8.m8.1.1.cmml" xref="S1.p3.8.m8.1.1"><csymbol cd="ambiguous" id="S1.p3.8.m8.1.1.1.cmml" xref="S1.p3.8.m8.1.1">superscript</csymbol><ci id="S1.p3.8.m8.1.1.2.cmml" xref="S1.p3.8.m8.1.1.2">𝐻</ci><infinity id="S1.p3.8.m8.1.1.3.cmml" xref="S1.p3.8.m8.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.8.m8.1c">H^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.8.m8.1d">italic_H start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>-calculus for the Laplacian on bounded <math alttext="C^{1,\lambda}" class="ltx_Math" display="inline" id="S1.p3.9.m9.2"><semantics id="S1.p3.9.m9.2a"><msup id="S1.p3.9.m9.2.3" xref="S1.p3.9.m9.2.3.cmml"><mi id="S1.p3.9.m9.2.3.2" xref="S1.p3.9.m9.2.3.2.cmml">C</mi><mrow id="S1.p3.9.m9.2.2.2.4" xref="S1.p3.9.m9.2.2.2.3.cmml"><mn id="S1.p3.9.m9.1.1.1.1" xref="S1.p3.9.m9.1.1.1.1.cmml">1</mn><mo id="S1.p3.9.m9.2.2.2.4.1" xref="S1.p3.9.m9.2.2.2.3.cmml">,</mo><mi id="S1.p3.9.m9.2.2.2.2" xref="S1.p3.9.m9.2.2.2.2.cmml">λ</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S1.p3.9.m9.2b"><apply id="S1.p3.9.m9.2.3.cmml" xref="S1.p3.9.m9.2.3"><csymbol cd="ambiguous" id="S1.p3.9.m9.2.3.1.cmml" xref="S1.p3.9.m9.2.3">superscript</csymbol><ci id="S1.p3.9.m9.2.3.2.cmml" xref="S1.p3.9.m9.2.3.2">𝐶</ci><list id="S1.p3.9.m9.2.2.2.3.cmml" xref="S1.p3.9.m9.2.2.2.4"><cn id="S1.p3.9.m9.1.1.1.1.cmml" type="integer" xref="S1.p3.9.m9.1.1.1.1">1</cn><ci id="S1.p3.9.m9.2.2.2.2.cmml" xref="S1.p3.9.m9.2.2.2.2">𝜆</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.9.m9.2c">C^{1,\lambda}</annotation><annotation encoding="application/x-llamapun" id="S1.p3.9.m9.2d">italic_C start_POSTSUPERSCRIPT 1 , italic_λ end_POSTSUPERSCRIPT</annotation></semantics></math>-domains for <math alttext="\lambda\in[0,1]" class="ltx_Math" display="inline" id="S1.p3.10.m10.2"><semantics id="S1.p3.10.m10.2a"><mrow id="S1.p3.10.m10.2.3" xref="S1.p3.10.m10.2.3.cmml"><mi id="S1.p3.10.m10.2.3.2" xref="S1.p3.10.m10.2.3.2.cmml">λ</mi><mo id="S1.p3.10.m10.2.3.1" xref="S1.p3.10.m10.2.3.1.cmml">∈</mo><mrow id="S1.p3.10.m10.2.3.3.2" xref="S1.p3.10.m10.2.3.3.1.cmml"><mo id="S1.p3.10.m10.2.3.3.2.1" stretchy="false" xref="S1.p3.10.m10.2.3.3.1.cmml">[</mo><mn id="S1.p3.10.m10.1.1" xref="S1.p3.10.m10.1.1.cmml">0</mn><mo id="S1.p3.10.m10.2.3.3.2.2" xref="S1.p3.10.m10.2.3.3.1.cmml">,</mo><mn id="S1.p3.10.m10.2.2" xref="S1.p3.10.m10.2.2.cmml">1</mn><mo id="S1.p3.10.m10.2.3.3.2.3" stretchy="false" xref="S1.p3.10.m10.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p3.10.m10.2b"><apply id="S1.p3.10.m10.2.3.cmml" xref="S1.p3.10.m10.2.3"><in id="S1.p3.10.m10.2.3.1.cmml" xref="S1.p3.10.m10.2.3.1"></in><ci id="S1.p3.10.m10.2.3.2.cmml" xref="S1.p3.10.m10.2.3.2">𝜆</ci><interval closure="closed" id="S1.p3.10.m10.2.3.3.1.cmml" xref="S1.p3.10.m10.2.3.3.2"><cn id="S1.p3.10.m10.1.1.cmml" type="integer" xref="S1.p3.10.m10.1.1">0</cn><cn id="S1.p3.10.m10.2.2.cmml" type="integer" xref="S1.p3.10.m10.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.10.m10.2c">\lambda\in[0,1]</annotation><annotation encoding="application/x-llamapun" id="S1.p3.10.m10.2d">italic_λ ∈ [ 0 , 1 ]</annotation></semantics></math> depending on <math alttext="\gamma" class="ltx_Math" display="inline" id="S1.p3.11.m11.1"><semantics id="S1.p3.11.m11.1a"><mi id="S1.p3.11.m11.1.1" xref="S1.p3.11.m11.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S1.p3.11.m11.1b"><ci id="S1.p3.11.m11.1.1.cmml" xref="S1.p3.11.m11.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p3.11.m11.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S1.p3.11.m11.1d">italic_γ</annotation></semantics></math>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S1.p4"> <p class="ltx_p" id="S1.p4.9">For <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S1.p4.1.m1.1"><semantics id="S1.p4.1.m1.1a"><mrow id="S1.p4.1.m1.1.1" xref="S1.p4.1.m1.1.1.cmml"><mi id="S1.p4.1.m1.1.1.2" xref="S1.p4.1.m1.1.1.2.cmml">m</mi><mo id="S1.p4.1.m1.1.1.1" xref="S1.p4.1.m1.1.1.1.cmml">∈</mo><msub id="S1.p4.1.m1.1.1.3" xref="S1.p4.1.m1.1.1.3.cmml"><mi id="S1.p4.1.m1.1.1.3.2" xref="S1.p4.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S1.p4.1.m1.1.1.3.3" xref="S1.p4.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.1.m1.1b"><apply id="S1.p4.1.m1.1.1.cmml" xref="S1.p4.1.m1.1.1"><in id="S1.p4.1.m1.1.1.1.cmml" xref="S1.p4.1.m1.1.1.1"></in><ci id="S1.p4.1.m1.1.1.2.cmml" xref="S1.p4.1.m1.1.1.2">𝑚</ci><apply id="S1.p4.1.m1.1.1.3.cmml" xref="S1.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S1.p4.1.m1.1.1.3.1.cmml" xref="S1.p4.1.m1.1.1.3">subscript</csymbol><ci id="S1.p4.1.m1.1.1.3.2.cmml" xref="S1.p4.1.m1.1.1.3.2">ℕ</ci><cn id="S1.p4.1.m1.1.1.3.3.cmml" type="integer" xref="S1.p4.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.1.m1.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.1.m1.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> we denote by <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S1.p4.2.m2.1"><semantics id="S1.p4.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S1.p4.2.m2.1.1" xref="S1.p4.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S1.p4.2.m2.1b"><ci id="S1.p4.2.m2.1.1.cmml" xref="S1.p4.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.2.m2.1d">caligraphic_B</annotation></semantics></math> an <em class="ltx_emph ltx_font_italic" id="S1.p4.3.1"><math alttext="m" class="ltx_Math" display="inline" id="S1.p4.3.1.m1.1"><semantics id="S1.p4.3.1.m1.1a"><mi id="S1.p4.3.1.m1.1.1" xref="S1.p4.3.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S1.p4.3.1.m1.1b"><ci id="S1.p4.3.1.m1.1.1.cmml" xref="S1.p4.3.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.3.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S1.p4.3.1.m1.1d">italic_m</annotation></semantics></math>-th order normal boundary operator</em> at <math alttext="\partial\mathbb{R}^{d}_{+}=\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}" class="ltx_Math" display="inline" id="S1.p4.4.m3.4"><semantics id="S1.p4.4.m3.4a"><mrow id="S1.p4.4.m3.4.4" xref="S1.p4.4.m3.4.4.cmml"><mrow id="S1.p4.4.m3.4.4.4" xref="S1.p4.4.m3.4.4.4.cmml"><mo id="S1.p4.4.m3.4.4.4.1" rspace="0em" xref="S1.p4.4.m3.4.4.4.1.cmml">∂</mo><msubsup id="S1.p4.4.m3.4.4.4.2" xref="S1.p4.4.m3.4.4.4.2.cmml"><mi id="S1.p4.4.m3.4.4.4.2.2.2" xref="S1.p4.4.m3.4.4.4.2.2.2.cmml">ℝ</mi><mo id="S1.p4.4.m3.4.4.4.2.3" xref="S1.p4.4.m3.4.4.4.2.3.cmml">+</mo><mi id="S1.p4.4.m3.4.4.4.2.2.3" xref="S1.p4.4.m3.4.4.4.2.2.3.cmml">d</mi></msubsup></mrow><mo id="S1.p4.4.m3.4.4.3" xref="S1.p4.4.m3.4.4.3.cmml">=</mo><mrow id="S1.p4.4.m3.4.4.2.2" xref="S1.p4.4.m3.4.4.2.3.cmml"><mo id="S1.p4.4.m3.4.4.2.2.3" stretchy="false" xref="S1.p4.4.m3.4.4.2.3.1.cmml">{</mo><mrow id="S1.p4.4.m3.3.3.1.1.1.2" xref="S1.p4.4.m3.3.3.1.1.1.1.cmml"><mo id="S1.p4.4.m3.3.3.1.1.1.2.1" stretchy="false" xref="S1.p4.4.m3.3.3.1.1.1.1.cmml">(</mo><mn id="S1.p4.4.m3.1.1" xref="S1.p4.4.m3.1.1.cmml">0</mn><mo id="S1.p4.4.m3.3.3.1.1.1.2.2" xref="S1.p4.4.m3.3.3.1.1.1.1.cmml">,</mo><mover accent="true" id="S1.p4.4.m3.2.2" xref="S1.p4.4.m3.2.2.cmml"><mi id="S1.p4.4.m3.2.2.2" xref="S1.p4.4.m3.2.2.2.cmml">x</mi><mo id="S1.p4.4.m3.2.2.1" xref="S1.p4.4.m3.2.2.1.cmml">~</mo></mover><mo id="S1.p4.4.m3.3.3.1.1.1.2.3" rspace="0.278em" stretchy="false" xref="S1.p4.4.m3.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S1.p4.4.m3.4.4.2.2.4" rspace="0.278em" xref="S1.p4.4.m3.4.4.2.3.1.cmml">:</mo><mrow id="S1.p4.4.m3.4.4.2.2.2" xref="S1.p4.4.m3.4.4.2.2.2.cmml"><mover accent="true" id="S1.p4.4.m3.4.4.2.2.2.2" xref="S1.p4.4.m3.4.4.2.2.2.2.cmml"><mi id="S1.p4.4.m3.4.4.2.2.2.2.2" xref="S1.p4.4.m3.4.4.2.2.2.2.2.cmml">x</mi><mo id="S1.p4.4.m3.4.4.2.2.2.2.1" xref="S1.p4.4.m3.4.4.2.2.2.2.1.cmml">~</mo></mover><mo id="S1.p4.4.m3.4.4.2.2.2.1" xref="S1.p4.4.m3.4.4.2.2.2.1.cmml">∈</mo><msup id="S1.p4.4.m3.4.4.2.2.2.3" xref="S1.p4.4.m3.4.4.2.2.2.3.cmml"><mi id="S1.p4.4.m3.4.4.2.2.2.3.2" xref="S1.p4.4.m3.4.4.2.2.2.3.2.cmml">ℝ</mi><mrow id="S1.p4.4.m3.4.4.2.2.2.3.3" xref="S1.p4.4.m3.4.4.2.2.2.3.3.cmml"><mi id="S1.p4.4.m3.4.4.2.2.2.3.3.2" xref="S1.p4.4.m3.4.4.2.2.2.3.3.2.cmml">d</mi><mo id="S1.p4.4.m3.4.4.2.2.2.3.3.1" xref="S1.p4.4.m3.4.4.2.2.2.3.3.1.cmml">−</mo><mn id="S1.p4.4.m3.4.4.2.2.2.3.3.3" xref="S1.p4.4.m3.4.4.2.2.2.3.3.3.cmml">1</mn></mrow></msup></mrow><mo id="S1.p4.4.m3.4.4.2.2.5" stretchy="false" xref="S1.p4.4.m3.4.4.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.4.m3.4b"><apply id="S1.p4.4.m3.4.4.cmml" xref="S1.p4.4.m3.4.4"><eq id="S1.p4.4.m3.4.4.3.cmml" xref="S1.p4.4.m3.4.4.3"></eq><apply id="S1.p4.4.m3.4.4.4.cmml" xref="S1.p4.4.m3.4.4.4"><partialdiff id="S1.p4.4.m3.4.4.4.1.cmml" xref="S1.p4.4.m3.4.4.4.1"></partialdiff><apply id="S1.p4.4.m3.4.4.4.2.cmml" xref="S1.p4.4.m3.4.4.4.2"><csymbol cd="ambiguous" id="S1.p4.4.m3.4.4.4.2.1.cmml" xref="S1.p4.4.m3.4.4.4.2">subscript</csymbol><apply id="S1.p4.4.m3.4.4.4.2.2.cmml" xref="S1.p4.4.m3.4.4.4.2"><csymbol cd="ambiguous" id="S1.p4.4.m3.4.4.4.2.2.1.cmml" xref="S1.p4.4.m3.4.4.4.2">superscript</csymbol><ci id="S1.p4.4.m3.4.4.4.2.2.2.cmml" xref="S1.p4.4.m3.4.4.4.2.2.2">ℝ</ci><ci id="S1.p4.4.m3.4.4.4.2.2.3.cmml" xref="S1.p4.4.m3.4.4.4.2.2.3">𝑑</ci></apply><plus id="S1.p4.4.m3.4.4.4.2.3.cmml" xref="S1.p4.4.m3.4.4.4.2.3"></plus></apply></apply><apply id="S1.p4.4.m3.4.4.2.3.cmml" xref="S1.p4.4.m3.4.4.2.2"><csymbol cd="latexml" id="S1.p4.4.m3.4.4.2.3.1.cmml" xref="S1.p4.4.m3.4.4.2.2.3">conditional-set</csymbol><interval closure="open" id="S1.p4.4.m3.3.3.1.1.1.1.cmml" xref="S1.p4.4.m3.3.3.1.1.1.2"><cn id="S1.p4.4.m3.1.1.cmml" type="integer" xref="S1.p4.4.m3.1.1">0</cn><apply id="S1.p4.4.m3.2.2.cmml" xref="S1.p4.4.m3.2.2"><ci id="S1.p4.4.m3.2.2.1.cmml" xref="S1.p4.4.m3.2.2.1">~</ci><ci id="S1.p4.4.m3.2.2.2.cmml" xref="S1.p4.4.m3.2.2.2">𝑥</ci></apply></interval><apply id="S1.p4.4.m3.4.4.2.2.2.cmml" xref="S1.p4.4.m3.4.4.2.2.2"><in id="S1.p4.4.m3.4.4.2.2.2.1.cmml" xref="S1.p4.4.m3.4.4.2.2.2.1"></in><apply id="S1.p4.4.m3.4.4.2.2.2.2.cmml" xref="S1.p4.4.m3.4.4.2.2.2.2"><ci id="S1.p4.4.m3.4.4.2.2.2.2.1.cmml" xref="S1.p4.4.m3.4.4.2.2.2.2.1">~</ci><ci id="S1.p4.4.m3.4.4.2.2.2.2.2.cmml" xref="S1.p4.4.m3.4.4.2.2.2.2.2">𝑥</ci></apply><apply id="S1.p4.4.m3.4.4.2.2.2.3.cmml" xref="S1.p4.4.m3.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S1.p4.4.m3.4.4.2.2.2.3.1.cmml" xref="S1.p4.4.m3.4.4.2.2.2.3">superscript</csymbol><ci id="S1.p4.4.m3.4.4.2.2.2.3.2.cmml" xref="S1.p4.4.m3.4.4.2.2.2.3.2">ℝ</ci><apply id="S1.p4.4.m3.4.4.2.2.2.3.3.cmml" xref="S1.p4.4.m3.4.4.2.2.2.3.3"><minus id="S1.p4.4.m3.4.4.2.2.2.3.3.1.cmml" xref="S1.p4.4.m3.4.4.2.2.2.3.3.1"></minus><ci id="S1.p4.4.m3.4.4.2.2.2.3.3.2.cmml" xref="S1.p4.4.m3.4.4.2.2.2.3.3.2">𝑑</ci><cn id="S1.p4.4.m3.4.4.2.2.2.3.3.3.cmml" type="integer" xref="S1.p4.4.m3.4.4.2.2.2.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.4.m3.4c">\partial\mathbb{R}^{d}_{+}=\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.4.m3.4d">∂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = { ( 0 , over~ start_ARG italic_x end_ARG ) : over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math>. Consider the power weight <math alttext="w_{\gamma}(x):=\operatorname{dist}(x,\partial\mathbb{R}^{d}_{+})^{\gamma}=|x_{% 1}|^{\gamma}" class="ltx_math_unparsed" display="inline" id="S1.p4.5.m4.3"><semantics id="S1.p4.5.m4.3a"><mrow id="S1.p4.5.m4.3b"><msub id="S1.p4.5.m4.3.4"><mi id="S1.p4.5.m4.3.4.2">w</mi><mi id="S1.p4.5.m4.3.4.3">γ</mi></msub><mrow id="S1.p4.5.m4.3.5"><mo id="S1.p4.5.m4.3.5.1" stretchy="false">(</mo><mi id="S1.p4.5.m4.1.1">x</mi><mo id="S1.p4.5.m4.3.5.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S1.p4.5.m4.3.6" rspace="0.278em">:=</mo><mi id="S1.p4.5.m4.2.2">dist</mi><msup id="S1.p4.5.m4.3.7"><mrow id="S1.p4.5.m4.3.7.2"><mo id="S1.p4.5.m4.3.7.2.1" stretchy="false">(</mo><mi id="S1.p4.5.m4.3.3">x</mi><mo id="S1.p4.5.m4.3.7.2.2">,</mo><mo id="S1.p4.5.m4.3.7.2.3" lspace="0em" rspace="0em">∂</mo><msubsup id="S1.p4.5.m4.3.7.2.4"><mi id="S1.p4.5.m4.3.7.2.4.2.2">ℝ</mi><mo id="S1.p4.5.m4.3.7.2.4.3">+</mo><mi id="S1.p4.5.m4.3.7.2.4.2.3">d</mi></msubsup><mo id="S1.p4.5.m4.3.7.2.5" stretchy="false">)</mo></mrow><mi id="S1.p4.5.m4.3.7.3">γ</mi></msup><mo id="S1.p4.5.m4.3.8" rspace="0em">=</mo><mo fence="false" id="S1.p4.5.m4.3.9" rspace="0.167em" stretchy="false">|</mo><msub id="S1.p4.5.m4.3.10"><mi id="S1.p4.5.m4.3.10.2">x</mi><mn id="S1.p4.5.m4.3.10.3">1</mn></msub><msup id="S1.p4.5.m4.3.11"><mo fence="false" id="S1.p4.5.m4.3.11.2" stretchy="false">|</mo><mi id="S1.p4.5.m4.3.11.3">γ</mi></msup></mrow><annotation encoding="application/x-tex" id="S1.p4.5.m4.3c">w_{\gamma}(x):=\operatorname{dist}(x,\partial\mathbb{R}^{d}_{+})^{\gamma}=|x_{% 1}|^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.5.m4.3d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_x ) := roman_dist ( italic_x , ∂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT = | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="x=(x_{1},\widetilde{x})\in\mathbb{R}_{+}\times\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S1.p4.6.m5.2"><semantics id="S1.p4.6.m5.2a"><mrow id="S1.p4.6.m5.2.2" xref="S1.p4.6.m5.2.2.cmml"><mi id="S1.p4.6.m5.2.2.3" xref="S1.p4.6.m5.2.2.3.cmml">x</mi><mo id="S1.p4.6.m5.2.2.4" xref="S1.p4.6.m5.2.2.4.cmml">=</mo><mrow id="S1.p4.6.m5.2.2.1.1" xref="S1.p4.6.m5.2.2.1.2.cmml"><mo id="S1.p4.6.m5.2.2.1.1.2" stretchy="false" xref="S1.p4.6.m5.2.2.1.2.cmml">(</mo><msub id="S1.p4.6.m5.2.2.1.1.1" xref="S1.p4.6.m5.2.2.1.1.1.cmml"><mi id="S1.p4.6.m5.2.2.1.1.1.2" xref="S1.p4.6.m5.2.2.1.1.1.2.cmml">x</mi><mn id="S1.p4.6.m5.2.2.1.1.1.3" xref="S1.p4.6.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S1.p4.6.m5.2.2.1.1.3" xref="S1.p4.6.m5.2.2.1.2.cmml">,</mo><mover accent="true" id="S1.p4.6.m5.1.1" xref="S1.p4.6.m5.1.1.cmml"><mi id="S1.p4.6.m5.1.1.2" xref="S1.p4.6.m5.1.1.2.cmml">x</mi><mo id="S1.p4.6.m5.1.1.1" xref="S1.p4.6.m5.1.1.1.cmml">~</mo></mover><mo id="S1.p4.6.m5.2.2.1.1.4" stretchy="false" xref="S1.p4.6.m5.2.2.1.2.cmml">)</mo></mrow><mo id="S1.p4.6.m5.2.2.5" xref="S1.p4.6.m5.2.2.5.cmml">∈</mo><mrow id="S1.p4.6.m5.2.2.6" xref="S1.p4.6.m5.2.2.6.cmml"><msub id="S1.p4.6.m5.2.2.6.2" xref="S1.p4.6.m5.2.2.6.2.cmml"><mi id="S1.p4.6.m5.2.2.6.2.2" xref="S1.p4.6.m5.2.2.6.2.2.cmml">ℝ</mi><mo id="S1.p4.6.m5.2.2.6.2.3" xref="S1.p4.6.m5.2.2.6.2.3.cmml">+</mo></msub><mo id="S1.p4.6.m5.2.2.6.1" lspace="0.222em" rspace="0.222em" xref="S1.p4.6.m5.2.2.6.1.cmml">×</mo><msup id="S1.p4.6.m5.2.2.6.3" xref="S1.p4.6.m5.2.2.6.3.cmml"><mi id="S1.p4.6.m5.2.2.6.3.2" xref="S1.p4.6.m5.2.2.6.3.2.cmml">ℝ</mi><mrow id="S1.p4.6.m5.2.2.6.3.3" xref="S1.p4.6.m5.2.2.6.3.3.cmml"><mi id="S1.p4.6.m5.2.2.6.3.3.2" xref="S1.p4.6.m5.2.2.6.3.3.2.cmml">d</mi><mo id="S1.p4.6.m5.2.2.6.3.3.1" xref="S1.p4.6.m5.2.2.6.3.3.1.cmml">−</mo><mn id="S1.p4.6.m5.2.2.6.3.3.3" xref="S1.p4.6.m5.2.2.6.3.3.3.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.6.m5.2b"><apply id="S1.p4.6.m5.2.2.cmml" xref="S1.p4.6.m5.2.2"><and id="S1.p4.6.m5.2.2a.cmml" xref="S1.p4.6.m5.2.2"></and><apply id="S1.p4.6.m5.2.2b.cmml" xref="S1.p4.6.m5.2.2"><eq id="S1.p4.6.m5.2.2.4.cmml" xref="S1.p4.6.m5.2.2.4"></eq><ci id="S1.p4.6.m5.2.2.3.cmml" xref="S1.p4.6.m5.2.2.3">𝑥</ci><interval closure="open" id="S1.p4.6.m5.2.2.1.2.cmml" xref="S1.p4.6.m5.2.2.1.1"><apply id="S1.p4.6.m5.2.2.1.1.1.cmml" xref="S1.p4.6.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.p4.6.m5.2.2.1.1.1.1.cmml" xref="S1.p4.6.m5.2.2.1.1.1">subscript</csymbol><ci id="S1.p4.6.m5.2.2.1.1.1.2.cmml" xref="S1.p4.6.m5.2.2.1.1.1.2">𝑥</ci><cn id="S1.p4.6.m5.2.2.1.1.1.3.cmml" type="integer" xref="S1.p4.6.m5.2.2.1.1.1.3">1</cn></apply><apply id="S1.p4.6.m5.1.1.cmml" xref="S1.p4.6.m5.1.1"><ci id="S1.p4.6.m5.1.1.1.cmml" xref="S1.p4.6.m5.1.1.1">~</ci><ci id="S1.p4.6.m5.1.1.2.cmml" xref="S1.p4.6.m5.1.1.2">𝑥</ci></apply></interval></apply><apply id="S1.p4.6.m5.2.2c.cmml" xref="S1.p4.6.m5.2.2"><in id="S1.p4.6.m5.2.2.5.cmml" xref="S1.p4.6.m5.2.2.5"></in><share href="https://arxiv.org/html/2503.14636v1#S1.p4.6.m5.2.2.1.cmml" id="S1.p4.6.m5.2.2d.cmml" xref="S1.p4.6.m5.2.2"></share><apply id="S1.p4.6.m5.2.2.6.cmml" xref="S1.p4.6.m5.2.2.6"><times id="S1.p4.6.m5.2.2.6.1.cmml" xref="S1.p4.6.m5.2.2.6.1"></times><apply id="S1.p4.6.m5.2.2.6.2.cmml" xref="S1.p4.6.m5.2.2.6.2"><csymbol cd="ambiguous" id="S1.p4.6.m5.2.2.6.2.1.cmml" xref="S1.p4.6.m5.2.2.6.2">subscript</csymbol><ci id="S1.p4.6.m5.2.2.6.2.2.cmml" xref="S1.p4.6.m5.2.2.6.2.2">ℝ</ci><plus id="S1.p4.6.m5.2.2.6.2.3.cmml" xref="S1.p4.6.m5.2.2.6.2.3"></plus></apply><apply id="S1.p4.6.m5.2.2.6.3.cmml" xref="S1.p4.6.m5.2.2.6.3"><csymbol cd="ambiguous" id="S1.p4.6.m5.2.2.6.3.1.cmml" xref="S1.p4.6.m5.2.2.6.3">superscript</csymbol><ci id="S1.p4.6.m5.2.2.6.3.2.cmml" xref="S1.p4.6.m5.2.2.6.3.2">ℝ</ci><apply id="S1.p4.6.m5.2.2.6.3.3.cmml" xref="S1.p4.6.m5.2.2.6.3.3"><minus id="S1.p4.6.m5.2.2.6.3.3.1.cmml" xref="S1.p4.6.m5.2.2.6.3.3.1"></minus><ci id="S1.p4.6.m5.2.2.6.3.3.2.cmml" xref="S1.p4.6.m5.2.2.6.3.3.2">𝑑</ci><cn id="S1.p4.6.m5.2.2.6.3.3.3.cmml" type="integer" xref="S1.p4.6.m5.2.2.6.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.6.m5.2c">x=(x_{1},\widetilde{x})\in\mathbb{R}_{+}\times\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S1.p4.6.m5.2d">italic_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) ∈ blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and let <math alttext="W^{k,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma})" class="ltx_Math" display="inline" id="S1.p4.7.m6.4"><semantics id="S1.p4.7.m6.4a"><mrow id="S1.p4.7.m6.4.4" xref="S1.p4.7.m6.4.4.cmml"><msubsup id="S1.p4.7.m6.4.4.4" xref="S1.p4.7.m6.4.4.4.cmml"><mi id="S1.p4.7.m6.4.4.4.2.2" xref="S1.p4.7.m6.4.4.4.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S1.p4.7.m6.4.4.4.3" xref="S1.p4.7.m6.4.4.4.3.cmml">ℬ</mi><mrow id="S1.p4.7.m6.2.2.2.4" xref="S1.p4.7.m6.2.2.2.3.cmml"><mi id="S1.p4.7.m6.1.1.1.1" xref="S1.p4.7.m6.1.1.1.1.cmml">k</mi><mo id="S1.p4.7.m6.2.2.2.4.1" xref="S1.p4.7.m6.2.2.2.3.cmml">,</mo><mi id="S1.p4.7.m6.2.2.2.2" xref="S1.p4.7.m6.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S1.p4.7.m6.4.4.3" xref="S1.p4.7.m6.4.4.3.cmml">⁢</mo><mrow id="S1.p4.7.m6.4.4.2.2" xref="S1.p4.7.m6.4.4.2.3.cmml"><mo id="S1.p4.7.m6.4.4.2.2.3" stretchy="false" xref="S1.p4.7.m6.4.4.2.3.cmml">(</mo><msubsup id="S1.p4.7.m6.3.3.1.1.1" xref="S1.p4.7.m6.3.3.1.1.1.cmml"><mi id="S1.p4.7.m6.3.3.1.1.1.2.2" xref="S1.p4.7.m6.3.3.1.1.1.2.2.cmml">ℝ</mi><mo id="S1.p4.7.m6.3.3.1.1.1.3" xref="S1.p4.7.m6.3.3.1.1.1.3.cmml">+</mo><mi id="S1.p4.7.m6.3.3.1.1.1.2.3" xref="S1.p4.7.m6.3.3.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S1.p4.7.m6.4.4.2.2.4" xref="S1.p4.7.m6.4.4.2.3.cmml">,</mo><msub id="S1.p4.7.m6.4.4.2.2.2" xref="S1.p4.7.m6.4.4.2.2.2.cmml"><mi id="S1.p4.7.m6.4.4.2.2.2.2" xref="S1.p4.7.m6.4.4.2.2.2.2.cmml">w</mi><mi id="S1.p4.7.m6.4.4.2.2.2.3" xref="S1.p4.7.m6.4.4.2.2.2.3.cmml">γ</mi></msub><mo id="S1.p4.7.m6.4.4.2.2.5" stretchy="false" xref="S1.p4.7.m6.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.7.m6.4b"><apply id="S1.p4.7.m6.4.4.cmml" xref="S1.p4.7.m6.4.4"><times id="S1.p4.7.m6.4.4.3.cmml" xref="S1.p4.7.m6.4.4.3"></times><apply id="S1.p4.7.m6.4.4.4.cmml" xref="S1.p4.7.m6.4.4.4"><csymbol cd="ambiguous" id="S1.p4.7.m6.4.4.4.1.cmml" xref="S1.p4.7.m6.4.4.4">subscript</csymbol><apply id="S1.p4.7.m6.4.4.4.2.cmml" xref="S1.p4.7.m6.4.4.4"><csymbol cd="ambiguous" id="S1.p4.7.m6.4.4.4.2.1.cmml" xref="S1.p4.7.m6.4.4.4">superscript</csymbol><ci id="S1.p4.7.m6.4.4.4.2.2.cmml" xref="S1.p4.7.m6.4.4.4.2.2">𝑊</ci><list id="S1.p4.7.m6.2.2.2.3.cmml" xref="S1.p4.7.m6.2.2.2.4"><ci id="S1.p4.7.m6.1.1.1.1.cmml" xref="S1.p4.7.m6.1.1.1.1">𝑘</ci><ci id="S1.p4.7.m6.2.2.2.2.cmml" xref="S1.p4.7.m6.2.2.2.2">𝑝</ci></list></apply><ci id="S1.p4.7.m6.4.4.4.3.cmml" xref="S1.p4.7.m6.4.4.4.3">ℬ</ci></apply><interval closure="open" id="S1.p4.7.m6.4.4.2.3.cmml" xref="S1.p4.7.m6.4.4.2.2"><apply id="S1.p4.7.m6.3.3.1.1.1.cmml" xref="S1.p4.7.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.p4.7.m6.3.3.1.1.1.1.cmml" xref="S1.p4.7.m6.3.3.1.1.1">subscript</csymbol><apply id="S1.p4.7.m6.3.3.1.1.1.2.cmml" xref="S1.p4.7.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.p4.7.m6.3.3.1.1.1.2.1.cmml" xref="S1.p4.7.m6.3.3.1.1.1">superscript</csymbol><ci id="S1.p4.7.m6.3.3.1.1.1.2.2.cmml" xref="S1.p4.7.m6.3.3.1.1.1.2.2">ℝ</ci><ci id="S1.p4.7.m6.3.3.1.1.1.2.3.cmml" xref="S1.p4.7.m6.3.3.1.1.1.2.3">𝑑</ci></apply><plus id="S1.p4.7.m6.3.3.1.1.1.3.cmml" xref="S1.p4.7.m6.3.3.1.1.1.3"></plus></apply><apply id="S1.p4.7.m6.4.4.2.2.2.cmml" xref="S1.p4.7.m6.4.4.2.2.2"><csymbol cd="ambiguous" id="S1.p4.7.m6.4.4.2.2.2.1.cmml" xref="S1.p4.7.m6.4.4.2.2.2">subscript</csymbol><ci id="S1.p4.7.m6.4.4.2.2.2.2.cmml" xref="S1.p4.7.m6.4.4.2.2.2.2">𝑤</ci><ci id="S1.p4.7.m6.4.4.2.2.2.3.cmml" xref="S1.p4.7.m6.4.4.2.2.2.3">𝛾</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.7.m6.4c">W^{k,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma})</annotation><annotation encoding="application/x-llamapun" id="S1.p4.7.m6.4d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT )</annotation></semantics></math> be the closed subspace of functions <math alttext="f\in W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma})" class="ltx_Math" display="inline" id="S1.p4.8.m7.4"><semantics id="S1.p4.8.m7.4a"><mrow id="S1.p4.8.m7.4.4" xref="S1.p4.8.m7.4.4.cmml"><mi id="S1.p4.8.m7.4.4.4" xref="S1.p4.8.m7.4.4.4.cmml">f</mi><mo id="S1.p4.8.m7.4.4.3" xref="S1.p4.8.m7.4.4.3.cmml">∈</mo><mrow id="S1.p4.8.m7.4.4.2" xref="S1.p4.8.m7.4.4.2.cmml"><msup id="S1.p4.8.m7.4.4.2.4" xref="S1.p4.8.m7.4.4.2.4.cmml"><mi id="S1.p4.8.m7.4.4.2.4.2" xref="S1.p4.8.m7.4.4.2.4.2.cmml">W</mi><mrow id="S1.p4.8.m7.2.2.2.4" xref="S1.p4.8.m7.2.2.2.3.cmml"><mi id="S1.p4.8.m7.1.1.1.1" xref="S1.p4.8.m7.1.1.1.1.cmml">k</mi><mo id="S1.p4.8.m7.2.2.2.4.1" xref="S1.p4.8.m7.2.2.2.3.cmml">,</mo><mi id="S1.p4.8.m7.2.2.2.2" xref="S1.p4.8.m7.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S1.p4.8.m7.4.4.2.3" xref="S1.p4.8.m7.4.4.2.3.cmml">⁢</mo><mrow id="S1.p4.8.m7.4.4.2.2.2" xref="S1.p4.8.m7.4.4.2.2.3.cmml"><mo id="S1.p4.8.m7.4.4.2.2.2.3" stretchy="false" xref="S1.p4.8.m7.4.4.2.2.3.cmml">(</mo><msubsup id="S1.p4.8.m7.3.3.1.1.1.1" xref="S1.p4.8.m7.3.3.1.1.1.1.cmml"><mi id="S1.p4.8.m7.3.3.1.1.1.1.2.2" xref="S1.p4.8.m7.3.3.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S1.p4.8.m7.3.3.1.1.1.1.3" xref="S1.p4.8.m7.3.3.1.1.1.1.3.cmml">+</mo><mi id="S1.p4.8.m7.3.3.1.1.1.1.2.3" xref="S1.p4.8.m7.3.3.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S1.p4.8.m7.4.4.2.2.2.4" xref="S1.p4.8.m7.4.4.2.2.3.cmml">,</mo><msub id="S1.p4.8.m7.4.4.2.2.2.2" xref="S1.p4.8.m7.4.4.2.2.2.2.cmml"><mi id="S1.p4.8.m7.4.4.2.2.2.2.2" xref="S1.p4.8.m7.4.4.2.2.2.2.2.cmml">w</mi><mi id="S1.p4.8.m7.4.4.2.2.2.2.3" xref="S1.p4.8.m7.4.4.2.2.2.2.3.cmml">γ</mi></msub><mo id="S1.p4.8.m7.4.4.2.2.2.5" stretchy="false" xref="S1.p4.8.m7.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.8.m7.4b"><apply id="S1.p4.8.m7.4.4.cmml" xref="S1.p4.8.m7.4.4"><in id="S1.p4.8.m7.4.4.3.cmml" xref="S1.p4.8.m7.4.4.3"></in><ci id="S1.p4.8.m7.4.4.4.cmml" xref="S1.p4.8.m7.4.4.4">𝑓</ci><apply id="S1.p4.8.m7.4.4.2.cmml" xref="S1.p4.8.m7.4.4.2"><times id="S1.p4.8.m7.4.4.2.3.cmml" xref="S1.p4.8.m7.4.4.2.3"></times><apply id="S1.p4.8.m7.4.4.2.4.cmml" xref="S1.p4.8.m7.4.4.2.4"><csymbol cd="ambiguous" id="S1.p4.8.m7.4.4.2.4.1.cmml" xref="S1.p4.8.m7.4.4.2.4">superscript</csymbol><ci id="S1.p4.8.m7.4.4.2.4.2.cmml" xref="S1.p4.8.m7.4.4.2.4.2">𝑊</ci><list id="S1.p4.8.m7.2.2.2.3.cmml" xref="S1.p4.8.m7.2.2.2.4"><ci id="S1.p4.8.m7.1.1.1.1.cmml" xref="S1.p4.8.m7.1.1.1.1">𝑘</ci><ci id="S1.p4.8.m7.2.2.2.2.cmml" xref="S1.p4.8.m7.2.2.2.2">𝑝</ci></list></apply><interval closure="open" id="S1.p4.8.m7.4.4.2.2.3.cmml" xref="S1.p4.8.m7.4.4.2.2.2"><apply id="S1.p4.8.m7.3.3.1.1.1.1.cmml" xref="S1.p4.8.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S1.p4.8.m7.3.3.1.1.1.1.1.cmml" xref="S1.p4.8.m7.3.3.1.1.1.1">subscript</csymbol><apply id="S1.p4.8.m7.3.3.1.1.1.1.2.cmml" xref="S1.p4.8.m7.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S1.p4.8.m7.3.3.1.1.1.1.2.1.cmml" xref="S1.p4.8.m7.3.3.1.1.1.1">superscript</csymbol><ci id="S1.p4.8.m7.3.3.1.1.1.1.2.2.cmml" xref="S1.p4.8.m7.3.3.1.1.1.1.2.2">ℝ</ci><ci id="S1.p4.8.m7.3.3.1.1.1.1.2.3.cmml" xref="S1.p4.8.m7.3.3.1.1.1.1.2.3">𝑑</ci></apply><plus id="S1.p4.8.m7.3.3.1.1.1.1.3.cmml" xref="S1.p4.8.m7.3.3.1.1.1.1.3"></plus></apply><apply id="S1.p4.8.m7.4.4.2.2.2.2.cmml" xref="S1.p4.8.m7.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S1.p4.8.m7.4.4.2.2.2.2.1.cmml" xref="S1.p4.8.m7.4.4.2.2.2.2">subscript</csymbol><ci id="S1.p4.8.m7.4.4.2.2.2.2.2.cmml" xref="S1.p4.8.m7.4.4.2.2.2.2.2">𝑤</ci><ci id="S1.p4.8.m7.4.4.2.2.2.2.3.cmml" xref="S1.p4.8.m7.4.4.2.2.2.2.3">𝛾</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.8.m7.4c">f\in W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma})</annotation><annotation encoding="application/x-llamapun" id="S1.p4.8.m7.4d">italic_f ∈ italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT )</annotation></semantics></math> such that <math alttext="\mathcal{B}f=0" class="ltx_Math" display="inline" id="S1.p4.9.m8.1"><semantics id="S1.p4.9.m8.1a"><mrow id="S1.p4.9.m8.1.1" xref="S1.p4.9.m8.1.1.cmml"><mrow id="S1.p4.9.m8.1.1.2" xref="S1.p4.9.m8.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.p4.9.m8.1.1.2.2" xref="S1.p4.9.m8.1.1.2.2.cmml">ℬ</mi><mo id="S1.p4.9.m8.1.1.2.1" xref="S1.p4.9.m8.1.1.2.1.cmml">⁢</mo><mi id="S1.p4.9.m8.1.1.2.3" xref="S1.p4.9.m8.1.1.2.3.cmml">f</mi></mrow><mo id="S1.p4.9.m8.1.1.1" xref="S1.p4.9.m8.1.1.1.cmml">=</mo><mn id="S1.p4.9.m8.1.1.3" xref="S1.p4.9.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p4.9.m8.1b"><apply id="S1.p4.9.m8.1.1.cmml" xref="S1.p4.9.m8.1.1"><eq id="S1.p4.9.m8.1.1.1.cmml" xref="S1.p4.9.m8.1.1.1"></eq><apply id="S1.p4.9.m8.1.1.2.cmml" xref="S1.p4.9.m8.1.1.2"><times id="S1.p4.9.m8.1.1.2.1.cmml" xref="S1.p4.9.m8.1.1.2.1"></times><ci id="S1.p4.9.m8.1.1.2.2.cmml" xref="S1.p4.9.m8.1.1.2.2">ℬ</ci><ci id="S1.p4.9.m8.1.1.2.3.cmml" xref="S1.p4.9.m8.1.1.2.3">𝑓</ci></apply><cn id="S1.p4.9.m8.1.1.3.cmml" type="integer" xref="S1.p4.9.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p4.9.m8.1c">\mathcal{B}f=0</annotation><annotation encoding="application/x-llamapun" id="S1.p4.9.m8.1d">caligraphic_B italic_f = 0</annotation></semantics></math>, whenever the traces exists. We refer to Sections <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5" title="5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6" title="6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6</span></a> for the precise definitions. Our main result includes the following characterisation of complex interpolation spaces (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS2" title="6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.1.1.1">Theorem 1.1</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem1.p1"> <p class="ltx_p" id="S1.Thmtheorem1.p1.4"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem1.p1.4.4">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.1.1.m1.2"><semantics id="S1.Thmtheorem1.p1.1.1.m1.2a"><mrow id="S1.Thmtheorem1.p1.1.1.m1.2.3" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="S1.Thmtheorem1.p1.1.1.m1.2.3.2" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S1.Thmtheorem1.p1.1.1.m1.2.3.1" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S1.Thmtheorem1.p1.1.1.m1.2.3.3.2" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml"><mo id="S1.Thmtheorem1.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S1.Thmtheorem1.p1.1.1.m1.1.1" xref="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml">1</mn><mo id="S1.Thmtheorem1.p1.1.1.m1.2.3.3.2.2" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S1.Thmtheorem1.p1.1.1.m1.2.2" mathvariant="normal" xref="S1.Thmtheorem1.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S1.Thmtheorem1.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.1.1.m1.2b"><apply id="S1.Thmtheorem1.p1.1.1.m1.2.3.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.2.3"><in id="S1.Thmtheorem1.p1.1.1.m1.2.3.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.1"></in><ci id="S1.Thmtheorem1.p1.1.1.m1.2.3.2.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S1.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.2.3.3.2"><cn id="S1.Thmtheorem1.p1.1.1.m1.1.1.cmml" type="integer" xref="S1.Thmtheorem1.p1.1.1.m1.1.1">1</cn><infinity id="S1.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="S1.Thmtheorem1.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="k\in\{2,3,\dots\}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.2.2.m2.3"><semantics id="S1.Thmtheorem1.p1.2.2.m2.3a"><mrow id="S1.Thmtheorem1.p1.2.2.m2.3.4" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.cmml"><mi id="S1.Thmtheorem1.p1.2.2.m2.3.4.2" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.2.cmml">k</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.3.4.1" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.1.cmml">∈</mo><mrow id="S1.Thmtheorem1.p1.2.2.m2.3.4.3.2" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.3.1.cmml"><mo id="S1.Thmtheorem1.p1.2.2.m2.3.4.3.2.1" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.3.1.cmml">{</mo><mn id="S1.Thmtheorem1.p1.2.2.m2.1.1" xref="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml">2</mn><mo id="S1.Thmtheorem1.p1.2.2.m2.3.4.3.2.2" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.3.1.cmml">,</mo><mn id="S1.Thmtheorem1.p1.2.2.m2.2.2" xref="S1.Thmtheorem1.p1.2.2.m2.2.2.cmml">3</mn><mo id="S1.Thmtheorem1.p1.2.2.m2.3.4.3.2.3" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.3.1.cmml">,</mo><mi id="S1.Thmtheorem1.p1.2.2.m2.3.3" mathvariant="normal" xref="S1.Thmtheorem1.p1.2.2.m2.3.3.cmml">…</mi><mo id="S1.Thmtheorem1.p1.2.2.m2.3.4.3.2.4" stretchy="false" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.2.2.m2.3b"><apply id="S1.Thmtheorem1.p1.2.2.m2.3.4.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.4"><in id="S1.Thmtheorem1.p1.2.2.m2.3.4.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.1"></in><ci id="S1.Thmtheorem1.p1.2.2.m2.3.4.2.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.2">𝑘</ci><set id="S1.Thmtheorem1.p1.2.2.m2.3.4.3.1.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.4.3.2"><cn id="S1.Thmtheorem1.p1.2.2.m2.1.1.cmml" type="integer" xref="S1.Thmtheorem1.p1.2.2.m2.1.1">2</cn><cn id="S1.Thmtheorem1.p1.2.2.m2.2.2.cmml" type="integer" xref="S1.Thmtheorem1.p1.2.2.m2.2.2">3</cn><ci id="S1.Thmtheorem1.p1.2.2.m2.3.3.cmml" xref="S1.Thmtheorem1.p1.2.2.m2.3.3">…</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.2.2.m2.3c">k\in\{2,3,\dots\}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.2.2.m2.3d">italic_k ∈ { 2 , 3 , … }</annotation></semantics></math> and <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.3.3.m3.4"><semantics id="S1.Thmtheorem1.p1.3.3.m3.4a"><mrow id="S1.Thmtheorem1.p1.3.3.m3.4.4" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.cmml"><mi id="S1.Thmtheorem1.p1.3.3.m3.4.4.5" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.5.cmml">γ</mi><mo id="S1.Thmtheorem1.p1.3.3.m3.4.4.4" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.4.cmml">∈</mo><mrow id="S1.Thmtheorem1.p1.3.3.m3.4.4.3" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.cmml"><mrow id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml"><mo id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.2" stretchy="false" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml">(</mo><mrow id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.cmml"><mo id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1a" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.2" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.3" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml">,</mo><mi id="S1.Thmtheorem1.p1.3.3.m3.1.1" mathvariant="normal" xref="S1.Thmtheorem1.p1.3.3.m3.1.1.cmml">∞</mi><mo id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.4" stretchy="false" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml">)</mo></mrow><mo id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.4" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.4.cmml">∖</mo><mrow id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.3.cmml"><mo id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.3" stretchy="false" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.3.1.cmml">{</mo><mrow id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.cmml"><mrow id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.cmml"><mi id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.2" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.1" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.3" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.1" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.3" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.3.1.cmml">:</mo><mrow id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.cmml"><mi id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.2" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.2.cmml">j</mi><mo id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.1" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.cmml"><mi id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.2" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.3" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.5" stretchy="false" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.3.3.m3.4b"><apply id="S1.Thmtheorem1.p1.3.3.m3.4.4.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4"><in id="S1.Thmtheorem1.p1.3.3.m3.4.4.4.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.4"></in><ci id="S1.Thmtheorem1.p1.3.3.m3.4.4.5.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.5">𝛾</ci><apply id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3"><setdiff id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.4.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.4"></setdiff><interval closure="open" id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.2.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1"><apply id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1"><minus id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1"></minus><cn id="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.2.cmml" type="integer" xref="S1.Thmtheorem1.p1.3.3.m3.2.2.1.1.1.1.2">1</cn></apply><infinity id="S1.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.1.1"></infinity></interval><apply id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.3.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2"><csymbol cd="latexml" id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.3.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.3">conditional-set</csymbol><apply id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1"><minus id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.1"></minus><apply id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2"><times id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.1"></times><ci id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.2.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.3.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.3.3.m3.3.3.2.2.1.1.3">1</cn></apply><apply id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2"><in id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.1"></in><ci id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.2.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.2">𝑗</ci><apply id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.1.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3">subscript</csymbol><ci id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.2.cmml" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.3.3.m3.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.3.3.m3.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.3.3.m3.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>. Then for <math alttext="\ell\in\{1,\dots,k-1\}" class="ltx_Math" display="inline" id="S1.Thmtheorem1.p1.4.4.m4.3"><semantics id="S1.Thmtheorem1.p1.4.4.m4.3a"><mrow id="S1.Thmtheorem1.p1.4.4.m4.3.3" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.cmml"><mi id="S1.Thmtheorem1.p1.4.4.m4.3.3.3" mathvariant="normal" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.3.cmml">ℓ</mi><mo id="S1.Thmtheorem1.p1.4.4.m4.3.3.2" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.2.cmml">∈</mo><mrow id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.2.cmml"><mo id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.2" stretchy="false" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.2.cmml">{</mo><mn id="S1.Thmtheorem1.p1.4.4.m4.1.1" xref="S1.Thmtheorem1.p1.4.4.m4.1.1.cmml">1</mn><mo id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.3" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.2.cmml">,</mo><mi id="S1.Thmtheorem1.p1.4.4.m4.2.2" mathvariant="normal" xref="S1.Thmtheorem1.p1.4.4.m4.2.2.cmml">…</mi><mo id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.4" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.2.cmml">,</mo><mrow id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.cmml"><mi id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.2" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.2.cmml">k</mi><mo id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.1" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.1.cmml">−</mo><mn id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.3" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.5" stretchy="false" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem1.p1.4.4.m4.3b"><apply id="S1.Thmtheorem1.p1.4.4.m4.3.3.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.3.3"><in id="S1.Thmtheorem1.p1.4.4.m4.3.3.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.2"></in><ci id="S1.Thmtheorem1.p1.4.4.m4.3.3.3.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.3">ℓ</ci><set id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1"><cn id="S1.Thmtheorem1.p1.4.4.m4.1.1.cmml" type="integer" xref="S1.Thmtheorem1.p1.4.4.m4.1.1">1</cn><ci id="S1.Thmtheorem1.p1.4.4.m4.2.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.2.2">…</ci><apply id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1"><minus id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.1.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.1"></minus><ci id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.2.cmml" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.2">𝑘</ci><cn id="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.3.cmml" type="integer" xref="S1.Thmtheorem1.p1.4.4.m4.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem1.p1.4.4.m4.3c">\ell\in\{1,\dots,k-1\}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem1.p1.4.4.m4.3d">roman_ℓ ∈ { 1 , … , italic_k - 1 }</annotation></semantics></math> we have</span></p> <table class="ltx_equationgroup ltx_eqn_table" id="S1.E1"> <tbody> <tr class="ltx_eqn_row" id="S6.EGx1"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S1.E1.1"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma}),W^{k,p}(\mathbb{R}^{d% }_{+},w_{\gamma})\big{]}_{\frac{\ell}{k}}" class="ltx_Math" display="inline" id="S1.E1.1.m1.4"><semantics id="S1.E1.1.m1.4a"><msub id="S1.E1.1.m1.4.4" xref="S1.E1.1.m1.4.4.cmml"><mrow id="S1.E1.1.m1.4.4.2.2" xref="S1.E1.1.m1.4.4.2.3.cmml"><mo id="S1.E1.1.m1.4.4.2.2.3" maxsize="120%" minsize="120%" xref="S1.E1.1.m1.4.4.2.3.cmml">[</mo><mrow id="S1.E1.1.m1.3.3.1.1.1" xref="S1.E1.1.m1.3.3.1.1.1.cmml"><msup id="S1.E1.1.m1.3.3.1.1.1.4" xref="S1.E1.1.m1.3.3.1.1.1.4.cmml"><mi id="S1.E1.1.m1.3.3.1.1.1.4.2" xref="S1.E1.1.m1.3.3.1.1.1.4.2.cmml">L</mi><mi id="S1.E1.1.m1.3.3.1.1.1.4.3" xref="S1.E1.1.m1.3.3.1.1.1.4.3.cmml">p</mi></msup><mo id="S1.E1.1.m1.3.3.1.1.1.3" xref="S1.E1.1.m1.3.3.1.1.1.3.cmml">⁢</mo><mrow id="S1.E1.1.m1.3.3.1.1.1.2.2" xref="S1.E1.1.m1.3.3.1.1.1.2.3.cmml"><mo id="S1.E1.1.m1.3.3.1.1.1.2.2.3" stretchy="false" xref="S1.E1.1.m1.3.3.1.1.1.2.3.cmml">(</mo><msubsup id="S1.E1.1.m1.3.3.1.1.1.1.1.1" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.2" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S1.E1.1.m1.3.3.1.1.1.1.1.1.3" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1.3.cmml">+</mo><mi id="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.3" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S1.E1.1.m1.3.3.1.1.1.2.2.4" xref="S1.E1.1.m1.3.3.1.1.1.2.3.cmml">,</mo><msub id="S1.E1.1.m1.3.3.1.1.1.2.2.2" xref="S1.E1.1.m1.3.3.1.1.1.2.2.2.cmml"><mi id="S1.E1.1.m1.3.3.1.1.1.2.2.2.2" xref="S1.E1.1.m1.3.3.1.1.1.2.2.2.2.cmml">w</mi><mi id="S1.E1.1.m1.3.3.1.1.1.2.2.2.3" xref="S1.E1.1.m1.3.3.1.1.1.2.2.2.3.cmml">γ</mi></msub><mo id="S1.E1.1.m1.3.3.1.1.1.2.2.5" stretchy="false" xref="S1.E1.1.m1.3.3.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S1.E1.1.m1.4.4.2.2.4" xref="S1.E1.1.m1.4.4.2.3.cmml">,</mo><mrow id="S1.E1.1.m1.4.4.2.2.2" xref="S1.E1.1.m1.4.4.2.2.2.cmml"><msup id="S1.E1.1.m1.4.4.2.2.2.4" xref="S1.E1.1.m1.4.4.2.2.2.4.cmml"><mi id="S1.E1.1.m1.4.4.2.2.2.4.2" xref="S1.E1.1.m1.4.4.2.2.2.4.2.cmml">W</mi><mrow id="S1.E1.1.m1.2.2.2.4" xref="S1.E1.1.m1.2.2.2.3.cmml"><mi id="S1.E1.1.m1.1.1.1.1" xref="S1.E1.1.m1.1.1.1.1.cmml">k</mi><mo id="S1.E1.1.m1.2.2.2.4.1" xref="S1.E1.1.m1.2.2.2.3.cmml">,</mo><mi id="S1.E1.1.m1.2.2.2.2" xref="S1.E1.1.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S1.E1.1.m1.4.4.2.2.2.3" xref="S1.E1.1.m1.4.4.2.2.2.3.cmml">⁢</mo><mrow id="S1.E1.1.m1.4.4.2.2.2.2.2" xref="S1.E1.1.m1.4.4.2.2.2.2.3.cmml"><mo id="S1.E1.1.m1.4.4.2.2.2.2.2.3" stretchy="false" xref="S1.E1.1.m1.4.4.2.2.2.2.3.cmml">(</mo><msubsup id="S1.E1.1.m1.4.4.2.2.2.1.1.1" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1.cmml"><mi id="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.2" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.2.cmml">ℝ</mi><mo id="S1.E1.1.m1.4.4.2.2.2.1.1.1.3" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1.3.cmml">+</mo><mi id="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.3" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S1.E1.1.m1.4.4.2.2.2.2.2.4" xref="S1.E1.1.m1.4.4.2.2.2.2.3.cmml">,</mo><msub id="S1.E1.1.m1.4.4.2.2.2.2.2.2" xref="S1.E1.1.m1.4.4.2.2.2.2.2.2.cmml"><mi id="S1.E1.1.m1.4.4.2.2.2.2.2.2.2" xref="S1.E1.1.m1.4.4.2.2.2.2.2.2.2.cmml">w</mi><mi id="S1.E1.1.m1.4.4.2.2.2.2.2.2.3" xref="S1.E1.1.m1.4.4.2.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S1.E1.1.m1.4.4.2.2.2.2.2.5" stretchy="false" xref="S1.E1.1.m1.4.4.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S1.E1.1.m1.4.4.2.2.5" maxsize="120%" minsize="120%" xref="S1.E1.1.m1.4.4.2.3.cmml">]</mo></mrow><mfrac id="S1.E1.1.m1.4.4.4" xref="S1.E1.1.m1.4.4.4.cmml"><mi id="S1.E1.1.m1.4.4.4.2" mathvariant="normal" xref="S1.E1.1.m1.4.4.4.2.cmml">ℓ</mi><mi id="S1.E1.1.m1.4.4.4.3" xref="S1.E1.1.m1.4.4.4.3.cmml">k</mi></mfrac></msub><annotation-xml encoding="MathML-Content" id="S1.E1.1.m1.4b"><apply id="S1.E1.1.m1.4.4.cmml" xref="S1.E1.1.m1.4.4"><csymbol cd="ambiguous" id="S1.E1.1.m1.4.4.3.cmml" xref="S1.E1.1.m1.4.4">subscript</csymbol><interval closure="closed" id="S1.E1.1.m1.4.4.2.3.cmml" xref="S1.E1.1.m1.4.4.2.2"><apply id="S1.E1.1.m1.3.3.1.1.1.cmml" xref="S1.E1.1.m1.3.3.1.1.1"><times id="S1.E1.1.m1.3.3.1.1.1.3.cmml" xref="S1.E1.1.m1.3.3.1.1.1.3"></times><apply id="S1.E1.1.m1.3.3.1.1.1.4.cmml" xref="S1.E1.1.m1.3.3.1.1.1.4"><csymbol cd="ambiguous" id="S1.E1.1.m1.3.3.1.1.1.4.1.cmml" xref="S1.E1.1.m1.3.3.1.1.1.4">superscript</csymbol><ci id="S1.E1.1.m1.3.3.1.1.1.4.2.cmml" xref="S1.E1.1.m1.3.3.1.1.1.4.2">𝐿</ci><ci id="S1.E1.1.m1.3.3.1.1.1.4.3.cmml" xref="S1.E1.1.m1.3.3.1.1.1.4.3">𝑝</ci></apply><interval closure="open" id="S1.E1.1.m1.3.3.1.1.1.2.3.cmml" xref="S1.E1.1.m1.3.3.1.1.1.2.2"><apply id="S1.E1.1.m1.3.3.1.1.1.1.1.1.cmml" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.E1.1.m1.3.3.1.1.1.1.1.1.1.cmml" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1">subscript</csymbol><apply id="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.cmml" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.1.cmml" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.2.cmml" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.2">ℝ</ci><ci id="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.3.cmml" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1.2.3">𝑑</ci></apply><plus id="S1.E1.1.m1.3.3.1.1.1.1.1.1.3.cmml" xref="S1.E1.1.m1.3.3.1.1.1.1.1.1.3"></plus></apply><apply id="S1.E1.1.m1.3.3.1.1.1.2.2.2.cmml" xref="S1.E1.1.m1.3.3.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S1.E1.1.m1.3.3.1.1.1.2.2.2.1.cmml" xref="S1.E1.1.m1.3.3.1.1.1.2.2.2">subscript</csymbol><ci id="S1.E1.1.m1.3.3.1.1.1.2.2.2.2.cmml" xref="S1.E1.1.m1.3.3.1.1.1.2.2.2.2">𝑤</ci><ci id="S1.E1.1.m1.3.3.1.1.1.2.2.2.3.cmml" xref="S1.E1.1.m1.3.3.1.1.1.2.2.2.3">𝛾</ci></apply></interval></apply><apply id="S1.E1.1.m1.4.4.2.2.2.cmml" xref="S1.E1.1.m1.4.4.2.2.2"><times id="S1.E1.1.m1.4.4.2.2.2.3.cmml" xref="S1.E1.1.m1.4.4.2.2.2.3"></times><apply id="S1.E1.1.m1.4.4.2.2.2.4.cmml" xref="S1.E1.1.m1.4.4.2.2.2.4"><csymbol cd="ambiguous" id="S1.E1.1.m1.4.4.2.2.2.4.1.cmml" xref="S1.E1.1.m1.4.4.2.2.2.4">superscript</csymbol><ci id="S1.E1.1.m1.4.4.2.2.2.4.2.cmml" xref="S1.E1.1.m1.4.4.2.2.2.4.2">𝑊</ci><list id="S1.E1.1.m1.2.2.2.3.cmml" xref="S1.E1.1.m1.2.2.2.4"><ci id="S1.E1.1.m1.1.1.1.1.cmml" xref="S1.E1.1.m1.1.1.1.1">𝑘</ci><ci id="S1.E1.1.m1.2.2.2.2.cmml" xref="S1.E1.1.m1.2.2.2.2">𝑝</ci></list></apply><interval closure="open" id="S1.E1.1.m1.4.4.2.2.2.2.3.cmml" xref="S1.E1.1.m1.4.4.2.2.2.2.2"><apply id="S1.E1.1.m1.4.4.2.2.2.1.1.1.cmml" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.E1.1.m1.4.4.2.2.2.1.1.1.1.cmml" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1">subscript</csymbol><apply id="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.cmml" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.1.cmml" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1">superscript</csymbol><ci id="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.2.cmml" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.2">ℝ</ci><ci id="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.3.cmml" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1.2.3">𝑑</ci></apply><plus id="S1.E1.1.m1.4.4.2.2.2.1.1.1.3.cmml" xref="S1.E1.1.m1.4.4.2.2.2.1.1.1.3"></plus></apply><apply id="S1.E1.1.m1.4.4.2.2.2.2.2.2.cmml" xref="S1.E1.1.m1.4.4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S1.E1.1.m1.4.4.2.2.2.2.2.2.1.cmml" xref="S1.E1.1.m1.4.4.2.2.2.2.2.2">subscript</csymbol><ci id="S1.E1.1.m1.4.4.2.2.2.2.2.2.2.cmml" xref="S1.E1.1.m1.4.4.2.2.2.2.2.2.2">𝑤</ci><ci id="S1.E1.1.m1.4.4.2.2.2.2.2.2.3.cmml" xref="S1.E1.1.m1.4.4.2.2.2.2.2.2.3">𝛾</ci></apply></interval></apply></interval><apply id="S1.E1.1.m1.4.4.4.cmml" xref="S1.E1.1.m1.4.4.4"><divide id="S1.E1.1.m1.4.4.4.1.cmml" xref="S1.E1.1.m1.4.4.4"></divide><ci id="S1.E1.1.m1.4.4.4.2.cmml" xref="S1.E1.1.m1.4.4.4.2">ℓ</ci><ci id="S1.E1.1.m1.4.4.4.3.cmml" xref="S1.E1.1.m1.4.4.4.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.1.m1.4c">\displaystyle\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma}),W^{k,p}(\mathbb{R}^{d% }_{+},w_{\gamma})\big{]}_{\frac{\ell}{k}}</annotation><annotation encoding="application/x-llamapun" id="S1.E1.1.m1.4d">[ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma})," class="ltx_Math" display="inline" id="S1.E1.1.m2.3"><semantics id="S1.E1.1.m2.3a"><mrow id="S1.E1.1.m2.3.3.1" xref="S1.E1.1.m2.3.3.1.1.cmml"><mrow id="S1.E1.1.m2.3.3.1.1" xref="S1.E1.1.m2.3.3.1.1.cmml"><mi id="S1.E1.1.m2.3.3.1.1.4" xref="S1.E1.1.m2.3.3.1.1.4.cmml"></mi><mo id="S1.E1.1.m2.3.3.1.1.3" xref="S1.E1.1.m2.3.3.1.1.3.cmml">=</mo><mrow id="S1.E1.1.m2.3.3.1.1.2" xref="S1.E1.1.m2.3.3.1.1.2.cmml"><msup id="S1.E1.1.m2.3.3.1.1.2.4" xref="S1.E1.1.m2.3.3.1.1.2.4.cmml"><mi id="S1.E1.1.m2.3.3.1.1.2.4.2" xref="S1.E1.1.m2.3.3.1.1.2.4.2.cmml">W</mi><mrow id="S1.E1.1.m2.2.2.2.4" xref="S1.E1.1.m2.2.2.2.3.cmml"><mi id="S1.E1.1.m2.1.1.1.1" mathvariant="normal" xref="S1.E1.1.m2.1.1.1.1.cmml">ℓ</mi><mo id="S1.E1.1.m2.2.2.2.4.1" xref="S1.E1.1.m2.2.2.2.3.cmml">,</mo><mi id="S1.E1.1.m2.2.2.2.2" xref="S1.E1.1.m2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S1.E1.1.m2.3.3.1.1.2.3" xref="S1.E1.1.m2.3.3.1.1.2.3.cmml">⁢</mo><mrow id="S1.E1.1.m2.3.3.1.1.2.2.2" xref="S1.E1.1.m2.3.3.1.1.2.2.3.cmml"><mo id="S1.E1.1.m2.3.3.1.1.2.2.2.3" stretchy="false" xref="S1.E1.1.m2.3.3.1.1.2.2.3.cmml">(</mo><msubsup id="S1.E1.1.m2.3.3.1.1.1.1.1.1" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1.cmml"><mi id="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.2" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S1.E1.1.m2.3.3.1.1.1.1.1.1.3" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1.3.cmml">+</mo><mi id="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.3" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S1.E1.1.m2.3.3.1.1.2.2.2.4" xref="S1.E1.1.m2.3.3.1.1.2.2.3.cmml">,</mo><msub id="S1.E1.1.m2.3.3.1.1.2.2.2.2" xref="S1.E1.1.m2.3.3.1.1.2.2.2.2.cmml"><mi id="S1.E1.1.m2.3.3.1.1.2.2.2.2.2" xref="S1.E1.1.m2.3.3.1.1.2.2.2.2.2.cmml">w</mi><mi id="S1.E1.1.m2.3.3.1.1.2.2.2.2.3" xref="S1.E1.1.m2.3.3.1.1.2.2.2.2.3.cmml">γ</mi></msub><mo id="S1.E1.1.m2.3.3.1.1.2.2.2.5" stretchy="false" xref="S1.E1.1.m2.3.3.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S1.E1.1.m2.3.3.1.2" xref="S1.E1.1.m2.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S1.E1.1.m2.3b"><apply id="S1.E1.1.m2.3.3.1.1.cmml" xref="S1.E1.1.m2.3.3.1"><eq id="S1.E1.1.m2.3.3.1.1.3.cmml" xref="S1.E1.1.m2.3.3.1.1.3"></eq><csymbol cd="latexml" id="S1.E1.1.m2.3.3.1.1.4.cmml" xref="S1.E1.1.m2.3.3.1.1.4">absent</csymbol><apply id="S1.E1.1.m2.3.3.1.1.2.cmml" xref="S1.E1.1.m2.3.3.1.1.2"><times id="S1.E1.1.m2.3.3.1.1.2.3.cmml" xref="S1.E1.1.m2.3.3.1.1.2.3"></times><apply id="S1.E1.1.m2.3.3.1.1.2.4.cmml" xref="S1.E1.1.m2.3.3.1.1.2.4"><csymbol cd="ambiguous" id="S1.E1.1.m2.3.3.1.1.2.4.1.cmml" xref="S1.E1.1.m2.3.3.1.1.2.4">superscript</csymbol><ci id="S1.E1.1.m2.3.3.1.1.2.4.2.cmml" xref="S1.E1.1.m2.3.3.1.1.2.4.2">𝑊</ci><list id="S1.E1.1.m2.2.2.2.3.cmml" xref="S1.E1.1.m2.2.2.2.4"><ci id="S1.E1.1.m2.1.1.1.1.cmml" xref="S1.E1.1.m2.1.1.1.1">ℓ</ci><ci id="S1.E1.1.m2.2.2.2.2.cmml" xref="S1.E1.1.m2.2.2.2.2">𝑝</ci></list></apply><interval closure="open" id="S1.E1.1.m2.3.3.1.1.2.2.3.cmml" xref="S1.E1.1.m2.3.3.1.1.2.2.2"><apply id="S1.E1.1.m2.3.3.1.1.1.1.1.1.cmml" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.E1.1.m2.3.3.1.1.1.1.1.1.1.cmml" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1">subscript</csymbol><apply id="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.cmml" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.1.cmml" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1">superscript</csymbol><ci id="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.2.cmml" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.2">ℝ</ci><ci id="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.3.cmml" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1.2.3">𝑑</ci></apply><plus id="S1.E1.1.m2.3.3.1.1.1.1.1.1.3.cmml" xref="S1.E1.1.m2.3.3.1.1.1.1.1.1.3"></plus></apply><apply id="S1.E1.1.m2.3.3.1.1.2.2.2.2.cmml" xref="S1.E1.1.m2.3.3.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S1.E1.1.m2.3.3.1.1.2.2.2.2.1.cmml" xref="S1.E1.1.m2.3.3.1.1.2.2.2.2">subscript</csymbol><ci id="S1.E1.1.m2.3.3.1.1.2.2.2.2.2.cmml" xref="S1.E1.1.m2.3.3.1.1.2.2.2.2.2">𝑤</ci><ci id="S1.E1.1.m2.3.3.1.1.2.2.2.2.3.cmml" xref="S1.E1.1.m2.3.3.1.1.2.2.2.2.3">𝛾</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.1.m2.3c">\displaystyle=W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma}),</annotation><annotation encoding="application/x-llamapun" id="S1.E1.1.m2.3d">= italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1.1a)</span></td> 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xref="S1.E1.2.m1.4.4.2.2.2.2.2.2.2">𝑤</ci><ci id="S1.E1.2.m1.4.4.2.2.2.2.2.2.3.cmml" xref="S1.E1.2.m1.4.4.2.2.2.2.2.2.3">𝛾</ci></apply></interval></apply></interval><apply id="S1.E1.2.m1.4.4.4.cmml" xref="S1.E1.2.m1.4.4.4"><divide id="S1.E1.2.m1.4.4.4.1.cmml" xref="S1.E1.2.m1.4.4.4"></divide><ci id="S1.E1.2.m1.4.4.4.2.cmml" xref="S1.E1.2.m1.4.4.4.2">ℓ</ci><ci id="S1.E1.2.m1.4.4.4.3.cmml" xref="S1.E1.2.m1.4.4.4.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.2.m1.4c">\displaystyle\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma}),W^{k,p}_{\mathcal{B}}% (\mathbb{R}^{d}_{+},w_{\gamma})\big{]}_{\frac{\ell}{k}}</annotation><annotation encoding="application/x-llamapun" id="S1.E1.2.m1.4d">[ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT italic_k , 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xref="S1.E1.2.m2.3.3.1.1.2.2.2.2.3">𝛾</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.E1.2.m2.3c">\displaystyle=W_{\mathcal{B}}^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma}).</annotation><annotation encoding="application/x-llamapun" id="S1.E1.2.m2.3d">= italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1.1b)</span></td> </tr> </tbody> </table> <p class="ltx_p" id="S1.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem1.p1.5.1">Moreover, by localisation, the results also hold for smooth bounded domains.</span></p> </div> </div> <div class="ltx_para" id="S1.p5"> <p class="ltx_p" id="S1.p5.5">The main novelty of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a>, compared to the existing literature, is the range of weights <math alttext="w_{\gamma}" class="ltx_Math" display="inline" id="S1.p5.1.m1.1"><semantics id="S1.p5.1.m1.1a"><msub id="S1.p5.1.m1.1.1" xref="S1.p5.1.m1.1.1.cmml"><mi id="S1.p5.1.m1.1.1.2" xref="S1.p5.1.m1.1.1.2.cmml">w</mi><mi id="S1.p5.1.m1.1.1.3" xref="S1.p5.1.m1.1.1.3.cmml">γ</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p5.1.m1.1b"><apply id="S1.p5.1.m1.1.1.cmml" xref="S1.p5.1.m1.1.1"><csymbol cd="ambiguous" id="S1.p5.1.m1.1.1.1.cmml" xref="S1.p5.1.m1.1.1">subscript</csymbol><ci id="S1.p5.1.m1.1.1.2.cmml" xref="S1.p5.1.m1.1.1.2">𝑤</ci><ci id="S1.p5.1.m1.1.1.3.cmml" xref="S1.p5.1.m1.1.1.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.1.m1.1c">w_{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S1.p5.1.m1.1d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S1.p5.2.m2.1"><semantics id="S1.p5.2.m2.1a"><mrow id="S1.p5.2.m2.1.1" xref="S1.p5.2.m2.1.1.cmml"><mi id="S1.p5.2.m2.1.1.2" xref="S1.p5.2.m2.1.1.2.cmml">γ</mi><mo id="S1.p5.2.m2.1.1.1" xref="S1.p5.2.m2.1.1.1.cmml">></mo><mrow id="S1.p5.2.m2.1.1.3" xref="S1.p5.2.m2.1.1.3.cmml"><mo id="S1.p5.2.m2.1.1.3a" xref="S1.p5.2.m2.1.1.3.cmml">−</mo><mn id="S1.p5.2.m2.1.1.3.2" xref="S1.p5.2.m2.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.2.m2.1b"><apply id="S1.p5.2.m2.1.1.cmml" xref="S1.p5.2.m2.1.1"><gt id="S1.p5.2.m2.1.1.1.cmml" xref="S1.p5.2.m2.1.1.1"></gt><ci id="S1.p5.2.m2.1.1.2.cmml" xref="S1.p5.2.m2.1.1.2">𝛾</ci><apply id="S1.p5.2.m2.1.1.3.cmml" xref="S1.p5.2.m2.1.1.3"><minus id="S1.p5.2.m2.1.1.3.1.cmml" xref="S1.p5.2.m2.1.1.3"></minus><cn id="S1.p5.2.m2.1.1.3.2.cmml" type="integer" xref="S1.p5.2.m2.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.2.m2.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S1.p5.2.m2.1d">italic_γ > - 1</annotation></semantics></math>. Unfortunately, the proof in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>]</cite> for the unweighted case <math alttext="\gamma=0" class="ltx_Math" display="inline" id="S1.p5.3.m3.1"><semantics id="S1.p5.3.m3.1a"><mrow id="S1.p5.3.m3.1.1" xref="S1.p5.3.m3.1.1.cmml"><mi id="S1.p5.3.m3.1.1.2" xref="S1.p5.3.m3.1.1.2.cmml">γ</mi><mo id="S1.p5.3.m3.1.1.1" xref="S1.p5.3.m3.1.1.1.cmml">=</mo><mn id="S1.p5.3.m3.1.1.3" xref="S1.p5.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p5.3.m3.1b"><apply id="S1.p5.3.m3.1.1.cmml" xref="S1.p5.3.m3.1.1"><eq id="S1.p5.3.m3.1.1.1.cmml" xref="S1.p5.3.m3.1.1.1"></eq><ci id="S1.p5.3.m3.1.1.2.cmml" xref="S1.p5.3.m3.1.1.2">𝛾</ci><cn id="S1.p5.3.m3.1.1.3.cmml" type="integer" xref="S1.p5.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.3.m3.1c">\gamma=0</annotation><annotation encoding="application/x-llamapun" id="S1.p5.3.m3.1d">italic_γ = 0</annotation></semantics></math> does not carry over immediately to the weighted setting. In the unweighted case the result (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.E1.1" title="In 1.1 ‣ Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1a</span></a>) is well known and is used to prove (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.E1.2" title="In 1.1 ‣ Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1b</span></a>) in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>]</cite>. In the weighted setting we can only prove the inclusion “<math alttext="\hookrightarrow" class="ltx_Math" display="inline" id="S1.p5.4.m4.1"><semantics id="S1.p5.4.m4.1a"><mo id="S1.p5.4.m4.1.1" stretchy="false" xref="S1.p5.4.m4.1.1.cmml">↪</mo><annotation-xml encoding="MathML-Content" id="S1.p5.4.m4.1b"><ci id="S1.p5.4.m4.1.1.cmml" xref="S1.p5.4.m4.1.1">↪</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.4.m4.1c">\hookrightarrow</annotation><annotation encoding="application/x-llamapun" id="S1.p5.4.m4.1d">↪</annotation></semantics></math>” in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.E1.1" title="In 1.1 ‣ Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1a</span></a>) directly with the aid of Wolff interpolation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib71" title="">71</a>]</cite> (see Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem3" title="Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a>). Nevertheless, this embedding is sufficient to prove (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.E1.2" title="In 1.1 ‣ Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1b</span></a>) using the characterisation of the trace spaces (see Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.2</span></a>) and similar arguments as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>]</cite>. Then, the converse inclusion “<math alttext="\hookleftarrow" class="ltx_Math" display="inline" id="S1.p5.5.m5.1"><semantics id="S1.p5.5.m5.1a"><mo id="S1.p5.5.m5.1.1" stretchy="false" xref="S1.p5.5.m5.1.1.cmml">↩</mo><annotation-xml encoding="MathML-Content" id="S1.p5.5.m5.1b"><ci id="S1.p5.5.m5.1.1.cmml" xref="S1.p5.5.m5.1.1">↩</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p5.5.m5.1c">\hookleftarrow</annotation><annotation encoding="application/x-llamapun" id="S1.p5.5.m5.1d">↩</annotation></semantics></math>” in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.E1.1" title="In 1.1 ‣ Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1a</span></a>) also follows. Furthermore, we make the following remarks concerning Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a>.</p> <ol class="ltx_enumerate" id="S1.I1"> <li class="ltx_item" id="S1.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S1.I1.i1.p1"> <p class="ltx_p" id="S1.I1.i1.p1.1">The class of normal boundary operators is considered in, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib20" title="">20</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib27" title="">27</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib65" title="">65</a>]</cite> and includes the important cases of Dirichlet and Neumann boundary conditions, but also mixed boundary conditions with space-dependent coefficients are allowed. Actually, we prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a> for systems of normal boundary operators. Differential operators with these types of boundary operators are studied in, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib3" title="">3</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib4" title="">4</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib5" title="">5</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib19" title="">19</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib25" title="">25</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib26" title="">26</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib55" title="">55</a>]</cite>.</p> </div> </li> <li class="ltx_item" id="S1.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S1.I1.i2.p1"> <p class="ltx_p" id="S1.I1.i2.p1.11">If <math alttext="\gamma\in(-1,p-1)" 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id="S1.I1.i2.p1.1.m1.2.2.2.2.2.1" xref="S1.I1.i2.p1.1.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S1.I1.i2.p1.1.m1.2.2.2.2.2.3" xref="S1.I1.i2.p1.1.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S1.I1.i2.p1.1.m1.2.2.2.2.5" stretchy="false" xref="S1.I1.i2.p1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.1.m1.2b"><apply id="S1.I1.i2.p1.1.m1.2.2.cmml" xref="S1.I1.i2.p1.1.m1.2.2"><in id="S1.I1.i2.p1.1.m1.2.2.3.cmml" xref="S1.I1.i2.p1.1.m1.2.2.3"></in><ci id="S1.I1.i2.p1.1.m1.2.2.4.cmml" xref="S1.I1.i2.p1.1.m1.2.2.4">𝛾</ci><interval closure="open" id="S1.I1.i2.p1.1.m1.2.2.2.3.cmml" xref="S1.I1.i2.p1.1.m1.2.2.2.2"><apply id="S1.I1.i2.p1.1.m1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1.1.1.1"><minus id="S1.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.1.m1.1.1.1.1.1"></minus><cn id="S1.I1.i2.p1.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S1.I1.i2.p1.1.m1.1.1.1.1.1.2">1</cn></apply><apply id="S1.I1.i2.p1.1.m1.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.1.m1.2.2.2.2.2"><minus id="S1.I1.i2.p1.1.m1.2.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.1.m1.2.2.2.2.2.1"></minus><ci id="S1.I1.i2.p1.1.m1.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.1.m1.2.2.2.2.2.2">𝑝</ci><cn id="S1.I1.i2.p1.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S1.I1.i2.p1.1.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.1.m1.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.1.m1.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>, i.e., <math alttext="w_{\gamma}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.2.m2.1"><semantics id="S1.I1.i2.p1.2.m2.1a"><msub id="S1.I1.i2.p1.2.m2.1.1" xref="S1.I1.i2.p1.2.m2.1.1.cmml"><mi id="S1.I1.i2.p1.2.m2.1.1.2" xref="S1.I1.i2.p1.2.m2.1.1.2.cmml">w</mi><mi id="S1.I1.i2.p1.2.m2.1.1.3" xref="S1.I1.i2.p1.2.m2.1.1.3.cmml">γ</mi></msub><annotation-xml encoding="MathML-Content" 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start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.5.m5.1"><semantics id="S1.I1.i2.p1.5.m5.1a"><mrow id="S1.I1.i2.p1.5.m5.1.1" xref="S1.I1.i2.p1.5.m5.1.1.cmml"><mi id="S1.I1.i2.p1.5.m5.1.1.2" xref="S1.I1.i2.p1.5.m5.1.1.2.cmml">k</mi><mo id="S1.I1.i2.p1.5.m5.1.1.1" xref="S1.I1.i2.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S1.I1.i2.p1.5.m5.1.1.3" xref="S1.I1.i2.p1.5.m5.1.1.3.cmml"><mi id="S1.I1.i2.p1.5.m5.1.1.3.2" xref="S1.I1.i2.p1.5.m5.1.1.3.2.cmml">ℕ</mi><mn id="S1.I1.i2.p1.5.m5.1.1.3.3" xref="S1.I1.i2.p1.5.m5.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.5.m5.1b"><apply id="S1.I1.i2.p1.5.m5.1.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1"><in id="S1.I1.i2.p1.5.m5.1.1.1.cmml" xref="S1.I1.i2.p1.5.m5.1.1.1"></in><ci id="S1.I1.i2.p1.5.m5.1.1.2.cmml" xref="S1.I1.i2.p1.5.m5.1.1.2">𝑘</ci><apply 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xref="S1.I1.i2.p1.6.m6.1.1.1.1">𝑠</ci><ci id="S1.I1.i2.p1.6.m6.2.2.2.2.cmml" xref="S1.I1.i2.p1.6.m6.2.2.2.2">𝑝</ci></list></apply><interval closure="open" id="S1.I1.i2.p1.6.m6.4.4.2.3.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2"><apply id="S1.I1.i2.p1.6.m6.3.3.1.1.1.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.6.m6.3.3.1.1.1.1.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1">subscript</csymbol><apply id="S1.I1.i2.p1.6.m6.3.3.1.1.1.2.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p1.6.m6.3.3.1.1.1.2.1.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1">superscript</csymbol><ci id="S1.I1.i2.p1.6.m6.3.3.1.1.1.2.2.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1.2.2">ℝ</ci><ci id="S1.I1.i2.p1.6.m6.3.3.1.1.1.2.3.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1.2.3">𝑑</ci></apply><plus id="S1.I1.i2.p1.6.m6.3.3.1.1.1.3.cmml" xref="S1.I1.i2.p1.6.m6.3.3.1.1.1.3"></plus></apply><apply id="S1.I1.i2.p1.6.m6.4.4.2.2.2.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.6.m6.4.4.2.2.2.1.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2">subscript</csymbol><ci id="S1.I1.i2.p1.6.m6.4.4.2.2.2.2.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.2">𝑤</ci><ci id="S1.I1.i2.p1.6.m6.4.4.2.2.2.3.cmml" xref="S1.I1.i2.p1.6.m6.4.4.2.2.2.3">𝛾</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.6.m6.4c">H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma})</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.6.m6.4d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S1.I1.i2.p1.7.m7.1"><semantics id="S1.I1.i2.p1.7.m7.1a"><mrow id="S1.I1.i2.p1.7.m7.1.1" xref="S1.I1.i2.p1.7.m7.1.1.cmml"><mi id="S1.I1.i2.p1.7.m7.1.1.2" xref="S1.I1.i2.p1.7.m7.1.1.2.cmml">s</mi><mo id="S1.I1.i2.p1.7.m7.1.1.1" xref="S1.I1.i2.p1.7.m7.1.1.1.cmml">∈</mo><mi id="S1.I1.i2.p1.7.m7.1.1.3" xref="S1.I1.i2.p1.7.m7.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.7.m7.1b"><apply id="S1.I1.i2.p1.7.m7.1.1.cmml" xref="S1.I1.i2.p1.7.m7.1.1"><in id="S1.I1.i2.p1.7.m7.1.1.1.cmml" xref="S1.I1.i2.p1.7.m7.1.1.1"></in><ci id="S1.I1.i2.p1.7.m7.1.1.2.cmml" xref="S1.I1.i2.p1.7.m7.1.1.2">𝑠</ci><ci id="S1.I1.i2.p1.7.m7.1.1.3.cmml" xref="S1.I1.i2.p1.7.m7.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.7.m7.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.7.m7.1d">italic_s ∈ blackboard_R</annotation></semantics></math> forms a complex interpolation scale, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.5 & 5.6]</cite>. In the case <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S1.I1.i2.p1.8.m8.2"><semantics id="S1.I1.i2.p1.8.m8.2a"><mrow id="S1.I1.i2.p1.8.m8.2.2" xref="S1.I1.i2.p1.8.m8.2.2.cmml"><mi id="S1.I1.i2.p1.8.m8.2.2.4" xref="S1.I1.i2.p1.8.m8.2.2.4.cmml">γ</mi><mo id="S1.I1.i2.p1.8.m8.2.2.3" xref="S1.I1.i2.p1.8.m8.2.2.3.cmml">∈</mo><mrow id="S1.I1.i2.p1.8.m8.2.2.2.2" xref="S1.I1.i2.p1.8.m8.2.2.2.3.cmml"><mo id="S1.I1.i2.p1.8.m8.2.2.2.2.3" stretchy="false" xref="S1.I1.i2.p1.8.m8.2.2.2.3.cmml">(</mo><mrow id="S1.I1.i2.p1.8.m8.1.1.1.1.1" xref="S1.I1.i2.p1.8.m8.1.1.1.1.1.cmml"><mo id="S1.I1.i2.p1.8.m8.1.1.1.1.1a" xref="S1.I1.i2.p1.8.m8.1.1.1.1.1.cmml">−</mo><mn id="S1.I1.i2.p1.8.m8.1.1.1.1.1.2" xref="S1.I1.i2.p1.8.m8.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.I1.i2.p1.8.m8.2.2.2.2.4" xref="S1.I1.i2.p1.8.m8.2.2.2.3.cmml">,</mo><mrow id="S1.I1.i2.p1.8.m8.2.2.2.2.2" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2.cmml"><mi id="S1.I1.i2.p1.8.m8.2.2.2.2.2.2" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2.2.cmml">p</mi><mo id="S1.I1.i2.p1.8.m8.2.2.2.2.2.1" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2.1.cmml">−</mo><mn id="S1.I1.i2.p1.8.m8.2.2.2.2.2.3" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S1.I1.i2.p1.8.m8.2.2.2.2.5" stretchy="false" xref="S1.I1.i2.p1.8.m8.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.8.m8.2b"><apply id="S1.I1.i2.p1.8.m8.2.2.cmml" xref="S1.I1.i2.p1.8.m8.2.2"><in id="S1.I1.i2.p1.8.m8.2.2.3.cmml" xref="S1.I1.i2.p1.8.m8.2.2.3"></in><ci id="S1.I1.i2.p1.8.m8.2.2.4.cmml" xref="S1.I1.i2.p1.8.m8.2.2.4">𝛾</ci><interval closure="open" id="S1.I1.i2.p1.8.m8.2.2.2.3.cmml" xref="S1.I1.i2.p1.8.m8.2.2.2.2"><apply id="S1.I1.i2.p1.8.m8.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.8.m8.1.1.1.1.1"><minus id="S1.I1.i2.p1.8.m8.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.8.m8.1.1.1.1.1"></minus><cn id="S1.I1.i2.p1.8.m8.1.1.1.1.1.2.cmml" type="integer" xref="S1.I1.i2.p1.8.m8.1.1.1.1.1.2">1</cn></apply><apply id="S1.I1.i2.p1.8.m8.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2"><minus id="S1.I1.i2.p1.8.m8.2.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2.1"></minus><ci id="S1.I1.i2.p1.8.m8.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2.2">𝑝</ci><cn id="S1.I1.i2.p1.8.m8.2.2.2.2.2.3.cmml" type="integer" xref="S1.I1.i2.p1.8.m8.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.8.m8.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.8.m8.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> we obtain Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a> for Bessel potential spaces with boundary conditions and fractional smoothness as well, see Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4" title="Theorem 6.4 (Complex interpolation of weighted Bessel potential spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a>. This partially extends the result in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorem VIII.2.4.8]</cite>. Outside the range <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S1.I1.i2.p1.9.m9.2"><semantics id="S1.I1.i2.p1.9.m9.2a"><mrow id="S1.I1.i2.p1.9.m9.2.2" xref="S1.I1.i2.p1.9.m9.2.2.cmml"><mi id="S1.I1.i2.p1.9.m9.2.2.4" xref="S1.I1.i2.p1.9.m9.2.2.4.cmml">γ</mi><mo id="S1.I1.i2.p1.9.m9.2.2.3" xref="S1.I1.i2.p1.9.m9.2.2.3.cmml">∈</mo><mrow id="S1.I1.i2.p1.9.m9.2.2.2.2" xref="S1.I1.i2.p1.9.m9.2.2.2.3.cmml"><mo id="S1.I1.i2.p1.9.m9.2.2.2.2.3" stretchy="false" xref="S1.I1.i2.p1.9.m9.2.2.2.3.cmml">(</mo><mrow id="S1.I1.i2.p1.9.m9.1.1.1.1.1" xref="S1.I1.i2.p1.9.m9.1.1.1.1.1.cmml"><mo id="S1.I1.i2.p1.9.m9.1.1.1.1.1a" xref="S1.I1.i2.p1.9.m9.1.1.1.1.1.cmml">−</mo><mn id="S1.I1.i2.p1.9.m9.1.1.1.1.1.2" xref="S1.I1.i2.p1.9.m9.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.I1.i2.p1.9.m9.2.2.2.2.4" xref="S1.I1.i2.p1.9.m9.2.2.2.3.cmml">,</mo><mrow id="S1.I1.i2.p1.9.m9.2.2.2.2.2" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2.cmml"><mi id="S1.I1.i2.p1.9.m9.2.2.2.2.2.2" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2.2.cmml">p</mi><mo id="S1.I1.i2.p1.9.m9.2.2.2.2.2.1" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2.1.cmml">−</mo><mn id="S1.I1.i2.p1.9.m9.2.2.2.2.2.3" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S1.I1.i2.p1.9.m9.2.2.2.2.5" stretchy="false" xref="S1.I1.i2.p1.9.m9.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p1.9.m9.2b"><apply id="S1.I1.i2.p1.9.m9.2.2.cmml" xref="S1.I1.i2.p1.9.m9.2.2"><in id="S1.I1.i2.p1.9.m9.2.2.3.cmml" xref="S1.I1.i2.p1.9.m9.2.2.3"></in><ci id="S1.I1.i2.p1.9.m9.2.2.4.cmml" xref="S1.I1.i2.p1.9.m9.2.2.4">𝛾</ci><interval closure="open" id="S1.I1.i2.p1.9.m9.2.2.2.3.cmml" xref="S1.I1.i2.p1.9.m9.2.2.2.2"><apply id="S1.I1.i2.p1.9.m9.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.9.m9.1.1.1.1.1"><minus id="S1.I1.i2.p1.9.m9.1.1.1.1.1.1.cmml" xref="S1.I1.i2.p1.9.m9.1.1.1.1.1"></minus><cn id="S1.I1.i2.p1.9.m9.1.1.1.1.1.2.cmml" type="integer" xref="S1.I1.i2.p1.9.m9.1.1.1.1.1.2">1</cn></apply><apply id="S1.I1.i2.p1.9.m9.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2"><minus id="S1.I1.i2.p1.9.m9.2.2.2.2.2.1.cmml" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2.1"></minus><ci id="S1.I1.i2.p1.9.m9.2.2.2.2.2.2.cmml" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2.2">𝑝</ci><cn id="S1.I1.i2.p1.9.m9.2.2.2.2.2.3.cmml" type="integer" xref="S1.I1.i2.p1.9.m9.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.9.m9.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.9.m9.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> we expect that our result also holds if one defines <math alttext="H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma})" class="ltx_Math" display="inline" id="S1.I1.i2.p1.10.m10.4"><semantics id="S1.I1.i2.p1.10.m10.4a"><mrow id="S1.I1.i2.p1.10.m10.4.4" xref="S1.I1.i2.p1.10.m10.4.4.cmml"><msup id="S1.I1.i2.p1.10.m10.4.4.4" xref="S1.I1.i2.p1.10.m10.4.4.4.cmml"><mi 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interpolation of <math alttext="W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma})" class="ltx_Math" display="inline" id="S1.I1.i2.p1.11.m11.4"><semantics id="S1.I1.i2.p1.11.m11.4a"><mrow id="S1.I1.i2.p1.11.m11.4.4" xref="S1.I1.i2.p1.11.m11.4.4.cmml"><msup id="S1.I1.i2.p1.11.m11.4.4.4" xref="S1.I1.i2.p1.11.m11.4.4.4.cmml"><mi id="S1.I1.i2.p1.11.m11.4.4.4.2" xref="S1.I1.i2.p1.11.m11.4.4.4.2.cmml">W</mi><mrow id="S1.I1.i2.p1.11.m11.2.2.2.4" xref="S1.I1.i2.p1.11.m11.2.2.2.3.cmml"><mi id="S1.I1.i2.p1.11.m11.1.1.1.1" xref="S1.I1.i2.p1.11.m11.1.1.1.1.cmml">k</mi><mo id="S1.I1.i2.p1.11.m11.2.2.2.4.1" xref="S1.I1.i2.p1.11.m11.2.2.2.3.cmml">,</mo><mi id="S1.I1.i2.p1.11.m11.2.2.2.2" xref="S1.I1.i2.p1.11.m11.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S1.I1.i2.p1.11.m11.4.4.3" xref="S1.I1.i2.p1.11.m11.4.4.3.cmml">⁢</mo><mrow id="S1.I1.i2.p1.11.m11.4.4.2.2" xref="S1.I1.i2.p1.11.m11.4.4.2.3.cmml"><mo id="S1.I1.i2.p1.11.m11.4.4.2.2.3" stretchy="false" xref="S1.I1.i2.p1.11.m11.4.4.2.3.cmml">(</mo><msubsup 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cd="ambiguous" id="S1.I1.i2.p1.11.m11.3.3.1.1.1.2.1.cmml" xref="S1.I1.i2.p1.11.m11.3.3.1.1.1">superscript</csymbol><ci id="S1.I1.i2.p1.11.m11.3.3.1.1.1.2.2.cmml" xref="S1.I1.i2.p1.11.m11.3.3.1.1.1.2.2">ℝ</ci><ci id="S1.I1.i2.p1.11.m11.3.3.1.1.1.2.3.cmml" xref="S1.I1.i2.p1.11.m11.3.3.1.1.1.2.3">𝑑</ci></apply><plus id="S1.I1.i2.p1.11.m11.3.3.1.1.1.3.cmml" xref="S1.I1.i2.p1.11.m11.3.3.1.1.1.3"></plus></apply><apply id="S1.I1.i2.p1.11.m11.4.4.2.2.2.cmml" xref="S1.I1.i2.p1.11.m11.4.4.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i2.p1.11.m11.4.4.2.2.2.1.cmml" xref="S1.I1.i2.p1.11.m11.4.4.2.2.2">subscript</csymbol><ci id="S1.I1.i2.p1.11.m11.4.4.2.2.2.2.cmml" xref="S1.I1.i2.p1.11.m11.4.4.2.2.2.2">𝑤</ci><ci id="S1.I1.i2.p1.11.m11.4.4.2.2.2.3.cmml" xref="S1.I1.i2.p1.11.m11.4.4.2.2.2.3">𝛾</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p1.11.m11.4c">W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma})</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p1.11.m11.4d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S1.I1.i2.p2"> <p class="ltx_p" id="S1.I1.i2.p2.2">The interpolation result in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a> without boundary conditions is new for <math alttext="\gamma>p-1" class="ltx_Math" display="inline" id="S1.I1.i2.p2.1.m1.1"><semantics id="S1.I1.i2.p2.1.m1.1a"><mrow id="S1.I1.i2.p2.1.m1.1.1" xref="S1.I1.i2.p2.1.m1.1.1.cmml"><mi id="S1.I1.i2.p2.1.m1.1.1.2" xref="S1.I1.i2.p2.1.m1.1.1.2.cmml">γ</mi><mo id="S1.I1.i2.p2.1.m1.1.1.1" xref="S1.I1.i2.p2.1.m1.1.1.1.cmml">></mo><mrow id="S1.I1.i2.p2.1.m1.1.1.3" xref="S1.I1.i2.p2.1.m1.1.1.3.cmml"><mi id="S1.I1.i2.p2.1.m1.1.1.3.2" xref="S1.I1.i2.p2.1.m1.1.1.3.2.cmml">p</mi><mo id="S1.I1.i2.p2.1.m1.1.1.3.1" xref="S1.I1.i2.p2.1.m1.1.1.3.1.cmml">−</mo><mn id="S1.I1.i2.p2.1.m1.1.1.3.3" xref="S1.I1.i2.p2.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p2.1.m1.1b"><apply id="S1.I1.i2.p2.1.m1.1.1.cmml" xref="S1.I1.i2.p2.1.m1.1.1"><gt id="S1.I1.i2.p2.1.m1.1.1.1.cmml" xref="S1.I1.i2.p2.1.m1.1.1.1"></gt><ci id="S1.I1.i2.p2.1.m1.1.1.2.cmml" xref="S1.I1.i2.p2.1.m1.1.1.2">𝛾</ci><apply id="S1.I1.i2.p2.1.m1.1.1.3.cmml" xref="S1.I1.i2.p2.1.m1.1.1.3"><minus id="S1.I1.i2.p2.1.m1.1.1.3.1.cmml" xref="S1.I1.i2.p2.1.m1.1.1.3.1"></minus><ci id="S1.I1.i2.p2.1.m1.1.1.3.2.cmml" xref="S1.I1.i2.p2.1.m1.1.1.3.2">𝑝</ci><cn id="S1.I1.i2.p2.1.m1.1.1.3.3.cmml" type="integer" xref="S1.I1.i2.p2.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p2.1.m1.1c">\gamma>p-1</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p2.1.m1.1d">italic_γ > italic_p - 1</annotation></semantics></math>. We refer to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib16" title="">16</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib66" title="">66</a>]</cite> for results on complex interpolation of Besov and Triebel-Lizorkin spaces with weights outside the class of <math alttext="A_{p}" class="ltx_Math" display="inline" id="S1.I1.i2.p2.2.m2.1"><semantics id="S1.I1.i2.p2.2.m2.1a"><msub id="S1.I1.i2.p2.2.m2.1.1" xref="S1.I1.i2.p2.2.m2.1.1.cmml"><mi id="S1.I1.i2.p2.2.m2.1.1.2" xref="S1.I1.i2.p2.2.m2.1.1.2.cmml">A</mi><mi id="S1.I1.i2.p2.2.m2.1.1.3" xref="S1.I1.i2.p2.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S1.I1.i2.p2.2.m2.1b"><apply id="S1.I1.i2.p2.2.m2.1.1.cmml" xref="S1.I1.i2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S1.I1.i2.p2.2.m2.1.1.1.cmml" xref="S1.I1.i2.p2.2.m2.1.1">subscript</csymbol><ci id="S1.I1.i2.p2.2.m2.1.1.2.cmml" xref="S1.I1.i2.p2.2.m2.1.1.2">𝐴</ci><ci id="S1.I1.i2.p2.2.m2.1.1.3.cmml" xref="S1.I1.i2.p2.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i2.p2.2.m2.1c">A_{p}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i2.p2.2.m2.1d">italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> weights.</p> </div> </li> <li class="ltx_item" id="S1.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S1.I1.i3.p1"> <p class="ltx_p" id="S1.I1.i3.p1.3">We can allow for vector-valued weighted Sobolev spaces <math alttext="W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S1.I1.i3.p1.1.m1.5"><semantics id="S1.I1.i3.p1.1.m1.5a"><mrow id="S1.I1.i3.p1.1.m1.5.5" xref="S1.I1.i3.p1.1.m1.5.5.cmml"><msup id="S1.I1.i3.p1.1.m1.5.5.4" xref="S1.I1.i3.p1.1.m1.5.5.4.cmml"><mi id="S1.I1.i3.p1.1.m1.5.5.4.2" xref="S1.I1.i3.p1.1.m1.5.5.4.2.cmml">W</mi><mrow 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id="S1.I1.i3.p1.1.m1.5.5.2.2.2.cmml" xref="S1.I1.i3.p1.1.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S1.I1.i3.p1.1.m1.5.5.2.2.2.1.cmml" xref="S1.I1.i3.p1.1.m1.5.5.2.2.2">subscript</csymbol><ci id="S1.I1.i3.p1.1.m1.5.5.2.2.2.2.cmml" xref="S1.I1.i3.p1.1.m1.5.5.2.2.2.2">𝑤</ci><ci id="S1.I1.i3.p1.1.m1.5.5.2.2.2.3.cmml" xref="S1.I1.i3.p1.1.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S1.I1.i3.p1.1.m1.3.3.cmml" xref="S1.I1.i3.p1.1.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.1.m1.5c">W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.1.m1.5d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>, where <math alttext="X" class="ltx_Math" display="inline" id="S1.I1.i3.p1.2.m2.1"><semantics id="S1.I1.i3.p1.2.m2.1a"><mi id="S1.I1.i3.p1.2.m2.1.1" xref="S1.I1.i3.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.2.m2.1b"><ci id="S1.I1.i3.p1.2.m2.1.1.cmml" xref="S1.I1.i3.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.2.m2.1d">italic_X</annotation></semantics></math> is a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S1.I1.i3.p1.3.m3.1"><semantics id="S1.I1.i3.p1.3.m3.1a"><mi id="S1.I1.i3.p1.3.m3.1.1" xref="S1.I1.i3.p1.3.m3.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S1.I1.i3.p1.3.m3.1b"><ci id="S1.I1.i3.p1.3.m3.1.1.cmml" xref="S1.I1.i3.p1.3.m3.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.I1.i3.p1.3.m3.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S1.I1.i3.p1.3.m3.1d">roman_UMD</annotation></semantics></math> Banach space.</p> </div> </li> </ol> </div> <div class="ltx_para" id="S1.p6"> <p class="ltx_p" id="S1.p6.1">A key ingredient for proving Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a> is the characterisation of the trace spaces of weighted vector-valued Besov, Triebel-Lizorkin, Bessel potential and Sobolev spaces on the half-space. The results are summarised in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.2</span></a> (see Sections <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3" title="3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS1" title="4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a>).</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S1.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.1.1.1">Theorem 1.2</span></span><span class="ltx_text ltx_font_bold" id="S1.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S1.Thmtheorem2.p1"> <p class="ltx_p" id="S1.Thmtheorem2.p1.9"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.9.9">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.1.1.m1.2"><semantics id="S1.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S1.Thmtheorem2.p1.1.1.m1.2.3" xref="S1.Thmtheorem2.p1.1.1.m1.2.3.cmml"><mi id="S1.Thmtheorem2.p1.1.1.m1.2.3.2" xref="S1.Thmtheorem2.p1.1.1.m1.2.3.2.cmml">p</mi><mo 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xref="S1.Thmtheorem2.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S1.Thmtheorem2.p1.2.2.m2.2.3.3.2" xref="S1.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml"><mo id="S1.Thmtheorem2.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S1.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml">[</mo><mn id="S1.Thmtheorem2.p1.2.2.m2.1.1" xref="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml">1</mn><mo id="S1.Thmtheorem2.p1.2.2.m2.2.3.3.2.2" xref="S1.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S1.Thmtheorem2.p1.2.2.m2.2.2" mathvariant="normal" xref="S1.Thmtheorem2.p1.2.2.m2.2.2.cmml">∞</mi><mo id="S1.Thmtheorem2.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S1.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.2.2.m2.2b"><apply id="S1.Thmtheorem2.p1.2.2.m2.2.3.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.2.3"><in id="S1.Thmtheorem2.p1.2.2.m2.2.3.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.2.3.1"></in><ci id="S1.Thmtheorem2.p1.2.2.m2.2.3.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S1.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.2.3.3.2"><cn id="S1.Thmtheorem2.p1.2.2.m2.1.1.cmml" type="integer" xref="S1.Thmtheorem2.p1.2.2.m2.1.1">1</cn><infinity id="S1.Thmtheorem2.p1.2.2.m2.2.2.cmml" xref="S1.Thmtheorem2.p1.2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.3.3.m3.1"><semantics id="S1.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S1.Thmtheorem2.p1.3.3.m3.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">m</mi><mo id="S1.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S1.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml"><mi id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.3.3.m3.1b"><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1"><in id="S1.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.1"></in><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.2">𝑚</ci><apply id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2.cmml" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.2">ℕ</ci><cn id="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.3.3.m3.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.3.3.m3.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.4.4.m4.1"><semantics id="S1.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="S1.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">γ</mi><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1.cmml">></mo><mrow id="S1.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml"><mo id="S1.Thmtheorem2.p1.4.4.m4.1.1.3a" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">−</mo><mn id="S1.Thmtheorem2.p1.4.4.m4.1.1.3.2" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.4.4.m4.1b"><apply id="S1.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1"><gt id="S1.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.1"></gt><ci id="S1.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.2">𝛾</ci><apply id="S1.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3"><minus id="S1.Thmtheorem2.p1.4.4.m4.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3"></minus><cn id="S1.Thmtheorem2.p1.4.4.m4.1.1.3.2.cmml" type="integer" xref="S1.Thmtheorem2.p1.4.4.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.4.4.m4.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.4.4.m4.1d">italic_γ > - 1</annotation></semantics></math>, <math alttext="s>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.5.5.m5.1"><semantics id="S1.Thmtheorem2.p1.5.5.m5.1a"><mrow id="S1.Thmtheorem2.p1.5.5.m5.1.1" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.cmml"><mi id="S1.Thmtheorem2.p1.5.5.m5.1.1.2" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.2.cmml">s</mi><mo id="S1.Thmtheorem2.p1.5.5.m5.1.1.1" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.1.cmml">></mo><mrow id="S1.Thmtheorem2.p1.5.5.m5.1.1.3" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.cmml"><mi id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.2" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.2.cmml">m</mi><mo id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.1" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.1.cmml">+</mo><mfrac id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.cmml"><mrow id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.cmml"><mi id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.2" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.1" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.3" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.3" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.5.5.m5.1b"><apply id="S1.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1"><gt id="S1.Thmtheorem2.p1.5.5.m5.1.1.1.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.1"></gt><ci id="S1.Thmtheorem2.p1.5.5.m5.1.1.2.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.2">𝑠</ci><apply id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3"><plus id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.1"></plus><ci id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.2.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.2">𝑚</ci><apply id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3"><divide id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.1.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3"></divide><apply id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2"><plus id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.1.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.1"></plus><ci id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.2.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.2">𝛾</ci><cn id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.2.3">1</cn></apply><ci id="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.3.cmml" xref="S1.Thmtheorem2.p1.5.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.5.5.m5.1c">s>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.5.5.m5.1d">italic_s > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.6.6.m6.1"><semantics id="S1.Thmtheorem2.p1.6.6.m6.1a"><mi id="S1.Thmtheorem2.p1.6.6.m6.1.1" xref="S1.Thmtheorem2.p1.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.6.6.m6.1b"><ci id="S1.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S1.Thmtheorem2.p1.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.6.6.m6.1d">italic_X</annotation></semantics></math> be a Banach space. Let <math alttext="\operatorname{Tr}_{m}=\operatorname{Tr}\circ\partial_{1}^{m}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.7.7.m7.1"><semantics id="S1.Thmtheorem2.p1.7.7.m7.1a"><mrow id="S1.Thmtheorem2.p1.7.7.m7.1.1" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.cmml"><msub id="S1.Thmtheorem2.p1.7.7.m7.1.1.2" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.2.cmml"><mi id="S1.Thmtheorem2.p1.7.7.m7.1.1.2.2" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.2.2.cmml">Tr</mi><mi id="S1.Thmtheorem2.p1.7.7.m7.1.1.2.3" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.2.3.cmml">m</mi></msub><mo id="S1.Thmtheorem2.p1.7.7.m7.1.1.1" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.1.cmml">=</mo><mrow id="S1.Thmtheorem2.p1.7.7.m7.1.1.3" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.cmml"><mi id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.2" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.2.cmml">Tr</mi><mo id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.1" lspace="0em" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.1.cmml">∘</mo><msubsup id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.cmml"><mo id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.2" lspace="0.055em" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.2.cmml">∂</mo><mn id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.3" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.3.cmml">1</mn><mi id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.3" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.3.cmml">m</mi></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.7.7.m7.1b"><apply id="S1.Thmtheorem2.p1.7.7.m7.1.1.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1"><eq id="S1.Thmtheorem2.p1.7.7.m7.1.1.1.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.1"></eq><apply id="S1.Thmtheorem2.p1.7.7.m7.1.1.2.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.2"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.7.7.m7.1.1.2.1.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.2">subscript</csymbol><ci id="S1.Thmtheorem2.p1.7.7.m7.1.1.2.2.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.2.2">Tr</ci><ci id="S1.Thmtheorem2.p1.7.7.m7.1.1.2.3.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.2.3">𝑚</ci></apply><apply id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3"><compose id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.1"></compose><ci id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.2.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.2">Tr</ci><apply id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.1.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3">superscript</csymbol><apply id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.1.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3">subscript</csymbol><partialdiff id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.2.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.2"></partialdiff><cn id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.2.3">1</cn></apply><ci id="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.3.cmml" xref="S1.Thmtheorem2.p1.7.7.m7.1.1.3.3.3">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.7.7.m7.1c">\operatorname{Tr}_{m}=\operatorname{Tr}\circ\partial_{1}^{m}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.7.7.m7.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = roman_Tr ∘ ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> be the <math alttext="m" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.8.8.m8.1"><semantics id="S1.Thmtheorem2.p1.8.8.m8.1a"><mi id="S1.Thmtheorem2.p1.8.8.m8.1.1" xref="S1.Thmtheorem2.p1.8.8.m8.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.8.8.m8.1b"><ci id="S1.Thmtheorem2.p1.8.8.m8.1.1.cmml" xref="S1.Thmtheorem2.p1.8.8.m8.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.8.8.m8.1c">m</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.8.8.m8.1d">italic_m</annotation></semantics></math>-th order trace operator at <math alttext="\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.9.9.m9.4"><semantics id="S1.Thmtheorem2.p1.9.9.m9.4a"><mrow id="S1.Thmtheorem2.p1.9.9.m9.4.4.2" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.3.cmml"><mo id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.3" stretchy="false" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.3.1.cmml">{</mo><mrow id="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.2" xref="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.cmml"><mo id="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.1" stretchy="false" xref="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.cmml">(</mo><mn id="S1.Thmtheorem2.p1.9.9.m9.1.1" xref="S1.Thmtheorem2.p1.9.9.m9.1.1.cmml">0</mn><mo id="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.2" xref="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.cmml">,</mo><mover accent="true" id="S1.Thmtheorem2.p1.9.9.m9.2.2" xref="S1.Thmtheorem2.p1.9.9.m9.2.2.cmml"><mi id="S1.Thmtheorem2.p1.9.9.m9.2.2.2" xref="S1.Thmtheorem2.p1.9.9.m9.2.2.2.cmml">x</mi><mo id="S1.Thmtheorem2.p1.9.9.m9.2.2.1" xref="S1.Thmtheorem2.p1.9.9.m9.2.2.1.cmml">~</mo></mover><mo id="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.2.3" rspace="0.278em" stretchy="false" xref="S1.Thmtheorem2.p1.9.9.m9.3.3.1.1.1.cmml">)</mo></mrow><mo id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.4" rspace="0.278em" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.3.1.cmml">:</mo><mrow id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.cmml"><mover accent="true" id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.cmml"><mi id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.2" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.2.cmml">x</mi><mo id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.1" 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id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.1.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.1">~</ci><ci id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.2.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.2.2">𝑥</ci></apply><apply id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.1.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3">superscript</csymbol><ci id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.2.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.2">ℝ</ci><apply id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3"><minus id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3.1.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3.1"></minus><ci id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3.2.cmml" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3.2">𝑑</ci><cn id="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.9.9.m9.4.4.2.2.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.9.9.m9.4c">\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.9.9.m9.4d">{ ( 0 , over~ start_ARG italic_x end_ARG ) : over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math>. Then</span></p> <ol class="ltx_enumerate" id="S1.I2"> <li class="ltx_item" id="S1.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S1.I2.i1.p1"> <p class="ltx_p" id="S1.I2.i1.p1.1"><math alttext="\operatorname{Tr}_{m}\big{(}B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\big{)}=B^% {s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S1.I2.i1.p1.1.m1.9"><semantics id="S1.I2.i1.p1.1.m1.9a"><mrow id="S1.I2.i1.p1.1.m1.9.9" xref="S1.I2.i1.p1.1.m1.9.9.cmml"><mrow id="S1.I2.i1.p1.1.m1.8.8.2.2" xref="S1.I2.i1.p1.1.m1.8.8.2.3.cmml"><msub id="S1.I2.i1.p1.1.m1.7.7.1.1.1" xref="S1.I2.i1.p1.1.m1.7.7.1.1.1.cmml"><mi id="S1.I2.i1.p1.1.m1.7.7.1.1.1.2" xref="S1.I2.i1.p1.1.m1.7.7.1.1.1.2.cmml">Tr</mi><mi id="S1.I2.i1.p1.1.m1.7.7.1.1.1.3" xref="S1.I2.i1.p1.1.m1.7.7.1.1.1.3.cmml">m</mi></msub><mo id="S1.I2.i1.p1.1.m1.8.8.2.2a" xref="S1.I2.i1.p1.1.m1.8.8.2.3.cmml">⁡</mo><mrow id="S1.I2.i1.p1.1.m1.8.8.2.2.2" 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start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) = italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i1.p1.1.1">,</span></p> </div> </li> <li class="ltx_item" id="S1.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S1.I2.i2.p1"> <p class="ltx_p" id="S1.I2.i2.p1.1"><math alttext="\operatorname{Tr}_{m}\big{(}F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\big{)}=B^% 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xref="S1.I2.i2.p1.1.m1.9.9.3.1.1.1.3.1"></minus><ci id="S1.I2.i2.p1.1.m1.9.9.3.1.1.1.3.2.cmml" xref="S1.I2.i2.p1.1.m1.9.9.3.1.1.1.3.2">𝑑</ci><cn id="S1.I2.i2.p1.1.m1.9.9.3.1.1.1.3.3.cmml" type="integer" xref="S1.I2.i2.p1.1.m1.9.9.3.1.1.1.3.3">1</cn></apply></apply><ci id="S1.I2.i2.p1.1.m1.6.6.cmml" xref="S1.I2.i2.p1.1.m1.6.6">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i2.p1.1.m1.9c">\operatorname{Tr}_{m}\big{(}F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\big{)}=B^% {s-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i2.p1.1.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) = italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i2.p1.1.1">,</span></p> </div> </li> <li class="ltx_item" id="S1.I2.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S1.I2.i3.p1"> <p class="ltx_p" id="S1.I2.i3.p1.2"><math alttext="\operatorname{Tr}_{m}\big{(}H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{)}=B^% {s-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S1.I2.i3.p1.1.m1.9"><semantics id="S1.I2.i3.p1.1.m1.9a"><mrow id="S1.I2.i3.p1.1.m1.9.9" xref="S1.I2.i3.p1.1.m1.9.9.cmml"><mrow id="S1.I2.i3.p1.1.m1.8.8.2.2" xref="S1.I2.i3.p1.1.m1.8.8.2.3.cmml"><msub id="S1.I2.i3.p1.1.m1.7.7.1.1.1" 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xref="S1.I2.i3.p1.1.m1.6.6">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i3.p1.1.m1.9c">\operatorname{Tr}_{m}\big{(}H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{)}=B^% {s-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i3.p1.1.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) = italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i3.p1.2.1"> if </span><math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S1.I2.i3.p1.2.m2.2"><semantics id="S1.I2.i3.p1.2.m2.2a"><mrow id="S1.I2.i3.p1.2.m2.2.2" xref="S1.I2.i3.p1.2.m2.2.2.cmml"><mi id="S1.I2.i3.p1.2.m2.2.2.4" xref="S1.I2.i3.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S1.I2.i3.p1.2.m2.2.2.3" xref="S1.I2.i3.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S1.I2.i3.p1.2.m2.2.2.2.2" xref="S1.I2.i3.p1.2.m2.2.2.2.3.cmml"><mo id="S1.I2.i3.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S1.I2.i3.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S1.I2.i3.p1.2.m2.1.1.1.1.1" xref="S1.I2.i3.p1.2.m2.1.1.1.1.1.cmml"><mo id="S1.I2.i3.p1.2.m2.1.1.1.1.1a" xref="S1.I2.i3.p1.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S1.I2.i3.p1.2.m2.1.1.1.1.1.2" xref="S1.I2.i3.p1.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.I2.i3.p1.2.m2.2.2.2.2.4" xref="S1.I2.i3.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="S1.I2.i3.p1.2.m2.2.2.2.2.2" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2.cmml"><mi id="S1.I2.i3.p1.2.m2.2.2.2.2.2.2" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2.2.cmml">p</mi><mo id="S1.I2.i3.p1.2.m2.2.2.2.2.2.1" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2.1.cmml">−</mo><mn id="S1.I2.i3.p1.2.m2.2.2.2.2.2.3" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S1.I2.i3.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S1.I2.i3.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i3.p1.2.m2.2b"><apply id="S1.I2.i3.p1.2.m2.2.2.cmml" xref="S1.I2.i3.p1.2.m2.2.2"><in id="S1.I2.i3.p1.2.m2.2.2.3.cmml" xref="S1.I2.i3.p1.2.m2.2.2.3"></in><ci id="S1.I2.i3.p1.2.m2.2.2.4.cmml" xref="S1.I2.i3.p1.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S1.I2.i3.p1.2.m2.2.2.2.3.cmml" xref="S1.I2.i3.p1.2.m2.2.2.2.2"><apply id="S1.I2.i3.p1.2.m2.1.1.1.1.1.cmml" xref="S1.I2.i3.p1.2.m2.1.1.1.1.1"><minus id="S1.I2.i3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S1.I2.i3.p1.2.m2.1.1.1.1.1"></minus><cn id="S1.I2.i3.p1.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S1.I2.i3.p1.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S1.I2.i3.p1.2.m2.2.2.2.2.2.cmml" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2"><minus id="S1.I2.i3.p1.2.m2.2.2.2.2.2.1.cmml" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2.1"></minus><ci id="S1.I2.i3.p1.2.m2.2.2.2.2.2.2.cmml" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2.2">𝑝</ci><cn id="S1.I2.i3.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S1.I2.i3.p1.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i3.p1.2.m2.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i3.p1.2.m2.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i3.p1.2.2">,</span></p> </div> </li> <li class="ltx_item" id="S1.I2.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iv)</span> <div class="ltx_para" id="S1.I2.i4.p1"> <p class="ltx_p" id="S1.I2.i4.p1.3"><math alttext="\operatorname{Tr}_{m}\big{(}W^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{)}=B^% {s-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S1.I2.i4.p1.1.m1.9"><semantics id="S1.I2.i4.p1.1.m1.9a"><mrow id="S1.I2.i4.p1.1.m1.9.9" xref="S1.I2.i4.p1.1.m1.9.9.cmml"><mrow id="S1.I2.i4.p1.1.m1.8.8.2.2" xref="S1.I2.i4.p1.1.m1.8.8.2.3.cmml"><msub id="S1.I2.i4.p1.1.m1.7.7.1.1.1" xref="S1.I2.i4.p1.1.m1.7.7.1.1.1.cmml"><mi id="S1.I2.i4.p1.1.m1.7.7.1.1.1.2" xref="S1.I2.i4.p1.1.m1.7.7.1.1.1.2.cmml">Tr</mi><mi id="S1.I2.i4.p1.1.m1.7.7.1.1.1.3" xref="S1.I2.i4.p1.1.m1.7.7.1.1.1.3.cmml">m</mi></msub><mo id="S1.I2.i4.p1.1.m1.8.8.2.2a" xref="S1.I2.i4.p1.1.m1.8.8.2.3.cmml">⁡</mo><mrow id="S1.I2.i4.p1.1.m1.8.8.2.2.2" xref="S1.I2.i4.p1.1.m1.8.8.2.3.cmml"><mo id="S1.I2.i4.p1.1.m1.8.8.2.2.2.2" maxsize="120%" minsize="120%" xref="S1.I2.i4.p1.1.m1.8.8.2.3.cmml">(</mo><mrow id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.cmml"><msup id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.4" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.4.cmml"><mi id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.4.2" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.4.2.cmml">W</mi><mrow id="S1.I2.i4.p1.1.m1.2.2.2.4" xref="S1.I2.i4.p1.1.m1.2.2.2.3.cmml"><mi id="S1.I2.i4.p1.1.m1.1.1.1.1" xref="S1.I2.i4.p1.1.m1.1.1.1.1.cmml">s</mi><mo id="S1.I2.i4.p1.1.m1.2.2.2.4.1" xref="S1.I2.i4.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S1.I2.i4.p1.1.m1.2.2.2.2" xref="S1.I2.i4.p1.1.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.3" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.3.cmml">⁢</mo><mrow id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.2.2" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.2.3.cmml"><mo id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.2.2.3" stretchy="false" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.2.3.cmml">(</mo><msubsup id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.1.1.1" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.1.1.1.cmml"><mi id="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.1.1.1.2.2" xref="S1.I2.i4.p1.1.m1.8.8.2.2.2.1.1.1.1.2.2.cmml">ℝ</mi><mo 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xref="S1.I2.i4.p1.1.m1.8.8.2.3.cmml">)</mo></mrow></mrow><mo id="S1.I2.i4.p1.1.m1.9.9.4" xref="S1.I2.i4.p1.1.m1.9.9.4.cmml">=</mo><mrow id="S1.I2.i4.p1.1.m1.9.9.3" xref="S1.I2.i4.p1.1.m1.9.9.3.cmml"><msubsup id="S1.I2.i4.p1.1.m1.9.9.3.3" xref="S1.I2.i4.p1.1.m1.9.9.3.3.cmml"><mi id="S1.I2.i4.p1.1.m1.9.9.3.3.2.2" xref="S1.I2.i4.p1.1.m1.9.9.3.3.2.2.cmml">B</mi><mrow id="S1.I2.i4.p1.1.m1.4.4.2.4" xref="S1.I2.i4.p1.1.m1.4.4.2.3.cmml"><mi id="S1.I2.i4.p1.1.m1.3.3.1.1" xref="S1.I2.i4.p1.1.m1.3.3.1.1.cmml">p</mi><mo id="S1.I2.i4.p1.1.m1.4.4.2.4.1" xref="S1.I2.i4.p1.1.m1.4.4.2.3.cmml">,</mo><mi id="S1.I2.i4.p1.1.m1.4.4.2.2" xref="S1.I2.i4.p1.1.m1.4.4.2.2.cmml">p</mi></mrow><mrow id="S1.I2.i4.p1.1.m1.9.9.3.3.2.3" xref="S1.I2.i4.p1.1.m1.9.9.3.3.2.3.cmml"><mi id="S1.I2.i4.p1.1.m1.9.9.3.3.2.3.2" xref="S1.I2.i4.p1.1.m1.9.9.3.3.2.3.2.cmml">s</mi><mo id="S1.I2.i4.p1.1.m1.9.9.3.3.2.3.1" xref="S1.I2.i4.p1.1.m1.9.9.3.3.2.3.1.cmml">−</mo><mi id="S1.I2.i4.p1.1.m1.9.9.3.3.2.3.3" 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italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) = italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i4.p1.3.1"> if </span><math alttext="s\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S1.I2.i4.p1.2.m2.1"><semantics id="S1.I2.i4.p1.2.m2.1a"><mrow id="S1.I2.i4.p1.2.m2.1.1" xref="S1.I2.i4.p1.2.m2.1.1.cmml"><mi id="S1.I2.i4.p1.2.m2.1.1.2" xref="S1.I2.i4.p1.2.m2.1.1.2.cmml">s</mi><mo id="S1.I2.i4.p1.2.m2.1.1.1" xref="S1.I2.i4.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S1.I2.i4.p1.2.m2.1.1.3" xref="S1.I2.i4.p1.2.m2.1.1.3.cmml"><mi id="S1.I2.i4.p1.2.m2.1.1.3.2" xref="S1.I2.i4.p1.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S1.I2.i4.p1.2.m2.1.1.3.3" xref="S1.I2.i4.p1.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i4.p1.2.m2.1b"><apply id="S1.I2.i4.p1.2.m2.1.1.cmml" xref="S1.I2.i4.p1.2.m2.1.1"><in id="S1.I2.i4.p1.2.m2.1.1.1.cmml" xref="S1.I2.i4.p1.2.m2.1.1.1"></in><ci id="S1.I2.i4.p1.2.m2.1.1.2.cmml" xref="S1.I2.i4.p1.2.m2.1.1.2">𝑠</ci><apply id="S1.I2.i4.p1.2.m2.1.1.3.cmml" xref="S1.I2.i4.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.I2.i4.p1.2.m2.1.1.3.1.cmml" xref="S1.I2.i4.p1.2.m2.1.1.3">subscript</csymbol><ci id="S1.I2.i4.p1.2.m2.1.1.3.2.cmml" xref="S1.I2.i4.p1.2.m2.1.1.3.2">ℕ</ci><cn id="S1.I2.i4.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S1.I2.i4.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i4.p1.2.m2.1c">s\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i4.p1.2.m2.1d">italic_s ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i4.p1.3.2"> and </span><math alttext="\gamma\notin\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S1.I2.i4.p1.3.m3.2"><semantics id="S1.I2.i4.p1.3.m3.2a"><mrow id="S1.I2.i4.p1.3.m3.2.2" xref="S1.I2.i4.p1.3.m3.2.2.cmml"><mi id="S1.I2.i4.p1.3.m3.2.2.4" xref="S1.I2.i4.p1.3.m3.2.2.4.cmml">γ</mi><mo id="S1.I2.i4.p1.3.m3.2.2.3" xref="S1.I2.i4.p1.3.m3.2.2.3.cmml">∉</mo><mrow id="S1.I2.i4.p1.3.m3.2.2.2.2" xref="S1.I2.i4.p1.3.m3.2.2.2.3.cmml"><mo id="S1.I2.i4.p1.3.m3.2.2.2.2.3" stretchy="false" xref="S1.I2.i4.p1.3.m3.2.2.2.3.1.cmml">{</mo><mrow id="S1.I2.i4.p1.3.m3.1.1.1.1.1" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.cmml"><mrow id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.2" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.2.cmml">j</mi><mo id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.1" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.3" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.3.cmml">p</mi></mrow><mo id="S1.I2.i4.p1.3.m3.1.1.1.1.1.1" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.1.cmml">−</mo><mn id="S1.I2.i4.p1.3.m3.1.1.1.1.1.3" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S1.I2.i4.p1.3.m3.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S1.I2.i4.p1.3.m3.2.2.2.3.1.cmml">:</mo><mrow id="S1.I2.i4.p1.3.m3.2.2.2.2.2" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.cmml"><mi id="S1.I2.i4.p1.3.m3.2.2.2.2.2.2" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.2.cmml">j</mi><mo id="S1.I2.i4.p1.3.m3.2.2.2.2.2.1" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.1.cmml">∈</mo><msub id="S1.I2.i4.p1.3.m3.2.2.2.2.2.3" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.cmml"><mi id="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.2" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.2.cmml">ℕ</mi><mn id="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.3" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S1.I2.i4.p1.3.m3.2.2.2.2.5" stretchy="false" xref="S1.I2.i4.p1.3.m3.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.I2.i4.p1.3.m3.2b"><apply id="S1.I2.i4.p1.3.m3.2.2.cmml" xref="S1.I2.i4.p1.3.m3.2.2"><notin id="S1.I2.i4.p1.3.m3.2.2.3.cmml" xref="S1.I2.i4.p1.3.m3.2.2.3"></notin><ci id="S1.I2.i4.p1.3.m3.2.2.4.cmml" xref="S1.I2.i4.p1.3.m3.2.2.4">𝛾</ci><apply id="S1.I2.i4.p1.3.m3.2.2.2.3.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2"><csymbol cd="latexml" id="S1.I2.i4.p1.3.m3.2.2.2.3.1.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2.3">conditional-set</csymbol><apply id="S1.I2.i4.p1.3.m3.1.1.1.1.1.cmml" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1"><minus id="S1.I2.i4.p1.3.m3.1.1.1.1.1.1.cmml" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.1"></minus><apply id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.cmml" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2"><times id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.1"></times><ci id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.2">𝑗</ci><ci id="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.2.3">𝑝</ci></apply><cn id="S1.I2.i4.p1.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S1.I2.i4.p1.3.m3.1.1.1.1.1.3">1</cn></apply><apply id="S1.I2.i4.p1.3.m3.2.2.2.2.2.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2"><in id="S1.I2.i4.p1.3.m3.2.2.2.2.2.1.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.1"></in><ci id="S1.I2.i4.p1.3.m3.2.2.2.2.2.2.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.2">𝑗</ci><apply id="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.1.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.3">subscript</csymbol><ci id="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.2.cmml" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.2">ℕ</ci><cn id="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.3.cmml" type="integer" xref="S1.I2.i4.p1.3.m3.2.2.2.2.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.I2.i4.p1.3.m3.2c">\gamma\notin\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S1.I2.i4.p1.3.m3.2d">italic_γ ∉ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S1.I2.i4.p1.3.3">,</span></p> </div> </li> </ol> <p class="ltx_p" id="S1.Thmtheorem2.p1.12"><span class="ltx_text ltx_font_italic" id="S1.Thmtheorem2.p1.12.3">where the notation <math alttext="\operatorname{Tr}_{m}(\mathcal{A}_{1})=\mathcal{A}_{2}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.10.1.m1.2"><semantics id="S1.Thmtheorem2.p1.10.1.m1.2a"><mrow id="S1.Thmtheorem2.p1.10.1.m1.2.2" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.cmml"><mrow id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.3.cmml"><msub id="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1" xref="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.cmml"><mi id="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.2" xref="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.2.cmml">Tr</mi><mi id="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.3" xref="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.3.cmml">m</mi></msub><mo id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2a" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.3.cmml">⁡</mo><mrow id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.3.cmml"><mo id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.2" stretchy="false" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.3.cmml">(</mo><msub id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.2" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.2.cmml">𝒜</mi><mn id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.3" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.3" stretchy="false" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S1.Thmtheorem2.p1.10.1.m1.2.2.3" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.3.cmml">=</mo><msub id="S1.Thmtheorem2.p1.10.1.m1.2.2.4" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.10.1.m1.2.2.4.2" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.4.2.cmml">𝒜</mi><mn id="S1.Thmtheorem2.p1.10.1.m1.2.2.4.3" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.4.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.10.1.m1.2b"><apply id="S1.Thmtheorem2.p1.10.1.m1.2.2.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2"><eq id="S1.Thmtheorem2.p1.10.1.m1.2.2.3.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.3"></eq><apply id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.3.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2"><apply id="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.1.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.2.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.2">Tr</ci><ci id="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.3.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.1.1.1.1.1.3">𝑚</ci></apply><apply id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.1.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1">subscript</csymbol><ci id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.2.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.2">𝒜</ci><cn id="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.2.2.2.1.3">1</cn></apply></apply><apply id="S1.Thmtheorem2.p1.10.1.m1.2.2.4.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.4"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.10.1.m1.2.2.4.1.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.4">subscript</csymbol><ci id="S1.Thmtheorem2.p1.10.1.m1.2.2.4.2.cmml" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.4.2">𝒜</ci><cn id="S1.Thmtheorem2.p1.10.1.m1.2.2.4.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.10.1.m1.2.2.4.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.10.1.m1.2c">\operatorname{Tr}_{m}(\mathcal{A}_{1})=\mathcal{A}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.10.1.m1.2d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> means that <math alttext="\operatorname{Tr}_{m}:\mathcal{A}_{1}\to\mathcal{A}_{2}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.11.2.m2.1"><semantics id="S1.Thmtheorem2.p1.11.2.m2.1a"><mrow id="S1.Thmtheorem2.p1.11.2.m2.1.1" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.cmml"><msub id="S1.Thmtheorem2.p1.11.2.m2.1.1.2" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.2.cmml"><mi id="S1.Thmtheorem2.p1.11.2.m2.1.1.2.2" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.2.2.cmml">Tr</mi><mi id="S1.Thmtheorem2.p1.11.2.m2.1.1.2.3" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.2.3.cmml">m</mi></msub><mo id="S1.Thmtheorem2.p1.11.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.1.cmml">:</mo><mrow id="S1.Thmtheorem2.p1.11.2.m2.1.1.3" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.cmml"><msub id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.2" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.2.cmml">𝒜</mi><mn id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.3" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.3.cmml">1</mn></msub><mo id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.1" stretchy="false" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.1.cmml">→</mo><msub id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.2" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.2.cmml">𝒜</mi><mn id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.3" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.11.2.m2.1b"><apply id="S1.Thmtheorem2.p1.11.2.m2.1.1.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1"><ci id="S1.Thmtheorem2.p1.11.2.m2.1.1.1.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.1">:</ci><apply id="S1.Thmtheorem2.p1.11.2.m2.1.1.2.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.2"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.11.2.m2.1.1.2.1.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.2">subscript</csymbol><ci id="S1.Thmtheorem2.p1.11.2.m2.1.1.2.2.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.2.2">Tr</ci><ci id="S1.Thmtheorem2.p1.11.2.m2.1.1.2.3.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.2.3">𝑚</ci></apply><apply id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3"><ci id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.1.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.1">→</ci><apply id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.1.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2">subscript</csymbol><ci id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.2.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.2">𝒜</ci><cn id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.2.3">1</cn></apply><apply id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.1.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3">subscript</csymbol><ci id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.2.cmml" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.2">𝒜</ci><cn id="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.3.cmml" type="integer" xref="S1.Thmtheorem2.p1.11.2.m2.1.1.3.3.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.11.2.m2.1c">\operatorname{Tr}_{m}:\mathcal{A}_{1}\to\mathcal{A}_{2}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.11.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → caligraphic_A start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> is continuous and surjective, and that it has a continuous right inverse (the so-called extension operator <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S1.Thmtheorem2.p1.12.3.m3.1"><semantics id="S1.Thmtheorem2.p1.12.3.m3.1a"><msub id="S1.Thmtheorem2.p1.12.3.m3.1.1" xref="S1.Thmtheorem2.p1.12.3.m3.1.1.cmml"><mi id="S1.Thmtheorem2.p1.12.3.m3.1.1.2" xref="S1.Thmtheorem2.p1.12.3.m3.1.1.2.cmml">ext</mi><mi id="S1.Thmtheorem2.p1.12.3.m3.1.1.3" xref="S1.Thmtheorem2.p1.12.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S1.Thmtheorem2.p1.12.3.m3.1b"><apply id="S1.Thmtheorem2.p1.12.3.m3.1.1.cmml" xref="S1.Thmtheorem2.p1.12.3.m3.1.1"><csymbol cd="ambiguous" id="S1.Thmtheorem2.p1.12.3.m3.1.1.1.cmml" xref="S1.Thmtheorem2.p1.12.3.m3.1.1">subscript</csymbol><ci id="S1.Thmtheorem2.p1.12.3.m3.1.1.2.cmml" xref="S1.Thmtheorem2.p1.12.3.m3.1.1.2">ext</ci><ci id="S1.Thmtheorem2.p1.12.3.m3.1.1.3.cmml" xref="S1.Thmtheorem2.p1.12.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Thmtheorem2.p1.12.3.m3.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S1.Thmtheorem2.p1.12.3.m3.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>).</span></p> </div> </div> <div class="ltx_para" id="S1.p7"> <p class="ltx_p" id="S1.p7.1">The innovative part of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.2</span></a> is the extension of the range of weights for Sobolev spaces, which eventually allows us to prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a> for these weights as well. To prove our weighted results, we partially extend the arguments in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>]</cite> for the unweighted case. Moreover, we apply a Sobolev embedding for Triebel-Lizorkin spaces to deduce Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.2</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.I2.i2" title="item ii ‣ Theorem 1.2. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>. This is in contrast to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>]</cite>, where an atomic approach was used.</p> </div> <div class="ltx_para" id="S1.p8"> <p class="ltx_p" id="S1.p8.2">In the unweighted scalar-valued setting, i.e., <math alttext="\gamma=0" class="ltx_Math" display="inline" id="S1.p8.1.m1.1"><semantics id="S1.p8.1.m1.1a"><mrow id="S1.p8.1.m1.1.1" xref="S1.p8.1.m1.1.1.cmml"><mi id="S1.p8.1.m1.1.1.2" xref="S1.p8.1.m1.1.1.2.cmml">γ</mi><mo id="S1.p8.1.m1.1.1.1" xref="S1.p8.1.m1.1.1.1.cmml">=</mo><mn id="S1.p8.1.m1.1.1.3" xref="S1.p8.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.1.m1.1b"><apply id="S1.p8.1.m1.1.1.cmml" xref="S1.p8.1.m1.1.1"><eq id="S1.p8.1.m1.1.1.1.cmml" xref="S1.p8.1.m1.1.1.1"></eq><ci id="S1.p8.1.m1.1.1.2.cmml" xref="S1.p8.1.m1.1.1.2">𝛾</ci><cn id="S1.p8.1.m1.1.1.3.cmml" type="integer" xref="S1.p8.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.1.m1.1c">\gamma=0</annotation><annotation encoding="application/x-llamapun" id="S1.p8.1.m1.1d">italic_γ = 0</annotation></semantics></math> and <math alttext="X=\mathbb{C}" class="ltx_Math" display="inline" id="S1.p8.2.m2.1"><semantics id="S1.p8.2.m2.1a"><mrow id="S1.p8.2.m2.1.1" xref="S1.p8.2.m2.1.1.cmml"><mi id="S1.p8.2.m2.1.1.2" xref="S1.p8.2.m2.1.1.2.cmml">X</mi><mo id="S1.p8.2.m2.1.1.1" xref="S1.p8.2.m2.1.1.1.cmml">=</mo><mi id="S1.p8.2.m2.1.1.3" xref="S1.p8.2.m2.1.1.3.cmml">ℂ</mi></mrow><annotation-xml encoding="MathML-Content" id="S1.p8.2.m2.1b"><apply id="S1.p8.2.m2.1.1.cmml" xref="S1.p8.2.m2.1.1"><eq id="S1.p8.2.m2.1.1.1.cmml" xref="S1.p8.2.m2.1.1.1"></eq><ci id="S1.p8.2.m2.1.1.2.cmml" xref="S1.p8.2.m2.1.1.2">𝑋</ci><ci id="S1.p8.2.m2.1.1.3.cmml" xref="S1.p8.2.m2.1.1.3">ℂ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p8.2.m2.1c">X=\mathbb{C}</annotation><annotation encoding="application/x-llamapun" id="S1.p8.2.m2.1d">italic_X = blackboard_C</annotation></semantics></math>, the trace spaces are well known, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib63" title="">63</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib68" title="">68</a>]</cite>. In the unweighted vector-valued setting, the trace spaces are characterised in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>]</cite>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S1.p9"> <p class="ltx_p" id="S1.p9.1">As a consequence of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.2</span></a>, we can derive a trace theorem for the vector <math alttext="\overline{\operatorname{Tr}}_{m}=(\operatorname{Tr}_{0},\dots,\operatorname{Tr% }_{m})" class="ltx_Math" display="inline" id="S1.p9.1.m1.3"><semantics id="S1.p9.1.m1.3a"><mrow id="S1.p9.1.m1.3.3" xref="S1.p9.1.m1.3.3.cmml"><msub id="S1.p9.1.m1.3.3.4" xref="S1.p9.1.m1.3.3.4.cmml"><mover accent="true" id="S1.p9.1.m1.3.3.4.2" xref="S1.p9.1.m1.3.3.4.2.cmml"><mi id="S1.p9.1.m1.3.3.4.2.2" xref="S1.p9.1.m1.3.3.4.2.2.cmml">Tr</mi><mo id="S1.p9.1.m1.3.3.4.2.1" xref="S1.p9.1.m1.3.3.4.2.1.cmml">¯</mo></mover><mi id="S1.p9.1.m1.3.3.4.3" xref="S1.p9.1.m1.3.3.4.3.cmml">m</mi></msub><mo id="S1.p9.1.m1.3.3.3" xref="S1.p9.1.m1.3.3.3.cmml">=</mo><mrow id="S1.p9.1.m1.3.3.2.2" xref="S1.p9.1.m1.3.3.2.3.cmml"><mo id="S1.p9.1.m1.3.3.2.2.3" stretchy="false" xref="S1.p9.1.m1.3.3.2.3.cmml">(</mo><msub id="S1.p9.1.m1.2.2.1.1.1" xref="S1.p9.1.m1.2.2.1.1.1.cmml"><mi id="S1.p9.1.m1.2.2.1.1.1.2" xref="S1.p9.1.m1.2.2.1.1.1.2.cmml">Tr</mi><mn id="S1.p9.1.m1.2.2.1.1.1.3" xref="S1.p9.1.m1.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S1.p9.1.m1.3.3.2.2.4" xref="S1.p9.1.m1.3.3.2.3.cmml">,</mo><mi id="S1.p9.1.m1.1.1" mathvariant="normal" xref="S1.p9.1.m1.1.1.cmml">…</mi><mo id="S1.p9.1.m1.3.3.2.2.5" xref="S1.p9.1.m1.3.3.2.3.cmml">,</mo><msub id="S1.p9.1.m1.3.3.2.2.2" xref="S1.p9.1.m1.3.3.2.2.2.cmml"><mi id="S1.p9.1.m1.3.3.2.2.2.2" xref="S1.p9.1.m1.3.3.2.2.2.2.cmml">Tr</mi><mi id="S1.p9.1.m1.3.3.2.2.2.3" xref="S1.p9.1.m1.3.3.2.2.2.3.cmml">m</mi></msub><mo id="S1.p9.1.m1.3.3.2.2.6" stretchy="false" xref="S1.p9.1.m1.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p9.1.m1.3b"><apply id="S1.p9.1.m1.3.3.cmml" xref="S1.p9.1.m1.3.3"><eq id="S1.p9.1.m1.3.3.3.cmml" xref="S1.p9.1.m1.3.3.3"></eq><apply id="S1.p9.1.m1.3.3.4.cmml" xref="S1.p9.1.m1.3.3.4"><csymbol cd="ambiguous" id="S1.p9.1.m1.3.3.4.1.cmml" xref="S1.p9.1.m1.3.3.4">subscript</csymbol><apply id="S1.p9.1.m1.3.3.4.2.cmml" xref="S1.p9.1.m1.3.3.4.2"><ci id="S1.p9.1.m1.3.3.4.2.1.cmml" xref="S1.p9.1.m1.3.3.4.2.1">¯</ci><ci id="S1.p9.1.m1.3.3.4.2.2.cmml" xref="S1.p9.1.m1.3.3.4.2.2">Tr</ci></apply><ci id="S1.p9.1.m1.3.3.4.3.cmml" xref="S1.p9.1.m1.3.3.4.3">𝑚</ci></apply><vector id="S1.p9.1.m1.3.3.2.3.cmml" xref="S1.p9.1.m1.3.3.2.2"><apply id="S1.p9.1.m1.2.2.1.1.1.cmml" xref="S1.p9.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S1.p9.1.m1.2.2.1.1.1.1.cmml" xref="S1.p9.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S1.p9.1.m1.2.2.1.1.1.2.cmml" xref="S1.p9.1.m1.2.2.1.1.1.2">Tr</ci><cn id="S1.p9.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S1.p9.1.m1.2.2.1.1.1.3">0</cn></apply><ci id="S1.p9.1.m1.1.1.cmml" xref="S1.p9.1.m1.1.1">…</ci><apply id="S1.p9.1.m1.3.3.2.2.2.cmml" xref="S1.p9.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S1.p9.1.m1.3.3.2.2.2.1.cmml" xref="S1.p9.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S1.p9.1.m1.3.3.2.2.2.2.cmml" xref="S1.p9.1.m1.3.3.2.2.2.2">Tr</ci><ci id="S1.p9.1.m1.3.3.2.2.2.3.cmml" xref="S1.p9.1.m1.3.3.2.2.2.3">𝑚</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p9.1.m1.3c">\overline{\operatorname{Tr}}_{m}=(\operatorname{Tr}_{0},\dots,\operatorname{Tr% }_{m})</annotation><annotation encoding="application/x-llamapun" id="S1.p9.1.m1.3d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = ( roman_Tr start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )</annotation></semantics></math> as well. For weighted scalar-valued Sobolev spaces this is proved in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>, Section 2.9.2]</cite> via different techniques.</p> </div> <div class="ltx_para" id="S1.p10"> <p class="ltx_p" id="S1.p10.4">Moreover, from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem2" title="Theorem 1.2. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.2</span></a> we derive some density results in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS2" title="4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.2</span></a>. In particular, we show that the space of test functions</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\bigl{\{}f\in C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X):(% \partial_{1}^{m}f)|_{\partial\mathbb{R}^{d}_{+}}=0\bigr{\}}" class="ltx_Math" display="block" id="S1.Ex1.m1.5"><semantics id="S1.Ex1.m1.5a"><mrow id="S1.Ex1.m1.5.5.2" xref="S1.Ex1.m1.5.5.3.cmml"><mo id="S1.Ex1.m1.5.5.2.3" maxsize="120%" minsize="120%" xref="S1.Ex1.m1.5.5.3.1.cmml">{</mo><mrow id="S1.Ex1.m1.4.4.1.1" xref="S1.Ex1.m1.4.4.1.1.cmml"><mi id="S1.Ex1.m1.4.4.1.1.2" xref="S1.Ex1.m1.4.4.1.1.2.cmml">f</mi><mo id="S1.Ex1.m1.4.4.1.1.1" xref="S1.Ex1.m1.4.4.1.1.1.cmml">∈</mo><mrow id="S1.Ex1.m1.4.4.1.1.3" xref="S1.Ex1.m1.4.4.1.1.3.cmml"><msubsup id="S1.Ex1.m1.4.4.1.1.3.2" xref="S1.Ex1.m1.4.4.1.1.3.2.cmml"><mi 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cd="ambiguous" id="S1.p10.1.m1.5.5.2.2.2.1.cmml" xref="S1.p10.1.m1.5.5.2.2.2">subscript</csymbol><ci id="S1.p10.1.m1.5.5.2.2.2.2.cmml" xref="S1.p10.1.m1.5.5.2.2.2.2">𝑤</ci><ci id="S1.p10.1.m1.5.5.2.2.2.3.cmml" xref="S1.p10.1.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S1.p10.1.m1.3.3.cmml" xref="S1.p10.1.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.1.m1.5c">W^{k,p}_{\operatorname{Tr}_{m}}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S1.p10.1.m1.5d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. This density result will be needed in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib48" title="">48</a>]</cite> to find trace characterisations for weighted Sobolev spaces on bounded <math alttext="C^{1}" class="ltx_Math" display="inline" id="S1.p10.2.m2.1"><semantics id="S1.p10.2.m2.1a"><msup id="S1.p10.2.m2.1.1" xref="S1.p10.2.m2.1.1.cmml"><mi id="S1.p10.2.m2.1.1.2" xref="S1.p10.2.m2.1.1.2.cmml">C</mi><mn id="S1.p10.2.m2.1.1.3" xref="S1.p10.2.m2.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S1.p10.2.m2.1b"><apply id="S1.p10.2.m2.1.1.cmml" xref="S1.p10.2.m2.1.1"><csymbol cd="ambiguous" id="S1.p10.2.m2.1.1.1.cmml" xref="S1.p10.2.m2.1.1">superscript</csymbol><ci id="S1.p10.2.m2.1.1.2.cmml" xref="S1.p10.2.m2.1.1.2">𝐶</ci><cn id="S1.p10.2.m2.1.1.3.cmml" type="integer" xref="S1.p10.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.2.m2.1c">C^{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p10.2.m2.1d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math>-domains. Application of a standard mollification procedure would result in all traces <math alttext="\overline{\operatorname{Tr}}_{m}" class="ltx_Math" display="inline" id="S1.p10.3.m3.1"><semantics id="S1.p10.3.m3.1a"><msub id="S1.p10.3.m3.1.1" xref="S1.p10.3.m3.1.1.cmml"><mover accent="true" id="S1.p10.3.m3.1.1.2" xref="S1.p10.3.m3.1.1.2.cmml"><mi id="S1.p10.3.m3.1.1.2.2" xref="S1.p10.3.m3.1.1.2.2.cmml">Tr</mi><mo id="S1.p10.3.m3.1.1.2.1" xref="S1.p10.3.m3.1.1.2.1.cmml">¯</mo></mover><mi id="S1.p10.3.m3.1.1.3" xref="S1.p10.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S1.p10.3.m3.1b"><apply id="S1.p10.3.m3.1.1.cmml" xref="S1.p10.3.m3.1.1"><csymbol cd="ambiguous" id="S1.p10.3.m3.1.1.1.cmml" xref="S1.p10.3.m3.1.1">subscript</csymbol><apply id="S1.p10.3.m3.1.1.2.cmml" xref="S1.p10.3.m3.1.1.2"><ci id="S1.p10.3.m3.1.1.2.1.cmml" xref="S1.p10.3.m3.1.1.2.1">¯</ci><ci id="S1.p10.3.m3.1.1.2.2.cmml" xref="S1.p10.3.m3.1.1.2.2">Tr</ci></apply><ci id="S1.p10.3.m3.1.1.3.cmml" xref="S1.p10.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p10.3.m3.1c">\overline{\operatorname{Tr}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S1.p10.3.m3.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> being zero instead of only one trace. We remark that results for mollification with preservation of boundary values are contained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib12" title="">12</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib30" title="">30</a>]</cite>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S1.p11"> <p class="ltx_p" id="S1.p11.2">We comment on some open and related problems. Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a> only deals with interpolation of the smoothness parameter, while it might be possible to interpolate in other parameters (such as <math alttext="p" class="ltx_Math" display="inline" id="S1.p11.1.m1.1"><semantics id="S1.p11.1.m1.1a"><mi id="S1.p11.1.m1.1.1" xref="S1.p11.1.m1.1.1.cmml">p</mi><annotation-xml encoding="MathML-Content" id="S1.p11.1.m1.1b"><ci id="S1.p11.1.m1.1.1.cmml" xref="S1.p11.1.m1.1.1">𝑝</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p11.1.m1.1c">p</annotation><annotation encoding="application/x-llamapun" id="S1.p11.1.m1.1d">italic_p</annotation></semantics></math> and <math alttext="\gamma" class="ltx_Math" display="inline" id="S1.p11.2.m2.1"><semantics id="S1.p11.2.m2.1a"><mi id="S1.p11.2.m2.1.1" xref="S1.p11.2.m2.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S1.p11.2.m2.1b"><ci id="S1.p11.2.m2.1.1.cmml" xref="S1.p11.2.m2.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S1.p11.2.m2.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S1.p11.2.m2.1d">italic_γ</annotation></semantics></math>) as well. Interpolation results for Sobolev spaces with different weights (but without complicated boundary conditions) are contained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib13" title="">13</a>]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib35" title="">35</a>, Section 5.3]</cite>, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Proposition 3.9]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib52" title="">52</a>]</cite>.</p> </div> <div class="ltx_para" id="S1.p12"> <p class="ltx_p" id="S1.p12.3">Besides complex interpolation, the real interpolation method plays an important role in the theory of evolution equations as well (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib18" title="">18</a>, Section 4.4]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Section 17.2.b]</cite>). Some related results for real interpolation of weighted spaces are contained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib11" title="">11</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib23" title="">23</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib53" title="">53</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>]</cite>. We note that if we have interpolation results for Besov spaces with boundary conditions, then we can obtain Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S1.Thmtheorem1" title="Theorem 1.1. ‣ 1. Introduction ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">1.1</span></a> for real interpolation, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorem VIII.2.4.5]</cite>. We expect that our results also hold for Besov spaces with boundary conditions if <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S1.p12.1.m1.2"><semantics id="S1.p12.1.m1.2a"><mrow id="S1.p12.1.m1.2.2" xref="S1.p12.1.m1.2.2.cmml"><mi id="S1.p12.1.m1.2.2.4" xref="S1.p12.1.m1.2.2.4.cmml">γ</mi><mo id="S1.p12.1.m1.2.2.3" xref="S1.p12.1.m1.2.2.3.cmml">∈</mo><mrow id="S1.p12.1.m1.2.2.2.2" xref="S1.p12.1.m1.2.2.2.3.cmml"><mo id="S1.p12.1.m1.2.2.2.2.3" stretchy="false" xref="S1.p12.1.m1.2.2.2.3.cmml">(</mo><mrow id="S1.p12.1.m1.1.1.1.1.1" xref="S1.p12.1.m1.1.1.1.1.1.cmml"><mo id="S1.p12.1.m1.1.1.1.1.1a" xref="S1.p12.1.m1.1.1.1.1.1.cmml">−</mo><mn id="S1.p12.1.m1.1.1.1.1.1.2" xref="S1.p12.1.m1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.p12.1.m1.2.2.2.2.4" xref="S1.p12.1.m1.2.2.2.3.cmml">,</mo><mrow id="S1.p12.1.m1.2.2.2.2.2" xref="S1.p12.1.m1.2.2.2.2.2.cmml"><mi id="S1.p12.1.m1.2.2.2.2.2.2" xref="S1.p12.1.m1.2.2.2.2.2.2.cmml">p</mi><mo id="S1.p12.1.m1.2.2.2.2.2.1" xref="S1.p12.1.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S1.p12.1.m1.2.2.2.2.2.3" xref="S1.p12.1.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S1.p12.1.m1.2.2.2.2.5" stretchy="false" xref="S1.p12.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p12.1.m1.2b"><apply id="S1.p12.1.m1.2.2.cmml" xref="S1.p12.1.m1.2.2"><in id="S1.p12.1.m1.2.2.3.cmml" xref="S1.p12.1.m1.2.2.3"></in><ci id="S1.p12.1.m1.2.2.4.cmml" xref="S1.p12.1.m1.2.2.4">𝛾</ci><interval closure="open" id="S1.p12.1.m1.2.2.2.3.cmml" xref="S1.p12.1.m1.2.2.2.2"><apply id="S1.p12.1.m1.1.1.1.1.1.cmml" xref="S1.p12.1.m1.1.1.1.1.1"><minus id="S1.p12.1.m1.1.1.1.1.1.1.cmml" xref="S1.p12.1.m1.1.1.1.1.1"></minus><cn id="S1.p12.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S1.p12.1.m1.1.1.1.1.1.2">1</cn></apply><apply id="S1.p12.1.m1.2.2.2.2.2.cmml" xref="S1.p12.1.m1.2.2.2.2.2"><minus id="S1.p12.1.m1.2.2.2.2.2.1.cmml" xref="S1.p12.1.m1.2.2.2.2.2.1"></minus><ci id="S1.p12.1.m1.2.2.2.2.2.2.cmml" xref="S1.p12.1.m1.2.2.2.2.2.2">𝑝</ci><cn id="S1.p12.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S1.p12.1.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p12.1.m1.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S1.p12.1.m1.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>. However, for <math alttext="\gamma>p-1" class="ltx_Math" display="inline" id="S1.p12.2.m2.1"><semantics id="S1.p12.2.m2.1a"><mrow id="S1.p12.2.m2.1.1" xref="S1.p12.2.m2.1.1.cmml"><mi id="S1.p12.2.m2.1.1.2" xref="S1.p12.2.m2.1.1.2.cmml">γ</mi><mo id="S1.p12.2.m2.1.1.1" xref="S1.p12.2.m2.1.1.1.cmml">></mo><mrow id="S1.p12.2.m2.1.1.3" xref="S1.p12.2.m2.1.1.3.cmml"><mi id="S1.p12.2.m2.1.1.3.2" xref="S1.p12.2.m2.1.1.3.2.cmml">p</mi><mo id="S1.p12.2.m2.1.1.3.1" xref="S1.p12.2.m2.1.1.3.1.cmml">−</mo><mn id="S1.p12.2.m2.1.1.3.3" xref="S1.p12.2.m2.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p12.2.m2.1b"><apply id="S1.p12.2.m2.1.1.cmml" xref="S1.p12.2.m2.1.1"><gt id="S1.p12.2.m2.1.1.1.cmml" xref="S1.p12.2.m2.1.1.1"></gt><ci id="S1.p12.2.m2.1.1.2.cmml" xref="S1.p12.2.m2.1.1.2">𝛾</ci><apply id="S1.p12.2.m2.1.1.3.cmml" xref="S1.p12.2.m2.1.1.3"><minus id="S1.p12.2.m2.1.1.3.1.cmml" xref="S1.p12.2.m2.1.1.3.1"></minus><ci id="S1.p12.2.m2.1.1.3.2.cmml" xref="S1.p12.2.m2.1.1.3.2">𝑝</ci><cn id="S1.p12.2.m2.1.1.3.3.cmml" type="integer" xref="S1.p12.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p12.2.m2.1c">\gamma>p-1</annotation><annotation encoding="application/x-llamapun" id="S1.p12.2.m2.1d">italic_γ > italic_p - 1</annotation></semantics></math> a more detailed study about interpolation of Besov spaces and their relation to Sobolev spaces is required. A particular problem is that for <math alttext="\gamma>p-1" class="ltx_Math" display="inline" id="S1.p12.3.m3.1"><semantics id="S1.p12.3.m3.1a"><mrow id="S1.p12.3.m3.1.1" xref="S1.p12.3.m3.1.1.cmml"><mi id="S1.p12.3.m3.1.1.2" xref="S1.p12.3.m3.1.1.2.cmml">γ</mi><mo id="S1.p12.3.m3.1.1.1" xref="S1.p12.3.m3.1.1.1.cmml">></mo><mrow id="S1.p12.3.m3.1.1.3" xref="S1.p12.3.m3.1.1.3.cmml"><mi id="S1.p12.3.m3.1.1.3.2" xref="S1.p12.3.m3.1.1.3.2.cmml">p</mi><mo id="S1.p12.3.m3.1.1.3.1" xref="S1.p12.3.m3.1.1.3.1.cmml">−</mo><mn id="S1.p12.3.m3.1.1.3.3" xref="S1.p12.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p12.3.m3.1b"><apply id="S1.p12.3.m3.1.1.cmml" xref="S1.p12.3.m3.1.1"><gt id="S1.p12.3.m3.1.1.1.cmml" xref="S1.p12.3.m3.1.1.1"></gt><ci id="S1.p12.3.m3.1.1.2.cmml" xref="S1.p12.3.m3.1.1.2">𝛾</ci><apply id="S1.p12.3.m3.1.1.3.cmml" xref="S1.p12.3.m3.1.1.3"><minus id="S1.p12.3.m3.1.1.3.1.cmml" xref="S1.p12.3.m3.1.1.3.1"></minus><ci id="S1.p12.3.m3.1.1.3.2.cmml" xref="S1.p12.3.m3.1.1.3.2">𝑝</ci><cn id="S1.p12.3.m3.1.1.3.3.cmml" type="integer" xref="S1.p12.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p12.3.m3.1c">\gamma>p-1</annotation><annotation encoding="application/x-llamapun" id="S1.p12.3.m3.1d">italic_γ > italic_p - 1</annotation></semantics></math> we only have the embedding</p> <table class="ltx_equation ltx_eqn_table" id="S1.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\qquad B^{n+\theta}_{p,q}(\mathbb{R}^{d}_{+},w_{\gamma})\hookrightarrow\big{(}% W^{n,p}(\mathbb{R}^{d}_{+},w_{\gamma}),W^{n+1,p}(\mathbb{R}^{d}_{+},w_{\gamma}% )\big{)}_{\theta,q},\quad n\in\mathbb{N}_{0},\,\theta\in(0,1)," class="ltx_Math" display="block" id="S1.Ex2.m1.11"><semantics id="S1.Ex2.m1.11a"><mrow id="S1.Ex2.m1.11.11.1"><mrow 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closure="open" id="S1.Ex2.m1.11.11.1.1.2.2.2.2.3.1.cmml" xref="S1.Ex2.m1.11.11.1.1.2.2.2.2.3.2"><cn id="S1.Ex2.m1.9.9.cmml" type="integer" xref="S1.Ex2.m1.9.9">0</cn><cn id="S1.Ex2.m1.10.10.cmml" type="integer" xref="S1.Ex2.m1.10.10">1</cn></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.Ex2.m1.11c">\qquad B^{n+\theta}_{p,q}(\mathbb{R}^{d}_{+},w_{\gamma})\hookrightarrow\big{(}% W^{n,p}(\mathbb{R}^{d}_{+},w_{\gamma}),W^{n+1,p}(\mathbb{R}^{d}_{+},w_{\gamma}% )\big{)}_{\theta,q},\quad n\in\mathbb{N}_{0},\,\theta\in(0,1),</annotation><annotation encoding="application/x-llamapun" id="S1.Ex2.m1.11d">italic_B start_POSTSUPERSCRIPT italic_n + italic_θ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ↪ ( italic_W start_POSTSUPERSCRIPT italic_n , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT italic_n + 1 , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ) start_POSTSUBSCRIPT italic_θ , italic_q end_POSTSUBSCRIPT , italic_n ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_θ ∈ ( 0 , 1 ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S1.p12.4">while equality holds for <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S1.p12.4.m1.2"><semantics id="S1.p12.4.m1.2a"><mrow id="S1.p12.4.m1.2.2" xref="S1.p12.4.m1.2.2.cmml"><mi id="S1.p12.4.m1.2.2.4" xref="S1.p12.4.m1.2.2.4.cmml">γ</mi><mo id="S1.p12.4.m1.2.2.3" xref="S1.p12.4.m1.2.2.3.cmml">∈</mo><mrow id="S1.p12.4.m1.2.2.2.2" xref="S1.p12.4.m1.2.2.2.3.cmml"><mo id="S1.p12.4.m1.2.2.2.2.3" stretchy="false" xref="S1.p12.4.m1.2.2.2.3.cmml">(</mo><mrow id="S1.p12.4.m1.1.1.1.1.1" xref="S1.p12.4.m1.1.1.1.1.1.cmml"><mo id="S1.p12.4.m1.1.1.1.1.1a" xref="S1.p12.4.m1.1.1.1.1.1.cmml">−</mo><mn id="S1.p12.4.m1.1.1.1.1.1.2" xref="S1.p12.4.m1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S1.p12.4.m1.2.2.2.2.4" xref="S1.p12.4.m1.2.2.2.3.cmml">,</mo><mrow id="S1.p12.4.m1.2.2.2.2.2" xref="S1.p12.4.m1.2.2.2.2.2.cmml"><mi id="S1.p12.4.m1.2.2.2.2.2.2" xref="S1.p12.4.m1.2.2.2.2.2.2.cmml">p</mi><mo id="S1.p12.4.m1.2.2.2.2.2.1" xref="S1.p12.4.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S1.p12.4.m1.2.2.2.2.2.3" xref="S1.p12.4.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S1.p12.4.m1.2.2.2.2.5" stretchy="false" xref="S1.p12.4.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p12.4.m1.2b"><apply id="S1.p12.4.m1.2.2.cmml" xref="S1.p12.4.m1.2.2"><in id="S1.p12.4.m1.2.2.3.cmml" xref="S1.p12.4.m1.2.2.3"></in><ci id="S1.p12.4.m1.2.2.4.cmml" xref="S1.p12.4.m1.2.2.4">𝛾</ci><interval closure="open" id="S1.p12.4.m1.2.2.2.3.cmml" xref="S1.p12.4.m1.2.2.2.2"><apply id="S1.p12.4.m1.1.1.1.1.1.cmml" xref="S1.p12.4.m1.1.1.1.1.1"><minus id="S1.p12.4.m1.1.1.1.1.1.1.cmml" xref="S1.p12.4.m1.1.1.1.1.1"></minus><cn id="S1.p12.4.m1.1.1.1.1.1.2.cmml" type="integer" xref="S1.p12.4.m1.1.1.1.1.1.2">1</cn></apply><apply id="S1.p12.4.m1.2.2.2.2.2.cmml" xref="S1.p12.4.m1.2.2.2.2.2"><minus id="S1.p12.4.m1.2.2.2.2.2.1.cmml" xref="S1.p12.4.m1.2.2.2.2.2.1"></minus><ci id="S1.p12.4.m1.2.2.2.2.2.2.cmml" xref="S1.p12.4.m1.2.2.2.2.2.2">𝑝</ci><cn id="S1.p12.4.m1.2.2.2.2.2.3.cmml" type="integer" xref="S1.p12.4.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p12.4.m1.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S1.p12.4.m1.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>, Section 7]</cite>.</p> </div> <div class="ltx_para" id="S1.p13"> <p class="ltx_p" id="S1.p13.5">Finally, we note that the special values <math alttext="\gamma=jp-1" class="ltx_Math" display="inline" id="S1.p13.1.m1.1"><semantics id="S1.p13.1.m1.1a"><mrow id="S1.p13.1.m1.1.1" xref="S1.p13.1.m1.1.1.cmml"><mi id="S1.p13.1.m1.1.1.2" xref="S1.p13.1.m1.1.1.2.cmml">γ</mi><mo id="S1.p13.1.m1.1.1.1" xref="S1.p13.1.m1.1.1.1.cmml">=</mo><mrow id="S1.p13.1.m1.1.1.3" xref="S1.p13.1.m1.1.1.3.cmml"><mrow id="S1.p13.1.m1.1.1.3.2" xref="S1.p13.1.m1.1.1.3.2.cmml"><mi id="S1.p13.1.m1.1.1.3.2.2" xref="S1.p13.1.m1.1.1.3.2.2.cmml">j</mi><mo id="S1.p13.1.m1.1.1.3.2.1" xref="S1.p13.1.m1.1.1.3.2.1.cmml">⁢</mo><mi id="S1.p13.1.m1.1.1.3.2.3" xref="S1.p13.1.m1.1.1.3.2.3.cmml">p</mi></mrow><mo id="S1.p13.1.m1.1.1.3.1" xref="S1.p13.1.m1.1.1.3.1.cmml">−</mo><mn id="S1.p13.1.m1.1.1.3.3" xref="S1.p13.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p13.1.m1.1b"><apply id="S1.p13.1.m1.1.1.cmml" xref="S1.p13.1.m1.1.1"><eq id="S1.p13.1.m1.1.1.1.cmml" xref="S1.p13.1.m1.1.1.1"></eq><ci id="S1.p13.1.m1.1.1.2.cmml" xref="S1.p13.1.m1.1.1.2">𝛾</ci><apply id="S1.p13.1.m1.1.1.3.cmml" xref="S1.p13.1.m1.1.1.3"><minus id="S1.p13.1.m1.1.1.3.1.cmml" xref="S1.p13.1.m1.1.1.3.1"></minus><apply id="S1.p13.1.m1.1.1.3.2.cmml" xref="S1.p13.1.m1.1.1.3.2"><times id="S1.p13.1.m1.1.1.3.2.1.cmml" xref="S1.p13.1.m1.1.1.3.2.1"></times><ci id="S1.p13.1.m1.1.1.3.2.2.cmml" xref="S1.p13.1.m1.1.1.3.2.2">𝑗</ci><ci id="S1.p13.1.m1.1.1.3.2.3.cmml" xref="S1.p13.1.m1.1.1.3.2.3">𝑝</ci></apply><cn id="S1.p13.1.m1.1.1.3.3.cmml" type="integer" xref="S1.p13.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p13.1.m1.1c">\gamma=jp-1</annotation><annotation encoding="application/x-llamapun" id="S1.p13.1.m1.1d">italic_γ = italic_j italic_p - 1</annotation></semantics></math> with <math alttext="j\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S1.p13.2.m2.1"><semantics id="S1.p13.2.m2.1a"><mrow id="S1.p13.2.m2.1.1" xref="S1.p13.2.m2.1.1.cmml"><mi id="S1.p13.2.m2.1.1.2" xref="S1.p13.2.m2.1.1.2.cmml">j</mi><mo id="S1.p13.2.m2.1.1.1" xref="S1.p13.2.m2.1.1.1.cmml">∈</mo><msub id="S1.p13.2.m2.1.1.3" xref="S1.p13.2.m2.1.1.3.cmml"><mi id="S1.p13.2.m2.1.1.3.2" xref="S1.p13.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S1.p13.2.m2.1.1.3.3" xref="S1.p13.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S1.p13.2.m2.1b"><apply id="S1.p13.2.m2.1.1.cmml" xref="S1.p13.2.m2.1.1"><in id="S1.p13.2.m2.1.1.1.cmml" xref="S1.p13.2.m2.1.1.1"></in><ci id="S1.p13.2.m2.1.1.2.cmml" xref="S1.p13.2.m2.1.1.2">𝑗</ci><apply id="S1.p13.2.m2.1.1.3.cmml" xref="S1.p13.2.m2.1.1.3"><csymbol cd="ambiguous" id="S1.p13.2.m2.1.1.3.1.cmml" xref="S1.p13.2.m2.1.1.3">subscript</csymbol><ci id="S1.p13.2.m2.1.1.3.2.cmml" xref="S1.p13.2.m2.1.1.3.2">ℕ</ci><cn id="S1.p13.2.m2.1.1.3.3.cmml" type="integer" xref="S1.p13.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p13.2.m2.1c">j\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S1.p13.2.m2.1d">italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> are excluded, since in these cases the study of the trace operator is more involved, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib35" title="">35</a>]</cite>. Moreover, we do not have certain trace characterisations available which play a crucial role in the proof of the main theorem. For <math alttext="\gamma\leq-1" class="ltx_Math" display="inline" id="S1.p13.3.m3.1"><semantics id="S1.p13.3.m3.1a"><mrow id="S1.p13.3.m3.1.1" xref="S1.p13.3.m3.1.1.cmml"><mi id="S1.p13.3.m3.1.1.2" xref="S1.p13.3.m3.1.1.2.cmml">γ</mi><mo id="S1.p13.3.m3.1.1.1" xref="S1.p13.3.m3.1.1.1.cmml">≤</mo><mrow id="S1.p13.3.m3.1.1.3" xref="S1.p13.3.m3.1.1.3.cmml"><mo id="S1.p13.3.m3.1.1.3a" xref="S1.p13.3.m3.1.1.3.cmml">−</mo><mn id="S1.p13.3.m3.1.1.3.2" xref="S1.p13.3.m3.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p13.3.m3.1b"><apply id="S1.p13.3.m3.1.1.cmml" xref="S1.p13.3.m3.1.1"><leq id="S1.p13.3.m3.1.1.1.cmml" xref="S1.p13.3.m3.1.1.1"></leq><ci id="S1.p13.3.m3.1.1.2.cmml" xref="S1.p13.3.m3.1.1.2">𝛾</ci><apply id="S1.p13.3.m3.1.1.3.cmml" xref="S1.p13.3.m3.1.1.3"><minus id="S1.p13.3.m3.1.1.3.1.cmml" xref="S1.p13.3.m3.1.1.3"></minus><cn id="S1.p13.3.m3.1.1.3.2.cmml" type="integer" xref="S1.p13.3.m3.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p13.3.m3.1c">\gamma\leq-1</annotation><annotation encoding="application/x-llamapun" id="S1.p13.3.m3.1d">italic_γ ≤ - 1</annotation></semantics></math> we have that <math alttext="C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+})" class="ltx_Math" display="inline" id="S1.p13.4.m4.1"><semantics id="S1.p13.4.m4.1a"><mrow id="S1.p13.4.m4.1.1" xref="S1.p13.4.m4.1.1.cmml"><msubsup id="S1.p13.4.m4.1.1.3" xref="S1.p13.4.m4.1.1.3.cmml"><mi id="S1.p13.4.m4.1.1.3.2.2" xref="S1.p13.4.m4.1.1.3.2.2.cmml">C</mi><mi id="S1.p13.4.m4.1.1.3.2.3" mathvariant="normal" xref="S1.p13.4.m4.1.1.3.2.3.cmml">c</mi><mi id="S1.p13.4.m4.1.1.3.3" mathvariant="normal" xref="S1.p13.4.m4.1.1.3.3.cmml">∞</mi></msubsup><mo id="S1.p13.4.m4.1.1.2" xref="S1.p13.4.m4.1.1.2.cmml">⁢</mo><mrow id="S1.p13.4.m4.1.1.1.1" xref="S1.p13.4.m4.1.1.1.1.1.cmml"><mo id="S1.p13.4.m4.1.1.1.1.2" stretchy="false" xref="S1.p13.4.m4.1.1.1.1.1.cmml">(</mo><msubsup id="S1.p13.4.m4.1.1.1.1.1" xref="S1.p13.4.m4.1.1.1.1.1.cmml"><mi id="S1.p13.4.m4.1.1.1.1.1.2.2" xref="S1.p13.4.m4.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S1.p13.4.m4.1.1.1.1.1.3" xref="S1.p13.4.m4.1.1.1.1.1.3.cmml">+</mo><mi id="S1.p13.4.m4.1.1.1.1.1.2.3" xref="S1.p13.4.m4.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S1.p13.4.m4.1.1.1.1.3" stretchy="false" xref="S1.p13.4.m4.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p13.4.m4.1b"><apply id="S1.p13.4.m4.1.1.cmml" xref="S1.p13.4.m4.1.1"><times id="S1.p13.4.m4.1.1.2.cmml" xref="S1.p13.4.m4.1.1.2"></times><apply id="S1.p13.4.m4.1.1.3.cmml" xref="S1.p13.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.p13.4.m4.1.1.3.1.cmml" xref="S1.p13.4.m4.1.1.3">superscript</csymbol><apply id="S1.p13.4.m4.1.1.3.2.cmml" xref="S1.p13.4.m4.1.1.3"><csymbol cd="ambiguous" id="S1.p13.4.m4.1.1.3.2.1.cmml" xref="S1.p13.4.m4.1.1.3">subscript</csymbol><ci id="S1.p13.4.m4.1.1.3.2.2.cmml" xref="S1.p13.4.m4.1.1.3.2.2">𝐶</ci><ci id="S1.p13.4.m4.1.1.3.2.3.cmml" xref="S1.p13.4.m4.1.1.3.2.3">c</ci></apply><infinity id="S1.p13.4.m4.1.1.3.3.cmml" xref="S1.p13.4.m4.1.1.3.3"></infinity></apply><apply id="S1.p13.4.m4.1.1.1.1.1.cmml" xref="S1.p13.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S1.p13.4.m4.1.1.1.1.1.1.cmml" xref="S1.p13.4.m4.1.1.1.1">subscript</csymbol><apply id="S1.p13.4.m4.1.1.1.1.1.2.cmml" xref="S1.p13.4.m4.1.1.1.1"><csymbol cd="ambiguous" id="S1.p13.4.m4.1.1.1.1.1.2.1.cmml" xref="S1.p13.4.m4.1.1.1.1">superscript</csymbol><ci id="S1.p13.4.m4.1.1.1.1.1.2.2.cmml" xref="S1.p13.4.m4.1.1.1.1.1.2.2">ℝ</ci><ci id="S1.p13.4.m4.1.1.1.1.1.2.3.cmml" xref="S1.p13.4.m4.1.1.1.1.1.2.3">𝑑</ci></apply><plus id="S1.p13.4.m4.1.1.1.1.1.3.cmml" xref="S1.p13.4.m4.1.1.1.1.1.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p13.4.m4.1c">C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+})</annotation><annotation encoding="application/x-llamapun" id="S1.p13.4.m4.1d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT )</annotation></semantics></math> is dense in <math alttext="W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma})" class="ltx_Math" display="inline" id="S1.p13.5.m5.4"><semantics id="S1.p13.5.m5.4a"><mrow id="S1.p13.5.m5.4.4" xref="S1.p13.5.m5.4.4.cmml"><msup id="S1.p13.5.m5.4.4.4" xref="S1.p13.5.m5.4.4.4.cmml"><mi id="S1.p13.5.m5.4.4.4.2" xref="S1.p13.5.m5.4.4.4.2.cmml">W</mi><mrow id="S1.p13.5.m5.2.2.2.4" xref="S1.p13.5.m5.2.2.2.3.cmml"><mi id="S1.p13.5.m5.1.1.1.1" xref="S1.p13.5.m5.1.1.1.1.cmml">k</mi><mo id="S1.p13.5.m5.2.2.2.4.1" xref="S1.p13.5.m5.2.2.2.3.cmml">,</mo><mi id="S1.p13.5.m5.2.2.2.2" xref="S1.p13.5.m5.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S1.p13.5.m5.4.4.3" xref="S1.p13.5.m5.4.4.3.cmml">⁢</mo><mrow id="S1.p13.5.m5.4.4.2.2" xref="S1.p13.5.m5.4.4.2.3.cmml"><mo id="S1.p13.5.m5.4.4.2.2.3" stretchy="false" xref="S1.p13.5.m5.4.4.2.3.cmml">(</mo><msubsup id="S1.p13.5.m5.3.3.1.1.1" xref="S1.p13.5.m5.3.3.1.1.1.cmml"><mi id="S1.p13.5.m5.3.3.1.1.1.2.2" xref="S1.p13.5.m5.3.3.1.1.1.2.2.cmml">ℝ</mi><mo id="S1.p13.5.m5.3.3.1.1.1.3" xref="S1.p13.5.m5.3.3.1.1.1.3.cmml">+</mo><mi id="S1.p13.5.m5.3.3.1.1.1.2.3" xref="S1.p13.5.m5.3.3.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S1.p13.5.m5.4.4.2.2.4" xref="S1.p13.5.m5.4.4.2.3.cmml">,</mo><msub id="S1.p13.5.m5.4.4.2.2.2" xref="S1.p13.5.m5.4.4.2.2.2.cmml"><mi id="S1.p13.5.m5.4.4.2.2.2.2" xref="S1.p13.5.m5.4.4.2.2.2.2.cmml">w</mi><mi id="S1.p13.5.m5.4.4.2.2.2.3" xref="S1.p13.5.m5.4.4.2.2.2.3.cmml">γ</mi></msub><mo id="S1.p13.5.m5.4.4.2.2.5" stretchy="false" xref="S1.p13.5.m5.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S1.p13.5.m5.4b"><apply id="S1.p13.5.m5.4.4.cmml" xref="S1.p13.5.m5.4.4"><times id="S1.p13.5.m5.4.4.3.cmml" xref="S1.p13.5.m5.4.4.3"></times><apply id="S1.p13.5.m5.4.4.4.cmml" xref="S1.p13.5.m5.4.4.4"><csymbol cd="ambiguous" id="S1.p13.5.m5.4.4.4.1.cmml" xref="S1.p13.5.m5.4.4.4">superscript</csymbol><ci id="S1.p13.5.m5.4.4.4.2.cmml" xref="S1.p13.5.m5.4.4.4.2">𝑊</ci><list id="S1.p13.5.m5.2.2.2.3.cmml" xref="S1.p13.5.m5.2.2.2.4"><ci id="S1.p13.5.m5.1.1.1.1.cmml" xref="S1.p13.5.m5.1.1.1.1">𝑘</ci><ci id="S1.p13.5.m5.2.2.2.2.cmml" xref="S1.p13.5.m5.2.2.2.2">𝑝</ci></list></apply><interval closure="open" id="S1.p13.5.m5.4.4.2.3.cmml" xref="S1.p13.5.m5.4.4.2.2"><apply id="S1.p13.5.m5.3.3.1.1.1.cmml" xref="S1.p13.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.p13.5.m5.3.3.1.1.1.1.cmml" xref="S1.p13.5.m5.3.3.1.1.1">subscript</csymbol><apply id="S1.p13.5.m5.3.3.1.1.1.2.cmml" xref="S1.p13.5.m5.3.3.1.1.1"><csymbol cd="ambiguous" id="S1.p13.5.m5.3.3.1.1.1.2.1.cmml" xref="S1.p13.5.m5.3.3.1.1.1">superscript</csymbol><ci id="S1.p13.5.m5.3.3.1.1.1.2.2.cmml" xref="S1.p13.5.m5.3.3.1.1.1.2.2">ℝ</ci><ci id="S1.p13.5.m5.3.3.1.1.1.2.3.cmml" xref="S1.p13.5.m5.3.3.1.1.1.2.3">𝑑</ci></apply><plus id="S1.p13.5.m5.3.3.1.1.1.3.cmml" xref="S1.p13.5.m5.3.3.1.1.1.3"></plus></apply><apply id="S1.p13.5.m5.4.4.2.2.2.cmml" xref="S1.p13.5.m5.4.4.2.2.2"><csymbol cd="ambiguous" id="S1.p13.5.m5.4.4.2.2.2.1.cmml" xref="S1.p13.5.m5.4.4.2.2.2">subscript</csymbol><ci id="S1.p13.5.m5.4.4.2.2.2.2.cmml" xref="S1.p13.5.m5.4.4.2.2.2.2">𝑤</ci><ci id="S1.p13.5.m5.4.4.2.2.2.3.cmml" xref="S1.p13.5.m5.4.4.2.2.2.3">𝛾</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S1.p13.5.m5.4c">W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma})</annotation><annotation encoding="application/x-llamapun" id="S1.p13.5.m5.4d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT )</annotation></semantics></math> and thus all traces will be zero.</p> </div> <section class="ltx_subsection" id="S1.SSx1"> <h3 class="ltx_title ltx_title_subsection">Outline</h3> <div class="ltx_para" id="S1.SSx1.p1"> <p class="ltx_p" id="S1.SSx1.p1.1">The outline of this paper is as follows. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2" title="2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2</span></a> we introduce some concepts and preliminary results needed throughout the paper. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3" title="3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3</span></a> we determine the trace spaces for weighted Besov and Triebel-Lizorkin spaces. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4" title="4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4</span></a> we extend the results to weighted Bessel potential and Sobolev spaces. As a consequence, we prove some density results. Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5" title="5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5</span></a> deals with the characterisation of the trace space for boundary operators. Finally, in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6" title="6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6</span></a> we prove the main result about complex interpolation of weighted Sobolev spaces with and without boundary conditions.</p> </div> </section> </section> <section class="ltx_section" id="S2"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">2. </span>Preliminaries</h2> <section class="ltx_subsection" id="S2.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.1. </span>Notation</h3> <div class="ltx_para" id="S2.SS1.p1"> <p class="ltx_p" id="S2.SS1.p1.11">We denote by <math alttext="\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S2.SS1.p1.1.m1.1"><semantics id="S2.SS1.p1.1.m1.1a"><msub id="S2.SS1.p1.1.m1.1.1" xref="S2.SS1.p1.1.m1.1.1.cmml"><mi id="S2.SS1.p1.1.m1.1.1.2" xref="S2.SS1.p1.1.m1.1.1.2.cmml">ℕ</mi><mn id="S2.SS1.p1.1.m1.1.1.3" xref="S2.SS1.p1.1.m1.1.1.3.cmml">0</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.1.m1.1b"><apply id="S2.SS1.p1.1.m1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.1.m1.1.1.1.cmml" xref="S2.SS1.p1.1.m1.1.1">subscript</csymbol><ci id="S2.SS1.p1.1.m1.1.1.2.cmml" xref="S2.SS1.p1.1.m1.1.1.2">ℕ</ci><cn id="S2.SS1.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.1.m1.1c">\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.1.m1.1d">blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.2.m2.1"><semantics id="S2.SS1.p1.2.m2.1a"><msub id="S2.SS1.p1.2.m2.1.1" xref="S2.SS1.p1.2.m2.1.1.cmml"><mi id="S2.SS1.p1.2.m2.1.1.2" xref="S2.SS1.p1.2.m2.1.1.2.cmml">ℕ</mi><mn id="S2.SS1.p1.2.m2.1.1.3" xref="S2.SS1.p1.2.m2.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.2.m2.1b"><apply id="S2.SS1.p1.2.m2.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.2.m2.1.1.1.cmml" xref="S2.SS1.p1.2.m2.1.1">subscript</csymbol><ci id="S2.SS1.p1.2.m2.1.1.2.cmml" xref="S2.SS1.p1.2.m2.1.1.2">ℕ</ci><cn id="S2.SS1.p1.2.m2.1.1.3.cmml" type="integer" xref="S2.SS1.p1.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.2.m2.1c">\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.2.m2.1d">blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> the set of natural numbers starting at <math alttext="0" class="ltx_Math" display="inline" id="S2.SS1.p1.3.m3.1"><semantics id="S2.SS1.p1.3.m3.1a"><mn id="S2.SS1.p1.3.m3.1.1" xref="S2.SS1.p1.3.m3.1.1.cmml">0</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.3.m3.1b"><cn id="S2.SS1.p1.3.m3.1.1.cmml" type="integer" xref="S2.SS1.p1.3.m3.1.1">0</cn></annotation-xml></semantics></math> and <math alttext="1" class="ltx_Math" display="inline" id="S2.SS1.p1.4.m4.1"><semantics id="S2.SS1.p1.4.m4.1a"><mn id="S2.SS1.p1.4.m4.1.1" xref="S2.SS1.p1.4.m4.1.1.cmml">1</mn><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.4.m4.1b"><cn id="S2.SS1.p1.4.m4.1.1.cmml" type="integer" xref="S2.SS1.p1.4.m4.1.1">1</cn></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.4.m4.1c">1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.4.m4.1d">1</annotation></semantics></math>, respectively. For <math alttext="d\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S2.SS1.p1.5.m5.1"><semantics id="S2.SS1.p1.5.m5.1a"><mrow id="S2.SS1.p1.5.m5.1.1" xref="S2.SS1.p1.5.m5.1.1.cmml"><mi id="S2.SS1.p1.5.m5.1.1.2" xref="S2.SS1.p1.5.m5.1.1.2.cmml">d</mi><mo id="S2.SS1.p1.5.m5.1.1.1" xref="S2.SS1.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S2.SS1.p1.5.m5.1.1.3" xref="S2.SS1.p1.5.m5.1.1.3.cmml"><mi id="S2.SS1.p1.5.m5.1.1.3.2" xref="S2.SS1.p1.5.m5.1.1.3.2.cmml">ℕ</mi><mn id="S2.SS1.p1.5.m5.1.1.3.3" xref="S2.SS1.p1.5.m5.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.5.m5.1b"><apply id="S2.SS1.p1.5.m5.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1"><in id="S2.SS1.p1.5.m5.1.1.1.cmml" xref="S2.SS1.p1.5.m5.1.1.1"></in><ci id="S2.SS1.p1.5.m5.1.1.2.cmml" xref="S2.SS1.p1.5.m5.1.1.2">𝑑</ci><apply id="S2.SS1.p1.5.m5.1.1.3.cmml" xref="S2.SS1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p1.5.m5.1.1.3.1.cmml" xref="S2.SS1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS1.p1.5.m5.1.1.3.2.cmml" xref="S2.SS1.p1.5.m5.1.1.3.2">ℕ</ci><cn id="S2.SS1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S2.SS1.p1.5.m5.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.5.m5.1c">d\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.5.m5.1d">italic_d ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> the half-spaces are given by <math alttext="\mathbb{R}^{d}_{\pm}=\mathbb{R}_{\pm}\times\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S2.SS1.p1.6.m6.1"><semantics id="S2.SS1.p1.6.m6.1a"><mrow id="S2.SS1.p1.6.m6.1.1" xref="S2.SS1.p1.6.m6.1.1.cmml"><msubsup id="S2.SS1.p1.6.m6.1.1.2" xref="S2.SS1.p1.6.m6.1.1.2.cmml"><mi id="S2.SS1.p1.6.m6.1.1.2.2.2" xref="S2.SS1.p1.6.m6.1.1.2.2.2.cmml">ℝ</mi><mo id="S2.SS1.p1.6.m6.1.1.2.3" xref="S2.SS1.p1.6.m6.1.1.2.3.cmml">±</mo><mi id="S2.SS1.p1.6.m6.1.1.2.2.3" xref="S2.SS1.p1.6.m6.1.1.2.2.3.cmml">d</mi></msubsup><mo id="S2.SS1.p1.6.m6.1.1.1" xref="S2.SS1.p1.6.m6.1.1.1.cmml">=</mo><mrow id="S2.SS1.p1.6.m6.1.1.3" xref="S2.SS1.p1.6.m6.1.1.3.cmml"><msub id="S2.SS1.p1.6.m6.1.1.3.2" xref="S2.SS1.p1.6.m6.1.1.3.2.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.2.2" xref="S2.SS1.p1.6.m6.1.1.3.2.2.cmml">ℝ</mi><mo id="S2.SS1.p1.6.m6.1.1.3.2.3" xref="S2.SS1.p1.6.m6.1.1.3.2.3.cmml">±</mo></msub><mo id="S2.SS1.p1.6.m6.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.6.m6.1.1.3.1.cmml">×</mo><msup id="S2.SS1.p1.6.m6.1.1.3.3" xref="S2.SS1.p1.6.m6.1.1.3.3.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.3.2" xref="S2.SS1.p1.6.m6.1.1.3.3.2.cmml">ℝ</mi><mrow id="S2.SS1.p1.6.m6.1.1.3.3.3" xref="S2.SS1.p1.6.m6.1.1.3.3.3.cmml"><mi id="S2.SS1.p1.6.m6.1.1.3.3.3.2" xref="S2.SS1.p1.6.m6.1.1.3.3.3.2.cmml">d</mi><mo id="S2.SS1.p1.6.m6.1.1.3.3.3.1" xref="S2.SS1.p1.6.m6.1.1.3.3.3.1.cmml">−</mo><mn id="S2.SS1.p1.6.m6.1.1.3.3.3.3" xref="S2.SS1.p1.6.m6.1.1.3.3.3.3.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.6.m6.1b"><apply id="S2.SS1.p1.6.m6.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1"><eq id="S2.SS1.p1.6.m6.1.1.1.cmml" xref="S2.SS1.p1.6.m6.1.1.1"></eq><apply id="S2.SS1.p1.6.m6.1.1.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.2.1.cmml" xref="S2.SS1.p1.6.m6.1.1.2">subscript</csymbol><apply id="S2.SS1.p1.6.m6.1.1.2.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.2.2.1.cmml" xref="S2.SS1.p1.6.m6.1.1.2">superscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.2.2.2.cmml" xref="S2.SS1.p1.6.m6.1.1.2.2.2">ℝ</ci><ci id="S2.SS1.p1.6.m6.1.1.2.2.3.cmml" xref="S2.SS1.p1.6.m6.1.1.2.2.3">𝑑</ci></apply><csymbol cd="latexml" id="S2.SS1.p1.6.m6.1.1.2.3.cmml" xref="S2.SS1.p1.6.m6.1.1.2.3">plus-or-minus</csymbol></apply><apply id="S2.SS1.p1.6.m6.1.1.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3"><times id="S2.SS1.p1.6.m6.1.1.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.1"></times><apply id="S2.SS1.p1.6.m6.1.1.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.2.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2">subscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.2.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2.2">ℝ</ci><csymbol cd="latexml" id="S2.SS1.p1.6.m6.1.1.3.2.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.2.3">plus-or-minus</csymbol></apply><apply id="S2.SS1.p1.6.m6.1.1.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3"><csymbol cd="ambiguous" id="S2.SS1.p1.6.m6.1.1.3.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3">superscript</csymbol><ci id="S2.SS1.p1.6.m6.1.1.3.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3.2">ℝ</ci><apply id="S2.SS1.p1.6.m6.1.1.3.3.3.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3.3"><minus id="S2.SS1.p1.6.m6.1.1.3.3.3.1.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3.3.1"></minus><ci id="S2.SS1.p1.6.m6.1.1.3.3.3.2.cmml" xref="S2.SS1.p1.6.m6.1.1.3.3.3.2">𝑑</ci><cn id="S2.SS1.p1.6.m6.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS1.p1.6.m6.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.6.m6.1c">\mathbb{R}^{d}_{\pm}=\mathbb{R}_{\pm}\times\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT = blackboard_R start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, where <math alttext="\mathbb{R}_{+}=(0,\infty)" class="ltx_Math" display="inline" id="S2.SS1.p1.7.m7.2"><semantics id="S2.SS1.p1.7.m7.2a"><mrow id="S2.SS1.p1.7.m7.2.3" xref="S2.SS1.p1.7.m7.2.3.cmml"><msub id="S2.SS1.p1.7.m7.2.3.2" xref="S2.SS1.p1.7.m7.2.3.2.cmml"><mi id="S2.SS1.p1.7.m7.2.3.2.2" xref="S2.SS1.p1.7.m7.2.3.2.2.cmml">ℝ</mi><mo id="S2.SS1.p1.7.m7.2.3.2.3" xref="S2.SS1.p1.7.m7.2.3.2.3.cmml">+</mo></msub><mo id="S2.SS1.p1.7.m7.2.3.1" xref="S2.SS1.p1.7.m7.2.3.1.cmml">=</mo><mrow id="S2.SS1.p1.7.m7.2.3.3.2" xref="S2.SS1.p1.7.m7.2.3.3.1.cmml"><mo id="S2.SS1.p1.7.m7.2.3.3.2.1" stretchy="false" xref="S2.SS1.p1.7.m7.2.3.3.1.cmml">(</mo><mn id="S2.SS1.p1.7.m7.1.1" xref="S2.SS1.p1.7.m7.1.1.cmml">0</mn><mo id="S2.SS1.p1.7.m7.2.3.3.2.2" xref="S2.SS1.p1.7.m7.2.3.3.1.cmml">,</mo><mi id="S2.SS1.p1.7.m7.2.2" mathvariant="normal" xref="S2.SS1.p1.7.m7.2.2.cmml">∞</mi><mo id="S2.SS1.p1.7.m7.2.3.3.2.3" stretchy="false" xref="S2.SS1.p1.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.7.m7.2b"><apply id="S2.SS1.p1.7.m7.2.3.cmml" xref="S2.SS1.p1.7.m7.2.3"><eq id="S2.SS1.p1.7.m7.2.3.1.cmml" xref="S2.SS1.p1.7.m7.2.3.1"></eq><apply id="S2.SS1.p1.7.m7.2.3.2.cmml" xref="S2.SS1.p1.7.m7.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.7.m7.2.3.2.1.cmml" xref="S2.SS1.p1.7.m7.2.3.2">subscript</csymbol><ci id="S2.SS1.p1.7.m7.2.3.2.2.cmml" xref="S2.SS1.p1.7.m7.2.3.2.2">ℝ</ci><plus id="S2.SS1.p1.7.m7.2.3.2.3.cmml" xref="S2.SS1.p1.7.m7.2.3.2.3"></plus></apply><interval closure="open" id="S2.SS1.p1.7.m7.2.3.3.1.cmml" xref="S2.SS1.p1.7.m7.2.3.3.2"><cn id="S2.SS1.p1.7.m7.1.1.cmml" type="integer" xref="S2.SS1.p1.7.m7.1.1">0</cn><infinity id="S2.SS1.p1.7.m7.2.2.cmml" xref="S2.SS1.p1.7.m7.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.7.m7.2c">\mathbb{R}_{+}=(0,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.7.m7.2d">blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT = ( 0 , ∞ )</annotation></semantics></math> and <math alttext="\mathbb{R}_{-}=(-\infty,0)" class="ltx_Math" display="inline" id="S2.SS1.p1.8.m8.2"><semantics id="S2.SS1.p1.8.m8.2a"><mrow id="S2.SS1.p1.8.m8.2.2" xref="S2.SS1.p1.8.m8.2.2.cmml"><msub id="S2.SS1.p1.8.m8.2.2.3" xref="S2.SS1.p1.8.m8.2.2.3.cmml"><mi id="S2.SS1.p1.8.m8.2.2.3.2" xref="S2.SS1.p1.8.m8.2.2.3.2.cmml">ℝ</mi><mo id="S2.SS1.p1.8.m8.2.2.3.3" xref="S2.SS1.p1.8.m8.2.2.3.3.cmml">−</mo></msub><mo id="S2.SS1.p1.8.m8.2.2.2" xref="S2.SS1.p1.8.m8.2.2.2.cmml">=</mo><mrow id="S2.SS1.p1.8.m8.2.2.1.1" xref="S2.SS1.p1.8.m8.2.2.1.2.cmml"><mo id="S2.SS1.p1.8.m8.2.2.1.1.2" stretchy="false" xref="S2.SS1.p1.8.m8.2.2.1.2.cmml">(</mo><mrow id="S2.SS1.p1.8.m8.2.2.1.1.1" xref="S2.SS1.p1.8.m8.2.2.1.1.1.cmml"><mo id="S2.SS1.p1.8.m8.2.2.1.1.1a" xref="S2.SS1.p1.8.m8.2.2.1.1.1.cmml">−</mo><mi id="S2.SS1.p1.8.m8.2.2.1.1.1.2" mathvariant="normal" xref="S2.SS1.p1.8.m8.2.2.1.1.1.2.cmml">∞</mi></mrow><mo id="S2.SS1.p1.8.m8.2.2.1.1.3" xref="S2.SS1.p1.8.m8.2.2.1.2.cmml">,</mo><mn id="S2.SS1.p1.8.m8.1.1" xref="S2.SS1.p1.8.m8.1.1.cmml">0</mn><mo id="S2.SS1.p1.8.m8.2.2.1.1.4" stretchy="false" xref="S2.SS1.p1.8.m8.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.8.m8.2b"><apply id="S2.SS1.p1.8.m8.2.2.cmml" xref="S2.SS1.p1.8.m8.2.2"><eq id="S2.SS1.p1.8.m8.2.2.2.cmml" xref="S2.SS1.p1.8.m8.2.2.2"></eq><apply id="S2.SS1.p1.8.m8.2.2.3.cmml" xref="S2.SS1.p1.8.m8.2.2.3"><csymbol cd="ambiguous" id="S2.SS1.p1.8.m8.2.2.3.1.cmml" xref="S2.SS1.p1.8.m8.2.2.3">subscript</csymbol><ci id="S2.SS1.p1.8.m8.2.2.3.2.cmml" xref="S2.SS1.p1.8.m8.2.2.3.2">ℝ</ci><minus id="S2.SS1.p1.8.m8.2.2.3.3.cmml" xref="S2.SS1.p1.8.m8.2.2.3.3"></minus></apply><interval closure="open" id="S2.SS1.p1.8.m8.2.2.1.2.cmml" xref="S2.SS1.p1.8.m8.2.2.1.1"><apply id="S2.SS1.p1.8.m8.2.2.1.1.1.cmml" xref="S2.SS1.p1.8.m8.2.2.1.1.1"><minus id="S2.SS1.p1.8.m8.2.2.1.1.1.1.cmml" xref="S2.SS1.p1.8.m8.2.2.1.1.1"></minus><infinity id="S2.SS1.p1.8.m8.2.2.1.1.1.2.cmml" xref="S2.SS1.p1.8.m8.2.2.1.1.1.2"></infinity></apply><cn id="S2.SS1.p1.8.m8.1.1.cmml" type="integer" xref="S2.SS1.p1.8.m8.1.1">0</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.8.m8.2c">\mathbb{R}_{-}=(-\infty,0)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.8.m8.2d">blackboard_R start_POSTSUBSCRIPT - end_POSTSUBSCRIPT = ( - ∞ , 0 )</annotation></semantics></math>. We define the power weight <math alttext="w_{\gamma}(x):=|x_{1}|^{\gamma}" class="ltx_Math" display="inline" id="S2.SS1.p1.9.m9.2"><semantics id="S2.SS1.p1.9.m9.2a"><mrow id="S2.SS1.p1.9.m9.2.2" xref="S2.SS1.p1.9.m9.2.2.cmml"><mrow id="S2.SS1.p1.9.m9.2.2.3" xref="S2.SS1.p1.9.m9.2.2.3.cmml"><msub id="S2.SS1.p1.9.m9.2.2.3.2" xref="S2.SS1.p1.9.m9.2.2.3.2.cmml"><mi id="S2.SS1.p1.9.m9.2.2.3.2.2" xref="S2.SS1.p1.9.m9.2.2.3.2.2.cmml">w</mi><mi id="S2.SS1.p1.9.m9.2.2.3.2.3" xref="S2.SS1.p1.9.m9.2.2.3.2.3.cmml">γ</mi></msub><mo id="S2.SS1.p1.9.m9.2.2.3.1" xref="S2.SS1.p1.9.m9.2.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p1.9.m9.2.2.3.3.2" xref="S2.SS1.p1.9.m9.2.2.3.cmml"><mo id="S2.SS1.p1.9.m9.2.2.3.3.2.1" stretchy="false" xref="S2.SS1.p1.9.m9.2.2.3.cmml">(</mo><mi id="S2.SS1.p1.9.m9.1.1" xref="S2.SS1.p1.9.m9.1.1.cmml">x</mi><mo id="S2.SS1.p1.9.m9.2.2.3.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS1.p1.9.m9.2.2.3.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p1.9.m9.2.2.2" rspace="0.278em" xref="S2.SS1.p1.9.m9.2.2.2.cmml">:=</mo><msup id="S2.SS1.p1.9.m9.2.2.1" xref="S2.SS1.p1.9.m9.2.2.1.cmml"><mrow id="S2.SS1.p1.9.m9.2.2.1.1.1" xref="S2.SS1.p1.9.m9.2.2.1.1.2.cmml"><mo id="S2.SS1.p1.9.m9.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p1.9.m9.2.2.1.1.2.1.cmml">|</mo><msub id="S2.SS1.p1.9.m9.2.2.1.1.1.1" xref="S2.SS1.p1.9.m9.2.2.1.1.1.1.cmml"><mi id="S2.SS1.p1.9.m9.2.2.1.1.1.1.2" xref="S2.SS1.p1.9.m9.2.2.1.1.1.1.2.cmml">x</mi><mn id="S2.SS1.p1.9.m9.2.2.1.1.1.1.3" xref="S2.SS1.p1.9.m9.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p1.9.m9.2.2.1.1.1.3" stretchy="false" xref="S2.SS1.p1.9.m9.2.2.1.1.2.1.cmml">|</mo></mrow><mi id="S2.SS1.p1.9.m9.2.2.1.3" xref="S2.SS1.p1.9.m9.2.2.1.3.cmml">γ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.9.m9.2b"><apply id="S2.SS1.p1.9.m9.2.2.cmml" xref="S2.SS1.p1.9.m9.2.2"><csymbol cd="latexml" id="S2.SS1.p1.9.m9.2.2.2.cmml" xref="S2.SS1.p1.9.m9.2.2.2">assign</csymbol><apply id="S2.SS1.p1.9.m9.2.2.3.cmml" xref="S2.SS1.p1.9.m9.2.2.3"><times id="S2.SS1.p1.9.m9.2.2.3.1.cmml" xref="S2.SS1.p1.9.m9.2.2.3.1"></times><apply id="S2.SS1.p1.9.m9.2.2.3.2.cmml" xref="S2.SS1.p1.9.m9.2.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.2.2.3.2.1.cmml" xref="S2.SS1.p1.9.m9.2.2.3.2">subscript</csymbol><ci id="S2.SS1.p1.9.m9.2.2.3.2.2.cmml" xref="S2.SS1.p1.9.m9.2.2.3.2.2">𝑤</ci><ci id="S2.SS1.p1.9.m9.2.2.3.2.3.cmml" xref="S2.SS1.p1.9.m9.2.2.3.2.3">𝛾</ci></apply><ci id="S2.SS1.p1.9.m9.1.1.cmml" xref="S2.SS1.p1.9.m9.1.1">𝑥</ci></apply><apply id="S2.SS1.p1.9.m9.2.2.1.cmml" xref="S2.SS1.p1.9.m9.2.2.1"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.2.2.1.2.cmml" xref="S2.SS1.p1.9.m9.2.2.1">superscript</csymbol><apply id="S2.SS1.p1.9.m9.2.2.1.1.2.cmml" xref="S2.SS1.p1.9.m9.2.2.1.1.1"><abs id="S2.SS1.p1.9.m9.2.2.1.1.2.1.cmml" xref="S2.SS1.p1.9.m9.2.2.1.1.1.2"></abs><apply id="S2.SS1.p1.9.m9.2.2.1.1.1.1.cmml" xref="S2.SS1.p1.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p1.9.m9.2.2.1.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.9.m9.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p1.9.m9.2.2.1.1.1.1.2">𝑥</ci><cn id="S2.SS1.p1.9.m9.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.9.m9.2.2.1.1.1.1.3">1</cn></apply></apply><ci id="S2.SS1.p1.9.m9.2.2.1.3.cmml" xref="S2.SS1.p1.9.m9.2.2.1.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.9.m9.2c">w_{\gamma}(x):=|x_{1}|^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.9.m9.2d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_x ) := | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S2.SS1.p1.10.m10.1"><semantics id="S2.SS1.p1.10.m10.1a"><mrow id="S2.SS1.p1.10.m10.1.1" xref="S2.SS1.p1.10.m10.1.1.cmml"><mi id="S2.SS1.p1.10.m10.1.1.2" xref="S2.SS1.p1.10.m10.1.1.2.cmml">γ</mi><mo id="S2.SS1.p1.10.m10.1.1.1" xref="S2.SS1.p1.10.m10.1.1.1.cmml">></mo><mrow id="S2.SS1.p1.10.m10.1.1.3" xref="S2.SS1.p1.10.m10.1.1.3.cmml"><mo id="S2.SS1.p1.10.m10.1.1.3a" xref="S2.SS1.p1.10.m10.1.1.3.cmml">−</mo><mn id="S2.SS1.p1.10.m10.1.1.3.2" xref="S2.SS1.p1.10.m10.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.10.m10.1b"><apply id="S2.SS1.p1.10.m10.1.1.cmml" xref="S2.SS1.p1.10.m10.1.1"><gt id="S2.SS1.p1.10.m10.1.1.1.cmml" xref="S2.SS1.p1.10.m10.1.1.1"></gt><ci id="S2.SS1.p1.10.m10.1.1.2.cmml" xref="S2.SS1.p1.10.m10.1.1.2">𝛾</ci><apply id="S2.SS1.p1.10.m10.1.1.3.cmml" xref="S2.SS1.p1.10.m10.1.1.3"><minus id="S2.SS1.p1.10.m10.1.1.3.1.cmml" xref="S2.SS1.p1.10.m10.1.1.3"></minus><cn id="S2.SS1.p1.10.m10.1.1.3.2.cmml" type="integer" xref="S2.SS1.p1.10.m10.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.10.m10.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.10.m10.1d">italic_γ > - 1</annotation></semantics></math>, where <math alttext="x=(x_{1},\widetilde{x})\in\mathbb{R}\times\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S2.SS1.p1.11.m11.2"><semantics id="S2.SS1.p1.11.m11.2a"><mrow id="S2.SS1.p1.11.m11.2.2" xref="S2.SS1.p1.11.m11.2.2.cmml"><mi id="S2.SS1.p1.11.m11.2.2.3" xref="S2.SS1.p1.11.m11.2.2.3.cmml">x</mi><mo id="S2.SS1.p1.11.m11.2.2.4" xref="S2.SS1.p1.11.m11.2.2.4.cmml">=</mo><mrow id="S2.SS1.p1.11.m11.2.2.1.1" xref="S2.SS1.p1.11.m11.2.2.1.2.cmml"><mo id="S2.SS1.p1.11.m11.2.2.1.1.2" stretchy="false" xref="S2.SS1.p1.11.m11.2.2.1.2.cmml">(</mo><msub id="S2.SS1.p1.11.m11.2.2.1.1.1" xref="S2.SS1.p1.11.m11.2.2.1.1.1.cmml"><mi id="S2.SS1.p1.11.m11.2.2.1.1.1.2" xref="S2.SS1.p1.11.m11.2.2.1.1.1.2.cmml">x</mi><mn id="S2.SS1.p1.11.m11.2.2.1.1.1.3" xref="S2.SS1.p1.11.m11.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS1.p1.11.m11.2.2.1.1.3" xref="S2.SS1.p1.11.m11.2.2.1.2.cmml">,</mo><mover accent="true" id="S2.SS1.p1.11.m11.1.1" xref="S2.SS1.p1.11.m11.1.1.cmml"><mi id="S2.SS1.p1.11.m11.1.1.2" xref="S2.SS1.p1.11.m11.1.1.2.cmml">x</mi><mo id="S2.SS1.p1.11.m11.1.1.1" xref="S2.SS1.p1.11.m11.1.1.1.cmml">~</mo></mover><mo id="S2.SS1.p1.11.m11.2.2.1.1.4" stretchy="false" xref="S2.SS1.p1.11.m11.2.2.1.2.cmml">)</mo></mrow><mo id="S2.SS1.p1.11.m11.2.2.5" xref="S2.SS1.p1.11.m11.2.2.5.cmml">∈</mo><mrow id="S2.SS1.p1.11.m11.2.2.6" xref="S2.SS1.p1.11.m11.2.2.6.cmml"><mi id="S2.SS1.p1.11.m11.2.2.6.2" xref="S2.SS1.p1.11.m11.2.2.6.2.cmml">ℝ</mi><mo id="S2.SS1.p1.11.m11.2.2.6.1" lspace="0.222em" rspace="0.222em" xref="S2.SS1.p1.11.m11.2.2.6.1.cmml">×</mo><msup id="S2.SS1.p1.11.m11.2.2.6.3" xref="S2.SS1.p1.11.m11.2.2.6.3.cmml"><mi id="S2.SS1.p1.11.m11.2.2.6.3.2" xref="S2.SS1.p1.11.m11.2.2.6.3.2.cmml">ℝ</mi><mrow id="S2.SS1.p1.11.m11.2.2.6.3.3" xref="S2.SS1.p1.11.m11.2.2.6.3.3.cmml"><mi id="S2.SS1.p1.11.m11.2.2.6.3.3.2" xref="S2.SS1.p1.11.m11.2.2.6.3.3.2.cmml">d</mi><mo id="S2.SS1.p1.11.m11.2.2.6.3.3.1" xref="S2.SS1.p1.11.m11.2.2.6.3.3.1.cmml">−</mo><mn id="S2.SS1.p1.11.m11.2.2.6.3.3.3" xref="S2.SS1.p1.11.m11.2.2.6.3.3.3.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p1.11.m11.2b"><apply id="S2.SS1.p1.11.m11.2.2.cmml" xref="S2.SS1.p1.11.m11.2.2"><and id="S2.SS1.p1.11.m11.2.2a.cmml" xref="S2.SS1.p1.11.m11.2.2"></and><apply id="S2.SS1.p1.11.m11.2.2b.cmml" xref="S2.SS1.p1.11.m11.2.2"><eq id="S2.SS1.p1.11.m11.2.2.4.cmml" xref="S2.SS1.p1.11.m11.2.2.4"></eq><ci id="S2.SS1.p1.11.m11.2.2.3.cmml" xref="S2.SS1.p1.11.m11.2.2.3">𝑥</ci><interval closure="open" id="S2.SS1.p1.11.m11.2.2.1.2.cmml" xref="S2.SS1.p1.11.m11.2.2.1.1"><apply id="S2.SS1.p1.11.m11.2.2.1.1.1.cmml" xref="S2.SS1.p1.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p1.11.m11.2.2.1.1.1.1.cmml" xref="S2.SS1.p1.11.m11.2.2.1.1.1">subscript</csymbol><ci id="S2.SS1.p1.11.m11.2.2.1.1.1.2.cmml" xref="S2.SS1.p1.11.m11.2.2.1.1.1.2">𝑥</ci><cn id="S2.SS1.p1.11.m11.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS1.p1.11.m11.2.2.1.1.1.3">1</cn></apply><apply id="S2.SS1.p1.11.m11.1.1.cmml" xref="S2.SS1.p1.11.m11.1.1"><ci id="S2.SS1.p1.11.m11.1.1.1.cmml" xref="S2.SS1.p1.11.m11.1.1.1">~</ci><ci id="S2.SS1.p1.11.m11.1.1.2.cmml" xref="S2.SS1.p1.11.m11.1.1.2">𝑥</ci></apply></interval></apply><apply id="S2.SS1.p1.11.m11.2.2c.cmml" xref="S2.SS1.p1.11.m11.2.2"><in id="S2.SS1.p1.11.m11.2.2.5.cmml" xref="S2.SS1.p1.11.m11.2.2.5"></in><share href="https://arxiv.org/html/2503.14636v1#S2.SS1.p1.11.m11.2.2.1.cmml" id="S2.SS1.p1.11.m11.2.2d.cmml" xref="S2.SS1.p1.11.m11.2.2"></share><apply id="S2.SS1.p1.11.m11.2.2.6.cmml" xref="S2.SS1.p1.11.m11.2.2.6"><times id="S2.SS1.p1.11.m11.2.2.6.1.cmml" xref="S2.SS1.p1.11.m11.2.2.6.1"></times><ci id="S2.SS1.p1.11.m11.2.2.6.2.cmml" xref="S2.SS1.p1.11.m11.2.2.6.2">ℝ</ci><apply id="S2.SS1.p1.11.m11.2.2.6.3.cmml" xref="S2.SS1.p1.11.m11.2.2.6.3"><csymbol cd="ambiguous" id="S2.SS1.p1.11.m11.2.2.6.3.1.cmml" xref="S2.SS1.p1.11.m11.2.2.6.3">superscript</csymbol><ci id="S2.SS1.p1.11.m11.2.2.6.3.2.cmml" xref="S2.SS1.p1.11.m11.2.2.6.3.2">ℝ</ci><apply id="S2.SS1.p1.11.m11.2.2.6.3.3.cmml" xref="S2.SS1.p1.11.m11.2.2.6.3.3"><minus id="S2.SS1.p1.11.m11.2.2.6.3.3.1.cmml" xref="S2.SS1.p1.11.m11.2.2.6.3.3.1"></minus><ci id="S2.SS1.p1.11.m11.2.2.6.3.3.2.cmml" xref="S2.SS1.p1.11.m11.2.2.6.3.3.2">𝑑</ci><cn id="S2.SS1.p1.11.m11.2.2.6.3.3.3.cmml" type="integer" xref="S2.SS1.p1.11.m11.2.2.6.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p1.11.m11.2c">x=(x_{1},\widetilde{x})\in\mathbb{R}\times\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p1.11.m11.2d">italic_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) ∈ blackboard_R × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p2"> <p class="ltx_p" id="S2.SS1.p2.7">For two topological vector spaces <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p2.1.m1.1"><semantics id="S2.SS1.p2.1.m1.1a"><mi id="S2.SS1.p2.1.m1.1.1" xref="S2.SS1.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.1.m1.1b"><ci id="S2.SS1.p2.1.m1.1.1.cmml" xref="S2.SS1.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.1.m1.1d">italic_X</annotation></semantics></math> and <math alttext="Y" class="ltx_Math" display="inline" id="S2.SS1.p2.2.m2.1"><semantics id="S2.SS1.p2.2.m2.1a"><mi id="S2.SS1.p2.2.m2.1.1" xref="S2.SS1.p2.2.m2.1.1.cmml">Y</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.2.m2.1b"><ci id="S2.SS1.p2.2.m2.1.1.cmml" xref="S2.SS1.p2.2.m2.1.1">𝑌</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.2.m2.1c">Y</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.2.m2.1d">italic_Y</annotation></semantics></math>, the space of continuous linear operators is <math alttext="\mathcal{L}(X,Y)" class="ltx_Math" display="inline" id="S2.SS1.p2.3.m3.2"><semantics id="S2.SS1.p2.3.m3.2a"><mrow id="S2.SS1.p2.3.m3.2.3" xref="S2.SS1.p2.3.m3.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.3.m3.2.3.2" xref="S2.SS1.p2.3.m3.2.3.2.cmml">ℒ</mi><mo id="S2.SS1.p2.3.m3.2.3.1" xref="S2.SS1.p2.3.m3.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p2.3.m3.2.3.3.2" xref="S2.SS1.p2.3.m3.2.3.3.1.cmml"><mo id="S2.SS1.p2.3.m3.2.3.3.2.1" stretchy="false" xref="S2.SS1.p2.3.m3.2.3.3.1.cmml">(</mo><mi id="S2.SS1.p2.3.m3.1.1" xref="S2.SS1.p2.3.m3.1.1.cmml">X</mi><mo id="S2.SS1.p2.3.m3.2.3.3.2.2" xref="S2.SS1.p2.3.m3.2.3.3.1.cmml">,</mo><mi id="S2.SS1.p2.3.m3.2.2" xref="S2.SS1.p2.3.m3.2.2.cmml">Y</mi><mo id="S2.SS1.p2.3.m3.2.3.3.2.3" stretchy="false" xref="S2.SS1.p2.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.3.m3.2b"><apply id="S2.SS1.p2.3.m3.2.3.cmml" xref="S2.SS1.p2.3.m3.2.3"><times id="S2.SS1.p2.3.m3.2.3.1.cmml" xref="S2.SS1.p2.3.m3.2.3.1"></times><ci id="S2.SS1.p2.3.m3.2.3.2.cmml" xref="S2.SS1.p2.3.m3.2.3.2">ℒ</ci><interval closure="open" id="S2.SS1.p2.3.m3.2.3.3.1.cmml" xref="S2.SS1.p2.3.m3.2.3.3.2"><ci id="S2.SS1.p2.3.m3.1.1.cmml" xref="S2.SS1.p2.3.m3.1.1">𝑋</ci><ci id="S2.SS1.p2.3.m3.2.2.cmml" xref="S2.SS1.p2.3.m3.2.2">𝑌</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.3.m3.2c">\mathcal{L}(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.3.m3.2d">caligraphic_L ( italic_X , italic_Y )</annotation></semantics></math> and <math alttext="\mathcal{L}(X):=\mathcal{L}(X,X)" class="ltx_Math" display="inline" id="S2.SS1.p2.4.m4.3"><semantics id="S2.SS1.p2.4.m4.3a"><mrow id="S2.SS1.p2.4.m4.3.4" xref="S2.SS1.p2.4.m4.3.4.cmml"><mrow id="S2.SS1.p2.4.m4.3.4.2" xref="S2.SS1.p2.4.m4.3.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.3.4.2.2" xref="S2.SS1.p2.4.m4.3.4.2.2.cmml">ℒ</mi><mo id="S2.SS1.p2.4.m4.3.4.2.1" xref="S2.SS1.p2.4.m4.3.4.2.1.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m4.3.4.2.3.2" xref="S2.SS1.p2.4.m4.3.4.2.cmml"><mo id="S2.SS1.p2.4.m4.3.4.2.3.2.1" stretchy="false" xref="S2.SS1.p2.4.m4.3.4.2.cmml">(</mo><mi id="S2.SS1.p2.4.m4.1.1" xref="S2.SS1.p2.4.m4.1.1.cmml">X</mi><mo id="S2.SS1.p2.4.m4.3.4.2.3.2.2" rspace="0.278em" stretchy="false" xref="S2.SS1.p2.4.m4.3.4.2.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p2.4.m4.3.4.1" rspace="0.278em" xref="S2.SS1.p2.4.m4.3.4.1.cmml">:=</mo><mrow id="S2.SS1.p2.4.m4.3.4.3" xref="S2.SS1.p2.4.m4.3.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.4.m4.3.4.3.2" xref="S2.SS1.p2.4.m4.3.4.3.2.cmml">ℒ</mi><mo id="S2.SS1.p2.4.m4.3.4.3.1" xref="S2.SS1.p2.4.m4.3.4.3.1.cmml">⁢</mo><mrow id="S2.SS1.p2.4.m4.3.4.3.3.2" xref="S2.SS1.p2.4.m4.3.4.3.3.1.cmml"><mo id="S2.SS1.p2.4.m4.3.4.3.3.2.1" stretchy="false" xref="S2.SS1.p2.4.m4.3.4.3.3.1.cmml">(</mo><mi id="S2.SS1.p2.4.m4.2.2" xref="S2.SS1.p2.4.m4.2.2.cmml">X</mi><mo id="S2.SS1.p2.4.m4.3.4.3.3.2.2" xref="S2.SS1.p2.4.m4.3.4.3.3.1.cmml">,</mo><mi id="S2.SS1.p2.4.m4.3.3" xref="S2.SS1.p2.4.m4.3.3.cmml">X</mi><mo id="S2.SS1.p2.4.m4.3.4.3.3.2.3" stretchy="false" xref="S2.SS1.p2.4.m4.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.4.m4.3b"><apply id="S2.SS1.p2.4.m4.3.4.cmml" xref="S2.SS1.p2.4.m4.3.4"><csymbol cd="latexml" id="S2.SS1.p2.4.m4.3.4.1.cmml" xref="S2.SS1.p2.4.m4.3.4.1">assign</csymbol><apply id="S2.SS1.p2.4.m4.3.4.2.cmml" xref="S2.SS1.p2.4.m4.3.4.2"><times id="S2.SS1.p2.4.m4.3.4.2.1.cmml" xref="S2.SS1.p2.4.m4.3.4.2.1"></times><ci id="S2.SS1.p2.4.m4.3.4.2.2.cmml" xref="S2.SS1.p2.4.m4.3.4.2.2">ℒ</ci><ci id="S2.SS1.p2.4.m4.1.1.cmml" xref="S2.SS1.p2.4.m4.1.1">𝑋</ci></apply><apply id="S2.SS1.p2.4.m4.3.4.3.cmml" xref="S2.SS1.p2.4.m4.3.4.3"><times id="S2.SS1.p2.4.m4.3.4.3.1.cmml" xref="S2.SS1.p2.4.m4.3.4.3.1"></times><ci id="S2.SS1.p2.4.m4.3.4.3.2.cmml" xref="S2.SS1.p2.4.m4.3.4.3.2">ℒ</ci><interval closure="open" id="S2.SS1.p2.4.m4.3.4.3.3.1.cmml" xref="S2.SS1.p2.4.m4.3.4.3.3.2"><ci id="S2.SS1.p2.4.m4.2.2.cmml" xref="S2.SS1.p2.4.m4.2.2">𝑋</ci><ci id="S2.SS1.p2.4.m4.3.3.cmml" xref="S2.SS1.p2.4.m4.3.3">𝑋</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.4.m4.3c">\mathcal{L}(X):=\mathcal{L}(X,X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.4.m4.3d">caligraphic_L ( italic_X ) := caligraphic_L ( italic_X , italic_X )</annotation></semantics></math>. Unless specified otherwise, <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p2.5.m5.1"><semantics id="S2.SS1.p2.5.m5.1a"><mi id="S2.SS1.p2.5.m5.1.1" xref="S2.SS1.p2.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.5.m5.1b"><ci id="S2.SS1.p2.5.m5.1.1.cmml" xref="S2.SS1.p2.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.5.m5.1d">italic_X</annotation></semantics></math> will always denote a Banach space with norm <math alttext="\|\cdot\|_{X}" class="ltx_math_unparsed" display="inline" id="S2.SS1.p2.6.m6.1"><semantics id="S2.SS1.p2.6.m6.1a"><mrow id="S2.SS1.p2.6.m6.1b"><mo id="S2.SS1.p2.6.m6.1.1" rspace="0em">∥</mo><mo id="S2.SS1.p2.6.m6.1.2" lspace="0em" rspace="0em">⋅</mo><msub id="S2.SS1.p2.6.m6.1.3"><mo id="S2.SS1.p2.6.m6.1.3.2" lspace="0em">∥</mo><mi id="S2.SS1.p2.6.m6.1.3.3">X</mi></msub></mrow><annotation encoding="application/x-tex" id="S2.SS1.p2.6.m6.1c">\|\cdot\|_{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.6.m6.1d">∥ ⋅ ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT</annotation></semantics></math> and the dual space is <math alttext="X^{\prime}:=\mathcal{L}(X,\mathbb{C})" class="ltx_Math" display="inline" id="S2.SS1.p2.7.m7.2"><semantics id="S2.SS1.p2.7.m7.2a"><mrow id="S2.SS1.p2.7.m7.2.3" xref="S2.SS1.p2.7.m7.2.3.cmml"><msup id="S2.SS1.p2.7.m7.2.3.2" xref="S2.SS1.p2.7.m7.2.3.2.cmml"><mi id="S2.SS1.p2.7.m7.2.3.2.2" xref="S2.SS1.p2.7.m7.2.3.2.2.cmml">X</mi><mo id="S2.SS1.p2.7.m7.2.3.2.3" xref="S2.SS1.p2.7.m7.2.3.2.3.cmml">′</mo></msup><mo id="S2.SS1.p2.7.m7.2.3.1" lspace="0.278em" rspace="0.278em" xref="S2.SS1.p2.7.m7.2.3.1.cmml">:=</mo><mrow id="S2.SS1.p2.7.m7.2.3.3" xref="S2.SS1.p2.7.m7.2.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p2.7.m7.2.3.3.2" xref="S2.SS1.p2.7.m7.2.3.3.2.cmml">ℒ</mi><mo id="S2.SS1.p2.7.m7.2.3.3.1" xref="S2.SS1.p2.7.m7.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p2.7.m7.2.3.3.3.2" xref="S2.SS1.p2.7.m7.2.3.3.3.1.cmml"><mo id="S2.SS1.p2.7.m7.2.3.3.3.2.1" stretchy="false" xref="S2.SS1.p2.7.m7.2.3.3.3.1.cmml">(</mo><mi id="S2.SS1.p2.7.m7.1.1" xref="S2.SS1.p2.7.m7.1.1.cmml">X</mi><mo id="S2.SS1.p2.7.m7.2.3.3.3.2.2" xref="S2.SS1.p2.7.m7.2.3.3.3.1.cmml">,</mo><mi id="S2.SS1.p2.7.m7.2.2" xref="S2.SS1.p2.7.m7.2.2.cmml">ℂ</mi><mo id="S2.SS1.p2.7.m7.2.3.3.3.2.3" stretchy="false" xref="S2.SS1.p2.7.m7.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p2.7.m7.2b"><apply id="S2.SS1.p2.7.m7.2.3.cmml" xref="S2.SS1.p2.7.m7.2.3"><csymbol cd="latexml" id="S2.SS1.p2.7.m7.2.3.1.cmml" xref="S2.SS1.p2.7.m7.2.3.1">assign</csymbol><apply id="S2.SS1.p2.7.m7.2.3.2.cmml" xref="S2.SS1.p2.7.m7.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p2.7.m7.2.3.2.1.cmml" xref="S2.SS1.p2.7.m7.2.3.2">superscript</csymbol><ci id="S2.SS1.p2.7.m7.2.3.2.2.cmml" xref="S2.SS1.p2.7.m7.2.3.2.2">𝑋</ci><ci id="S2.SS1.p2.7.m7.2.3.2.3.cmml" xref="S2.SS1.p2.7.m7.2.3.2.3">′</ci></apply><apply id="S2.SS1.p2.7.m7.2.3.3.cmml" xref="S2.SS1.p2.7.m7.2.3.3"><times id="S2.SS1.p2.7.m7.2.3.3.1.cmml" xref="S2.SS1.p2.7.m7.2.3.3.1"></times><ci id="S2.SS1.p2.7.m7.2.3.3.2.cmml" xref="S2.SS1.p2.7.m7.2.3.3.2">ℒ</ci><interval closure="open" id="S2.SS1.p2.7.m7.2.3.3.3.1.cmml" xref="S2.SS1.p2.7.m7.2.3.3.3.2"><ci id="S2.SS1.p2.7.m7.1.1.cmml" xref="S2.SS1.p2.7.m7.1.1">𝑋</ci><ci id="S2.SS1.p2.7.m7.2.2.cmml" xref="S2.SS1.p2.7.m7.2.2">ℂ</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p2.7.m7.2c">X^{\prime}:=\mathcal{L}(X,\mathbb{C})</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p2.7.m7.2d">italic_X start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT := caligraphic_L ( italic_X , blackboard_C )</annotation></semantics></math>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S2.SS1.p3"> <p class="ltx_p" id="S2.SS1.p3.15">For an open and non-empty <math alttext="\mathcal{O}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S2.SS1.p3.1.m1.1"><semantics id="S2.SS1.p3.1.m1.1a"><mrow id="S2.SS1.p3.1.m1.1.1" xref="S2.SS1.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.1.m1.1.1.2" xref="S2.SS1.p3.1.m1.1.1.2.cmml">𝒪</mi><mo id="S2.SS1.p3.1.m1.1.1.1" xref="S2.SS1.p3.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.SS1.p3.1.m1.1.1.3" xref="S2.SS1.p3.1.m1.1.1.3.cmml"><mi id="S2.SS1.p3.1.m1.1.1.3.2" xref="S2.SS1.p3.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS1.p3.1.m1.1.1.3.3" xref="S2.SS1.p3.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.1.m1.1b"><apply id="S2.SS1.p3.1.m1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1"><subset id="S2.SS1.p3.1.m1.1.1.1.cmml" xref="S2.SS1.p3.1.m1.1.1.1"></subset><ci id="S2.SS1.p3.1.m1.1.1.2.cmml" xref="S2.SS1.p3.1.m1.1.1.2">𝒪</ci><apply id="S2.SS1.p3.1.m1.1.1.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.1.m1.1.1.3.1.cmml" xref="S2.SS1.p3.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS1.p3.1.m1.1.1.3.2.cmml" xref="S2.SS1.p3.1.m1.1.1.3.2">ℝ</ci><ci id="S2.SS1.p3.1.m1.1.1.3.3.cmml" xref="S2.SS1.p3.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.1.m1.1c">\mathcal{O}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.1.m1.1d">caligraphic_O ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="k\in\mathbb{N}_{0}\cup\{\infty\}" class="ltx_Math" display="inline" id="S2.SS1.p3.2.m2.1"><semantics id="S2.SS1.p3.2.m2.1a"><mrow id="S2.SS1.p3.2.m2.1.2" xref="S2.SS1.p3.2.m2.1.2.cmml"><mi id="S2.SS1.p3.2.m2.1.2.2" xref="S2.SS1.p3.2.m2.1.2.2.cmml">k</mi><mo id="S2.SS1.p3.2.m2.1.2.1" xref="S2.SS1.p3.2.m2.1.2.1.cmml">∈</mo><mrow id="S2.SS1.p3.2.m2.1.2.3" xref="S2.SS1.p3.2.m2.1.2.3.cmml"><msub id="S2.SS1.p3.2.m2.1.2.3.2" xref="S2.SS1.p3.2.m2.1.2.3.2.cmml"><mi id="S2.SS1.p3.2.m2.1.2.3.2.2" xref="S2.SS1.p3.2.m2.1.2.3.2.2.cmml">ℕ</mi><mn id="S2.SS1.p3.2.m2.1.2.3.2.3" xref="S2.SS1.p3.2.m2.1.2.3.2.3.cmml">0</mn></msub><mo id="S2.SS1.p3.2.m2.1.2.3.1" xref="S2.SS1.p3.2.m2.1.2.3.1.cmml">∪</mo><mrow id="S2.SS1.p3.2.m2.1.2.3.3.2" xref="S2.SS1.p3.2.m2.1.2.3.3.1.cmml"><mo id="S2.SS1.p3.2.m2.1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p3.2.m2.1.2.3.3.1.cmml">{</mo><mi id="S2.SS1.p3.2.m2.1.1" mathvariant="normal" xref="S2.SS1.p3.2.m2.1.1.cmml">∞</mi><mo id="S2.SS1.p3.2.m2.1.2.3.3.2.2" stretchy="false" xref="S2.SS1.p3.2.m2.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.2.m2.1b"><apply id="S2.SS1.p3.2.m2.1.2.cmml" xref="S2.SS1.p3.2.m2.1.2"><in id="S2.SS1.p3.2.m2.1.2.1.cmml" xref="S2.SS1.p3.2.m2.1.2.1"></in><ci id="S2.SS1.p3.2.m2.1.2.2.cmml" xref="S2.SS1.p3.2.m2.1.2.2">𝑘</ci><apply id="S2.SS1.p3.2.m2.1.2.3.cmml" xref="S2.SS1.p3.2.m2.1.2.3"><union id="S2.SS1.p3.2.m2.1.2.3.1.cmml" xref="S2.SS1.p3.2.m2.1.2.3.1"></union><apply id="S2.SS1.p3.2.m2.1.2.3.2.cmml" xref="S2.SS1.p3.2.m2.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.2.m2.1.2.3.2.1.cmml" xref="S2.SS1.p3.2.m2.1.2.3.2">subscript</csymbol><ci id="S2.SS1.p3.2.m2.1.2.3.2.2.cmml" xref="S2.SS1.p3.2.m2.1.2.3.2.2">ℕ</ci><cn id="S2.SS1.p3.2.m2.1.2.3.2.3.cmml" type="integer" xref="S2.SS1.p3.2.m2.1.2.3.2.3">0</cn></apply><set id="S2.SS1.p3.2.m2.1.2.3.3.1.cmml" xref="S2.SS1.p3.2.m2.1.2.3.3.2"><infinity id="S2.SS1.p3.2.m2.1.1.cmml" xref="S2.SS1.p3.2.m2.1.1"></infinity></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.2.m2.1c">k\in\mathbb{N}_{0}\cup\{\infty\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.2.m2.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ { ∞ }</annotation></semantics></math>, the space <math alttext="C^{k}(\mathcal{O};X)" class="ltx_Math" display="inline" id="S2.SS1.p3.3.m3.2"><semantics id="S2.SS1.p3.3.m3.2a"><mrow id="S2.SS1.p3.3.m3.2.3" xref="S2.SS1.p3.3.m3.2.3.cmml"><msup id="S2.SS1.p3.3.m3.2.3.2" xref="S2.SS1.p3.3.m3.2.3.2.cmml"><mi id="S2.SS1.p3.3.m3.2.3.2.2" xref="S2.SS1.p3.3.m3.2.3.2.2.cmml">C</mi><mi id="S2.SS1.p3.3.m3.2.3.2.3" xref="S2.SS1.p3.3.m3.2.3.2.3.cmml">k</mi></msup><mo id="S2.SS1.p3.3.m3.2.3.1" xref="S2.SS1.p3.3.m3.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.3.m3.2.3.3.2" xref="S2.SS1.p3.3.m3.2.3.3.1.cmml"><mo id="S2.SS1.p3.3.m3.2.3.3.2.1" stretchy="false" xref="S2.SS1.p3.3.m3.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.3.m3.1.1" xref="S2.SS1.p3.3.m3.1.1.cmml">𝒪</mi><mo id="S2.SS1.p3.3.m3.2.3.3.2.2" xref="S2.SS1.p3.3.m3.2.3.3.1.cmml">;</mo><mi id="S2.SS1.p3.3.m3.2.2" xref="S2.SS1.p3.3.m3.2.2.cmml">X</mi><mo id="S2.SS1.p3.3.m3.2.3.3.2.3" stretchy="false" xref="S2.SS1.p3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.3.m3.2b"><apply id="S2.SS1.p3.3.m3.2.3.cmml" xref="S2.SS1.p3.3.m3.2.3"><times id="S2.SS1.p3.3.m3.2.3.1.cmml" xref="S2.SS1.p3.3.m3.2.3.1"></times><apply id="S2.SS1.p3.3.m3.2.3.2.cmml" xref="S2.SS1.p3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.3.m3.2.3.2.1.cmml" xref="S2.SS1.p3.3.m3.2.3.2">superscript</csymbol><ci id="S2.SS1.p3.3.m3.2.3.2.2.cmml" xref="S2.SS1.p3.3.m3.2.3.2.2">𝐶</ci><ci id="S2.SS1.p3.3.m3.2.3.2.3.cmml" xref="S2.SS1.p3.3.m3.2.3.2.3">𝑘</ci></apply><list id="S2.SS1.p3.3.m3.2.3.3.1.cmml" xref="S2.SS1.p3.3.m3.2.3.3.2"><ci id="S2.SS1.p3.3.m3.1.1.cmml" xref="S2.SS1.p3.3.m3.1.1">𝒪</ci><ci id="S2.SS1.p3.3.m3.2.2.cmml" xref="S2.SS1.p3.3.m3.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.3.m3.2c">C^{k}(\mathcal{O};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.3.m3.2d">italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( caligraphic_O ; italic_X )</annotation></semantics></math> denotes the space of <math alttext="k" class="ltx_Math" display="inline" id="S2.SS1.p3.4.m4.1"><semantics id="S2.SS1.p3.4.m4.1a"><mi id="S2.SS1.p3.4.m4.1.1" xref="S2.SS1.p3.4.m4.1.1.cmml">k</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.4.m4.1b"><ci id="S2.SS1.p3.4.m4.1.1.cmml" xref="S2.SS1.p3.4.m4.1.1">𝑘</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.4.m4.1c">k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.4.m4.1d">italic_k</annotation></semantics></math>-times continuously differentiable functions from <math alttext="\mathcal{O}" class="ltx_Math" display="inline" id="S2.SS1.p3.5.m5.1"><semantics id="S2.SS1.p3.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.5.m5.1.1" xref="S2.SS1.p3.5.m5.1.1.cmml">𝒪</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.5.m5.1b"><ci id="S2.SS1.p3.5.m5.1.1.cmml" xref="S2.SS1.p3.5.m5.1.1">𝒪</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.5.m5.1c">\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.5.m5.1d">caligraphic_O</annotation></semantics></math> to some Banach space <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p3.6.m6.1"><semantics id="S2.SS1.p3.6.m6.1a"><mi id="S2.SS1.p3.6.m6.1.1" xref="S2.SS1.p3.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.6.m6.1b"><ci id="S2.SS1.p3.6.m6.1.1.cmml" xref="S2.SS1.p3.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.6.m6.1d">italic_X</annotation></semantics></math>. In the case <math alttext="k=0" class="ltx_Math" display="inline" id="S2.SS1.p3.7.m7.1"><semantics id="S2.SS1.p3.7.m7.1a"><mrow id="S2.SS1.p3.7.m7.1.1" xref="S2.SS1.p3.7.m7.1.1.cmml"><mi id="S2.SS1.p3.7.m7.1.1.2" xref="S2.SS1.p3.7.m7.1.1.2.cmml">k</mi><mo id="S2.SS1.p3.7.m7.1.1.1" xref="S2.SS1.p3.7.m7.1.1.1.cmml">=</mo><mn id="S2.SS1.p3.7.m7.1.1.3" xref="S2.SS1.p3.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.7.m7.1b"><apply id="S2.SS1.p3.7.m7.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1"><eq id="S2.SS1.p3.7.m7.1.1.1.cmml" xref="S2.SS1.p3.7.m7.1.1.1"></eq><ci id="S2.SS1.p3.7.m7.1.1.2.cmml" xref="S2.SS1.p3.7.m7.1.1.2">𝑘</ci><cn id="S2.SS1.p3.7.m7.1.1.3.cmml" type="integer" xref="S2.SS1.p3.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.7.m7.1c">k=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.7.m7.1d">italic_k = 0</annotation></semantics></math> we write <math alttext="C(\mathcal{O};X)" class="ltx_Math" display="inline" id="S2.SS1.p3.8.m8.2"><semantics id="S2.SS1.p3.8.m8.2a"><mrow id="S2.SS1.p3.8.m8.2.3" xref="S2.SS1.p3.8.m8.2.3.cmml"><mi id="S2.SS1.p3.8.m8.2.3.2" xref="S2.SS1.p3.8.m8.2.3.2.cmml">C</mi><mo id="S2.SS1.p3.8.m8.2.3.1" xref="S2.SS1.p3.8.m8.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.8.m8.2.3.3.2" xref="S2.SS1.p3.8.m8.2.3.3.1.cmml"><mo id="S2.SS1.p3.8.m8.2.3.3.2.1" stretchy="false" xref="S2.SS1.p3.8.m8.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.8.m8.1.1" xref="S2.SS1.p3.8.m8.1.1.cmml">𝒪</mi><mo id="S2.SS1.p3.8.m8.2.3.3.2.2" xref="S2.SS1.p3.8.m8.2.3.3.1.cmml">;</mo><mi id="S2.SS1.p3.8.m8.2.2" xref="S2.SS1.p3.8.m8.2.2.cmml">X</mi><mo id="S2.SS1.p3.8.m8.2.3.3.2.3" stretchy="false" xref="S2.SS1.p3.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.8.m8.2b"><apply id="S2.SS1.p3.8.m8.2.3.cmml" xref="S2.SS1.p3.8.m8.2.3"><times id="S2.SS1.p3.8.m8.2.3.1.cmml" xref="S2.SS1.p3.8.m8.2.3.1"></times><ci id="S2.SS1.p3.8.m8.2.3.2.cmml" xref="S2.SS1.p3.8.m8.2.3.2">𝐶</ci><list id="S2.SS1.p3.8.m8.2.3.3.1.cmml" xref="S2.SS1.p3.8.m8.2.3.3.2"><ci id="S2.SS1.p3.8.m8.1.1.cmml" xref="S2.SS1.p3.8.m8.1.1">𝒪</ci><ci id="S2.SS1.p3.8.m8.2.2.cmml" xref="S2.SS1.p3.8.m8.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.8.m8.2c">C(\mathcal{O};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.8.m8.2d">italic_C ( caligraphic_O ; italic_X )</annotation></semantics></math> for <math alttext="C^{0}(\mathcal{O};X)" class="ltx_Math" display="inline" id="S2.SS1.p3.9.m9.2"><semantics id="S2.SS1.p3.9.m9.2a"><mrow id="S2.SS1.p3.9.m9.2.3" xref="S2.SS1.p3.9.m9.2.3.cmml"><msup id="S2.SS1.p3.9.m9.2.3.2" xref="S2.SS1.p3.9.m9.2.3.2.cmml"><mi id="S2.SS1.p3.9.m9.2.3.2.2" xref="S2.SS1.p3.9.m9.2.3.2.2.cmml">C</mi><mn id="S2.SS1.p3.9.m9.2.3.2.3" xref="S2.SS1.p3.9.m9.2.3.2.3.cmml">0</mn></msup><mo id="S2.SS1.p3.9.m9.2.3.1" xref="S2.SS1.p3.9.m9.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.9.m9.2.3.3.2" xref="S2.SS1.p3.9.m9.2.3.3.1.cmml"><mo id="S2.SS1.p3.9.m9.2.3.3.2.1" stretchy="false" xref="S2.SS1.p3.9.m9.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.9.m9.1.1" xref="S2.SS1.p3.9.m9.1.1.cmml">𝒪</mi><mo id="S2.SS1.p3.9.m9.2.3.3.2.2" xref="S2.SS1.p3.9.m9.2.3.3.1.cmml">;</mo><mi id="S2.SS1.p3.9.m9.2.2" xref="S2.SS1.p3.9.m9.2.2.cmml">X</mi><mo id="S2.SS1.p3.9.m9.2.3.3.2.3" stretchy="false" xref="S2.SS1.p3.9.m9.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.9.m9.2b"><apply id="S2.SS1.p3.9.m9.2.3.cmml" xref="S2.SS1.p3.9.m9.2.3"><times id="S2.SS1.p3.9.m9.2.3.1.cmml" xref="S2.SS1.p3.9.m9.2.3.1"></times><apply id="S2.SS1.p3.9.m9.2.3.2.cmml" xref="S2.SS1.p3.9.m9.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.9.m9.2.3.2.1.cmml" xref="S2.SS1.p3.9.m9.2.3.2">superscript</csymbol><ci id="S2.SS1.p3.9.m9.2.3.2.2.cmml" xref="S2.SS1.p3.9.m9.2.3.2.2">𝐶</ci><cn id="S2.SS1.p3.9.m9.2.3.2.3.cmml" type="integer" xref="S2.SS1.p3.9.m9.2.3.2.3">0</cn></apply><list id="S2.SS1.p3.9.m9.2.3.3.1.cmml" xref="S2.SS1.p3.9.m9.2.3.3.2"><ci id="S2.SS1.p3.9.m9.1.1.cmml" xref="S2.SS1.p3.9.m9.1.1">𝒪</ci><ci id="S2.SS1.p3.9.m9.2.2.cmml" xref="S2.SS1.p3.9.m9.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.9.m9.2c">C^{0}(\mathcal{O};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.9.m9.2d">italic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ( caligraphic_O ; italic_X )</annotation></semantics></math>. Furthermore, we write <math alttext="C^{k}_{{\rm b}}(\mathcal{O};X)" class="ltx_Math" display="inline" id="S2.SS1.p3.10.m10.2"><semantics id="S2.SS1.p3.10.m10.2a"><mrow id="S2.SS1.p3.10.m10.2.3" xref="S2.SS1.p3.10.m10.2.3.cmml"><msubsup id="S2.SS1.p3.10.m10.2.3.2" xref="S2.SS1.p3.10.m10.2.3.2.cmml"><mi id="S2.SS1.p3.10.m10.2.3.2.2.2" xref="S2.SS1.p3.10.m10.2.3.2.2.2.cmml">C</mi><mi id="S2.SS1.p3.10.m10.2.3.2.3" mathvariant="normal" xref="S2.SS1.p3.10.m10.2.3.2.3.cmml">b</mi><mi id="S2.SS1.p3.10.m10.2.3.2.2.3" xref="S2.SS1.p3.10.m10.2.3.2.2.3.cmml">k</mi></msubsup><mo id="S2.SS1.p3.10.m10.2.3.1" xref="S2.SS1.p3.10.m10.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.10.m10.2.3.3.2" xref="S2.SS1.p3.10.m10.2.3.3.1.cmml"><mo id="S2.SS1.p3.10.m10.2.3.3.2.1" stretchy="false" xref="S2.SS1.p3.10.m10.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.10.m10.1.1" xref="S2.SS1.p3.10.m10.1.1.cmml">𝒪</mi><mo id="S2.SS1.p3.10.m10.2.3.3.2.2" xref="S2.SS1.p3.10.m10.2.3.3.1.cmml">;</mo><mi id="S2.SS1.p3.10.m10.2.2" xref="S2.SS1.p3.10.m10.2.2.cmml">X</mi><mo id="S2.SS1.p3.10.m10.2.3.3.2.3" stretchy="false" xref="S2.SS1.p3.10.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.10.m10.2b"><apply id="S2.SS1.p3.10.m10.2.3.cmml" xref="S2.SS1.p3.10.m10.2.3"><times id="S2.SS1.p3.10.m10.2.3.1.cmml" xref="S2.SS1.p3.10.m10.2.3.1"></times><apply id="S2.SS1.p3.10.m10.2.3.2.cmml" xref="S2.SS1.p3.10.m10.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.10.m10.2.3.2.1.cmml" xref="S2.SS1.p3.10.m10.2.3.2">subscript</csymbol><apply id="S2.SS1.p3.10.m10.2.3.2.2.cmml" xref="S2.SS1.p3.10.m10.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.10.m10.2.3.2.2.1.cmml" xref="S2.SS1.p3.10.m10.2.3.2">superscript</csymbol><ci id="S2.SS1.p3.10.m10.2.3.2.2.2.cmml" xref="S2.SS1.p3.10.m10.2.3.2.2.2">𝐶</ci><ci id="S2.SS1.p3.10.m10.2.3.2.2.3.cmml" xref="S2.SS1.p3.10.m10.2.3.2.2.3">𝑘</ci></apply><ci id="S2.SS1.p3.10.m10.2.3.2.3.cmml" xref="S2.SS1.p3.10.m10.2.3.2.3">b</ci></apply><list id="S2.SS1.p3.10.m10.2.3.3.1.cmml" xref="S2.SS1.p3.10.m10.2.3.3.2"><ci id="S2.SS1.p3.10.m10.1.1.cmml" xref="S2.SS1.p3.10.m10.1.1">𝒪</ci><ci id="S2.SS1.p3.10.m10.2.2.cmml" xref="S2.SS1.p3.10.m10.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.10.m10.2c">C^{k}_{{\rm b}}(\mathcal{O};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.10.m10.2d">italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT ( caligraphic_O ; italic_X )</annotation></semantics></math> for the space of all functions in <math alttext="f\in C^{k}(\mathcal{O};X)" class="ltx_Math" display="inline" id="S2.SS1.p3.11.m11.2"><semantics id="S2.SS1.p3.11.m11.2a"><mrow id="S2.SS1.p3.11.m11.2.3" xref="S2.SS1.p3.11.m11.2.3.cmml"><mi id="S2.SS1.p3.11.m11.2.3.2" xref="S2.SS1.p3.11.m11.2.3.2.cmml">f</mi><mo id="S2.SS1.p3.11.m11.2.3.1" xref="S2.SS1.p3.11.m11.2.3.1.cmml">∈</mo><mrow id="S2.SS1.p3.11.m11.2.3.3" xref="S2.SS1.p3.11.m11.2.3.3.cmml"><msup id="S2.SS1.p3.11.m11.2.3.3.2" xref="S2.SS1.p3.11.m11.2.3.3.2.cmml"><mi id="S2.SS1.p3.11.m11.2.3.3.2.2" xref="S2.SS1.p3.11.m11.2.3.3.2.2.cmml">C</mi><mi id="S2.SS1.p3.11.m11.2.3.3.2.3" xref="S2.SS1.p3.11.m11.2.3.3.2.3.cmml">k</mi></msup><mo id="S2.SS1.p3.11.m11.2.3.3.1" xref="S2.SS1.p3.11.m11.2.3.3.1.cmml">⁢</mo><mrow id="S2.SS1.p3.11.m11.2.3.3.3.2" xref="S2.SS1.p3.11.m11.2.3.3.3.1.cmml"><mo id="S2.SS1.p3.11.m11.2.3.3.3.2.1" stretchy="false" xref="S2.SS1.p3.11.m11.2.3.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.11.m11.1.1" xref="S2.SS1.p3.11.m11.1.1.cmml">𝒪</mi><mo id="S2.SS1.p3.11.m11.2.3.3.3.2.2" xref="S2.SS1.p3.11.m11.2.3.3.3.1.cmml">;</mo><mi id="S2.SS1.p3.11.m11.2.2" xref="S2.SS1.p3.11.m11.2.2.cmml">X</mi><mo id="S2.SS1.p3.11.m11.2.3.3.3.2.3" stretchy="false" xref="S2.SS1.p3.11.m11.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.11.m11.2b"><apply id="S2.SS1.p3.11.m11.2.3.cmml" xref="S2.SS1.p3.11.m11.2.3"><in id="S2.SS1.p3.11.m11.2.3.1.cmml" xref="S2.SS1.p3.11.m11.2.3.1"></in><ci id="S2.SS1.p3.11.m11.2.3.2.cmml" xref="S2.SS1.p3.11.m11.2.3.2">𝑓</ci><apply id="S2.SS1.p3.11.m11.2.3.3.cmml" xref="S2.SS1.p3.11.m11.2.3.3"><times id="S2.SS1.p3.11.m11.2.3.3.1.cmml" xref="S2.SS1.p3.11.m11.2.3.3.1"></times><apply id="S2.SS1.p3.11.m11.2.3.3.2.cmml" xref="S2.SS1.p3.11.m11.2.3.3.2"><csymbol cd="ambiguous" id="S2.SS1.p3.11.m11.2.3.3.2.1.cmml" xref="S2.SS1.p3.11.m11.2.3.3.2">superscript</csymbol><ci id="S2.SS1.p3.11.m11.2.3.3.2.2.cmml" xref="S2.SS1.p3.11.m11.2.3.3.2.2">𝐶</ci><ci id="S2.SS1.p3.11.m11.2.3.3.2.3.cmml" xref="S2.SS1.p3.11.m11.2.3.3.2.3">𝑘</ci></apply><list id="S2.SS1.p3.11.m11.2.3.3.3.1.cmml" xref="S2.SS1.p3.11.m11.2.3.3.3.2"><ci id="S2.SS1.p3.11.m11.1.1.cmml" xref="S2.SS1.p3.11.m11.1.1">𝒪</ci><ci id="S2.SS1.p3.11.m11.2.2.cmml" xref="S2.SS1.p3.11.m11.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.11.m11.2c">f\in C^{k}(\mathcal{O};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.11.m11.2d">italic_f ∈ italic_C start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( caligraphic_O ; italic_X )</annotation></semantics></math> such that <math alttext="\partial^{\alpha}f" class="ltx_Math" display="inline" id="S2.SS1.p3.12.m12.1"><semantics id="S2.SS1.p3.12.m12.1a"><mrow id="S2.SS1.p3.12.m12.1.1" xref="S2.SS1.p3.12.m12.1.1.cmml"><msup id="S2.SS1.p3.12.m12.1.1.1" xref="S2.SS1.p3.12.m12.1.1.1.cmml"><mo id="S2.SS1.p3.12.m12.1.1.1.2" xref="S2.SS1.p3.12.m12.1.1.1.2.cmml">∂</mo><mi id="S2.SS1.p3.12.m12.1.1.1.3" xref="S2.SS1.p3.12.m12.1.1.1.3.cmml">α</mi></msup><mi id="S2.SS1.p3.12.m12.1.1.2" xref="S2.SS1.p3.12.m12.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.12.m12.1b"><apply id="S2.SS1.p3.12.m12.1.1.cmml" xref="S2.SS1.p3.12.m12.1.1"><apply id="S2.SS1.p3.12.m12.1.1.1.cmml" xref="S2.SS1.p3.12.m12.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p3.12.m12.1.1.1.1.cmml" xref="S2.SS1.p3.12.m12.1.1.1">superscript</csymbol><partialdiff id="S2.SS1.p3.12.m12.1.1.1.2.cmml" xref="S2.SS1.p3.12.m12.1.1.1.2"></partialdiff><ci id="S2.SS1.p3.12.m12.1.1.1.3.cmml" xref="S2.SS1.p3.12.m12.1.1.1.3">𝛼</ci></apply><ci id="S2.SS1.p3.12.m12.1.1.2.cmml" xref="S2.SS1.p3.12.m12.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.12.m12.1c">\partial^{\alpha}f</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.12.m12.1d">∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_f</annotation></semantics></math> is bounded on <math alttext="\mathcal{O}" class="ltx_Math" display="inline" id="S2.SS1.p3.13.m13.1"><semantics id="S2.SS1.p3.13.m13.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p3.13.m13.1.1" xref="S2.SS1.p3.13.m13.1.1.cmml">𝒪</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.13.m13.1b"><ci id="S2.SS1.p3.13.m13.1.1.cmml" xref="S2.SS1.p3.13.m13.1.1">𝒪</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.13.m13.1c">\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.13.m13.1d">caligraphic_O</annotation></semantics></math> for all multi-indices <math alttext="\alpha\in\mathbb{N}_{0}^{d}" class="ltx_Math" display="inline" id="S2.SS1.p3.14.m14.1"><semantics id="S2.SS1.p3.14.m14.1a"><mrow id="S2.SS1.p3.14.m14.1.1" xref="S2.SS1.p3.14.m14.1.1.cmml"><mi id="S2.SS1.p3.14.m14.1.1.2" xref="S2.SS1.p3.14.m14.1.1.2.cmml">α</mi><mo id="S2.SS1.p3.14.m14.1.1.1" xref="S2.SS1.p3.14.m14.1.1.1.cmml">∈</mo><msubsup id="S2.SS1.p3.14.m14.1.1.3" xref="S2.SS1.p3.14.m14.1.1.3.cmml"><mi id="S2.SS1.p3.14.m14.1.1.3.2.2" xref="S2.SS1.p3.14.m14.1.1.3.2.2.cmml">ℕ</mi><mn id="S2.SS1.p3.14.m14.1.1.3.2.3" xref="S2.SS1.p3.14.m14.1.1.3.2.3.cmml">0</mn><mi id="S2.SS1.p3.14.m14.1.1.3.3" xref="S2.SS1.p3.14.m14.1.1.3.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.14.m14.1b"><apply id="S2.SS1.p3.14.m14.1.1.cmml" xref="S2.SS1.p3.14.m14.1.1"><in id="S2.SS1.p3.14.m14.1.1.1.cmml" xref="S2.SS1.p3.14.m14.1.1.1"></in><ci id="S2.SS1.p3.14.m14.1.1.2.cmml" xref="S2.SS1.p3.14.m14.1.1.2">𝛼</ci><apply id="S2.SS1.p3.14.m14.1.1.3.cmml" xref="S2.SS1.p3.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.14.m14.1.1.3.1.cmml" xref="S2.SS1.p3.14.m14.1.1.3">superscript</csymbol><apply id="S2.SS1.p3.14.m14.1.1.3.2.cmml" xref="S2.SS1.p3.14.m14.1.1.3"><csymbol cd="ambiguous" id="S2.SS1.p3.14.m14.1.1.3.2.1.cmml" xref="S2.SS1.p3.14.m14.1.1.3">subscript</csymbol><ci id="S2.SS1.p3.14.m14.1.1.3.2.2.cmml" xref="S2.SS1.p3.14.m14.1.1.3.2.2">ℕ</ci><cn id="S2.SS1.p3.14.m14.1.1.3.2.3.cmml" type="integer" xref="S2.SS1.p3.14.m14.1.1.3.2.3">0</cn></apply><ci id="S2.SS1.p3.14.m14.1.1.3.3.cmml" xref="S2.SS1.p3.14.m14.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.14.m14.1c">\alpha\in\mathbb{N}_{0}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.14.m14.1d">italic_α ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="|\alpha|\leq k" class="ltx_Math" display="inline" id="S2.SS1.p3.15.m15.1"><semantics id="S2.SS1.p3.15.m15.1a"><mrow id="S2.SS1.p3.15.m15.1.2" xref="S2.SS1.p3.15.m15.1.2.cmml"><mrow id="S2.SS1.p3.15.m15.1.2.2.2" xref="S2.SS1.p3.15.m15.1.2.2.1.cmml"><mo id="S2.SS1.p3.15.m15.1.2.2.2.1" stretchy="false" xref="S2.SS1.p3.15.m15.1.2.2.1.1.cmml">|</mo><mi id="S2.SS1.p3.15.m15.1.1" xref="S2.SS1.p3.15.m15.1.1.cmml">α</mi><mo id="S2.SS1.p3.15.m15.1.2.2.2.2" stretchy="false" xref="S2.SS1.p3.15.m15.1.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS1.p3.15.m15.1.2.1" xref="S2.SS1.p3.15.m15.1.2.1.cmml">≤</mo><mi id="S2.SS1.p3.15.m15.1.2.3" xref="S2.SS1.p3.15.m15.1.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p3.15.m15.1b"><apply id="S2.SS1.p3.15.m15.1.2.cmml" xref="S2.SS1.p3.15.m15.1.2"><leq id="S2.SS1.p3.15.m15.1.2.1.cmml" xref="S2.SS1.p3.15.m15.1.2.1"></leq><apply id="S2.SS1.p3.15.m15.1.2.2.1.cmml" xref="S2.SS1.p3.15.m15.1.2.2.2"><abs id="S2.SS1.p3.15.m15.1.2.2.1.1.cmml" xref="S2.SS1.p3.15.m15.1.2.2.2.1"></abs><ci id="S2.SS1.p3.15.m15.1.1.cmml" xref="S2.SS1.p3.15.m15.1.1">𝛼</ci></apply><ci id="S2.SS1.p3.15.m15.1.2.3.cmml" xref="S2.SS1.p3.15.m15.1.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p3.15.m15.1c">|\alpha|\leq k</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p3.15.m15.1d">| italic_α | ≤ italic_k</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p4"> <p class="ltx_p" id="S2.SS1.p4.6">Let <math alttext="C_{\mathrm{c}}^{\infty}(\mathcal{O};X)" class="ltx_Math" display="inline" id="S2.SS1.p4.1.m1.2"><semantics id="S2.SS1.p4.1.m1.2a"><mrow id="S2.SS1.p4.1.m1.2.3" xref="S2.SS1.p4.1.m1.2.3.cmml"><msubsup id="S2.SS1.p4.1.m1.2.3.2" xref="S2.SS1.p4.1.m1.2.3.2.cmml"><mi id="S2.SS1.p4.1.m1.2.3.2.2.2" xref="S2.SS1.p4.1.m1.2.3.2.2.2.cmml">C</mi><mi id="S2.SS1.p4.1.m1.2.3.2.2.3" mathvariant="normal" xref="S2.SS1.p4.1.m1.2.3.2.2.3.cmml">c</mi><mi id="S2.SS1.p4.1.m1.2.3.2.3" mathvariant="normal" xref="S2.SS1.p4.1.m1.2.3.2.3.cmml">∞</mi></msubsup><mo id="S2.SS1.p4.1.m1.2.3.1" xref="S2.SS1.p4.1.m1.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p4.1.m1.2.3.3.2" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml"><mo id="S2.SS1.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.1.m1.1.1" xref="S2.SS1.p4.1.m1.1.1.cmml">𝒪</mi><mo id="S2.SS1.p4.1.m1.2.3.3.2.2" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml">;</mo><mi id="S2.SS1.p4.1.m1.2.2" xref="S2.SS1.p4.1.m1.2.2.cmml">X</mi><mo id="S2.SS1.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS1.p4.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.1.m1.2b"><apply id="S2.SS1.p4.1.m1.2.3.cmml" xref="S2.SS1.p4.1.m1.2.3"><times id="S2.SS1.p4.1.m1.2.3.1.cmml" xref="S2.SS1.p4.1.m1.2.3.1"></times><apply id="S2.SS1.p4.1.m1.2.3.2.cmml" xref="S2.SS1.p4.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.2.3.2.1.cmml" xref="S2.SS1.p4.1.m1.2.3.2">superscript</csymbol><apply id="S2.SS1.p4.1.m1.2.3.2.2.cmml" xref="S2.SS1.p4.1.m1.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.1.m1.2.3.2.2.1.cmml" xref="S2.SS1.p4.1.m1.2.3.2">subscript</csymbol><ci id="S2.SS1.p4.1.m1.2.3.2.2.2.cmml" xref="S2.SS1.p4.1.m1.2.3.2.2.2">𝐶</ci><ci id="S2.SS1.p4.1.m1.2.3.2.2.3.cmml" xref="S2.SS1.p4.1.m1.2.3.2.2.3">c</ci></apply><infinity id="S2.SS1.p4.1.m1.2.3.2.3.cmml" xref="S2.SS1.p4.1.m1.2.3.2.3"></infinity></apply><list id="S2.SS1.p4.1.m1.2.3.3.1.cmml" xref="S2.SS1.p4.1.m1.2.3.3.2"><ci id="S2.SS1.p4.1.m1.1.1.cmml" xref="S2.SS1.p4.1.m1.1.1">𝒪</ci><ci id="S2.SS1.p4.1.m1.2.2.cmml" xref="S2.SS1.p4.1.m1.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.1.m1.2c">C_{\mathrm{c}}^{\infty}(\mathcal{O};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.1.m1.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( caligraphic_O ; italic_X )</annotation></semantics></math> be the space of compactly supported smooth functions on <math alttext="\mathcal{O}" class="ltx_Math" display="inline" id="S2.SS1.p4.2.m2.1"><semantics id="S2.SS1.p4.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.2.m2.1.1" xref="S2.SS1.p4.2.m2.1.1.cmml">𝒪</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.2.m2.1b"><ci id="S2.SS1.p4.2.m2.1.1.cmml" xref="S2.SS1.p4.2.m2.1.1">𝒪</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.2.m2.1c">\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.2.m2.1d">caligraphic_O</annotation></semantics></math> equipped with its usual inductive limit topology. The space of <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p4.3.m3.1"><semantics id="S2.SS1.p4.3.m3.1a"><mi id="S2.SS1.p4.3.m3.1.1" xref="S2.SS1.p4.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.3.m3.1b"><ci id="S2.SS1.p4.3.m3.1.1.cmml" xref="S2.SS1.p4.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.3.m3.1d">italic_X</annotation></semantics></math>-valued distributions is given by <math alttext="\mathcal{D}^{\prime}(\mathcal{O};X):=\mathcal{L}(C_{\mathrm{c}}^{\infty}(% \mathcal{O});X)" class="ltx_Math" display="inline" id="S2.SS1.p4.4.m4.5"><semantics id="S2.SS1.p4.4.m4.5a"><mrow id="S2.SS1.p4.4.m4.5.5" xref="S2.SS1.p4.4.m4.5.5.cmml"><mrow id="S2.SS1.p4.4.m4.5.5.3" xref="S2.SS1.p4.4.m4.5.5.3.cmml"><msup id="S2.SS1.p4.4.m4.5.5.3.2" xref="S2.SS1.p4.4.m4.5.5.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.4.m4.5.5.3.2.2" xref="S2.SS1.p4.4.m4.5.5.3.2.2.cmml">𝒟</mi><mo id="S2.SS1.p4.4.m4.5.5.3.2.3" xref="S2.SS1.p4.4.m4.5.5.3.2.3.cmml">′</mo></msup><mo id="S2.SS1.p4.4.m4.5.5.3.1" xref="S2.SS1.p4.4.m4.5.5.3.1.cmml">⁢</mo><mrow id="S2.SS1.p4.4.m4.5.5.3.3.2" xref="S2.SS1.p4.4.m4.5.5.3.3.1.cmml"><mo id="S2.SS1.p4.4.m4.5.5.3.3.2.1" stretchy="false" xref="S2.SS1.p4.4.m4.5.5.3.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.4.m4.1.1" xref="S2.SS1.p4.4.m4.1.1.cmml">𝒪</mi><mo id="S2.SS1.p4.4.m4.5.5.3.3.2.2" xref="S2.SS1.p4.4.m4.5.5.3.3.1.cmml">;</mo><mi id="S2.SS1.p4.4.m4.2.2" xref="S2.SS1.p4.4.m4.2.2.cmml">X</mi><mo id="S2.SS1.p4.4.m4.5.5.3.3.2.3" rspace="0.278em" stretchy="false" xref="S2.SS1.p4.4.m4.5.5.3.3.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p4.4.m4.5.5.2" rspace="0.278em" xref="S2.SS1.p4.4.m4.5.5.2.cmml">:=</mo><mrow id="S2.SS1.p4.4.m4.5.5.1" xref="S2.SS1.p4.4.m4.5.5.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.4.m4.5.5.1.3" xref="S2.SS1.p4.4.m4.5.5.1.3.cmml">ℒ</mi><mo id="S2.SS1.p4.4.m4.5.5.1.2" xref="S2.SS1.p4.4.m4.5.5.1.2.cmml">⁢</mo><mrow id="S2.SS1.p4.4.m4.5.5.1.1.1" xref="S2.SS1.p4.4.m4.5.5.1.1.2.cmml"><mo id="S2.SS1.p4.4.m4.5.5.1.1.1.2" stretchy="false" xref="S2.SS1.p4.4.m4.5.5.1.1.2.cmml">(</mo><mrow id="S2.SS1.p4.4.m4.5.5.1.1.1.1" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.cmml"><msubsup id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.cmml"><mi id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.2" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.2.cmml">C</mi><mi id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.3" mathvariant="normal" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.3.cmml">c</mi><mi id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.3" mathvariant="normal" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.3.cmml">∞</mi></msubsup><mo id="S2.SS1.p4.4.m4.5.5.1.1.1.1.1" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.SS1.p4.4.m4.5.5.1.1.1.1.3.2" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.cmml"><mo id="S2.SS1.p4.4.m4.5.5.1.1.1.1.3.2.1" stretchy="false" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.4.m4.3.3" xref="S2.SS1.p4.4.m4.3.3.cmml">𝒪</mi><mo id="S2.SS1.p4.4.m4.5.5.1.1.1.1.3.2.2" stretchy="false" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS1.p4.4.m4.5.5.1.1.1.3" xref="S2.SS1.p4.4.m4.5.5.1.1.2.cmml">;</mo><mi id="S2.SS1.p4.4.m4.4.4" xref="S2.SS1.p4.4.m4.4.4.cmml">X</mi><mo id="S2.SS1.p4.4.m4.5.5.1.1.1.4" stretchy="false" xref="S2.SS1.p4.4.m4.5.5.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.4.m4.5b"><apply id="S2.SS1.p4.4.m4.5.5.cmml" xref="S2.SS1.p4.4.m4.5.5"><csymbol cd="latexml" id="S2.SS1.p4.4.m4.5.5.2.cmml" xref="S2.SS1.p4.4.m4.5.5.2">assign</csymbol><apply id="S2.SS1.p4.4.m4.5.5.3.cmml" xref="S2.SS1.p4.4.m4.5.5.3"><times id="S2.SS1.p4.4.m4.5.5.3.1.cmml" xref="S2.SS1.p4.4.m4.5.5.3.1"></times><apply id="S2.SS1.p4.4.m4.5.5.3.2.cmml" xref="S2.SS1.p4.4.m4.5.5.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.4.m4.5.5.3.2.1.cmml" xref="S2.SS1.p4.4.m4.5.5.3.2">superscript</csymbol><ci id="S2.SS1.p4.4.m4.5.5.3.2.2.cmml" xref="S2.SS1.p4.4.m4.5.5.3.2.2">𝒟</ci><ci id="S2.SS1.p4.4.m4.5.5.3.2.3.cmml" xref="S2.SS1.p4.4.m4.5.5.3.2.3">′</ci></apply><list id="S2.SS1.p4.4.m4.5.5.3.3.1.cmml" xref="S2.SS1.p4.4.m4.5.5.3.3.2"><ci id="S2.SS1.p4.4.m4.1.1.cmml" xref="S2.SS1.p4.4.m4.1.1">𝒪</ci><ci id="S2.SS1.p4.4.m4.2.2.cmml" xref="S2.SS1.p4.4.m4.2.2">𝑋</ci></list></apply><apply id="S2.SS1.p4.4.m4.5.5.1.cmml" xref="S2.SS1.p4.4.m4.5.5.1"><times id="S2.SS1.p4.4.m4.5.5.1.2.cmml" xref="S2.SS1.p4.4.m4.5.5.1.2"></times><ci id="S2.SS1.p4.4.m4.5.5.1.3.cmml" xref="S2.SS1.p4.4.m4.5.5.1.3">ℒ</ci><list id="S2.SS1.p4.4.m4.5.5.1.1.2.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1"><apply id="S2.SS1.p4.4.m4.5.5.1.1.1.1.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1"><times id="S2.SS1.p4.4.m4.5.5.1.1.1.1.1.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.1"></times><apply id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.1.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2">superscript</csymbol><apply id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.1.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2">subscript</csymbol><ci id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.2.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.2">𝐶</ci><ci id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.3.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.2.3">c</ci></apply><infinity id="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.3.cmml" xref="S2.SS1.p4.4.m4.5.5.1.1.1.1.2.3"></infinity></apply><ci id="S2.SS1.p4.4.m4.3.3.cmml" xref="S2.SS1.p4.4.m4.3.3">𝒪</ci></apply><ci id="S2.SS1.p4.4.m4.4.4.cmml" xref="S2.SS1.p4.4.m4.4.4">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.4.m4.5c">\mathcal{D}^{\prime}(\mathcal{O};X):=\mathcal{L}(C_{\mathrm{c}}^{\infty}(% \mathcal{O});X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.4.m4.5d">caligraphic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( caligraphic_O ; italic_X ) := caligraphic_L ( italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( caligraphic_O ) ; italic_X )</annotation></semantics></math>. Moreover, <math alttext="C_{\mathrm{c}}^{\infty}(\overline{\mathcal{O}};X)" class="ltx_Math" display="inline" id="S2.SS1.p4.5.m5.2"><semantics id="S2.SS1.p4.5.m5.2a"><mrow id="S2.SS1.p4.5.m5.2.3" xref="S2.SS1.p4.5.m5.2.3.cmml"><msubsup id="S2.SS1.p4.5.m5.2.3.2" xref="S2.SS1.p4.5.m5.2.3.2.cmml"><mi id="S2.SS1.p4.5.m5.2.3.2.2.2" xref="S2.SS1.p4.5.m5.2.3.2.2.2.cmml">C</mi><mi id="S2.SS1.p4.5.m5.2.3.2.2.3" mathvariant="normal" xref="S2.SS1.p4.5.m5.2.3.2.2.3.cmml">c</mi><mi id="S2.SS1.p4.5.m5.2.3.2.3" mathvariant="normal" xref="S2.SS1.p4.5.m5.2.3.2.3.cmml">∞</mi></msubsup><mo id="S2.SS1.p4.5.m5.2.3.1" xref="S2.SS1.p4.5.m5.2.3.1.cmml">⁢</mo><mrow id="S2.SS1.p4.5.m5.2.3.3.2" xref="S2.SS1.p4.5.m5.2.3.3.1.cmml"><mo id="S2.SS1.p4.5.m5.2.3.3.2.1" stretchy="false" xref="S2.SS1.p4.5.m5.2.3.3.1.cmml">(</mo><mover accent="true" id="S2.SS1.p4.5.m5.1.1" xref="S2.SS1.p4.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.5.m5.1.1.2" xref="S2.SS1.p4.5.m5.1.1.2.cmml">𝒪</mi><mo id="S2.SS1.p4.5.m5.1.1.1" xref="S2.SS1.p4.5.m5.1.1.1.cmml">¯</mo></mover><mo id="S2.SS1.p4.5.m5.2.3.3.2.2" xref="S2.SS1.p4.5.m5.2.3.3.1.cmml">;</mo><mi id="S2.SS1.p4.5.m5.2.2" xref="S2.SS1.p4.5.m5.2.2.cmml">X</mi><mo id="S2.SS1.p4.5.m5.2.3.3.2.3" stretchy="false" xref="S2.SS1.p4.5.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.5.m5.2b"><apply id="S2.SS1.p4.5.m5.2.3.cmml" xref="S2.SS1.p4.5.m5.2.3"><times id="S2.SS1.p4.5.m5.2.3.1.cmml" xref="S2.SS1.p4.5.m5.2.3.1"></times><apply id="S2.SS1.p4.5.m5.2.3.2.cmml" xref="S2.SS1.p4.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.5.m5.2.3.2.1.cmml" xref="S2.SS1.p4.5.m5.2.3.2">superscript</csymbol><apply id="S2.SS1.p4.5.m5.2.3.2.2.cmml" xref="S2.SS1.p4.5.m5.2.3.2"><csymbol cd="ambiguous" id="S2.SS1.p4.5.m5.2.3.2.2.1.cmml" xref="S2.SS1.p4.5.m5.2.3.2">subscript</csymbol><ci id="S2.SS1.p4.5.m5.2.3.2.2.2.cmml" xref="S2.SS1.p4.5.m5.2.3.2.2.2">𝐶</ci><ci id="S2.SS1.p4.5.m5.2.3.2.2.3.cmml" xref="S2.SS1.p4.5.m5.2.3.2.2.3">c</ci></apply><infinity id="S2.SS1.p4.5.m5.2.3.2.3.cmml" xref="S2.SS1.p4.5.m5.2.3.2.3"></infinity></apply><list id="S2.SS1.p4.5.m5.2.3.3.1.cmml" xref="S2.SS1.p4.5.m5.2.3.3.2"><apply id="S2.SS1.p4.5.m5.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1"><ci id="S2.SS1.p4.5.m5.1.1.1.cmml" xref="S2.SS1.p4.5.m5.1.1.1">¯</ci><ci id="S2.SS1.p4.5.m5.1.1.2.cmml" xref="S2.SS1.p4.5.m5.1.1.2">𝒪</ci></apply><ci id="S2.SS1.p4.5.m5.2.2.cmml" xref="S2.SS1.p4.5.m5.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.5.m5.2c">C_{\mathrm{c}}^{\infty}(\overline{\mathcal{O}};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.5.m5.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG caligraphic_O end_ARG ; italic_X )</annotation></semantics></math> is the space of smooth functions with its support in a compact set contained in <math alttext="\overline{\mathcal{O}}" class="ltx_Math" display="inline" id="S2.SS1.p4.6.m6.1"><semantics id="S2.SS1.p4.6.m6.1a"><mover accent="true" id="S2.SS1.p4.6.m6.1.1" xref="S2.SS1.p4.6.m6.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS1.p4.6.m6.1.1.2" xref="S2.SS1.p4.6.m6.1.1.2.cmml">𝒪</mi><mo id="S2.SS1.p4.6.m6.1.1.1" xref="S2.SS1.p4.6.m6.1.1.1.cmml">¯</mo></mover><annotation-xml encoding="MathML-Content" id="S2.SS1.p4.6.m6.1b"><apply id="S2.SS1.p4.6.m6.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1"><ci id="S2.SS1.p4.6.m6.1.1.1.cmml" xref="S2.SS1.p4.6.m6.1.1.1">¯</ci><ci id="S2.SS1.p4.6.m6.1.1.2.cmml" xref="S2.SS1.p4.6.m6.1.1.2">𝒪</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p4.6.m6.1c">\overline{\mathcal{O}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p4.6.m6.1d">over¯ start_ARG caligraphic_O end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS1.p5"> <p class="ltx_p" id="S2.SS1.p5.5">We denote the Schwartz space by <math alttext="\SS(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS1.p5.1.m1.2"><semantics id="S2.SS1.p5.1.m1.2a"><mrow id="S2.SS1.p5.1.m1.2.2" xref="S2.SS1.p5.1.m1.2.2.cmml"><mi id="S2.SS1.p5.1.m1.2.2.3" xref="S2.SS1.p5.1.m1.2.2.3.cmml">SS</mi><mo id="S2.SS1.p5.1.m1.2.2.2" xref="S2.SS1.p5.1.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p5.1.m1.2.2.1.1" xref="S2.SS1.p5.1.m1.2.2.1.2.cmml"><mo id="S2.SS1.p5.1.m1.2.2.1.1.2" stretchy="false" xref="S2.SS1.p5.1.m1.2.2.1.2.cmml">(</mo><msup id="S2.SS1.p5.1.m1.2.2.1.1.1" xref="S2.SS1.p5.1.m1.2.2.1.1.1.cmml"><mi id="S2.SS1.p5.1.m1.2.2.1.1.1.2" xref="S2.SS1.p5.1.m1.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS1.p5.1.m1.2.2.1.1.1.3" xref="S2.SS1.p5.1.m1.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS1.p5.1.m1.2.2.1.1.3" xref="S2.SS1.p5.1.m1.2.2.1.2.cmml">;</mo><mi id="S2.SS1.p5.1.m1.1.1" xref="S2.SS1.p5.1.m1.1.1.cmml">X</mi><mo id="S2.SS1.p5.1.m1.2.2.1.1.4" stretchy="false" xref="S2.SS1.p5.1.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.1.m1.2b"><apply id="S2.SS1.p5.1.m1.2.2.cmml" xref="S2.SS1.p5.1.m1.2.2"><times id="S2.SS1.p5.1.m1.2.2.2.cmml" xref="S2.SS1.p5.1.m1.2.2.2"></times><ci id="S2.SS1.p5.1.m1.2.2.3.cmml" xref="S2.SS1.p5.1.m1.2.2.3">SS</ci><list id="S2.SS1.p5.1.m1.2.2.1.2.cmml" xref="S2.SS1.p5.1.m1.2.2.1.1"><apply id="S2.SS1.p5.1.m1.2.2.1.1.1.cmml" xref="S2.SS1.p5.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.1.m1.2.2.1.1.1.1.cmml" xref="S2.SS1.p5.1.m1.2.2.1.1.1">superscript</csymbol><ci id="S2.SS1.p5.1.m1.2.2.1.1.1.2.cmml" xref="S2.SS1.p5.1.m1.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS1.p5.1.m1.2.2.1.1.1.3.cmml" xref="S2.SS1.p5.1.m1.2.2.1.1.1.3">𝑑</ci></apply><ci id="S2.SS1.p5.1.m1.1.1.cmml" xref="S2.SS1.p5.1.m1.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.1.m1.2c">\SS(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.1.m1.2d">roman_SS ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> and <math alttext="\SS^{\prime}(\mathbb{R}^{d};X):=\mathcal{L}(\SS(\mathbb{R}^{d});X)" class="ltx_Math" display="inline" id="S2.SS1.p5.2.m2.4"><semantics id="S2.SS1.p5.2.m2.4a"><mrow id="S2.SS1.p5.2.m2.4.4" xref="S2.SS1.p5.2.m2.4.4.cmml"><mrow id="S2.SS1.p5.2.m2.3.3.1" xref="S2.SS1.p5.2.m2.3.3.1.cmml"><msup id="S2.SS1.p5.2.m2.3.3.1.3" xref="S2.SS1.p5.2.m2.3.3.1.3.cmml"><mi id="S2.SS1.p5.2.m2.3.3.1.3.2" xref="S2.SS1.p5.2.m2.3.3.1.3.2.cmml">SS</mi><mo id="S2.SS1.p5.2.m2.3.3.1.3.3" xref="S2.SS1.p5.2.m2.3.3.1.3.3.cmml">′</mo></msup><mo id="S2.SS1.p5.2.m2.3.3.1.2" xref="S2.SS1.p5.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS1.p5.2.m2.3.3.1.1.1" xref="S2.SS1.p5.2.m2.3.3.1.1.2.cmml"><mo id="S2.SS1.p5.2.m2.3.3.1.1.1.2" stretchy="false" xref="S2.SS1.p5.2.m2.3.3.1.1.2.cmml">(</mo><msup id="S2.SS1.p5.2.m2.3.3.1.1.1.1" xref="S2.SS1.p5.2.m2.3.3.1.1.1.1.cmml"><mi 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stretchy="false" xref="S2.SS1.p5.2.m2.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.2.m2.4b"><apply id="S2.SS1.p5.2.m2.4.4.cmml" xref="S2.SS1.p5.2.m2.4.4"><csymbol cd="latexml" id="S2.SS1.p5.2.m2.4.4.3.cmml" xref="S2.SS1.p5.2.m2.4.4.3">assign</csymbol><apply id="S2.SS1.p5.2.m2.3.3.1.cmml" xref="S2.SS1.p5.2.m2.3.3.1"><times id="S2.SS1.p5.2.m2.3.3.1.2.cmml" xref="S2.SS1.p5.2.m2.3.3.1.2"></times><apply id="S2.SS1.p5.2.m2.3.3.1.3.cmml" xref="S2.SS1.p5.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.3.3.1.3.1.cmml" xref="S2.SS1.p5.2.m2.3.3.1.3">superscript</csymbol><ci id="S2.SS1.p5.2.m2.3.3.1.3.2.cmml" xref="S2.SS1.p5.2.m2.3.3.1.3.2">SS</ci><ci id="S2.SS1.p5.2.m2.3.3.1.3.3.cmml" xref="S2.SS1.p5.2.m2.3.3.1.3.3">′</ci></apply><list id="S2.SS1.p5.2.m2.3.3.1.1.2.cmml" xref="S2.SS1.p5.2.m2.3.3.1.1.1"><apply id="S2.SS1.p5.2.m2.3.3.1.1.1.1.cmml" xref="S2.SS1.p5.2.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.3.3.1.1.1.1.1.cmml" xref="S2.SS1.p5.2.m2.3.3.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p5.2.m2.3.3.1.1.1.1.2.cmml" xref="S2.SS1.p5.2.m2.3.3.1.1.1.1.2">ℝ</ci><ci id="S2.SS1.p5.2.m2.3.3.1.1.1.1.3.cmml" xref="S2.SS1.p5.2.m2.3.3.1.1.1.1.3">𝑑</ci></apply><ci id="S2.SS1.p5.2.m2.1.1.cmml" xref="S2.SS1.p5.2.m2.1.1">𝑋</ci></list></apply><apply id="S2.SS1.p5.2.m2.4.4.2.cmml" xref="S2.SS1.p5.2.m2.4.4.2"><times id="S2.SS1.p5.2.m2.4.4.2.2.cmml" xref="S2.SS1.p5.2.m2.4.4.2.2"></times><ci id="S2.SS1.p5.2.m2.4.4.2.3.cmml" xref="S2.SS1.p5.2.m2.4.4.2.3">ℒ</ci><list id="S2.SS1.p5.2.m2.4.4.2.1.2.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1"><apply id="S2.SS1.p5.2.m2.4.4.2.1.1.1.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1.1"><times id="S2.SS1.p5.2.m2.4.4.2.1.1.1.2.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1.1.2"></times><ci id="S2.SS1.p5.2.m2.4.4.2.1.1.1.3.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1.1.3">SS</ci><apply id="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.1.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.2.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.3.cmml" xref="S2.SS1.p5.2.m2.4.4.2.1.1.1.1.1.1.3">𝑑</ci></apply></apply><ci id="S2.SS1.p5.2.m2.2.2.cmml" xref="S2.SS1.p5.2.m2.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.2.m2.4c">\SS^{\prime}(\mathbb{R}^{d};X):=\mathcal{L}(\SS(\mathbb{R}^{d});X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.2.m2.4d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X ) := caligraphic_L ( roman_SS ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) ; italic_X )</annotation></semantics></math> is the space of <math alttext="X" class="ltx_Math" display="inline" id="S2.SS1.p5.3.m3.1"><semantics id="S2.SS1.p5.3.m3.1a"><mi id="S2.SS1.p5.3.m3.1.1" xref="S2.SS1.p5.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.3.m3.1b"><ci id="S2.SS1.p5.3.m3.1.1.cmml" xref="S2.SS1.p5.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.3.m3.1d">italic_X</annotation></semantics></math>-valued tempered distributions. For <math alttext="f\in\SS(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS1.p5.4.m4.2"><semantics id="S2.SS1.p5.4.m4.2a"><mrow id="S2.SS1.p5.4.m4.2.2" xref="S2.SS1.p5.4.m4.2.2.cmml"><mi id="S2.SS1.p5.4.m4.2.2.3" xref="S2.SS1.p5.4.m4.2.2.3.cmml">f</mi><mo id="S2.SS1.p5.4.m4.2.2.2" xref="S2.SS1.p5.4.m4.2.2.2.cmml">∈</mo><mrow id="S2.SS1.p5.4.m4.2.2.1" xref="S2.SS1.p5.4.m4.2.2.1.cmml"><mi id="S2.SS1.p5.4.m4.2.2.1.3" xref="S2.SS1.p5.4.m4.2.2.1.3.cmml">SS</mi><mo id="S2.SS1.p5.4.m4.2.2.1.2" xref="S2.SS1.p5.4.m4.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS1.p5.4.m4.2.2.1.1.1" xref="S2.SS1.p5.4.m4.2.2.1.1.2.cmml"><mo id="S2.SS1.p5.4.m4.2.2.1.1.1.2" stretchy="false" xref="S2.SS1.p5.4.m4.2.2.1.1.2.cmml">(</mo><msup id="S2.SS1.p5.4.m4.2.2.1.1.1.1" xref="S2.SS1.p5.4.m4.2.2.1.1.1.1.cmml"><mi id="S2.SS1.p5.4.m4.2.2.1.1.1.1.2" xref="S2.SS1.p5.4.m4.2.2.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS1.p5.4.m4.2.2.1.1.1.1.3" xref="S2.SS1.p5.4.m4.2.2.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS1.p5.4.m4.2.2.1.1.1.3" xref="S2.SS1.p5.4.m4.2.2.1.1.2.cmml">;</mo><mi id="S2.SS1.p5.4.m4.1.1" xref="S2.SS1.p5.4.m4.1.1.cmml">X</mi><mo id="S2.SS1.p5.4.m4.2.2.1.1.1.4" stretchy="false" xref="S2.SS1.p5.4.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.4.m4.2b"><apply id="S2.SS1.p5.4.m4.2.2.cmml" xref="S2.SS1.p5.4.m4.2.2"><in id="S2.SS1.p5.4.m4.2.2.2.cmml" xref="S2.SS1.p5.4.m4.2.2.2"></in><ci id="S2.SS1.p5.4.m4.2.2.3.cmml" xref="S2.SS1.p5.4.m4.2.2.3">𝑓</ci><apply id="S2.SS1.p5.4.m4.2.2.1.cmml" xref="S2.SS1.p5.4.m4.2.2.1"><times id="S2.SS1.p5.4.m4.2.2.1.2.cmml" xref="S2.SS1.p5.4.m4.2.2.1.2"></times><ci id="S2.SS1.p5.4.m4.2.2.1.3.cmml" xref="S2.SS1.p5.4.m4.2.2.1.3">SS</ci><list id="S2.SS1.p5.4.m4.2.2.1.1.2.cmml" xref="S2.SS1.p5.4.m4.2.2.1.1.1"><apply id="S2.SS1.p5.4.m4.2.2.1.1.1.1.cmml" xref="S2.SS1.p5.4.m4.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS1.p5.4.m4.2.2.1.1.1.1.1.cmml" xref="S2.SS1.p5.4.m4.2.2.1.1.1.1">superscript</csymbol><ci id="S2.SS1.p5.4.m4.2.2.1.1.1.1.2.cmml" xref="S2.SS1.p5.4.m4.2.2.1.1.1.1.2">ℝ</ci><ci id="S2.SS1.p5.4.m4.2.2.1.1.1.1.3.cmml" xref="S2.SS1.p5.4.m4.2.2.1.1.1.1.3">𝑑</ci></apply><ci id="S2.SS1.p5.4.m4.1.1.cmml" xref="S2.SS1.p5.4.m4.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.4.m4.2c">f\in\SS(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.4.m4.2d">italic_f ∈ roman_SS ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> we define the <math alttext="d" class="ltx_Math" display="inline" id="S2.SS1.p5.5.m5.1"><semantics id="S2.SS1.p5.5.m5.1a"><mi id="S2.SS1.p5.5.m5.1.1" xref="S2.SS1.p5.5.m5.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S2.SS1.p5.5.m5.1b"><ci id="S2.SS1.p5.5.m5.1.1.cmml" xref="S2.SS1.p5.5.m5.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS1.p5.5.m5.1c">d</annotation><annotation encoding="application/x-llamapun" id="S2.SS1.p5.5.m5.1d">italic_d</annotation></semantics></math>-dimensional Fourier transform and its inverse by</p> <table class="ltx_equation ltx_eqn_table" id="S2.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\mathcal{F}f)(\xi)=\widehat{f}(\xi)=\int_{\mathbb{R}^{d}}f(x)e^{-{\rm i}x% \cdot\xi}\hskip 2.0pt\mathrm{d}x\quad\text{ and }\quad(\mathcal{F}^{-1}f)(x)=% \frac{1}{(2\pi)^{d}}\int_{\mathbb{R}^{d}}f(\xi)e^{{\rm i}x\cdot\xi}\hskip 2.0% pt\mathrm{d}\xi," class="ltx_Math" display="block" id="S2.E1.m1.8"><semantics id="S2.E1.m1.8a"><mrow id="S2.E1.m1.8.8.1"><mrow id="S2.E1.m1.8.8.1.1.2" xref="S2.E1.m1.8.8.1.1.3.cmml"><mrow id="S2.E1.m1.8.8.1.1.1.1" xref="S2.E1.m1.8.8.1.1.1.1.cmml"><mrow id="S2.E1.m1.8.8.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.1.1.1.cmml"><mrow id="S2.E1.m1.8.8.1.1.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.8.8.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.2" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.2.cmml">ℱ</mi><mo id="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.3" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.3.cmml">f</mi></mrow><mo id="S2.E1.m1.8.8.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E1.m1.8.8.1.1.1.1.1.2" xref="S2.E1.m1.8.8.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.1.3.2" xref="S2.E1.m1.8.8.1.1.1.1.1.cmml"><mo id="S2.E1.m1.8.8.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.1.cmml">(</mo><mi id="S2.E1.m1.2.2" xref="S2.E1.m1.2.2.cmml">ξ</mi><mo id="S2.E1.m1.8.8.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.8.8.1.1.1.1.3" xref="S2.E1.m1.8.8.1.1.1.1.3.cmml">=</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.4" xref="S2.E1.m1.8.8.1.1.1.1.4.cmml"><mover accent="true" id="S2.E1.m1.8.8.1.1.1.1.4.2" xref="S2.E1.m1.8.8.1.1.1.1.4.2.cmml"><mi id="S2.E1.m1.8.8.1.1.1.1.4.2.2" xref="S2.E1.m1.8.8.1.1.1.1.4.2.2.cmml">f</mi><mo id="S2.E1.m1.8.8.1.1.1.1.4.2.1" xref="S2.E1.m1.8.8.1.1.1.1.4.2.1.cmml">^</mo></mover><mo id="S2.E1.m1.8.8.1.1.1.1.4.1" xref="S2.E1.m1.8.8.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.4.3.2" xref="S2.E1.m1.8.8.1.1.1.1.4.cmml"><mo id="S2.E1.m1.8.8.1.1.1.1.4.3.2.1" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.4.cmml">(</mo><mi id="S2.E1.m1.3.3" xref="S2.E1.m1.3.3.cmml">ξ</mi><mo id="S2.E1.m1.8.8.1.1.1.1.4.3.2.2" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.4.cmml">)</mo></mrow></mrow><mo id="S2.E1.m1.8.8.1.1.1.1.5" rspace="0.111em" xref="S2.E1.m1.8.8.1.1.1.1.5.cmml">=</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.6" xref="S2.E1.m1.8.8.1.1.1.1.6.cmml"><msub id="S2.E1.m1.8.8.1.1.1.1.6.1" xref="S2.E1.m1.8.8.1.1.1.1.6.1.cmml"><mo id="S2.E1.m1.8.8.1.1.1.1.6.1.2" xref="S2.E1.m1.8.8.1.1.1.1.6.1.2.cmml">∫</mo><msup id="S2.E1.m1.8.8.1.1.1.1.6.1.3" xref="S2.E1.m1.8.8.1.1.1.1.6.1.3.cmml"><mi id="S2.E1.m1.8.8.1.1.1.1.6.1.3.2" xref="S2.E1.m1.8.8.1.1.1.1.6.1.3.2.cmml">ℝ</mi><mi id="S2.E1.m1.8.8.1.1.1.1.6.1.3.3" xref="S2.E1.m1.8.8.1.1.1.1.6.1.3.3.cmml">d</mi></msup></msub><mrow id="S2.E1.m1.8.8.1.1.1.1.6.2" xref="S2.E1.m1.8.8.1.1.1.1.6.2.cmml"><mi id="S2.E1.m1.8.8.1.1.1.1.6.2.2" xref="S2.E1.m1.8.8.1.1.1.1.6.2.2.cmml">f</mi><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.1" xref="S2.E1.m1.8.8.1.1.1.1.6.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.6.2.3.2" xref="S2.E1.m1.8.8.1.1.1.1.6.2.cmml"><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.3.2.1" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.6.2.cmml">(</mo><mi id="S2.E1.m1.4.4" xref="S2.E1.m1.4.4.cmml">x</mi><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.3.2.2" stretchy="false" xref="S2.E1.m1.8.8.1.1.1.1.6.2.cmml">)</mo></mrow><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.1a" xref="S2.E1.m1.8.8.1.1.1.1.6.2.1.cmml">⁢</mo><msup id="S2.E1.m1.8.8.1.1.1.1.6.2.4" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.cmml"><mi id="S2.E1.m1.8.8.1.1.1.1.6.2.4.2" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.2.cmml">e</mi><mrow id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.cmml"><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3a" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.cmml">−</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.cmml"><mrow id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2.cmml"><mi id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2.2" mathvariant="normal" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2.2.cmml">i</mi><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2.1" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2.1.cmml">⁢</mo><mi id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2.3" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.2.3.cmml">x</mi></mrow><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.1" lspace="0.222em" rspace="0.222em" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.1.cmml">⋅</mo><mi id="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.3" xref="S2.E1.m1.8.8.1.1.1.1.6.2.4.3.2.3.cmml">ξ</mi></mrow></mrow></msup><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.1b" lspace="0em" xref="S2.E1.m1.8.8.1.1.1.1.6.2.1.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.1.1.6.2.5" xref="S2.E1.m1.8.8.1.1.1.1.6.2.5.cmml"><mo id="S2.E1.m1.8.8.1.1.1.1.6.2.5.1" rspace="0em" xref="S2.E1.m1.8.8.1.1.1.1.6.2.5.1.cmml">d</mo><mi id="S2.E1.m1.8.8.1.1.1.1.6.2.5.2" xref="S2.E1.m1.8.8.1.1.1.1.6.2.5.2.cmml">x</mi></mrow></mrow></mrow></mrow><mspace id="S2.E1.m1.8.8.1.1.2.3" width="1em" xref="S2.E1.m1.8.8.1.1.3a.cmml"></mspace><mrow id="S2.E1.m1.8.8.1.1.2.2" xref="S2.E1.m1.8.8.1.1.2.2.cmml"><mrow id="S2.E1.m1.8.8.1.1.2.2.1.1" xref="S2.E1.m1.8.8.1.1.2.2.1.2.cmml"><mtext id="S2.E1.m1.7.7" xref="S2.E1.m1.7.7a.cmml"> and </mtext><mspace id="S2.E1.m1.8.8.1.1.2.2.1.1.2" width="1em" xref="S2.E1.m1.8.8.1.1.2.2.1.2.cmml"></mspace><mrow id="S2.E1.m1.8.8.1.1.2.2.1.1.1" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.cmml"><mrow id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.cmml"><mo id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.cmml"><msup id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.2" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.2.cmml">ℱ</mi><mrow id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.3" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.3.cmml"><mo id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.3a" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.3.2" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.1" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.3" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.3.cmml">f</mi></mrow><mo id="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.3" stretchy="false" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.E1.m1.8.8.1.1.2.2.1.1.1.2" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.E1.m1.8.8.1.1.2.2.1.1.1.3.2" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.cmml"><mo id="S2.E1.m1.8.8.1.1.2.2.1.1.1.3.2.1" stretchy="false" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.cmml">(</mo><mi id="S2.E1.m1.5.5" xref="S2.E1.m1.5.5.cmml">x</mi><mo id="S2.E1.m1.8.8.1.1.2.2.1.1.1.3.2.2" stretchy="false" xref="S2.E1.m1.8.8.1.1.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><mo id="S2.E1.m1.8.8.1.1.2.2.2" xref="S2.E1.m1.8.8.1.1.2.2.2.cmml">=</mo><mrow id="S2.E1.m1.8.8.1.1.2.2.3" xref="S2.E1.m1.8.8.1.1.2.2.3.cmml"><mfrac 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encoding="application/x-llamapun" id="S2.E1.m1.8d">( caligraphic_F italic_f ) ( italic_ξ ) = over^ start_ARG italic_f end_ARG ( italic_ξ ) = ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f ( italic_x ) italic_e start_POSTSUPERSCRIPT - roman_i italic_x ⋅ italic_ξ end_POSTSUPERSCRIPT roman_d italic_x and ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_f ) ( italic_x ) = divide start_ARG 1 end_ARG start_ARG ( 2 italic_π ) start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_ARG ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_f ( italic_ξ ) italic_e start_POSTSUPERSCRIPT roman_i italic_x ⋅ italic_ξ end_POSTSUPERSCRIPT roman_d italic_ξ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS1.p5.6">which extends to <math alttext="\SS^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS1.p5.6.m1.2"><semantics id="S2.SS1.p5.6.m1.2a"><mrow id="S2.SS1.p5.6.m1.2.2" xref="S2.SS1.p5.6.m1.2.2.cmml"><msup id="S2.SS1.p5.6.m1.2.2.3" xref="S2.SS1.p5.6.m1.2.2.3.cmml"><mi id="S2.SS1.p5.6.m1.2.2.3.2" xref="S2.SS1.p5.6.m1.2.2.3.2.cmml">SS</mi><mo id="S2.SS1.p5.6.m1.2.2.3.3" xref="S2.SS1.p5.6.m1.2.2.3.3.cmml">′</mo></msup><mo id="S2.SS1.p5.6.m1.2.2.2" xref="S2.SS1.p5.6.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS1.p5.6.m1.2.2.1.1" xref="S2.SS1.p5.6.m1.2.2.1.2.cmml"><mo id="S2.SS1.p5.6.m1.2.2.1.1.2" stretchy="false" xref="S2.SS1.p5.6.m1.2.2.1.2.cmml">(</mo><msup id="S2.SS1.p5.6.m1.2.2.1.1.1" xref="S2.SS1.p5.6.m1.2.2.1.1.1.cmml"><mi id="S2.SS1.p5.6.m1.2.2.1.1.1.2" xref="S2.SS1.p5.6.m1.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS1.p5.6.m1.2.2.1.1.1.3" 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id="S2.SS2.p1.2.m2.2.3.3.cmml" xref="S2.SS2.p1.2.m2.2.3.3"><ci id="S2.SS2.p1.2.m2.2.3.3.1.cmml" xref="S2.SS2.p1.2.m2.2.3.3.1">→</ci><ci id="S2.SS2.p1.2.m2.2.3.3.2.cmml" xref="S2.SS2.p1.2.m2.2.3.3.2">𝒪</ci><interval closure="open" id="S2.SS2.p1.2.m2.2.3.3.3.1.cmml" xref="S2.SS2.p1.2.m2.2.3.3.3.2"><cn id="S2.SS2.p1.2.m2.1.1.cmml" type="integer" xref="S2.SS2.p1.2.m2.1.1">0</cn><infinity id="S2.SS2.p1.2.m2.2.2.cmml" xref="S2.SS2.p1.2.m2.2.2"></infinity></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.2.m2.2c">w:\mathcal{O}\to(0,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.2.m2.2d">italic_w : caligraphic_O → ( 0 , ∞ )</annotation></semantics></math> is called a <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.6.1">weight</em>. For <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S2.SS2.p1.3.m3.2"><semantics id="S2.SS2.p1.3.m3.2a"><mrow id="S2.SS2.p1.3.m3.2.3" xref="S2.SS2.p1.3.m3.2.3.cmml"><mi id="S2.SS2.p1.3.m3.2.3.2" xref="S2.SS2.p1.3.m3.2.3.2.cmml">p</mi><mo id="S2.SS2.p1.3.m3.2.3.1" xref="S2.SS2.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S2.SS2.p1.3.m3.2.3.3.2" xref="S2.SS2.p1.3.m3.2.3.3.1.cmml"><mo id="S2.SS2.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S2.SS2.p1.3.m3.2.3.3.1.cmml">(</mo><mn id="S2.SS2.p1.3.m3.1.1" xref="S2.SS2.p1.3.m3.1.1.cmml">1</mn><mo id="S2.SS2.p1.3.m3.2.3.3.2.2" xref="S2.SS2.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S2.SS2.p1.3.m3.2.2" mathvariant="normal" xref="S2.SS2.p1.3.m3.2.2.cmml">∞</mi><mo id="S2.SS2.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S2.SS2.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.3.m3.2b"><apply id="S2.SS2.p1.3.m3.2.3.cmml" xref="S2.SS2.p1.3.m3.2.3"><in id="S2.SS2.p1.3.m3.2.3.1.cmml" xref="S2.SS2.p1.3.m3.2.3.1"></in><ci id="S2.SS2.p1.3.m3.2.3.2.cmml" xref="S2.SS2.p1.3.m3.2.3.2">𝑝</ci><interval closure="open" id="S2.SS2.p1.3.m3.2.3.3.1.cmml" xref="S2.SS2.p1.3.m3.2.3.3.2"><cn id="S2.SS2.p1.3.m3.1.1.cmml" type="integer" xref="S2.SS2.p1.3.m3.1.1">1</cn><infinity id="S2.SS2.p1.3.m3.2.2.cmml" xref="S2.SS2.p1.3.m3.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.3.m3.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.3.m3.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math> we denote the class of <em class="ltx_emph ltx_font_italic" id="S2.SS2.p1.6.2">Muckenhoupt weights</em> on <math alttext="\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}" class="ltx_Math" display="inline" id="S2.SS2.p1.4.m4.2"><semantics id="S2.SS2.p1.4.m4.2a"><mrow id="S2.SS2.p1.4.m4.2.2" xref="S2.SS2.p1.4.m4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.4.m4.2.2.4" xref="S2.SS2.p1.4.m4.2.2.4.cmml">𝒪</mi><mo id="S2.SS2.p1.4.m4.2.2.3" xref="S2.SS2.p1.4.m4.2.2.3.cmml">∈</mo><mrow id="S2.SS2.p1.4.m4.2.2.2.2" xref="S2.SS2.p1.4.m4.2.2.2.3.cmml"><mo id="S2.SS2.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S2.SS2.p1.4.m4.2.2.2.3.cmml">{</mo><msup id="S2.SS2.p1.4.m4.1.1.1.1.1" xref="S2.SS2.p1.4.m4.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.4.m4.1.1.1.1.1.2" xref="S2.SS2.p1.4.m4.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p1.4.m4.1.1.1.1.1.3" xref="S2.SS2.p1.4.m4.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p1.4.m4.2.2.2.2.4" xref="S2.SS2.p1.4.m4.2.2.2.3.cmml">,</mo><msubsup id="S2.SS2.p1.4.m4.2.2.2.2.2" xref="S2.SS2.p1.4.m4.2.2.2.2.2.cmml"><mi id="S2.SS2.p1.4.m4.2.2.2.2.2.2.2" xref="S2.SS2.p1.4.m4.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S2.SS2.p1.4.m4.2.2.2.2.2.3" xref="S2.SS2.p1.4.m4.2.2.2.2.2.3.cmml">+</mo><mi id="S2.SS2.p1.4.m4.2.2.2.2.2.2.3" xref="S2.SS2.p1.4.m4.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S2.SS2.p1.4.m4.2.2.2.2.5" stretchy="false" xref="S2.SS2.p1.4.m4.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.4.m4.2b"><apply id="S2.SS2.p1.4.m4.2.2.cmml" xref="S2.SS2.p1.4.m4.2.2"><in id="S2.SS2.p1.4.m4.2.2.3.cmml" xref="S2.SS2.p1.4.m4.2.2.3"></in><ci id="S2.SS2.p1.4.m4.2.2.4.cmml" xref="S2.SS2.p1.4.m4.2.2.4">𝒪</ci><set id="S2.SS2.p1.4.m4.2.2.2.3.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2"><apply id="S2.SS2.p1.4.m4.1.1.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p1.4.m4.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p1.4.m4.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.4.m4.1.1.1.1.1.3">𝑑</ci></apply><apply id="S2.SS2.p1.4.m4.2.2.2.2.2.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.2.2.2.2.2.1.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2.2">subscript</csymbol><apply id="S2.SS2.p1.4.m4.2.2.2.2.2.2.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2.2">superscript</csymbol><ci id="S2.SS2.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2.2.2.2">ℝ</ci><ci id="S2.SS2.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S2.SS2.p1.4.m4.2.2.2.2.2.3.cmml" xref="S2.SS2.p1.4.m4.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.4.m4.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.4.m4.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math> by <math alttext="A_{p}(\mathcal{O})" class="ltx_Math" display="inline" id="S2.SS2.p1.5.m5.1"><semantics id="S2.SS2.p1.5.m5.1a"><mrow id="S2.SS2.p1.5.m5.1.2" xref="S2.SS2.p1.5.m5.1.2.cmml"><msub id="S2.SS2.p1.5.m5.1.2.2" xref="S2.SS2.p1.5.m5.1.2.2.cmml"><mi id="S2.SS2.p1.5.m5.1.2.2.2" xref="S2.SS2.p1.5.m5.1.2.2.2.cmml">A</mi><mi id="S2.SS2.p1.5.m5.1.2.2.3" xref="S2.SS2.p1.5.m5.1.2.2.3.cmml">p</mi></msub><mo id="S2.SS2.p1.5.m5.1.2.1" xref="S2.SS2.p1.5.m5.1.2.1.cmml">⁢</mo><mrow id="S2.SS2.p1.5.m5.1.2.3.2" xref="S2.SS2.p1.5.m5.1.2.cmml"><mo id="S2.SS2.p1.5.m5.1.2.3.2.1" stretchy="false" xref="S2.SS2.p1.5.m5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.5.m5.1.1" xref="S2.SS2.p1.5.m5.1.1.cmml">𝒪</mi><mo id="S2.SS2.p1.5.m5.1.2.3.2.2" stretchy="false" xref="S2.SS2.p1.5.m5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.5.m5.1b"><apply id="S2.SS2.p1.5.m5.1.2.cmml" xref="S2.SS2.p1.5.m5.1.2"><times id="S2.SS2.p1.5.m5.1.2.1.cmml" xref="S2.SS2.p1.5.m5.1.2.1"></times><apply id="S2.SS2.p1.5.m5.1.2.2.cmml" xref="S2.SS2.p1.5.m5.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.5.m5.1.2.2.1.cmml" xref="S2.SS2.p1.5.m5.1.2.2">subscript</csymbol><ci id="S2.SS2.p1.5.m5.1.2.2.2.cmml" xref="S2.SS2.p1.5.m5.1.2.2.2">𝐴</ci><ci id="S2.SS2.p1.5.m5.1.2.2.3.cmml" xref="S2.SS2.p1.5.m5.1.2.2.3">𝑝</ci></apply><ci id="S2.SS2.p1.5.m5.1.1.cmml" xref="S2.SS2.p1.5.m5.1.1">𝒪</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.5.m5.1c">A_{p}(\mathcal{O})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.5.m5.1d">italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( caligraphic_O )</annotation></semantics></math>, i.e., <math alttext="w\in A_{p}(\mathcal{O})" class="ltx_Math" display="inline" id="S2.SS2.p1.6.m6.1"><semantics id="S2.SS2.p1.6.m6.1a"><mrow id="S2.SS2.p1.6.m6.1.2" xref="S2.SS2.p1.6.m6.1.2.cmml"><mi id="S2.SS2.p1.6.m6.1.2.2" xref="S2.SS2.p1.6.m6.1.2.2.cmml">w</mi><mo id="S2.SS2.p1.6.m6.1.2.1" xref="S2.SS2.p1.6.m6.1.2.1.cmml">∈</mo><mrow id="S2.SS2.p1.6.m6.1.2.3" xref="S2.SS2.p1.6.m6.1.2.3.cmml"><msub id="S2.SS2.p1.6.m6.1.2.3.2" xref="S2.SS2.p1.6.m6.1.2.3.2.cmml"><mi id="S2.SS2.p1.6.m6.1.2.3.2.2" xref="S2.SS2.p1.6.m6.1.2.3.2.2.cmml">A</mi><mi id="S2.SS2.p1.6.m6.1.2.3.2.3" xref="S2.SS2.p1.6.m6.1.2.3.2.3.cmml">p</mi></msub><mo id="S2.SS2.p1.6.m6.1.2.3.1" xref="S2.SS2.p1.6.m6.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p1.6.m6.1.2.3.3.2" xref="S2.SS2.p1.6.m6.1.2.3.cmml"><mo id="S2.SS2.p1.6.m6.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p1.6.m6.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.6.m6.1.1" xref="S2.SS2.p1.6.m6.1.1.cmml">𝒪</mi><mo id="S2.SS2.p1.6.m6.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.p1.6.m6.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.6.m6.1b"><apply id="S2.SS2.p1.6.m6.1.2.cmml" xref="S2.SS2.p1.6.m6.1.2"><in id="S2.SS2.p1.6.m6.1.2.1.cmml" xref="S2.SS2.p1.6.m6.1.2.1"></in><ci id="S2.SS2.p1.6.m6.1.2.2.cmml" xref="S2.SS2.p1.6.m6.1.2.2">𝑤</ci><apply id="S2.SS2.p1.6.m6.1.2.3.cmml" xref="S2.SS2.p1.6.m6.1.2.3"><times id="S2.SS2.p1.6.m6.1.2.3.1.cmml" xref="S2.SS2.p1.6.m6.1.2.3.1"></times><apply id="S2.SS2.p1.6.m6.1.2.3.2.cmml" xref="S2.SS2.p1.6.m6.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.6.m6.1.2.3.2.1.cmml" xref="S2.SS2.p1.6.m6.1.2.3.2">subscript</csymbol><ci id="S2.SS2.p1.6.m6.1.2.3.2.2.cmml" xref="S2.SS2.p1.6.m6.1.2.3.2.2">𝐴</ci><ci id="S2.SS2.p1.6.m6.1.2.3.2.3.cmml" xref="S2.SS2.p1.6.m6.1.2.3.2.3">𝑝</ci></apply><ci id="S2.SS2.p1.6.m6.1.1.cmml" xref="S2.SS2.p1.6.m6.1.1">𝒪</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.6.m6.1c">w\in A_{p}(\mathcal{O})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.6.m6.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( caligraphic_O )</annotation></semantics></math> if</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="[w]_{A_{p}(\mathcal{O})}:=\sup_{B}\Big{(}\frac{1}{|B|}\int_{B}w(x)\hskip 2.0pt% \mathrm{d}x\Big{)}\Big{(}\frac{1}{|B|}\int_{B}w(x)^{-\frac{1}{p-1}}\hskip 2.0% pt\mathrm{d}x\Big{)}^{p-1}<\infty," class="ltx_Math" display="block" id="S2.Ex1.m1.7"><semantics id="S2.Ex1.m1.7a"><mrow id="S2.Ex1.m1.7.7.1" xref="S2.Ex1.m1.7.7.1.1.cmml"><mrow id="S2.Ex1.m1.7.7.1.1" xref="S2.Ex1.m1.7.7.1.1.cmml"><msub id="S2.Ex1.m1.7.7.1.1.4" xref="S2.Ex1.m1.7.7.1.1.4.cmml"><mrow id="S2.Ex1.m1.7.7.1.1.4.2.2" xref="S2.Ex1.m1.7.7.1.1.4.2.1.cmml"><mo id="S2.Ex1.m1.7.7.1.1.4.2.2.1" stretchy="false" xref="S2.Ex1.m1.7.7.1.1.4.2.1.1.cmml">[</mo><mi id="S2.Ex1.m1.4.4" xref="S2.Ex1.m1.4.4.cmml">w</mi><mo id="S2.Ex1.m1.7.7.1.1.4.2.2.2" stretchy="false" xref="S2.Ex1.m1.7.7.1.1.4.2.1.1.cmml">]</mo></mrow><mrow id="S2.Ex1.m1.1.1.1" xref="S2.Ex1.m1.1.1.1.cmml"><msub id="S2.Ex1.m1.1.1.1.3" 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end_POSTSUBSCRIPT ( caligraphic_O ) end_POSTSUBSCRIPT := roman_sup start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( divide start_ARG 1 end_ARG start_ARG | italic_B | end_ARG ∫ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_w ( italic_x ) roman_d italic_x ) ( divide start_ARG 1 end_ARG start_ARG | italic_B | end_ARG ∫ start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_w ( italic_x ) start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG italic_p - 1 end_ARG end_POSTSUPERSCRIPT roman_d italic_x ) start_POSTSUPERSCRIPT italic_p - 1 end_POSTSUPERSCRIPT < ∞ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p1.15">where the supremum is taken over all balls <math alttext="B\subseteq\mathcal{O}" class="ltx_Math" display="inline" id="S2.SS2.p1.7.m1.1"><semantics id="S2.SS2.p1.7.m1.1a"><mrow id="S2.SS2.p1.7.m1.1.1" xref="S2.SS2.p1.7.m1.1.1.cmml"><mi id="S2.SS2.p1.7.m1.1.1.2" xref="S2.SS2.p1.7.m1.1.1.2.cmml">B</mi><mo id="S2.SS2.p1.7.m1.1.1.1" xref="S2.SS2.p1.7.m1.1.1.1.cmml">⊆</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.7.m1.1.1.3" xref="S2.SS2.p1.7.m1.1.1.3.cmml">𝒪</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.7.m1.1b"><apply id="S2.SS2.p1.7.m1.1.1.cmml" xref="S2.SS2.p1.7.m1.1.1"><subset id="S2.SS2.p1.7.m1.1.1.1.cmml" xref="S2.SS2.p1.7.m1.1.1.1"></subset><ci id="S2.SS2.p1.7.m1.1.1.2.cmml" xref="S2.SS2.p1.7.m1.1.1.2">𝐵</ci><ci id="S2.SS2.p1.7.m1.1.1.3.cmml" xref="S2.SS2.p1.7.m1.1.1.3">𝒪</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.7.m1.1c">B\subseteq\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.7.m1.1d">italic_B ⊆ caligraphic_O</annotation></semantics></math>. 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We will mainly focus on the case of power weights <math alttext="w_{\gamma}(x_{1},\widetilde{x})=|x_{1}|^{\gamma}" class="ltx_Math" display="inline" id="S2.SS2.p1.9.m3.3"><semantics id="S2.SS2.p1.9.m3.3a"><mrow id="S2.SS2.p1.9.m3.3.3" xref="S2.SS2.p1.9.m3.3.3.cmml"><mrow id="S2.SS2.p1.9.m3.2.2.1" xref="S2.SS2.p1.9.m3.2.2.1.cmml"><msub id="S2.SS2.p1.9.m3.2.2.1.3" xref="S2.SS2.p1.9.m3.2.2.1.3.cmml"><mi id="S2.SS2.p1.9.m3.2.2.1.3.2" xref="S2.SS2.p1.9.m3.2.2.1.3.2.cmml">w</mi><mi id="S2.SS2.p1.9.m3.2.2.1.3.3" xref="S2.SS2.p1.9.m3.2.2.1.3.3.cmml">γ</mi></msub><mo id="S2.SS2.p1.9.m3.2.2.1.2" xref="S2.SS2.p1.9.m3.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS2.p1.9.m3.2.2.1.1.1" xref="S2.SS2.p1.9.m3.2.2.1.1.2.cmml"><mo id="S2.SS2.p1.9.m3.2.2.1.1.1.2" stretchy="false" xref="S2.SS2.p1.9.m3.2.2.1.1.2.cmml">(</mo><msub id="S2.SS2.p1.9.m3.2.2.1.1.1.1" xref="S2.SS2.p1.9.m3.2.2.1.1.1.1.cmml"><mi id="S2.SS2.p1.9.m3.2.2.1.1.1.1.2" xref="S2.SS2.p1.9.m3.2.2.1.1.1.1.2.cmml">x</mi><mn id="S2.SS2.p1.9.m3.2.2.1.1.1.1.3" xref="S2.SS2.p1.9.m3.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.p1.9.m3.2.2.1.1.1.3" xref="S2.SS2.p1.9.m3.2.2.1.1.2.cmml">,</mo><mover accent="true" id="S2.SS2.p1.9.m3.1.1" xref="S2.SS2.p1.9.m3.1.1.cmml"><mi id="S2.SS2.p1.9.m3.1.1.2" xref="S2.SS2.p1.9.m3.1.1.2.cmml">x</mi><mo id="S2.SS2.p1.9.m3.1.1.1" xref="S2.SS2.p1.9.m3.1.1.1.cmml">~</mo></mover><mo id="S2.SS2.p1.9.m3.2.2.1.1.1.4" stretchy="false" xref="S2.SS2.p1.9.m3.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p1.9.m3.3.3.3" xref="S2.SS2.p1.9.m3.3.3.3.cmml">=</mo><msup id="S2.SS2.p1.9.m3.3.3.2" xref="S2.SS2.p1.9.m3.3.3.2.cmml"><mrow id="S2.SS2.p1.9.m3.3.3.2.1.1" xref="S2.SS2.p1.9.m3.3.3.2.1.2.cmml"><mo id="S2.SS2.p1.9.m3.3.3.2.1.1.2" stretchy="false" xref="S2.SS2.p1.9.m3.3.3.2.1.2.1.cmml">|</mo><msub id="S2.SS2.p1.9.m3.3.3.2.1.1.1" xref="S2.SS2.p1.9.m3.3.3.2.1.1.1.cmml"><mi id="S2.SS2.p1.9.m3.3.3.2.1.1.1.2" xref="S2.SS2.p1.9.m3.3.3.2.1.1.1.2.cmml">x</mi><mn 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id="S2.SS2.p1.9.m3.2.2.1.1.2.cmml" xref="S2.SS2.p1.9.m3.2.2.1.1.1"><apply id="S2.SS2.p1.9.m3.2.2.1.1.1.1.cmml" xref="S2.SS2.p1.9.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m3.2.2.1.1.1.1.1.cmml" xref="S2.SS2.p1.9.m3.2.2.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.9.m3.2.2.1.1.1.1.2.cmml" xref="S2.SS2.p1.9.m3.2.2.1.1.1.1.2">𝑥</ci><cn id="S2.SS2.p1.9.m3.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.SS2.p1.9.m3.2.2.1.1.1.1.3">1</cn></apply><apply id="S2.SS2.p1.9.m3.1.1.cmml" xref="S2.SS2.p1.9.m3.1.1"><ci id="S2.SS2.p1.9.m3.1.1.1.cmml" xref="S2.SS2.p1.9.m3.1.1.1">~</ci><ci id="S2.SS2.p1.9.m3.1.1.2.cmml" xref="S2.SS2.p1.9.m3.1.1.2">𝑥</ci></apply></interval></apply><apply id="S2.SS2.p1.9.m3.3.3.2.cmml" xref="S2.SS2.p1.9.m3.3.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m3.3.3.2.2.cmml" xref="S2.SS2.p1.9.m3.3.3.2">superscript</csymbol><apply id="S2.SS2.p1.9.m3.3.3.2.1.2.cmml" xref="S2.SS2.p1.9.m3.3.3.2.1.1"><abs id="S2.SS2.p1.9.m3.3.3.2.1.2.1.cmml" xref="S2.SS2.p1.9.m3.3.3.2.1.1.2"></abs><apply id="S2.SS2.p1.9.m3.3.3.2.1.1.1.cmml" xref="S2.SS2.p1.9.m3.3.3.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.9.m3.3.3.2.1.1.1.1.cmml" xref="S2.SS2.p1.9.m3.3.3.2.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.9.m3.3.3.2.1.1.1.2.cmml" xref="S2.SS2.p1.9.m3.3.3.2.1.1.1.2">𝑥</ci><cn id="S2.SS2.p1.9.m3.3.3.2.1.1.1.3.cmml" type="integer" xref="S2.SS2.p1.9.m3.3.3.2.1.1.1.3">1</cn></apply></apply><ci id="S2.SS2.p1.9.m3.3.3.2.3.cmml" xref="S2.SS2.p1.9.m3.3.3.2.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.9.m3.3c">w_{\gamma}(x_{1},\widetilde{x})=|x_{1}|^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.9.m3.3d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) = | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S2.SS2.p1.10.m4.1"><semantics id="S2.SS2.p1.10.m4.1a"><mrow id="S2.SS2.p1.10.m4.1.1" xref="S2.SS2.p1.10.m4.1.1.cmml"><mi id="S2.SS2.p1.10.m4.1.1.2" xref="S2.SS2.p1.10.m4.1.1.2.cmml">γ</mi><mo id="S2.SS2.p1.10.m4.1.1.1" xref="S2.SS2.p1.10.m4.1.1.1.cmml">></mo><mrow id="S2.SS2.p1.10.m4.1.1.3" xref="S2.SS2.p1.10.m4.1.1.3.cmml"><mo id="S2.SS2.p1.10.m4.1.1.3a" xref="S2.SS2.p1.10.m4.1.1.3.cmml">−</mo><mn id="S2.SS2.p1.10.m4.1.1.3.2" xref="S2.SS2.p1.10.m4.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.10.m4.1b"><apply id="S2.SS2.p1.10.m4.1.1.cmml" xref="S2.SS2.p1.10.m4.1.1"><gt id="S2.SS2.p1.10.m4.1.1.1.cmml" xref="S2.SS2.p1.10.m4.1.1.1"></gt><ci id="S2.SS2.p1.10.m4.1.1.2.cmml" xref="S2.SS2.p1.10.m4.1.1.2">𝛾</ci><apply id="S2.SS2.p1.10.m4.1.1.3.cmml" xref="S2.SS2.p1.10.m4.1.1.3"><minus id="S2.SS2.p1.10.m4.1.1.3.1.cmml" xref="S2.SS2.p1.10.m4.1.1.3"></minus><cn id="S2.SS2.p1.10.m4.1.1.3.2.cmml" type="integer" xref="S2.SS2.p1.10.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.10.m4.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.10.m4.1d">italic_γ > - 1</annotation></semantics></math>, where <math alttext="x=(x_{1},\widetilde{x})\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S2.SS2.p1.11.m5.2"><semantics id="S2.SS2.p1.11.m5.2a"><mrow id="S2.SS2.p1.11.m5.2.2" xref="S2.SS2.p1.11.m5.2.2.cmml"><mi id="S2.SS2.p1.11.m5.2.2.3" xref="S2.SS2.p1.11.m5.2.2.3.cmml">x</mi><mo id="S2.SS2.p1.11.m5.2.2.4" xref="S2.SS2.p1.11.m5.2.2.4.cmml">=</mo><mrow id="S2.SS2.p1.11.m5.2.2.1.1" xref="S2.SS2.p1.11.m5.2.2.1.2.cmml"><mo id="S2.SS2.p1.11.m5.2.2.1.1.2" stretchy="false" xref="S2.SS2.p1.11.m5.2.2.1.2.cmml">(</mo><msub id="S2.SS2.p1.11.m5.2.2.1.1.1" xref="S2.SS2.p1.11.m5.2.2.1.1.1.cmml"><mi id="S2.SS2.p1.11.m5.2.2.1.1.1.2" xref="S2.SS2.p1.11.m5.2.2.1.1.1.2.cmml">x</mi><mn id="S2.SS2.p1.11.m5.2.2.1.1.1.3" xref="S2.SS2.p1.11.m5.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS2.p1.11.m5.2.2.1.1.3" xref="S2.SS2.p1.11.m5.2.2.1.2.cmml">,</mo><mover accent="true" id="S2.SS2.p1.11.m5.1.1" xref="S2.SS2.p1.11.m5.1.1.cmml"><mi id="S2.SS2.p1.11.m5.1.1.2" xref="S2.SS2.p1.11.m5.1.1.2.cmml">x</mi><mo id="S2.SS2.p1.11.m5.1.1.1" xref="S2.SS2.p1.11.m5.1.1.1.cmml">~</mo></mover><mo id="S2.SS2.p1.11.m5.2.2.1.1.4" stretchy="false" xref="S2.SS2.p1.11.m5.2.2.1.2.cmml">)</mo></mrow><mo id="S2.SS2.p1.11.m5.2.2.5" xref="S2.SS2.p1.11.m5.2.2.5.cmml">∈</mo><msup id="S2.SS2.p1.11.m5.2.2.6" xref="S2.SS2.p1.11.m5.2.2.6.cmml"><mi id="S2.SS2.p1.11.m5.2.2.6.2" xref="S2.SS2.p1.11.m5.2.2.6.2.cmml">ℝ</mi><mi id="S2.SS2.p1.11.m5.2.2.6.3" xref="S2.SS2.p1.11.m5.2.2.6.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.11.m5.2b"><apply id="S2.SS2.p1.11.m5.2.2.cmml" xref="S2.SS2.p1.11.m5.2.2"><and id="S2.SS2.p1.11.m5.2.2a.cmml" xref="S2.SS2.p1.11.m5.2.2"></and><apply id="S2.SS2.p1.11.m5.2.2b.cmml" xref="S2.SS2.p1.11.m5.2.2"><eq id="S2.SS2.p1.11.m5.2.2.4.cmml" xref="S2.SS2.p1.11.m5.2.2.4"></eq><ci id="S2.SS2.p1.11.m5.2.2.3.cmml" xref="S2.SS2.p1.11.m5.2.2.3">𝑥</ci><interval closure="open" id="S2.SS2.p1.11.m5.2.2.1.2.cmml" xref="S2.SS2.p1.11.m5.2.2.1.1"><apply id="S2.SS2.p1.11.m5.2.2.1.1.1.cmml" xref="S2.SS2.p1.11.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.11.m5.2.2.1.1.1.1.cmml" xref="S2.SS2.p1.11.m5.2.2.1.1.1">subscript</csymbol><ci id="S2.SS2.p1.11.m5.2.2.1.1.1.2.cmml" xref="S2.SS2.p1.11.m5.2.2.1.1.1.2">𝑥</ci><cn id="S2.SS2.p1.11.m5.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS2.p1.11.m5.2.2.1.1.1.3">1</cn></apply><apply id="S2.SS2.p1.11.m5.1.1.cmml" xref="S2.SS2.p1.11.m5.1.1"><ci id="S2.SS2.p1.11.m5.1.1.1.cmml" xref="S2.SS2.p1.11.m5.1.1.1">~</ci><ci id="S2.SS2.p1.11.m5.1.1.2.cmml" xref="S2.SS2.p1.11.m5.1.1.2">𝑥</ci></apply></interval></apply><apply id="S2.SS2.p1.11.m5.2.2c.cmml" xref="S2.SS2.p1.11.m5.2.2"><in id="S2.SS2.p1.11.m5.2.2.5.cmml" xref="S2.SS2.p1.11.m5.2.2.5"></in><share href="https://arxiv.org/html/2503.14636v1#S2.SS2.p1.11.m5.2.2.1.cmml" id="S2.SS2.p1.11.m5.2.2d.cmml" xref="S2.SS2.p1.11.m5.2.2"></share><apply id="S2.SS2.p1.11.m5.2.2.6.cmml" xref="S2.SS2.p1.11.m5.2.2.6"><csymbol cd="ambiguous" id="S2.SS2.p1.11.m5.2.2.6.1.cmml" xref="S2.SS2.p1.11.m5.2.2.6">superscript</csymbol><ci id="S2.SS2.p1.11.m5.2.2.6.2.cmml" xref="S2.SS2.p1.11.m5.2.2.6.2">ℝ</ci><ci id="S2.SS2.p1.11.m5.2.2.6.3.cmml" xref="S2.SS2.p1.11.m5.2.2.6.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.11.m5.2c">x=(x_{1},\widetilde{x})\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.11.m5.2d">italic_x = ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="\widetilde{x}\in\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S2.SS2.p1.12.m6.1"><semantics id="S2.SS2.p1.12.m6.1a"><mrow id="S2.SS2.p1.12.m6.1.1" xref="S2.SS2.p1.12.m6.1.1.cmml"><mover accent="true" id="S2.SS2.p1.12.m6.1.1.2" xref="S2.SS2.p1.12.m6.1.1.2.cmml"><mi id="S2.SS2.p1.12.m6.1.1.2.2" xref="S2.SS2.p1.12.m6.1.1.2.2.cmml">x</mi><mo id="S2.SS2.p1.12.m6.1.1.2.1" xref="S2.SS2.p1.12.m6.1.1.2.1.cmml">~</mo></mover><mo id="S2.SS2.p1.12.m6.1.1.1" xref="S2.SS2.p1.12.m6.1.1.1.cmml">∈</mo><msup id="S2.SS2.p1.12.m6.1.1.3" xref="S2.SS2.p1.12.m6.1.1.3.cmml"><mi id="S2.SS2.p1.12.m6.1.1.3.2" xref="S2.SS2.p1.12.m6.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS2.p1.12.m6.1.1.3.3" xref="S2.SS2.p1.12.m6.1.1.3.3.cmml"><mi id="S2.SS2.p1.12.m6.1.1.3.3.2" xref="S2.SS2.p1.12.m6.1.1.3.3.2.cmml">d</mi><mo id="S2.SS2.p1.12.m6.1.1.3.3.1" xref="S2.SS2.p1.12.m6.1.1.3.3.1.cmml">−</mo><mn id="S2.SS2.p1.12.m6.1.1.3.3.3" xref="S2.SS2.p1.12.m6.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.12.m6.1b"><apply id="S2.SS2.p1.12.m6.1.1.cmml" xref="S2.SS2.p1.12.m6.1.1"><in id="S2.SS2.p1.12.m6.1.1.1.cmml" xref="S2.SS2.p1.12.m6.1.1.1"></in><apply id="S2.SS2.p1.12.m6.1.1.2.cmml" xref="S2.SS2.p1.12.m6.1.1.2"><ci id="S2.SS2.p1.12.m6.1.1.2.1.cmml" xref="S2.SS2.p1.12.m6.1.1.2.1">~</ci><ci id="S2.SS2.p1.12.m6.1.1.2.2.cmml" xref="S2.SS2.p1.12.m6.1.1.2.2">𝑥</ci></apply><apply id="S2.SS2.p1.12.m6.1.1.3.cmml" xref="S2.SS2.p1.12.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p1.12.m6.1.1.3.1.cmml" xref="S2.SS2.p1.12.m6.1.1.3">superscript</csymbol><ci id="S2.SS2.p1.12.m6.1.1.3.2.cmml" xref="S2.SS2.p1.12.m6.1.1.3.2">ℝ</ci><apply id="S2.SS2.p1.12.m6.1.1.3.3.cmml" xref="S2.SS2.p1.12.m6.1.1.3.3"><minus id="S2.SS2.p1.12.m6.1.1.3.3.1.cmml" xref="S2.SS2.p1.12.m6.1.1.3.3.1"></minus><ci id="S2.SS2.p1.12.m6.1.1.3.3.2.cmml" xref="S2.SS2.p1.12.m6.1.1.3.3.2">𝑑</ci><cn id="S2.SS2.p1.12.m6.1.1.3.3.3.cmml" type="integer" xref="S2.SS2.p1.12.m6.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.12.m6.1c">\widetilde{x}\in\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.12.m6.1d">over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. For <math alttext="\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}" class="ltx_Math" display="inline" id="S2.SS2.p1.13.m7.2"><semantics id="S2.SS2.p1.13.m7.2a"><mrow id="S2.SS2.p1.13.m7.2.2" xref="S2.SS2.p1.13.m7.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.13.m7.2.2.4" xref="S2.SS2.p1.13.m7.2.2.4.cmml">𝒪</mi><mo id="S2.SS2.p1.13.m7.2.2.3" xref="S2.SS2.p1.13.m7.2.2.3.cmml">∈</mo><mrow id="S2.SS2.p1.13.m7.2.2.2.2" xref="S2.SS2.p1.13.m7.2.2.2.3.cmml"><mo id="S2.SS2.p1.13.m7.2.2.2.2.3" stretchy="false" xref="S2.SS2.p1.13.m7.2.2.2.3.cmml">{</mo><msup id="S2.SS2.p1.13.m7.1.1.1.1.1" xref="S2.SS2.p1.13.m7.1.1.1.1.1.cmml"><mi id="S2.SS2.p1.13.m7.1.1.1.1.1.2" xref="S2.SS2.p1.13.m7.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p1.13.m7.1.1.1.1.1.3" xref="S2.SS2.p1.13.m7.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p1.13.m7.2.2.2.2.4" xref="S2.SS2.p1.13.m7.2.2.2.3.cmml">,</mo><msubsup id="S2.SS2.p1.13.m7.2.2.2.2.2" xref="S2.SS2.p1.13.m7.2.2.2.2.2.cmml"><mi id="S2.SS2.p1.13.m7.2.2.2.2.2.2.2" xref="S2.SS2.p1.13.m7.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S2.SS2.p1.13.m7.2.2.2.2.2.3" xref="S2.SS2.p1.13.m7.2.2.2.2.2.3.cmml">+</mo><mi id="S2.SS2.p1.13.m7.2.2.2.2.2.2.3" xref="S2.SS2.p1.13.m7.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S2.SS2.p1.13.m7.2.2.2.2.5" stretchy="false" xref="S2.SS2.p1.13.m7.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.13.m7.2b"><apply id="S2.SS2.p1.13.m7.2.2.cmml" xref="S2.SS2.p1.13.m7.2.2"><in id="S2.SS2.p1.13.m7.2.2.3.cmml" xref="S2.SS2.p1.13.m7.2.2.3"></in><ci id="S2.SS2.p1.13.m7.2.2.4.cmml" xref="S2.SS2.p1.13.m7.2.2.4">𝒪</ci><set id="S2.SS2.p1.13.m7.2.2.2.3.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2"><apply id="S2.SS2.p1.13.m7.1.1.1.1.1.cmml" xref="S2.SS2.p1.13.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p1.13.m7.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.13.m7.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p1.13.m7.1.1.1.1.1.2.cmml" xref="S2.SS2.p1.13.m7.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p1.13.m7.1.1.1.1.1.3.cmml" xref="S2.SS2.p1.13.m7.1.1.1.1.1.3">𝑑</ci></apply><apply id="S2.SS2.p1.13.m7.2.2.2.2.2.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.13.m7.2.2.2.2.2.1.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2.2">subscript</csymbol><apply id="S2.SS2.p1.13.m7.2.2.2.2.2.2.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.13.m7.2.2.2.2.2.2.1.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2.2">superscript</csymbol><ci id="S2.SS2.p1.13.m7.2.2.2.2.2.2.2.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2.2.2.2">ℝ</ci><ci id="S2.SS2.p1.13.m7.2.2.2.2.2.2.3.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S2.SS2.p1.13.m7.2.2.2.2.2.3.cmml" xref="S2.SS2.p1.13.m7.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.13.m7.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.13.m7.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math> it holds that <math alttext="w_{\gamma}\in A_{p}(\mathcal{O})" class="ltx_Math" display="inline" id="S2.SS2.p1.14.m8.1"><semantics id="S2.SS2.p1.14.m8.1a"><mrow id="S2.SS2.p1.14.m8.1.2" xref="S2.SS2.p1.14.m8.1.2.cmml"><msub id="S2.SS2.p1.14.m8.1.2.2" xref="S2.SS2.p1.14.m8.1.2.2.cmml"><mi id="S2.SS2.p1.14.m8.1.2.2.2" xref="S2.SS2.p1.14.m8.1.2.2.2.cmml">w</mi><mi id="S2.SS2.p1.14.m8.1.2.2.3" xref="S2.SS2.p1.14.m8.1.2.2.3.cmml">γ</mi></msub><mo id="S2.SS2.p1.14.m8.1.2.1" xref="S2.SS2.p1.14.m8.1.2.1.cmml">∈</mo><mrow id="S2.SS2.p1.14.m8.1.2.3" xref="S2.SS2.p1.14.m8.1.2.3.cmml"><msub id="S2.SS2.p1.14.m8.1.2.3.2" xref="S2.SS2.p1.14.m8.1.2.3.2.cmml"><mi id="S2.SS2.p1.14.m8.1.2.3.2.2" xref="S2.SS2.p1.14.m8.1.2.3.2.2.cmml">A</mi><mi id="S2.SS2.p1.14.m8.1.2.3.2.3" xref="S2.SS2.p1.14.m8.1.2.3.2.3.cmml">p</mi></msub><mo id="S2.SS2.p1.14.m8.1.2.3.1" xref="S2.SS2.p1.14.m8.1.2.3.1.cmml">⁢</mo><mrow id="S2.SS2.p1.14.m8.1.2.3.3.2" xref="S2.SS2.p1.14.m8.1.2.3.cmml"><mo id="S2.SS2.p1.14.m8.1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p1.14.m8.1.2.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p1.14.m8.1.1" xref="S2.SS2.p1.14.m8.1.1.cmml">𝒪</mi><mo id="S2.SS2.p1.14.m8.1.2.3.3.2.2" stretchy="false" xref="S2.SS2.p1.14.m8.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.14.m8.1b"><apply id="S2.SS2.p1.14.m8.1.2.cmml" xref="S2.SS2.p1.14.m8.1.2"><in id="S2.SS2.p1.14.m8.1.2.1.cmml" xref="S2.SS2.p1.14.m8.1.2.1"></in><apply id="S2.SS2.p1.14.m8.1.2.2.cmml" xref="S2.SS2.p1.14.m8.1.2.2"><csymbol cd="ambiguous" id="S2.SS2.p1.14.m8.1.2.2.1.cmml" xref="S2.SS2.p1.14.m8.1.2.2">subscript</csymbol><ci id="S2.SS2.p1.14.m8.1.2.2.2.cmml" xref="S2.SS2.p1.14.m8.1.2.2.2">𝑤</ci><ci id="S2.SS2.p1.14.m8.1.2.2.3.cmml" xref="S2.SS2.p1.14.m8.1.2.2.3">𝛾</ci></apply><apply id="S2.SS2.p1.14.m8.1.2.3.cmml" xref="S2.SS2.p1.14.m8.1.2.3"><times id="S2.SS2.p1.14.m8.1.2.3.1.cmml" xref="S2.SS2.p1.14.m8.1.2.3.1"></times><apply id="S2.SS2.p1.14.m8.1.2.3.2.cmml" xref="S2.SS2.p1.14.m8.1.2.3.2"><csymbol cd="ambiguous" id="S2.SS2.p1.14.m8.1.2.3.2.1.cmml" xref="S2.SS2.p1.14.m8.1.2.3.2">subscript</csymbol><ci id="S2.SS2.p1.14.m8.1.2.3.2.2.cmml" xref="S2.SS2.p1.14.m8.1.2.3.2.2">𝐴</ci><ci id="S2.SS2.p1.14.m8.1.2.3.2.3.cmml" xref="S2.SS2.p1.14.m8.1.2.3.2.3">𝑝</ci></apply><ci id="S2.SS2.p1.14.m8.1.1.cmml" xref="S2.SS2.p1.14.m8.1.1">𝒪</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.14.m8.1c">w_{\gamma}\in A_{p}(\mathcal{O})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.14.m8.1d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( caligraphic_O )</annotation></semantics></math> if and only if <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S2.SS2.p1.15.m9.2"><semantics id="S2.SS2.p1.15.m9.2a"><mrow id="S2.SS2.p1.15.m9.2.2" xref="S2.SS2.p1.15.m9.2.2.cmml"><mi id="S2.SS2.p1.15.m9.2.2.4" xref="S2.SS2.p1.15.m9.2.2.4.cmml">γ</mi><mo id="S2.SS2.p1.15.m9.2.2.3" xref="S2.SS2.p1.15.m9.2.2.3.cmml">∈</mo><mrow id="S2.SS2.p1.15.m9.2.2.2.2" xref="S2.SS2.p1.15.m9.2.2.2.3.cmml"><mo id="S2.SS2.p1.15.m9.2.2.2.2.3" stretchy="false" xref="S2.SS2.p1.15.m9.2.2.2.3.cmml">(</mo><mrow id="S2.SS2.p1.15.m9.1.1.1.1.1" xref="S2.SS2.p1.15.m9.1.1.1.1.1.cmml"><mo id="S2.SS2.p1.15.m9.1.1.1.1.1a" xref="S2.SS2.p1.15.m9.1.1.1.1.1.cmml">−</mo><mn id="S2.SS2.p1.15.m9.1.1.1.1.1.2" xref="S2.SS2.p1.15.m9.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S2.SS2.p1.15.m9.2.2.2.2.4" xref="S2.SS2.p1.15.m9.2.2.2.3.cmml">,</mo><mrow id="S2.SS2.p1.15.m9.2.2.2.2.2" xref="S2.SS2.p1.15.m9.2.2.2.2.2.cmml"><mi id="S2.SS2.p1.15.m9.2.2.2.2.2.2" xref="S2.SS2.p1.15.m9.2.2.2.2.2.2.cmml">p</mi><mo id="S2.SS2.p1.15.m9.2.2.2.2.2.1" xref="S2.SS2.p1.15.m9.2.2.2.2.2.1.cmml">−</mo><mn id="S2.SS2.p1.15.m9.2.2.2.2.2.3" xref="S2.SS2.p1.15.m9.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S2.SS2.p1.15.m9.2.2.2.2.5" stretchy="false" xref="S2.SS2.p1.15.m9.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p1.15.m9.2b"><apply id="S2.SS2.p1.15.m9.2.2.cmml" xref="S2.SS2.p1.15.m9.2.2"><in id="S2.SS2.p1.15.m9.2.2.3.cmml" xref="S2.SS2.p1.15.m9.2.2.3"></in><ci id="S2.SS2.p1.15.m9.2.2.4.cmml" xref="S2.SS2.p1.15.m9.2.2.4">𝛾</ci><interval closure="open" id="S2.SS2.p1.15.m9.2.2.2.3.cmml" xref="S2.SS2.p1.15.m9.2.2.2.2"><apply id="S2.SS2.p1.15.m9.1.1.1.1.1.cmml" xref="S2.SS2.p1.15.m9.1.1.1.1.1"><minus id="S2.SS2.p1.15.m9.1.1.1.1.1.1.cmml" xref="S2.SS2.p1.15.m9.1.1.1.1.1"></minus><cn id="S2.SS2.p1.15.m9.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS2.p1.15.m9.1.1.1.1.1.2">1</cn></apply><apply id="S2.SS2.p1.15.m9.2.2.2.2.2.cmml" xref="S2.SS2.p1.15.m9.2.2.2.2.2"><minus id="S2.SS2.p1.15.m9.2.2.2.2.2.1.cmml" xref="S2.SS2.p1.15.m9.2.2.2.2.2.1"></minus><ci id="S2.SS2.p1.15.m9.2.2.2.2.2.2.cmml" xref="S2.SS2.p1.15.m9.2.2.2.2.2.2">𝑝</ci><cn id="S2.SS2.p1.15.m9.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS2.p1.15.m9.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p1.15.m9.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p1.15.m9.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib21" title="">21</a>, Example 7.1.7]</cite>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S2.SS2.p2"> <p class="ltx_p" id="S2.SS2.p2.6">For <math alttext="\mathcal{O}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S2.SS2.p2.1.m1.1"><semantics id="S2.SS2.p2.1.m1.1a"><mrow id="S2.SS2.p2.1.m1.1.1" xref="S2.SS2.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.1.m1.1.1.2" xref="S2.SS2.p2.1.m1.1.1.2.cmml">𝒪</mi><mo id="S2.SS2.p2.1.m1.1.1.1" xref="S2.SS2.p2.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.SS2.p2.1.m1.1.1.3" xref="S2.SS2.p2.1.m1.1.1.3.cmml"><mi id="S2.SS2.p2.1.m1.1.1.3.2" xref="S2.SS2.p2.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS2.p2.1.m1.1.1.3.3" xref="S2.SS2.p2.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.1.m1.1b"><apply id="S2.SS2.p2.1.m1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1"><subset id="S2.SS2.p2.1.m1.1.1.1.cmml" xref="S2.SS2.p2.1.m1.1.1.1"></subset><ci id="S2.SS2.p2.1.m1.1.1.2.cmml" xref="S2.SS2.p2.1.m1.1.1.2">𝒪</ci><apply id="S2.SS2.p2.1.m1.1.1.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p2.1.m1.1.1.3.1.cmml" xref="S2.SS2.p2.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS2.p2.1.m1.1.1.3.2.cmml" xref="S2.SS2.p2.1.m1.1.1.3.2">ℝ</ci><ci id="S2.SS2.p2.1.m1.1.1.3.3.cmml" xref="S2.SS2.p2.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.1.m1.1c">\mathcal{O}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.1.m1.1d">caligraphic_O ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> open, <math alttext="p\in[1,\infty)" class="ltx_Math" display="inline" id="S2.SS2.p2.2.m2.2"><semantics id="S2.SS2.p2.2.m2.2a"><mrow id="S2.SS2.p2.2.m2.2.3" xref="S2.SS2.p2.2.m2.2.3.cmml"><mi id="S2.SS2.p2.2.m2.2.3.2" xref="S2.SS2.p2.2.m2.2.3.2.cmml">p</mi><mo id="S2.SS2.p2.2.m2.2.3.1" xref="S2.SS2.p2.2.m2.2.3.1.cmml">∈</mo><mrow id="S2.SS2.p2.2.m2.2.3.3.2" xref="S2.SS2.p2.2.m2.2.3.3.1.cmml"><mo id="S2.SS2.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS2.p2.2.m2.2.3.3.1.cmml">[</mo><mn id="S2.SS2.p2.2.m2.1.1" xref="S2.SS2.p2.2.m2.1.1.cmml">1</mn><mo id="S2.SS2.p2.2.m2.2.3.3.2.2" xref="S2.SS2.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S2.SS2.p2.2.m2.2.2" mathvariant="normal" xref="S2.SS2.p2.2.m2.2.2.cmml">∞</mi><mo id="S2.SS2.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS2.p2.2.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.2.m2.2b"><apply id="S2.SS2.p2.2.m2.2.3.cmml" xref="S2.SS2.p2.2.m2.2.3"><in id="S2.SS2.p2.2.m2.2.3.1.cmml" xref="S2.SS2.p2.2.m2.2.3.1"></in><ci id="S2.SS2.p2.2.m2.2.3.2.cmml" xref="S2.SS2.p2.2.m2.2.3.2">𝑝</ci><interval closure="closed-open" id="S2.SS2.p2.2.m2.2.3.3.1.cmml" xref="S2.SS2.p2.2.m2.2.3.3.2"><cn id="S2.SS2.p2.2.m2.1.1.cmml" type="integer" xref="S2.SS2.p2.2.m2.1.1">1</cn><infinity id="S2.SS2.p2.2.m2.2.2.cmml" xref="S2.SS2.p2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.2.m2.2c">p\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.2.m2.2d">italic_p ∈ [ 1 , ∞ )</annotation></semantics></math>, <math alttext="w" class="ltx_Math" display="inline" id="S2.SS2.p2.3.m3.1"><semantics id="S2.SS2.p2.3.m3.1a"><mi id="S2.SS2.p2.3.m3.1.1" xref="S2.SS2.p2.3.m3.1.1.cmml">w</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.3.m3.1b"><ci id="S2.SS2.p2.3.m3.1.1.cmml" xref="S2.SS2.p2.3.m3.1.1">𝑤</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.3.m3.1c">w</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.3.m3.1d">italic_w</annotation></semantics></math> a weight and <math alttext="X" class="ltx_Math" display="inline" id="S2.SS2.p2.4.m4.1"><semantics id="S2.SS2.p2.4.m4.1a"><mi id="S2.SS2.p2.4.m4.1.1" xref="S2.SS2.p2.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.4.m4.1b"><ci id="S2.SS2.p2.4.m4.1.1.cmml" xref="S2.SS2.p2.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.4.m4.1d">italic_X</annotation></semantics></math> a Banach space, we define the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p2.5.1">weighted Lebesgue space <math alttext="L^{p}(\mathcal{O},w;X)" class="ltx_Math" display="inline" id="S2.SS2.p2.5.1.m1.3"><semantics id="S2.SS2.p2.5.1.m1.3a"><mrow id="S2.SS2.p2.5.1.m1.3.4" xref="S2.SS2.p2.5.1.m1.3.4.cmml"><msup id="S2.SS2.p2.5.1.m1.3.4.2" xref="S2.SS2.p2.5.1.m1.3.4.2.cmml"><mi id="S2.SS2.p2.5.1.m1.3.4.2.2" xref="S2.SS2.p2.5.1.m1.3.4.2.2.cmml">L</mi><mi id="S2.SS2.p2.5.1.m1.3.4.2.3" xref="S2.SS2.p2.5.1.m1.3.4.2.3.cmml">p</mi></msup><mo id="S2.SS2.p2.5.1.m1.3.4.1" xref="S2.SS2.p2.5.1.m1.3.4.1.cmml">⁢</mo><mrow id="S2.SS2.p2.5.1.m1.3.4.3.2" xref="S2.SS2.p2.5.1.m1.3.4.3.1.cmml"><mo id="S2.SS2.p2.5.1.m1.3.4.3.2.1" stretchy="false" xref="S2.SS2.p2.5.1.m1.3.4.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.5.1.m1.1.1" xref="S2.SS2.p2.5.1.m1.1.1.cmml">𝒪</mi><mo id="S2.SS2.p2.5.1.m1.3.4.3.2.2" xref="S2.SS2.p2.5.1.m1.3.4.3.1.cmml">,</mo><mi id="S2.SS2.p2.5.1.m1.2.2" xref="S2.SS2.p2.5.1.m1.2.2.cmml">w</mi><mo id="S2.SS2.p2.5.1.m1.3.4.3.2.3" xref="S2.SS2.p2.5.1.m1.3.4.3.1.cmml">;</mo><mi id="S2.SS2.p2.5.1.m1.3.3" xref="S2.SS2.p2.5.1.m1.3.3.cmml">X</mi><mo id="S2.SS2.p2.5.1.m1.3.4.3.2.4" stretchy="false" xref="S2.SS2.p2.5.1.m1.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.5.1.m1.3b"><apply id="S2.SS2.p2.5.1.m1.3.4.cmml" xref="S2.SS2.p2.5.1.m1.3.4"><times id="S2.SS2.p2.5.1.m1.3.4.1.cmml" xref="S2.SS2.p2.5.1.m1.3.4.1"></times><apply id="S2.SS2.p2.5.1.m1.3.4.2.cmml" xref="S2.SS2.p2.5.1.m1.3.4.2"><csymbol cd="ambiguous" id="S2.SS2.p2.5.1.m1.3.4.2.1.cmml" xref="S2.SS2.p2.5.1.m1.3.4.2">superscript</csymbol><ci id="S2.SS2.p2.5.1.m1.3.4.2.2.cmml" xref="S2.SS2.p2.5.1.m1.3.4.2.2">𝐿</ci><ci id="S2.SS2.p2.5.1.m1.3.4.2.3.cmml" xref="S2.SS2.p2.5.1.m1.3.4.2.3">𝑝</ci></apply><vector id="S2.SS2.p2.5.1.m1.3.4.3.1.cmml" xref="S2.SS2.p2.5.1.m1.3.4.3.2"><ci id="S2.SS2.p2.5.1.m1.1.1.cmml" xref="S2.SS2.p2.5.1.m1.1.1">𝒪</ci><ci id="S2.SS2.p2.5.1.m1.2.2.cmml" xref="S2.SS2.p2.5.1.m1.2.2">𝑤</ci><ci id="S2.SS2.p2.5.1.m1.3.3.cmml" xref="S2.SS2.p2.5.1.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.5.1.m1.3c">L^{p}(\mathcal{O},w;X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.5.1.m1.3d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w ; italic_X )</annotation></semantics></math></em>, which consists of all strongly measurable <math alttext="f\colon\mathcal{O}\to X" class="ltx_Math" display="inline" id="S2.SS2.p2.6.m5.1"><semantics id="S2.SS2.p2.6.m5.1a"><mrow id="S2.SS2.p2.6.m5.1.1" xref="S2.SS2.p2.6.m5.1.1.cmml"><mi id="S2.SS2.p2.6.m5.1.1.2" xref="S2.SS2.p2.6.m5.1.1.2.cmml">f</mi><mo id="S2.SS2.p2.6.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p2.6.m5.1.1.1.cmml">:</mo><mrow id="S2.SS2.p2.6.m5.1.1.3" xref="S2.SS2.p2.6.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p2.6.m5.1.1.3.2" xref="S2.SS2.p2.6.m5.1.1.3.2.cmml">𝒪</mi><mo id="S2.SS2.p2.6.m5.1.1.3.1" stretchy="false" xref="S2.SS2.p2.6.m5.1.1.3.1.cmml">→</mo><mi id="S2.SS2.p2.6.m5.1.1.3.3" xref="S2.SS2.p2.6.m5.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p2.6.m5.1b"><apply id="S2.SS2.p2.6.m5.1.1.cmml" xref="S2.SS2.p2.6.m5.1.1"><ci id="S2.SS2.p2.6.m5.1.1.1.cmml" xref="S2.SS2.p2.6.m5.1.1.1">:</ci><ci id="S2.SS2.p2.6.m5.1.1.2.cmml" xref="S2.SS2.p2.6.m5.1.1.2">𝑓</ci><apply id="S2.SS2.p2.6.m5.1.1.3.cmml" xref="S2.SS2.p2.6.m5.1.1.3"><ci id="S2.SS2.p2.6.m5.1.1.3.1.cmml" xref="S2.SS2.p2.6.m5.1.1.3.1">→</ci><ci id="S2.SS2.p2.6.m5.1.1.3.2.cmml" xref="S2.SS2.p2.6.m5.1.1.3.2">𝒪</ci><ci id="S2.SS2.p2.6.m5.1.1.3.3.cmml" xref="S2.SS2.p2.6.m5.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.6.m5.1c">f\colon\mathcal{O}\to X</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.6.m5.1d">italic_f : caligraphic_O → italic_X</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lVert f\rVert_{L^{p}(\mathcal{O},w;X)}:=\Bigl{(}\int_{\mathcal{O}}\|f(x)\|^{p% }_{X}\>w(x)\hskip 2.0pt\mathrm{d}x\Bigr{)}^{1/p}<\infty." class="ltx_Math" display="block" id="S2.Ex2.m1.7"><semantics id="S2.Ex2.m1.7a"><mrow id="S2.Ex2.m1.7.7.1" xref="S2.Ex2.m1.7.7.1.1.cmml"><mrow id="S2.Ex2.m1.7.7.1.1" xref="S2.Ex2.m1.7.7.1.1.cmml"><msub id="S2.Ex2.m1.7.7.1.1.3" xref="S2.Ex2.m1.7.7.1.1.3.cmml"><mrow id="S2.Ex2.m1.7.7.1.1.3.2.2" xref="S2.Ex2.m1.7.7.1.1.3.2.1.cmml"><mo fence="true" id="S2.Ex2.m1.7.7.1.1.3.2.2.1" rspace="0em" xref="S2.Ex2.m1.7.7.1.1.3.2.1.1.cmml">∥</mo><mi id="S2.Ex2.m1.4.4" xref="S2.Ex2.m1.4.4.cmml">f</mi><mo fence="true" id="S2.Ex2.m1.7.7.1.1.3.2.2.2" lspace="0em" xref="S2.Ex2.m1.7.7.1.1.3.2.1.1.cmml">∥</mo></mrow><mrow id="S2.Ex2.m1.3.3.3" xref="S2.Ex2.m1.3.3.3.cmml"><msup id="S2.Ex2.m1.3.3.3.5" xref="S2.Ex2.m1.3.3.3.5.cmml"><mi id="S2.Ex2.m1.3.3.3.5.2" xref="S2.Ex2.m1.3.3.3.5.2.cmml">L</mi><mi id="S2.Ex2.m1.3.3.3.5.3" xref="S2.Ex2.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S2.Ex2.m1.3.3.3.4" xref="S2.Ex2.m1.3.3.3.4.cmml">⁢</mo><mrow id="S2.Ex2.m1.3.3.3.6.2" xref="S2.Ex2.m1.3.3.3.6.1.cmml"><mo id="S2.Ex2.m1.3.3.3.6.2.1" stretchy="false" xref="S2.Ex2.m1.3.3.3.6.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.1.1.1.1" xref="S2.Ex2.m1.1.1.1.1.cmml">𝒪</mi><mo id="S2.Ex2.m1.3.3.3.6.2.2" xref="S2.Ex2.m1.3.3.3.6.1.cmml">,</mo><mi id="S2.Ex2.m1.2.2.2.2" xref="S2.Ex2.m1.2.2.2.2.cmml">w</mi><mo id="S2.Ex2.m1.3.3.3.6.2.3" xref="S2.Ex2.m1.3.3.3.6.1.cmml">;</mo><mi id="S2.Ex2.m1.3.3.3.3" xref="S2.Ex2.m1.3.3.3.3.cmml">X</mi><mo id="S2.Ex2.m1.3.3.3.6.2.4" stretchy="false" xref="S2.Ex2.m1.3.3.3.6.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex2.m1.7.7.1.1.4" lspace="0.278em" rspace="0.278em" xref="S2.Ex2.m1.7.7.1.1.4.cmml">:=</mo><msup id="S2.Ex2.m1.7.7.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.cmml"><mrow id="S2.Ex2.m1.7.7.1.1.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S2.Ex2.m1.7.7.1.1.1.1.1.2" maxsize="160%" minsize="160%" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex2.m1.7.7.1.1.1.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.cmml"><msub id="S2.Ex2.m1.7.7.1.1.1.1.1.1.2" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.2.cmml"><mo id="S2.Ex2.m1.7.7.1.1.1.1.1.1.2.2" lspace="0em" rspace="0em" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.2.2.cmml">∫</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex2.m1.7.7.1.1.1.1.1.1.2.3" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.2.3.cmml">𝒪</mi></msub><mrow id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.cmml"><msubsup id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.cmml"><mrow id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.2.cmml">f</mi><mo id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.3.2" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mi id="S2.Ex2.m1.5.5" xref="S2.Ex2.m1.5.5.cmml">x</mi><mo id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.3.2.2" stretchy="false" xref="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex2.m1.7.7.1.1.1.1.1.1.1.1.1.1.1.3" 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encoding="application/x-llamapun" id="S2.Ex2.m1.7d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w ; italic_X ) end_POSTSUBSCRIPT := ( ∫ start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT ∥ italic_f ( italic_x ) ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT italic_w ( italic_x ) roman_d italic_x ) start_POSTSUPERSCRIPT 1 / italic_p end_POSTSUPERSCRIPT < ∞ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p2.10">Additionally, for <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S2.SS2.p2.7.m1.1"><semantics id="S2.SS2.p2.7.m1.1a"><mrow id="S2.SS2.p2.7.m1.1.1" xref="S2.SS2.p2.7.m1.1.1.cmml"><mi id="S2.SS2.p2.7.m1.1.1.2" xref="S2.SS2.p2.7.m1.1.1.2.cmml">k</mi><mo id="S2.SS2.p2.7.m1.1.1.1" xref="S2.SS2.p2.7.m1.1.1.1.cmml">∈</mo><msub 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divide start_ARG 1 end_ARG start_ARG italic_p - 1 end_ARG end_POSTSUPERSCRIPT ∈ italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_loc end_POSTSUBSCRIPT ( caligraphic_O )</annotation></semantics></math>, we define the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p2.10.1">weighted Sobolev space</em> consisting of <math alttext="f\in L^{p}(\mathcal{O},w;X)" class="ltx_Math" display="inline" id="S2.SS2.p2.10.m4.3"><semantics id="S2.SS2.p2.10.m4.3a"><mrow id="S2.SS2.p2.10.m4.3.4" xref="S2.SS2.p2.10.m4.3.4.cmml"><mi id="S2.SS2.p2.10.m4.3.4.2" xref="S2.SS2.p2.10.m4.3.4.2.cmml">f</mi><mo id="S2.SS2.p2.10.m4.3.4.1" xref="S2.SS2.p2.10.m4.3.4.1.cmml">∈</mo><mrow id="S2.SS2.p2.10.m4.3.4.3" xref="S2.SS2.p2.10.m4.3.4.3.cmml"><msup id="S2.SS2.p2.10.m4.3.4.3.2" xref="S2.SS2.p2.10.m4.3.4.3.2.cmml"><mi id="S2.SS2.p2.10.m4.3.4.3.2.2" xref="S2.SS2.p2.10.m4.3.4.3.2.2.cmml">L</mi><mi id="S2.SS2.p2.10.m4.3.4.3.2.3" xref="S2.SS2.p2.10.m4.3.4.3.2.3.cmml">p</mi></msup><mo 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xref="S2.SS2.p2.10.m4.3.4.2">𝑓</ci><apply id="S2.SS2.p2.10.m4.3.4.3.cmml" xref="S2.SS2.p2.10.m4.3.4.3"><times id="S2.SS2.p2.10.m4.3.4.3.1.cmml" xref="S2.SS2.p2.10.m4.3.4.3.1"></times><apply id="S2.SS2.p2.10.m4.3.4.3.2.cmml" xref="S2.SS2.p2.10.m4.3.4.3.2"><csymbol cd="ambiguous" id="S2.SS2.p2.10.m4.3.4.3.2.1.cmml" xref="S2.SS2.p2.10.m4.3.4.3.2">superscript</csymbol><ci id="S2.SS2.p2.10.m4.3.4.3.2.2.cmml" xref="S2.SS2.p2.10.m4.3.4.3.2.2">𝐿</ci><ci id="S2.SS2.p2.10.m4.3.4.3.2.3.cmml" xref="S2.SS2.p2.10.m4.3.4.3.2.3">𝑝</ci></apply><vector id="S2.SS2.p2.10.m4.3.4.3.3.1.cmml" xref="S2.SS2.p2.10.m4.3.4.3.3.2"><ci id="S2.SS2.p2.10.m4.1.1.cmml" xref="S2.SS2.p2.10.m4.1.1">𝒪</ci><ci id="S2.SS2.p2.10.m4.2.2.cmml" xref="S2.SS2.p2.10.m4.2.2">𝑤</ci><ci id="S2.SS2.p2.10.m4.3.3.cmml" xref="S2.SS2.p2.10.m4.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p2.10.m4.3c">f\in L^{p}(\mathcal{O},w;X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p2.10.m4.3d">italic_f ∈ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w ; italic_X )</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f\|_{W^{k,p}(\mathcal{O},w;X)}:=\sum_{|\alpha|\leq k}\|\partial^{\alpha}f\|_% {L^{p}(\mathcal{O},w;X)}<\infty." class="ltx_Math" display="block" id="S2.Ex3.m1.11"><semantics id="S2.Ex3.m1.11a"><mrow id="S2.Ex3.m1.11.11.1" xref="S2.Ex3.m1.11.11.1.1.cmml"><mrow id="S2.Ex3.m1.11.11.1.1" xref="S2.Ex3.m1.11.11.1.1.cmml"><msub id="S2.Ex3.m1.11.11.1.1.3" xref="S2.Ex3.m1.11.11.1.1.3.cmml"><mrow id="S2.Ex3.m1.11.11.1.1.3.2.2" xref="S2.Ex3.m1.11.11.1.1.3.2.1.cmml"><mo id="S2.Ex3.m1.11.11.1.1.3.2.2.1" stretchy="false" 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id="S2.Ex3.m1.5.5.5.8.2.2" xref="S2.Ex3.m1.5.5.5.8.1.cmml">,</mo><mi id="S2.Ex3.m1.4.4.4.4" xref="S2.Ex3.m1.4.4.4.4.cmml">w</mi><mo id="S2.Ex3.m1.5.5.5.8.2.3" xref="S2.Ex3.m1.5.5.5.8.1.cmml">;</mo><mi id="S2.Ex3.m1.5.5.5.5" xref="S2.Ex3.m1.5.5.5.5.cmml">X</mi><mo id="S2.Ex3.m1.5.5.5.8.2.4" stretchy="false" xref="S2.Ex3.m1.5.5.5.8.1.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex3.m1.11.11.1.1.4" lspace="0.278em" rspace="0.111em" xref="S2.Ex3.m1.11.11.1.1.4.cmml">:=</mo><mrow id="S2.Ex3.m1.11.11.1.1.1" xref="S2.Ex3.m1.11.11.1.1.1.cmml"><munder id="S2.Ex3.m1.11.11.1.1.1.2" xref="S2.Ex3.m1.11.11.1.1.1.2.cmml"><mo id="S2.Ex3.m1.11.11.1.1.1.2.2" movablelimits="false" rspace="0em" xref="S2.Ex3.m1.11.11.1.1.1.2.2.cmml">∑</mo><mrow id="S2.Ex3.m1.6.6.1" xref="S2.Ex3.m1.6.6.1.cmml"><mrow id="S2.Ex3.m1.6.6.1.3.2" xref="S2.Ex3.m1.6.6.1.3.1.cmml"><mo id="S2.Ex3.m1.6.6.1.3.2.1" stretchy="false" xref="S2.Ex3.m1.6.6.1.3.1.1.cmml">|</mo><mi id="S2.Ex3.m1.6.6.1.1" xref="S2.Ex3.m1.6.6.1.1.cmml">α</mi><mo 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k}\|\partial^{\alpha}f\|_% {L^{p}(\mathcal{O},w;X)}<\infty.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3.m1.11d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w ; italic_X ) end_POSTSUBSCRIPT := ∑ start_POSTSUBSCRIPT | italic_α | ≤ italic_k end_POSTSUBSCRIPT ∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w ; italic_X ) end_POSTSUBSCRIPT < ∞ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_theorem ltx_theorem_remark" id="S2.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.1.1.1">Remark 2.1</span></span><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem1.p1"> <p class="ltx_p" id="S2.Thmtheorem1.p1.14">The local <math alttext="L^{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.1.m1.1"><semantics id="S2.Thmtheorem1.p1.1.m1.1a"><msup id="S2.Thmtheorem1.p1.1.m1.1.1" xref="S2.Thmtheorem1.p1.1.m1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.1.m1.1.1.2" xref="S2.Thmtheorem1.p1.1.m1.1.1.2.cmml">L</mi><mn id="S2.Thmtheorem1.p1.1.m1.1.1.3" xref="S2.Thmtheorem1.p1.1.m1.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.1.m1.1b"><apply id="S2.Thmtheorem1.p1.1.m1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.1.m1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1">superscript</csymbol><ci id="S2.Thmtheorem1.p1.1.m1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.1.m1.1.1.2">𝐿</ci><cn id="S2.Thmtheorem1.p1.1.m1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.1.m1.1c">L^{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.1.m1.1d">italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math> condition for <math alttext="w^{-\frac{1}{p-1}}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.2.m2.1"><semantics id="S2.Thmtheorem1.p1.2.m2.1a"><msup id="S2.Thmtheorem1.p1.2.m2.1.1" xref="S2.Thmtheorem1.p1.2.m2.1.1.cmml"><mi id="S2.Thmtheorem1.p1.2.m2.1.1.2" xref="S2.Thmtheorem1.p1.2.m2.1.1.2.cmml">w</mi><mrow id="S2.Thmtheorem1.p1.2.m2.1.1.3" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.cmml"><mo id="S2.Thmtheorem1.p1.2.m2.1.1.3a" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.cmml">−</mo><mfrac id="S2.Thmtheorem1.p1.2.m2.1.1.3.2" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2.cmml"><mn id="S2.Thmtheorem1.p1.2.m2.1.1.3.2.2" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2.2.cmml">1</mn><mrow 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xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2"><divide id="S2.Thmtheorem1.p1.2.m2.1.1.3.2.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2"></divide><cn id="S2.Thmtheorem1.p1.2.m2.1.1.3.2.2.cmml" type="integer" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2.2">1</cn><apply id="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3"><minus id="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3.1.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3.1"></minus><ci id="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3.2.cmml" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3.2">𝑝</ci><cn id="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.2.m2.1.1.3.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.2.m2.1c">w^{-\frac{1}{p-1}}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.2.m2.1d">italic_w start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG italic_p - 1 end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math> ensures that all the derivatives <math alttext="\partial^{\alpha}f" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.3.m3.1"><semantics id="S2.Thmtheorem1.p1.3.m3.1a"><mrow id="S2.Thmtheorem1.p1.3.m3.1.1" xref="S2.Thmtheorem1.p1.3.m3.1.1.cmml"><msup id="S2.Thmtheorem1.p1.3.m3.1.1.1" xref="S2.Thmtheorem1.p1.3.m3.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.3.m3.1.1.1.2" xref="S2.Thmtheorem1.p1.3.m3.1.1.1.2.cmml">∂</mo><mi id="S2.Thmtheorem1.p1.3.m3.1.1.1.3" xref="S2.Thmtheorem1.p1.3.m3.1.1.1.3.cmml">α</mi></msup><mi id="S2.Thmtheorem1.p1.3.m3.1.1.2" xref="S2.Thmtheorem1.p1.3.m3.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.3.m3.1b"><apply id="S2.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1"><apply id="S2.Thmtheorem1.p1.3.m3.1.1.1.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.3.m3.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.1">superscript</csymbol><partialdiff id="S2.Thmtheorem1.p1.3.m3.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.1.2"></partialdiff><ci id="S2.Thmtheorem1.p1.3.m3.1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.1.3">𝛼</ci></apply><ci id="S2.Thmtheorem1.p1.3.m3.1.1.2.cmml" xref="S2.Thmtheorem1.p1.3.m3.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.3.m3.1c">\partial^{\alpha}f</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.3.m3.1d">∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_f</annotation></semantics></math> are locally integrable. For <math alttext="\mathcal{O}=\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.4.m4.1"><semantics id="S2.Thmtheorem1.p1.4.m4.1a"><mrow id="S2.Thmtheorem1.p1.4.m4.1.1" xref="S2.Thmtheorem1.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.4.m4.1.1.2" xref="S2.Thmtheorem1.p1.4.m4.1.1.2.cmml">𝒪</mi><mo id="S2.Thmtheorem1.p1.4.m4.1.1.1" xref="S2.Thmtheorem1.p1.4.m4.1.1.1.cmml">=</mo><msubsup id="S2.Thmtheorem1.p1.4.m4.1.1.3" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.4.m4.1.1.3.2.2" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.2.2.cmml">ℝ</mi><mo id="S2.Thmtheorem1.p1.4.m4.1.1.3.3" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.3.cmml">+</mo><mi id="S2.Thmtheorem1.p1.4.m4.1.1.3.2.3" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.4.m4.1b"><apply id="S2.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1"><eq id="S2.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.1"></eq><ci id="S2.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.2">𝒪</ci><apply id="S2.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.4.m4.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3">subscript</csymbol><apply id="S2.Thmtheorem1.p1.4.m4.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.4.m4.1.1.3.2.1.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem1.p1.4.m4.1.1.3.2.2.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.2.2">ℝ</ci><ci id="S2.Thmtheorem1.p1.4.m4.1.1.3.2.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.2.3">𝑑</ci></apply><plus id="S2.Thmtheorem1.p1.4.m4.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.4.m4.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.4.m4.1c">\mathcal{O}=\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.4.m4.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> this condition holds if <math alttext="w\in A_{p}(\mathbb{R}^{d}_{+})" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.5.m5.1"><semantics id="S2.Thmtheorem1.p1.5.m5.1a"><mrow id="S2.Thmtheorem1.p1.5.m5.1.1" xref="S2.Thmtheorem1.p1.5.m5.1.1.cmml"><mi id="S2.Thmtheorem1.p1.5.m5.1.1.3" xref="S2.Thmtheorem1.p1.5.m5.1.1.3.cmml">w</mi><mo id="S2.Thmtheorem1.p1.5.m5.1.1.2" xref="S2.Thmtheorem1.p1.5.m5.1.1.2.cmml">∈</mo><mrow id="S2.Thmtheorem1.p1.5.m5.1.1.1" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.cmml"><msub id="S2.Thmtheorem1.p1.5.m5.1.1.1.3" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.5.m5.1.1.1.3.2" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.3.2.cmml">A</mi><mi id="S2.Thmtheorem1.p1.5.m5.1.1.1.3.3" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.3.3.cmml">p</mi></msub><mo id="S2.Thmtheorem1.p1.5.m5.1.1.1.2" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.cmml">(</mo><msubsup id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.2" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.3" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.3.cmml">+</mo><mi id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.3" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.5.m5.1b"><apply id="S2.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1"><in id="S2.Thmtheorem1.p1.5.m5.1.1.2.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.2"></in><ci id="S2.Thmtheorem1.p1.5.m5.1.1.3.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.3">𝑤</ci><apply id="S2.Thmtheorem1.p1.5.m5.1.1.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1"><times id="S2.Thmtheorem1.p1.5.m5.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.2"></times><apply id="S2.Thmtheorem1.p1.5.m5.1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.m5.1.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.5.m5.1.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.3.2">𝐴</ci><ci id="S2.Thmtheorem1.p1.5.m5.1.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.3.3">𝑝</ci></apply><apply id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1">subscript</csymbol><apply id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.2">ℝ</ci><ci id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.3.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.2.3">𝑑</ci></apply><plus id="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.5.m5.1.1.1.1.1.1.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.5.m5.1c">w\in A_{p}(\mathbb{R}^{d}_{+})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.5.m5.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT )</annotation></semantics></math> or <math alttext="w=w_{\gamma}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.6.m6.1"><semantics id="S2.Thmtheorem1.p1.6.m6.1a"><mrow id="S2.Thmtheorem1.p1.6.m6.1.1" xref="S2.Thmtheorem1.p1.6.m6.1.1.cmml"><mi id="S2.Thmtheorem1.p1.6.m6.1.1.2" xref="S2.Thmtheorem1.p1.6.m6.1.1.2.cmml">w</mi><mo id="S2.Thmtheorem1.p1.6.m6.1.1.1" xref="S2.Thmtheorem1.p1.6.m6.1.1.1.cmml">=</mo><msub id="S2.Thmtheorem1.p1.6.m6.1.1.3" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.6.m6.1.1.3.2" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.2.cmml">w</mi><mi id="S2.Thmtheorem1.p1.6.m6.1.1.3.3" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.3.cmml">γ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.6.m6.1b"><apply id="S2.Thmtheorem1.p1.6.m6.1.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1"><eq id="S2.Thmtheorem1.p1.6.m6.1.1.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.1"></eq><ci id="S2.Thmtheorem1.p1.6.m6.1.1.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.2">𝑤</ci><apply id="S2.Thmtheorem1.p1.6.m6.1.1.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.6.m6.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.6.m6.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.2">𝑤</ci><ci id="S2.Thmtheorem1.p1.6.m6.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.6.m6.1.1.3.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.6.m6.1c">w=w_{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.6.m6.1d">italic_w = italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\gamma\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.7.m7.1"><semantics id="S2.Thmtheorem1.p1.7.m7.1a"><mrow id="S2.Thmtheorem1.p1.7.m7.1.1" xref="S2.Thmtheorem1.p1.7.m7.1.1.cmml"><mi id="S2.Thmtheorem1.p1.7.m7.1.1.2" xref="S2.Thmtheorem1.p1.7.m7.1.1.2.cmml">γ</mi><mo id="S2.Thmtheorem1.p1.7.m7.1.1.1" xref="S2.Thmtheorem1.p1.7.m7.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem1.p1.7.m7.1.1.3" xref="S2.Thmtheorem1.p1.7.m7.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.7.m7.1b"><apply id="S2.Thmtheorem1.p1.7.m7.1.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1"><in id="S2.Thmtheorem1.p1.7.m7.1.1.1.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.1"></in><ci id="S2.Thmtheorem1.p1.7.m7.1.1.2.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.2">𝛾</ci><ci id="S2.Thmtheorem1.p1.7.m7.1.1.3.cmml" xref="S2.Thmtheorem1.p1.7.m7.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.7.m7.1c">\gamma\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.7.m7.1d">italic_γ ∈ blackboard_R</annotation></semantics></math>. For <math alttext="\mathcal{O}=\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.8.m8.1"><semantics id="S2.Thmtheorem1.p1.8.m8.1a"><mrow id="S2.Thmtheorem1.p1.8.m8.1.1" xref="S2.Thmtheorem1.p1.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Thmtheorem1.p1.8.m8.1.1.2" xref="S2.Thmtheorem1.p1.8.m8.1.1.2.cmml">𝒪</mi><mo id="S2.Thmtheorem1.p1.8.m8.1.1.1" xref="S2.Thmtheorem1.p1.8.m8.1.1.1.cmml">=</mo><msup id="S2.Thmtheorem1.p1.8.m8.1.1.3" xref="S2.Thmtheorem1.p1.8.m8.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.8.m8.1.1.3.2" xref="S2.Thmtheorem1.p1.8.m8.1.1.3.2.cmml">ℝ</mi><mi id="S2.Thmtheorem1.p1.8.m8.1.1.3.3" xref="S2.Thmtheorem1.p1.8.m8.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.8.m8.1b"><apply id="S2.Thmtheorem1.p1.8.m8.1.1.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1"><eq id="S2.Thmtheorem1.p1.8.m8.1.1.1.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1.1"></eq><ci id="S2.Thmtheorem1.p1.8.m8.1.1.2.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1.2">𝒪</ci><apply id="S2.Thmtheorem1.p1.8.m8.1.1.3.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.8.m8.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1.3">superscript</csymbol><ci id="S2.Thmtheorem1.p1.8.m8.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1.3.2">ℝ</ci><ci id="S2.Thmtheorem1.p1.8.m8.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.8.m8.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.8.m8.1c">\mathcal{O}=\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.8.m8.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> the local <math alttext="L^{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.9.m9.1"><semantics id="S2.Thmtheorem1.p1.9.m9.1a"><msup id="S2.Thmtheorem1.p1.9.m9.1.1" xref="S2.Thmtheorem1.p1.9.m9.1.1.cmml"><mi id="S2.Thmtheorem1.p1.9.m9.1.1.2" xref="S2.Thmtheorem1.p1.9.m9.1.1.2.cmml">L</mi><mn id="S2.Thmtheorem1.p1.9.m9.1.1.3" xref="S2.Thmtheorem1.p1.9.m9.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.9.m9.1b"><apply id="S2.Thmtheorem1.p1.9.m9.1.1.cmml" xref="S2.Thmtheorem1.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.9.m9.1.1.1.cmml" xref="S2.Thmtheorem1.p1.9.m9.1.1">superscript</csymbol><ci id="S2.Thmtheorem1.p1.9.m9.1.1.2.cmml" xref="S2.Thmtheorem1.p1.9.m9.1.1.2">𝐿</ci><cn id="S2.Thmtheorem1.p1.9.m9.1.1.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.9.m9.1c">L^{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.9.m9.1d">italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math> condition holds if <math alttext="w\in A_{p}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.10.m10.1"><semantics id="S2.Thmtheorem1.p1.10.m10.1a"><mrow id="S2.Thmtheorem1.p1.10.m10.1.1" xref="S2.Thmtheorem1.p1.10.m10.1.1.cmml"><mi id="S2.Thmtheorem1.p1.10.m10.1.1.3" xref="S2.Thmtheorem1.p1.10.m10.1.1.3.cmml">w</mi><mo id="S2.Thmtheorem1.p1.10.m10.1.1.2" xref="S2.Thmtheorem1.p1.10.m10.1.1.2.cmml">∈</mo><mrow id="S2.Thmtheorem1.p1.10.m10.1.1.1" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.cmml"><msub id="S2.Thmtheorem1.p1.10.m10.1.1.1.3" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.10.m10.1.1.1.3.2" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.3.2.cmml">A</mi><mi id="S2.Thmtheorem1.p1.10.m10.1.1.1.3.3" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.3.3.cmml">p</mi></msub><mo id="S2.Thmtheorem1.p1.10.m10.1.1.1.2" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.10.m10.1b"><apply id="S2.Thmtheorem1.p1.10.m10.1.1.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1"><in id="S2.Thmtheorem1.p1.10.m10.1.1.2.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.2"></in><ci id="S2.Thmtheorem1.p1.10.m10.1.1.3.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.3">𝑤</ci><apply id="S2.Thmtheorem1.p1.10.m10.1.1.1.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1"><times id="S2.Thmtheorem1.p1.10.m10.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.2"></times><apply id="S2.Thmtheorem1.p1.10.m10.1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.10.m10.1.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.10.m10.1.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.3.2">𝐴</ci><ci id="S2.Thmtheorem1.p1.10.m10.1.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.3.3">𝑝</ci></apply><apply id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem1.p1.10.m10.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.10.m10.1c">w\in A_{p}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.10.m10.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> or <math alttext="w=w_{\gamma}" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.11.m11.1"><semantics id="S2.Thmtheorem1.p1.11.m11.1a"><mrow id="S2.Thmtheorem1.p1.11.m11.1.1" xref="S2.Thmtheorem1.p1.11.m11.1.1.cmml"><mi id="S2.Thmtheorem1.p1.11.m11.1.1.2" xref="S2.Thmtheorem1.p1.11.m11.1.1.2.cmml">w</mi><mo id="S2.Thmtheorem1.p1.11.m11.1.1.1" xref="S2.Thmtheorem1.p1.11.m11.1.1.1.cmml">=</mo><msub id="S2.Thmtheorem1.p1.11.m11.1.1.3" xref="S2.Thmtheorem1.p1.11.m11.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.11.m11.1.1.3.2" xref="S2.Thmtheorem1.p1.11.m11.1.1.3.2.cmml">w</mi><mi id="S2.Thmtheorem1.p1.11.m11.1.1.3.3" xref="S2.Thmtheorem1.p1.11.m11.1.1.3.3.cmml">γ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.11.m11.1b"><apply id="S2.Thmtheorem1.p1.11.m11.1.1.cmml" xref="S2.Thmtheorem1.p1.11.m11.1.1"><eq id="S2.Thmtheorem1.p1.11.m11.1.1.1.cmml" xref="S2.Thmtheorem1.p1.11.m11.1.1.1"></eq><ci id="S2.Thmtheorem1.p1.11.m11.1.1.2.cmml" xref="S2.Thmtheorem1.p1.11.m11.1.1.2">𝑤</ci><apply id="S2.Thmtheorem1.p1.11.m11.1.1.3.cmml" xref="S2.Thmtheorem1.p1.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.11.m11.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.11.m11.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem1.p1.11.m11.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.11.m11.1.1.3.2">𝑤</ci><ci id="S2.Thmtheorem1.p1.11.m11.1.1.3.3.cmml" xref="S2.Thmtheorem1.p1.11.m11.1.1.3.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.11.m11.1c">w=w_{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.11.m11.1d">italic_w = italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\gamma\in(-\infty,p-1)" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.12.m12.2"><semantics id="S2.Thmtheorem1.p1.12.m12.2a"><mrow id="S2.Thmtheorem1.p1.12.m12.2.2" xref="S2.Thmtheorem1.p1.12.m12.2.2.cmml"><mi id="S2.Thmtheorem1.p1.12.m12.2.2.4" xref="S2.Thmtheorem1.p1.12.m12.2.2.4.cmml">γ</mi><mo id="S2.Thmtheorem1.p1.12.m12.2.2.3" xref="S2.Thmtheorem1.p1.12.m12.2.2.3.cmml">∈</mo><mrow id="S2.Thmtheorem1.p1.12.m12.2.2.2.2" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.3.cmml"><mo id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.3.cmml">(</mo><mrow id="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1" xref="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1a" xref="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.cmml">−</mo><mi id="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.2" mathvariant="normal" xref="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.2.cmml">∞</mi></mrow><mo id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.4" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.3.cmml">,</mo><mrow id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.cmml"><mi id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.2" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.2.cmml">p</mi><mo id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.1" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.1.cmml">−</mo><mn id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.3" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.5" stretchy="false" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.12.m12.2b"><apply id="S2.Thmtheorem1.p1.12.m12.2.2.cmml" xref="S2.Thmtheorem1.p1.12.m12.2.2"><in id="S2.Thmtheorem1.p1.12.m12.2.2.3.cmml" xref="S2.Thmtheorem1.p1.12.m12.2.2.3"></in><ci id="S2.Thmtheorem1.p1.12.m12.2.2.4.cmml" xref="S2.Thmtheorem1.p1.12.m12.2.2.4">𝛾</ci><interval closure="open" id="S2.Thmtheorem1.p1.12.m12.2.2.2.3.cmml" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2"><apply id="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1"><minus id="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1"></minus><infinity id="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem1.p1.12.m12.1.1.1.1.1.2"></infinity></apply><apply id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2"><minus id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.1"></minus><ci id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.2">𝑝</ci><cn id="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.12.m12.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.12.m12.2c">\gamma\in(-\infty,p-1)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.12.m12.2d">italic_γ ∈ ( - ∞ , italic_p - 1 )</annotation></semantics></math>. For <math alttext="\gamma\geq p-1" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.13.m13.1"><semantics id="S2.Thmtheorem1.p1.13.m13.1a"><mrow id="S2.Thmtheorem1.p1.13.m13.1.1" xref="S2.Thmtheorem1.p1.13.m13.1.1.cmml"><mi id="S2.Thmtheorem1.p1.13.m13.1.1.2" xref="S2.Thmtheorem1.p1.13.m13.1.1.2.cmml">γ</mi><mo id="S2.Thmtheorem1.p1.13.m13.1.1.1" xref="S2.Thmtheorem1.p1.13.m13.1.1.1.cmml">≥</mo><mrow id="S2.Thmtheorem1.p1.13.m13.1.1.3" xref="S2.Thmtheorem1.p1.13.m13.1.1.3.cmml"><mi id="S2.Thmtheorem1.p1.13.m13.1.1.3.2" xref="S2.Thmtheorem1.p1.13.m13.1.1.3.2.cmml">p</mi><mo id="S2.Thmtheorem1.p1.13.m13.1.1.3.1" xref="S2.Thmtheorem1.p1.13.m13.1.1.3.1.cmml">−</mo><mn id="S2.Thmtheorem1.p1.13.m13.1.1.3.3" xref="S2.Thmtheorem1.p1.13.m13.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.13.m13.1b"><apply id="S2.Thmtheorem1.p1.13.m13.1.1.cmml" xref="S2.Thmtheorem1.p1.13.m13.1.1"><geq id="S2.Thmtheorem1.p1.13.m13.1.1.1.cmml" xref="S2.Thmtheorem1.p1.13.m13.1.1.1"></geq><ci id="S2.Thmtheorem1.p1.13.m13.1.1.2.cmml" xref="S2.Thmtheorem1.p1.13.m13.1.1.2">𝛾</ci><apply id="S2.Thmtheorem1.p1.13.m13.1.1.3.cmml" xref="S2.Thmtheorem1.p1.13.m13.1.1.3"><minus id="S2.Thmtheorem1.p1.13.m13.1.1.3.1.cmml" xref="S2.Thmtheorem1.p1.13.m13.1.1.3.1"></minus><ci id="S2.Thmtheorem1.p1.13.m13.1.1.3.2.cmml" xref="S2.Thmtheorem1.p1.13.m13.1.1.3.2">𝑝</ci><cn id="S2.Thmtheorem1.p1.13.m13.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.13.m13.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.13.m13.1c">\gamma\geq p-1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.13.m13.1d">italic_γ ≥ italic_p - 1</annotation></semantics></math> one has to be careful with defining the weighted Sobolev spaces on the full space because functions might not be locally integrable near <math alttext="x_{1}=0" class="ltx_Math" display="inline" id="S2.Thmtheorem1.p1.14.m14.1"><semantics id="S2.Thmtheorem1.p1.14.m14.1a"><mrow id="S2.Thmtheorem1.p1.14.m14.1.1" xref="S2.Thmtheorem1.p1.14.m14.1.1.cmml"><msub id="S2.Thmtheorem1.p1.14.m14.1.1.2" xref="S2.Thmtheorem1.p1.14.m14.1.1.2.cmml"><mi id="S2.Thmtheorem1.p1.14.m14.1.1.2.2" xref="S2.Thmtheorem1.p1.14.m14.1.1.2.2.cmml">x</mi><mn id="S2.Thmtheorem1.p1.14.m14.1.1.2.3" xref="S2.Thmtheorem1.p1.14.m14.1.1.2.3.cmml">1</mn></msub><mo id="S2.Thmtheorem1.p1.14.m14.1.1.1" xref="S2.Thmtheorem1.p1.14.m14.1.1.1.cmml">=</mo><mn id="S2.Thmtheorem1.p1.14.m14.1.1.3" xref="S2.Thmtheorem1.p1.14.m14.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem1.p1.14.m14.1b"><apply id="S2.Thmtheorem1.p1.14.m14.1.1.cmml" xref="S2.Thmtheorem1.p1.14.m14.1.1"><eq id="S2.Thmtheorem1.p1.14.m14.1.1.1.cmml" xref="S2.Thmtheorem1.p1.14.m14.1.1.1"></eq><apply id="S2.Thmtheorem1.p1.14.m14.1.1.2.cmml" xref="S2.Thmtheorem1.p1.14.m14.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem1.p1.14.m14.1.1.2.1.cmml" xref="S2.Thmtheorem1.p1.14.m14.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem1.p1.14.m14.1.1.2.2.cmml" xref="S2.Thmtheorem1.p1.14.m14.1.1.2.2">𝑥</ci><cn id="S2.Thmtheorem1.p1.14.m14.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.14.m14.1.1.2.3">1</cn></apply><cn id="S2.Thmtheorem1.p1.14.m14.1.1.3.cmml" type="integer" xref="S2.Thmtheorem1.p1.14.m14.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem1.p1.14.m14.1c">x_{1}=0</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem1.p1.14.m14.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib43" title="">43</a>]</cite>.</p> </div> </div> <div class="ltx_para" id="S2.SS2.p3"> <p class="ltx_p" id="S2.SS2.p3.14">We continue with the definitions of weighted Besov, Triebel-Lizorkin and Bessel potential spaces. We denote by <math alttext="\Phi(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS2.p3.1.m1.1"><semantics id="S2.SS2.p3.1.m1.1a"><mrow id="S2.SS2.p3.1.m1.1.1" xref="S2.SS2.p3.1.m1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.3" mathvariant="normal" xref="S2.SS2.p3.1.m1.1.1.3.cmml">Φ</mi><mo id="S2.SS2.p3.1.m1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p3.1.m1.1.1.1.1" xref="S2.SS2.p3.1.m1.1.1.1.1.1.cmml"><mo id="S2.SS2.p3.1.m1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p3.1.m1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS2.p3.1.m1.1.1.1.1.1" xref="S2.SS2.p3.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS2.p3.1.m1.1.1.1.1.1.2" xref="S2.SS2.p3.1.m1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p3.1.m1.1.1.1.1.1.3" xref="S2.SS2.p3.1.m1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p3.1.m1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.1.m1.1b"><apply id="S2.SS2.p3.1.m1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1"><times id="S2.SS2.p3.1.m1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.2"></times><ci id="S2.SS2.p3.1.m1.1.1.3.cmml" xref="S2.SS2.p3.1.m1.1.1.3">Φ</ci><apply id="S2.SS2.p3.1.m1.1.1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.1.m1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p3.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS2.p3.1.m1.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p3.1.m1.1.1.1.1.1.3.cmml" xref="S2.SS2.p3.1.m1.1.1.1.1.1.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.1.m1.1c">\Phi(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.1.m1.1d">roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> the set of <em class="ltx_emph ltx_font_italic" id="S2.SS2.p3.14.1">inhomogeneous Littlewood-Paley sequences</em> consisting of sequences <math alttext="(\varphi_{n})_{n\geq 0}\subseteq\mathcal{S}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS2.p3.2.m2.2"><semantics id="S2.SS2.p3.2.m2.2a"><mrow id="S2.SS2.p3.2.m2.2.2" xref="S2.SS2.p3.2.m2.2.2.cmml"><msub id="S2.SS2.p3.2.m2.1.1.1" xref="S2.SS2.p3.2.m2.1.1.1.cmml"><mrow id="S2.SS2.p3.2.m2.1.1.1.1.1" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p3.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p3.2.m2.1.1.1.1.1.1" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p3.2.m2.1.1.1.1.1.1.2" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.2.cmml">φ</mi><mi id="S2.SS2.p3.2.m2.1.1.1.1.1.1.3" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.SS2.p3.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS2.p3.2.m2.1.1.1.3" xref="S2.SS2.p3.2.m2.1.1.1.3.cmml"><mi id="S2.SS2.p3.2.m2.1.1.1.3.2" xref="S2.SS2.p3.2.m2.1.1.1.3.2.cmml">n</mi><mo id="S2.SS2.p3.2.m2.1.1.1.3.1" xref="S2.SS2.p3.2.m2.1.1.1.3.1.cmml">≥</mo><mn id="S2.SS2.p3.2.m2.1.1.1.3.3" xref="S2.SS2.p3.2.m2.1.1.1.3.3.cmml">0</mn></mrow></msub><mo id="S2.SS2.p3.2.m2.2.2.3" xref="S2.SS2.p3.2.m2.2.2.3.cmml">⊆</mo><mrow id="S2.SS2.p3.2.m2.2.2.2" xref="S2.SS2.p3.2.m2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.2.m2.2.2.2.3" xref="S2.SS2.p3.2.m2.2.2.2.3.cmml">𝒮</mi><mo id="S2.SS2.p3.2.m2.2.2.2.2" xref="S2.SS2.p3.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p3.2.m2.2.2.2.1.1" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.cmml"><mo id="S2.SS2.p3.2.m2.2.2.2.1.1.2" stretchy="false" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.cmml">(</mo><msup id="S2.SS2.p3.2.m2.2.2.2.1.1.1" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.cmml"><mi id="S2.SS2.p3.2.m2.2.2.2.1.1.1.2" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p3.2.m2.2.2.2.1.1.1.3" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p3.2.m2.2.2.2.1.1.3" stretchy="false" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.2.m2.2b"><apply id="S2.SS2.p3.2.m2.2.2.cmml" xref="S2.SS2.p3.2.m2.2.2"><subset id="S2.SS2.p3.2.m2.2.2.3.cmml" xref="S2.SS2.p3.2.m2.2.2.3"></subset><apply id="S2.SS2.p3.2.m2.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.2.m2.1.1.1.2.cmml" xref="S2.SS2.p3.2.m2.1.1.1">subscript</csymbol><apply id="S2.SS2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p3.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.2">𝜑</ci><ci id="S2.SS2.p3.2.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p3.2.m2.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S2.SS2.p3.2.m2.1.1.1.3.cmml" xref="S2.SS2.p3.2.m2.1.1.1.3"><geq id="S2.SS2.p3.2.m2.1.1.1.3.1.cmml" xref="S2.SS2.p3.2.m2.1.1.1.3.1"></geq><ci id="S2.SS2.p3.2.m2.1.1.1.3.2.cmml" xref="S2.SS2.p3.2.m2.1.1.1.3.2">𝑛</ci><cn id="S2.SS2.p3.2.m2.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p3.2.m2.1.1.1.3.3">0</cn></apply></apply><apply id="S2.SS2.p3.2.m2.2.2.2.cmml" xref="S2.SS2.p3.2.m2.2.2.2"><times id="S2.SS2.p3.2.m2.2.2.2.2.cmml" xref="S2.SS2.p3.2.m2.2.2.2.2"></times><ci id="S2.SS2.p3.2.m2.2.2.2.3.cmml" xref="S2.SS2.p3.2.m2.2.2.2.3">𝒮</ci><apply id="S2.SS2.p3.2.m2.2.2.2.1.1.1.cmml" xref="S2.SS2.p3.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.2.m2.2.2.2.1.1.1.1.cmml" xref="S2.SS2.p3.2.m2.2.2.2.1.1">superscript</csymbol><ci id="S2.SS2.p3.2.m2.2.2.2.1.1.1.2.cmml" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS2.p3.2.m2.2.2.2.1.1.1.3.cmml" xref="S2.SS2.p3.2.m2.2.2.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.2.m2.2c">(\varphi_{n})_{n\geq 0}\subseteq\mathcal{S}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.2.m2.2d">( italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ⊆ caligraphic_S ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> such that <math alttext="\widehat{\varphi}_{0}:=\widehat{\varphi}" class="ltx_Math" display="inline" id="S2.SS2.p3.3.m3.1"><semantics id="S2.SS2.p3.3.m3.1a"><mrow id="S2.SS2.p3.3.m3.1.1" xref="S2.SS2.p3.3.m3.1.1.cmml"><msub id="S2.SS2.p3.3.m3.1.1.2" xref="S2.SS2.p3.3.m3.1.1.2.cmml"><mover accent="true" id="S2.SS2.p3.3.m3.1.1.2.2" xref="S2.SS2.p3.3.m3.1.1.2.2.cmml"><mi id="S2.SS2.p3.3.m3.1.1.2.2.2" xref="S2.SS2.p3.3.m3.1.1.2.2.2.cmml">φ</mi><mo id="S2.SS2.p3.3.m3.1.1.2.2.1" xref="S2.SS2.p3.3.m3.1.1.2.2.1.cmml">^</mo></mover><mn id="S2.SS2.p3.3.m3.1.1.2.3" xref="S2.SS2.p3.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS2.p3.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p3.3.m3.1.1.1.cmml">:=</mo><mover accent="true" id="S2.SS2.p3.3.m3.1.1.3" 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xref="S2.SS2.p3.4.m4.3.3.2.2.1.1.1.2.3.2.2">𝑛</ci></apply><cn id="S2.SS2.p3.4.m4.3.3.2.2.1.1.1.2.3.3.cmml" type="integer" xref="S2.SS2.p3.4.m4.3.3.2.2.1.1.1.2.3.3">1</cn></apply></apply><ci id="S2.SS2.p3.4.m4.3.3.2.2.1.1.1.3.cmml" xref="S2.SS2.p3.4.m4.3.3.2.2.1.1.1.3">𝜉</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.4.m4.3c">\widehat{\varphi}_{n}(\xi):=\widehat{\varphi}(2^{-n}\xi)-\widehat{\varphi}(2^{% -n+1}\xi)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.4.m4.3d">over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_ξ ) := over^ start_ARG italic_φ end_ARG ( 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT italic_ξ ) - over^ start_ARG italic_φ end_ARG ( 2 start_POSTSUPERSCRIPT - italic_n + 1 end_POSTSUPERSCRIPT italic_ξ )</annotation></semantics></math> for <math alttext="\xi\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S2.SS2.p3.5.m5.1"><semantics id="S2.SS2.p3.5.m5.1a"><mrow id="S2.SS2.p3.5.m5.1.1" xref="S2.SS2.p3.5.m5.1.1.cmml"><mi id="S2.SS2.p3.5.m5.1.1.2" xref="S2.SS2.p3.5.m5.1.1.2.cmml">ξ</mi><mo id="S2.SS2.p3.5.m5.1.1.1" xref="S2.SS2.p3.5.m5.1.1.1.cmml">∈</mo><msup id="S2.SS2.p3.5.m5.1.1.3" xref="S2.SS2.p3.5.m5.1.1.3.cmml"><mi id="S2.SS2.p3.5.m5.1.1.3.2" xref="S2.SS2.p3.5.m5.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS2.p3.5.m5.1.1.3.3" xref="S2.SS2.p3.5.m5.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.5.m5.1b"><apply id="S2.SS2.p3.5.m5.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1"><in id="S2.SS2.p3.5.m5.1.1.1.cmml" xref="S2.SS2.p3.5.m5.1.1.1"></in><ci id="S2.SS2.p3.5.m5.1.1.2.cmml" xref="S2.SS2.p3.5.m5.1.1.2">𝜉</ci><apply id="S2.SS2.p3.5.m5.1.1.3.cmml" xref="S2.SS2.p3.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.5.m5.1.1.3.1.cmml" xref="S2.SS2.p3.5.m5.1.1.3">superscript</csymbol><ci id="S2.SS2.p3.5.m5.1.1.3.2.cmml" xref="S2.SS2.p3.5.m5.1.1.3.2">ℝ</ci><ci id="S2.SS2.p3.5.m5.1.1.3.3.cmml" xref="S2.SS2.p3.5.m5.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.5.m5.1c">\xi\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.5.m5.1d">italic_ξ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="n\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S2.SS2.p3.6.m6.1"><semantics id="S2.SS2.p3.6.m6.1a"><mrow id="S2.SS2.p3.6.m6.1.1" xref="S2.SS2.p3.6.m6.1.1.cmml"><mi id="S2.SS2.p3.6.m6.1.1.2" xref="S2.SS2.p3.6.m6.1.1.2.cmml">n</mi><mo id="S2.SS2.p3.6.m6.1.1.1" xref="S2.SS2.p3.6.m6.1.1.1.cmml">∈</mo><msub id="S2.SS2.p3.6.m6.1.1.3" xref="S2.SS2.p3.6.m6.1.1.3.cmml"><mi id="S2.SS2.p3.6.m6.1.1.3.2" xref="S2.SS2.p3.6.m6.1.1.3.2.cmml">ℕ</mi><mn id="S2.SS2.p3.6.m6.1.1.3.3" xref="S2.SS2.p3.6.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.6.m6.1b"><apply id="S2.SS2.p3.6.m6.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1"><in id="S2.SS2.p3.6.m6.1.1.1.cmml" xref="S2.SS2.p3.6.m6.1.1.1"></in><ci id="S2.SS2.p3.6.m6.1.1.2.cmml" xref="S2.SS2.p3.6.m6.1.1.2">𝑛</ci><apply id="S2.SS2.p3.6.m6.1.1.3.cmml" xref="S2.SS2.p3.6.m6.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.6.m6.1.1.3.1.cmml" xref="S2.SS2.p3.6.m6.1.1.3">subscript</csymbol><ci id="S2.SS2.p3.6.m6.1.1.3.2.cmml" xref="S2.SS2.p3.6.m6.1.1.3.2">ℕ</ci><cn id="S2.SS2.p3.6.m6.1.1.3.3.cmml" type="integer" xref="S2.SS2.p3.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.6.m6.1c">n\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.6.m6.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, where the generating function <math alttext="\varphi" class="ltx_Math" display="inline" id="S2.SS2.p3.7.m7.1"><semantics id="S2.SS2.p3.7.m7.1a"><mi id="S2.SS2.p3.7.m7.1.1" xref="S2.SS2.p3.7.m7.1.1.cmml">φ</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.7.m7.1b"><ci id="S2.SS2.p3.7.m7.1.1.cmml" xref="S2.SS2.p3.7.m7.1.1">𝜑</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.7.m7.1c">\varphi</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.7.m7.1d">italic_φ</annotation></semantics></math> satisfies <math alttext="0\leq\widehat{\varphi}(\xi)\leq 1" class="ltx_Math" display="inline" id="S2.SS2.p3.8.m8.1"><semantics id="S2.SS2.p3.8.m8.1a"><mrow id="S2.SS2.p3.8.m8.1.2" xref="S2.SS2.p3.8.m8.1.2.cmml"><mn id="S2.SS2.p3.8.m8.1.2.2" xref="S2.SS2.p3.8.m8.1.2.2.cmml">0</mn><mo id="S2.SS2.p3.8.m8.1.2.3" xref="S2.SS2.p3.8.m8.1.2.3.cmml">≤</mo><mrow id="S2.SS2.p3.8.m8.1.2.4" xref="S2.SS2.p3.8.m8.1.2.4.cmml"><mover accent="true" id="S2.SS2.p3.8.m8.1.2.4.2" xref="S2.SS2.p3.8.m8.1.2.4.2.cmml"><mi id="S2.SS2.p3.8.m8.1.2.4.2.2" xref="S2.SS2.p3.8.m8.1.2.4.2.2.cmml">φ</mi><mo id="S2.SS2.p3.8.m8.1.2.4.2.1" xref="S2.SS2.p3.8.m8.1.2.4.2.1.cmml">^</mo></mover><mo id="S2.SS2.p3.8.m8.1.2.4.1" xref="S2.SS2.p3.8.m8.1.2.4.1.cmml">⁢</mo><mrow id="S2.SS2.p3.8.m8.1.2.4.3.2" xref="S2.SS2.p3.8.m8.1.2.4.cmml"><mo id="S2.SS2.p3.8.m8.1.2.4.3.2.1" stretchy="false" xref="S2.SS2.p3.8.m8.1.2.4.cmml">(</mo><mi id="S2.SS2.p3.8.m8.1.1" xref="S2.SS2.p3.8.m8.1.1.cmml">ξ</mi><mo id="S2.SS2.p3.8.m8.1.2.4.3.2.2" stretchy="false" xref="S2.SS2.p3.8.m8.1.2.4.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p3.8.m8.1.2.5" xref="S2.SS2.p3.8.m8.1.2.5.cmml">≤</mo><mn id="S2.SS2.p3.8.m8.1.2.6" xref="S2.SS2.p3.8.m8.1.2.6.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.8.m8.1b"><apply id="S2.SS2.p3.8.m8.1.2.cmml" xref="S2.SS2.p3.8.m8.1.2"><and id="S2.SS2.p3.8.m8.1.2a.cmml" xref="S2.SS2.p3.8.m8.1.2"></and><apply id="S2.SS2.p3.8.m8.1.2b.cmml" xref="S2.SS2.p3.8.m8.1.2"><leq id="S2.SS2.p3.8.m8.1.2.3.cmml" xref="S2.SS2.p3.8.m8.1.2.3"></leq><cn id="S2.SS2.p3.8.m8.1.2.2.cmml" type="integer" xref="S2.SS2.p3.8.m8.1.2.2">0</cn><apply id="S2.SS2.p3.8.m8.1.2.4.cmml" xref="S2.SS2.p3.8.m8.1.2.4"><times id="S2.SS2.p3.8.m8.1.2.4.1.cmml" xref="S2.SS2.p3.8.m8.1.2.4.1"></times><apply id="S2.SS2.p3.8.m8.1.2.4.2.cmml" xref="S2.SS2.p3.8.m8.1.2.4.2"><ci id="S2.SS2.p3.8.m8.1.2.4.2.1.cmml" xref="S2.SS2.p3.8.m8.1.2.4.2.1">^</ci><ci id="S2.SS2.p3.8.m8.1.2.4.2.2.cmml" xref="S2.SS2.p3.8.m8.1.2.4.2.2">𝜑</ci></apply><ci id="S2.SS2.p3.8.m8.1.1.cmml" xref="S2.SS2.p3.8.m8.1.1">𝜉</ci></apply></apply><apply id="S2.SS2.p3.8.m8.1.2c.cmml" xref="S2.SS2.p3.8.m8.1.2"><leq id="S2.SS2.p3.8.m8.1.2.5.cmml" xref="S2.SS2.p3.8.m8.1.2.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S2.SS2.p3.8.m8.1.2.4.cmml" id="S2.SS2.p3.8.m8.1.2d.cmml" xref="S2.SS2.p3.8.m8.1.2"></share><cn id="S2.SS2.p3.8.m8.1.2.6.cmml" type="integer" xref="S2.SS2.p3.8.m8.1.2.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.8.m8.1c">0\leq\widehat{\varphi}(\xi)\leq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.8.m8.1d">0 ≤ over^ start_ARG italic_φ end_ARG ( italic_ξ ) ≤ 1</annotation></semantics></math> for <math alttext="\xi\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S2.SS2.p3.9.m9.1"><semantics id="S2.SS2.p3.9.m9.1a"><mrow id="S2.SS2.p3.9.m9.1.1" xref="S2.SS2.p3.9.m9.1.1.cmml"><mi id="S2.SS2.p3.9.m9.1.1.2" xref="S2.SS2.p3.9.m9.1.1.2.cmml">ξ</mi><mo id="S2.SS2.p3.9.m9.1.1.1" xref="S2.SS2.p3.9.m9.1.1.1.cmml">∈</mo><msup id="S2.SS2.p3.9.m9.1.1.3" xref="S2.SS2.p3.9.m9.1.1.3.cmml"><mi id="S2.SS2.p3.9.m9.1.1.3.2" xref="S2.SS2.p3.9.m9.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS2.p3.9.m9.1.1.3.3" xref="S2.SS2.p3.9.m9.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.9.m9.1b"><apply id="S2.SS2.p3.9.m9.1.1.cmml" xref="S2.SS2.p3.9.m9.1.1"><in id="S2.SS2.p3.9.m9.1.1.1.cmml" xref="S2.SS2.p3.9.m9.1.1.1"></in><ci id="S2.SS2.p3.9.m9.1.1.2.cmml" xref="S2.SS2.p3.9.m9.1.1.2">𝜉</ci><apply id="S2.SS2.p3.9.m9.1.1.3.cmml" xref="S2.SS2.p3.9.m9.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p3.9.m9.1.1.3.1.cmml" xref="S2.SS2.p3.9.m9.1.1.3">superscript</csymbol><ci id="S2.SS2.p3.9.m9.1.1.3.2.cmml" xref="S2.SS2.p3.9.m9.1.1.3.2">ℝ</ci><ci id="S2.SS2.p3.9.m9.1.1.3.3.cmml" xref="S2.SS2.p3.9.m9.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.9.m9.1c">\xi\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.9.m9.1d">italic_ξ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, <math alttext="\widehat{\varphi}(\xi)=1" class="ltx_Math" display="inline" id="S2.SS2.p3.10.m10.1"><semantics id="S2.SS2.p3.10.m10.1a"><mrow id="S2.SS2.p3.10.m10.1.2" xref="S2.SS2.p3.10.m10.1.2.cmml"><mrow id="S2.SS2.p3.10.m10.1.2.2" xref="S2.SS2.p3.10.m10.1.2.2.cmml"><mover accent="true" id="S2.SS2.p3.10.m10.1.2.2.2" xref="S2.SS2.p3.10.m10.1.2.2.2.cmml"><mi id="S2.SS2.p3.10.m10.1.2.2.2.2" xref="S2.SS2.p3.10.m10.1.2.2.2.2.cmml">φ</mi><mo id="S2.SS2.p3.10.m10.1.2.2.2.1" xref="S2.SS2.p3.10.m10.1.2.2.2.1.cmml">^</mo></mover><mo id="S2.SS2.p3.10.m10.1.2.2.1" xref="S2.SS2.p3.10.m10.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p3.10.m10.1.2.2.3.2" xref="S2.SS2.p3.10.m10.1.2.2.cmml"><mo id="S2.SS2.p3.10.m10.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p3.10.m10.1.2.2.cmml">(</mo><mi id="S2.SS2.p3.10.m10.1.1" xref="S2.SS2.p3.10.m10.1.1.cmml">ξ</mi><mo id="S2.SS2.p3.10.m10.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p3.10.m10.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p3.10.m10.1.2.1" xref="S2.SS2.p3.10.m10.1.2.1.cmml">=</mo><mn id="S2.SS2.p3.10.m10.1.2.3" xref="S2.SS2.p3.10.m10.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.10.m10.1b"><apply id="S2.SS2.p3.10.m10.1.2.cmml" xref="S2.SS2.p3.10.m10.1.2"><eq id="S2.SS2.p3.10.m10.1.2.1.cmml" xref="S2.SS2.p3.10.m10.1.2.1"></eq><apply id="S2.SS2.p3.10.m10.1.2.2.cmml" xref="S2.SS2.p3.10.m10.1.2.2"><times id="S2.SS2.p3.10.m10.1.2.2.1.cmml" xref="S2.SS2.p3.10.m10.1.2.2.1"></times><apply id="S2.SS2.p3.10.m10.1.2.2.2.cmml" xref="S2.SS2.p3.10.m10.1.2.2.2"><ci id="S2.SS2.p3.10.m10.1.2.2.2.1.cmml" xref="S2.SS2.p3.10.m10.1.2.2.2.1">^</ci><ci id="S2.SS2.p3.10.m10.1.2.2.2.2.cmml" xref="S2.SS2.p3.10.m10.1.2.2.2.2">𝜑</ci></apply><ci id="S2.SS2.p3.10.m10.1.1.cmml" xref="S2.SS2.p3.10.m10.1.1">𝜉</ci></apply><cn id="S2.SS2.p3.10.m10.1.2.3.cmml" type="integer" xref="S2.SS2.p3.10.m10.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.10.m10.1c">\widehat{\varphi}(\xi)=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.10.m10.1d">over^ start_ARG italic_φ end_ARG ( italic_ξ ) = 1</annotation></semantics></math> for <math alttext="|\xi|\leq 1" class="ltx_Math" display="inline" id="S2.SS2.p3.11.m11.1"><semantics id="S2.SS2.p3.11.m11.1a"><mrow id="S2.SS2.p3.11.m11.1.2" xref="S2.SS2.p3.11.m11.1.2.cmml"><mrow id="S2.SS2.p3.11.m11.1.2.2.2" xref="S2.SS2.p3.11.m11.1.2.2.1.cmml"><mo id="S2.SS2.p3.11.m11.1.2.2.2.1" stretchy="false" xref="S2.SS2.p3.11.m11.1.2.2.1.1.cmml">|</mo><mi id="S2.SS2.p3.11.m11.1.1" xref="S2.SS2.p3.11.m11.1.1.cmml">ξ</mi><mo id="S2.SS2.p3.11.m11.1.2.2.2.2" stretchy="false" xref="S2.SS2.p3.11.m11.1.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS2.p3.11.m11.1.2.1" xref="S2.SS2.p3.11.m11.1.2.1.cmml">≤</mo><mn id="S2.SS2.p3.11.m11.1.2.3" xref="S2.SS2.p3.11.m11.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.11.m11.1b"><apply id="S2.SS2.p3.11.m11.1.2.cmml" xref="S2.SS2.p3.11.m11.1.2"><leq id="S2.SS2.p3.11.m11.1.2.1.cmml" xref="S2.SS2.p3.11.m11.1.2.1"></leq><apply id="S2.SS2.p3.11.m11.1.2.2.1.cmml" xref="S2.SS2.p3.11.m11.1.2.2.2"><abs id="S2.SS2.p3.11.m11.1.2.2.1.1.cmml" xref="S2.SS2.p3.11.m11.1.2.2.2.1"></abs><ci id="S2.SS2.p3.11.m11.1.1.cmml" xref="S2.SS2.p3.11.m11.1.1">𝜉</ci></apply><cn id="S2.SS2.p3.11.m11.1.2.3.cmml" type="integer" xref="S2.SS2.p3.11.m11.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.11.m11.1c">|\xi|\leq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.11.m11.1d">| italic_ξ | ≤ 1</annotation></semantics></math> and <math alttext="\widehat{\varphi}(\xi)=0" class="ltx_Math" display="inline" id="S2.SS2.p3.12.m12.1"><semantics id="S2.SS2.p3.12.m12.1a"><mrow id="S2.SS2.p3.12.m12.1.2" xref="S2.SS2.p3.12.m12.1.2.cmml"><mrow id="S2.SS2.p3.12.m12.1.2.2" xref="S2.SS2.p3.12.m12.1.2.2.cmml"><mover accent="true" id="S2.SS2.p3.12.m12.1.2.2.2" xref="S2.SS2.p3.12.m12.1.2.2.2.cmml"><mi id="S2.SS2.p3.12.m12.1.2.2.2.2" xref="S2.SS2.p3.12.m12.1.2.2.2.2.cmml">φ</mi><mo id="S2.SS2.p3.12.m12.1.2.2.2.1" xref="S2.SS2.p3.12.m12.1.2.2.2.1.cmml">^</mo></mover><mo id="S2.SS2.p3.12.m12.1.2.2.1" xref="S2.SS2.p3.12.m12.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS2.p3.12.m12.1.2.2.3.2" xref="S2.SS2.p3.12.m12.1.2.2.cmml"><mo id="S2.SS2.p3.12.m12.1.2.2.3.2.1" stretchy="false" xref="S2.SS2.p3.12.m12.1.2.2.cmml">(</mo><mi id="S2.SS2.p3.12.m12.1.1" xref="S2.SS2.p3.12.m12.1.1.cmml">ξ</mi><mo id="S2.SS2.p3.12.m12.1.2.2.3.2.2" stretchy="false" xref="S2.SS2.p3.12.m12.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS2.p3.12.m12.1.2.1" xref="S2.SS2.p3.12.m12.1.2.1.cmml">=</mo><mn id="S2.SS2.p3.12.m12.1.2.3" xref="S2.SS2.p3.12.m12.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.12.m12.1b"><apply id="S2.SS2.p3.12.m12.1.2.cmml" xref="S2.SS2.p3.12.m12.1.2"><eq id="S2.SS2.p3.12.m12.1.2.1.cmml" xref="S2.SS2.p3.12.m12.1.2.1"></eq><apply id="S2.SS2.p3.12.m12.1.2.2.cmml" xref="S2.SS2.p3.12.m12.1.2.2"><times id="S2.SS2.p3.12.m12.1.2.2.1.cmml" xref="S2.SS2.p3.12.m12.1.2.2.1"></times><apply id="S2.SS2.p3.12.m12.1.2.2.2.cmml" xref="S2.SS2.p3.12.m12.1.2.2.2"><ci id="S2.SS2.p3.12.m12.1.2.2.2.1.cmml" xref="S2.SS2.p3.12.m12.1.2.2.2.1">^</ci><ci id="S2.SS2.p3.12.m12.1.2.2.2.2.cmml" xref="S2.SS2.p3.12.m12.1.2.2.2.2">𝜑</ci></apply><ci id="S2.SS2.p3.12.m12.1.1.cmml" xref="S2.SS2.p3.12.m12.1.1">𝜉</ci></apply><cn id="S2.SS2.p3.12.m12.1.2.3.cmml" type="integer" xref="S2.SS2.p3.12.m12.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.12.m12.1c">\widehat{\varphi}(\xi)=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.12.m12.1d">over^ start_ARG italic_φ end_ARG ( italic_ξ ) = 0</annotation></semantics></math> for <math alttext="|\xi|\geq\frac{3}{2}" class="ltx_Math" display="inline" id="S2.SS2.p3.13.m13.1"><semantics id="S2.SS2.p3.13.m13.1a"><mrow id="S2.SS2.p3.13.m13.1.2" xref="S2.SS2.p3.13.m13.1.2.cmml"><mrow id="S2.SS2.p3.13.m13.1.2.2.2" xref="S2.SS2.p3.13.m13.1.2.2.1.cmml"><mo id="S2.SS2.p3.13.m13.1.2.2.2.1" stretchy="false" xref="S2.SS2.p3.13.m13.1.2.2.1.1.cmml">|</mo><mi id="S2.SS2.p3.13.m13.1.1" xref="S2.SS2.p3.13.m13.1.1.cmml">ξ</mi><mo id="S2.SS2.p3.13.m13.1.2.2.2.2" stretchy="false" xref="S2.SS2.p3.13.m13.1.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS2.p3.13.m13.1.2.1" xref="S2.SS2.p3.13.m13.1.2.1.cmml">≥</mo><mfrac id="S2.SS2.p3.13.m13.1.2.3" xref="S2.SS2.p3.13.m13.1.2.3.cmml"><mn id="S2.SS2.p3.13.m13.1.2.3.2" xref="S2.SS2.p3.13.m13.1.2.3.2.cmml">3</mn><mn id="S2.SS2.p3.13.m13.1.2.3.3" xref="S2.SS2.p3.13.m13.1.2.3.3.cmml">2</mn></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.13.m13.1b"><apply id="S2.SS2.p3.13.m13.1.2.cmml" xref="S2.SS2.p3.13.m13.1.2"><geq id="S2.SS2.p3.13.m13.1.2.1.cmml" xref="S2.SS2.p3.13.m13.1.2.1"></geq><apply id="S2.SS2.p3.13.m13.1.2.2.1.cmml" xref="S2.SS2.p3.13.m13.1.2.2.2"><abs id="S2.SS2.p3.13.m13.1.2.2.1.1.cmml" xref="S2.SS2.p3.13.m13.1.2.2.2.1"></abs><ci id="S2.SS2.p3.13.m13.1.1.cmml" xref="S2.SS2.p3.13.m13.1.1">𝜉</ci></apply><apply id="S2.SS2.p3.13.m13.1.2.3.cmml" xref="S2.SS2.p3.13.m13.1.2.3"><divide id="S2.SS2.p3.13.m13.1.2.3.1.cmml" xref="S2.SS2.p3.13.m13.1.2.3"></divide><cn id="S2.SS2.p3.13.m13.1.2.3.2.cmml" type="integer" xref="S2.SS2.p3.13.m13.1.2.3.2">3</cn><cn id="S2.SS2.p3.13.m13.1.2.3.3.cmml" type="integer" xref="S2.SS2.p3.13.m13.1.2.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.13.m13.1c">|\xi|\geq\frac{3}{2}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.13.m13.1d">| italic_ξ | ≥ divide start_ARG 3 end_ARG start_ARG 2 end_ARG</annotation></semantics></math>. For <math alttext="f\in\mathcal{S}^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS2.p3.14.m14.2"><semantics id="S2.SS2.p3.14.m14.2a"><mrow id="S2.SS2.p3.14.m14.2.2" xref="S2.SS2.p3.14.m14.2.2.cmml"><mi id="S2.SS2.p3.14.m14.2.2.3" xref="S2.SS2.p3.14.m14.2.2.3.cmml">f</mi><mo id="S2.SS2.p3.14.m14.2.2.2" xref="S2.SS2.p3.14.m14.2.2.2.cmml">∈</mo><mrow id="S2.SS2.p3.14.m14.2.2.1" xref="S2.SS2.p3.14.m14.2.2.1.cmml"><msup id="S2.SS2.p3.14.m14.2.2.1.3" xref="S2.SS2.p3.14.m14.2.2.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p3.14.m14.2.2.1.3.2" xref="S2.SS2.p3.14.m14.2.2.1.3.2.cmml">𝒮</mi><mo id="S2.SS2.p3.14.m14.2.2.1.3.3" xref="S2.SS2.p3.14.m14.2.2.1.3.3.cmml">′</mo></msup><mo id="S2.SS2.p3.14.m14.2.2.1.2" xref="S2.SS2.p3.14.m14.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS2.p3.14.m14.2.2.1.1.1" xref="S2.SS2.p3.14.m14.2.2.1.1.2.cmml"><mo id="S2.SS2.p3.14.m14.2.2.1.1.1.2" stretchy="false" xref="S2.SS2.p3.14.m14.2.2.1.1.2.cmml">(</mo><msup id="S2.SS2.p3.14.m14.2.2.1.1.1.1" xref="S2.SS2.p3.14.m14.2.2.1.1.1.1.cmml"><mi id="S2.SS2.p3.14.m14.2.2.1.1.1.1.2" xref="S2.SS2.p3.14.m14.2.2.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p3.14.m14.2.2.1.1.1.1.3" xref="S2.SS2.p3.14.m14.2.2.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p3.14.m14.2.2.1.1.1.3" xref="S2.SS2.p3.14.m14.2.2.1.1.2.cmml">;</mo><mi id="S2.SS2.p3.14.m14.1.1" xref="S2.SS2.p3.14.m14.1.1.cmml">X</mi><mo id="S2.SS2.p3.14.m14.2.2.1.1.1.4" stretchy="false" xref="S2.SS2.p3.14.m14.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.14.m14.2b"><apply id="S2.SS2.p3.14.m14.2.2.cmml" xref="S2.SS2.p3.14.m14.2.2"><in id="S2.SS2.p3.14.m14.2.2.2.cmml" xref="S2.SS2.p3.14.m14.2.2.2"></in><ci id="S2.SS2.p3.14.m14.2.2.3.cmml" xref="S2.SS2.p3.14.m14.2.2.3">𝑓</ci><apply id="S2.SS2.p3.14.m14.2.2.1.cmml" xref="S2.SS2.p3.14.m14.2.2.1"><times id="S2.SS2.p3.14.m14.2.2.1.2.cmml" xref="S2.SS2.p3.14.m14.2.2.1.2"></times><apply 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id="S2.SS2.p3.14.m14.2c">f\in\mathcal{S}^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.14.m14.2d">italic_f ∈ caligraphic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> we use the notation</p> <table class="ltx_equation ltx_eqn_table" id="S2.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S_{n}f:=\varphi_{n}\ast f\quad\text{ for }n\in\mathbb{N}_{0}\quad\text{ and }% \quad S_{-1}f=0," class="ltx_Math" display="block" id="S2.E2.m1.2"><semantics id="S2.E2.m1.2a"><mrow id="S2.E2.m1.2.2.1"><mrow id="S2.E2.m1.2.2.1.1.2" xref="S2.E2.m1.2.2.1.1.3.cmml"><mrow id="S2.E2.m1.2.2.1.1.1.1" xref="S2.E2.m1.2.2.1.1.1.1.cmml"><mrow id="S2.E2.m1.2.2.1.1.1.1.2" xref="S2.E2.m1.2.2.1.1.1.1.2.cmml"><msub 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xref="S2.E2.m1.2.2.1.1.2.2.2.2.2.2">subscript</csymbol><ci id="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.2.cmml" xref="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.2">𝑆</ci><apply id="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.3.cmml" xref="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.3"><minus id="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.3.1.cmml" xref="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.3"></minus><cn id="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.3.2.cmml" type="integer" xref="S2.E2.m1.2.2.1.1.2.2.2.2.2.2.3.2">1</cn></apply></apply><ci id="S2.E2.m1.2.2.1.1.2.2.2.2.2.3.cmml" xref="S2.E2.m1.2.2.1.1.2.2.2.2.2.3">𝑓</ci></apply><cn id="S2.E2.m1.2.2.1.1.2.2.2.2.3.cmml" type="integer" xref="S2.E2.m1.2.2.1.1.2.2.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E2.m1.2c">S_{n}f:=\varphi_{n}\ast f\quad\text{ for }n\in\mathbb{N}_{0}\quad\text{ and }% \quad S_{-1}f=0,</annotation><annotation encoding="application/x-llamapun" id="S2.E2.m1.2d">italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f := italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_f for italic_n ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT and italic_S start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_f = 0 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p3.15">where <math alttext="(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS2.p3.15.m1.2"><semantics id="S2.SS2.p3.15.m1.2a"><mrow id="S2.SS2.p3.15.m1.2.2" xref="S2.SS2.p3.15.m1.2.2.cmml"><msub id="S2.SS2.p3.15.m1.1.1.1" xref="S2.SS2.p3.15.m1.1.1.1.cmml"><mrow id="S2.SS2.p3.15.m1.1.1.1.1.1" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p3.15.m1.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS2.p3.15.m1.1.1.1.1.1.1" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p3.15.m1.1.1.1.1.1.1.2" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.2.cmml">φ</mi><mi id="S2.SS2.p3.15.m1.1.1.1.1.1.1.3" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.SS2.p3.15.m1.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS2.p3.15.m1.1.1.1.3" xref="S2.SS2.p3.15.m1.1.1.1.3.cmml"><mi id="S2.SS2.p3.15.m1.1.1.1.3.2" xref="S2.SS2.p3.15.m1.1.1.1.3.2.cmml">n</mi><mo id="S2.SS2.p3.15.m1.1.1.1.3.1" xref="S2.SS2.p3.15.m1.1.1.1.3.1.cmml">≥</mo><mn id="S2.SS2.p3.15.m1.1.1.1.3.3" xref="S2.SS2.p3.15.m1.1.1.1.3.3.cmml">0</mn></mrow></msub><mo id="S2.SS2.p3.15.m1.2.2.3" xref="S2.SS2.p3.15.m1.2.2.3.cmml">∈</mo><mrow id="S2.SS2.p3.15.m1.2.2.2" xref="S2.SS2.p3.15.m1.2.2.2.cmml"><mi id="S2.SS2.p3.15.m1.2.2.2.3" mathvariant="normal" xref="S2.SS2.p3.15.m1.2.2.2.3.cmml">Φ</mi><mo id="S2.SS2.p3.15.m1.2.2.2.2" xref="S2.SS2.p3.15.m1.2.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p3.15.m1.2.2.2.1.1" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.cmml"><mo id="S2.SS2.p3.15.m1.2.2.2.1.1.2" stretchy="false" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.cmml">(</mo><msup id="S2.SS2.p3.15.m1.2.2.2.1.1.1" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.cmml"><mi id="S2.SS2.p3.15.m1.2.2.2.1.1.1.2" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p3.15.m1.2.2.2.1.1.1.3" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p3.15.m1.2.2.2.1.1.3" stretchy="false" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p3.15.m1.2b"><apply id="S2.SS2.p3.15.m1.2.2.cmml" xref="S2.SS2.p3.15.m1.2.2"><in id="S2.SS2.p3.15.m1.2.2.3.cmml" xref="S2.SS2.p3.15.m1.2.2.3"></in><apply id="S2.SS2.p3.15.m1.1.1.1.cmml" xref="S2.SS2.p3.15.m1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.15.m1.1.1.1.2.cmml" xref="S2.SS2.p3.15.m1.1.1.1">subscript</csymbol><apply id="S2.SS2.p3.15.m1.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.15.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.15.m1.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p3.15.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS2.p3.15.m1.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.2">𝜑</ci><ci id="S2.SS2.p3.15.m1.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p3.15.m1.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S2.SS2.p3.15.m1.1.1.1.3.cmml" xref="S2.SS2.p3.15.m1.1.1.1.3"><geq id="S2.SS2.p3.15.m1.1.1.1.3.1.cmml" xref="S2.SS2.p3.15.m1.1.1.1.3.1"></geq><ci id="S2.SS2.p3.15.m1.1.1.1.3.2.cmml" xref="S2.SS2.p3.15.m1.1.1.1.3.2">𝑛</ci><cn id="S2.SS2.p3.15.m1.1.1.1.3.3.cmml" type="integer" xref="S2.SS2.p3.15.m1.1.1.1.3.3">0</cn></apply></apply><apply id="S2.SS2.p3.15.m1.2.2.2.cmml" xref="S2.SS2.p3.15.m1.2.2.2"><times id="S2.SS2.p3.15.m1.2.2.2.2.cmml" xref="S2.SS2.p3.15.m1.2.2.2.2"></times><ci id="S2.SS2.p3.15.m1.2.2.2.3.cmml" xref="S2.SS2.p3.15.m1.2.2.2.3">Φ</ci><apply id="S2.SS2.p3.15.m1.2.2.2.1.1.1.cmml" xref="S2.SS2.p3.15.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS2.p3.15.m1.2.2.2.1.1.1.1.cmml" xref="S2.SS2.p3.15.m1.2.2.2.1.1">superscript</csymbol><ci id="S2.SS2.p3.15.m1.2.2.2.1.1.1.2.cmml" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS2.p3.15.m1.2.2.2.1.1.1.3.cmml" xref="S2.SS2.p3.15.m1.2.2.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p3.15.m1.2c">(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p3.15.m1.2d">( italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS2.p4"> <p class="ltx_p" id="S2.SS2.p4.6">We record the following weighted and vector-valued extension of the multiplier result in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib68" title="">68</a>, Section 1.6.3]</cite>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 2.4]</cite>. Let <math alttext="p\in[1,\infty)" class="ltx_Math" display="inline" id="S2.SS2.p4.1.m1.2"><semantics id="S2.SS2.p4.1.m1.2a"><mrow id="S2.SS2.p4.1.m1.2.3" xref="S2.SS2.p4.1.m1.2.3.cmml"><mi id="S2.SS2.p4.1.m1.2.3.2" xref="S2.SS2.p4.1.m1.2.3.2.cmml">p</mi><mo id="S2.SS2.p4.1.m1.2.3.1" xref="S2.SS2.p4.1.m1.2.3.1.cmml">∈</mo><mrow id="S2.SS2.p4.1.m1.2.3.3.2" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml"><mo id="S2.SS2.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml">[</mo><mn id="S2.SS2.p4.1.m1.1.1" xref="S2.SS2.p4.1.m1.1.1.cmml">1</mn><mo id="S2.SS2.p4.1.m1.2.3.3.2.2" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS2.p4.1.m1.2.2" mathvariant="normal" xref="S2.SS2.p4.1.m1.2.2.cmml">∞</mi><mo id="S2.SS2.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS2.p4.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.1.m1.2b"><apply id="S2.SS2.p4.1.m1.2.3.cmml" xref="S2.SS2.p4.1.m1.2.3"><in id="S2.SS2.p4.1.m1.2.3.1.cmml" xref="S2.SS2.p4.1.m1.2.3.1"></in><ci id="S2.SS2.p4.1.m1.2.3.2.cmml" xref="S2.SS2.p4.1.m1.2.3.2">𝑝</ci><interval closure="closed-open" id="S2.SS2.p4.1.m1.2.3.3.1.cmml" xref="S2.SS2.p4.1.m1.2.3.3.2"><cn id="S2.SS2.p4.1.m1.1.1.cmml" type="integer" xref="S2.SS2.p4.1.m1.1.1">1</cn><infinity id="S2.SS2.p4.1.m1.2.2.cmml" xref="S2.SS2.p4.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.1.m1.2c">p\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.1.m1.2d">italic_p ∈ [ 1 , ∞ )</annotation></semantics></math>, <math alttext="w\in A_{\infty}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS2.p4.2.m2.1"><semantics id="S2.SS2.p4.2.m2.1a"><mrow id="S2.SS2.p4.2.m2.1.1" xref="S2.SS2.p4.2.m2.1.1.cmml"><mi id="S2.SS2.p4.2.m2.1.1.3" xref="S2.SS2.p4.2.m2.1.1.3.cmml">w</mi><mo id="S2.SS2.p4.2.m2.1.1.2" xref="S2.SS2.p4.2.m2.1.1.2.cmml">∈</mo><mrow id="S2.SS2.p4.2.m2.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.cmml"><msub id="S2.SS2.p4.2.m2.1.1.1.3" xref="S2.SS2.p4.2.m2.1.1.1.3.cmml"><mi id="S2.SS2.p4.2.m2.1.1.1.3.2" xref="S2.SS2.p4.2.m2.1.1.1.3.2.cmml">A</mi><mi id="S2.SS2.p4.2.m2.1.1.1.3.3" mathvariant="normal" xref="S2.SS2.p4.2.m2.1.1.1.3.3.cmml">∞</mi></msub><mo id="S2.SS2.p4.2.m2.1.1.1.2" xref="S2.SS2.p4.2.m2.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.2.m2.1.1.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p4.2.m2.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS2.p4.2.m2.1.1.1.1.1.1" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.2.m2.1.1.1.1.1.1.2" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p4.2.m2.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.2.m2.1b"><apply id="S2.SS2.p4.2.m2.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1"><in id="S2.SS2.p4.2.m2.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.2"></in><ci id="S2.SS2.p4.2.m2.1.1.3.cmml" xref="S2.SS2.p4.2.m2.1.1.3">𝑤</ci><apply id="S2.SS2.p4.2.m2.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1"><times id="S2.SS2.p4.2.m2.1.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1.2"></times><apply id="S2.SS2.p4.2.m2.1.1.1.3.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m2.1.1.1.3.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.2.m2.1.1.1.3.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3.2">𝐴</ci><infinity id="S2.SS2.p4.2.m2.1.1.1.3.3.cmml" xref="S2.SS2.p4.2.m2.1.1.1.3.3"></infinity></apply><apply id="S2.SS2.p4.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.2.m2.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.2.m2.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p4.2.m2.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.2.m2.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.2.m2.1c">w\in A_{\infty}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.2.m2.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S2.SS2.p4.3.m3.1"><semantics id="S2.SS2.p4.3.m3.1a"><mi id="S2.SS2.p4.3.m3.1.1" xref="S2.SS2.p4.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.3.m3.1b"><ci id="S2.SS2.p4.3.m3.1.1.cmml" xref="S2.SS2.p4.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.3.m3.1d">italic_X</annotation></semantics></math> a Banach space. Fix <math alttext="n\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S2.SS2.p4.4.m4.1"><semantics id="S2.SS2.p4.4.m4.1a"><mrow id="S2.SS2.p4.4.m4.1.1" xref="S2.SS2.p4.4.m4.1.1.cmml"><mi id="S2.SS2.p4.4.m4.1.1.2" xref="S2.SS2.p4.4.m4.1.1.2.cmml">n</mi><mo id="S2.SS2.p4.4.m4.1.1.1" xref="S2.SS2.p4.4.m4.1.1.1.cmml">∈</mo><msub id="S2.SS2.p4.4.m4.1.1.3" xref="S2.SS2.p4.4.m4.1.1.3.cmml"><mi id="S2.SS2.p4.4.m4.1.1.3.2" xref="S2.SS2.p4.4.m4.1.1.3.2.cmml">ℕ</mi><mn id="S2.SS2.p4.4.m4.1.1.3.3" xref="S2.SS2.p4.4.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.4.m4.1b"><apply id="S2.SS2.p4.4.m4.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1"><in id="S2.SS2.p4.4.m4.1.1.1.cmml" xref="S2.SS2.p4.4.m4.1.1.1"></in><ci id="S2.SS2.p4.4.m4.1.1.2.cmml" xref="S2.SS2.p4.4.m4.1.1.2">𝑛</ci><apply id="S2.SS2.p4.4.m4.1.1.3.cmml" xref="S2.SS2.p4.4.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.4.m4.1.1.3.1.cmml" xref="S2.SS2.p4.4.m4.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.4.m4.1.1.3.2.cmml" xref="S2.SS2.p4.4.m4.1.1.3.2">ℕ</ci><cn id="S2.SS2.p4.4.m4.1.1.3.3.cmml" type="integer" xref="S2.SS2.p4.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.4.m4.1c">n\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.4.m4.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, then for all <math alttext="f\in L^{p}(\mathbb{R}^{d},w;X)" class="ltx_Math" display="inline" id="S2.SS2.p4.5.m5.3"><semantics id="S2.SS2.p4.5.m5.3a"><mrow id="S2.SS2.p4.5.m5.3.3" xref="S2.SS2.p4.5.m5.3.3.cmml"><mi id="S2.SS2.p4.5.m5.3.3.3" xref="S2.SS2.p4.5.m5.3.3.3.cmml">f</mi><mo id="S2.SS2.p4.5.m5.3.3.2" xref="S2.SS2.p4.5.m5.3.3.2.cmml">∈</mo><mrow id="S2.SS2.p4.5.m5.3.3.1" xref="S2.SS2.p4.5.m5.3.3.1.cmml"><msup id="S2.SS2.p4.5.m5.3.3.1.3" xref="S2.SS2.p4.5.m5.3.3.1.3.cmml"><mi id="S2.SS2.p4.5.m5.3.3.1.3.2" xref="S2.SS2.p4.5.m5.3.3.1.3.2.cmml">L</mi><mi id="S2.SS2.p4.5.m5.3.3.1.3.3" xref="S2.SS2.p4.5.m5.3.3.1.3.3.cmml">p</mi></msup><mo id="S2.SS2.p4.5.m5.3.3.1.2" xref="S2.SS2.p4.5.m5.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.5.m5.3.3.1.1.1" xref="S2.SS2.p4.5.m5.3.3.1.1.2.cmml"><mo id="S2.SS2.p4.5.m5.3.3.1.1.1.2" stretchy="false" xref="S2.SS2.p4.5.m5.3.3.1.1.2.cmml">(</mo><msup id="S2.SS2.p4.5.m5.3.3.1.1.1.1" xref="S2.SS2.p4.5.m5.3.3.1.1.1.1.cmml"><mi id="S2.SS2.p4.5.m5.3.3.1.1.1.1.2" xref="S2.SS2.p4.5.m5.3.3.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p4.5.m5.3.3.1.1.1.1.3" xref="S2.SS2.p4.5.m5.3.3.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p4.5.m5.3.3.1.1.1.3" xref="S2.SS2.p4.5.m5.3.3.1.1.2.cmml">,</mo><mi id="S2.SS2.p4.5.m5.1.1" xref="S2.SS2.p4.5.m5.1.1.cmml">w</mi><mo id="S2.SS2.p4.5.m5.3.3.1.1.1.4" xref="S2.SS2.p4.5.m5.3.3.1.1.2.cmml">;</mo><mi id="S2.SS2.p4.5.m5.2.2" xref="S2.SS2.p4.5.m5.2.2.cmml">X</mi><mo id="S2.SS2.p4.5.m5.3.3.1.1.1.5" stretchy="false" xref="S2.SS2.p4.5.m5.3.3.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.5.m5.3b"><apply id="S2.SS2.p4.5.m5.3.3.cmml" xref="S2.SS2.p4.5.m5.3.3"><in id="S2.SS2.p4.5.m5.3.3.2.cmml" xref="S2.SS2.p4.5.m5.3.3.2"></in><ci id="S2.SS2.p4.5.m5.3.3.3.cmml" xref="S2.SS2.p4.5.m5.3.3.3">𝑓</ci><apply id="S2.SS2.p4.5.m5.3.3.1.cmml" xref="S2.SS2.p4.5.m5.3.3.1"><times id="S2.SS2.p4.5.m5.3.3.1.2.cmml" xref="S2.SS2.p4.5.m5.3.3.1.2"></times><apply id="S2.SS2.p4.5.m5.3.3.1.3.cmml" xref="S2.SS2.p4.5.m5.3.3.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.3.3.1.3.1.cmml" xref="S2.SS2.p4.5.m5.3.3.1.3">superscript</csymbol><ci id="S2.SS2.p4.5.m5.3.3.1.3.2.cmml" xref="S2.SS2.p4.5.m5.3.3.1.3.2">𝐿</ci><ci id="S2.SS2.p4.5.m5.3.3.1.3.3.cmml" xref="S2.SS2.p4.5.m5.3.3.1.3.3">𝑝</ci></apply><vector id="S2.SS2.p4.5.m5.3.3.1.1.2.cmml" xref="S2.SS2.p4.5.m5.3.3.1.1.1"><apply id="S2.SS2.p4.5.m5.3.3.1.1.1.1.cmml" xref="S2.SS2.p4.5.m5.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.5.m5.3.3.1.1.1.1.1.cmml" xref="S2.SS2.p4.5.m5.3.3.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.5.m5.3.3.1.1.1.1.2.cmml" xref="S2.SS2.p4.5.m5.3.3.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p4.5.m5.3.3.1.1.1.1.3.cmml" xref="S2.SS2.p4.5.m5.3.3.1.1.1.1.3">𝑑</ci></apply><ci id="S2.SS2.p4.5.m5.1.1.cmml" xref="S2.SS2.p4.5.m5.1.1">𝑤</ci><ci id="S2.SS2.p4.5.m5.2.2.cmml" xref="S2.SS2.p4.5.m5.2.2">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.5.m5.3c">f\in L^{p}(\mathbb{R}^{d},w;X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.5.m5.3d">italic_f ∈ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X )</annotation></semantics></math> such that <math alttext="\text{\rm supp\,}\widehat{f}\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq 2^{n}K% \text{ for some }K>0\}" class="ltx_Math" display="inline" id="S2.SS2.p4.6.m6.3"><semantics id="S2.SS2.p4.6.m6.3a"><mrow id="S2.SS2.p4.6.m6.3.3" xref="S2.SS2.p4.6.m6.3.3.cmml"><mrow id="S2.SS2.p4.6.m6.3.3.4" xref="S2.SS2.p4.6.m6.3.3.4.cmml"><mtext id="S2.SS2.p4.6.m6.3.3.4.2" xref="S2.SS2.p4.6.m6.3.3.4.2a.cmml">supp </mtext><mo id="S2.SS2.p4.6.m6.3.3.4.1" xref="S2.SS2.p4.6.m6.3.3.4.1.cmml">⁢</mo><mover accent="true" id="S2.SS2.p4.6.m6.3.3.4.3" xref="S2.SS2.p4.6.m6.3.3.4.3.cmml"><mi id="S2.SS2.p4.6.m6.3.3.4.3.2" xref="S2.SS2.p4.6.m6.3.3.4.3.2.cmml">f</mi><mo id="S2.SS2.p4.6.m6.3.3.4.3.1" xref="S2.SS2.p4.6.m6.3.3.4.3.1.cmml">^</mo></mover></mrow><mo id="S2.SS2.p4.6.m6.3.3.3" xref="S2.SS2.p4.6.m6.3.3.3.cmml">⊆</mo><mrow id="S2.SS2.p4.6.m6.3.3.2.2" xref="S2.SS2.p4.6.m6.3.3.2.3.cmml"><mo id="S2.SS2.p4.6.m6.3.3.2.2.3" stretchy="false" xref="S2.SS2.p4.6.m6.3.3.2.3.1.cmml">{</mo><mrow id="S2.SS2.p4.6.m6.2.2.1.1.1" xref="S2.SS2.p4.6.m6.2.2.1.1.1.cmml"><mi id="S2.SS2.p4.6.m6.2.2.1.1.1.2" xref="S2.SS2.p4.6.m6.2.2.1.1.1.2.cmml">ξ</mi><mo id="S2.SS2.p4.6.m6.2.2.1.1.1.1" xref="S2.SS2.p4.6.m6.2.2.1.1.1.1.cmml">∈</mo><msup id="S2.SS2.p4.6.m6.2.2.1.1.1.3" xref="S2.SS2.p4.6.m6.2.2.1.1.1.3.cmml"><mi id="S2.SS2.p4.6.m6.2.2.1.1.1.3.2" xref="S2.SS2.p4.6.m6.2.2.1.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS2.p4.6.m6.2.2.1.1.1.3.3" xref="S2.SS2.p4.6.m6.2.2.1.1.1.3.3.cmml">d</mi></msup></mrow><mo id="S2.SS2.p4.6.m6.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.SS2.p4.6.m6.3.3.2.3.1.cmml">:</mo><mrow id="S2.SS2.p4.6.m6.3.3.2.2.2" xref="S2.SS2.p4.6.m6.3.3.2.2.2.cmml"><mrow id="S2.SS2.p4.6.m6.3.3.2.2.2.2.2" xref="S2.SS2.p4.6.m6.3.3.2.2.2.2.1.cmml"><mo id="S2.SS2.p4.6.m6.3.3.2.2.2.2.2.1" stretchy="false" xref="S2.SS2.p4.6.m6.3.3.2.2.2.2.1.1.cmml">|</mo><mi id="S2.SS2.p4.6.m6.1.1" xref="S2.SS2.p4.6.m6.1.1.cmml">ξ</mi><mo id="S2.SS2.p4.6.m6.3.3.2.2.2.2.2.2" stretchy="false" xref="S2.SS2.p4.6.m6.3.3.2.2.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS2.p4.6.m6.3.3.2.2.2.3" xref="S2.SS2.p4.6.m6.3.3.2.2.2.3.cmml">≤</mo><mrow 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xref="S2.SS2.p4.6.m6.3.3.2.2.2.6.cmml">0</mn></mrow><mo id="S2.SS2.p4.6.m6.3.3.2.2.5" stretchy="false" xref="S2.SS2.p4.6.m6.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.6.m6.3b"><apply id="S2.SS2.p4.6.m6.3.3.cmml" xref="S2.SS2.p4.6.m6.3.3"><subset id="S2.SS2.p4.6.m6.3.3.3.cmml" xref="S2.SS2.p4.6.m6.3.3.3"></subset><apply id="S2.SS2.p4.6.m6.3.3.4.cmml" xref="S2.SS2.p4.6.m6.3.3.4"><times id="S2.SS2.p4.6.m6.3.3.4.1.cmml" xref="S2.SS2.p4.6.m6.3.3.4.1"></times><ci id="S2.SS2.p4.6.m6.3.3.4.2a.cmml" xref="S2.SS2.p4.6.m6.3.3.4.2"><mtext id="S2.SS2.p4.6.m6.3.3.4.2.cmml" xref="S2.SS2.p4.6.m6.3.3.4.2">supp </mtext></ci><apply id="S2.SS2.p4.6.m6.3.3.4.3.cmml" xref="S2.SS2.p4.6.m6.3.3.4.3"><ci id="S2.SS2.p4.6.m6.3.3.4.3.1.cmml" xref="S2.SS2.p4.6.m6.3.3.4.3.1">^</ci><ci id="S2.SS2.p4.6.m6.3.3.4.3.2.cmml" xref="S2.SS2.p4.6.m6.3.3.4.3.2">𝑓</ci></apply></apply><apply id="S2.SS2.p4.6.m6.3.3.2.3.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2"><csymbol cd="latexml" 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xref="S2.SS2.p4.6.m6.3.3.2.2.2.3"></leq><apply id="S2.SS2.p4.6.m6.3.3.2.2.2.2.1.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.2.2"><abs id="S2.SS2.p4.6.m6.3.3.2.2.2.2.1.1.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.2.2.1"></abs><ci id="S2.SS2.p4.6.m6.1.1.cmml" xref="S2.SS2.p4.6.m6.1.1">𝜉</ci></apply><apply id="S2.SS2.p4.6.m6.3.3.2.2.2.4.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4"><times id="S2.SS2.p4.6.m6.3.3.2.2.2.4.1.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.1"></times><apply id="S2.SS2.p4.6.m6.3.3.2.2.2.4.2.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.2"><csymbol cd="ambiguous" id="S2.SS2.p4.6.m6.3.3.2.2.2.4.2.1.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.2">superscript</csymbol><cn id="S2.SS2.p4.6.m6.3.3.2.2.2.4.2.2.cmml" type="integer" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.2.2">2</cn><ci id="S2.SS2.p4.6.m6.3.3.2.2.2.4.2.3.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.2.3">𝑛</ci></apply><ci id="S2.SS2.p4.6.m6.3.3.2.2.2.4.3.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.3">𝐾</ci><ci id="S2.SS2.p4.6.m6.3.3.2.2.2.4.4a.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.4"><mtext id="S2.SS2.p4.6.m6.3.3.2.2.2.4.4.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.4"> for some </mtext></ci><ci id="S2.SS2.p4.6.m6.3.3.2.2.2.4.5.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.4.5">𝐾</ci></apply></apply><apply id="S2.SS2.p4.6.m6.3.3.2.2.2c.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2"><gt id="S2.SS2.p4.6.m6.3.3.2.2.2.5.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2.5"></gt><share href="https://arxiv.org/html/2503.14636v1#S2.SS2.p4.6.m6.3.3.2.2.2.4.cmml" id="S2.SS2.p4.6.m6.3.3.2.2.2d.cmml" xref="S2.SS2.p4.6.m6.3.3.2.2.2"></share><cn id="S2.SS2.p4.6.m6.3.3.2.2.2.6.cmml" type="integer" xref="S2.SS2.p4.6.m6.3.3.2.2.2.6">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.6.m6.3c">\text{\rm supp\,}\widehat{f}\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq 2^{n}K% \text{ for some }K>0\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.6.m6.3d">supp over^ start_ARG italic_f end_ARG ⊆ { italic_ξ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : | italic_ξ | ≤ 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_K for some italic_K > 0 }</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S2.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|S_{n}f\|_{L^{p}(\mathbb{R}^{d},w;X)}\leq C\|f\|_{L^{p}(\mathbb{R}^{d},w;X)}," class="ltx_Math" display="block" id="S2.E3.m1.8"><semantics id="S2.E3.m1.8a"><mrow id="S2.E3.m1.8.8.1" xref="S2.E3.m1.8.8.1.1.cmml"><mrow id="S2.E3.m1.8.8.1.1" xref="S2.E3.m1.8.8.1.1.cmml"><msub id="S2.E3.m1.8.8.1.1.1" xref="S2.E3.m1.8.8.1.1.1.cmml"><mrow id="S2.E3.m1.8.8.1.1.1.1.1" xref="S2.E3.m1.8.8.1.1.1.1.2.cmml"><mo id="S2.E3.m1.8.8.1.1.1.1.1.2" stretchy="false" xref="S2.E3.m1.8.8.1.1.1.1.2.1.cmml">‖</mo><mrow id="S2.E3.m1.8.8.1.1.1.1.1.1" xref="S2.E3.m1.8.8.1.1.1.1.1.1.cmml"><msub id="S2.E3.m1.8.8.1.1.1.1.1.1.2" xref="S2.E3.m1.8.8.1.1.1.1.1.1.2.cmml"><mi id="S2.E3.m1.8.8.1.1.1.1.1.1.2.2" xref="S2.E3.m1.8.8.1.1.1.1.1.1.2.2.cmml">S</mi><mi id="S2.E3.m1.8.8.1.1.1.1.1.1.2.3" xref="S2.E3.m1.8.8.1.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S2.E3.m1.8.8.1.1.1.1.1.1.1" xref="S2.E3.m1.8.8.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S2.E3.m1.8.8.1.1.1.1.1.1.3" xref="S2.E3.m1.8.8.1.1.1.1.1.1.3.cmml">f</mi></mrow><mo id="S2.E3.m1.8.8.1.1.1.1.1.3" stretchy="false" xref="S2.E3.m1.8.8.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S2.E3.m1.3.3.3" xref="S2.E3.m1.3.3.3.cmml"><msup id="S2.E3.m1.3.3.3.5" xref="S2.E3.m1.3.3.3.5.cmml"><mi id="S2.E3.m1.3.3.3.5.2" xref="S2.E3.m1.3.3.3.5.2.cmml">L</mi><mi id="S2.E3.m1.3.3.3.5.3" xref="S2.E3.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S2.E3.m1.3.3.3.4" xref="S2.E3.m1.3.3.3.4.cmml">⁢</mo><mrow id="S2.E3.m1.3.3.3.3.1" xref="S2.E3.m1.3.3.3.3.2.cmml"><mo id="S2.E3.m1.3.3.3.3.1.2" stretchy="false" xref="S2.E3.m1.3.3.3.3.2.cmml">(</mo><msup id="S2.E3.m1.3.3.3.3.1.1" xref="S2.E3.m1.3.3.3.3.1.1.cmml"><mi id="S2.E3.m1.3.3.3.3.1.1.2" xref="S2.E3.m1.3.3.3.3.1.1.2.cmml">ℝ</mi><mi id="S2.E3.m1.3.3.3.3.1.1.3" xref="S2.E3.m1.3.3.3.3.1.1.3.cmml">d</mi></msup><mo id="S2.E3.m1.3.3.3.3.1.3" xref="S2.E3.m1.3.3.3.3.2.cmml">,</mo><mi id="S2.E3.m1.1.1.1.1" xref="S2.E3.m1.1.1.1.1.cmml">w</mi><mo id="S2.E3.m1.3.3.3.3.1.4" xref="S2.E3.m1.3.3.3.3.2.cmml">;</mo><mi id="S2.E3.m1.2.2.2.2" xref="S2.E3.m1.2.2.2.2.cmml">X</mi><mo id="S2.E3.m1.3.3.3.3.1.5" stretchy="false" xref="S2.E3.m1.3.3.3.3.2.cmml">)</mo></mrow></mrow></msub><mo id="S2.E3.m1.8.8.1.1.2" xref="S2.E3.m1.8.8.1.1.2.cmml">≤</mo><mrow id="S2.E3.m1.8.8.1.1.3" xref="S2.E3.m1.8.8.1.1.3.cmml"><mi id="S2.E3.m1.8.8.1.1.3.2" xref="S2.E3.m1.8.8.1.1.3.2.cmml">C</mi><mo id="S2.E3.m1.8.8.1.1.3.1" xref="S2.E3.m1.8.8.1.1.3.1.cmml">⁢</mo><msub id="S2.E3.m1.8.8.1.1.3.3" xref="S2.E3.m1.8.8.1.1.3.3.cmml"><mrow id="S2.E3.m1.8.8.1.1.3.3.2.2" xref="S2.E3.m1.8.8.1.1.3.3.2.1.cmml"><mo id="S2.E3.m1.8.8.1.1.3.3.2.2.1" stretchy="false" xref="S2.E3.m1.8.8.1.1.3.3.2.1.1.cmml">‖</mo><mi id="S2.E3.m1.7.7" xref="S2.E3.m1.7.7.cmml">f</mi><mo id="S2.E3.m1.8.8.1.1.3.3.2.2.2" stretchy="false" xref="S2.E3.m1.8.8.1.1.3.3.2.1.1.cmml">‖</mo></mrow><mrow id="S2.E3.m1.6.6.3" xref="S2.E3.m1.6.6.3.cmml"><msup id="S2.E3.m1.6.6.3.5" xref="S2.E3.m1.6.6.3.5.cmml"><mi id="S2.E3.m1.6.6.3.5.2" xref="S2.E3.m1.6.6.3.5.2.cmml">L</mi><mi id="S2.E3.m1.6.6.3.5.3" xref="S2.E3.m1.6.6.3.5.3.cmml">p</mi></msup><mo id="S2.E3.m1.6.6.3.4" xref="S2.E3.m1.6.6.3.4.cmml">⁢</mo><mrow id="S2.E3.m1.6.6.3.3.1" xref="S2.E3.m1.6.6.3.3.2.cmml"><mo id="S2.E3.m1.6.6.3.3.1.2" stretchy="false" xref="S2.E3.m1.6.6.3.3.2.cmml">(</mo><msup id="S2.E3.m1.6.6.3.3.1.1" xref="S2.E3.m1.6.6.3.3.1.1.cmml"><mi id="S2.E3.m1.6.6.3.3.1.1.2" xref="S2.E3.m1.6.6.3.3.1.1.2.cmml">ℝ</mi><mi id="S2.E3.m1.6.6.3.3.1.1.3" xref="S2.E3.m1.6.6.3.3.1.1.3.cmml">d</mi></msup><mo id="S2.E3.m1.6.6.3.3.1.3" xref="S2.E3.m1.6.6.3.3.2.cmml">,</mo><mi id="S2.E3.m1.4.4.1.1" xref="S2.E3.m1.4.4.1.1.cmml">w</mi><mo id="S2.E3.m1.6.6.3.3.1.4" xref="S2.E3.m1.6.6.3.3.2.cmml">;</mo><mi id="S2.E3.m1.5.5.2.2" xref="S2.E3.m1.5.5.2.2.cmml">X</mi><mo id="S2.E3.m1.6.6.3.3.1.5" stretchy="false" xref="S2.E3.m1.6.6.3.3.2.cmml">)</mo></mrow></mrow></msub></mrow></mrow><mo id="S2.E3.m1.8.8.1.2" xref="S2.E3.m1.8.8.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.E3.m1.8b"><apply id="S2.E3.m1.8.8.1.1.cmml" xref="S2.E3.m1.8.8.1"><leq id="S2.E3.m1.8.8.1.1.2.cmml" xref="S2.E3.m1.8.8.1.1.2"></leq><apply id="S2.E3.m1.8.8.1.1.1.cmml" xref="S2.E3.m1.8.8.1.1.1"><csymbol cd="ambiguous" id="S2.E3.m1.8.8.1.1.1.2.cmml" xref="S2.E3.m1.8.8.1.1.1">subscript</csymbol><apply id="S2.E3.m1.8.8.1.1.1.1.2.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1"><csymbol cd="latexml" id="S2.E3.m1.8.8.1.1.1.1.2.1.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.2">norm</csymbol><apply id="S2.E3.m1.8.8.1.1.1.1.1.1.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.1"><times id="S2.E3.m1.8.8.1.1.1.1.1.1.1.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.1.1"></times><apply id="S2.E3.m1.8.8.1.1.1.1.1.1.2.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.E3.m1.8.8.1.1.1.1.1.1.2.1.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.1.2">subscript</csymbol><ci id="S2.E3.m1.8.8.1.1.1.1.1.1.2.2.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.1.2.2">𝑆</ci><ci id="S2.E3.m1.8.8.1.1.1.1.1.1.2.3.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S2.E3.m1.8.8.1.1.1.1.1.1.3.cmml" xref="S2.E3.m1.8.8.1.1.1.1.1.1.3">𝑓</ci></apply></apply><apply id="S2.E3.m1.3.3.3.cmml" xref="S2.E3.m1.3.3.3"><times id="S2.E3.m1.3.3.3.4.cmml" xref="S2.E3.m1.3.3.3.4"></times><apply id="S2.E3.m1.3.3.3.5.cmml" xref="S2.E3.m1.3.3.3.5"><csymbol cd="ambiguous" id="S2.E3.m1.3.3.3.5.1.cmml" xref="S2.E3.m1.3.3.3.5">superscript</csymbol><ci id="S2.E3.m1.3.3.3.5.2.cmml" xref="S2.E3.m1.3.3.3.5.2">𝐿</ci><ci id="S2.E3.m1.3.3.3.5.3.cmml" xref="S2.E3.m1.3.3.3.5.3">𝑝</ci></apply><vector id="S2.E3.m1.3.3.3.3.2.cmml" xref="S2.E3.m1.3.3.3.3.1"><apply id="S2.E3.m1.3.3.3.3.1.1.cmml" xref="S2.E3.m1.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S2.E3.m1.3.3.3.3.1.1.1.cmml" xref="S2.E3.m1.3.3.3.3.1.1">superscript</csymbol><ci id="S2.E3.m1.3.3.3.3.1.1.2.cmml" xref="S2.E3.m1.3.3.3.3.1.1.2">ℝ</ci><ci id="S2.E3.m1.3.3.3.3.1.1.3.cmml" xref="S2.E3.m1.3.3.3.3.1.1.3">𝑑</ci></apply><ci id="S2.E3.m1.1.1.1.1.cmml" xref="S2.E3.m1.1.1.1.1">𝑤</ci><ci id="S2.E3.m1.2.2.2.2.cmml" xref="S2.E3.m1.2.2.2.2">𝑋</ci></vector></apply></apply><apply id="S2.E3.m1.8.8.1.1.3.cmml" xref="S2.E3.m1.8.8.1.1.3"><times id="S2.E3.m1.8.8.1.1.3.1.cmml" xref="S2.E3.m1.8.8.1.1.3.1"></times><ci id="S2.E3.m1.8.8.1.1.3.2.cmml" xref="S2.E3.m1.8.8.1.1.3.2">𝐶</ci><apply id="S2.E3.m1.8.8.1.1.3.3.cmml" xref="S2.E3.m1.8.8.1.1.3.3"><csymbol cd="ambiguous" id="S2.E3.m1.8.8.1.1.3.3.1.cmml" xref="S2.E3.m1.8.8.1.1.3.3">subscript</csymbol><apply id="S2.E3.m1.8.8.1.1.3.3.2.1.cmml" xref="S2.E3.m1.8.8.1.1.3.3.2.2"><csymbol cd="latexml" id="S2.E3.m1.8.8.1.1.3.3.2.1.1.cmml" xref="S2.E3.m1.8.8.1.1.3.3.2.2.1">norm</csymbol><ci id="S2.E3.m1.7.7.cmml" xref="S2.E3.m1.7.7">𝑓</ci></apply><apply id="S2.E3.m1.6.6.3.cmml" xref="S2.E3.m1.6.6.3"><times id="S2.E3.m1.6.6.3.4.cmml" xref="S2.E3.m1.6.6.3.4"></times><apply id="S2.E3.m1.6.6.3.5.cmml" xref="S2.E3.m1.6.6.3.5"><csymbol cd="ambiguous" id="S2.E3.m1.6.6.3.5.1.cmml" xref="S2.E3.m1.6.6.3.5">superscript</csymbol><ci id="S2.E3.m1.6.6.3.5.2.cmml" xref="S2.E3.m1.6.6.3.5.2">𝐿</ci><ci id="S2.E3.m1.6.6.3.5.3.cmml" xref="S2.E3.m1.6.6.3.5.3">𝑝</ci></apply><vector id="S2.E3.m1.6.6.3.3.2.cmml" xref="S2.E3.m1.6.6.3.3.1"><apply id="S2.E3.m1.6.6.3.3.1.1.cmml" xref="S2.E3.m1.6.6.3.3.1.1"><csymbol cd="ambiguous" id="S2.E3.m1.6.6.3.3.1.1.1.cmml" xref="S2.E3.m1.6.6.3.3.1.1">superscript</csymbol><ci id="S2.E3.m1.6.6.3.3.1.1.2.cmml" xref="S2.E3.m1.6.6.3.3.1.1.2">ℝ</ci><ci id="S2.E3.m1.6.6.3.3.1.1.3.cmml" xref="S2.E3.m1.6.6.3.3.1.1.3">𝑑</ci></apply><ci id="S2.E3.m1.4.4.1.1.cmml" xref="S2.E3.m1.4.4.1.1">𝑤</ci><ci id="S2.E3.m1.5.5.2.2.cmml" xref="S2.E3.m1.5.5.2.2">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E3.m1.8c">\|S_{n}f\|_{L^{p}(\mathbb{R}^{d},w;X)}\leq C\|f\|_{L^{p}(\mathbb{R}^{d},w;X)},</annotation><annotation encoding="application/x-llamapun" id="S2.E3.m1.8d">∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p4.13">where the constant <math alttext="C>0" class="ltx_Math" display="inline" id="S2.SS2.p4.7.m1.1"><semantics id="S2.SS2.p4.7.m1.1a"><mrow id="S2.SS2.p4.7.m1.1.1" xref="S2.SS2.p4.7.m1.1.1.cmml"><mi id="S2.SS2.p4.7.m1.1.1.2" xref="S2.SS2.p4.7.m1.1.1.2.cmml">C</mi><mo id="S2.SS2.p4.7.m1.1.1.1" xref="S2.SS2.p4.7.m1.1.1.1.cmml">></mo><mn id="S2.SS2.p4.7.m1.1.1.3" xref="S2.SS2.p4.7.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.7.m1.1b"><apply id="S2.SS2.p4.7.m1.1.1.cmml" xref="S2.SS2.p4.7.m1.1.1"><gt id="S2.SS2.p4.7.m1.1.1.1.cmml" xref="S2.SS2.p4.7.m1.1.1.1"></gt><ci id="S2.SS2.p4.7.m1.1.1.2.cmml" xref="S2.SS2.p4.7.m1.1.1.2">𝐶</ci><cn id="S2.SS2.p4.7.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p4.7.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.7.m1.1c">C>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.7.m1.1d">italic_C > 0</annotation></semantics></math> is independent of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS2.p4.8.m2.1"><semantics id="S2.SS2.p4.8.m2.1a"><mi id="S2.SS2.p4.8.m2.1.1" xref="S2.SS2.p4.8.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.8.m2.1b"><ci id="S2.SS2.p4.8.m2.1.1.cmml" xref="S2.SS2.p4.8.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.8.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.8.m2.1d">italic_f</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S2.SS2.p4.9.m3.1"><semantics id="S2.SS2.p4.9.m3.1a"><mi id="S2.SS2.p4.9.m3.1.1" xref="S2.SS2.p4.9.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.9.m3.1b"><ci id="S2.SS2.p4.9.m3.1.1.cmml" xref="S2.SS2.p4.9.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.9.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.9.m3.1d">italic_n</annotation></semantics></math>. Indeed, since <math alttext="w\in A_{\infty}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS2.p4.10.m4.1"><semantics id="S2.SS2.p4.10.m4.1a"><mrow id="S2.SS2.p4.10.m4.1.1" xref="S2.SS2.p4.10.m4.1.1.cmml"><mi id="S2.SS2.p4.10.m4.1.1.3" xref="S2.SS2.p4.10.m4.1.1.3.cmml">w</mi><mo id="S2.SS2.p4.10.m4.1.1.2" xref="S2.SS2.p4.10.m4.1.1.2.cmml">∈</mo><mrow id="S2.SS2.p4.10.m4.1.1.1" xref="S2.SS2.p4.10.m4.1.1.1.cmml"><msub id="S2.SS2.p4.10.m4.1.1.1.3" xref="S2.SS2.p4.10.m4.1.1.1.3.cmml"><mi id="S2.SS2.p4.10.m4.1.1.1.3.2" xref="S2.SS2.p4.10.m4.1.1.1.3.2.cmml">A</mi><mi id="S2.SS2.p4.10.m4.1.1.1.3.3" mathvariant="normal" xref="S2.SS2.p4.10.m4.1.1.1.3.3.cmml">∞</mi></msub><mo id="S2.SS2.p4.10.m4.1.1.1.2" xref="S2.SS2.p4.10.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.10.m4.1.1.1.1.1" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p4.10.m4.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS2.p4.10.m4.1.1.1.1.1.1" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.10.m4.1.1.1.1.1.1.2" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p4.10.m4.1.1.1.1.1.1.3" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p4.10.m4.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.10.m4.1b"><apply id="S2.SS2.p4.10.m4.1.1.cmml" xref="S2.SS2.p4.10.m4.1.1"><in id="S2.SS2.p4.10.m4.1.1.2.cmml" xref="S2.SS2.p4.10.m4.1.1.2"></in><ci id="S2.SS2.p4.10.m4.1.1.3.cmml" xref="S2.SS2.p4.10.m4.1.1.3">𝑤</ci><apply id="S2.SS2.p4.10.m4.1.1.1.cmml" xref="S2.SS2.p4.10.m4.1.1.1"><times id="S2.SS2.p4.10.m4.1.1.1.2.cmml" xref="S2.SS2.p4.10.m4.1.1.1.2"></times><apply id="S2.SS2.p4.10.m4.1.1.1.3.cmml" xref="S2.SS2.p4.10.m4.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.10.m4.1.1.1.3.1.cmml" xref="S2.SS2.p4.10.m4.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.10.m4.1.1.1.3.2.cmml" xref="S2.SS2.p4.10.m4.1.1.1.3.2">𝐴</ci><infinity id="S2.SS2.p4.10.m4.1.1.1.3.3.cmml" xref="S2.SS2.p4.10.m4.1.1.1.3.3"></infinity></apply><apply id="S2.SS2.p4.10.m4.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.10.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.10.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.10.m4.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.10.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p4.10.m4.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.10.m4.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.10.m4.1c">w\in A_{\infty}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.10.m4.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math>, we can find an <math alttext="r\in(0,\min\{p,q\})" class="ltx_Math" display="inline" id="S2.SS2.p4.11.m5.5"><semantics id="S2.SS2.p4.11.m5.5a"><mrow id="S2.SS2.p4.11.m5.5.5" xref="S2.SS2.p4.11.m5.5.5.cmml"><mi id="S2.SS2.p4.11.m5.5.5.3" xref="S2.SS2.p4.11.m5.5.5.3.cmml">r</mi><mo id="S2.SS2.p4.11.m5.5.5.2" xref="S2.SS2.p4.11.m5.5.5.2.cmml">∈</mo><mrow id="S2.SS2.p4.11.m5.5.5.1.1" xref="S2.SS2.p4.11.m5.5.5.1.2.cmml"><mo id="S2.SS2.p4.11.m5.5.5.1.1.2" stretchy="false" xref="S2.SS2.p4.11.m5.5.5.1.2.cmml">(</mo><mn id="S2.SS2.p4.11.m5.4.4" xref="S2.SS2.p4.11.m5.4.4.cmml">0</mn><mo id="S2.SS2.p4.11.m5.5.5.1.1.3" xref="S2.SS2.p4.11.m5.5.5.1.2.cmml">,</mo><mrow id="S2.SS2.p4.11.m5.5.5.1.1.1.2" xref="S2.SS2.p4.11.m5.5.5.1.1.1.1.cmml"><mi id="S2.SS2.p4.11.m5.1.1" xref="S2.SS2.p4.11.m5.1.1.cmml">min</mi><mo id="S2.SS2.p4.11.m5.5.5.1.1.1.2a" xref="S2.SS2.p4.11.m5.5.5.1.1.1.1.cmml">⁡</mo><mrow id="S2.SS2.p4.11.m5.5.5.1.1.1.2.1" xref="S2.SS2.p4.11.m5.5.5.1.1.1.1.cmml"><mo id="S2.SS2.p4.11.m5.5.5.1.1.1.2.1.1" stretchy="false" xref="S2.SS2.p4.11.m5.5.5.1.1.1.1.cmml">{</mo><mi id="S2.SS2.p4.11.m5.2.2" xref="S2.SS2.p4.11.m5.2.2.cmml">p</mi><mo id="S2.SS2.p4.11.m5.5.5.1.1.1.2.1.2" xref="S2.SS2.p4.11.m5.5.5.1.1.1.1.cmml">,</mo><mi id="S2.SS2.p4.11.m5.3.3" xref="S2.SS2.p4.11.m5.3.3.cmml">q</mi><mo id="S2.SS2.p4.11.m5.5.5.1.1.1.2.1.3" stretchy="false" xref="S2.SS2.p4.11.m5.5.5.1.1.1.1.cmml">}</mo></mrow></mrow><mo id="S2.SS2.p4.11.m5.5.5.1.1.4" stretchy="false" xref="S2.SS2.p4.11.m5.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.11.m5.5b"><apply id="S2.SS2.p4.11.m5.5.5.cmml" xref="S2.SS2.p4.11.m5.5.5"><in id="S2.SS2.p4.11.m5.5.5.2.cmml" xref="S2.SS2.p4.11.m5.5.5.2"></in><ci id="S2.SS2.p4.11.m5.5.5.3.cmml" xref="S2.SS2.p4.11.m5.5.5.3">𝑟</ci><interval closure="open" id="S2.SS2.p4.11.m5.5.5.1.2.cmml" xref="S2.SS2.p4.11.m5.5.5.1.1"><cn id="S2.SS2.p4.11.m5.4.4.cmml" type="integer" xref="S2.SS2.p4.11.m5.4.4">0</cn><apply id="S2.SS2.p4.11.m5.5.5.1.1.1.1.cmml" xref="S2.SS2.p4.11.m5.5.5.1.1.1.2"><min id="S2.SS2.p4.11.m5.1.1.cmml" xref="S2.SS2.p4.11.m5.1.1"></min><ci id="S2.SS2.p4.11.m5.2.2.cmml" xref="S2.SS2.p4.11.m5.2.2">𝑝</ci><ci id="S2.SS2.p4.11.m5.3.3.cmml" xref="S2.SS2.p4.11.m5.3.3">𝑞</ci></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.11.m5.5c">r\in(0,\min\{p,q\})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.11.m5.5d">italic_r ∈ ( 0 , roman_min { italic_p , italic_q } )</annotation></semantics></math> for any <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S2.SS2.p4.12.m6.2"><semantics id="S2.SS2.p4.12.m6.2a"><mrow id="S2.SS2.p4.12.m6.2.3" xref="S2.SS2.p4.12.m6.2.3.cmml"><mi id="S2.SS2.p4.12.m6.2.3.2" xref="S2.SS2.p4.12.m6.2.3.2.cmml">q</mi><mo id="S2.SS2.p4.12.m6.2.3.1" xref="S2.SS2.p4.12.m6.2.3.1.cmml">∈</mo><mrow id="S2.SS2.p4.12.m6.2.3.3.2" xref="S2.SS2.p4.12.m6.2.3.3.1.cmml"><mo id="S2.SS2.p4.12.m6.2.3.3.2.1" stretchy="false" xref="S2.SS2.p4.12.m6.2.3.3.1.cmml">[</mo><mn id="S2.SS2.p4.12.m6.1.1" xref="S2.SS2.p4.12.m6.1.1.cmml">1</mn><mo id="S2.SS2.p4.12.m6.2.3.3.2.2" xref="S2.SS2.p4.12.m6.2.3.3.1.cmml">,</mo><mi id="S2.SS2.p4.12.m6.2.2" mathvariant="normal" xref="S2.SS2.p4.12.m6.2.2.cmml">∞</mi><mo id="S2.SS2.p4.12.m6.2.3.3.2.3" stretchy="false" xref="S2.SS2.p4.12.m6.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.12.m6.2b"><apply id="S2.SS2.p4.12.m6.2.3.cmml" xref="S2.SS2.p4.12.m6.2.3"><in id="S2.SS2.p4.12.m6.2.3.1.cmml" xref="S2.SS2.p4.12.m6.2.3.1"></in><ci id="S2.SS2.p4.12.m6.2.3.2.cmml" xref="S2.SS2.p4.12.m6.2.3.2">𝑞</ci><interval closure="closed" id="S2.SS2.p4.12.m6.2.3.3.1.cmml" xref="S2.SS2.p4.12.m6.2.3.3.2"><cn id="S2.SS2.p4.12.m6.1.1.cmml" type="integer" xref="S2.SS2.p4.12.m6.1.1">1</cn><infinity id="S2.SS2.p4.12.m6.2.2.cmml" xref="S2.SS2.p4.12.m6.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.12.m6.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.12.m6.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math> such that <math alttext="w\in A_{\frac{p}{r}}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS2.p4.13.m7.1"><semantics id="S2.SS2.p4.13.m7.1a"><mrow id="S2.SS2.p4.13.m7.1.1" xref="S2.SS2.p4.13.m7.1.1.cmml"><mi id="S2.SS2.p4.13.m7.1.1.3" xref="S2.SS2.p4.13.m7.1.1.3.cmml">w</mi><mo id="S2.SS2.p4.13.m7.1.1.2" xref="S2.SS2.p4.13.m7.1.1.2.cmml">∈</mo><mrow id="S2.SS2.p4.13.m7.1.1.1" xref="S2.SS2.p4.13.m7.1.1.1.cmml"><msub id="S2.SS2.p4.13.m7.1.1.1.3" xref="S2.SS2.p4.13.m7.1.1.1.3.cmml"><mi id="S2.SS2.p4.13.m7.1.1.1.3.2" xref="S2.SS2.p4.13.m7.1.1.1.3.2.cmml">A</mi><mfrac id="S2.SS2.p4.13.m7.1.1.1.3.3" xref="S2.SS2.p4.13.m7.1.1.1.3.3.cmml"><mi id="S2.SS2.p4.13.m7.1.1.1.3.3.2" xref="S2.SS2.p4.13.m7.1.1.1.3.3.2.cmml">p</mi><mi id="S2.SS2.p4.13.m7.1.1.1.3.3.3" xref="S2.SS2.p4.13.m7.1.1.1.3.3.3.cmml">r</mi></mfrac></msub><mo id="S2.SS2.p4.13.m7.1.1.1.2" xref="S2.SS2.p4.13.m7.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS2.p4.13.m7.1.1.1.1.1" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.cmml"><mo id="S2.SS2.p4.13.m7.1.1.1.1.1.2" stretchy="false" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS2.p4.13.m7.1.1.1.1.1.1" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.cmml"><mi id="S2.SS2.p4.13.m7.1.1.1.1.1.1.2" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p4.13.m7.1.1.1.1.1.1.3" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p4.13.m7.1.1.1.1.1.3" stretchy="false" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.13.m7.1b"><apply id="S2.SS2.p4.13.m7.1.1.cmml" xref="S2.SS2.p4.13.m7.1.1"><in id="S2.SS2.p4.13.m7.1.1.2.cmml" xref="S2.SS2.p4.13.m7.1.1.2"></in><ci id="S2.SS2.p4.13.m7.1.1.3.cmml" xref="S2.SS2.p4.13.m7.1.1.3">𝑤</ci><apply id="S2.SS2.p4.13.m7.1.1.1.cmml" xref="S2.SS2.p4.13.m7.1.1.1"><times id="S2.SS2.p4.13.m7.1.1.1.2.cmml" xref="S2.SS2.p4.13.m7.1.1.1.2"></times><apply id="S2.SS2.p4.13.m7.1.1.1.3.cmml" xref="S2.SS2.p4.13.m7.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p4.13.m7.1.1.1.3.1.cmml" xref="S2.SS2.p4.13.m7.1.1.1.3">subscript</csymbol><ci id="S2.SS2.p4.13.m7.1.1.1.3.2.cmml" xref="S2.SS2.p4.13.m7.1.1.1.3.2">𝐴</ci><apply id="S2.SS2.p4.13.m7.1.1.1.3.3.cmml" xref="S2.SS2.p4.13.m7.1.1.1.3.3"><divide id="S2.SS2.p4.13.m7.1.1.1.3.3.1.cmml" xref="S2.SS2.p4.13.m7.1.1.1.3.3"></divide><ci id="S2.SS2.p4.13.m7.1.1.1.3.3.2.cmml" xref="S2.SS2.p4.13.m7.1.1.1.3.3.2">𝑝</ci><ci id="S2.SS2.p4.13.m7.1.1.1.3.3.3.cmml" xref="S2.SS2.p4.13.m7.1.1.1.3.3.3">𝑟</ci></apply></apply><apply id="S2.SS2.p4.13.m7.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.13.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p4.13.m7.1.1.1.1.1.1.1.cmml" xref="S2.SS2.p4.13.m7.1.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p4.13.m7.1.1.1.1.1.1.2.cmml" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p4.13.m7.1.1.1.1.1.1.3.cmml" xref="S2.SS2.p4.13.m7.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.13.m7.1c">w\in A_{\frac{p}{r}}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.13.m7.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT divide start_ARG italic_p end_ARG start_ARG italic_r end_ARG end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 2.4]</cite> implies</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx2"> <tbody id="S2.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|S_{n}f\|_{L^{p}(\mathbb{R}^{d},w;X)}" class="ltx_Math" display="inline" id="S2.Ex4.m1.4"><semantics id="S2.Ex4.m1.4a"><msub id="S2.Ex4.m1.4.4" xref="S2.Ex4.m1.4.4.cmml"><mrow id="S2.Ex4.m1.4.4.1.1" xref="S2.Ex4.m1.4.4.1.2.cmml"><mo id="S2.Ex4.m1.4.4.1.1.2" stretchy="false" xref="S2.Ex4.m1.4.4.1.2.1.cmml">‖</mo><mrow id="S2.Ex4.m1.4.4.1.1.1" xref="S2.Ex4.m1.4.4.1.1.1.cmml"><msub id="S2.Ex4.m1.4.4.1.1.1.2" xref="S2.Ex4.m1.4.4.1.1.1.2.cmml"><mi id="S2.Ex4.m1.4.4.1.1.1.2.2" xref="S2.Ex4.m1.4.4.1.1.1.2.2.cmml">S</mi><mi 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xref="S2.Ex4.m2.3.3.3.5"><csymbol cd="ambiguous" id="S2.Ex4.m2.3.3.3.5.1.cmml" xref="S2.Ex4.m2.3.3.3.5">superscript</csymbol><ci id="S2.Ex4.m2.3.3.3.5.2.cmml" xref="S2.Ex4.m2.3.3.3.5.2">𝐿</ci><ci id="S2.Ex4.m2.3.3.3.5.3.cmml" xref="S2.Ex4.m2.3.3.3.5.3">𝑝</ci></apply><vector id="S2.Ex4.m2.3.3.3.3.2.cmml" xref="S2.Ex4.m2.3.3.3.3.1"><apply id="S2.Ex4.m2.3.3.3.3.1.1.cmml" xref="S2.Ex4.m2.3.3.3.3.1.1"><csymbol cd="ambiguous" id="S2.Ex4.m2.3.3.3.3.1.1.1.cmml" xref="S2.Ex4.m2.3.3.3.3.1.1">superscript</csymbol><ci id="S2.Ex4.m2.3.3.3.3.1.1.2.cmml" xref="S2.Ex4.m2.3.3.3.3.1.1.2">ℝ</ci><ci id="S2.Ex4.m2.3.3.3.3.1.1.3.cmml" xref="S2.Ex4.m2.3.3.3.3.1.1.3">𝑑</ci></apply><ci id="S2.Ex4.m2.1.1.1.1.cmml" xref="S2.Ex4.m2.1.1.1.1">𝑤</ci><ci id="S2.Ex4.m2.2.2.2.2.cmml" xref="S2.Ex4.m2.2.2.2.2">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex4.m2.4c">\displaystyle=\|\mathcal{F}^{-1}(\widehat{\varphi}_{n}\widehat{f}\,)\|_{L^{p}(% \mathbb{R}^{d},w;X)}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4.m2.4d">= ∥ caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT over^ start_ARG italic_f end_ARG ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|(1+|\cdot|^{\frac{d}{r}})\mathcal{F}^{-1}[\widehat{% \varphi}_{n}(2^{n}K\cdot)]\|_{L^{1}(\mathbb{R}^{d})}\|f\|_{L^{p}(\mathbb{R}^{d% },w;X)}," class="ltx_math_unparsed" display="inline" id="S2.Ex5.m1.4"><semantics id="S2.Ex5.m1.4a"><mrow id="S2.Ex5.m1.4b"><mo id="S2.Ex5.m1.4.5">≤</mo><mi id="S2.Ex5.m1.4.6">C</mi><mo id="S2.Ex5.m1.4.7" lspace="0em" rspace="0.167em">∥</mo><mrow id="S2.Ex5.m1.4.8"><mo id="S2.Ex5.m1.4.8.1" stretchy="false">(</mo><mn id="S2.Ex5.m1.4.8.2">1</mn><mo id="S2.Ex5.m1.4.8.3" rspace="0em">+</mo><mo fence="false" id="S2.Ex5.m1.4.8.4" stretchy="false">|</mo><mo id="S2.Ex5.m1.4.8.5" lspace="0em" rspace="0em">⋅</mo><msup id="S2.Ex5.m1.4.8.6"><mo fence="false" id="S2.Ex5.m1.4.8.6.2" rspace="0.167em" stretchy="false">|</mo><mfrac id="S2.Ex5.m1.4.8.6.3"><mi id="S2.Ex5.m1.4.8.6.3.2">d</mi><mi id="S2.Ex5.m1.4.8.6.3.3">r</mi></mfrac></msup><mo id="S2.Ex5.m1.4.8.7" stretchy="false">)</mo></mrow><msup id="S2.Ex5.m1.4.9"><mi class="ltx_font_mathcaligraphic" id="S2.Ex5.m1.4.9.2">ℱ</mi><mrow id="S2.Ex5.m1.4.9.3"><mo id="S2.Ex5.m1.4.9.3a">−</mo><mn id="S2.Ex5.m1.4.9.3.2">1</mn></mrow></msup><mrow id="S2.Ex5.m1.4.10"><mo id="S2.Ex5.m1.4.10.1" stretchy="false">[</mo><msub id="S2.Ex5.m1.4.10.2"><mover accent="true" id="S2.Ex5.m1.4.10.2.2"><mi id="S2.Ex5.m1.4.10.2.2.2">φ</mi><mo id="S2.Ex5.m1.4.10.2.2.1">^</mo></mover><mi id="S2.Ex5.m1.4.10.2.3">n</mi></msub><mrow id="S2.Ex5.m1.4.10.3"><mo id="S2.Ex5.m1.4.10.3.1" stretchy="false">(</mo><msup id="S2.Ex5.m1.4.10.3.2"><mn id="S2.Ex5.m1.4.10.3.2.2">2</mn><mi id="S2.Ex5.m1.4.10.3.2.3">n</mi></msup><mi id="S2.Ex5.m1.4.10.3.3">K</mi><mo id="S2.Ex5.m1.4.10.3.4" lspace="0.222em" rspace="0em">⋅</mo><mo id="S2.Ex5.m1.4.10.3.5" stretchy="false">)</mo></mrow><mo id="S2.Ex5.m1.4.10.4" stretchy="false">]</mo></mrow><msub id="S2.Ex5.m1.4.11"><mo id="S2.Ex5.m1.4.11.2" lspace="0em" rspace="0.0835em">∥</mo><mrow id="S2.Ex5.m1.1.1.1"><msup id="S2.Ex5.m1.1.1.1.3"><mi id="S2.Ex5.m1.1.1.1.3.2">L</mi><mn id="S2.Ex5.m1.1.1.1.3.3">1</mn></msup><mo id="S2.Ex5.m1.1.1.1.2">⁢</mo><mrow id="S2.Ex5.m1.1.1.1.1.1"><mo id="S2.Ex5.m1.1.1.1.1.1.2" stretchy="false">(</mo><msup id="S2.Ex5.m1.1.1.1.1.1.1"><mi id="S2.Ex5.m1.1.1.1.1.1.1.2">ℝ</mi><mi id="S2.Ex5.m1.1.1.1.1.1.1.3">d</mi></msup><mo id="S2.Ex5.m1.1.1.1.1.1.3" stretchy="false">)</mo></mrow></mrow></msub><mo id="S2.Ex5.m1.4.12" lspace="0.0835em" rspace="0.167em">∥</mo><mi id="S2.Ex5.m1.4.13">f</mi><msub id="S2.Ex5.m1.4.14"><mo id="S2.Ex5.m1.4.14.2" lspace="0em" rspace="0.167em">∥</mo><mrow id="S2.Ex5.m1.4.4.3"><msup id="S2.Ex5.m1.4.4.3.5"><mi id="S2.Ex5.m1.4.4.3.5.2">L</mi><mi id="S2.Ex5.m1.4.4.3.5.3">p</mi></msup><mo id="S2.Ex5.m1.4.4.3.4">⁢</mo><mrow id="S2.Ex5.m1.4.4.3.3.1"><mo id="S2.Ex5.m1.4.4.3.3.1.2" stretchy="false">(</mo><msup id="S2.Ex5.m1.4.4.3.3.1.1"><mi id="S2.Ex5.m1.4.4.3.3.1.1.2">ℝ</mi><mi id="S2.Ex5.m1.4.4.3.3.1.1.3">d</mi></msup><mo id="S2.Ex5.m1.4.4.3.3.1.3">,</mo><mi id="S2.Ex5.m1.2.2.1.1">w</mi><mo id="S2.Ex5.m1.4.4.3.3.1.4">;</mo><mi id="S2.Ex5.m1.3.3.2.2">X</mi><mo id="S2.Ex5.m1.4.4.3.3.1.5" stretchy="false">)</mo></mrow></mrow></msub><mo id="S2.Ex5.m1.4.15">,</mo></mrow><annotation encoding="application/x-tex" id="S2.Ex5.m1.4c">\displaystyle\leq C\|(1+|\cdot|^{\frac{d}{r}})\mathcal{F}^{-1}[\widehat{% \varphi}_{n}(2^{n}K\cdot)]\|_{L^{1}(\mathbb{R}^{d})}\|f\|_{L^{p}(\mathbb{R}^{d% },w;X)},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5.m1.4d">≤ italic_C ∥ ( 1 + | ⋅ | start_POSTSUPERSCRIPT divide start_ARG italic_d end_ARG start_ARG italic_r end_ARG end_POSTSUPERSCRIPT ) caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT [ over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_K ⋅ ) ] ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) end_POSTSUBSCRIPT ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p4.20">where <math alttext="C>0" class="ltx_Math" display="inline" id="S2.SS2.p4.14.m1.1"><semantics id="S2.SS2.p4.14.m1.1a"><mrow id="S2.SS2.p4.14.m1.1.1" xref="S2.SS2.p4.14.m1.1.1.cmml"><mi id="S2.SS2.p4.14.m1.1.1.2" xref="S2.SS2.p4.14.m1.1.1.2.cmml">C</mi><mo id="S2.SS2.p4.14.m1.1.1.1" xref="S2.SS2.p4.14.m1.1.1.1.cmml">></mo><mn id="S2.SS2.p4.14.m1.1.1.3" xref="S2.SS2.p4.14.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.14.m1.1b"><apply id="S2.SS2.p4.14.m1.1.1.cmml" xref="S2.SS2.p4.14.m1.1.1"><gt id="S2.SS2.p4.14.m1.1.1.1.cmml" xref="S2.SS2.p4.14.m1.1.1.1"></gt><ci id="S2.SS2.p4.14.m1.1.1.2.cmml" xref="S2.SS2.p4.14.m1.1.1.2">𝐶</ci><cn id="S2.SS2.p4.14.m1.1.1.3.cmml" type="integer" xref="S2.SS2.p4.14.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.14.m1.1c">C>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.14.m1.1d">italic_C > 0</annotation></semantics></math> is independent of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS2.p4.15.m2.1"><semantics id="S2.SS2.p4.15.m2.1a"><mi id="S2.SS2.p4.15.m2.1.1" xref="S2.SS2.p4.15.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.15.m2.1b"><ci id="S2.SS2.p4.15.m2.1.1.cmml" xref="S2.SS2.p4.15.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.15.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.15.m2.1d">italic_f</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S2.SS2.p4.16.m3.1"><semantics id="S2.SS2.p4.16.m3.1a"><mi id="S2.SS2.p4.16.m3.1.1" xref="S2.SS2.p4.16.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.16.m3.1b"><ci id="S2.SS2.p4.16.m3.1.1.cmml" xref="S2.SS2.p4.16.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.16.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.16.m3.1d">italic_n</annotation></semantics></math>. Using the scaling property <math alttext="\widehat{\varphi}_{n}(\cdot)=\widehat{\varphi}_{1}(2^{-n+1}\cdot)" class="ltx_math_unparsed" display="inline" id="S2.SS2.p4.17.m4.1"><semantics id="S2.SS2.p4.17.m4.1a"><mrow id="S2.SS2.p4.17.m4.1b"><msub id="S2.SS2.p4.17.m4.1.2"><mover accent="true" id="S2.SS2.p4.17.m4.1.2.2"><mi id="S2.SS2.p4.17.m4.1.2.2.2">φ</mi><mo id="S2.SS2.p4.17.m4.1.2.2.1">^</mo></mover><mi id="S2.SS2.p4.17.m4.1.2.3">n</mi></msub><mrow id="S2.SS2.p4.17.m4.1.3"><mo id="S2.SS2.p4.17.m4.1.3.1" stretchy="false">(</mo><mo id="S2.SS2.p4.17.m4.1.1" lspace="0em" rspace="0em">⋅</mo><mo id="S2.SS2.p4.17.m4.1.3.2" stretchy="false">)</mo></mrow><mo id="S2.SS2.p4.17.m4.1.4">=</mo><msub id="S2.SS2.p4.17.m4.1.5"><mover accent="true" id="S2.SS2.p4.17.m4.1.5.2"><mi id="S2.SS2.p4.17.m4.1.5.2.2">φ</mi><mo id="S2.SS2.p4.17.m4.1.5.2.1">^</mo></mover><mn id="S2.SS2.p4.17.m4.1.5.3">1</mn></msub><mrow id="S2.SS2.p4.17.m4.1.6"><mo id="S2.SS2.p4.17.m4.1.6.1" stretchy="false">(</mo><msup id="S2.SS2.p4.17.m4.1.6.2"><mn id="S2.SS2.p4.17.m4.1.6.2.2">2</mn><mrow id="S2.SS2.p4.17.m4.1.6.2.3"><mrow id="S2.SS2.p4.17.m4.1.6.2.3.2"><mo id="S2.SS2.p4.17.m4.1.6.2.3.2a">−</mo><mi id="S2.SS2.p4.17.m4.1.6.2.3.2.2">n</mi></mrow><mo id="S2.SS2.p4.17.m4.1.6.2.3.1">+</mo><mn id="S2.SS2.p4.17.m4.1.6.2.3.3">1</mn></mrow></msup><mo id="S2.SS2.p4.17.m4.1.6.3" lspace="0.222em" rspace="0em">⋅</mo><mo id="S2.SS2.p4.17.m4.1.6.4" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS2.p4.17.m4.1c">\widehat{\varphi}_{n}(\cdot)=\widehat{\varphi}_{1}(2^{-n+1}\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.17.m4.1d">over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( ⋅ ) = over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 2 start_POSTSUPERSCRIPT - italic_n + 1 end_POSTSUPERSCRIPT ⋅ )</annotation></semantics></math> for <math alttext="n\geq 1" class="ltx_Math" display="inline" id="S2.SS2.p4.18.m5.1"><semantics id="S2.SS2.p4.18.m5.1a"><mrow id="S2.SS2.p4.18.m5.1.1" xref="S2.SS2.p4.18.m5.1.1.cmml"><mi id="S2.SS2.p4.18.m5.1.1.2" xref="S2.SS2.p4.18.m5.1.1.2.cmml">n</mi><mo id="S2.SS2.p4.18.m5.1.1.1" xref="S2.SS2.p4.18.m5.1.1.1.cmml">≥</mo><mn id="S2.SS2.p4.18.m5.1.1.3" xref="S2.SS2.p4.18.m5.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.18.m5.1b"><apply id="S2.SS2.p4.18.m5.1.1.cmml" xref="S2.SS2.p4.18.m5.1.1"><geq id="S2.SS2.p4.18.m5.1.1.1.cmml" xref="S2.SS2.p4.18.m5.1.1.1"></geq><ci id="S2.SS2.p4.18.m5.1.1.2.cmml" xref="S2.SS2.p4.18.m5.1.1.2">𝑛</ci><cn id="S2.SS2.p4.18.m5.1.1.3.cmml" type="integer" xref="S2.SS2.p4.18.m5.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.18.m5.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.18.m5.1d">italic_n ≥ 1</annotation></semantics></math> gives that <math alttext="\widehat{\varphi}_{n}(2^{n}K\cdot)" class="ltx_math_unparsed" display="inline" id="S2.SS2.p4.19.m6.1"><semantics id="S2.SS2.p4.19.m6.1a"><mrow id="S2.SS2.p4.19.m6.1b"><msub id="S2.SS2.p4.19.m6.1.1"><mover accent="true" id="S2.SS2.p4.19.m6.1.1.2"><mi id="S2.SS2.p4.19.m6.1.1.2.2">φ</mi><mo id="S2.SS2.p4.19.m6.1.1.2.1">^</mo></mover><mi id="S2.SS2.p4.19.m6.1.1.3">n</mi></msub><mrow id="S2.SS2.p4.19.m6.1.2"><mo id="S2.SS2.p4.19.m6.1.2.1" stretchy="false">(</mo><msup id="S2.SS2.p4.19.m6.1.2.2"><mn id="S2.SS2.p4.19.m6.1.2.2.2">2</mn><mi id="S2.SS2.p4.19.m6.1.2.2.3">n</mi></msup><mi id="S2.SS2.p4.19.m6.1.2.3">K</mi><mo id="S2.SS2.p4.19.m6.1.2.4" lspace="0.222em" rspace="0em">⋅</mo><mo id="S2.SS2.p4.19.m6.1.2.5" stretchy="false">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S2.SS2.p4.19.m6.1c">\widehat{\varphi}_{n}(2^{n}K\cdot)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.19.m6.1d">over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_K ⋅ )</annotation></semantics></math> has its support in a ball with a radius that is independent of <math alttext="n" class="ltx_Math" display="inline" id="S2.SS2.p4.20.m7.1"><semantics id="S2.SS2.p4.20.m7.1a"><mi id="S2.SS2.p4.20.m7.1.1" xref="S2.SS2.p4.20.m7.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p4.20.m7.1b"><ci id="S2.SS2.p4.20.m7.1.1.cmml" xref="S2.SS2.p4.20.m7.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p4.20.m7.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p4.20.m7.1d">italic_n</annotation></semantics></math>. Now, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Lemma 14.2.12]</cite> gives (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E3" title="In 2.2. Weighted function spaces ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>). <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S2.SS2.p5"> <p class="ltx_p" id="S2.SS2.p5.4">For <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S2.SS2.p5.1.m1.2"><semantics id="S2.SS2.p5.1.m1.2a"><mrow id="S2.SS2.p5.1.m1.2.3" xref="S2.SS2.p5.1.m1.2.3.cmml"><mi id="S2.SS2.p5.1.m1.2.3.2" xref="S2.SS2.p5.1.m1.2.3.2.cmml">p</mi><mo id="S2.SS2.p5.1.m1.2.3.1" xref="S2.SS2.p5.1.m1.2.3.1.cmml">∈</mo><mrow id="S2.SS2.p5.1.m1.2.3.3.2" xref="S2.SS2.p5.1.m1.2.3.3.1.cmml"><mo id="S2.SS2.p5.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS2.p5.1.m1.2.3.3.1.cmml">(</mo><mn id="S2.SS2.p5.1.m1.1.1" xref="S2.SS2.p5.1.m1.1.1.cmml">1</mn><mo id="S2.SS2.p5.1.m1.2.3.3.2.2" xref="S2.SS2.p5.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS2.p5.1.m1.2.2" mathvariant="normal" xref="S2.SS2.p5.1.m1.2.2.cmml">∞</mi><mo id="S2.SS2.p5.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS2.p5.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.1.m1.2b"><apply id="S2.SS2.p5.1.m1.2.3.cmml" xref="S2.SS2.p5.1.m1.2.3"><in id="S2.SS2.p5.1.m1.2.3.1.cmml" xref="S2.SS2.p5.1.m1.2.3.1"></in><ci id="S2.SS2.p5.1.m1.2.3.2.cmml" xref="S2.SS2.p5.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S2.SS2.p5.1.m1.2.3.3.1.cmml" xref="S2.SS2.p5.1.m1.2.3.3.2"><cn id="S2.SS2.p5.1.m1.1.1.cmml" type="integer" xref="S2.SS2.p5.1.m1.1.1">1</cn><infinity id="S2.SS2.p5.1.m1.2.2.cmml" xref="S2.SS2.p5.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty],s\in\mathbb{R},w\in A_{\infty}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS2.p5.2.m2.4"><semantics 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xref="S2.SS2.p5.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.3.m3.1b"><ci id="S2.SS2.p5.3.m3.1.1.cmml" xref="S2.SS2.p5.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.3.m3.1d">italic_X</annotation></semantics></math> a Banach space, we define the <em class="ltx_emph ltx_font_italic" id="S2.SS2.p5.4.1"> weighted Besov, Triebel-Lizorkin and Bessel potential spaces</em> as the space of all <math alttext="f\in\mathcal{S}^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS2.p5.4.m4.2"><semantics id="S2.SS2.p5.4.m4.2a"><mrow id="S2.SS2.p5.4.m4.2.2" xref="S2.SS2.p5.4.m4.2.2.cmml"><mi id="S2.SS2.p5.4.m4.2.2.3" xref="S2.SS2.p5.4.m4.2.2.3.cmml">f</mi><mo id="S2.SS2.p5.4.m4.2.2.2" xref="S2.SS2.p5.4.m4.2.2.2.cmml">∈</mo><mrow id="S2.SS2.p5.4.m4.2.2.1" xref="S2.SS2.p5.4.m4.2.2.1.cmml"><msup id="S2.SS2.p5.4.m4.2.2.1.3" 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id="S2.SS2.p5.4.m4.2.2.1.1.1.1.1.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1.1.1">superscript</csymbol><ci id="S2.SS2.p5.4.m4.2.2.1.1.1.1.2.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1.1.1.2">ℝ</ci><ci id="S2.SS2.p5.4.m4.2.2.1.1.1.1.3.cmml" xref="S2.SS2.p5.4.m4.2.2.1.1.1.1.3">𝑑</ci></apply><ci id="S2.SS2.p5.4.m4.1.1.cmml" xref="S2.SS2.p5.4.m4.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.4.m4.2c">f\in\mathcal{S}^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.4.m4.2d">italic_f ∈ caligraphic_S start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> for which, respectively,</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx3"> <tbody id="S2.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|f\|_{B^{s}_{p,q}(\mathbb{R}^{d},w;X)}" class="ltx_Math" display="inline" id="S2.Ex6.m1.6"><semantics id="S2.Ex6.m1.6a"><msub id="S2.Ex6.m1.6.7" xref="S2.Ex6.m1.6.7.cmml"><mrow id="S2.Ex6.m1.6.7.2.2" xref="S2.Ex6.m1.6.7.2.1.cmml"><mo id="S2.Ex6.m1.6.7.2.2.1" stretchy="false" xref="S2.Ex6.m1.6.7.2.1.1.cmml">‖</mo><mi id="S2.Ex6.m1.6.6" xref="S2.Ex6.m1.6.6.cmml">f</mi><mo id="S2.Ex6.m1.6.7.2.2.2" stretchy="false" xref="S2.Ex6.m1.6.7.2.1.1.cmml">‖</mo></mrow><mrow id="S2.Ex6.m1.5.5.5" xref="S2.Ex6.m1.5.5.5.cmml"><msubsup id="S2.Ex6.m1.5.5.5.7" xref="S2.Ex6.m1.5.5.5.7.cmml"><mi id="S2.Ex6.m1.5.5.5.7.2.2" xref="S2.Ex6.m1.5.5.5.7.2.2.cmml">B</mi><mrow id="S2.Ex6.m1.2.2.2.2.2.4" xref="S2.Ex6.m1.2.2.2.2.2.3.cmml"><mi id="S2.Ex6.m1.1.1.1.1.1.1" xref="S2.Ex6.m1.1.1.1.1.1.1.cmml">p</mi><mo id="S2.Ex6.m1.2.2.2.2.2.4.1" xref="S2.Ex6.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S2.Ex6.m1.2.2.2.2.2.2" xref="S2.Ex6.m1.2.2.2.2.2.2.cmml">q</mi></mrow><mi id="S2.Ex6.m1.5.5.5.7.2.3" xref="S2.Ex6.m1.5.5.5.7.2.3.cmml">s</mi></msubsup><mo id="S2.Ex6.m1.5.5.5.6" xref="S2.Ex6.m1.5.5.5.6.cmml">⁢</mo><mrow id="S2.Ex6.m1.5.5.5.5.1" xref="S2.Ex6.m1.5.5.5.5.2.cmml"><mo id="S2.Ex6.m1.5.5.5.5.1.2" stretchy="false" xref="S2.Ex6.m1.5.5.5.5.2.cmml">(</mo><msup id="S2.Ex6.m1.5.5.5.5.1.1" xref="S2.Ex6.m1.5.5.5.5.1.1.cmml"><mi id="S2.Ex6.m1.5.5.5.5.1.1.2" xref="S2.Ex6.m1.5.5.5.5.1.1.2.cmml">ℝ</mi><mi id="S2.Ex6.m1.5.5.5.5.1.1.3" xref="S2.Ex6.m1.5.5.5.5.1.1.3.cmml">d</mi></msup><mo id="S2.Ex6.m1.5.5.5.5.1.3" xref="S2.Ex6.m1.5.5.5.5.2.cmml">,</mo><mi id="S2.Ex6.m1.3.3.3.3" xref="S2.Ex6.m1.3.3.3.3.cmml">w</mi><mo id="S2.Ex6.m1.5.5.5.5.1.4" xref="S2.Ex6.m1.5.5.5.5.2.cmml">;</mo><mi id="S2.Ex6.m1.4.4.4.4" xref="S2.Ex6.m1.4.4.4.4.cmml">X</mi><mo id="S2.Ex6.m1.5.5.5.5.1.5" stretchy="false" xref="S2.Ex6.m1.5.5.5.5.2.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.Ex6.m1.6b"><apply id="S2.Ex6.m1.6.7.cmml" 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id="S2.Ex7.m2.5c">\displaystyle:=\|(2^{ns}\varphi_{n}\ast f)_{n\geq 0}\|_{L^{p}(\mathbb{R}^{d},w% ;\ell^{q}(X))}<\infty,</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7.m2.5d">:= ∥ ( 2 start_POSTSUPERSCRIPT italic_n italic_s end_POSTSUPERSCRIPT italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_f ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; roman_ℓ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ( italic_X ) ) end_POSTSUBSCRIPT < ∞ ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S2.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|f\|_{H^{s,p}(\mathbb{R}^{d},w;X)}" 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xref="S2.Ex8.m1.5.5.5.5.1.1.2">ℝ</ci><ci id="S2.Ex8.m1.5.5.5.5.1.1.3.cmml" xref="S2.Ex8.m1.5.5.5.5.1.1.3">𝑑</ci></apply><ci id="S2.Ex8.m1.3.3.3.3.cmml" xref="S2.Ex8.m1.3.3.3.3">𝑤</ci><ci id="S2.Ex8.m1.4.4.4.4.cmml" xref="S2.Ex8.m1.4.4.4.4">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8.m1.6c">\displaystyle\|f\|_{H^{s,p}(\mathbb{R}^{d},w;X)}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m1.6d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\|\mathcal{F}^{-1}((1+|\cdot|^{2})^{s/2}\mathcal{F}f)\|_{L^{p}(% \mathbb{R}^{d},w;X)}<\infty,\quad w\in A_{p}(\mathbb{R}^{d})." class="ltx_math_unparsed" display="inline" id="S2.Ex8.m2.3"><semantics 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id="S2.Ex8.m2.3.10" mathvariant="normal">∞</mi><mo id="S2.Ex8.m2.3.11" rspace="1.167em">,</mo><mi id="S2.Ex8.m2.3.12">w</mi><mo id="S2.Ex8.m2.3.13">∈</mo><msub id="S2.Ex8.m2.3.14"><mi id="S2.Ex8.m2.3.14.2">A</mi><mi id="S2.Ex8.m2.3.14.3">p</mi></msub><mrow id="S2.Ex8.m2.3.15"><mo id="S2.Ex8.m2.3.15.1" stretchy="false">(</mo><msup id="S2.Ex8.m2.3.15.2"><mi id="S2.Ex8.m2.3.15.2.2">ℝ</mi><mi id="S2.Ex8.m2.3.15.2.3">d</mi></msup><mo id="S2.Ex8.m2.3.15.3" stretchy="false">)</mo></mrow><mo id="S2.Ex8.m2.3.16" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S2.Ex8.m2.3c">\displaystyle:=\|\mathcal{F}^{-1}((1+|\cdot|^{2})^{s/2}\mathcal{F}f)\|_{L^{p}(% \mathbb{R}^{d},w;X)}<\infty,\quad w\in A_{p}(\mathbb{R}^{d}).</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8.m2.3d">:= ∥ caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( ( 1 + | ⋅ | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT italic_s / 2 end_POSTSUPERSCRIPT caligraphic_F italic_f ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) end_POSTSUBSCRIPT < ∞ , italic_w ∈ italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p5.7">The definitions of the Besov and Triebel-Lizorkin spaces are independent of the chosen Littlewood-Paley sequence up to an equivalent norm, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 3.4]</cite>. We note that all the spaces defined above embed continuously into <math alttext="\SS^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS2.p5.5.m1.2"><semantics id="S2.SS2.p5.5.m1.2a"><mrow id="S2.SS2.p5.5.m1.2.2" xref="S2.SS2.p5.5.m1.2.2.cmml"><msup id="S2.SS2.p5.5.m1.2.2.3" xref="S2.SS2.p5.5.m1.2.2.3.cmml"><mi id="S2.SS2.p5.5.m1.2.2.3.2" xref="S2.SS2.p5.5.m1.2.2.3.2.cmml">SS</mi><mo id="S2.SS2.p5.5.m1.2.2.3.3" xref="S2.SS2.p5.5.m1.2.2.3.3.cmml">′</mo></msup><mo id="S2.SS2.p5.5.m1.2.2.2" xref="S2.SS2.p5.5.m1.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p5.5.m1.2.2.1.1" xref="S2.SS2.p5.5.m1.2.2.1.2.cmml"><mo id="S2.SS2.p5.5.m1.2.2.1.1.2" stretchy="false" xref="S2.SS2.p5.5.m1.2.2.1.2.cmml">(</mo><msup id="S2.SS2.p5.5.m1.2.2.1.1.1" xref="S2.SS2.p5.5.m1.2.2.1.1.1.cmml"><mi id="S2.SS2.p5.5.m1.2.2.1.1.1.2" xref="S2.SS2.p5.5.m1.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p5.5.m1.2.2.1.1.1.3" xref="S2.SS2.p5.5.m1.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p5.5.m1.2.2.1.1.3" xref="S2.SS2.p5.5.m1.2.2.1.2.cmml">;</mo><mi id="S2.SS2.p5.5.m1.1.1" xref="S2.SS2.p5.5.m1.1.1.cmml">X</mi><mo id="S2.SS2.p5.5.m1.2.2.1.1.4" stretchy="false" xref="S2.SS2.p5.5.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.5.m1.2b"><apply id="S2.SS2.p5.5.m1.2.2.cmml" xref="S2.SS2.p5.5.m1.2.2"><times id="S2.SS2.p5.5.m1.2.2.2.cmml" xref="S2.SS2.p5.5.m1.2.2.2"></times><apply id="S2.SS2.p5.5.m1.2.2.3.cmml" xref="S2.SS2.p5.5.m1.2.2.3"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m1.2.2.3.1.cmml" xref="S2.SS2.p5.5.m1.2.2.3">superscript</csymbol><ci id="S2.SS2.p5.5.m1.2.2.3.2.cmml" xref="S2.SS2.p5.5.m1.2.2.3.2">SS</ci><ci id="S2.SS2.p5.5.m1.2.2.3.3.cmml" xref="S2.SS2.p5.5.m1.2.2.3.3">′</ci></apply><list id="S2.SS2.p5.5.m1.2.2.1.2.cmml" xref="S2.SS2.p5.5.m1.2.2.1.1"><apply id="S2.SS2.p5.5.m1.2.2.1.1.1.cmml" xref="S2.SS2.p5.5.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.5.m1.2.2.1.1.1.1.cmml" xref="S2.SS2.p5.5.m1.2.2.1.1.1">superscript</csymbol><ci id="S2.SS2.p5.5.m1.2.2.1.1.1.2.cmml" xref="S2.SS2.p5.5.m1.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS2.p5.5.m1.2.2.1.1.1.3.cmml" xref="S2.SS2.p5.5.m1.2.2.1.1.1.3">𝑑</ci></apply><ci id="S2.SS2.p5.5.m1.1.1.cmml" xref="S2.SS2.p5.5.m1.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.5.m1.2c">\SS^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.5.m1.2d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib45" title="">45</a>, Section 5.2.1e]</cite>. Furthermore, <math alttext="\SS(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS2.p5.6.m2.2"><semantics id="S2.SS2.p5.6.m2.2a"><mrow id="S2.SS2.p5.6.m2.2.2" xref="S2.SS2.p5.6.m2.2.2.cmml"><mi id="S2.SS2.p5.6.m2.2.2.3" xref="S2.SS2.p5.6.m2.2.2.3.cmml">SS</mi><mo id="S2.SS2.p5.6.m2.2.2.2" xref="S2.SS2.p5.6.m2.2.2.2.cmml">⁢</mo><mrow id="S2.SS2.p5.6.m2.2.2.1.1" xref="S2.SS2.p5.6.m2.2.2.1.2.cmml"><mo id="S2.SS2.p5.6.m2.2.2.1.1.2" stretchy="false" xref="S2.SS2.p5.6.m2.2.2.1.2.cmml">(</mo><msup id="S2.SS2.p5.6.m2.2.2.1.1.1" xref="S2.SS2.p5.6.m2.2.2.1.1.1.cmml"><mi id="S2.SS2.p5.6.m2.2.2.1.1.1.2" xref="S2.SS2.p5.6.m2.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS2.p5.6.m2.2.2.1.1.1.3" xref="S2.SS2.p5.6.m2.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS2.p5.6.m2.2.2.1.1.3" xref="S2.SS2.p5.6.m2.2.2.1.2.cmml">;</mo><mi id="S2.SS2.p5.6.m2.1.1" xref="S2.SS2.p5.6.m2.1.1.cmml">X</mi><mo id="S2.SS2.p5.6.m2.2.2.1.1.4" stretchy="false" xref="S2.SS2.p5.6.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.6.m2.2b"><apply id="S2.SS2.p5.6.m2.2.2.cmml" xref="S2.SS2.p5.6.m2.2.2"><times id="S2.SS2.p5.6.m2.2.2.2.cmml" xref="S2.SS2.p5.6.m2.2.2.2"></times><ci id="S2.SS2.p5.6.m2.2.2.3.cmml" xref="S2.SS2.p5.6.m2.2.2.3">SS</ci><list id="S2.SS2.p5.6.m2.2.2.1.2.cmml" xref="S2.SS2.p5.6.m2.2.2.1.1"><apply id="S2.SS2.p5.6.m2.2.2.1.1.1.cmml" xref="S2.SS2.p5.6.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS2.p5.6.m2.2.2.1.1.1.1.cmml" xref="S2.SS2.p5.6.m2.2.2.1.1.1">superscript</csymbol><ci id="S2.SS2.p5.6.m2.2.2.1.1.1.2.cmml" xref="S2.SS2.p5.6.m2.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS2.p5.6.m2.2.2.1.1.1.3.cmml" xref="S2.SS2.p5.6.m2.2.2.1.1.1.3">𝑑</ci></apply><ci id="S2.SS2.p5.6.m2.1.1.cmml" xref="S2.SS2.p5.6.m2.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.6.m2.2c">\SS(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.6.m2.2d">roman_SS ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> embeds continuously into all the above spaces and this embedding is dense if <math alttext="p,q<\infty" class="ltx_Math" display="inline" id="S2.SS2.p5.7.m3.2"><semantics id="S2.SS2.p5.7.m3.2a"><mrow id="S2.SS2.p5.7.m3.2.3" xref="S2.SS2.p5.7.m3.2.3.cmml"><mrow id="S2.SS2.p5.7.m3.2.3.2.2" xref="S2.SS2.p5.7.m3.2.3.2.1.cmml"><mi id="S2.SS2.p5.7.m3.1.1" xref="S2.SS2.p5.7.m3.1.1.cmml">p</mi><mo id="S2.SS2.p5.7.m3.2.3.2.2.1" xref="S2.SS2.p5.7.m3.2.3.2.1.cmml">,</mo><mi id="S2.SS2.p5.7.m3.2.2" xref="S2.SS2.p5.7.m3.2.2.cmml">q</mi></mrow><mo id="S2.SS2.p5.7.m3.2.3.1" xref="S2.SS2.p5.7.m3.2.3.1.cmml"><</mo><mi id="S2.SS2.p5.7.m3.2.3.3" mathvariant="normal" xref="S2.SS2.p5.7.m3.2.3.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p5.7.m3.2b"><apply id="S2.SS2.p5.7.m3.2.3.cmml" xref="S2.SS2.p5.7.m3.2.3"><lt id="S2.SS2.p5.7.m3.2.3.1.cmml" xref="S2.SS2.p5.7.m3.2.3.1"></lt><list id="S2.SS2.p5.7.m3.2.3.2.1.cmml" xref="S2.SS2.p5.7.m3.2.3.2.2"><ci id="S2.SS2.p5.7.m3.1.1.cmml" xref="S2.SS2.p5.7.m3.1.1">𝑝</ci><ci id="S2.SS2.p5.7.m3.2.2.cmml" xref="S2.SS2.p5.7.m3.2.2">𝑞</ci></list><infinity id="S2.SS2.p5.7.m3.2.3.3.cmml" xref="S2.SS2.p5.7.m3.2.3.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p5.7.m3.2c">p,q<\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p5.7.m3.2d">italic_p , italic_q < ∞</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Lemma 3.8]</cite>.</p> </div> <div class="ltx_para" id="S2.SS2.p6"> <p class="ltx_p" id="S2.SS2.p6.5">Let <math alttext="\mathcal{O}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S2.SS2.p6.1.m1.1"><semantics id="S2.SS2.p6.1.m1.1a"><mrow id="S2.SS2.p6.1.m1.1.1" xref="S2.SS2.p6.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.1.m1.1.1.2" xref="S2.SS2.p6.1.m1.1.1.2.cmml">𝒪</mi><mo id="S2.SS2.p6.1.m1.1.1.1" xref="S2.SS2.p6.1.m1.1.1.1.cmml">⊆</mo><msup id="S2.SS2.p6.1.m1.1.1.3" xref="S2.SS2.p6.1.m1.1.1.3.cmml"><mi id="S2.SS2.p6.1.m1.1.1.3.2" xref="S2.SS2.p6.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS2.p6.1.m1.1.1.3.3" xref="S2.SS2.p6.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.1.m1.1b"><apply id="S2.SS2.p6.1.m1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1"><subset id="S2.SS2.p6.1.m1.1.1.1.cmml" xref="S2.SS2.p6.1.m1.1.1.1"></subset><ci id="S2.SS2.p6.1.m1.1.1.2.cmml" xref="S2.SS2.p6.1.m1.1.1.2">𝒪</ci><apply id="S2.SS2.p6.1.m1.1.1.3.cmml" xref="S2.SS2.p6.1.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS2.p6.1.m1.1.1.3.1.cmml" xref="S2.SS2.p6.1.m1.1.1.3">superscript</csymbol><ci id="S2.SS2.p6.1.m1.1.1.3.2.cmml" xref="S2.SS2.p6.1.m1.1.1.3.2">ℝ</ci><ci id="S2.SS2.p6.1.m1.1.1.3.3.cmml" xref="S2.SS2.p6.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.1.m1.1c">\mathcal{O}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.1.m1.1d">caligraphic_O ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be open. To define the <math alttext="B,F" class="ltx_Math" display="inline" id="S2.SS2.p6.2.m2.2"><semantics id="S2.SS2.p6.2.m2.2a"><mrow id="S2.SS2.p6.2.m2.2.3.2" xref="S2.SS2.p6.2.m2.2.3.1.cmml"><mi id="S2.SS2.p6.2.m2.1.1" xref="S2.SS2.p6.2.m2.1.1.cmml">B</mi><mo id="S2.SS2.p6.2.m2.2.3.2.1" xref="S2.SS2.p6.2.m2.2.3.1.cmml">,</mo><mi id="S2.SS2.p6.2.m2.2.2" xref="S2.SS2.p6.2.m2.2.2.cmml">F</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.2.m2.2b"><list id="S2.SS2.p6.2.m2.2.3.1.cmml" xref="S2.SS2.p6.2.m2.2.3.2"><ci id="S2.SS2.p6.2.m2.1.1.cmml" xref="S2.SS2.p6.2.m2.1.1">𝐵</ci><ci id="S2.SS2.p6.2.m2.2.2.cmml" xref="S2.SS2.p6.2.m2.2.2">𝐹</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.2.m2.2c">B,F</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.2.m2.2d">italic_B , italic_F</annotation></semantics></math> and <math alttext="H" class="ltx_Math" display="inline" id="S2.SS2.p6.3.m3.1"><semantics id="S2.SS2.p6.3.m3.1a"><mi id="S2.SS2.p6.3.m3.1.1" xref="S2.SS2.p6.3.m3.1.1.cmml">H</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.3.m3.1b"><ci id="S2.SS2.p6.3.m3.1.1.cmml" xref="S2.SS2.p6.3.m3.1.1">𝐻</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.3.m3.1c">H</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.3.m3.1d">italic_H</annotation></semantics></math> spaces on <math alttext="\mathcal{O}" class="ltx_Math" display="inline" id="S2.SS2.p6.4.m4.1"><semantics id="S2.SS2.p6.4.m4.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS2.p6.4.m4.1.1" xref="S2.SS2.p6.4.m4.1.1.cmml">𝒪</mi><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.4.m4.1b"><ci id="S2.SS2.p6.4.m4.1.1.cmml" xref="S2.SS2.p6.4.m4.1.1">𝒪</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.4.m4.1c">\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.4.m4.1d">caligraphic_O</annotation></semantics></math> we use <em class="ltx_emph ltx_font_italic" id="S2.SS2.p6.5.1">restriction/factor spaces</em>. Let <math alttext="{\mathbb{F}}\in\{B,F,H\}" class="ltx_Math" display="inline" id="S2.SS2.p6.5.m5.3"><semantics id="S2.SS2.p6.5.m5.3a"><mrow id="S2.SS2.p6.5.m5.3.4" xref="S2.SS2.p6.5.m5.3.4.cmml"><mi id="S2.SS2.p6.5.m5.3.4.2" xref="S2.SS2.p6.5.m5.3.4.2.cmml">𝔽</mi><mo id="S2.SS2.p6.5.m5.3.4.1" xref="S2.SS2.p6.5.m5.3.4.1.cmml">∈</mo><mrow id="S2.SS2.p6.5.m5.3.4.3.2" xref="S2.SS2.p6.5.m5.3.4.3.1.cmml"><mo id="S2.SS2.p6.5.m5.3.4.3.2.1" stretchy="false" xref="S2.SS2.p6.5.m5.3.4.3.1.cmml">{</mo><mi id="S2.SS2.p6.5.m5.1.1" xref="S2.SS2.p6.5.m5.1.1.cmml">B</mi><mo id="S2.SS2.p6.5.m5.3.4.3.2.2" xref="S2.SS2.p6.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.SS2.p6.5.m5.2.2" xref="S2.SS2.p6.5.m5.2.2.cmml">F</mi><mo id="S2.SS2.p6.5.m5.3.4.3.2.3" xref="S2.SS2.p6.5.m5.3.4.3.1.cmml">,</mo><mi id="S2.SS2.p6.5.m5.3.3" xref="S2.SS2.p6.5.m5.3.3.cmml">H</mi><mo id="S2.SS2.p6.5.m5.3.4.3.2.4" stretchy="false" xref="S2.SS2.p6.5.m5.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.5.m5.3b"><apply id="S2.SS2.p6.5.m5.3.4.cmml" xref="S2.SS2.p6.5.m5.3.4"><in id="S2.SS2.p6.5.m5.3.4.1.cmml" xref="S2.SS2.p6.5.m5.3.4.1"></in><ci id="S2.SS2.p6.5.m5.3.4.2.cmml" xref="S2.SS2.p6.5.m5.3.4.2">𝔽</ci><set id="S2.SS2.p6.5.m5.3.4.3.1.cmml" xref="S2.SS2.p6.5.m5.3.4.3.2"><ci id="S2.SS2.p6.5.m5.1.1.cmml" xref="S2.SS2.p6.5.m5.1.1">𝐵</ci><ci id="S2.SS2.p6.5.m5.2.2.cmml" xref="S2.SS2.p6.5.m5.2.2">𝐹</ci><ci id="S2.SS2.p6.5.m5.3.3.cmml" xref="S2.SS2.p6.5.m5.3.3">𝐻</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.5.m5.3c">{\mathbb{F}}\in\{B,F,H\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.5.m5.3d">blackboard_F ∈ { italic_B , italic_F , italic_H }</annotation></semantics></math>, then we define</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\mathbb{F}}(\mathcal{O},w;X):=\big{\{}f\in\mathcal{D}^{\prime}(\mathbb{R}^{d}% ;X):\exists g\in{\mathbb{F}}(\mathbb{R}^{d},w;X),g|_{\mathcal{O}}=f\big{\}}" class="ltx_Math" display="block" id="S2.Ex9.m1.10"><semantics id="S2.Ex9.m1.10a"><mrow id="S2.Ex9.m1.10.10" xref="S2.Ex9.m1.10.10.cmml"><mrow id="S2.Ex9.m1.10.10.4" xref="S2.Ex9.m1.10.10.4.cmml"><mi id="S2.Ex9.m1.10.10.4.2" xref="S2.Ex9.m1.10.10.4.2.cmml">𝔽</mi><mo id="S2.Ex9.m1.10.10.4.1" xref="S2.Ex9.m1.10.10.4.1.cmml">⁢</mo><mrow id="S2.Ex9.m1.10.10.4.3.2" xref="S2.Ex9.m1.10.10.4.3.1.cmml"><mo id="S2.Ex9.m1.10.10.4.3.2.1" stretchy="false" xref="S2.Ex9.m1.10.10.4.3.1.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S2.Ex9.m1.1.1" xref="S2.Ex9.m1.1.1.cmml">𝒪</mi><mo id="S2.Ex9.m1.10.10.4.3.2.2" xref="S2.Ex9.m1.10.10.4.3.1.cmml">,</mo><mi id="S2.Ex9.m1.2.2" xref="S2.Ex9.m1.2.2.cmml">w</mi><mo id="S2.Ex9.m1.10.10.4.3.2.3" xref="S2.Ex9.m1.10.10.4.3.1.cmml">;</mo><mi id="S2.Ex9.m1.3.3" xref="S2.Ex9.m1.3.3.cmml">X</mi><mo id="S2.Ex9.m1.10.10.4.3.2.4" rspace="0.278em" stretchy="false" xref="S2.Ex9.m1.10.10.4.3.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex9.m1.10.10.3" rspace="0.278em" xref="S2.Ex9.m1.10.10.3.cmml">:=</mo><mrow id="S2.Ex9.m1.10.10.2.2" xref="S2.Ex9.m1.10.10.2.3.cmml"><mo id="S2.Ex9.m1.10.10.2.2.3" maxsize="120%" minsize="120%" xref="S2.Ex9.m1.10.10.2.3.1.cmml">{</mo><mrow id="S2.Ex9.m1.9.9.1.1.1" xref="S2.Ex9.m1.9.9.1.1.1.cmml"><mi id="S2.Ex9.m1.9.9.1.1.1.3" xref="S2.Ex9.m1.9.9.1.1.1.3.cmml">f</mi><mo id="S2.Ex9.m1.9.9.1.1.1.2" xref="S2.Ex9.m1.9.9.1.1.1.2.cmml">∈</mo><mrow id="S2.Ex9.m1.9.9.1.1.1.1" xref="S2.Ex9.m1.9.9.1.1.1.1.cmml"><msup id="S2.Ex9.m1.9.9.1.1.1.1.3" xref="S2.Ex9.m1.9.9.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex9.m1.9.9.1.1.1.1.3.2" xref="S2.Ex9.m1.9.9.1.1.1.1.3.2.cmml">𝒟</mi><mo id="S2.Ex9.m1.9.9.1.1.1.1.3.3" xref="S2.Ex9.m1.9.9.1.1.1.1.3.3.cmml">′</mo></msup><mo id="S2.Ex9.m1.9.9.1.1.1.1.2" xref="S2.Ex9.m1.9.9.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex9.m1.9.9.1.1.1.1.1.1" xref="S2.Ex9.m1.9.9.1.1.1.1.1.2.cmml"><mo id="S2.Ex9.m1.9.9.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex9.m1.9.9.1.1.1.1.1.2.cmml">(</mo><msup id="S2.Ex9.m1.9.9.1.1.1.1.1.1.1" xref="S2.Ex9.m1.9.9.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex9.m1.9.9.1.1.1.1.1.1.1.2" xref="S2.Ex9.m1.9.9.1.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.Ex9.m1.9.9.1.1.1.1.1.1.1.3" xref="S2.Ex9.m1.9.9.1.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.Ex9.m1.9.9.1.1.1.1.1.1.3" xref="S2.Ex9.m1.9.9.1.1.1.1.1.2.cmml">;</mo><mi id="S2.Ex9.m1.4.4" xref="S2.Ex9.m1.4.4.cmml">X</mi><mo id="S2.Ex9.m1.9.9.1.1.1.1.1.1.4" rspace="0.278em" stretchy="false" xref="S2.Ex9.m1.9.9.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex9.m1.10.10.2.2.4" rspace="0.278em" xref="S2.Ex9.m1.10.10.2.3.1.cmml">:</mo><mrow id="S2.Ex9.m1.10.10.2.2.2.2" xref="S2.Ex9.m1.10.10.2.2.2.3.cmml"><mrow id="S2.Ex9.m1.10.10.2.2.2.1.1" xref="S2.Ex9.m1.10.10.2.2.2.1.1.cmml"><mrow id="S2.Ex9.m1.10.10.2.2.2.1.1.3" xref="S2.Ex9.m1.10.10.2.2.2.1.1.3.cmml"><mo id="S2.Ex9.m1.10.10.2.2.2.1.1.3.1" rspace="0.167em" xref="S2.Ex9.m1.10.10.2.2.2.1.1.3.1.cmml">∃</mo><mi id="S2.Ex9.m1.10.10.2.2.2.1.1.3.2" xref="S2.Ex9.m1.10.10.2.2.2.1.1.3.2.cmml">g</mi></mrow><mo id="S2.Ex9.m1.10.10.2.2.2.1.1.2" xref="S2.Ex9.m1.10.10.2.2.2.1.1.2.cmml">∈</mo><mrow id="S2.Ex9.m1.10.10.2.2.2.1.1.1" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.cmml"><mi id="S2.Ex9.m1.10.10.2.2.2.1.1.1.3" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.3.cmml">𝔽</mi><mo id="S2.Ex9.m1.10.10.2.2.2.1.1.1.2" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.2.cmml"><mo id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.2" stretchy="false" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.2.cmml">(</mo><msup id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.1" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.1.cmml"><mi id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.1.2" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.1.3" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.3" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.2.cmml">,</mo><mi id="S2.Ex9.m1.5.5" xref="S2.Ex9.m1.5.5.cmml">w</mi><mo id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.4" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.2.cmml">;</mo><mi id="S2.Ex9.m1.6.6" xref="S2.Ex9.m1.6.6.cmml">X</mi><mo id="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.1.5" stretchy="false" xref="S2.Ex9.m1.10.10.2.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S2.Ex9.m1.10.10.2.2.2.2.3" xref="S2.Ex9.m1.10.10.2.2.2.3a.cmml">,</mo><mrow id="S2.Ex9.m1.10.10.2.2.2.2.2" xref="S2.Ex9.m1.10.10.2.2.2.2.2.cmml"><msub id="S2.Ex9.m1.10.10.2.2.2.2.2.2.2" xref="S2.Ex9.m1.10.10.2.2.2.2.2.2.1.cmml"><mrow id="S2.Ex9.m1.10.10.2.2.2.2.2.2.2.2" xref="S2.Ex9.m1.10.10.2.2.2.2.2.2.1.cmml"><mi id="S2.Ex9.m1.7.7" xref="S2.Ex9.m1.7.7.cmml">g</mi><mo id="S2.Ex9.m1.10.10.2.2.2.2.2.2.2.2.1" stretchy="false" xref="S2.Ex9.m1.10.10.2.2.2.2.2.2.1.1.cmml">|</mo></mrow><mi class="ltx_font_mathcaligraphic" id="S2.Ex9.m1.8.8.1" xref="S2.Ex9.m1.8.8.1.cmml">𝒪</mi></msub><mo id="S2.Ex9.m1.10.10.2.2.2.2.2.1" xref="S2.Ex9.m1.10.10.2.2.2.2.2.1.cmml">=</mo><mi id="S2.Ex9.m1.10.10.2.2.2.2.2.3" xref="S2.Ex9.m1.10.10.2.2.2.2.2.3.cmml">f</mi></mrow></mrow><mo id="S2.Ex9.m1.10.10.2.2.5" maxsize="120%" minsize="120%" xref="S2.Ex9.m1.10.10.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex9.m1.10b"><apply id="S2.Ex9.m1.10.10.cmml" xref="S2.Ex9.m1.10.10"><csymbol cd="latexml" id="S2.Ex9.m1.10.10.3.cmml" xref="S2.Ex9.m1.10.10.3">assign</csymbol><apply id="S2.Ex9.m1.10.10.4.cmml" xref="S2.Ex9.m1.10.10.4"><times id="S2.Ex9.m1.10.10.4.1.cmml" xref="S2.Ex9.m1.10.10.4.1"></times><ci id="S2.Ex9.m1.10.10.4.2.cmml" xref="S2.Ex9.m1.10.10.4.2">𝔽</ci><vector id="S2.Ex9.m1.10.10.4.3.1.cmml" xref="S2.Ex9.m1.10.10.4.3.2"><ci id="S2.Ex9.m1.1.1.cmml" xref="S2.Ex9.m1.1.1">𝒪</ci><ci 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g\in{\mathbb{F}}(\mathbb{R}^{d},w;X),g|_{\mathcal{O}}=f\big{\}}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9.m1.10d">blackboard_F ( caligraphic_O , italic_w ; italic_X ) := { italic_f ∈ caligraphic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X ) : ∃ italic_g ∈ blackboard_F ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) , italic_g | start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT = italic_f }</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p6.9">endowed with the norm</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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id="S2.Ex10.m1.8.13.6.3">𝒪</mi></msub><mo id="S2.Ex10.m1.8.13.7" lspace="0.167em">=</mo><mi id="S2.Ex10.m1.8.13.8">f</mi><mo id="S2.Ex10.m1.8.13.9" stretchy="false">}</mo></mrow><mo id="S2.Ex10.m1.8.14" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S2.Ex10.m1.8c">\|f\|_{{\mathbb{F}}(\mathcal{O},w;X)}:=\inf\{\|g\|_{{\mathbb{F}}(\mathbb{R}^{d% },w;X)}:g|_{\mathcal{O}}=f\}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex10.m1.8d">∥ italic_f ∥ start_POSTSUBSCRIPT blackboard_F ( caligraphic_O , italic_w ; italic_X ) end_POSTSUBSCRIPT := roman_inf { ∥ italic_g ∥ start_POSTSUBSCRIPT blackboard_F ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) end_POSTSUBSCRIPT : italic_g | start_POSTSUBSCRIPT caligraphic_O end_POSTSUBSCRIPT = italic_f } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS2.p6.8">Similarly, we can define the factor space for weighted Sobolev spaces. 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blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT )</annotation></semantics></math>, then the weighted Sobolev spaces on <math alttext="\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S2.SS2.p6.8.m3.1"><semantics id="S2.SS2.p6.8.m3.1a"><msubsup id="S2.SS2.p6.8.m3.1.1" xref="S2.SS2.p6.8.m3.1.1.cmml"><mi id="S2.SS2.p6.8.m3.1.1.2.2" xref="S2.SS2.p6.8.m3.1.1.2.2.cmml">ℝ</mi><mo id="S2.SS2.p6.8.m3.1.1.3" xref="S2.SS2.p6.8.m3.1.1.3.cmml">+</mo><mi id="S2.SS2.p6.8.m3.1.1.2.3" xref="S2.SS2.p6.8.m3.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S2.SS2.p6.8.m3.1b"><apply id="S2.SS2.p6.8.m3.1.1.cmml" xref="S2.SS2.p6.8.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p6.8.m3.1.1.1.cmml" xref="S2.SS2.p6.8.m3.1.1">subscript</csymbol><apply id="S2.SS2.p6.8.m3.1.1.2.cmml" xref="S2.SS2.p6.8.m3.1.1"><csymbol cd="ambiguous" id="S2.SS2.p6.8.m3.1.1.2.1.cmml" xref="S2.SS2.p6.8.m3.1.1">superscript</csymbol><ci id="S2.SS2.p6.8.m3.1.1.2.2.cmml" xref="S2.SS2.p6.8.m3.1.1.2.2">ℝ</ci><ci id="S2.SS2.p6.8.m3.1.1.2.3.cmml" xref="S2.SS2.p6.8.m3.1.1.2.3">𝑑</ci></apply><plus id="S2.SS2.p6.8.m3.1.1.3.cmml" xref="S2.SS2.p6.8.m3.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS2.p6.8.m3.1c">\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S2.SS2.p6.8.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> as defined above coincide with their factor spaces, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Section 5]</cite>. The proof is based on the existence of an extension operator (in the sense of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Definition 5.2]</cite>) and can also be extended to sufficiently smooth domains, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib2" title="">2</a>, Theorem 5.22]</cite>. For a discussion of extension operators on weighted spaces, we refer to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib70" title="">70</a>, Section 2.1.2]</cite>.</p> </div> </section> <section class="ltx_subsection" id="S2.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.3. </span>Properties of Banach spaces</h3> <div class="ltx_para" id="S2.SS3.p1"> <p class="ltx_p" id="S2.SS3.p1.9">We recall the Fatou and <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.SS3.p1.1.m1.1"><semantics id="S2.SS3.p1.1.m1.1a"><mi id="S2.SS3.p1.1.m1.1.1" xref="S2.SS3.p1.1.m1.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.1.m1.1b"><ci id="S2.SS3.p1.1.m1.1.1.cmml" xref="S2.SS3.p1.1.m1.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.1.m1.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.1.m1.1d">roman_UMD</annotation></semantics></math> property for Banach spaces. If <math alttext="X,\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.2.m2.2"><semantics id="S2.SS3.p1.2.m2.2a"><mrow id="S2.SS3.p1.2.m2.2.3.2" xref="S2.SS3.p1.2.m2.2.3.1.cmml"><mi id="S2.SS3.p1.2.m2.1.1" xref="S2.SS3.p1.2.m2.1.1.cmml">X</mi><mo id="S2.SS3.p1.2.m2.2.3.2.1" xref="S2.SS3.p1.2.m2.2.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.2.m2.2.2" xref="S2.SS3.p1.2.m2.2.2.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.2.m2.2b"><list id="S2.SS3.p1.2.m2.2.3.1.cmml" xref="S2.SS3.p1.2.m2.2.3.2"><ci id="S2.SS3.p1.2.m2.1.1.cmml" xref="S2.SS3.p1.2.m2.1.1">𝑋</ci><ci id="S2.SS3.p1.2.m2.2.2.cmml" xref="S2.SS3.p1.2.m2.2.2">𝒳</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.2.m2.2c">X,\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.2.m2.2d">italic_X , caligraphic_X</annotation></semantics></math> are Banach spaces such that <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.3.m3.1"><semantics id="S2.SS3.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.3.m3.1.1" xref="S2.SS3.p1.3.m3.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.3.m3.1b"><ci id="S2.SS3.p1.3.m3.1.1.cmml" xref="S2.SS3.p1.3.m3.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.3.m3.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.3.m3.1d">caligraphic_X</annotation></semantics></math> embeds continuously into <math alttext="\SS^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS3.p1.4.m4.2"><semantics id="S2.SS3.p1.4.m4.2a"><mrow id="S2.SS3.p1.4.m4.2.2" xref="S2.SS3.p1.4.m4.2.2.cmml"><msup id="S2.SS3.p1.4.m4.2.2.3" xref="S2.SS3.p1.4.m4.2.2.3.cmml"><mi id="S2.SS3.p1.4.m4.2.2.3.2" xref="S2.SS3.p1.4.m4.2.2.3.2.cmml">SS</mi><mo id="S2.SS3.p1.4.m4.2.2.3.3" xref="S2.SS3.p1.4.m4.2.2.3.3.cmml">′</mo></msup><mo id="S2.SS3.p1.4.m4.2.2.2" xref="S2.SS3.p1.4.m4.2.2.2.cmml">⁢</mo><mrow id="S2.SS3.p1.4.m4.2.2.1.1" xref="S2.SS3.p1.4.m4.2.2.1.2.cmml"><mo id="S2.SS3.p1.4.m4.2.2.1.1.2" stretchy="false" xref="S2.SS3.p1.4.m4.2.2.1.2.cmml">(</mo><msup id="S2.SS3.p1.4.m4.2.2.1.1.1" xref="S2.SS3.p1.4.m4.2.2.1.1.1.cmml"><mi id="S2.SS3.p1.4.m4.2.2.1.1.1.2" xref="S2.SS3.p1.4.m4.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS3.p1.4.m4.2.2.1.1.1.3" xref="S2.SS3.p1.4.m4.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS3.p1.4.m4.2.2.1.1.3" xref="S2.SS3.p1.4.m4.2.2.1.2.cmml">;</mo><mi id="S2.SS3.p1.4.m4.1.1" xref="S2.SS3.p1.4.m4.1.1.cmml">X</mi><mo id="S2.SS3.p1.4.m4.2.2.1.1.4" stretchy="false" xref="S2.SS3.p1.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.4.m4.2b"><apply id="S2.SS3.p1.4.m4.2.2.cmml" xref="S2.SS3.p1.4.m4.2.2"><times id="S2.SS3.p1.4.m4.2.2.2.cmml" xref="S2.SS3.p1.4.m4.2.2.2"></times><apply id="S2.SS3.p1.4.m4.2.2.3.cmml" xref="S2.SS3.p1.4.m4.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.p1.4.m4.2.2.3.1.cmml" xref="S2.SS3.p1.4.m4.2.2.3">superscript</csymbol><ci id="S2.SS3.p1.4.m4.2.2.3.2.cmml" xref="S2.SS3.p1.4.m4.2.2.3.2">SS</ci><ci id="S2.SS3.p1.4.m4.2.2.3.3.cmml" xref="S2.SS3.p1.4.m4.2.2.3.3">′</ci></apply><list id="S2.SS3.p1.4.m4.2.2.1.2.cmml" xref="S2.SS3.p1.4.m4.2.2.1.1"><apply id="S2.SS3.p1.4.m4.2.2.1.1.1.cmml" xref="S2.SS3.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.4.m4.2.2.1.1.1.1.cmml" xref="S2.SS3.p1.4.m4.2.2.1.1.1">superscript</csymbol><ci id="S2.SS3.p1.4.m4.2.2.1.1.1.2.cmml" xref="S2.SS3.p1.4.m4.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS3.p1.4.m4.2.2.1.1.1.3.cmml" xref="S2.SS3.p1.4.m4.2.2.1.1.1.3">𝑑</ci></apply><ci id="S2.SS3.p1.4.m4.1.1.cmml" xref="S2.SS3.p1.4.m4.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.4.m4.2c">\SS^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.4.m4.2d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>, then we say that <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.5.m5.1"><semantics id="S2.SS3.p1.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.5.m5.1.1" xref="S2.SS3.p1.5.m5.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.5.m5.1b"><ci id="S2.SS3.p1.5.m5.1.1.cmml" xref="S2.SS3.p1.5.m5.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.5.m5.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.5.m5.1d">caligraphic_X</annotation></semantics></math> has the <em class="ltx_emph ltx_font_italic" id="S2.SS3.p1.9.1">Fatou property</em> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib10" title="">10</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib17" title="">17</a>]</cite>) if for all <math alttext="(f_{n})_{n\geq 0}\subseteq\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS3.p1.6.m6.1"><semantics id="S2.SS3.p1.6.m6.1a"><mrow id="S2.SS3.p1.6.m6.1.1" xref="S2.SS3.p1.6.m6.1.1.cmml"><msub id="S2.SS3.p1.6.m6.1.1.1" xref="S2.SS3.p1.6.m6.1.1.1.cmml"><mrow id="S2.SS3.p1.6.m6.1.1.1.1.1" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.cmml"><mo id="S2.SS3.p1.6.m6.1.1.1.1.1.2" stretchy="false" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.cmml">(</mo><msub id="S2.SS3.p1.6.m6.1.1.1.1.1.1" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.cmml"><mi id="S2.SS3.p1.6.m6.1.1.1.1.1.1.2" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.2.cmml">f</mi><mi id="S2.SS3.p1.6.m6.1.1.1.1.1.1.3" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.SS3.p1.6.m6.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S2.SS3.p1.6.m6.1.1.1.3" xref="S2.SS3.p1.6.m6.1.1.1.3.cmml"><mi id="S2.SS3.p1.6.m6.1.1.1.3.2" xref="S2.SS3.p1.6.m6.1.1.1.3.2.cmml">n</mi><mo id="S2.SS3.p1.6.m6.1.1.1.3.1" xref="S2.SS3.p1.6.m6.1.1.1.3.1.cmml">≥</mo><mn id="S2.SS3.p1.6.m6.1.1.1.3.3" xref="S2.SS3.p1.6.m6.1.1.1.3.3.cmml">0</mn></mrow></msub><mo id="S2.SS3.p1.6.m6.1.1.2" xref="S2.SS3.p1.6.m6.1.1.2.cmml">⊆</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.6.m6.1.1.3" xref="S2.SS3.p1.6.m6.1.1.3.cmml">𝒳</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.6.m6.1b"><apply id="S2.SS3.p1.6.m6.1.1.cmml" xref="S2.SS3.p1.6.m6.1.1"><subset id="S2.SS3.p1.6.m6.1.1.2.cmml" xref="S2.SS3.p1.6.m6.1.1.2"></subset><apply id="S2.SS3.p1.6.m6.1.1.1.cmml" xref="S2.SS3.p1.6.m6.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.6.m6.1.1.1.2.cmml" xref="S2.SS3.p1.6.m6.1.1.1">subscript</csymbol><apply id="S2.SS3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S2.SS3.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="S2.SS3.p1.6.m6.1.1.1.1.1.1.2.cmml" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.2">𝑓</ci><ci id="S2.SS3.p1.6.m6.1.1.1.1.1.1.3.cmml" xref="S2.SS3.p1.6.m6.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S2.SS3.p1.6.m6.1.1.1.3.cmml" xref="S2.SS3.p1.6.m6.1.1.1.3"><geq id="S2.SS3.p1.6.m6.1.1.1.3.1.cmml" xref="S2.SS3.p1.6.m6.1.1.1.3.1"></geq><ci id="S2.SS3.p1.6.m6.1.1.1.3.2.cmml" xref="S2.SS3.p1.6.m6.1.1.1.3.2">𝑛</ci><cn id="S2.SS3.p1.6.m6.1.1.1.3.3.cmml" type="integer" xref="S2.SS3.p1.6.m6.1.1.1.3.3">0</cn></apply></apply><ci id="S2.SS3.p1.6.m6.1.1.3.cmml" xref="S2.SS3.p1.6.m6.1.1.3">𝒳</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.6.m6.1c">(f_{n})_{n\geq 0}\subseteq\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.6.m6.1d">( italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ⊆ caligraphic_X</annotation></semantics></math> such that <math alttext="\lim_{n\to\infty}f_{n}=f" class="ltx_Math" display="inline" id="S2.SS3.p1.7.m7.1"><semantics id="S2.SS3.p1.7.m7.1a"><mrow id="S2.SS3.p1.7.m7.1.1" xref="S2.SS3.p1.7.m7.1.1.cmml"><mrow id="S2.SS3.p1.7.m7.1.1.2" xref="S2.SS3.p1.7.m7.1.1.2.cmml"><msub id="S2.SS3.p1.7.m7.1.1.2.1" xref="S2.SS3.p1.7.m7.1.1.2.1.cmml"><mo id="S2.SS3.p1.7.m7.1.1.2.1.2" xref="S2.SS3.p1.7.m7.1.1.2.1.2.cmml">lim</mo><mrow id="S2.SS3.p1.7.m7.1.1.2.1.3" xref="S2.SS3.p1.7.m7.1.1.2.1.3.cmml"><mi id="S2.SS3.p1.7.m7.1.1.2.1.3.2" xref="S2.SS3.p1.7.m7.1.1.2.1.3.2.cmml">n</mi><mo id="S2.SS3.p1.7.m7.1.1.2.1.3.1" stretchy="false" xref="S2.SS3.p1.7.m7.1.1.2.1.3.1.cmml">→</mo><mi id="S2.SS3.p1.7.m7.1.1.2.1.3.3" mathvariant="normal" xref="S2.SS3.p1.7.m7.1.1.2.1.3.3.cmml">∞</mi></mrow></msub><msub id="S2.SS3.p1.7.m7.1.1.2.2" xref="S2.SS3.p1.7.m7.1.1.2.2.cmml"><mi id="S2.SS3.p1.7.m7.1.1.2.2.2" xref="S2.SS3.p1.7.m7.1.1.2.2.2.cmml">f</mi><mi id="S2.SS3.p1.7.m7.1.1.2.2.3" xref="S2.SS3.p1.7.m7.1.1.2.2.3.cmml">n</mi></msub></mrow><mo id="S2.SS3.p1.7.m7.1.1.1" xref="S2.SS3.p1.7.m7.1.1.1.cmml">=</mo><mi id="S2.SS3.p1.7.m7.1.1.3" xref="S2.SS3.p1.7.m7.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.7.m7.1b"><apply id="S2.SS3.p1.7.m7.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1"><eq id="S2.SS3.p1.7.m7.1.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1.1"></eq><apply id="S2.SS3.p1.7.m7.1.1.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2"><apply id="S2.SS3.p1.7.m7.1.1.2.1.cmml" xref="S2.SS3.p1.7.m7.1.1.2.1"><csymbol cd="ambiguous" id="S2.SS3.p1.7.m7.1.1.2.1.1.cmml" xref="S2.SS3.p1.7.m7.1.1.2.1">subscript</csymbol><limit id="S2.SS3.p1.7.m7.1.1.2.1.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2.1.2"></limit><apply id="S2.SS3.p1.7.m7.1.1.2.1.3.cmml" xref="S2.SS3.p1.7.m7.1.1.2.1.3"><ci id="S2.SS3.p1.7.m7.1.1.2.1.3.1.cmml" xref="S2.SS3.p1.7.m7.1.1.2.1.3.1">→</ci><ci id="S2.SS3.p1.7.m7.1.1.2.1.3.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2.1.3.2">𝑛</ci><infinity id="S2.SS3.p1.7.m7.1.1.2.1.3.3.cmml" xref="S2.SS3.p1.7.m7.1.1.2.1.3.3"></infinity></apply></apply><apply id="S2.SS3.p1.7.m7.1.1.2.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS3.p1.7.m7.1.1.2.2.1.cmml" xref="S2.SS3.p1.7.m7.1.1.2.2">subscript</csymbol><ci id="S2.SS3.p1.7.m7.1.1.2.2.2.cmml" xref="S2.SS3.p1.7.m7.1.1.2.2.2">𝑓</ci><ci id="S2.SS3.p1.7.m7.1.1.2.2.3.cmml" xref="S2.SS3.p1.7.m7.1.1.2.2.3">𝑛</ci></apply></apply><ci id="S2.SS3.p1.7.m7.1.1.3.cmml" xref="S2.SS3.p1.7.m7.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.7.m7.1c">\lim_{n\to\infty}f_{n}=f</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.7.m7.1d">roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = italic_f</annotation></semantics></math> in <math alttext="\SS^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS3.p1.8.m8.2"><semantics id="S2.SS3.p1.8.m8.2a"><mrow id="S2.SS3.p1.8.m8.2.2" xref="S2.SS3.p1.8.m8.2.2.cmml"><msup id="S2.SS3.p1.8.m8.2.2.3" xref="S2.SS3.p1.8.m8.2.2.3.cmml"><mi id="S2.SS3.p1.8.m8.2.2.3.2" xref="S2.SS3.p1.8.m8.2.2.3.2.cmml">SS</mi><mo id="S2.SS3.p1.8.m8.2.2.3.3" xref="S2.SS3.p1.8.m8.2.2.3.3.cmml">′</mo></msup><mo id="S2.SS3.p1.8.m8.2.2.2" xref="S2.SS3.p1.8.m8.2.2.2.cmml">⁢</mo><mrow id="S2.SS3.p1.8.m8.2.2.1.1" xref="S2.SS3.p1.8.m8.2.2.1.2.cmml"><mo id="S2.SS3.p1.8.m8.2.2.1.1.2" stretchy="false" xref="S2.SS3.p1.8.m8.2.2.1.2.cmml">(</mo><msup id="S2.SS3.p1.8.m8.2.2.1.1.1" xref="S2.SS3.p1.8.m8.2.2.1.1.1.cmml"><mi id="S2.SS3.p1.8.m8.2.2.1.1.1.2" xref="S2.SS3.p1.8.m8.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS3.p1.8.m8.2.2.1.1.1.3" xref="S2.SS3.p1.8.m8.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS3.p1.8.m8.2.2.1.1.3" xref="S2.SS3.p1.8.m8.2.2.1.2.cmml">;</mo><mi id="S2.SS3.p1.8.m8.1.1" xref="S2.SS3.p1.8.m8.1.1.cmml">X</mi><mo id="S2.SS3.p1.8.m8.2.2.1.1.4" stretchy="false" xref="S2.SS3.p1.8.m8.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.8.m8.2b"><apply id="S2.SS3.p1.8.m8.2.2.cmml" xref="S2.SS3.p1.8.m8.2.2"><times id="S2.SS3.p1.8.m8.2.2.2.cmml" xref="S2.SS3.p1.8.m8.2.2.2"></times><apply id="S2.SS3.p1.8.m8.2.2.3.cmml" xref="S2.SS3.p1.8.m8.2.2.3"><csymbol cd="ambiguous" id="S2.SS3.p1.8.m8.2.2.3.1.cmml" xref="S2.SS3.p1.8.m8.2.2.3">superscript</csymbol><ci id="S2.SS3.p1.8.m8.2.2.3.2.cmml" xref="S2.SS3.p1.8.m8.2.2.3.2">SS</ci><ci id="S2.SS3.p1.8.m8.2.2.3.3.cmml" xref="S2.SS3.p1.8.m8.2.2.3.3">′</ci></apply><list id="S2.SS3.p1.8.m8.2.2.1.2.cmml" xref="S2.SS3.p1.8.m8.2.2.1.1"><apply id="S2.SS3.p1.8.m8.2.2.1.1.1.cmml" xref="S2.SS3.p1.8.m8.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.8.m8.2.2.1.1.1.1.cmml" xref="S2.SS3.p1.8.m8.2.2.1.1.1">superscript</csymbol><ci id="S2.SS3.p1.8.m8.2.2.1.1.1.2.cmml" xref="S2.SS3.p1.8.m8.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS3.p1.8.m8.2.2.1.1.1.3.cmml" xref="S2.SS3.p1.8.m8.2.2.1.1.1.3">𝑑</ci></apply><ci id="S2.SS3.p1.8.m8.1.1.cmml" xref="S2.SS3.p1.8.m8.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.8.m8.2c">\SS^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.8.m8.2d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> and <math alttext="\liminf_{n\to\infty}\|f_{n}\|_{\mathcal{X}}<\infty" class="ltx_Math" display="inline" id="S2.SS3.p1.9.m9.1"><semantics id="S2.SS3.p1.9.m9.1a"><mrow id="S2.SS3.p1.9.m9.1.1" xref="S2.SS3.p1.9.m9.1.1.cmml"><mrow id="S2.SS3.p1.9.m9.1.1.1" xref="S2.SS3.p1.9.m9.1.1.1.cmml"><msub id="S2.SS3.p1.9.m9.1.1.1.2" xref="S2.SS3.p1.9.m9.1.1.1.2.cmml"><mo id="S2.SS3.p1.9.m9.1.1.1.2.2" xref="S2.SS3.p1.9.m9.1.1.1.2.2.cmml">lim inf</mo><mrow id="S2.SS3.p1.9.m9.1.1.1.2.3" xref="S2.SS3.p1.9.m9.1.1.1.2.3.cmml"><mi id="S2.SS3.p1.9.m9.1.1.1.2.3.2" xref="S2.SS3.p1.9.m9.1.1.1.2.3.2.cmml">n</mi><mo id="S2.SS3.p1.9.m9.1.1.1.2.3.1" stretchy="false" xref="S2.SS3.p1.9.m9.1.1.1.2.3.1.cmml">→</mo><mi id="S2.SS3.p1.9.m9.1.1.1.2.3.3" mathvariant="normal" xref="S2.SS3.p1.9.m9.1.1.1.2.3.3.cmml">∞</mi></mrow></msub><msub id="S2.SS3.p1.9.m9.1.1.1.1" xref="S2.SS3.p1.9.m9.1.1.1.1.cmml"><mrow id="S2.SS3.p1.9.m9.1.1.1.1.1.1" xref="S2.SS3.p1.9.m9.1.1.1.1.1.2.cmml"><mo id="S2.SS3.p1.9.m9.1.1.1.1.1.1.2" lspace="0em" stretchy="false" xref="S2.SS3.p1.9.m9.1.1.1.1.1.2.1.cmml">‖</mo><msub id="S2.SS3.p1.9.m9.1.1.1.1.1.1.1" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.cmml"><mi id="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.2" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.2.cmml">f</mi><mi id="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.3" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.SS3.p1.9.m9.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS3.p1.9.m9.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mi class="ltx_font_mathcaligraphic" id="S2.SS3.p1.9.m9.1.1.1.1.3" xref="S2.SS3.p1.9.m9.1.1.1.1.3.cmml">𝒳</mi></msub></mrow><mo id="S2.SS3.p1.9.m9.1.1.2" xref="S2.SS3.p1.9.m9.1.1.2.cmml"><</mo><mi id="S2.SS3.p1.9.m9.1.1.3" mathvariant="normal" xref="S2.SS3.p1.9.m9.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p1.9.m9.1b"><apply id="S2.SS3.p1.9.m9.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1"><lt id="S2.SS3.p1.9.m9.1.1.2.cmml" xref="S2.SS3.p1.9.m9.1.1.2"></lt><apply id="S2.SS3.p1.9.m9.1.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1"><apply id="S2.SS3.p1.9.m9.1.1.1.2.cmml" xref="S2.SS3.p1.9.m9.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS3.p1.9.m9.1.1.1.2.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1.2">subscript</csymbol><csymbol cd="latexml" id="S2.SS3.p1.9.m9.1.1.1.2.2.cmml" xref="S2.SS3.p1.9.m9.1.1.1.2.2">limit-infimum</csymbol><apply id="S2.SS3.p1.9.m9.1.1.1.2.3.cmml" xref="S2.SS3.p1.9.m9.1.1.1.2.3"><ci id="S2.SS3.p1.9.m9.1.1.1.2.3.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1.2.3.1">→</ci><ci id="S2.SS3.p1.9.m9.1.1.1.2.3.2.cmml" xref="S2.SS3.p1.9.m9.1.1.1.2.3.2">𝑛</ci><infinity id="S2.SS3.p1.9.m9.1.1.1.2.3.3.cmml" xref="S2.SS3.p1.9.m9.1.1.1.2.3.3"></infinity></apply></apply><apply id="S2.SS3.p1.9.m9.1.1.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.9.m9.1.1.1.1.2.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1">subscript</csymbol><apply id="S2.SS3.p1.9.m9.1.1.1.1.1.2.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1"><csymbol cd="latexml" id="S2.SS3.p1.9.m9.1.1.1.1.1.2.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.2">norm</csymbol><apply id="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.1.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.2.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.2">𝑓</ci><ci id="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.3.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1.1.1.1.3">𝑛</ci></apply></apply><ci id="S2.SS3.p1.9.m9.1.1.1.1.3.cmml" xref="S2.SS3.p1.9.m9.1.1.1.1.3">𝒳</ci></apply></apply><infinity id="S2.SS3.p1.9.m9.1.1.3.cmml" xref="S2.SS3.p1.9.m9.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p1.9.m9.1c">\liminf_{n\to\infty}\|f_{n}\|_{\mathcal{X}}<\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p1.9.m9.1d">lim inf start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ∥ italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT < ∞</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f\in\mathcal{X}\quad\text{ and }\quad\|f\|_{\mathcal{X}}\leq\liminf_{n\to% \infty}\|f_{n}\|_{\mathcal{X}}." class="ltx_Math" display="block" id="S2.Ex11.m1.4"><semantics id="S2.Ex11.m1.4a"><mrow id="S2.Ex11.m1.4.4.1"><mrow id="S2.Ex11.m1.4.4.1.1.2" xref="S2.Ex11.m1.4.4.1.1.3.cmml"><mrow id="S2.Ex11.m1.4.4.1.1.1.1" xref="S2.Ex11.m1.4.4.1.1.1.1.cmml"><mi id="S2.Ex11.m1.4.4.1.1.1.1.2" xref="S2.Ex11.m1.4.4.1.1.1.1.2.cmml">f</mi><mo id="S2.Ex11.m1.4.4.1.1.1.1.1" xref="S2.Ex11.m1.4.4.1.1.1.1.1.cmml">∈</mo><mrow id="S2.Ex11.m1.4.4.1.1.1.1.3.2" xref="S2.Ex11.m1.4.4.1.1.1.1.3.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.Ex11.m1.2.2" xref="S2.Ex11.m1.2.2.cmml">𝒳</mi><mspace id="S2.Ex11.m1.4.4.1.1.1.1.3.2.1" width="1em" xref="S2.Ex11.m1.4.4.1.1.1.1.3.1.cmml"></mspace><mtext id="S2.Ex11.m1.3.3" xref="S2.Ex11.m1.3.3a.cmml"> and </mtext></mrow></mrow><mspace id="S2.Ex11.m1.4.4.1.1.2.3" width="1em" xref="S2.Ex11.m1.4.4.1.1.3a.cmml"></mspace><mrow id="S2.Ex11.m1.4.4.1.1.2.2" xref="S2.Ex11.m1.4.4.1.1.2.2.cmml"><msub id="S2.Ex11.m1.4.4.1.1.2.2.3" xref="S2.Ex11.m1.4.4.1.1.2.2.3.cmml"><mrow id="S2.Ex11.m1.4.4.1.1.2.2.3.2.2" xref="S2.Ex11.m1.4.4.1.1.2.2.3.2.1.cmml"><mo id="S2.Ex11.m1.4.4.1.1.2.2.3.2.2.1" stretchy="false" xref="S2.Ex11.m1.4.4.1.1.2.2.3.2.1.1.cmml">‖</mo><mi id="S2.Ex11.m1.1.1" xref="S2.Ex11.m1.1.1.cmml">f</mi><mo id="S2.Ex11.m1.4.4.1.1.2.2.3.2.2.2" stretchy="false" xref="S2.Ex11.m1.4.4.1.1.2.2.3.2.1.1.cmml">‖</mo></mrow><mi class="ltx_font_mathcaligraphic" id="S2.Ex11.m1.4.4.1.1.2.2.3.3" xref="S2.Ex11.m1.4.4.1.1.2.2.3.3.cmml">𝒳</mi></msub><mo id="S2.Ex11.m1.4.4.1.1.2.2.2" rspace="0.1389em" xref="S2.Ex11.m1.4.4.1.1.2.2.2.cmml">≤</mo><mrow id="S2.Ex11.m1.4.4.1.1.2.2.1" xref="S2.Ex11.m1.4.4.1.1.2.2.1.cmml"><munder id="S2.Ex11.m1.4.4.1.1.2.2.1.2" xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.cmml"><mo id="S2.Ex11.m1.4.4.1.1.2.2.1.2.2" lspace="0.1389em" movablelimits="false" rspace="0em" xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.2.cmml">lim inf</mo><mrow id="S2.Ex11.m1.4.4.1.1.2.2.1.2.3" xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.cmml"><mi id="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.2" xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.2.cmml">n</mi><mo id="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.1" stretchy="false" xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.1.cmml">→</mo><mi id="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.3" mathvariant="normal" xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.3.cmml">∞</mi></mrow></munder><msub id="S2.Ex11.m1.4.4.1.1.2.2.1.1" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.cmml"><mrow id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.2.cmml"><mo id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.2" stretchy="false" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.2.1.cmml">‖</mo><msub id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.cmml"><mi id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.2" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.2.cmml">f</mi><mi id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.3" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.3" stretchy="false" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.2.1.cmml">‖</mo></mrow><mi class="ltx_font_mathcaligraphic" id="S2.Ex11.m1.4.4.1.1.2.2.1.1.3" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.3.cmml">𝒳</mi></msub></mrow></mrow></mrow><mo id="S2.Ex11.m1.4.4.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex11.m1.4b"><apply id="S2.Ex11.m1.4.4.1.1.3.cmml" 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xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.2">𝑛</ci><infinity id="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.3.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.2.3.3"></infinity></apply></apply><apply id="S2.Ex11.m1.4.4.1.1.2.2.1.1.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1"><csymbol cd="ambiguous" id="S2.Ex11.m1.4.4.1.1.2.2.1.1.2.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1">subscript</csymbol><apply id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.2.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1"><csymbol cd="latexml" id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.2.1.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.2">norm</csymbol><apply id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.1.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1">subscript</csymbol><ci id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.2.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.2">𝑓</ci><ci id="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.3.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.1.1.1.3">𝑛</ci></apply></apply><ci id="S2.Ex11.m1.4.4.1.1.2.2.1.1.3.cmml" xref="S2.Ex11.m1.4.4.1.1.2.2.1.1.3">𝒳</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex11.m1.4c">f\in\mathcal{X}\quad\text{ and }\quad\|f\|_{\mathcal{X}}\leq\liminf_{n\to% \infty}\|f_{n}\|_{\mathcal{X}}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex11.m1.4d">italic_f ∈ caligraphic_X and ∥ italic_f ∥ start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT ≤ lim inf start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT ∥ italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT caligraphic_X end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS3.p1.10">The proof of the Fatou property for Besov and Triebel-Lizorkin spaces in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib17" title="">17</a>]</cite> carries over to the vector-valued case (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Proposition 2.18]</cite>) and to the weighted case as well.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S2.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.1.1.1">Proposition 2.2</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem2.p1"> <p class="ltx_p" id="S2.Thmtheorem2.p1.7"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem2.p1.7.7">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.1.1.m1.2"><semantics id="S2.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S2.Thmtheorem2.p1.1.1.m1.2.3" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.cmml"><mi id="S2.Thmtheorem2.p1.1.1.m1.2.3.2" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S2.Thmtheorem2.p1.1.1.m1.2.3.1" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S2.Thmtheorem2.p1.1.1.m1.2.3.3.2" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml"><mo id="S2.Thmtheorem2.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S2.Thmtheorem2.p1.1.1.m1.1.1" xref="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml">1</mn><mo id="S2.Thmtheorem2.p1.1.1.m1.2.3.3.2.2" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem2.p1.1.1.m1.2.2" mathvariant="normal" xref="S2.Thmtheorem2.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S2.Thmtheorem2.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.1.1.m1.2b"><apply id="S2.Thmtheorem2.p1.1.1.m1.2.3.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.2.3"><in id="S2.Thmtheorem2.p1.1.1.m1.2.3.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.1"></in><ci id="S2.Thmtheorem2.p1.1.1.m1.2.3.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S2.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.2.3.3.2"><cn id="S2.Thmtheorem2.p1.1.1.m1.1.1.cmml" type="integer" xref="S2.Thmtheorem2.p1.1.1.m1.1.1">1</cn><infinity id="S2.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S2.Thmtheorem2.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.2.2.m2.2"><semantics id="S2.Thmtheorem2.p1.2.2.m2.2a"><mrow id="S2.Thmtheorem2.p1.2.2.m2.2.3" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.cmml"><mi id="S2.Thmtheorem2.p1.2.2.m2.2.3.2" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.2.cmml">q</mi><mo id="S2.Thmtheorem2.p1.2.2.m2.2.3.1" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S2.Thmtheorem2.p1.2.2.m2.2.3.3.2" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml"><mo id="S2.Thmtheorem2.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml">[</mo><mn id="S2.Thmtheorem2.p1.2.2.m2.1.1" xref="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml">1</mn><mo id="S2.Thmtheorem2.p1.2.2.m2.2.3.3.2.2" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S2.Thmtheorem2.p1.2.2.m2.2.2" mathvariant="normal" xref="S2.Thmtheorem2.p1.2.2.m2.2.2.cmml">∞</mi><mo id="S2.Thmtheorem2.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.2.2.m2.2b"><apply id="S2.Thmtheorem2.p1.2.2.m2.2.3.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.3"><in id="S2.Thmtheorem2.p1.2.2.m2.2.3.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.1"></in><ci id="S2.Thmtheorem2.p1.2.2.m2.2.3.2.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S2.Thmtheorem2.p1.2.2.m2.2.3.3.1.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.3.3.2"><cn id="S2.Thmtheorem2.p1.2.2.m2.1.1.cmml" type="integer" xref="S2.Thmtheorem2.p1.2.2.m2.1.1">1</cn><infinity id="S2.Thmtheorem2.p1.2.2.m2.2.2.cmml" xref="S2.Thmtheorem2.p1.2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.3.3.m3.1"><semantics id="S2.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem2.p1.3.3.m3.1.1" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml"><mi id="S2.Thmtheorem2.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2.cmml">s</mi><mo id="S2.Thmtheorem2.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.1.cmml">∈</mo><mi id="S2.Thmtheorem2.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.3.3.m3.1b"><apply id="S2.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1"><in id="S2.Thmtheorem2.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.1"></in><ci id="S2.Thmtheorem2.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.2">𝑠</ci><ci id="S2.Thmtheorem2.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem2.p1.3.3.m3.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.3.3.m3.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.3.3.m3.1d">italic_s ∈ blackboard_R</annotation></semantics></math>, <math alttext="w\in A_{\infty}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.4.4.m4.1"><semantics id="S2.Thmtheorem2.p1.4.4.m4.1a"><mrow id="S2.Thmtheorem2.p1.4.4.m4.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml">w</mi><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml">∈</mo><mrow id="S2.Thmtheorem2.p1.4.4.m4.1.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.cmml"><msub id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.2.cmml">A</mi><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.3" mathvariant="normal" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.3.cmml">∞</mi></msub><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml"><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.4.4.m4.1b"><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1"><in id="S2.Thmtheorem2.p1.4.4.m4.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.2"></in><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.3">𝑤</ci><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1"><times id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.2"></times><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.2">𝐴</ci><infinity id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.3.3"></infinity></apply><apply id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.4.4.m4.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.4.4.m4.1c">w\in A_{\infty}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.4.4.m4.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.5.5.m5.1"><semantics id="S2.Thmtheorem2.p1.5.5.m5.1a"><mi id="S2.Thmtheorem2.p1.5.5.m5.1.1" xref="S2.Thmtheorem2.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.5.5.m5.1b"><ci id="S2.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem2.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Then <math alttext="B^{s}_{p,q}(\mathbb{R}^{d},w;X)" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.6.6.m6.5"><semantics id="S2.Thmtheorem2.p1.6.6.m6.5a"><mrow id="S2.Thmtheorem2.p1.6.6.m6.5.5" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.cmml"><msubsup id="S2.Thmtheorem2.p1.6.6.m6.5.5.3" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3.cmml"><mi id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.2" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.2.cmml">B</mi><mrow id="S2.Thmtheorem2.p1.6.6.m6.2.2.2.4" xref="S2.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml"><mi id="S2.Thmtheorem2.p1.6.6.m6.1.1.1.1" xref="S2.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml">p</mi><mo id="S2.Thmtheorem2.p1.6.6.m6.2.2.2.4.1" xref="S2.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml">,</mo><mi id="S2.Thmtheorem2.p1.6.6.m6.2.2.2.2" xref="S2.Thmtheorem2.p1.6.6.m6.2.2.2.2.cmml">q</mi></mrow><mi id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.3" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.3.cmml">s</mi></msubsup><mo id="S2.Thmtheorem2.p1.6.6.m6.5.5.2" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.2.cmml">⁢</mo><mrow id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.2.cmml"><mo id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.2" stretchy="false" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.2.cmml">(</mo><msup id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.2" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.2.cmml">ℝ</mi><mi id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.3" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.3.cmml">d</mi></msup><mo id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.3" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.2.cmml">,</mo><mi id="S2.Thmtheorem2.p1.6.6.m6.3.3" xref="S2.Thmtheorem2.p1.6.6.m6.3.3.cmml">w</mi><mo id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.4" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.2.cmml">;</mo><mi id="S2.Thmtheorem2.p1.6.6.m6.4.4" xref="S2.Thmtheorem2.p1.6.6.m6.4.4.cmml">X</mi><mo id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.5" stretchy="false" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.6.6.m6.5b"><apply id="S2.Thmtheorem2.p1.6.6.m6.5.5.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5"><times id="S2.Thmtheorem2.p1.6.6.m6.5.5.2.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.2"></times><apply id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.1.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3">subscript</csymbol><apply id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.1.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3">superscript</csymbol><ci id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.2.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.2">𝐵</ci><ci id="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.3.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.3.2.3">𝑠</ci></apply><list id="S2.Thmtheorem2.p1.6.6.m6.2.2.2.3.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.2.2.2.4"><ci id="S2.Thmtheorem2.p1.6.6.m6.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.1.1.1.1">𝑝</ci><ci id="S2.Thmtheorem2.p1.6.6.m6.2.2.2.2.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.2.2.2.2">𝑞</ci></list></apply><vector id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.2.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1"><apply id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.2">ℝ</ci><ci id="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.5.5.1.1.1.3">𝑑</ci></apply><ci id="S2.Thmtheorem2.p1.6.6.m6.3.3.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.3.3">𝑤</ci><ci id="S2.Thmtheorem2.p1.6.6.m6.4.4.cmml" xref="S2.Thmtheorem2.p1.6.6.m6.4.4">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.6.6.m6.5c">B^{s}_{p,q}(\mathbb{R}^{d},w;X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.6.6.m6.5d">italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X )</annotation></semantics></math> and <math alttext="F^{s}_{p,q}(\mathbb{R}^{d},w;X)" class="ltx_Math" display="inline" id="S2.Thmtheorem2.p1.7.7.m7.5"><semantics id="S2.Thmtheorem2.p1.7.7.m7.5a"><mrow id="S2.Thmtheorem2.p1.7.7.m7.5.5" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.cmml"><msubsup id="S2.Thmtheorem2.p1.7.7.m7.5.5.3" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3.cmml"><mi id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.2" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.2.cmml">F</mi><mrow id="S2.Thmtheorem2.p1.7.7.m7.2.2.2.4" xref="S2.Thmtheorem2.p1.7.7.m7.2.2.2.3.cmml"><mi id="S2.Thmtheorem2.p1.7.7.m7.1.1.1.1" xref="S2.Thmtheorem2.p1.7.7.m7.1.1.1.1.cmml">p</mi><mo id="S2.Thmtheorem2.p1.7.7.m7.2.2.2.4.1" xref="S2.Thmtheorem2.p1.7.7.m7.2.2.2.3.cmml">,</mo><mi id="S2.Thmtheorem2.p1.7.7.m7.2.2.2.2" xref="S2.Thmtheorem2.p1.7.7.m7.2.2.2.2.cmml">q</mi></mrow><mi id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.3" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.3.cmml">s</mi></msubsup><mo id="S2.Thmtheorem2.p1.7.7.m7.5.5.2" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.2.cmml">⁢</mo><mrow id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.2.cmml"><mo id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.2" stretchy="false" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.2.cmml">(</mo><msup id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.cmml"><mi id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.2" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.2.cmml">ℝ</mi><mi id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.3" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.3.cmml">d</mi></msup><mo id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.3" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.2.cmml">,</mo><mi id="S2.Thmtheorem2.p1.7.7.m7.3.3" xref="S2.Thmtheorem2.p1.7.7.m7.3.3.cmml">w</mi><mo id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.4" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.2.cmml">;</mo><mi id="S2.Thmtheorem2.p1.7.7.m7.4.4" xref="S2.Thmtheorem2.p1.7.7.m7.4.4.cmml">X</mi><mo id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.5" stretchy="false" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem2.p1.7.7.m7.5b"><apply id="S2.Thmtheorem2.p1.7.7.m7.5.5.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5"><times id="S2.Thmtheorem2.p1.7.7.m7.5.5.2.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.2"></times><apply id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.1.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3">subscript</csymbol><apply id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.1.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3">superscript</csymbol><ci id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.2.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.2">𝐹</ci><ci id="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.3.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.3.2.3">𝑠</ci></apply><list id="S2.Thmtheorem2.p1.7.7.m7.2.2.2.3.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.2.2.2.4"><ci id="S2.Thmtheorem2.p1.7.7.m7.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.1.1.1.1">𝑝</ci><ci id="S2.Thmtheorem2.p1.7.7.m7.2.2.2.2.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.2.2.2.2">𝑞</ci></list></apply><vector id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.2.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1"><apply id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.1.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1">superscript</csymbol><ci id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.2.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.2">ℝ</ci><ci id="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.3.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.5.5.1.1.1.3">𝑑</ci></apply><ci id="S2.Thmtheorem2.p1.7.7.m7.3.3.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.3.3">𝑤</ci><ci id="S2.Thmtheorem2.p1.7.7.m7.4.4.cmml" xref="S2.Thmtheorem2.p1.7.7.m7.4.4">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem2.p1.7.7.m7.5c">F^{s}_{p,q}(\mathbb{R}^{d},w;X)</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem2.p1.7.7.m7.5d">italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X )</annotation></semantics></math> have the Fatou property.</span></p> </div> </div> <div class="ltx_para" id="S2.SS3.p2"> <p class="ltx_p" id="S2.SS3.p2.4">A Banach space <math alttext="X" class="ltx_Math" display="inline" id="S2.SS3.p2.1.m1.1"><semantics id="S2.SS3.p2.1.m1.1a"><mi id="S2.SS3.p2.1.m1.1.1" xref="S2.SS3.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.1.m1.1b"><ci id="S2.SS3.p2.1.m1.1.1.cmml" xref="S2.SS3.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.1.m1.1d">italic_X</annotation></semantics></math> satisfies the condition <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.SS3.p2.2.m2.1"><semantics id="S2.SS3.p2.2.m2.1a"><mi id="S2.SS3.p2.2.m2.1.1" xref="S2.SS3.p2.2.m2.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.2.m2.1b"><ci id="S2.SS3.p2.2.m2.1.1.cmml" xref="S2.SS3.p2.2.m2.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.2.m2.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.2.m2.1d">roman_UMD</annotation></semantics></math> (unconditional martingale differences) if and only if the Hilbert transform extends to a bounded operator on <math alttext="L^{p}(\mathbb{R};X)" class="ltx_Math" display="inline" id="S2.SS3.p2.3.m3.2"><semantics id="S2.SS3.p2.3.m3.2a"><mrow id="S2.SS3.p2.3.m3.2.3" xref="S2.SS3.p2.3.m3.2.3.cmml"><msup id="S2.SS3.p2.3.m3.2.3.2" xref="S2.SS3.p2.3.m3.2.3.2.cmml"><mi id="S2.SS3.p2.3.m3.2.3.2.2" xref="S2.SS3.p2.3.m3.2.3.2.2.cmml">L</mi><mi id="S2.SS3.p2.3.m3.2.3.2.3" xref="S2.SS3.p2.3.m3.2.3.2.3.cmml">p</mi></msup><mo id="S2.SS3.p2.3.m3.2.3.1" xref="S2.SS3.p2.3.m3.2.3.1.cmml">⁢</mo><mrow id="S2.SS3.p2.3.m3.2.3.3.2" xref="S2.SS3.p2.3.m3.2.3.3.1.cmml"><mo id="S2.SS3.p2.3.m3.2.3.3.2.1" stretchy="false" xref="S2.SS3.p2.3.m3.2.3.3.1.cmml">(</mo><mi id="S2.SS3.p2.3.m3.1.1" xref="S2.SS3.p2.3.m3.1.1.cmml">ℝ</mi><mo id="S2.SS3.p2.3.m3.2.3.3.2.2" xref="S2.SS3.p2.3.m3.2.3.3.1.cmml">;</mo><mi id="S2.SS3.p2.3.m3.2.2" xref="S2.SS3.p2.3.m3.2.2.cmml">X</mi><mo id="S2.SS3.p2.3.m3.2.3.3.2.3" stretchy="false" xref="S2.SS3.p2.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.3.m3.2b"><apply id="S2.SS3.p2.3.m3.2.3.cmml" xref="S2.SS3.p2.3.m3.2.3"><times id="S2.SS3.p2.3.m3.2.3.1.cmml" xref="S2.SS3.p2.3.m3.2.3.1"></times><apply id="S2.SS3.p2.3.m3.2.3.2.cmml" xref="S2.SS3.p2.3.m3.2.3.2"><csymbol cd="ambiguous" id="S2.SS3.p2.3.m3.2.3.2.1.cmml" xref="S2.SS3.p2.3.m3.2.3.2">superscript</csymbol><ci id="S2.SS3.p2.3.m3.2.3.2.2.cmml" xref="S2.SS3.p2.3.m3.2.3.2.2">𝐿</ci><ci id="S2.SS3.p2.3.m3.2.3.2.3.cmml" xref="S2.SS3.p2.3.m3.2.3.2.3">𝑝</ci></apply><list id="S2.SS3.p2.3.m3.2.3.3.1.cmml" xref="S2.SS3.p2.3.m3.2.3.3.2"><ci id="S2.SS3.p2.3.m3.1.1.cmml" xref="S2.SS3.p2.3.m3.1.1">ℝ</ci><ci id="S2.SS3.p2.3.m3.2.2.cmml" xref="S2.SS3.p2.3.m3.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.3.m3.2c">L^{p}(\mathbb{R};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.3.m3.2d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R ; italic_X )</annotation></semantics></math>. We recall the following properties of <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.SS3.p2.4.m4.1"><semantics id="S2.SS3.p2.4.m4.1a"><mi id="S2.SS3.p2.4.m4.1.1" xref="S2.SS3.p2.4.m4.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.SS3.p2.4.m4.1b"><ci id="S2.SS3.p2.4.m4.1.1.cmml" xref="S2.SS3.p2.4.m4.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS3.p2.4.m4.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.SS3.p2.4.m4.1d">roman_UMD</annotation></semantics></math> spaces, see for instance <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib31" title="">31</a>, Chapter 4 & 5]</cite></p> <ol class="ltx_enumerate" id="S2.I1"> <li class="ltx_item" id="S2.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S2.I1.i1.p1"> <p class="ltx_p" id="S2.I1.i1.p1.3">Hilbert spaces are <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.I1.i1.p1.1.m1.1"><semantics id="S2.I1.i1.p1.1.m1.1a"><mi id="S2.I1.i1.p1.1.m1.1.1" xref="S2.I1.i1.p1.1.m1.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.1.m1.1b"><ci id="S2.I1.i1.p1.1.m1.1.1.cmml" xref="S2.I1.i1.p1.1.m1.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.1.m1.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.1.m1.1d">roman_UMD</annotation></semantics></math> spaces. In particular, <math alttext="\mathbb{C}" class="ltx_Math" display="inline" id="S2.I1.i1.p1.2.m2.1"><semantics id="S2.I1.i1.p1.2.m2.1a"><mi id="S2.I1.i1.p1.2.m2.1.1" xref="S2.I1.i1.p1.2.m2.1.1.cmml">ℂ</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.2.m2.1b"><ci id="S2.I1.i1.p1.2.m2.1.1.cmml" xref="S2.I1.i1.p1.2.m2.1.1">ℂ</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.2.m2.1c">\mathbb{C}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.2.m2.1d">blackboard_C</annotation></semantics></math> is a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.I1.i1.p1.3.m3.1"><semantics id="S2.I1.i1.p1.3.m3.1a"><mi id="S2.I1.i1.p1.3.m3.1.1" xref="S2.I1.i1.p1.3.m3.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i1.p1.3.m3.1b"><ci id="S2.I1.i1.p1.3.m3.1.1.cmml" xref="S2.I1.i1.p1.3.m3.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i1.p1.3.m3.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i1.p1.3.m3.1d">roman_UMD</annotation></semantics></math> Banach space.</p> </div> </li> <li class="ltx_item" id="S2.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S2.I1.i2.p1"> <p class="ltx_p" id="S2.I1.i2.p1.7">If <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S2.I1.i2.p1.1.m1.2"><semantics id="S2.I1.i2.p1.1.m1.2a"><mrow id="S2.I1.i2.p1.1.m1.2.3" xref="S2.I1.i2.p1.1.m1.2.3.cmml"><mi id="S2.I1.i2.p1.1.m1.2.3.2" xref="S2.I1.i2.p1.1.m1.2.3.2.cmml">p</mi><mo id="S2.I1.i2.p1.1.m1.2.3.1" xref="S2.I1.i2.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S2.I1.i2.p1.1.m1.2.3.3.2" xref="S2.I1.i2.p1.1.m1.2.3.3.1.cmml"><mo id="S2.I1.i2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.I1.i2.p1.1.m1.2.3.3.1.cmml">(</mo><mn id="S2.I1.i2.p1.1.m1.1.1" xref="S2.I1.i2.p1.1.m1.1.1.cmml">1</mn><mo id="S2.I1.i2.p1.1.m1.2.3.3.2.2" xref="S2.I1.i2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.I1.i2.p1.1.m1.2.2" mathvariant="normal" xref="S2.I1.i2.p1.1.m1.2.2.cmml">∞</mi><mo id="S2.I1.i2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.I1.i2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.1.m1.2b"><apply id="S2.I1.i2.p1.1.m1.2.3.cmml" xref="S2.I1.i2.p1.1.m1.2.3"><in id="S2.I1.i2.p1.1.m1.2.3.1.cmml" xref="S2.I1.i2.p1.1.m1.2.3.1"></in><ci id="S2.I1.i2.p1.1.m1.2.3.2.cmml" xref="S2.I1.i2.p1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S2.I1.i2.p1.1.m1.2.3.3.1.cmml" xref="S2.I1.i2.p1.1.m1.2.3.3.2"><cn id="S2.I1.i2.p1.1.m1.1.1.cmml" type="integer" xref="S2.I1.i2.p1.1.m1.1.1">1</cn><infinity id="S2.I1.i2.p1.1.m1.2.2.cmml" xref="S2.I1.i2.p1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="(S,\Sigma,\mu)" class="ltx_Math" display="inline" id="S2.I1.i2.p1.2.m2.3"><semantics id="S2.I1.i2.p1.2.m2.3a"><mrow id="S2.I1.i2.p1.2.m2.3.4.2" xref="S2.I1.i2.p1.2.m2.3.4.1.cmml"><mo id="S2.I1.i2.p1.2.m2.3.4.2.1" stretchy="false" xref="S2.I1.i2.p1.2.m2.3.4.1.cmml">(</mo><mi id="S2.I1.i2.p1.2.m2.1.1" xref="S2.I1.i2.p1.2.m2.1.1.cmml">S</mi><mo id="S2.I1.i2.p1.2.m2.3.4.2.2" xref="S2.I1.i2.p1.2.m2.3.4.1.cmml">,</mo><mi id="S2.I1.i2.p1.2.m2.2.2" mathvariant="normal" xref="S2.I1.i2.p1.2.m2.2.2.cmml">Σ</mi><mo id="S2.I1.i2.p1.2.m2.3.4.2.3" xref="S2.I1.i2.p1.2.m2.3.4.1.cmml">,</mo><mi id="S2.I1.i2.p1.2.m2.3.3" xref="S2.I1.i2.p1.2.m2.3.3.cmml">μ</mi><mo id="S2.I1.i2.p1.2.m2.3.4.2.4" stretchy="false" xref="S2.I1.i2.p1.2.m2.3.4.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.2.m2.3b"><vector id="S2.I1.i2.p1.2.m2.3.4.1.cmml" xref="S2.I1.i2.p1.2.m2.3.4.2"><ci id="S2.I1.i2.p1.2.m2.1.1.cmml" xref="S2.I1.i2.p1.2.m2.1.1">𝑆</ci><ci id="S2.I1.i2.p1.2.m2.2.2.cmml" xref="S2.I1.i2.p1.2.m2.2.2">Σ</ci><ci id="S2.I1.i2.p1.2.m2.3.3.cmml" xref="S2.I1.i2.p1.2.m2.3.3">𝜇</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.2.m2.3c">(S,\Sigma,\mu)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.2.m2.3d">( italic_S , roman_Σ , italic_μ )</annotation></semantics></math> is a <math alttext="\sigma" class="ltx_Math" display="inline" id="S2.I1.i2.p1.3.m3.1"><semantics id="S2.I1.i2.p1.3.m3.1a"><mi id="S2.I1.i2.p1.3.m3.1.1" xref="S2.I1.i2.p1.3.m3.1.1.cmml">σ</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.3.m3.1b"><ci id="S2.I1.i2.p1.3.m3.1.1.cmml" xref="S2.I1.i2.p1.3.m3.1.1">𝜎</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.3.m3.1c">\sigma</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.3.m3.1d">italic_σ</annotation></semantics></math>-finite measure space and <math alttext="X" class="ltx_Math" display="inline" id="S2.I1.i2.p1.4.m4.1"><semantics id="S2.I1.i2.p1.4.m4.1a"><mi id="S2.I1.i2.p1.4.m4.1.1" xref="S2.I1.i2.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.4.m4.1b"><ci id="S2.I1.i2.p1.4.m4.1.1.cmml" xref="S2.I1.i2.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.4.m4.1d">italic_X</annotation></semantics></math> is a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.5.m5.1"><semantics id="S2.I1.i2.p1.5.m5.1a"><mi id="S2.I1.i2.p1.5.m5.1.1" xref="S2.I1.i2.p1.5.m5.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.5.m5.1b"><ci id="S2.I1.i2.p1.5.m5.1.1.cmml" xref="S2.I1.i2.p1.5.m5.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.5.m5.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.5.m5.1d">roman_UMD</annotation></semantics></math> Banach space, then <math alttext="L^{p}(S;X)" class="ltx_Math" display="inline" id="S2.I1.i2.p1.6.m6.2"><semantics id="S2.I1.i2.p1.6.m6.2a"><mrow id="S2.I1.i2.p1.6.m6.2.3" xref="S2.I1.i2.p1.6.m6.2.3.cmml"><msup id="S2.I1.i2.p1.6.m6.2.3.2" xref="S2.I1.i2.p1.6.m6.2.3.2.cmml"><mi id="S2.I1.i2.p1.6.m6.2.3.2.2" xref="S2.I1.i2.p1.6.m6.2.3.2.2.cmml">L</mi><mi id="S2.I1.i2.p1.6.m6.2.3.2.3" xref="S2.I1.i2.p1.6.m6.2.3.2.3.cmml">p</mi></msup><mo id="S2.I1.i2.p1.6.m6.2.3.1" xref="S2.I1.i2.p1.6.m6.2.3.1.cmml">⁢</mo><mrow id="S2.I1.i2.p1.6.m6.2.3.3.2" xref="S2.I1.i2.p1.6.m6.2.3.3.1.cmml"><mo id="S2.I1.i2.p1.6.m6.2.3.3.2.1" stretchy="false" xref="S2.I1.i2.p1.6.m6.2.3.3.1.cmml">(</mo><mi id="S2.I1.i2.p1.6.m6.1.1" xref="S2.I1.i2.p1.6.m6.1.1.cmml">S</mi><mo id="S2.I1.i2.p1.6.m6.2.3.3.2.2" xref="S2.I1.i2.p1.6.m6.2.3.3.1.cmml">;</mo><mi id="S2.I1.i2.p1.6.m6.2.2" xref="S2.I1.i2.p1.6.m6.2.2.cmml">X</mi><mo id="S2.I1.i2.p1.6.m6.2.3.3.2.3" stretchy="false" xref="S2.I1.i2.p1.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.6.m6.2b"><apply id="S2.I1.i2.p1.6.m6.2.3.cmml" xref="S2.I1.i2.p1.6.m6.2.3"><times id="S2.I1.i2.p1.6.m6.2.3.1.cmml" xref="S2.I1.i2.p1.6.m6.2.3.1"></times><apply id="S2.I1.i2.p1.6.m6.2.3.2.cmml" xref="S2.I1.i2.p1.6.m6.2.3.2"><csymbol cd="ambiguous" id="S2.I1.i2.p1.6.m6.2.3.2.1.cmml" xref="S2.I1.i2.p1.6.m6.2.3.2">superscript</csymbol><ci id="S2.I1.i2.p1.6.m6.2.3.2.2.cmml" xref="S2.I1.i2.p1.6.m6.2.3.2.2">𝐿</ci><ci id="S2.I1.i2.p1.6.m6.2.3.2.3.cmml" xref="S2.I1.i2.p1.6.m6.2.3.2.3">𝑝</ci></apply><list id="S2.I1.i2.p1.6.m6.2.3.3.1.cmml" xref="S2.I1.i2.p1.6.m6.2.3.3.2"><ci id="S2.I1.i2.p1.6.m6.1.1.cmml" xref="S2.I1.i2.p1.6.m6.1.1">𝑆</ci><ci id="S2.I1.i2.p1.6.m6.2.2.cmml" xref="S2.I1.i2.p1.6.m6.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.6.m6.2c">L^{p}(S;X)</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.6.m6.2d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_S ; italic_X )</annotation></semantics></math> is a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.I1.i2.p1.7.m7.1"><semantics id="S2.I1.i2.p1.7.m7.1a"><mi id="S2.I1.i2.p1.7.m7.1.1" xref="S2.I1.i2.p1.7.m7.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i2.p1.7.m7.1b"><ci id="S2.I1.i2.p1.7.m7.1.1.cmml" xref="S2.I1.i2.p1.7.m7.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i2.p1.7.m7.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i2.p1.7.m7.1d">roman_UMD</annotation></semantics></math> Banach space.</p> </div> </li> <li class="ltx_item" id="S2.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S2.I1.i3.p1"> <p class="ltx_p" id="S2.I1.i3.p1.1"><math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S2.I1.i3.p1.1.m1.1"><semantics id="S2.I1.i3.p1.1.m1.1a"><mi id="S2.I1.i3.p1.1.m1.1.1" xref="S2.I1.i3.p1.1.m1.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S2.I1.i3.p1.1.m1.1b"><ci id="S2.I1.i3.p1.1.m1.1.1.cmml" xref="S2.I1.i3.p1.1.m1.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.I1.i3.p1.1.m1.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S2.I1.i3.p1.1.m1.1d">roman_UMD</annotation></semantics></math> Banach spaces are reflexive.</p> </div> </li> </ol> </div> </section> <section class="ltx_subsection" id="S2.SS4"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.4. </span>Embeddings</h3> <div class="ltx_para" id="S2.SS4.p1"> <p class="ltx_p" id="S2.SS4.p1.5">From <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Propositions 3.11 & 3.12]</cite> we recall the following embeddings. Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S2.SS4.p1.1.m1.2"><semantics id="S2.SS4.p1.1.m1.2a"><mrow id="S2.SS4.p1.1.m1.2.3" xref="S2.SS4.p1.1.m1.2.3.cmml"><mi id="S2.SS4.p1.1.m1.2.3.2" xref="S2.SS4.p1.1.m1.2.3.2.cmml">p</mi><mo id="S2.SS4.p1.1.m1.2.3.1" xref="S2.SS4.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S2.SS4.p1.1.m1.2.3.3.2" xref="S2.SS4.p1.1.m1.2.3.3.1.cmml"><mo id="S2.SS4.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS4.p1.1.m1.2.3.3.1.cmml">(</mo><mn id="S2.SS4.p1.1.m1.1.1" xref="S2.SS4.p1.1.m1.1.1.cmml">1</mn><mo id="S2.SS4.p1.1.m1.2.3.3.2.2" xref="S2.SS4.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p1.1.m1.2.2" mathvariant="normal" xref="S2.SS4.p1.1.m1.2.2.cmml">∞</mi><mo id="S2.SS4.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS4.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.1.m1.2b"><apply id="S2.SS4.p1.1.m1.2.3.cmml" xref="S2.SS4.p1.1.m1.2.3"><in id="S2.SS4.p1.1.m1.2.3.1.cmml" xref="S2.SS4.p1.1.m1.2.3.1"></in><ci id="S2.SS4.p1.1.m1.2.3.2.cmml" xref="S2.SS4.p1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S2.SS4.p1.1.m1.2.3.3.1.cmml" xref="S2.SS4.p1.1.m1.2.3.3.2"><cn id="S2.SS4.p1.1.m1.1.1.cmml" type="integer" xref="S2.SS4.p1.1.m1.1.1">1</cn><infinity id="S2.SS4.p1.1.m1.2.2.cmml" xref="S2.SS4.p1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS4.p1.2.m2.1"><semantics id="S2.SS4.p1.2.m2.1a"><mrow id="S2.SS4.p1.2.m2.1.1" xref="S2.SS4.p1.2.m2.1.1.cmml"><mi id="S2.SS4.p1.2.m2.1.1.2" xref="S2.SS4.p1.2.m2.1.1.2.cmml">s</mi><mo id="S2.SS4.p1.2.m2.1.1.1" xref="S2.SS4.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S2.SS4.p1.2.m2.1.1.3" xref="S2.SS4.p1.2.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.2.m2.1b"><apply id="S2.SS4.p1.2.m2.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1"><in id="S2.SS4.p1.2.m2.1.1.1.cmml" xref="S2.SS4.p1.2.m2.1.1.1"></in><ci id="S2.SS4.p1.2.m2.1.1.2.cmml" xref="S2.SS4.p1.2.m2.1.1.2">𝑠</ci><ci id="S2.SS4.p1.2.m2.1.1.3.cmml" xref="S2.SS4.p1.2.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.2.m2.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.2.m2.1d">italic_s ∈ blackboard_R</annotation></semantics></math>, <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S2.SS4.p1.3.m3.1"><semantics id="S2.SS4.p1.3.m3.1a"><mrow id="S2.SS4.p1.3.m3.1.1" xref="S2.SS4.p1.3.m3.1.1.cmml"><mi id="S2.SS4.p1.3.m3.1.1.2" xref="S2.SS4.p1.3.m3.1.1.2.cmml">k</mi><mo id="S2.SS4.p1.3.m3.1.1.1" xref="S2.SS4.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S2.SS4.p1.3.m3.1.1.3" xref="S2.SS4.p1.3.m3.1.1.3.cmml"><mi id="S2.SS4.p1.3.m3.1.1.3.2" xref="S2.SS4.p1.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S2.SS4.p1.3.m3.1.1.3.3" xref="S2.SS4.p1.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.3.m3.1b"><apply id="S2.SS4.p1.3.m3.1.1.cmml" xref="S2.SS4.p1.3.m3.1.1"><in id="S2.SS4.p1.3.m3.1.1.1.cmml" xref="S2.SS4.p1.3.m3.1.1.1"></in><ci id="S2.SS4.p1.3.m3.1.1.2.cmml" xref="S2.SS4.p1.3.m3.1.1.2">𝑘</ci><apply id="S2.SS4.p1.3.m3.1.1.3.cmml" xref="S2.SS4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p1.3.m3.1.1.3.1.cmml" xref="S2.SS4.p1.3.m3.1.1.3">subscript</csymbol><ci id="S2.SS4.p1.3.m3.1.1.3.2.cmml" xref="S2.SS4.p1.3.m3.1.1.3.2">ℕ</ci><cn id="S2.SS4.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S2.SS4.p1.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.3.m3.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.3.m3.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="w\in A_{p}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS4.p1.4.m4.1"><semantics id="S2.SS4.p1.4.m4.1a"><mrow id="S2.SS4.p1.4.m4.1.1" xref="S2.SS4.p1.4.m4.1.1.cmml"><mi id="S2.SS4.p1.4.m4.1.1.3" xref="S2.SS4.p1.4.m4.1.1.3.cmml">w</mi><mo id="S2.SS4.p1.4.m4.1.1.2" xref="S2.SS4.p1.4.m4.1.1.2.cmml">∈</mo><mrow id="S2.SS4.p1.4.m4.1.1.1" xref="S2.SS4.p1.4.m4.1.1.1.cmml"><msub id="S2.SS4.p1.4.m4.1.1.1.3" xref="S2.SS4.p1.4.m4.1.1.1.3.cmml"><mi id="S2.SS4.p1.4.m4.1.1.1.3.2" xref="S2.SS4.p1.4.m4.1.1.1.3.2.cmml">A</mi><mi id="S2.SS4.p1.4.m4.1.1.1.3.3" xref="S2.SS4.p1.4.m4.1.1.1.3.3.cmml">p</mi></msub><mo id="S2.SS4.p1.4.m4.1.1.1.2" xref="S2.SS4.p1.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.p1.4.m4.1.1.1.1.1" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.cmml"><mo id="S2.SS4.p1.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS4.p1.4.m4.1.1.1.1.1.1" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.cmml"><mi id="S2.SS4.p1.4.m4.1.1.1.1.1.1.2" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS4.p1.4.m4.1.1.1.1.1.1.3" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS4.p1.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.4.m4.1b"><apply id="S2.SS4.p1.4.m4.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1"><in id="S2.SS4.p1.4.m4.1.1.2.cmml" xref="S2.SS4.p1.4.m4.1.1.2"></in><ci id="S2.SS4.p1.4.m4.1.1.3.cmml" xref="S2.SS4.p1.4.m4.1.1.3">𝑤</ci><apply id="S2.SS4.p1.4.m4.1.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1.1"><times id="S2.SS4.p1.4.m4.1.1.1.2.cmml" xref="S2.SS4.p1.4.m4.1.1.1.2"></times><apply id="S2.SS4.p1.4.m4.1.1.1.3.cmml" xref="S2.SS4.p1.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m4.1.1.1.3.1.cmml" xref="S2.SS4.p1.4.m4.1.1.1.3">subscript</csymbol><ci id="S2.SS4.p1.4.m4.1.1.1.3.2.cmml" xref="S2.SS4.p1.4.m4.1.1.1.3.2">𝐴</ci><ci id="S2.SS4.p1.4.m4.1.1.1.3.3.cmml" xref="S2.SS4.p1.4.m4.1.1.1.3.3">𝑝</ci></apply><apply id="S2.SS4.p1.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p1.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S2.SS4.p1.4.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS4.p1.4.m4.1.1.1.1.1.1.3.cmml" xref="S2.SS4.p1.4.m4.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.4.m4.1c">w\in A_{p}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.4.m4.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S2.SS4.p1.5.m5.1"><semantics id="S2.SS4.p1.5.m5.1a"><mi id="S2.SS4.p1.5.m5.1.1" xref="S2.SS4.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.5.m5.1b"><ci id="S2.SS4.p1.5.m5.1.1.cmml" xref="S2.SS4.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Then</p> <table class="ltx_equationgroup ltx_eqn_table" id="S2.E4"> <tbody> <tr class="ltx_eqn_row" id="S6.EGx4"><td class="ltx_eqn_cell" colspan="5"></td></tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E4.1"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle F^{s}_{p,1}(\mathbb{R}^{d},w;X)" class="ltx_Math" display="inline" id="S2.E4.1.m1.5"><semantics id="S2.E4.1.m1.5a"><mrow id="S2.E4.1.m1.5.5" xref="S2.E4.1.m1.5.5.cmml"><msubsup id="S2.E4.1.m1.5.5.3" xref="S2.E4.1.m1.5.5.3.cmml"><mi id="S2.E4.1.m1.5.5.3.2.2" xref="S2.E4.1.m1.5.5.3.2.2.cmml">F</mi><mrow id="S2.E4.1.m1.2.2.2.4" xref="S2.E4.1.m1.2.2.2.3.cmml"><mi id="S2.E4.1.m1.1.1.1.1" xref="S2.E4.1.m1.1.1.1.1.cmml">p</mi><mo id="S2.E4.1.m1.2.2.2.4.1" xref="S2.E4.1.m1.2.2.2.3.cmml">,</mo><mn id="S2.E4.1.m1.2.2.2.2" xref="S2.E4.1.m1.2.2.2.2.cmml">1</mn></mrow><mi id="S2.E4.1.m1.5.5.3.2.3" xref="S2.E4.1.m1.5.5.3.2.3.cmml">s</mi></msubsup><mo id="S2.E4.1.m1.5.5.2" xref="S2.E4.1.m1.5.5.2.cmml">⁢</mo><mrow id="S2.E4.1.m1.5.5.1.1" xref="S2.E4.1.m1.5.5.1.2.cmml"><mo id="S2.E4.1.m1.5.5.1.1.2" stretchy="false" xref="S2.E4.1.m1.5.5.1.2.cmml">(</mo><msup id="S2.E4.1.m1.5.5.1.1.1" xref="S2.E4.1.m1.5.5.1.1.1.cmml"><mi id="S2.E4.1.m1.5.5.1.1.1.2" xref="S2.E4.1.m1.5.5.1.1.1.2.cmml">ℝ</mi><mi id="S2.E4.1.m1.5.5.1.1.1.3" xref="S2.E4.1.m1.5.5.1.1.1.3.cmml">d</mi></msup><mo id="S2.E4.1.m1.5.5.1.1.3" xref="S2.E4.1.m1.5.5.1.2.cmml">,</mo><mi id="S2.E4.1.m1.3.3" xref="S2.E4.1.m1.3.3.cmml">w</mi><mo id="S2.E4.1.m1.5.5.1.1.4" xref="S2.E4.1.m1.5.5.1.2.cmml">;</mo><mi id="S2.E4.1.m1.4.4" xref="S2.E4.1.m1.4.4.cmml">X</mi><mo id="S2.E4.1.m1.5.5.1.1.5" stretchy="false" xref="S2.E4.1.m1.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.E4.1.m1.5b"><apply id="S2.E4.1.m1.5.5.cmml" xref="S2.E4.1.m1.5.5"><times id="S2.E4.1.m1.5.5.2.cmml" 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start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) ↪ italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.4a)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E4.2"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle F^{k}_{p,1}(\mathbb{R}^{d},w;X)" class="ltx_Math" display="inline" id="S2.E4.2.m1.5"><semantics id="S2.E4.2.m1.5a"><mrow id="S2.E4.2.m1.5.5" xref="S2.E4.2.m1.5.5.cmml"><msubsup 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xref="S2.E4.2.m2.9.9.1.1.2.3">superscript</csymbol><ci id="S2.E4.2.m2.9.9.1.1.2.3.2.2.cmml" xref="S2.E4.2.m2.9.9.1.1.2.3.2.2">𝐹</ci><ci id="S2.E4.2.m2.9.9.1.1.2.3.2.3.cmml" xref="S2.E4.2.m2.9.9.1.1.2.3.2.3">𝑘</ci></apply><list id="S2.E4.2.m2.4.4.2.3.cmml" xref="S2.E4.2.m2.4.4.2.4"><ci id="S2.E4.2.m2.3.3.1.1.cmml" xref="S2.E4.2.m2.3.3.1.1">𝑝</ci><infinity id="S2.E4.2.m2.4.4.2.2.cmml" xref="S2.E4.2.m2.4.4.2.2"></infinity></list></apply><vector id="S2.E4.2.m2.9.9.1.1.2.1.2.cmml" xref="S2.E4.2.m2.9.9.1.1.2.1.1"><apply id="S2.E4.2.m2.9.9.1.1.2.1.1.1.cmml" xref="S2.E4.2.m2.9.9.1.1.2.1.1.1"><csymbol cd="ambiguous" id="S2.E4.2.m2.9.9.1.1.2.1.1.1.1.cmml" xref="S2.E4.2.m2.9.9.1.1.2.1.1.1">superscript</csymbol><ci id="S2.E4.2.m2.9.9.1.1.2.1.1.1.2.cmml" xref="S2.E4.2.m2.9.9.1.1.2.1.1.1.2">ℝ</ci><ci id="S2.E4.2.m2.9.9.1.1.2.1.1.1.3.cmml" xref="S2.E4.2.m2.9.9.1.1.2.1.1.1.3">𝑑</ci></apply><ci id="S2.E4.2.m2.7.7.cmml" xref="S2.E4.2.m2.7.7">𝑤</ci><ci id="S2.E4.2.m2.8.8.cmml" xref="S2.E4.2.m2.8.8">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E4.2.m2.9c">\displaystyle\hookrightarrow W^{k,p}(\mathbb{R}^{d},w;X)\hookrightarrow F^{k}_% {p,\infty}(\mathbb{R}^{d},w;X).</annotation><annotation encoding="application/x-llamapun" id="S2.E4.2.m2.9d">↪ italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) ↪ italic_F start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.4b)</span></td> </tr> </tbody> </table> <p class="ltx_p" 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id="S2.SS4.p1.6.m1.3.3.1.1" xref="S2.SS4.p1.6.m1.3.3.1.1.cmml">s</mi><mo id="S2.SS4.p1.6.m1.4.4.2.4.1" xref="S2.SS4.p1.6.m1.4.4.2.3.cmml">,</mo><mi id="S2.SS4.p1.6.m1.4.4.2.2" xref="S2.SS4.p1.6.m1.4.4.2.2.cmml">p</mi></mrow></msup><mo id="S2.SS4.p1.6.m1.8.8.2.2" xref="S2.SS4.p1.6.m1.8.8.2.2.cmml">⁢</mo><mrow id="S2.SS4.p1.6.m1.8.8.2.1.1" xref="S2.SS4.p1.6.m1.8.8.2.1.2.cmml"><mo id="S2.SS4.p1.6.m1.8.8.2.1.1.2" stretchy="false" xref="S2.SS4.p1.6.m1.8.8.2.1.2.cmml">(</mo><msup id="S2.SS4.p1.6.m1.8.8.2.1.1.1" xref="S2.SS4.p1.6.m1.8.8.2.1.1.1.cmml"><mi id="S2.SS4.p1.6.m1.8.8.2.1.1.1.2" xref="S2.SS4.p1.6.m1.8.8.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS4.p1.6.m1.8.8.2.1.1.1.3" xref="S2.SS4.p1.6.m1.8.8.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS4.p1.6.m1.8.8.2.1.1.3" xref="S2.SS4.p1.6.m1.8.8.2.1.2.cmml">;</mo><mi id="S2.SS4.p1.6.m1.6.6" xref="S2.SS4.p1.6.m1.6.6.cmml">X</mi><mo id="S2.SS4.p1.6.m1.8.8.2.1.1.4" stretchy="false" 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id="S2.SS4.p1.6.m1.8c">F^{s}_{p,2}(\mathbb{R}^{d};X)=H^{s,p}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.6.m1.8d">italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 2 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X ) = italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> if and only if <math alttext="X" class="ltx_Math" display="inline" id="S2.SS4.p1.7.m2.1"><semantics id="S2.SS4.p1.7.m2.1a"><mi id="S2.SS4.p1.7.m2.1.1" xref="S2.SS4.p1.7.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.7.m2.1b"><ci id="S2.SS4.p1.7.m2.1.1.cmml" xref="S2.SS4.p1.7.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.7.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.7.m2.1d">italic_X</annotation></semantics></math> is isomorphic to a Hilbert space, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib28" title="">28</a>]</cite> or <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Theorem 14.7.9]</cite> for a more direct proof. For general Banach spaces <math alttext="X" class="ltx_Math" display="inline" id="S2.SS4.p1.8.m3.1"><semantics id="S2.SS4.p1.8.m3.1a"><mi id="S2.SS4.p1.8.m3.1.1" xref="S2.SS4.p1.8.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.p1.8.m3.1b"><ci id="S2.SS4.p1.8.m3.1.1.cmml" xref="S2.SS4.p1.8.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p1.8.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p1.8.m3.1d">italic_X</annotation></semantics></math> the Bessel potential and Sobolev spaces are not equal to a Triebel-Lizorkin space, but instead, we have the embeddings (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E4" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.4</span></a>).</p> </div> <div class="ltx_para" id="S2.SS4.p2"> <p class="ltx_p" id="S2.SS4.p2.4">Moreover, for all <math alttext="p\in[1,\infty)" class="ltx_Math" display="inline" id="S2.SS4.p2.1.m1.2"><semantics id="S2.SS4.p2.1.m1.2a"><mrow id="S2.SS4.p2.1.m1.2.3" xref="S2.SS4.p2.1.m1.2.3.cmml"><mi id="S2.SS4.p2.1.m1.2.3.2" xref="S2.SS4.p2.1.m1.2.3.2.cmml">p</mi><mo id="S2.SS4.p2.1.m1.2.3.1" xref="S2.SS4.p2.1.m1.2.3.1.cmml">∈</mo><mrow id="S2.SS4.p2.1.m1.2.3.3.2" xref="S2.SS4.p2.1.m1.2.3.3.1.cmml"><mo id="S2.SS4.p2.1.m1.2.3.3.2.1" stretchy="false" xref="S2.SS4.p2.1.m1.2.3.3.1.cmml">[</mo><mn id="S2.SS4.p2.1.m1.1.1" xref="S2.SS4.p2.1.m1.1.1.cmml">1</mn><mo id="S2.SS4.p2.1.m1.2.3.3.2.2" xref="S2.SS4.p2.1.m1.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p2.1.m1.2.2" mathvariant="normal" xref="S2.SS4.p2.1.m1.2.2.cmml">∞</mi><mo id="S2.SS4.p2.1.m1.2.3.3.2.3" stretchy="false" xref="S2.SS4.p2.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.1.m1.2b"><apply id="S2.SS4.p2.1.m1.2.3.cmml" xref="S2.SS4.p2.1.m1.2.3"><in id="S2.SS4.p2.1.m1.2.3.1.cmml" xref="S2.SS4.p2.1.m1.2.3.1"></in><ci id="S2.SS4.p2.1.m1.2.3.2.cmml" xref="S2.SS4.p2.1.m1.2.3.2">𝑝</ci><interval closure="closed-open" id="S2.SS4.p2.1.m1.2.3.3.1.cmml" xref="S2.SS4.p2.1.m1.2.3.3.2"><cn id="S2.SS4.p2.1.m1.1.1.cmml" type="integer" xref="S2.SS4.p2.1.m1.1.1">1</cn><infinity id="S2.SS4.p2.1.m1.2.2.cmml" xref="S2.SS4.p2.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.1.m1.2c">p\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.1.m1.2d">italic_p ∈ [ 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S2.SS4.p2.2.m2.2"><semantics id="S2.SS4.p2.2.m2.2a"><mrow id="S2.SS4.p2.2.m2.2.3" xref="S2.SS4.p2.2.m2.2.3.cmml"><mi id="S2.SS4.p2.2.m2.2.3.2" xref="S2.SS4.p2.2.m2.2.3.2.cmml">q</mi><mo id="S2.SS4.p2.2.m2.2.3.1" xref="S2.SS4.p2.2.m2.2.3.1.cmml">∈</mo><mrow id="S2.SS4.p2.2.m2.2.3.3.2" xref="S2.SS4.p2.2.m2.2.3.3.1.cmml"><mo id="S2.SS4.p2.2.m2.2.3.3.2.1" stretchy="false" xref="S2.SS4.p2.2.m2.2.3.3.1.cmml">[</mo><mn id="S2.SS4.p2.2.m2.1.1" xref="S2.SS4.p2.2.m2.1.1.cmml">1</mn><mo id="S2.SS4.p2.2.m2.2.3.3.2.2" xref="S2.SS4.p2.2.m2.2.3.3.1.cmml">,</mo><mi id="S2.SS4.p2.2.m2.2.2" mathvariant="normal" xref="S2.SS4.p2.2.m2.2.2.cmml">∞</mi><mo id="S2.SS4.p2.2.m2.2.3.3.2.3" stretchy="false" xref="S2.SS4.p2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.2.m2.2b"><apply id="S2.SS4.p2.2.m2.2.3.cmml" xref="S2.SS4.p2.2.m2.2.3"><in id="S2.SS4.p2.2.m2.2.3.1.cmml" xref="S2.SS4.p2.2.m2.2.3.1"></in><ci id="S2.SS4.p2.2.m2.2.3.2.cmml" xref="S2.SS4.p2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S2.SS4.p2.2.m2.2.3.3.1.cmml" xref="S2.SS4.p2.2.m2.2.3.3.2"><cn id="S2.SS4.p2.2.m2.1.1.cmml" type="integer" xref="S2.SS4.p2.2.m2.1.1">1</cn><infinity id="S2.SS4.p2.2.m2.2.2.cmml" xref="S2.SS4.p2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS4.p2.3.m3.1"><semantics id="S2.SS4.p2.3.m3.1a"><mrow id="S2.SS4.p2.3.m3.1.1" xref="S2.SS4.p2.3.m3.1.1.cmml"><mi id="S2.SS4.p2.3.m3.1.1.2" xref="S2.SS4.p2.3.m3.1.1.2.cmml">s</mi><mo id="S2.SS4.p2.3.m3.1.1.1" xref="S2.SS4.p2.3.m3.1.1.1.cmml">∈</mo><mi id="S2.SS4.p2.3.m3.1.1.3" xref="S2.SS4.p2.3.m3.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.3.m3.1b"><apply id="S2.SS4.p2.3.m3.1.1.cmml" xref="S2.SS4.p2.3.m3.1.1"><in id="S2.SS4.p2.3.m3.1.1.1.cmml" xref="S2.SS4.p2.3.m3.1.1.1"></in><ci id="S2.SS4.p2.3.m3.1.1.2.cmml" xref="S2.SS4.p2.3.m3.1.1.2">𝑠</ci><ci id="S2.SS4.p2.3.m3.1.1.3.cmml" xref="S2.SS4.p2.3.m3.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.3.m3.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.3.m3.1d">italic_s ∈ blackboard_R</annotation></semantics></math>, <math alttext="w\in A_{\infty}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S2.SS4.p2.4.m4.1"><semantics id="S2.SS4.p2.4.m4.1a"><mrow id="S2.SS4.p2.4.m4.1.1" xref="S2.SS4.p2.4.m4.1.1.cmml"><mi id="S2.SS4.p2.4.m4.1.1.3" xref="S2.SS4.p2.4.m4.1.1.3.cmml">w</mi><mo id="S2.SS4.p2.4.m4.1.1.2" xref="S2.SS4.p2.4.m4.1.1.2.cmml">∈</mo><mrow id="S2.SS4.p2.4.m4.1.1.1" xref="S2.SS4.p2.4.m4.1.1.1.cmml"><msub id="S2.SS4.p2.4.m4.1.1.1.3" xref="S2.SS4.p2.4.m4.1.1.1.3.cmml"><mi id="S2.SS4.p2.4.m4.1.1.1.3.2" xref="S2.SS4.p2.4.m4.1.1.1.3.2.cmml">A</mi><mi id="S2.SS4.p2.4.m4.1.1.1.3.3" mathvariant="normal" xref="S2.SS4.p2.4.m4.1.1.1.3.3.cmml">∞</mi></msub><mo id="S2.SS4.p2.4.m4.1.1.1.2" xref="S2.SS4.p2.4.m4.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS4.p2.4.m4.1.1.1.1.1" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.cmml"><mo id="S2.SS4.p2.4.m4.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.cmml">(</mo><msup id="S2.SS4.p2.4.m4.1.1.1.1.1.1" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.cmml"><mi id="S2.SS4.p2.4.m4.1.1.1.1.1.1.2" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS4.p2.4.m4.1.1.1.1.1.1.3" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS4.p2.4.m4.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.4.m4.1b"><apply id="S2.SS4.p2.4.m4.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1"><in id="S2.SS4.p2.4.m4.1.1.2.cmml" xref="S2.SS4.p2.4.m4.1.1.2"></in><ci id="S2.SS4.p2.4.m4.1.1.3.cmml" xref="S2.SS4.p2.4.m4.1.1.3">𝑤</ci><apply id="S2.SS4.p2.4.m4.1.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1.1"><times id="S2.SS4.p2.4.m4.1.1.1.2.cmml" xref="S2.SS4.p2.4.m4.1.1.1.2"></times><apply id="S2.SS4.p2.4.m4.1.1.1.3.cmml" xref="S2.SS4.p2.4.m4.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.p2.4.m4.1.1.1.3.1.cmml" xref="S2.SS4.p2.4.m4.1.1.1.3">subscript</csymbol><ci id="S2.SS4.p2.4.m4.1.1.1.3.2.cmml" xref="S2.SS4.p2.4.m4.1.1.1.3.2">𝐴</ci><infinity id="S2.SS4.p2.4.m4.1.1.1.3.3.cmml" xref="S2.SS4.p2.4.m4.1.1.1.3.3"></infinity></apply><apply id="S2.SS4.p2.4.m4.1.1.1.1.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p2.4.m4.1.1.1.1.1.1.1.cmml" xref="S2.SS4.p2.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S2.SS4.p2.4.m4.1.1.1.1.1.1.2.cmml" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.2">ℝ</ci><ci id="S2.SS4.p2.4.m4.1.1.1.1.1.1.3.cmml" xref="S2.SS4.p2.4.m4.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.4.m4.1c">w\in A_{\infty}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.4.m4.1d">italic_w ∈ italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math>, it holds</p> <table class="ltx_equation ltx_eqn_table" id="S2.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="B^{s}_{p,\min\{p,q\}}(\mathbb{R}^{d},w;X)\hookrightarrow F^{s}_{p,q}(\mathbb{R% }^{d},w;X)\hookrightarrow B^{s}_{p,\max\{p,q\}}(\mathbb{R}^{d},w;X)," class="ltx_Math" display="block" id="S2.E5.m1.19"><semantics id="S2.E5.m1.19a"><mrow id="S2.E5.m1.19.19.1" xref="S2.E5.m1.19.19.1.1.cmml"><mrow id="S2.E5.m1.19.19.1.1" xref="S2.E5.m1.19.19.1.1.cmml"><mrow id="S2.E5.m1.19.19.1.1.1" xref="S2.E5.m1.19.19.1.1.1.cmml"><msubsup id="S2.E5.m1.19.19.1.1.1.3" xref="S2.E5.m1.19.19.1.1.1.3.cmml"><mi id="S2.E5.m1.19.19.1.1.1.3.2.2" xref="S2.E5.m1.19.19.1.1.1.3.2.2.cmml">B</mi><mrow id="S2.E5.m1.5.5.5.5" xref="S2.E5.m1.5.5.5.6.cmml"><mi id="S2.E5.m1.4.4.4.4" xref="S2.E5.m1.4.4.4.4.cmml">p</mi><mo id="S2.E5.m1.5.5.5.5.2" xref="S2.E5.m1.5.5.5.6.cmml">,</mo><mrow id="S2.E5.m1.5.5.5.5.1.2" xref="S2.E5.m1.5.5.5.5.1.1.cmml"><mi id="S2.E5.m1.1.1.1.1" xref="S2.E5.m1.1.1.1.1.cmml">min</mi><mo id="S2.E5.m1.5.5.5.5.1.2a" xref="S2.E5.m1.5.5.5.5.1.1.cmml">⁡</mo><mrow id="S2.E5.m1.5.5.5.5.1.2.1" xref="S2.E5.m1.5.5.5.5.1.1.cmml"><mo id="S2.E5.m1.5.5.5.5.1.2.1.1" stretchy="false" xref="S2.E5.m1.5.5.5.5.1.1.cmml">{</mo><mi id="S2.E5.m1.2.2.2.2" xref="S2.E5.m1.2.2.2.2.cmml">p</mi><mo id="S2.E5.m1.5.5.5.5.1.2.1.2" xref="S2.E5.m1.5.5.5.5.1.1.cmml">,</mo><mi id="S2.E5.m1.3.3.3.3" xref="S2.E5.m1.3.3.3.3.cmml">q</mi><mo id="S2.E5.m1.5.5.5.5.1.2.1.3" stretchy="false" xref="S2.E5.m1.5.5.5.5.1.1.cmml">}</mo></mrow></mrow></mrow><mi id="S2.E5.m1.19.19.1.1.1.3.2.3" xref="S2.E5.m1.19.19.1.1.1.3.2.3.cmml">s</mi></msubsup><mo 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xref="S2.E5.m1.19.19.1.1.3.1.1.1.3">𝑑</ci></apply><ci id="S2.E5.m1.17.17.cmml" xref="S2.E5.m1.17.17">𝑤</ci><ci id="S2.E5.m1.18.18.cmml" xref="S2.E5.m1.18.18">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E5.m1.19c">B^{s}_{p,\min\{p,q\}}(\mathbb{R}^{d},w;X)\hookrightarrow F^{s}_{p,q}(\mathbb{R% }^{d},w;X)\hookrightarrow B^{s}_{p,\max\{p,q\}}(\mathbb{R}^{d},w;X),</annotation><annotation encoding="application/x-llamapun" id="S2.E5.m1.19d">italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , roman_min { italic_p , italic_q } end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) ↪ italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) ↪ italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , roman_max { italic_p , italic_q } end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.p2.5">and if, in addition, <math alttext="1\leq q_{0}\leq q_{1}\leq\infty" class="ltx_Math" display="inline" id="S2.SS4.p2.5.m1.1"><semantics id="S2.SS4.p2.5.m1.1a"><mrow id="S2.SS4.p2.5.m1.1.1" xref="S2.SS4.p2.5.m1.1.1.cmml"><mn id="S2.SS4.p2.5.m1.1.1.2" xref="S2.SS4.p2.5.m1.1.1.2.cmml">1</mn><mo id="S2.SS4.p2.5.m1.1.1.3" xref="S2.SS4.p2.5.m1.1.1.3.cmml">≤</mo><msub id="S2.SS4.p2.5.m1.1.1.4" xref="S2.SS4.p2.5.m1.1.1.4.cmml"><mi id="S2.SS4.p2.5.m1.1.1.4.2" xref="S2.SS4.p2.5.m1.1.1.4.2.cmml">q</mi><mn id="S2.SS4.p2.5.m1.1.1.4.3" xref="S2.SS4.p2.5.m1.1.1.4.3.cmml">0</mn></msub><mo id="S2.SS4.p2.5.m1.1.1.5" xref="S2.SS4.p2.5.m1.1.1.5.cmml">≤</mo><msub id="S2.SS4.p2.5.m1.1.1.6" xref="S2.SS4.p2.5.m1.1.1.6.cmml"><mi id="S2.SS4.p2.5.m1.1.1.6.2" xref="S2.SS4.p2.5.m1.1.1.6.2.cmml">q</mi><mn id="S2.SS4.p2.5.m1.1.1.6.3" xref="S2.SS4.p2.5.m1.1.1.6.3.cmml">1</mn></msub><mo id="S2.SS4.p2.5.m1.1.1.7" xref="S2.SS4.p2.5.m1.1.1.7.cmml">≤</mo><mi id="S2.SS4.p2.5.m1.1.1.8" mathvariant="normal" xref="S2.SS4.p2.5.m1.1.1.8.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.p2.5.m1.1b"><apply id="S2.SS4.p2.5.m1.1.1.cmml" xref="S2.SS4.p2.5.m1.1.1"><and id="S2.SS4.p2.5.m1.1.1a.cmml" xref="S2.SS4.p2.5.m1.1.1"></and><apply id="S2.SS4.p2.5.m1.1.1b.cmml" xref="S2.SS4.p2.5.m1.1.1"><leq id="S2.SS4.p2.5.m1.1.1.3.cmml" xref="S2.SS4.p2.5.m1.1.1.3"></leq><cn id="S2.SS4.p2.5.m1.1.1.2.cmml" type="integer" xref="S2.SS4.p2.5.m1.1.1.2">1</cn><apply id="S2.SS4.p2.5.m1.1.1.4.cmml" xref="S2.SS4.p2.5.m1.1.1.4"><csymbol cd="ambiguous" id="S2.SS4.p2.5.m1.1.1.4.1.cmml" xref="S2.SS4.p2.5.m1.1.1.4">subscript</csymbol><ci id="S2.SS4.p2.5.m1.1.1.4.2.cmml" xref="S2.SS4.p2.5.m1.1.1.4.2">𝑞</ci><cn id="S2.SS4.p2.5.m1.1.1.4.3.cmml" type="integer" xref="S2.SS4.p2.5.m1.1.1.4.3">0</cn></apply></apply><apply id="S2.SS4.p2.5.m1.1.1c.cmml" xref="S2.SS4.p2.5.m1.1.1"><leq id="S2.SS4.p2.5.m1.1.1.5.cmml" xref="S2.SS4.p2.5.m1.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S2.SS4.p2.5.m1.1.1.4.cmml" id="S2.SS4.p2.5.m1.1.1d.cmml" xref="S2.SS4.p2.5.m1.1.1"></share><apply id="S2.SS4.p2.5.m1.1.1.6.cmml" xref="S2.SS4.p2.5.m1.1.1.6"><csymbol cd="ambiguous" id="S2.SS4.p2.5.m1.1.1.6.1.cmml" xref="S2.SS4.p2.5.m1.1.1.6">subscript</csymbol><ci id="S2.SS4.p2.5.m1.1.1.6.2.cmml" xref="S2.SS4.p2.5.m1.1.1.6.2">𝑞</ci><cn id="S2.SS4.p2.5.m1.1.1.6.3.cmml" type="integer" xref="S2.SS4.p2.5.m1.1.1.6.3">1</cn></apply></apply><apply id="S2.SS4.p2.5.m1.1.1e.cmml" xref="S2.SS4.p2.5.m1.1.1"><leq id="S2.SS4.p2.5.m1.1.1.7.cmml" xref="S2.SS4.p2.5.m1.1.1.7"></leq><share href="https://arxiv.org/html/2503.14636v1#S2.SS4.p2.5.m1.1.1.6.cmml" id="S2.SS4.p2.5.m1.1.1f.cmml" xref="S2.SS4.p2.5.m1.1.1"></share><infinity id="S2.SS4.p2.5.m1.1.1.8.cmml" xref="S2.SS4.p2.5.m1.1.1.8"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p2.5.m1.1c">1\leq q_{0}\leq q_{1}\leq\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p2.5.m1.1d">1 ≤ italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ ∞</annotation></semantics></math>, then</p> <table class="ltx_equation ltx_eqn_table" id="S2.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="B^{s}_{p,q_{0}}(\mathbb{R}^{d},w;X)\hookrightarrow B^{s}_{p,q_{1}}(\mathbb{R}^% {d},w;X)\quad\text{ and }\quad 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↪ italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) and italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) ↪ italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.6)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S2.SS4.p3"> <p class="ltx_p" id="S2.SS4.p3.1">Finally, we state the following theorem on Sobolev embeddings for Besov and Triebel-Lizorkin spaces. Its proof is based on a similar result for radial weights, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Theorems 1.1 & 1.2]</cite>. Moreover, in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib58" title="">58</a>]</cite> general <math alttext="A_{\infty}" class="ltx_Math" display="inline" id="S2.SS4.p3.1.m1.1"><semantics id="S2.SS4.p3.1.m1.1a"><msub id="S2.SS4.p3.1.m1.1.1" xref="S2.SS4.p3.1.m1.1.1.cmml"><mi id="S2.SS4.p3.1.m1.1.1.2" xref="S2.SS4.p3.1.m1.1.1.2.cmml">A</mi><mi id="S2.SS4.p3.1.m1.1.1.3" mathvariant="normal" xref="S2.SS4.p3.1.m1.1.1.3.cmml">∞</mi></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.p3.1.m1.1b"><apply id="S2.SS4.p3.1.m1.1.1.cmml" xref="S2.SS4.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.p3.1.m1.1.1.1.cmml" xref="S2.SS4.p3.1.m1.1.1">subscript</csymbol><ci id="S2.SS4.p3.1.m1.1.1.2.cmml" xref="S2.SS4.p3.1.m1.1.1.2">𝐴</ci><infinity id="S2.SS4.p3.1.m1.1.1.3.cmml" xref="S2.SS4.p3.1.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.p3.1.m1.1c">A_{\infty}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.p3.1.m1.1d">italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT</annotation></semantics></math> weights are considered in the scalar-valued setting. The embedding with scalar-valued spaces can be extended to vector-valued spaces only for Muckenhoupt weights, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib58" title="">58</a>, Theorem 1.3]</cite>. However, this is not sufficient for our purposes.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S2.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.1.1.1">Theorem 2.3</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem3.p1"> <p class="ltx_p" id="S2.Thmtheorem3.p1.5"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p1.5.5">Let <math alttext="1<p_{0}\leq p_{1}<\infty" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.1.1.m1.1"><semantics id="S2.Thmtheorem3.p1.1.1.m1.1a"><mrow id="S2.Thmtheorem3.p1.1.1.m1.1.1" xref="S2.Thmtheorem3.p1.1.1.m1.1.1.cmml"><mn id="S2.Thmtheorem3.p1.1.1.m1.1.1.2" xref="S2.Thmtheorem3.p1.1.1.m1.1.1.2.cmml">1</mn><mo id="S2.Thmtheorem3.p1.1.1.m1.1.1.3" xref="S2.Thmtheorem3.p1.1.1.m1.1.1.3.cmml"><</mo><msub id="S2.Thmtheorem3.p1.1.1.m1.1.1.4" 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start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="s_{0}>s_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.3.3.m3.1"><semantics id="S2.Thmtheorem3.p1.3.3.m3.1a"><mrow id="S2.Thmtheorem3.p1.3.3.m3.1.1" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.cmml"><msub id="S2.Thmtheorem3.p1.3.3.m3.1.1.2" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2.cmml"><mi id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.2" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2.2.cmml">s</mi><mn id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.3" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p1.3.3.m3.1.1.1" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.1.cmml">></mo><msub id="S2.Thmtheorem3.p1.3.3.m3.1.1.3" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.cmml"><mi id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml">s</mi><mn id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.3.3.m3.1b"><apply id="S2.Thmtheorem3.p1.3.3.m3.1.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1"><gt id="S2.Thmtheorem3.p1.3.3.m3.1.1.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.1"></gt><apply id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2">subscript</csymbol><ci id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.2.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2.2">𝑠</ci><cn id="S2.Thmtheorem3.p1.3.3.m3.1.1.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.2.3">0</cn></apply><apply id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.1.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2.cmml" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.2">𝑠</ci><cn id="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.3.3.m3.1c">s_{0}>s_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.3.3.m3.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma_{0},\gamma_{1}>-1" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.4.4.m4.2"><semantics id="S2.Thmtheorem3.p1.4.4.m4.2a"><mrow id="S2.Thmtheorem3.p1.4.4.m4.2.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.cmml"><mrow id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.3.cmml"><msub id="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.2" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.2.cmml">γ</mi><mn id="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.3" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.3" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.3.cmml">,</mo><msub id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.cmml"><mi id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.2.cmml">γ</mi><mn id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.3" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.3.cmml">1</mn></msub></mrow><mo id="S2.Thmtheorem3.p1.4.4.m4.2.2.3" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.3.cmml">></mo><mrow id="S2.Thmtheorem3.p1.4.4.m4.2.2.4" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.4.cmml"><mo id="S2.Thmtheorem3.p1.4.4.m4.2.2.4a" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.4.cmml">−</mo><mn id="S2.Thmtheorem3.p1.4.4.m4.2.2.4.2" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.4.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.4.4.m4.2b"><apply id="S2.Thmtheorem3.p1.4.4.m4.2.2.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2"><gt id="S2.Thmtheorem3.p1.4.4.m4.2.2.3.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.3"></gt><list id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.3.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2"><apply id="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.2">𝛾</ci><cn id="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.4.4.m4.1.1.1.1.1.3">0</cn></apply><apply id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.2">𝛾</ci><cn id="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.2.2.2.3">1</cn></apply></list><apply id="S2.Thmtheorem3.p1.4.4.m4.2.2.4.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.4"><minus id="S2.Thmtheorem3.p1.4.4.m4.2.2.4.1.cmml" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.4"></minus><cn id="S2.Thmtheorem3.p1.4.4.m4.2.2.4.2.cmml" type="integer" xref="S2.Thmtheorem3.p1.4.4.m4.2.2.4.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.4.4.m4.2c">\gamma_{0},\gamma_{1}>-1</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.4.4.m4.2d">italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > - 1</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S2.Thmtheorem3.p1.5.5.m5.1"><semantics id="S2.Thmtheorem3.p1.5.5.m5.1a"><mi id="S2.Thmtheorem3.p1.5.5.m5.1.1" xref="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem3.p1.5.5.m5.1b"><ci id="S2.Thmtheorem3.p1.5.5.m5.1.1.cmml" xref="S2.Thmtheorem3.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem3.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem3.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Suppose that</span></p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex1a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{\gamma_{1}}{p_{1}}\leq\frac{\gamma_{0}}{p_{0}}\quad\text{ and }\quad s_{% 0}-\frac{d+\gamma_{0}}{p_{0}}=s_{1}-\frac{d+\gamma_{1}}{p_{1}}." class="ltx_Math" display="block" id="S2.Ex1a.m1.3"><semantics id="S2.Ex1a.m1.3a"><mrow id="S2.Ex1a.m1.3.3.1"><mrow id="S2.Ex1a.m1.3.3.1.1.2" xref="S2.Ex1a.m1.3.3.1.1.3.cmml"><mrow id="S2.Ex1a.m1.3.3.1.1.1.1" xref="S2.Ex1a.m1.3.3.1.1.1.1.cmml"><mfrac id="S2.Ex1a.m1.3.3.1.1.1.1.2" xref="S2.Ex1a.m1.3.3.1.1.1.1.2.cmml"><msub id="S2.Ex1a.m1.3.3.1.1.1.1.2.2" xref="S2.Ex1a.m1.3.3.1.1.1.1.2.2.cmml"><mi id="S2.Ex1a.m1.3.3.1.1.1.1.2.2.2" xref="S2.Ex1a.m1.3.3.1.1.1.1.2.2.2.cmml">γ</mi><mn id="S2.Ex1a.m1.3.3.1.1.1.1.2.2.3" xref="S2.Ex1a.m1.3.3.1.1.1.1.2.2.3.cmml">1</mn></msub><msub id="S2.Ex1a.m1.3.3.1.1.1.1.2.3" xref="S2.Ex1a.m1.3.3.1.1.1.1.2.3.cmml"><mi id="S2.Ex1a.m1.3.3.1.1.1.1.2.3.2" xref="S2.Ex1a.m1.3.3.1.1.1.1.2.3.2.cmml">p</mi><mn id="S2.Ex1a.m1.3.3.1.1.1.1.2.3.3" xref="S2.Ex1a.m1.3.3.1.1.1.1.2.3.3.cmml">1</mn></msub></mfrac><mo id="S2.Ex1a.m1.3.3.1.1.1.1.1" xref="S2.Ex1a.m1.3.3.1.1.1.1.1.cmml">≤</mo><mrow id="S2.Ex1a.m1.3.3.1.1.1.1.3.2" xref="S2.Ex1a.m1.3.3.1.1.1.1.3.1.cmml"><mfrac id="S2.Ex1a.m1.1.1" xref="S2.Ex1a.m1.1.1.cmml"><msub id="S2.Ex1a.m1.1.1.2" xref="S2.Ex1a.m1.1.1.2.cmml"><mi id="S2.Ex1a.m1.1.1.2.2" xref="S2.Ex1a.m1.1.1.2.2.cmml">γ</mi><mn id="S2.Ex1a.m1.1.1.2.3" xref="S2.Ex1a.m1.1.1.2.3.cmml">0</mn></msub><msub id="S2.Ex1a.m1.1.1.3" xref="S2.Ex1a.m1.1.1.3.cmml"><mi id="S2.Ex1a.m1.1.1.3.2" xref="S2.Ex1a.m1.1.1.3.2.cmml">p</mi><mn id="S2.Ex1a.m1.1.1.3.3" xref="S2.Ex1a.m1.1.1.3.3.cmml">0</mn></msub></mfrac><mspace id="S2.Ex1a.m1.3.3.1.1.1.1.3.2.1" width="1em" xref="S2.Ex1a.m1.3.3.1.1.1.1.3.1.cmml"></mspace><mtext class="ltx_mathvariant_italic" 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xref="S2.Ex1a.m1.3.3.1.1.2.2.3.3.3">subscript</csymbol><ci id="S2.Ex1a.m1.3.3.1.1.2.2.3.3.3.2.cmml" xref="S2.Ex1a.m1.3.3.1.1.2.2.3.3.3.2">𝑝</ci><cn id="S2.Ex1a.m1.3.3.1.1.2.2.3.3.3.3.cmml" type="integer" xref="S2.Ex1a.m1.3.3.1.1.2.2.3.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex1a.m1.3c">\frac{\gamma_{1}}{p_{1}}\leq\frac{\gamma_{0}}{p_{0}}\quad\text{ and }\quad s_{% 0}-\frac{d+\gamma_{0}}{p_{0}}=s_{1}-\frac{d+\gamma_{1}}{p_{1}}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex1a.m1.3d">divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ≤ divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG and italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - divide start_ARG italic_d + italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - divide start_ARG italic_d + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.Thmtheorem3.p1.6"><span class="ltx_text ltx_font_italic" id="S2.Thmtheorem3.p1.6.1">Then</span></p> <ol class="ltx_enumerate" id="S2.I2"> <li class="ltx_item" id="S2.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S2.I2.i1.p1"> <p class="ltx_p" id="S2.I2.i1.p1.1"><math alttext="F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)\hookrightarrow F^{s_{% 1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)" class="ltx_Math" display="inline" id="S2.I2.i1.p1.1.m1.10"><semantics 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xref="S2.I2.i1.p1.1.m1.9.9.3.1.1.1">superscript</csymbol><ci id="S2.I2.i1.p1.1.m1.9.9.3.1.1.1.2.cmml" xref="S2.I2.i1.p1.1.m1.9.9.3.1.1.1.2">ℝ</ci><ci id="S2.I2.i1.p1.1.m1.9.9.3.1.1.1.3.cmml" xref="S2.I2.i1.p1.1.m1.9.9.3.1.1.1.3">𝑑</ci></apply><apply id="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.cmml" xref="S2.I2.i1.p1.1.m1.10.10.4.2.2.2"><csymbol cd="ambiguous" id="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.1.cmml" xref="S2.I2.i1.p1.1.m1.10.10.4.2.2.2">subscript</csymbol><ci id="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.2.cmml" xref="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.2">𝑤</ci><apply id="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3.cmml" xref="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3"><csymbol cd="ambiguous" id="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3.1.cmml" xref="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3">subscript</csymbol><ci id="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3.2.cmml" xref="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3.2">𝛾</ci><cn id="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3.3.cmml" type="integer" xref="S2.I2.i1.p1.1.m1.10.10.4.2.2.2.3.3">1</cn></apply></apply><ci id="S2.I2.i1.p1.1.m1.6.6.cmml" xref="S2.I2.i1.p1.1.m1.6.6">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i1.p1.1.m1.10c">F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)\hookrightarrow F^{s_{% 1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i1.p1.1.m1.10d">italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) ↪ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i1.p1.1.1">,</span></p> </div> </li> <li class="ltx_item" id="S2.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S2.I2.i2.p1"> <p class="ltx_p" id="S2.I2.i2.p1.2"><math alttext="B^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)\hookrightarrow B^{s_{% 1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)" class="ltx_Math" display="inline" id="S2.I2.i2.p1.1.m1.10"><semantics id="S2.I2.i2.p1.1.m1.10a"><mrow id="S2.I2.i2.p1.1.m1.10.10" xref="S2.I2.i2.p1.1.m1.10.10.cmml"><mrow id="S2.I2.i2.p1.1.m1.8.8.2" xref="S2.I2.i2.p1.1.m1.8.8.2.cmml"><msubsup id="S2.I2.i2.p1.1.m1.8.8.2.4" 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1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i2.p1.1.m1.10d">italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) ↪ italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.2.1"> if, in addition, </span><math alttext="q_{0}\leq q_{1}" class="ltx_Math" display="inline" id="S2.I2.i2.p1.2.m2.1"><semantics id="S2.I2.i2.p1.2.m2.1a"><mrow id="S2.I2.i2.p1.2.m2.1.1" xref="S2.I2.i2.p1.2.m2.1.1.cmml"><msub id="S2.I2.i2.p1.2.m2.1.1.2" xref="S2.I2.i2.p1.2.m2.1.1.2.cmml"><mi id="S2.I2.i2.p1.2.m2.1.1.2.2" xref="S2.I2.i2.p1.2.m2.1.1.2.2.cmml">q</mi><mn id="S2.I2.i2.p1.2.m2.1.1.2.3" xref="S2.I2.i2.p1.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S2.I2.i2.p1.2.m2.1.1.1" xref="S2.I2.i2.p1.2.m2.1.1.1.cmml">≤</mo><msub id="S2.I2.i2.p1.2.m2.1.1.3" xref="S2.I2.i2.p1.2.m2.1.1.3.cmml"><mi id="S2.I2.i2.p1.2.m2.1.1.3.2" xref="S2.I2.i2.p1.2.m2.1.1.3.2.cmml">q</mi><mn id="S2.I2.i2.p1.2.m2.1.1.3.3" xref="S2.I2.i2.p1.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.I2.i2.p1.2.m2.1b"><apply id="S2.I2.i2.p1.2.m2.1.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1"><leq id="S2.I2.i2.p1.2.m2.1.1.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.1"></leq><apply id="S2.I2.i2.p1.2.m2.1.1.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.1.1.2.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S2.I2.i2.p1.2.m2.1.1.2.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.2.2">𝑞</ci><cn id="S2.I2.i2.p1.2.m2.1.1.2.3.cmml" type="integer" xref="S2.I2.i2.p1.2.m2.1.1.2.3">0</cn></apply><apply id="S2.I2.i2.p1.2.m2.1.1.3.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S2.I2.i2.p1.2.m2.1.1.3.1.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S2.I2.i2.p1.2.m2.1.1.3.2.cmml" xref="S2.I2.i2.p1.2.m2.1.1.3.2">𝑞</ci><cn id="S2.I2.i2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S2.I2.i2.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.I2.i2.p1.2.m2.1c">q_{0}\leq q_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.I2.i2.p1.2.m2.1d">italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S2.I2.i2.p1.2.2">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S2.SS4.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S2.SS4.1.p1"> <p class="ltx_p" id="S2.SS4.1.p1.5"><span class="ltx_text ltx_font_italic" id="S2.SS4.1.p1.5.1">Step 1: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.I2.i2" title="item ii ‣ Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>.</span> By (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E6" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.6</span></a>) it suffices to consider <math alttext="q:=q_{0}=q_{1}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.1.m1.1"><semantics id="S2.SS4.1.p1.1.m1.1a"><mrow id="S2.SS4.1.p1.1.m1.1.1" xref="S2.SS4.1.p1.1.m1.1.1.cmml"><mi id="S2.SS4.1.p1.1.m1.1.1.2" xref="S2.SS4.1.p1.1.m1.1.1.2.cmml">q</mi><mo id="S2.SS4.1.p1.1.m1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S2.SS4.1.p1.1.m1.1.1.3.cmml">:=</mo><msub id="S2.SS4.1.p1.1.m1.1.1.4" xref="S2.SS4.1.p1.1.m1.1.1.4.cmml"><mi id="S2.SS4.1.p1.1.m1.1.1.4.2" xref="S2.SS4.1.p1.1.m1.1.1.4.2.cmml">q</mi><mn id="S2.SS4.1.p1.1.m1.1.1.4.3" xref="S2.SS4.1.p1.1.m1.1.1.4.3.cmml">0</mn></msub><mo id="S2.SS4.1.p1.1.m1.1.1.5" xref="S2.SS4.1.p1.1.m1.1.1.5.cmml">=</mo><msub id="S2.SS4.1.p1.1.m1.1.1.6" xref="S2.SS4.1.p1.1.m1.1.1.6.cmml"><mi id="S2.SS4.1.p1.1.m1.1.1.6.2" xref="S2.SS4.1.p1.1.m1.1.1.6.2.cmml">q</mi><mn id="S2.SS4.1.p1.1.m1.1.1.6.3" xref="S2.SS4.1.p1.1.m1.1.1.6.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.1.m1.1b"><apply id="S2.SS4.1.p1.1.m1.1.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1"><and id="S2.SS4.1.p1.1.m1.1.1a.cmml" xref="S2.SS4.1.p1.1.m1.1.1"></and><apply id="S2.SS4.1.p1.1.m1.1.1b.cmml" xref="S2.SS4.1.p1.1.m1.1.1"><csymbol cd="latexml" id="S2.SS4.1.p1.1.m1.1.1.3.cmml" xref="S2.SS4.1.p1.1.m1.1.1.3">assign</csymbol><ci id="S2.SS4.1.p1.1.m1.1.1.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.2">𝑞</ci><apply id="S2.SS4.1.p1.1.m1.1.1.4.cmml" xref="S2.SS4.1.p1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S2.SS4.1.p1.1.m1.1.1.4.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.4">subscript</csymbol><ci id="S2.SS4.1.p1.1.m1.1.1.4.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.4.2">𝑞</ci><cn id="S2.SS4.1.p1.1.m1.1.1.4.3.cmml" type="integer" xref="S2.SS4.1.p1.1.m1.1.1.4.3">0</cn></apply></apply><apply id="S2.SS4.1.p1.1.m1.1.1c.cmml" xref="S2.SS4.1.p1.1.m1.1.1"><eq id="S2.SS4.1.p1.1.m1.1.1.5.cmml" xref="S2.SS4.1.p1.1.m1.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S2.SS4.1.p1.1.m1.1.1.4.cmml" id="S2.SS4.1.p1.1.m1.1.1d.cmml" xref="S2.SS4.1.p1.1.m1.1.1"></share><apply id="S2.SS4.1.p1.1.m1.1.1.6.cmml" xref="S2.SS4.1.p1.1.m1.1.1.6"><csymbol cd="ambiguous" id="S2.SS4.1.p1.1.m1.1.1.6.1.cmml" xref="S2.SS4.1.p1.1.m1.1.1.6">subscript</csymbol><ci id="S2.SS4.1.p1.1.m1.1.1.6.2.cmml" xref="S2.SS4.1.p1.1.m1.1.1.6.2">𝑞</ci><cn id="S2.SS4.1.p1.1.m1.1.1.6.3.cmml" type="integer" xref="S2.SS4.1.p1.1.m1.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.1.m1.1c">q:=q_{0}=q_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.1.m1.1d">italic_q := italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="f\in B^{s_{0}}_{p_{0},q}(\mathbb{R}^{d},w_{\gamma_{0}};X)" class="ltx_Math" display="inline" id="S2.SS4.1.p1.2.m2.5"><semantics id="S2.SS4.1.p1.2.m2.5a"><mrow id="S2.SS4.1.p1.2.m2.5.5" xref="S2.SS4.1.p1.2.m2.5.5.cmml"><mi id="S2.SS4.1.p1.2.m2.5.5.4" xref="S2.SS4.1.p1.2.m2.5.5.4.cmml">f</mi><mo id="S2.SS4.1.p1.2.m2.5.5.3" xref="S2.SS4.1.p1.2.m2.5.5.3.cmml">∈</mo><mrow id="S2.SS4.1.p1.2.m2.5.5.2" xref="S2.SS4.1.p1.2.m2.5.5.2.cmml"><msubsup id="S2.SS4.1.p1.2.m2.5.5.2.4" xref="S2.SS4.1.p1.2.m2.5.5.2.4.cmml"><mi id="S2.SS4.1.p1.2.m2.5.5.2.4.2.2" xref="S2.SS4.1.p1.2.m2.5.5.2.4.2.2.cmml">B</mi><mrow id="S2.SS4.1.p1.2.m2.2.2.2.2" xref="S2.SS4.1.p1.2.m2.2.2.2.3.cmml"><msub id="S2.SS4.1.p1.2.m2.2.2.2.2.1" xref="S2.SS4.1.p1.2.m2.2.2.2.2.1.cmml"><mi id="S2.SS4.1.p1.2.m2.2.2.2.2.1.2" xref="S2.SS4.1.p1.2.m2.2.2.2.2.1.2.cmml">p</mi><mn id="S2.SS4.1.p1.2.m2.2.2.2.2.1.3" xref="S2.SS4.1.p1.2.m2.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S2.SS4.1.p1.2.m2.2.2.2.2.2" xref="S2.SS4.1.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S2.SS4.1.p1.2.m2.1.1.1.1" xref="S2.SS4.1.p1.2.m2.1.1.1.1.cmml">q</mi></mrow><msub id="S2.SS4.1.p1.2.m2.5.5.2.4.2.3" xref="S2.SS4.1.p1.2.m2.5.5.2.4.2.3.cmml"><mi id="S2.SS4.1.p1.2.m2.5.5.2.4.2.3.2" xref="S2.SS4.1.p1.2.m2.5.5.2.4.2.3.2.cmml">s</mi><mn id="S2.SS4.1.p1.2.m2.5.5.2.4.2.3.3" xref="S2.SS4.1.p1.2.m2.5.5.2.4.2.3.3.cmml">0</mn></msub></msubsup><mo id="S2.SS4.1.p1.2.m2.5.5.2.3" xref="S2.SS4.1.p1.2.m2.5.5.2.3.cmml">⁢</mo><mrow id="S2.SS4.1.p1.2.m2.5.5.2.2.2" xref="S2.SS4.1.p1.2.m2.5.5.2.2.3.cmml"><mo id="S2.SS4.1.p1.2.m2.5.5.2.2.2.3" stretchy="false" xref="S2.SS4.1.p1.2.m2.5.5.2.2.3.cmml">(</mo><msup id="S2.SS4.1.p1.2.m2.4.4.1.1.1.1" xref="S2.SS4.1.p1.2.m2.4.4.1.1.1.1.cmml"><mi id="S2.SS4.1.p1.2.m2.4.4.1.1.1.1.2" xref="S2.SS4.1.p1.2.m2.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS4.1.p1.2.m2.4.4.1.1.1.1.3" xref="S2.SS4.1.p1.2.m2.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS4.1.p1.2.m2.5.5.2.2.2.4" 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xref="S2.SS4.1.p1.3.m3.2.2.2.3.cmml">Φ</mi><mo id="S2.SS4.1.p1.3.m3.2.2.2.2" xref="S2.SS4.1.p1.3.m3.2.2.2.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.3.m3.2.2.2.1.1" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.cmml"><mo id="S2.SS4.1.p1.3.m3.2.2.2.1.1.2" stretchy="false" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.cmml">(</mo><msup id="S2.SS4.1.p1.3.m3.2.2.2.1.1.1" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.cmml"><mi id="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.2" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.3" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS4.1.p1.3.m3.2.2.2.1.1.3" stretchy="false" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.3.m3.2b"><apply id="S2.SS4.1.p1.3.m3.2.2.cmml" xref="S2.SS4.1.p1.3.m3.2.2"><in id="S2.SS4.1.p1.3.m3.2.2.3.cmml" xref="S2.SS4.1.p1.3.m3.2.2.3"></in><apply id="S2.SS4.1.p1.3.m3.1.1.1.cmml" xref="S2.SS4.1.p1.3.m3.1.1.1"><csymbol cd="ambiguous" 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id="S2.SS4.1.p1.3.m3.2.2.2.3.cmml" xref="S2.SS4.1.p1.3.m3.2.2.2.3">Φ</ci><apply id="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.cmml" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.1.cmml" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1">superscript</csymbol><ci id="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.2.cmml" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.3.cmml" xref="S2.SS4.1.p1.3.m3.2.2.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.3.m3.2c">(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.3.m3.2d">( italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> and define <math alttext="S_{n}f:=\varphi_{n}\ast f" class="ltx_Math" display="inline" id="S2.SS4.1.p1.4.m4.1"><semantics id="S2.SS4.1.p1.4.m4.1a"><mrow id="S2.SS4.1.p1.4.m4.1.1" xref="S2.SS4.1.p1.4.m4.1.1.cmml"><mrow id="S2.SS4.1.p1.4.m4.1.1.2" xref="S2.SS4.1.p1.4.m4.1.1.2.cmml"><msub id="S2.SS4.1.p1.4.m4.1.1.2.2" xref="S2.SS4.1.p1.4.m4.1.1.2.2.cmml"><mi id="S2.SS4.1.p1.4.m4.1.1.2.2.2" xref="S2.SS4.1.p1.4.m4.1.1.2.2.2.cmml">S</mi><mi id="S2.SS4.1.p1.4.m4.1.1.2.2.3" xref="S2.SS4.1.p1.4.m4.1.1.2.2.3.cmml">n</mi></msub><mo id="S2.SS4.1.p1.4.m4.1.1.2.1" xref="S2.SS4.1.p1.4.m4.1.1.2.1.cmml">⁢</mo><mi id="S2.SS4.1.p1.4.m4.1.1.2.3" xref="S2.SS4.1.p1.4.m4.1.1.2.3.cmml">f</mi></mrow><mo id="S2.SS4.1.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS4.1.p1.4.m4.1.1.1.cmml">:=</mo><mrow id="S2.SS4.1.p1.4.m4.1.1.3" xref="S2.SS4.1.p1.4.m4.1.1.3.cmml"><msub id="S2.SS4.1.p1.4.m4.1.1.3.2" xref="S2.SS4.1.p1.4.m4.1.1.3.2.cmml"><mi id="S2.SS4.1.p1.4.m4.1.1.3.2.2" xref="S2.SS4.1.p1.4.m4.1.1.3.2.2.cmml">φ</mi><mi id="S2.SS4.1.p1.4.m4.1.1.3.2.3" xref="S2.SS4.1.p1.4.m4.1.1.3.2.3.cmml">n</mi></msub><mo id="S2.SS4.1.p1.4.m4.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS4.1.p1.4.m4.1.1.3.1.cmml">∗</mo><mi id="S2.SS4.1.p1.4.m4.1.1.3.3" xref="S2.SS4.1.p1.4.m4.1.1.3.3.cmml">f</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.4.m4.1b"><apply id="S2.SS4.1.p1.4.m4.1.1.cmml" xref="S2.SS4.1.p1.4.m4.1.1"><csymbol cd="latexml" id="S2.SS4.1.p1.4.m4.1.1.1.cmml" xref="S2.SS4.1.p1.4.m4.1.1.1">assign</csymbol><apply id="S2.SS4.1.p1.4.m4.1.1.2.cmml" xref="S2.SS4.1.p1.4.m4.1.1.2"><times id="S2.SS4.1.p1.4.m4.1.1.2.1.cmml" xref="S2.SS4.1.p1.4.m4.1.1.2.1"></times><apply id="S2.SS4.1.p1.4.m4.1.1.2.2.cmml" xref="S2.SS4.1.p1.4.m4.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.1.1.2.2.1.cmml" xref="S2.SS4.1.p1.4.m4.1.1.2.2">subscript</csymbol><ci id="S2.SS4.1.p1.4.m4.1.1.2.2.2.cmml" xref="S2.SS4.1.p1.4.m4.1.1.2.2.2">𝑆</ci><ci id="S2.SS4.1.p1.4.m4.1.1.2.2.3.cmml" xref="S2.SS4.1.p1.4.m4.1.1.2.2.3">𝑛</ci></apply><ci id="S2.SS4.1.p1.4.m4.1.1.2.3.cmml" xref="S2.SS4.1.p1.4.m4.1.1.2.3">𝑓</ci></apply><apply id="S2.SS4.1.p1.4.m4.1.1.3.cmml" xref="S2.SS4.1.p1.4.m4.1.1.3"><ci id="S2.SS4.1.p1.4.m4.1.1.3.1.cmml" xref="S2.SS4.1.p1.4.m4.1.1.3.1">∗</ci><apply id="S2.SS4.1.p1.4.m4.1.1.3.2.cmml" xref="S2.SS4.1.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.4.m4.1.1.3.2.1.cmml" xref="S2.SS4.1.p1.4.m4.1.1.3.2">subscript</csymbol><ci id="S2.SS4.1.p1.4.m4.1.1.3.2.2.cmml" xref="S2.SS4.1.p1.4.m4.1.1.3.2.2">𝜑</ci><ci id="S2.SS4.1.p1.4.m4.1.1.3.2.3.cmml" xref="S2.SS4.1.p1.4.m4.1.1.3.2.3">𝑛</ci></apply><ci id="S2.SS4.1.p1.4.m4.1.1.3.3.cmml" xref="S2.SS4.1.p1.4.m4.1.1.3.3">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.4.m4.1c">S_{n}f:=\varphi_{n}\ast f</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.4.m4.1d">italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f := italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_f</annotation></semantics></math> as in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E2" title="In 2.2. Weighted function spaces ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.2</span></a>). We write <math alttext="s_{1}+\delta=s_{0}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.5.m5.1"><semantics id="S2.SS4.1.p1.5.m5.1a"><mrow id="S2.SS4.1.p1.5.m5.1.1" xref="S2.SS4.1.p1.5.m5.1.1.cmml"><mrow id="S2.SS4.1.p1.5.m5.1.1.2" xref="S2.SS4.1.p1.5.m5.1.1.2.cmml"><msub id="S2.SS4.1.p1.5.m5.1.1.2.2" xref="S2.SS4.1.p1.5.m5.1.1.2.2.cmml"><mi id="S2.SS4.1.p1.5.m5.1.1.2.2.2" xref="S2.SS4.1.p1.5.m5.1.1.2.2.2.cmml">s</mi><mn id="S2.SS4.1.p1.5.m5.1.1.2.2.3" xref="S2.SS4.1.p1.5.m5.1.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS4.1.p1.5.m5.1.1.2.1" xref="S2.SS4.1.p1.5.m5.1.1.2.1.cmml">+</mo><mi id="S2.SS4.1.p1.5.m5.1.1.2.3" xref="S2.SS4.1.p1.5.m5.1.1.2.3.cmml">δ</mi></mrow><mo id="S2.SS4.1.p1.5.m5.1.1.1" xref="S2.SS4.1.p1.5.m5.1.1.1.cmml">=</mo><msub id="S2.SS4.1.p1.5.m5.1.1.3" xref="S2.SS4.1.p1.5.m5.1.1.3.cmml"><mi id="S2.SS4.1.p1.5.m5.1.1.3.2" xref="S2.SS4.1.p1.5.m5.1.1.3.2.cmml">s</mi><mn id="S2.SS4.1.p1.5.m5.1.1.3.3" xref="S2.SS4.1.p1.5.m5.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.5.m5.1b"><apply id="S2.SS4.1.p1.5.m5.1.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1"><eq id="S2.SS4.1.p1.5.m5.1.1.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1.1"></eq><apply id="S2.SS4.1.p1.5.m5.1.1.2.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2"><plus id="S2.SS4.1.p1.5.m5.1.1.2.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2.1"></plus><apply id="S2.SS4.1.p1.5.m5.1.1.2.2.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.5.m5.1.1.2.2.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2.2">subscript</csymbol><ci id="S2.SS4.1.p1.5.m5.1.1.2.2.2.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2.2.2">𝑠</ci><cn id="S2.SS4.1.p1.5.m5.1.1.2.2.3.cmml" type="integer" xref="S2.SS4.1.p1.5.m5.1.1.2.2.3">1</cn></apply><ci id="S2.SS4.1.p1.5.m5.1.1.2.3.cmml" xref="S2.SS4.1.p1.5.m5.1.1.2.3">𝛿</ci></apply><apply id="S2.SS4.1.p1.5.m5.1.1.3.cmml" xref="S2.SS4.1.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.5.m5.1.1.3.1.cmml" xref="S2.SS4.1.p1.5.m5.1.1.3">subscript</csymbol><ci id="S2.SS4.1.p1.5.m5.1.1.3.2.cmml" xref="S2.SS4.1.p1.5.m5.1.1.3.2">𝑠</ci><cn id="S2.SS4.1.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S2.SS4.1.p1.5.m5.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.5.m5.1c">s_{1}+\delta=s_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.5.m5.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_δ = italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, where</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex2a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\delta=\delta_{0}+\delta_{1}\quad\text{ with }\quad\delta_{0}:=\frac{\gamma_{0% }+1}{p_{0}}-\frac{\gamma_{1}+1}{p_{1}}\quad\text{ and }\quad\delta_{1}:=\frac{% d-1}{p_{0}}-\frac{d-1}{p_{1}}." class="ltx_Math" display="block" id="S2.Ex2a.m1.3"><semantics id="S2.Ex2a.m1.3a"><mrow id="S2.Ex2a.m1.3.3.1"><mrow id="S2.Ex2a.m1.3.3.1.1.2" xref="S2.Ex2a.m1.3.3.1.1.3.cmml"><mrow id="S2.Ex2a.m1.3.3.1.1.1.1" xref="S2.Ex2a.m1.3.3.1.1.1.1.cmml"><mi id="S2.Ex2a.m1.3.3.1.1.1.1.3" xref="S2.Ex2a.m1.3.3.1.1.1.1.3.cmml">δ</mi><mo id="S2.Ex2a.m1.3.3.1.1.1.1.2" xref="S2.Ex2a.m1.3.3.1.1.1.1.2.cmml">=</mo><mrow id="S2.Ex2a.m1.3.3.1.1.1.1.1.1" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.2.cmml"><mrow id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.cmml"><msub id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.2" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.2.cmml"><mi id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.2.2" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.2.2.cmml">δ</mi><mn id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.2.3" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.1" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.1.cmml">+</mo><msub id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.3" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.3.cmml"><mi id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.3.2" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.3.2.cmml">δ</mi><mn id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.3.3" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mspace id="S2.Ex2a.m1.3.3.1.1.1.1.1.1.2" width="1em" xref="S2.Ex2a.m1.3.3.1.1.1.1.1.2.cmml"></mspace><mtext id="S2.Ex2a.m1.2.2" xref="S2.Ex2a.m1.2.2a.cmml"> with </mtext></mrow></mrow><mspace id="S2.Ex2a.m1.3.3.1.1.2.3" width="1em" xref="S2.Ex2a.m1.3.3.1.1.3a.cmml"></mspace><mrow id="S2.Ex2a.m1.3.3.1.1.2.2.2" xref="S2.Ex2a.m1.3.3.1.1.2.2.3.cmml"><mrow id="S2.Ex2a.m1.3.3.1.1.2.2.1.1" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.cmml"><msub id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.3" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.3.cmml"><mi id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.3.2" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.3.2.cmml">δ</mi><mn id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.3.3" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.3.3.cmml">0</mn></msub><mo id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.2" lspace="0.278em" rspace="0.278em" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.2.cmml">:=</mo><mrow id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.2.cmml"><mrow id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.cmml"><mfrac id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.cmml"><mrow id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.cmml"><msub id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.2" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.2.cmml"><mi id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.2.2" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.2.2.cmml">γ</mi><mn id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.2.3" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.2.3.cmml">0</mn></msub><mo id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.1" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.1.cmml">+</mo><mn id="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.3" xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.2.2.3.cmml">1</mn></mrow><msub 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xref="S2.Ex2a.m1.3.3.1.1.2.2.1.1.1.1.1.3.3.3">1</cn></apply></apply></apply><ci id="S2.Ex2a.m1.1.1a.cmml" xref="S2.Ex2a.m1.1.1"><mtext id="S2.Ex2a.m1.1.1.cmml" xref="S2.Ex2a.m1.1.1"> and </mtext></ci></list></apply><apply id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2"><csymbol cd="latexml" id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.1.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.1">assign</csymbol><apply id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2.1.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2">subscript</csymbol><ci id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2.2.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2.2">𝛿</ci><cn id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2.3.cmml" type="integer" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.2.3">1</cn></apply><apply id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3"><minus id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.1.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.1"></minus><apply 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xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.2.3.3">0</cn></apply></apply><apply id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3"><divide id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.1.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3"></divide><apply id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2"><minus id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2.1.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2.1"></minus><ci id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2.2.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2.2">𝑑</ci><cn id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2.3.cmml" type="integer" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.2.3">1</cn></apply><apply id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3"><csymbol cd="ambiguous" id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3.1.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3">subscript</csymbol><ci id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3.2.cmml" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3.2">𝑝</ci><cn id="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3.3.cmml" type="integer" xref="S2.Ex2a.m1.3.3.1.1.2.2.2.2.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex2a.m1.3c">\delta=\delta_{0}+\delta_{1}\quad\text{ with }\quad\delta_{0}:=\frac{\gamma_{0% }+1}{p_{0}}-\frac{\gamma_{1}+1}{p_{1}}\quad\text{ and }\quad\delta_{1}:=\frac{% d-1}{p_{0}}-\frac{d-1}{p_{1}}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex2a.m1.3d">italic_δ = italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT with italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG - divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG and italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := divide start_ARG italic_d - 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG - divide start_ARG italic_d - 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.1.p1.14">Let <math alttext="\mathcal{F}_{x_{1}}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.6.m1.1"><semantics id="S2.SS4.1.p1.6.m1.1a"><msub id="S2.SS4.1.p1.6.m1.1.1" xref="S2.SS4.1.p1.6.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.1.p1.6.m1.1.1.2" xref="S2.SS4.1.p1.6.m1.1.1.2.cmml">ℱ</mi><msub id="S2.SS4.1.p1.6.m1.1.1.3" xref="S2.SS4.1.p1.6.m1.1.1.3.cmml"><mi id="S2.SS4.1.p1.6.m1.1.1.3.2" xref="S2.SS4.1.p1.6.m1.1.1.3.2.cmml">x</mi><mn id="S2.SS4.1.p1.6.m1.1.1.3.3" xref="S2.SS4.1.p1.6.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.6.m1.1b"><apply id="S2.SS4.1.p1.6.m1.1.1.cmml" xref="S2.SS4.1.p1.6.m1.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.6.m1.1.1.1.cmml" xref="S2.SS4.1.p1.6.m1.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.6.m1.1.1.2.cmml" xref="S2.SS4.1.p1.6.m1.1.1.2">ℱ</ci><apply id="S2.SS4.1.p1.6.m1.1.1.3.cmml" xref="S2.SS4.1.p1.6.m1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.6.m1.1.1.3.1.cmml" xref="S2.SS4.1.p1.6.m1.1.1.3">subscript</csymbol><ci id="S2.SS4.1.p1.6.m1.1.1.3.2.cmml" xref="S2.SS4.1.p1.6.m1.1.1.3.2">𝑥</ci><cn id="S2.SS4.1.p1.6.m1.1.1.3.3.cmml" type="integer" xref="S2.SS4.1.p1.6.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.6.m1.1c">\mathcal{F}_{x_{1}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.6.m1.1d">caligraphic_F start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{F}_{\widetilde{x}}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.7.m2.1"><semantics id="S2.SS4.1.p1.7.m2.1a"><msub id="S2.SS4.1.p1.7.m2.1.1" xref="S2.SS4.1.p1.7.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.1.p1.7.m2.1.1.2" xref="S2.SS4.1.p1.7.m2.1.1.2.cmml">ℱ</mi><mover accent="true" id="S2.SS4.1.p1.7.m2.1.1.3" xref="S2.SS4.1.p1.7.m2.1.1.3.cmml"><mi id="S2.SS4.1.p1.7.m2.1.1.3.2" xref="S2.SS4.1.p1.7.m2.1.1.3.2.cmml">x</mi><mo id="S2.SS4.1.p1.7.m2.1.1.3.1" xref="S2.SS4.1.p1.7.m2.1.1.3.1.cmml">~</mo></mover></msub><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.7.m2.1b"><apply id="S2.SS4.1.p1.7.m2.1.1.cmml" xref="S2.SS4.1.p1.7.m2.1.1"><csymbol cd="ambiguous" id="S2.SS4.1.p1.7.m2.1.1.1.cmml" xref="S2.SS4.1.p1.7.m2.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.7.m2.1.1.2.cmml" xref="S2.SS4.1.p1.7.m2.1.1.2">ℱ</ci><apply id="S2.SS4.1.p1.7.m2.1.1.3.cmml" xref="S2.SS4.1.p1.7.m2.1.1.3"><ci id="S2.SS4.1.p1.7.m2.1.1.3.1.cmml" xref="S2.SS4.1.p1.7.m2.1.1.3.1">~</ci><ci id="S2.SS4.1.p1.7.m2.1.1.3.2.cmml" xref="S2.SS4.1.p1.7.m2.1.1.3.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.7.m2.1c">\mathcal{F}_{\widetilde{x}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.7.m2.1d">caligraphic_F start_POSTSUBSCRIPT over~ start_ARG italic_x end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> be the Fourier transform with respect to <math alttext="x_{1}\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.8.m3.1"><semantics id="S2.SS4.1.p1.8.m3.1a"><mrow id="S2.SS4.1.p1.8.m3.1.1" xref="S2.SS4.1.p1.8.m3.1.1.cmml"><msub id="S2.SS4.1.p1.8.m3.1.1.2" xref="S2.SS4.1.p1.8.m3.1.1.2.cmml"><mi id="S2.SS4.1.p1.8.m3.1.1.2.2" xref="S2.SS4.1.p1.8.m3.1.1.2.2.cmml">x</mi><mn id="S2.SS4.1.p1.8.m3.1.1.2.3" xref="S2.SS4.1.p1.8.m3.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS4.1.p1.8.m3.1.1.1" xref="S2.SS4.1.p1.8.m3.1.1.1.cmml">∈</mo><mi id="S2.SS4.1.p1.8.m3.1.1.3" xref="S2.SS4.1.p1.8.m3.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.8.m3.1b"><apply id="S2.SS4.1.p1.8.m3.1.1.cmml" xref="S2.SS4.1.p1.8.m3.1.1"><in id="S2.SS4.1.p1.8.m3.1.1.1.cmml" xref="S2.SS4.1.p1.8.m3.1.1.1"></in><apply id="S2.SS4.1.p1.8.m3.1.1.2.cmml" xref="S2.SS4.1.p1.8.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.8.m3.1.1.2.1.cmml" xref="S2.SS4.1.p1.8.m3.1.1.2">subscript</csymbol><ci id="S2.SS4.1.p1.8.m3.1.1.2.2.cmml" xref="S2.SS4.1.p1.8.m3.1.1.2.2">𝑥</ci><cn id="S2.SS4.1.p1.8.m3.1.1.2.3.cmml" type="integer" xref="S2.SS4.1.p1.8.m3.1.1.2.3">1</cn></apply><ci id="S2.SS4.1.p1.8.m3.1.1.3.cmml" xref="S2.SS4.1.p1.8.m3.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.8.m3.1c">x_{1}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.8.m3.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math> and <math alttext="\widetilde{x}\in\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.9.m4.1"><semantics id="S2.SS4.1.p1.9.m4.1a"><mrow id="S2.SS4.1.p1.9.m4.1.1" xref="S2.SS4.1.p1.9.m4.1.1.cmml"><mover accent="true" id="S2.SS4.1.p1.9.m4.1.1.2" xref="S2.SS4.1.p1.9.m4.1.1.2.cmml"><mi id="S2.SS4.1.p1.9.m4.1.1.2.2" xref="S2.SS4.1.p1.9.m4.1.1.2.2.cmml">x</mi><mo id="S2.SS4.1.p1.9.m4.1.1.2.1" xref="S2.SS4.1.p1.9.m4.1.1.2.1.cmml">~</mo></mover><mo id="S2.SS4.1.p1.9.m4.1.1.1" xref="S2.SS4.1.p1.9.m4.1.1.1.cmml">∈</mo><msup id="S2.SS4.1.p1.9.m4.1.1.3" xref="S2.SS4.1.p1.9.m4.1.1.3.cmml"><mi id="S2.SS4.1.p1.9.m4.1.1.3.2" xref="S2.SS4.1.p1.9.m4.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS4.1.p1.9.m4.1.1.3.3" xref="S2.SS4.1.p1.9.m4.1.1.3.3.cmml"><mi id="S2.SS4.1.p1.9.m4.1.1.3.3.2" xref="S2.SS4.1.p1.9.m4.1.1.3.3.2.cmml">d</mi><mo id="S2.SS4.1.p1.9.m4.1.1.3.3.1" xref="S2.SS4.1.p1.9.m4.1.1.3.3.1.cmml">−</mo><mn id="S2.SS4.1.p1.9.m4.1.1.3.3.3" xref="S2.SS4.1.p1.9.m4.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.9.m4.1b"><apply id="S2.SS4.1.p1.9.m4.1.1.cmml" xref="S2.SS4.1.p1.9.m4.1.1"><in id="S2.SS4.1.p1.9.m4.1.1.1.cmml" xref="S2.SS4.1.p1.9.m4.1.1.1"></in><apply id="S2.SS4.1.p1.9.m4.1.1.2.cmml" xref="S2.SS4.1.p1.9.m4.1.1.2"><ci id="S2.SS4.1.p1.9.m4.1.1.2.1.cmml" xref="S2.SS4.1.p1.9.m4.1.1.2.1">~</ci><ci id="S2.SS4.1.p1.9.m4.1.1.2.2.cmml" xref="S2.SS4.1.p1.9.m4.1.1.2.2">𝑥</ci></apply><apply id="S2.SS4.1.p1.9.m4.1.1.3.cmml" xref="S2.SS4.1.p1.9.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.9.m4.1.1.3.1.cmml" xref="S2.SS4.1.p1.9.m4.1.1.3">superscript</csymbol><ci id="S2.SS4.1.p1.9.m4.1.1.3.2.cmml" xref="S2.SS4.1.p1.9.m4.1.1.3.2">ℝ</ci><apply id="S2.SS4.1.p1.9.m4.1.1.3.3.cmml" xref="S2.SS4.1.p1.9.m4.1.1.3.3"><minus id="S2.SS4.1.p1.9.m4.1.1.3.3.1.cmml" xref="S2.SS4.1.p1.9.m4.1.1.3.3.1"></minus><ci id="S2.SS4.1.p1.9.m4.1.1.3.3.2.cmml" xref="S2.SS4.1.p1.9.m4.1.1.3.3.2">𝑑</ci><cn id="S2.SS4.1.p1.9.m4.1.1.3.3.3.cmml" type="integer" xref="S2.SS4.1.p1.9.m4.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.9.m4.1c">\widetilde{x}\in\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.9.m4.1d">over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, respectively. Note that <math alttext="\text{\rm supp\,}\mathcal{F}(S_{n}f)\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq 3% \cdot 2^{n-1}\}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.10.m5.4"><semantics id="S2.SS4.1.p1.10.m5.4a"><mrow id="S2.SS4.1.p1.10.m5.4.4" xref="S2.SS4.1.p1.10.m5.4.4.cmml"><mrow id="S2.SS4.1.p1.10.m5.2.2.1" xref="S2.SS4.1.p1.10.m5.2.2.1.cmml"><mtext id="S2.SS4.1.p1.10.m5.2.2.1.3" xref="S2.SS4.1.p1.10.m5.2.2.1.3a.cmml">supp </mtext><mo id="S2.SS4.1.p1.10.m5.2.2.1.2" xref="S2.SS4.1.p1.10.m5.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S2.SS4.1.p1.10.m5.2.2.1.4" xref="S2.SS4.1.p1.10.m5.2.2.1.4.cmml">ℱ</mi><mo id="S2.SS4.1.p1.10.m5.2.2.1.2a" xref="S2.SS4.1.p1.10.m5.2.2.1.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.10.m5.2.2.1.1.1" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.cmml"><mo id="S2.SS4.1.p1.10.m5.2.2.1.1.1.2" stretchy="false" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.cmml"><msub id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.cmml"><mi id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.2" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.2.cmml">S</mi><mi id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.3" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.1" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.3" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.3.cmml">f</mi></mrow><mo id="S2.SS4.1.p1.10.m5.2.2.1.1.1.3" stretchy="false" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.SS4.1.p1.10.m5.4.4.4" xref="S2.SS4.1.p1.10.m5.4.4.4.cmml">⊆</mo><mrow id="S2.SS4.1.p1.10.m5.4.4.3.2" xref="S2.SS4.1.p1.10.m5.4.4.3.3.cmml"><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.3" stretchy="false" xref="S2.SS4.1.p1.10.m5.4.4.3.3.1.cmml">{</mo><mrow id="S2.SS4.1.p1.10.m5.3.3.2.1.1" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.cmml"><mi id="S2.SS4.1.p1.10.m5.3.3.2.1.1.2" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.2.cmml">ξ</mi><mo id="S2.SS4.1.p1.10.m5.3.3.2.1.1.1" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.1.cmml">∈</mo><msup id="S2.SS4.1.p1.10.m5.3.3.2.1.1.3" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.cmml"><mi id="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.2" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.2.cmml">ℝ</mi><mi id="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.3" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.3.cmml">d</mi></msup></mrow><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.4" lspace="0.278em" rspace="0.278em" xref="S2.SS4.1.p1.10.m5.4.4.3.3.1.cmml">:</mo><mrow id="S2.SS4.1.p1.10.m5.4.4.3.2.2" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.cmml"><mrow id="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.2" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.1.cmml"><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.2.1" stretchy="false" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.1.1.cmml">|</mo><mi id="S2.SS4.1.p1.10.m5.1.1" xref="S2.SS4.1.p1.10.m5.1.1.cmml">ξ</mi><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.2.2" stretchy="false" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.1.1.cmml">|</mo></mrow><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.2.1" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.1.cmml">≤</mo><mrow id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.cmml"><mn id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.2" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.2.cmml">3</mn><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.1" lspace="0.222em" rspace="0.222em" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.1.cmml">⋅</mo><msup id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.cmml"><mn id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.2" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.2.cmml">2</mn><mrow id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.cmml"><mi id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.2" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.2.cmml">n</mi><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.1" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.1.cmml">−</mo><mn id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.3" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.3.cmml">1</mn></mrow></msup></mrow></mrow><mo id="S2.SS4.1.p1.10.m5.4.4.3.2.5" stretchy="false" xref="S2.SS4.1.p1.10.m5.4.4.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.10.m5.4b"><apply id="S2.SS4.1.p1.10.m5.4.4.cmml" xref="S2.SS4.1.p1.10.m5.4.4"><subset id="S2.SS4.1.p1.10.m5.4.4.4.cmml" xref="S2.SS4.1.p1.10.m5.4.4.4"></subset><apply id="S2.SS4.1.p1.10.m5.2.2.1.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1"><times id="S2.SS4.1.p1.10.m5.2.2.1.2.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.2"></times><ci id="S2.SS4.1.p1.10.m5.2.2.1.3a.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.3"><mtext id="S2.SS4.1.p1.10.m5.2.2.1.3.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.3">supp </mtext></ci><ci id="S2.SS4.1.p1.10.m5.2.2.1.4.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.4">ℱ</ci><apply id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1"><times id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.1.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.1"></times><apply id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.1.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2">subscript</csymbol><ci id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.2.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.2">𝑆</ci><ci id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.3.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.2.3">𝑛</ci></apply><ci id="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.3.cmml" xref="S2.SS4.1.p1.10.m5.2.2.1.1.1.1.3">𝑓</ci></apply></apply><apply id="S2.SS4.1.p1.10.m5.4.4.3.3.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2"><csymbol cd="latexml" id="S2.SS4.1.p1.10.m5.4.4.3.3.1.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.3">conditional-set</csymbol><apply id="S2.SS4.1.p1.10.m5.3.3.2.1.1.cmml" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1"><in id="S2.SS4.1.p1.10.m5.3.3.2.1.1.1.cmml" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.1"></in><ci id="S2.SS4.1.p1.10.m5.3.3.2.1.1.2.cmml" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.2">𝜉</ci><apply id="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.cmml" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.1.cmml" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.3">superscript</csymbol><ci id="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.2.cmml" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.2">ℝ</ci><ci id="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.3.cmml" xref="S2.SS4.1.p1.10.m5.3.3.2.1.1.3.3">𝑑</ci></apply></apply><apply id="S2.SS4.1.p1.10.m5.4.4.3.2.2.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2"><leq id="S2.SS4.1.p1.10.m5.4.4.3.2.2.1.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.1"></leq><apply id="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.1.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.2"><abs id="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.1.1.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.2.2.1"></abs><ci id="S2.SS4.1.p1.10.m5.1.1.cmml" xref="S2.SS4.1.p1.10.m5.1.1">𝜉</ci></apply><apply id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3"><ci id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.1.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.1">⋅</ci><cn id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.2.cmml" type="integer" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.2">3</cn><apply id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.1.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3">superscript</csymbol><cn id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.2">2</cn><apply id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3"><minus id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.1.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.1"></minus><ci id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.2.cmml" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.2">𝑛</ci><cn id="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.3.cmml" type="integer" xref="S2.SS4.1.p1.10.m5.4.4.3.2.2.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.10.m5.4c">\text{\rm supp\,}\mathcal{F}(S_{n}f)\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq 3% \cdot 2^{n-1}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.10.m5.4d">supp caligraphic_F ( italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ) ⊆ { italic_ξ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : | italic_ξ | ≤ 3 ⋅ 2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math>, so that <math alttext="\text{\rm supp\,}\mathcal{F}_{x_{1}}(S_{n}f(\cdot,\widetilde{x}))\subseteq\{% \xi_{1}\in\mathbb{R}:|\xi_{1}|\leq 3\cdot 2^{n-1}\}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.11.m6.5"><semantics id="S2.SS4.1.p1.11.m6.5a"><mrow id="S2.SS4.1.p1.11.m6.5.5" xref="S2.SS4.1.p1.11.m6.5.5.cmml"><mrow id="S2.SS4.1.p1.11.m6.3.3.1" xref="S2.SS4.1.p1.11.m6.3.3.1.cmml"><mtext id="S2.SS4.1.p1.11.m6.3.3.1.3" xref="S2.SS4.1.p1.11.m6.3.3.1.3a.cmml">supp </mtext><mo id="S2.SS4.1.p1.11.m6.3.3.1.2" xref="S2.SS4.1.p1.11.m6.3.3.1.2.cmml">⁢</mo><msub id="S2.SS4.1.p1.11.m6.3.3.1.4" xref="S2.SS4.1.p1.11.m6.3.3.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S2.SS4.1.p1.11.m6.3.3.1.4.2" xref="S2.SS4.1.p1.11.m6.3.3.1.4.2.cmml">ℱ</mi><msub id="S2.SS4.1.p1.11.m6.3.3.1.4.3" xref="S2.SS4.1.p1.11.m6.3.3.1.4.3.cmml"><mi id="S2.SS4.1.p1.11.m6.3.3.1.4.3.2" xref="S2.SS4.1.p1.11.m6.3.3.1.4.3.2.cmml">x</mi><mn id="S2.SS4.1.p1.11.m6.3.3.1.4.3.3" xref="S2.SS4.1.p1.11.m6.3.3.1.4.3.3.cmml">1</mn></msub></msub><mo id="S2.SS4.1.p1.11.m6.3.3.1.2a" xref="S2.SS4.1.p1.11.m6.3.3.1.2.cmml">⁢</mo><mrow id="S2.SS4.1.p1.11.m6.3.3.1.1.1" xref="S2.SS4.1.p1.11.m6.3.3.1.1.1.1.cmml"><mo id="S2.SS4.1.p1.11.m6.3.3.1.1.1.2" stretchy="false" xref="S2.SS4.1.p1.11.m6.3.3.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.1.p1.11.m6.3.3.1.1.1.1" 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cd="ambiguous" id="S2.SS4.1.p1.11.m6.5.5.3.2.2.1.1.1.1.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.1.1.1">subscript</csymbol><ci id="S2.SS4.1.p1.11.m6.5.5.3.2.2.1.1.1.2.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.1.1.1.2">𝜉</ci><cn id="S2.SS4.1.p1.11.m6.5.5.3.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.1.1.1.3">1</cn></apply></apply><apply id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3"><ci id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.1.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.1">⋅</ci><cn id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.2.cmml" type="integer" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.2">3</cn><apply id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.1.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3">superscript</csymbol><cn id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.2.cmml" type="integer" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.2">2</cn><apply id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3"><minus id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3.1.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3.1"></minus><ci id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3.2.cmml" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3.2">𝑛</ci><cn id="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3.3.cmml" type="integer" xref="S2.SS4.1.p1.11.m6.5.5.3.2.2.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.11.m6.5c">\text{\rm supp\,}\mathcal{F}_{x_{1}}(S_{n}f(\cdot,\widetilde{x}))\subseteq\{% \xi_{1}\in\mathbb{R}:|\xi_{1}|\leq 3\cdot 2^{n-1}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.11.m6.5d">supp caligraphic_F start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( ⋅ , over~ start_ARG italic_x end_ARG ) ) ⊆ { italic_ξ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R : | italic_ξ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ 3 ⋅ 2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math> for fixed <math alttext="\widetilde{x}\in\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.12.m7.1"><semantics id="S2.SS4.1.p1.12.m7.1a"><mrow id="S2.SS4.1.p1.12.m7.1.1" xref="S2.SS4.1.p1.12.m7.1.1.cmml"><mover accent="true" id="S2.SS4.1.p1.12.m7.1.1.2" xref="S2.SS4.1.p1.12.m7.1.1.2.cmml"><mi id="S2.SS4.1.p1.12.m7.1.1.2.2" xref="S2.SS4.1.p1.12.m7.1.1.2.2.cmml">x</mi><mo id="S2.SS4.1.p1.12.m7.1.1.2.1" xref="S2.SS4.1.p1.12.m7.1.1.2.1.cmml">~</mo></mover><mo id="S2.SS4.1.p1.12.m7.1.1.1" xref="S2.SS4.1.p1.12.m7.1.1.1.cmml">∈</mo><msup id="S2.SS4.1.p1.12.m7.1.1.3" xref="S2.SS4.1.p1.12.m7.1.1.3.cmml"><mi id="S2.SS4.1.p1.12.m7.1.1.3.2" xref="S2.SS4.1.p1.12.m7.1.1.3.2.cmml">ℝ</mi><mrow id="S2.SS4.1.p1.12.m7.1.1.3.3" xref="S2.SS4.1.p1.12.m7.1.1.3.3.cmml"><mi id="S2.SS4.1.p1.12.m7.1.1.3.3.2" xref="S2.SS4.1.p1.12.m7.1.1.3.3.2.cmml">d</mi><mo id="S2.SS4.1.p1.12.m7.1.1.3.3.1" xref="S2.SS4.1.p1.12.m7.1.1.3.3.1.cmml">−</mo><mn id="S2.SS4.1.p1.12.m7.1.1.3.3.3" xref="S2.SS4.1.p1.12.m7.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.12.m7.1b"><apply id="S2.SS4.1.p1.12.m7.1.1.cmml" xref="S2.SS4.1.p1.12.m7.1.1"><in id="S2.SS4.1.p1.12.m7.1.1.1.cmml" xref="S2.SS4.1.p1.12.m7.1.1.1"></in><apply id="S2.SS4.1.p1.12.m7.1.1.2.cmml" xref="S2.SS4.1.p1.12.m7.1.1.2"><ci id="S2.SS4.1.p1.12.m7.1.1.2.1.cmml" xref="S2.SS4.1.p1.12.m7.1.1.2.1">~</ci><ci id="S2.SS4.1.p1.12.m7.1.1.2.2.cmml" xref="S2.SS4.1.p1.12.m7.1.1.2.2">𝑥</ci></apply><apply id="S2.SS4.1.p1.12.m7.1.1.3.cmml" xref="S2.SS4.1.p1.12.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.12.m7.1.1.3.1.cmml" xref="S2.SS4.1.p1.12.m7.1.1.3">superscript</csymbol><ci id="S2.SS4.1.p1.12.m7.1.1.3.2.cmml" xref="S2.SS4.1.p1.12.m7.1.1.3.2">ℝ</ci><apply id="S2.SS4.1.p1.12.m7.1.1.3.3.cmml" xref="S2.SS4.1.p1.12.m7.1.1.3.3"><minus id="S2.SS4.1.p1.12.m7.1.1.3.3.1.cmml" xref="S2.SS4.1.p1.12.m7.1.1.3.3.1"></minus><ci id="S2.SS4.1.p1.12.m7.1.1.3.3.2.cmml" xref="S2.SS4.1.p1.12.m7.1.1.3.3.2">𝑑</ci><cn id="S2.SS4.1.p1.12.m7.1.1.3.3.3.cmml" type="integer" xref="S2.SS4.1.p1.12.m7.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.12.m7.1c">\widetilde{x}\in\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.12.m7.1d">over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\text{\rm supp\,}\mathcal{F}_{\widetilde{x}}(S_{n}f(x_{1},\cdot))\subseteq\{% \widetilde{\xi}\in\mathbb{R}^{d-1}:|\widetilde{\xi}|\leq 3\cdot 2^{n-1}\}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.13.m8.5"><semantics id="S2.SS4.1.p1.13.m8.5a"><mrow id="S2.SS4.1.p1.13.m8.5.5" 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xref="S2.SS4.1.p1.13.m8.5.5.3.2.2.3.3.3"><minus id="S2.SS4.1.p1.13.m8.5.5.3.2.2.3.3.3.1.cmml" xref="S2.SS4.1.p1.13.m8.5.5.3.2.2.3.3.3.1"></minus><ci id="S2.SS4.1.p1.13.m8.5.5.3.2.2.3.3.3.2.cmml" xref="S2.SS4.1.p1.13.m8.5.5.3.2.2.3.3.3.2">𝑛</ci><cn id="S2.SS4.1.p1.13.m8.5.5.3.2.2.3.3.3.3.cmml" type="integer" xref="S2.SS4.1.p1.13.m8.5.5.3.2.2.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.13.m8.5c">\text{\rm supp\,}\mathcal{F}_{\widetilde{x}}(S_{n}f(x_{1},\cdot))\subseteq\{% \widetilde{\xi}\in\mathbb{R}^{d-1}:|\widetilde{\xi}|\leq 3\cdot 2^{n-1}\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.13.m8.5d">supp caligraphic_F start_POSTSUBSCRIPT over~ start_ARG italic_x end_ARG end_POSTSUBSCRIPT ( italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) ) ⊆ { over~ start_ARG italic_ξ end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT : | over~ start_ARG italic_ξ end_ARG | ≤ 3 ⋅ 2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math> for fixed <math alttext="x_{1}\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.14.m9.1"><semantics id="S2.SS4.1.p1.14.m9.1a"><mrow id="S2.SS4.1.p1.14.m9.1.1" xref="S2.SS4.1.p1.14.m9.1.1.cmml"><msub id="S2.SS4.1.p1.14.m9.1.1.2" xref="S2.SS4.1.p1.14.m9.1.1.2.cmml"><mi id="S2.SS4.1.p1.14.m9.1.1.2.2" xref="S2.SS4.1.p1.14.m9.1.1.2.2.cmml">x</mi><mn id="S2.SS4.1.p1.14.m9.1.1.2.3" xref="S2.SS4.1.p1.14.m9.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS4.1.p1.14.m9.1.1.1" xref="S2.SS4.1.p1.14.m9.1.1.1.cmml">∈</mo><mi id="S2.SS4.1.p1.14.m9.1.1.3" xref="S2.SS4.1.p1.14.m9.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.14.m9.1b"><apply id="S2.SS4.1.p1.14.m9.1.1.cmml" xref="S2.SS4.1.p1.14.m9.1.1"><in id="S2.SS4.1.p1.14.m9.1.1.1.cmml" xref="S2.SS4.1.p1.14.m9.1.1.1"></in><apply id="S2.SS4.1.p1.14.m9.1.1.2.cmml" xref="S2.SS4.1.p1.14.m9.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.14.m9.1.1.2.1.cmml" xref="S2.SS4.1.p1.14.m9.1.1.2">subscript</csymbol><ci id="S2.SS4.1.p1.14.m9.1.1.2.2.cmml" xref="S2.SS4.1.p1.14.m9.1.1.2.2">𝑥</ci><cn id="S2.SS4.1.p1.14.m9.1.1.2.3.cmml" type="integer" xref="S2.SS4.1.p1.14.m9.1.1.2.3">1</cn></apply><ci id="S2.SS4.1.p1.14.m9.1.1.3.cmml" xref="S2.SS4.1.p1.14.m9.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.14.m9.1c">x_{1}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.14.m9.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Theorem 4.9, Step 1 & 2]</cite>). Now, the Plancherel-Polya-Nikol’skij type inequality from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 4.1]</cite> twice and Minkowski’s inequality, yield</p> <table class="ltx_equationgroup ltx_eqn_table" id="S2.E7"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E7X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|S_{n}f\|_{L^{p_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}" class="ltx_Math" display="inline" id="S2.E7X.2.1.1.m1.4"><semantics id="S2.E7X.2.1.1.m1.4a"><msub id="S2.E7X.2.1.1.m1.4.4" xref="S2.E7X.2.1.1.m1.4.4.cmml"><mrow id="S2.E7X.2.1.1.m1.4.4.1.1" xref="S2.E7X.2.1.1.m1.4.4.1.2.cmml"><mo id="S2.E7X.2.1.1.m1.4.4.1.1.2" stretchy="false" xref="S2.E7X.2.1.1.m1.4.4.1.2.1.cmml">‖</mo><mrow id="S2.E7X.2.1.1.m1.4.4.1.1.1" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.cmml"><msub id="S2.E7X.2.1.1.m1.4.4.1.1.1.2" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.2.cmml"><mi id="S2.E7X.2.1.1.m1.4.4.1.1.1.2.2" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.2.2.cmml">S</mi><mi id="S2.E7X.2.1.1.m1.4.4.1.1.1.2.3" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.2.3.cmml">n</mi></msub><mo id="S2.E7X.2.1.1.m1.4.4.1.1.1.1" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.1.cmml">⁢</mo><mi id="S2.E7X.2.1.1.m1.4.4.1.1.1.3" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.3.cmml">f</mi></mrow><mo id="S2.E7X.2.1.1.m1.4.4.1.1.3" stretchy="false" xref="S2.E7X.2.1.1.m1.4.4.1.2.1.cmml">‖</mo></mrow><mrow id="S2.E7X.2.1.1.m1.3.3.3" xref="S2.E7X.2.1.1.m1.3.3.3.cmml"><msup id="S2.E7X.2.1.1.m1.3.3.3.5" xref="S2.E7X.2.1.1.m1.3.3.3.5.cmml"><mi id="S2.E7X.2.1.1.m1.3.3.3.5.2" xref="S2.E7X.2.1.1.m1.3.3.3.5.2.cmml">L</mi><msub id="S2.E7X.2.1.1.m1.3.3.3.5.3" xref="S2.E7X.2.1.1.m1.3.3.3.5.3.cmml"><mi id="S2.E7X.2.1.1.m1.3.3.3.5.3.2" xref="S2.E7X.2.1.1.m1.3.3.3.5.3.2.cmml">p</mi><mn id="S2.E7X.2.1.1.m1.3.3.3.5.3.3" xref="S2.E7X.2.1.1.m1.3.3.3.5.3.3.cmml">1</mn></msub></msup><mo id="S2.E7X.2.1.1.m1.3.3.3.4" xref="S2.E7X.2.1.1.m1.3.3.3.4.cmml">⁢</mo><mrow id="S2.E7X.2.1.1.m1.3.3.3.3.2" xref="S2.E7X.2.1.1.m1.3.3.3.3.3.cmml"><mo id="S2.E7X.2.1.1.m1.3.3.3.3.2.3" stretchy="false" xref="S2.E7X.2.1.1.m1.3.3.3.3.3.cmml">(</mo><msup id="S2.E7X.2.1.1.m1.2.2.2.2.1.1" xref="S2.E7X.2.1.1.m1.2.2.2.2.1.1.cmml"><mi id="S2.E7X.2.1.1.m1.2.2.2.2.1.1.2" xref="S2.E7X.2.1.1.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mi id="S2.E7X.2.1.1.m1.2.2.2.2.1.1.3" xref="S2.E7X.2.1.1.m1.2.2.2.2.1.1.3.cmml">d</mi></msup><mo id="S2.E7X.2.1.1.m1.3.3.3.3.2.4" xref="S2.E7X.2.1.1.m1.3.3.3.3.3.cmml">,</mo><msub id="S2.E7X.2.1.1.m1.3.3.3.3.2.2" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.cmml"><mi id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.2" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.2.cmml">w</mi><msub id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.cmml"><mi id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.2" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.2.cmml">γ</mi><mn id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.3" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.3.cmml">1</mn></msub></msub><mo id="S2.E7X.2.1.1.m1.3.3.3.3.2.5" xref="S2.E7X.2.1.1.m1.3.3.3.3.3.cmml">;</mo><mi id="S2.E7X.2.1.1.m1.1.1.1.1" xref="S2.E7X.2.1.1.m1.1.1.1.1.cmml">X</mi><mo id="S2.E7X.2.1.1.m1.3.3.3.3.2.6" stretchy="false" xref="S2.E7X.2.1.1.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S2.E7X.2.1.1.m1.4b"><apply id="S2.E7X.2.1.1.m1.4.4.cmml" xref="S2.E7X.2.1.1.m1.4.4"><csymbol cd="ambiguous" id="S2.E7X.2.1.1.m1.4.4.2.cmml" xref="S2.E7X.2.1.1.m1.4.4">subscript</csymbol><apply id="S2.E7X.2.1.1.m1.4.4.1.2.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1"><csymbol cd="latexml" id="S2.E7X.2.1.1.m1.4.4.1.2.1.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.2">norm</csymbol><apply id="S2.E7X.2.1.1.m1.4.4.1.1.1.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.1"><times id="S2.E7X.2.1.1.m1.4.4.1.1.1.1.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.1"></times><apply id="S2.E7X.2.1.1.m1.4.4.1.1.1.2.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S2.E7X.2.1.1.m1.4.4.1.1.1.2.1.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.2">subscript</csymbol><ci id="S2.E7X.2.1.1.m1.4.4.1.1.1.2.2.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.2.2">𝑆</ci><ci id="S2.E7X.2.1.1.m1.4.4.1.1.1.2.3.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.2.3">𝑛</ci></apply><ci id="S2.E7X.2.1.1.m1.4.4.1.1.1.3.cmml" xref="S2.E7X.2.1.1.m1.4.4.1.1.1.3">𝑓</ci></apply></apply><apply id="S2.E7X.2.1.1.m1.3.3.3.cmml" xref="S2.E7X.2.1.1.m1.3.3.3"><times id="S2.E7X.2.1.1.m1.3.3.3.4.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.4"></times><apply id="S2.E7X.2.1.1.m1.3.3.3.5.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.5"><csymbol cd="ambiguous" id="S2.E7X.2.1.1.m1.3.3.3.5.1.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.5">superscript</csymbol><ci id="S2.E7X.2.1.1.m1.3.3.3.5.2.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.5.2">𝐿</ci><apply id="S2.E7X.2.1.1.m1.3.3.3.5.3.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.5.3"><csymbol cd="ambiguous" id="S2.E7X.2.1.1.m1.3.3.3.5.3.1.cmml" 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id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3"><csymbol cd="ambiguous" id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.1.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3">subscript</csymbol><ci id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.2.cmml" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.2">𝛾</ci><cn id="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.3.cmml" type="integer" xref="S2.E7X.2.1.1.m1.3.3.3.3.2.2.3.3">1</cn></apply></apply><ci id="S2.E7X.2.1.1.m1.1.1.1.1.cmml" xref="S2.E7X.2.1.1.m1.1.1.1.1">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7X.2.1.1.m1.4c">\displaystyle\|S_{n}f\|_{L^{p_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}</annotation><annotation encoding="application/x-llamapun" id="S2.E7X.2.1.1.m1.4d">∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\,2^{\delta_{0}n}\Big{(}\int_{\mathbb{R}^{d-1}}\Big{(}\int_% {\mathbb{R}}|x_{1}|^{\gamma_{0}}\|S_{n}f(x_{1},\widetilde{x})\|^{p_{0}}\hskip 2% .0pt\mathrm{d}x_{1}\Big{)}^{\frac{p_{1}}{p_{0}}}\hskip 2.0pt\mathrm{d}% \widetilde{x}\Big{)}^{\frac{1}{p_{1}}}" class="ltx_Math" display="inline" id="S2.E7X.3.2.2.m1.2"><semantics id="S2.E7X.3.2.2.m1.2a"><mrow id="S2.E7X.3.2.2.m1.2.2" xref="S2.E7X.3.2.2.m1.2.2.cmml"><mi id="S2.E7X.3.2.2.m1.2.2.3" xref="S2.E7X.3.2.2.m1.2.2.3.cmml"></mi><mo id="S2.E7X.3.2.2.m1.2.2.2" xref="S2.E7X.3.2.2.m1.2.2.2.cmml">≤</mo><mrow id="S2.E7X.3.2.2.m1.2.2.1" xref="S2.E7X.3.2.2.m1.2.2.1.cmml"><mi id="S2.E7X.3.2.2.m1.2.2.1.3" xref="S2.E7X.3.2.2.m1.2.2.1.3.cmml">C</mi><mo id="S2.E7X.3.2.2.m1.2.2.1.2" 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id="S2.E7X.3.2.2.m1.2.2.1.1.3.3.2.cmml" xref="S2.E7X.3.2.2.m1.2.2.1.1.3.3.2">𝑝</ci><cn id="S2.E7X.3.2.2.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S2.E7X.3.2.2.m1.2.2.1.1.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7X.3.2.2.m1.2c">\displaystyle\leq C\,2^{\delta_{0}n}\Big{(}\int_{\mathbb{R}^{d-1}}\Big{(}\int_% {\mathbb{R}}|x_{1}|^{\gamma_{0}}\|S_{n}f(x_{1},\widetilde{x})\|^{p_{0}}\hskip 2% .0pt\mathrm{d}x_{1}\Big{)}^{\frac{p_{1}}{p_{0}}}\hskip 2.0pt\mathrm{d}% \widetilde{x}\Big{)}^{\frac{1}{p_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S2.E7X.3.2.2.m1.2d">≤ italic_C 2 start_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT ( ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ( ∫ start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) ∥ start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="3"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(2.7)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E7Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\,2^{\delta_{0}n}\Big{(}\int_{\mathbb{R}}|x_{1}|^{\gamma_{0% }}\Big{(}\int_{\mathbb{R}^{d-1}}\|S_{n}f(x_{1},\widetilde{x})\|^{p_{1}}\hskip 2% .0pt\mathrm{d}\widetilde{x}\Big{)}^{\frac{p_{0}}{p_{1}}}\hskip 2.0pt\mathrm{d}% x_{1}\Big{)}^{\frac{1}{p_{0}}}" class="ltx_Math" display="inline" id="S2.E7Xa.2.1.1.m1.2"><semantics id="S2.E7Xa.2.1.1.m1.2a"><mrow id="S2.E7Xa.2.1.1.m1.2.2" xref="S2.E7Xa.2.1.1.m1.2.2.cmml"><mi id="S2.E7Xa.2.1.1.m1.2.2.3" xref="S2.E7Xa.2.1.1.m1.2.2.3.cmml"></mi><mo id="S2.E7Xa.2.1.1.m1.2.2.2" xref="S2.E7Xa.2.1.1.m1.2.2.2.cmml">≤</mo><mrow id="S2.E7Xa.2.1.1.m1.2.2.1" xref="S2.E7Xa.2.1.1.m1.2.2.1.cmml"><mi id="S2.E7Xa.2.1.1.m1.2.2.1.3" xref="S2.E7Xa.2.1.1.m1.2.2.1.3.cmml">C</mi><mo 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xref="S2.E7Xa.2.1.1.m1.2.2.1.1.3.3">subscript</csymbol><ci id="S2.E7Xa.2.1.1.m1.2.2.1.1.3.3.2.cmml" xref="S2.E7Xa.2.1.1.m1.2.2.1.1.3.3.2">𝑝</ci><cn id="S2.E7Xa.2.1.1.m1.2.2.1.1.3.3.3.cmml" type="integer" xref="S2.E7Xa.2.1.1.m1.2.2.1.1.3.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7Xa.2.1.1.m1.2c">\displaystyle\leq C\,2^{\delta_{0}n}\Big{(}\int_{\mathbb{R}}|x_{1}|^{\gamma_{0% }}\Big{(}\int_{\mathbb{R}^{d-1}}\|S_{n}f(x_{1},\widetilde{x})\|^{p_{1}}\hskip 2% .0pt\mathrm{d}\widetilde{x}\Big{)}^{\frac{p_{0}}{p_{1}}}\hskip 2.0pt\mathrm{d}% x_{1}\Big{)}^{\frac{1}{p_{0}}}</annotation><annotation encoding="application/x-llamapun" id="S2.E7Xa.2.1.1.m1.2d">≤ italic_C 2 start_POSTSUPERSCRIPT italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_n end_POSTSUPERSCRIPT ( ∫ start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) ∥ start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S2.E7Xb"> <td 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xref="S2.E7Xb.2.1.1.m1.2.2.1.1">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E7Xb.2.1.1.m1.5c">\displaystyle\leq C\,2^{(\delta_{0}+\delta_{1})n}\|S_{n}f\|_{L^{p_{0}}(\mathbb% {R}^{d},w_{\gamma_{0}};X)},</annotation><annotation encoding="application/x-llamapun" id="S2.E7Xb.2.1.1.m1.5d">≤ italic_C 2 start_POSTSUPERSCRIPT ( italic_δ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_n end_POSTSUPERSCRIPT ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S2.SS4.1.p1.18">where <math alttext="C>0" class="ltx_Math" display="inline" id="S2.SS4.1.p1.15.m1.1"><semantics id="S2.SS4.1.p1.15.m1.1a"><mrow id="S2.SS4.1.p1.15.m1.1.1" xref="S2.SS4.1.p1.15.m1.1.1.cmml"><mi id="S2.SS4.1.p1.15.m1.1.1.2" xref="S2.SS4.1.p1.15.m1.1.1.2.cmml">C</mi><mo id="S2.SS4.1.p1.15.m1.1.1.1" xref="S2.SS4.1.p1.15.m1.1.1.1.cmml">></mo><mn id="S2.SS4.1.p1.15.m1.1.1.3" xref="S2.SS4.1.p1.15.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.15.m1.1b"><apply id="S2.SS4.1.p1.15.m1.1.1.cmml" xref="S2.SS4.1.p1.15.m1.1.1"><gt id="S2.SS4.1.p1.15.m1.1.1.1.cmml" xref="S2.SS4.1.p1.15.m1.1.1.1"></gt><ci id="S2.SS4.1.p1.15.m1.1.1.2.cmml" xref="S2.SS4.1.p1.15.m1.1.1.2">𝐶</ci><cn id="S2.SS4.1.p1.15.m1.1.1.3.cmml" type="integer" xref="S2.SS4.1.p1.15.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.15.m1.1c">C>0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.15.m1.1d">italic_C > 0</annotation></semantics></math> is independent of <math alttext="f" class="ltx_Math" display="inline" id="S2.SS4.1.p1.16.m2.1"><semantics id="S2.SS4.1.p1.16.m2.1a"><mi id="S2.SS4.1.p1.16.m2.1.1" xref="S2.SS4.1.p1.16.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.16.m2.1b"><ci id="S2.SS4.1.p1.16.m2.1.1.cmml" xref="S2.SS4.1.p1.16.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.16.m2.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.16.m2.1d">italic_f</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S2.SS4.1.p1.17.m3.1"><semantics id="S2.SS4.1.p1.17.m3.1a"><mi id="S2.SS4.1.p1.17.m3.1.1" xref="S2.SS4.1.p1.17.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.17.m3.1b"><ci id="S2.SS4.1.p1.17.m3.1.1.cmml" xref="S2.SS4.1.p1.17.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.17.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.17.m3.1d">italic_n</annotation></semantics></math>. Statement <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.I2.i2" title="item ii ‣ Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> now follows from the definition of the Besov norms and the fact that <math alttext="s_{1}+\delta=s_{0}" class="ltx_Math" display="inline" id="S2.SS4.1.p1.18.m4.1"><semantics id="S2.SS4.1.p1.18.m4.1a"><mrow id="S2.SS4.1.p1.18.m4.1.1" xref="S2.SS4.1.p1.18.m4.1.1.cmml"><mrow id="S2.SS4.1.p1.18.m4.1.1.2" xref="S2.SS4.1.p1.18.m4.1.1.2.cmml"><msub id="S2.SS4.1.p1.18.m4.1.1.2.2" xref="S2.SS4.1.p1.18.m4.1.1.2.2.cmml"><mi id="S2.SS4.1.p1.18.m4.1.1.2.2.2" xref="S2.SS4.1.p1.18.m4.1.1.2.2.2.cmml">s</mi><mn id="S2.SS4.1.p1.18.m4.1.1.2.2.3" xref="S2.SS4.1.p1.18.m4.1.1.2.2.3.cmml">1</mn></msub><mo id="S2.SS4.1.p1.18.m4.1.1.2.1" xref="S2.SS4.1.p1.18.m4.1.1.2.1.cmml">+</mo><mi id="S2.SS4.1.p1.18.m4.1.1.2.3" xref="S2.SS4.1.p1.18.m4.1.1.2.3.cmml">δ</mi></mrow><mo id="S2.SS4.1.p1.18.m4.1.1.1" xref="S2.SS4.1.p1.18.m4.1.1.1.cmml">=</mo><msub id="S2.SS4.1.p1.18.m4.1.1.3" xref="S2.SS4.1.p1.18.m4.1.1.3.cmml"><mi id="S2.SS4.1.p1.18.m4.1.1.3.2" xref="S2.SS4.1.p1.18.m4.1.1.3.2.cmml">s</mi><mn id="S2.SS4.1.p1.18.m4.1.1.3.3" xref="S2.SS4.1.p1.18.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.1.p1.18.m4.1b"><apply id="S2.SS4.1.p1.18.m4.1.1.cmml" xref="S2.SS4.1.p1.18.m4.1.1"><eq id="S2.SS4.1.p1.18.m4.1.1.1.cmml" xref="S2.SS4.1.p1.18.m4.1.1.1"></eq><apply id="S2.SS4.1.p1.18.m4.1.1.2.cmml" xref="S2.SS4.1.p1.18.m4.1.1.2"><plus id="S2.SS4.1.p1.18.m4.1.1.2.1.cmml" xref="S2.SS4.1.p1.18.m4.1.1.2.1"></plus><apply id="S2.SS4.1.p1.18.m4.1.1.2.2.cmml" xref="S2.SS4.1.p1.18.m4.1.1.2.2"><csymbol cd="ambiguous" id="S2.SS4.1.p1.18.m4.1.1.2.2.1.cmml" xref="S2.SS4.1.p1.18.m4.1.1.2.2">subscript</csymbol><ci id="S2.SS4.1.p1.18.m4.1.1.2.2.2.cmml" xref="S2.SS4.1.p1.18.m4.1.1.2.2.2">𝑠</ci><cn id="S2.SS4.1.p1.18.m4.1.1.2.2.3.cmml" type="integer" xref="S2.SS4.1.p1.18.m4.1.1.2.2.3">1</cn></apply><ci id="S2.SS4.1.p1.18.m4.1.1.2.3.cmml" xref="S2.SS4.1.p1.18.m4.1.1.2.3">𝛿</ci></apply><apply id="S2.SS4.1.p1.18.m4.1.1.3.cmml" xref="S2.SS4.1.p1.18.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.1.p1.18.m4.1.1.3.1.cmml" xref="S2.SS4.1.p1.18.m4.1.1.3">subscript</csymbol><ci id="S2.SS4.1.p1.18.m4.1.1.3.2.cmml" xref="S2.SS4.1.p1.18.m4.1.1.3.2">𝑠</ci><cn id="S2.SS4.1.p1.18.m4.1.1.3.3.cmml" type="integer" xref="S2.SS4.1.p1.18.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.1.p1.18.m4.1c">s_{1}+\delta=s_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.1.p1.18.m4.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_δ = italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS4.2.p2"> <p class="ltx_p" id="S2.SS4.2.p2.5"><span class="ltx_text ltx_font_italic" id="S2.SS4.2.p2.5.1">Step 2: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.I2.i1" title="item i ‣ Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>.</span> We follow the proof of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Theorem 1.2]</cite> for radial weights to deduce <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.I2.i1" title="item i ‣ Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>. Let <math alttext="f\in F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)\cap F^{s_{1}}_{p% _{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)" class="ltx_Math" display="inline" id="S2.SS4.2.p2.1.m1.10"><semantics id="S2.SS4.2.p2.1.m1.10a"><mrow id="S2.SS4.2.p2.1.m1.10.10" xref="S2.SS4.2.p2.1.m1.10.10.cmml"><mi id="S2.SS4.2.p2.1.m1.10.10.6" xref="S2.SS4.2.p2.1.m1.10.10.6.cmml">f</mi><mo id="S2.SS4.2.p2.1.m1.10.10.5" xref="S2.SS4.2.p2.1.m1.10.10.5.cmml">∈</mo><mrow id="S2.SS4.2.p2.1.m1.10.10.4" xref="S2.SS4.2.p2.1.m1.10.10.4.cmml"><mrow id="S2.SS4.2.p2.1.m1.8.8.2.2" xref="S2.SS4.2.p2.1.m1.8.8.2.2.cmml"><msubsup id="S2.SS4.2.p2.1.m1.8.8.2.2.4" xref="S2.SS4.2.p2.1.m1.8.8.2.2.4.cmml"><mi id="S2.SS4.2.p2.1.m1.8.8.2.2.4.2.2" xref="S2.SS4.2.p2.1.m1.8.8.2.2.4.2.2.cmml">F</mi><mrow id="S2.SS4.2.p2.1.m1.2.2.2.2" xref="S2.SS4.2.p2.1.m1.2.2.2.3.cmml"><msub id="S2.SS4.2.p2.1.m1.1.1.1.1.1" xref="S2.SS4.2.p2.1.m1.1.1.1.1.1.cmml"><mi id="S2.SS4.2.p2.1.m1.1.1.1.1.1.2" 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cd="ambiguous" id="S2.SS4.2.p2.1.m1.3.3.1.1.1.1.cmml" xref="S2.SS4.2.p2.1.m1.3.3.1.1.1">subscript</csymbol><ci id="S2.SS4.2.p2.1.m1.3.3.1.1.1.2.cmml" xref="S2.SS4.2.p2.1.m1.3.3.1.1.1.2">𝑝</ci><cn id="S2.SS4.2.p2.1.m1.3.3.1.1.1.3.cmml" type="integer" xref="S2.SS4.2.p2.1.m1.3.3.1.1.1.3">1</cn></apply><apply id="S2.SS4.2.p2.1.m1.4.4.2.2.2.cmml" xref="S2.SS4.2.p2.1.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.2.p2.1.m1.4.4.2.2.2.1.cmml" xref="S2.SS4.2.p2.1.m1.4.4.2.2.2">subscript</csymbol><ci id="S2.SS4.2.p2.1.m1.4.4.2.2.2.2.cmml" xref="S2.SS4.2.p2.1.m1.4.4.2.2.2.2">𝑞</ci><cn id="S2.SS4.2.p2.1.m1.4.4.2.2.2.3.cmml" type="integer" xref="S2.SS4.2.p2.1.m1.4.4.2.2.2.3">1</cn></apply></list></apply><vector id="S2.SS4.2.p2.1.m1.10.10.4.4.2.3.cmml" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2"><apply id="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1.cmml" xref="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1.1.cmml" xref="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1">superscript</csymbol><ci id="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1.2.cmml" xref="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1.2">ℝ</ci><ci id="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1.3.cmml" xref="S2.SS4.2.p2.1.m1.9.9.3.3.1.1.1.3">𝑑</ci></apply><apply id="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.cmml" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.1.cmml" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2">subscript</csymbol><ci id="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.2.cmml" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.2">𝑤</ci><apply id="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3.cmml" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3.1.cmml" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3">subscript</csymbol><ci id="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3.2.cmml" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3.2">𝛾</ci><cn id="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3.3.cmml" type="integer" xref="S2.SS4.2.p2.1.m1.10.10.4.4.2.2.2.3.3">1</cn></apply></apply><ci id="S2.SS4.2.p2.1.m1.6.6.cmml" xref="S2.SS4.2.p2.1.m1.6.6">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.1.m1.10c">f\in F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)\cap F^{s_{1}}_{p% _{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.1.m1.10d">italic_f ∈ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) ∩ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. By (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E6" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.6</span></a>) it suffices to prove the required estimate for <math alttext="q_{1}=1" class="ltx_Math" display="inline" id="S2.SS4.2.p2.2.m2.1"><semantics id="S2.SS4.2.p2.2.m2.1a"><mrow id="S2.SS4.2.p2.2.m2.1.1" xref="S2.SS4.2.p2.2.m2.1.1.cmml"><msub id="S2.SS4.2.p2.2.m2.1.1.2" xref="S2.SS4.2.p2.2.m2.1.1.2.cmml"><mi id="S2.SS4.2.p2.2.m2.1.1.2.2" xref="S2.SS4.2.p2.2.m2.1.1.2.2.cmml">q</mi><mn id="S2.SS4.2.p2.2.m2.1.1.2.3" xref="S2.SS4.2.p2.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS4.2.p2.2.m2.1.1.1" xref="S2.SS4.2.p2.2.m2.1.1.1.cmml">=</mo><mn id="S2.SS4.2.p2.2.m2.1.1.3" xref="S2.SS4.2.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.2.m2.1b"><apply id="S2.SS4.2.p2.2.m2.1.1.cmml" xref="S2.SS4.2.p2.2.m2.1.1"><eq id="S2.SS4.2.p2.2.m2.1.1.1.cmml" xref="S2.SS4.2.p2.2.m2.1.1.1"></eq><apply id="S2.SS4.2.p2.2.m2.1.1.2.cmml" xref="S2.SS4.2.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.2.p2.2.m2.1.1.2.1.cmml" xref="S2.SS4.2.p2.2.m2.1.1.2">subscript</csymbol><ci id="S2.SS4.2.p2.2.m2.1.1.2.2.cmml" xref="S2.SS4.2.p2.2.m2.1.1.2.2">𝑞</ci><cn id="S2.SS4.2.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S2.SS4.2.p2.2.m2.1.1.2.3">1</cn></apply><cn id="S2.SS4.2.p2.2.m2.1.1.3.cmml" type="integer" xref="S2.SS4.2.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.2.m2.1c">q_{1}=1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.2.m2.1d">italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 1</annotation></semantics></math>. Let <math alttext="\theta_{0}\in[0,1)" class="ltx_Math" display="inline" id="S2.SS4.2.p2.3.m3.2"><semantics id="S2.SS4.2.p2.3.m3.2a"><mrow id="S2.SS4.2.p2.3.m3.2.3" xref="S2.SS4.2.p2.3.m3.2.3.cmml"><msub id="S2.SS4.2.p2.3.m3.2.3.2" xref="S2.SS4.2.p2.3.m3.2.3.2.cmml"><mi id="S2.SS4.2.p2.3.m3.2.3.2.2" xref="S2.SS4.2.p2.3.m3.2.3.2.2.cmml">θ</mi><mn id="S2.SS4.2.p2.3.m3.2.3.2.3" xref="S2.SS4.2.p2.3.m3.2.3.2.3.cmml">0</mn></msub><mo id="S2.SS4.2.p2.3.m3.2.3.1" xref="S2.SS4.2.p2.3.m3.2.3.1.cmml">∈</mo><mrow id="S2.SS4.2.p2.3.m3.2.3.3.2" xref="S2.SS4.2.p2.3.m3.2.3.3.1.cmml"><mo id="S2.SS4.2.p2.3.m3.2.3.3.2.1" stretchy="false" xref="S2.SS4.2.p2.3.m3.2.3.3.1.cmml">[</mo><mn id="S2.SS4.2.p2.3.m3.1.1" xref="S2.SS4.2.p2.3.m3.1.1.cmml">0</mn><mo id="S2.SS4.2.p2.3.m3.2.3.3.2.2" xref="S2.SS4.2.p2.3.m3.2.3.3.1.cmml">,</mo><mn id="S2.SS4.2.p2.3.m3.2.2" xref="S2.SS4.2.p2.3.m3.2.2.cmml">1</mn><mo id="S2.SS4.2.p2.3.m3.2.3.3.2.3" stretchy="false" xref="S2.SS4.2.p2.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.3.m3.2b"><apply id="S2.SS4.2.p2.3.m3.2.3.cmml" xref="S2.SS4.2.p2.3.m3.2.3"><in id="S2.SS4.2.p2.3.m3.2.3.1.cmml" xref="S2.SS4.2.p2.3.m3.2.3.1"></in><apply id="S2.SS4.2.p2.3.m3.2.3.2.cmml" xref="S2.SS4.2.p2.3.m3.2.3.2"><csymbol cd="ambiguous" id="S2.SS4.2.p2.3.m3.2.3.2.1.cmml" xref="S2.SS4.2.p2.3.m3.2.3.2">subscript</csymbol><ci id="S2.SS4.2.p2.3.m3.2.3.2.2.cmml" xref="S2.SS4.2.p2.3.m3.2.3.2.2">𝜃</ci><cn id="S2.SS4.2.p2.3.m3.2.3.2.3.cmml" type="integer" xref="S2.SS4.2.p2.3.m3.2.3.2.3">0</cn></apply><interval closure="closed-open" id="S2.SS4.2.p2.3.m3.2.3.3.1.cmml" xref="S2.SS4.2.p2.3.m3.2.3.3.2"><cn id="S2.SS4.2.p2.3.m3.1.1.cmml" type="integer" xref="S2.SS4.2.p2.3.m3.1.1">0</cn><cn id="S2.SS4.2.p2.3.m3.2.2.cmml" type="integer" xref="S2.SS4.2.p2.3.m3.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.3.m3.2c">\theta_{0}\in[0,1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.3.m3.2d">italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ [ 0 , 1 )</annotation></semantics></math> be such that <math alttext="\frac{1}{p_{1}}-\frac{1-\theta_{0}}{p_{0}}=0" class="ltx_Math" display="inline" id="S2.SS4.2.p2.4.m4.1"><semantics id="S2.SS4.2.p2.4.m4.1a"><mrow id="S2.SS4.2.p2.4.m4.1.1" xref="S2.SS4.2.p2.4.m4.1.1.cmml"><mrow id="S2.SS4.2.p2.4.m4.1.1.2" xref="S2.SS4.2.p2.4.m4.1.1.2.cmml"><mfrac id="S2.SS4.2.p2.4.m4.1.1.2.2" xref="S2.SS4.2.p2.4.m4.1.1.2.2.cmml"><mn id="S2.SS4.2.p2.4.m4.1.1.2.2.2" xref="S2.SS4.2.p2.4.m4.1.1.2.2.2.cmml">1</mn><msub id="S2.SS4.2.p2.4.m4.1.1.2.2.3" xref="S2.SS4.2.p2.4.m4.1.1.2.2.3.cmml"><mi id="S2.SS4.2.p2.4.m4.1.1.2.2.3.2" xref="S2.SS4.2.p2.4.m4.1.1.2.2.3.2.cmml">p</mi><mn id="S2.SS4.2.p2.4.m4.1.1.2.2.3.3" xref="S2.SS4.2.p2.4.m4.1.1.2.2.3.3.cmml">1</mn></msub></mfrac><mo id="S2.SS4.2.p2.4.m4.1.1.2.1" xref="S2.SS4.2.p2.4.m4.1.1.2.1.cmml">−</mo><mfrac id="S2.SS4.2.p2.4.m4.1.1.2.3" xref="S2.SS4.2.p2.4.m4.1.1.2.3.cmml"><mrow id="S2.SS4.2.p2.4.m4.1.1.2.3.2" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.cmml"><mn id="S2.SS4.2.p2.4.m4.1.1.2.3.2.2" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.2.cmml">1</mn><mo id="S2.SS4.2.p2.4.m4.1.1.2.3.2.1" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.1.cmml">−</mo><msub id="S2.SS4.2.p2.4.m4.1.1.2.3.2.3" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.cmml"><mi id="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.2" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.2.cmml">θ</mi><mn id="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.3" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.3.cmml">0</mn></msub></mrow><msub id="S2.SS4.2.p2.4.m4.1.1.2.3.3" xref="S2.SS4.2.p2.4.m4.1.1.2.3.3.cmml"><mi id="S2.SS4.2.p2.4.m4.1.1.2.3.3.2" xref="S2.SS4.2.p2.4.m4.1.1.2.3.3.2.cmml">p</mi><mn id="S2.SS4.2.p2.4.m4.1.1.2.3.3.3" xref="S2.SS4.2.p2.4.m4.1.1.2.3.3.3.cmml">0</mn></msub></mfrac></mrow><mo id="S2.SS4.2.p2.4.m4.1.1.1" xref="S2.SS4.2.p2.4.m4.1.1.1.cmml">=</mo><mn id="S2.SS4.2.p2.4.m4.1.1.3" xref="S2.SS4.2.p2.4.m4.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.4.m4.1b"><apply id="S2.SS4.2.p2.4.m4.1.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1"><eq id="S2.SS4.2.p2.4.m4.1.1.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.1"></eq><apply id="S2.SS4.2.p2.4.m4.1.1.2.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2"><minus id="S2.SS4.2.p2.4.m4.1.1.2.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.1"></minus><apply id="S2.SS4.2.p2.4.m4.1.1.2.2.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.2"><divide id="S2.SS4.2.p2.4.m4.1.1.2.2.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.2"></divide><cn id="S2.SS4.2.p2.4.m4.1.1.2.2.2.cmml" type="integer" xref="S2.SS4.2.p2.4.m4.1.1.2.2.2">1</cn><apply id="S2.SS4.2.p2.4.m4.1.1.2.2.3.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.2.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.4.m4.1.1.2.2.3.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.2.3">subscript</csymbol><ci id="S2.SS4.2.p2.4.m4.1.1.2.2.3.2.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.2.3.2">𝑝</ci><cn id="S2.SS4.2.p2.4.m4.1.1.2.2.3.3.cmml" type="integer" xref="S2.SS4.2.p2.4.m4.1.1.2.2.3.3">1</cn></apply></apply><apply id="S2.SS4.2.p2.4.m4.1.1.2.3.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3"><divide id="S2.SS4.2.p2.4.m4.1.1.2.3.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3"></divide><apply id="S2.SS4.2.p2.4.m4.1.1.2.3.2.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2"><minus id="S2.SS4.2.p2.4.m4.1.1.2.3.2.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.1"></minus><cn id="S2.SS4.2.p2.4.m4.1.1.2.3.2.2.cmml" type="integer" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.2">1</cn><apply id="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.3">subscript</csymbol><ci id="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.2.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.2">𝜃</ci><cn id="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.3.cmml" type="integer" xref="S2.SS4.2.p2.4.m4.1.1.2.3.2.3.3">0</cn></apply></apply><apply id="S2.SS4.2.p2.4.m4.1.1.2.3.3.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.4.m4.1.1.2.3.3.1.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.3">subscript</csymbol><ci id="S2.SS4.2.p2.4.m4.1.1.2.3.3.2.cmml" xref="S2.SS4.2.p2.4.m4.1.1.2.3.3.2">𝑝</ci><cn id="S2.SS4.2.p2.4.m4.1.1.2.3.3.3.cmml" type="integer" xref="S2.SS4.2.p2.4.m4.1.1.2.3.3.3">0</cn></apply></apply></apply><cn id="S2.SS4.2.p2.4.m4.1.1.3.cmml" type="integer" xref="S2.SS4.2.p2.4.m4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.4.m4.1c">\frac{1}{p_{1}}-\frac{1-\theta_{0}}{p_{0}}=0</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.4.m4.1d">divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG - divide start_ARG 1 - italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG = 0</annotation></semantics></math> and define the continuous function <math alttext="g:(\theta_{0},1]\to\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS4.2.p2.5.m5.2"><semantics id="S2.SS4.2.p2.5.m5.2a"><mrow id="S2.SS4.2.p2.5.m5.2.2" xref="S2.SS4.2.p2.5.m5.2.2.cmml"><mi id="S2.SS4.2.p2.5.m5.2.2.3" xref="S2.SS4.2.p2.5.m5.2.2.3.cmml">g</mi><mo id="S2.SS4.2.p2.5.m5.2.2.2" lspace="0.278em" rspace="0.278em" xref="S2.SS4.2.p2.5.m5.2.2.2.cmml">:</mo><mrow id="S2.SS4.2.p2.5.m5.2.2.1" xref="S2.SS4.2.p2.5.m5.2.2.1.cmml"><mrow id="S2.SS4.2.p2.5.m5.2.2.1.1.1" xref="S2.SS4.2.p2.5.m5.2.2.1.1.2.cmml"><mo id="S2.SS4.2.p2.5.m5.2.2.1.1.1.2" stretchy="false" xref="S2.SS4.2.p2.5.m5.2.2.1.1.2.cmml">(</mo><msub id="S2.SS4.2.p2.5.m5.2.2.1.1.1.1" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.cmml"><mi id="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.2" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.2.cmml">θ</mi><mn id="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.3" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS4.2.p2.5.m5.2.2.1.1.1.3" xref="S2.SS4.2.p2.5.m5.2.2.1.1.2.cmml">,</mo><mn id="S2.SS4.2.p2.5.m5.1.1" xref="S2.SS4.2.p2.5.m5.1.1.cmml">1</mn><mo id="S2.SS4.2.p2.5.m5.2.2.1.1.1.4" stretchy="false" xref="S2.SS4.2.p2.5.m5.2.2.1.1.2.cmml">]</mo></mrow><mo id="S2.SS4.2.p2.5.m5.2.2.1.2" stretchy="false" xref="S2.SS4.2.p2.5.m5.2.2.1.2.cmml">→</mo><mi id="S2.SS4.2.p2.5.m5.2.2.1.3" xref="S2.SS4.2.p2.5.m5.2.2.1.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.5.m5.2b"><apply id="S2.SS4.2.p2.5.m5.2.2.cmml" xref="S2.SS4.2.p2.5.m5.2.2"><ci id="S2.SS4.2.p2.5.m5.2.2.2.cmml" xref="S2.SS4.2.p2.5.m5.2.2.2">:</ci><ci id="S2.SS4.2.p2.5.m5.2.2.3.cmml" xref="S2.SS4.2.p2.5.m5.2.2.3">𝑔</ci><apply id="S2.SS4.2.p2.5.m5.2.2.1.cmml" xref="S2.SS4.2.p2.5.m5.2.2.1"><ci id="S2.SS4.2.p2.5.m5.2.2.1.2.cmml" xref="S2.SS4.2.p2.5.m5.2.2.1.2">→</ci><interval closure="open-closed" id="S2.SS4.2.p2.5.m5.2.2.1.1.2.cmml" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1"><apply id="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.cmml" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.1.cmml" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1.1">subscript</csymbol><ci id="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.2.cmml" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.2">𝜃</ci><cn id="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.2.p2.5.m5.2.2.1.1.1.1.3">0</cn></apply><cn id="S2.SS4.2.p2.5.m5.1.1.cmml" type="integer" xref="S2.SS4.2.p2.5.m5.1.1">1</cn></interval><ci id="S2.SS4.2.p2.5.m5.2.2.1.3.cmml" xref="S2.SS4.2.p2.5.m5.2.2.1.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.5.m5.2c">g:(\theta_{0},1]\to\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.5.m5.2d">italic_g : ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 ] → blackboard_R</annotation></semantics></math> by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex3a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g(\theta)=\frac{\frac{\gamma_{1}}{p_{1}}-\frac{(1-\theta)\gamma_{0}}{p_{0}}}{% \frac{1}{p_{1}}-\frac{1-\theta}{p_{0}}}." class="ltx_Math" display="block" id="S2.Ex3a.m1.3"><semantics id="S2.Ex3a.m1.3a"><mrow id="S2.Ex3a.m1.3.3.1" xref="S2.Ex3a.m1.3.3.1.1.cmml"><mrow id="S2.Ex3a.m1.3.3.1.1" xref="S2.Ex3a.m1.3.3.1.1.cmml"><mrow id="S2.Ex3a.m1.3.3.1.1.2" xref="S2.Ex3a.m1.3.3.1.1.2.cmml"><mi id="S2.Ex3a.m1.3.3.1.1.2.2" xref="S2.Ex3a.m1.3.3.1.1.2.2.cmml">g</mi><mo id="S2.Ex3a.m1.3.3.1.1.2.1" xref="S2.Ex3a.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S2.Ex3a.m1.3.3.1.1.2.3.2" xref="S2.Ex3a.m1.3.3.1.1.2.cmml"><mo id="S2.Ex3a.m1.3.3.1.1.2.3.2.1" stretchy="false" xref="S2.Ex3a.m1.3.3.1.1.2.cmml">(</mo><mi id="S2.Ex3a.m1.2.2" xref="S2.Ex3a.m1.2.2.cmml">θ</mi><mo id="S2.Ex3a.m1.3.3.1.1.2.3.2.2" stretchy="false" xref="S2.Ex3a.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S2.Ex3a.m1.3.3.1.1.1" xref="S2.Ex3a.m1.3.3.1.1.1.cmml">=</mo><mfrac id="S2.Ex3a.m1.1.1" xref="S2.Ex3a.m1.1.1.cmml"><mrow id="S2.Ex3a.m1.1.1.1" xref="S2.Ex3a.m1.1.1.1.cmml"><mfrac id="S2.Ex3a.m1.1.1.1.3" xref="S2.Ex3a.m1.1.1.1.3.cmml"><msub id="S2.Ex3a.m1.1.1.1.3.2" xref="S2.Ex3a.m1.1.1.1.3.2.cmml"><mi id="S2.Ex3a.m1.1.1.1.3.2.2" xref="S2.Ex3a.m1.1.1.1.3.2.2.cmml">γ</mi><mn id="S2.Ex3a.m1.1.1.1.3.2.3" xref="S2.Ex3a.m1.1.1.1.3.2.3.cmml">1</mn></msub><msub id="S2.Ex3a.m1.1.1.1.3.3" xref="S2.Ex3a.m1.1.1.1.3.3.cmml"><mi id="S2.Ex3a.m1.1.1.1.3.3.2" xref="S2.Ex3a.m1.1.1.1.3.3.2.cmml">p</mi><mn id="S2.Ex3a.m1.1.1.1.3.3.3" xref="S2.Ex3a.m1.1.1.1.3.3.3.cmml">1</mn></msub></mfrac><mo id="S2.Ex3a.m1.1.1.1.2" xref="S2.Ex3a.m1.1.1.1.2.cmml">−</mo><mfrac id="S2.Ex3a.m1.1.1.1.1" xref="S2.Ex3a.m1.1.1.1.1.cmml"><mrow id="S2.Ex3a.m1.1.1.1.1.1" xref="S2.Ex3a.m1.1.1.1.1.1.cmml"><mrow id="S2.Ex3a.m1.1.1.1.1.1.1.1" xref="S2.Ex3a.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex3a.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex3a.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex3a.m1.1.1.1.1.1.1.1.1" xref="S2.Ex3a.m1.1.1.1.1.1.1.1.1.cmml"><mn id="S2.Ex3a.m1.1.1.1.1.1.1.1.1.2" xref="S2.Ex3a.m1.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.Ex3a.m1.1.1.1.1.1.1.1.1.1" xref="S2.Ex3a.m1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S2.Ex3a.m1.1.1.1.1.1.1.1.1.3" xref="S2.Ex3a.m1.1.1.1.1.1.1.1.1.3.cmml">θ</mi></mrow><mo id="S2.Ex3a.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex3a.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.Ex3a.m1.1.1.1.1.1.2" xref="S2.Ex3a.m1.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.Ex3a.m1.1.1.1.1.1.3" xref="S2.Ex3a.m1.1.1.1.1.1.3.cmml"><mi id="S2.Ex3a.m1.1.1.1.1.1.3.2" xref="S2.Ex3a.m1.1.1.1.1.1.3.2.cmml">γ</mi><mn id="S2.Ex3a.m1.1.1.1.1.1.3.3" xref="S2.Ex3a.m1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><msub id="S2.Ex3a.m1.1.1.1.1.3" xref="S2.Ex3a.m1.1.1.1.1.3.cmml"><mi id="S2.Ex3a.m1.1.1.1.1.3.2" xref="S2.Ex3a.m1.1.1.1.1.3.2.cmml">p</mi><mn id="S2.Ex3a.m1.1.1.1.1.3.3" xref="S2.Ex3a.m1.1.1.1.1.3.3.cmml">0</mn></msub></mfrac></mrow><mrow id="S2.Ex3a.m1.1.1.3" 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xref="S2.Ex3a.m1.1.1.3.3.3.3.cmml">0</mn></msub></mfrac></mrow></mfrac></mrow><mo id="S2.Ex3a.m1.3.3.1.2" lspace="0em" xref="S2.Ex3a.m1.3.3.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex3a.m1.3b"><apply id="S2.Ex3a.m1.3.3.1.1.cmml" xref="S2.Ex3a.m1.3.3.1"><eq id="S2.Ex3a.m1.3.3.1.1.1.cmml" xref="S2.Ex3a.m1.3.3.1.1.1"></eq><apply id="S2.Ex3a.m1.3.3.1.1.2.cmml" xref="S2.Ex3a.m1.3.3.1.1.2"><times id="S2.Ex3a.m1.3.3.1.1.2.1.cmml" xref="S2.Ex3a.m1.3.3.1.1.2.1"></times><ci id="S2.Ex3a.m1.3.3.1.1.2.2.cmml" xref="S2.Ex3a.m1.3.3.1.1.2.2">𝑔</ci><ci id="S2.Ex3a.m1.2.2.cmml" xref="S2.Ex3a.m1.2.2">𝜃</ci></apply><apply id="S2.Ex3a.m1.1.1.cmml" xref="S2.Ex3a.m1.1.1"><divide id="S2.Ex3a.m1.1.1.2.cmml" xref="S2.Ex3a.m1.1.1"></divide><apply id="S2.Ex3a.m1.1.1.1.cmml" xref="S2.Ex3a.m1.1.1.1"><minus id="S2.Ex3a.m1.1.1.1.2.cmml" xref="S2.Ex3a.m1.1.1.1.2"></minus><apply id="S2.Ex3a.m1.1.1.1.3.cmml" xref="S2.Ex3a.m1.1.1.1.3"><divide id="S2.Ex3a.m1.1.1.1.3.1.cmml" 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xref="S2.Ex3a.m1.1.1.1.1.3.2">𝑝</ci><cn id="S2.Ex3a.m1.1.1.1.1.3.3.cmml" type="integer" xref="S2.Ex3a.m1.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.Ex3a.m1.1.1.3.cmml" xref="S2.Ex3a.m1.1.1.3"><minus id="S2.Ex3a.m1.1.1.3.1.cmml" xref="S2.Ex3a.m1.1.1.3.1"></minus><apply id="S2.Ex3a.m1.1.1.3.2.cmml" xref="S2.Ex3a.m1.1.1.3.2"><divide id="S2.Ex3a.m1.1.1.3.2.1.cmml" xref="S2.Ex3a.m1.1.1.3.2"></divide><cn id="S2.Ex3a.m1.1.1.3.2.2.cmml" type="integer" xref="S2.Ex3a.m1.1.1.3.2.2">1</cn><apply id="S2.Ex3a.m1.1.1.3.2.3.cmml" xref="S2.Ex3a.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S2.Ex3a.m1.1.1.3.2.3.1.cmml" xref="S2.Ex3a.m1.1.1.3.2.3">subscript</csymbol><ci id="S2.Ex3a.m1.1.1.3.2.3.2.cmml" xref="S2.Ex3a.m1.1.1.3.2.3.2">𝑝</ci><cn id="S2.Ex3a.m1.1.1.3.2.3.3.cmml" type="integer" xref="S2.Ex3a.m1.1.1.3.2.3.3">1</cn></apply></apply><apply id="S2.Ex3a.m1.1.1.3.3.cmml" xref="S2.Ex3a.m1.1.1.3.3"><divide id="S2.Ex3a.m1.1.1.3.3.1.cmml" xref="S2.Ex3a.m1.1.1.3.3"></divide><apply id="S2.Ex3a.m1.1.1.3.3.2.cmml" xref="S2.Ex3a.m1.1.1.3.3.2"><minus id="S2.Ex3a.m1.1.1.3.3.2.1.cmml" xref="S2.Ex3a.m1.1.1.3.3.2.1"></minus><cn id="S2.Ex3a.m1.1.1.3.3.2.2.cmml" type="integer" xref="S2.Ex3a.m1.1.1.3.3.2.2">1</cn><ci id="S2.Ex3a.m1.1.1.3.3.2.3.cmml" xref="S2.Ex3a.m1.1.1.3.3.2.3">𝜃</ci></apply><apply id="S2.Ex3a.m1.1.1.3.3.3.cmml" xref="S2.Ex3a.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.Ex3a.m1.1.1.3.3.3.1.cmml" xref="S2.Ex3a.m1.1.1.3.3.3">subscript</csymbol><ci id="S2.Ex3a.m1.1.1.3.3.3.2.cmml" xref="S2.Ex3a.m1.1.1.3.3.3.2">𝑝</ci><cn id="S2.Ex3a.m1.1.1.3.3.3.3.cmml" type="integer" xref="S2.Ex3a.m1.1.1.3.3.3.3">0</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex3a.m1.3c">g(\theta)=\frac{\frac{\gamma_{1}}{p_{1}}-\frac{(1-\theta)\gamma_{0}}{p_{0}}}{% \frac{1}{p_{1}}-\frac{1-\theta}{p_{0}}}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex3a.m1.3d">italic_g ( italic_θ ) = divide start_ARG divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG - divide start_ARG ( 1 - italic_θ ) italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_ARG start_ARG divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG - divide start_ARG 1 - italic_θ end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.2.p2.11">Note that <math alttext="g(1)=\gamma_{1}" class="ltx_Math" display="inline" id="S2.SS4.2.p2.6.m1.1"><semantics id="S2.SS4.2.p2.6.m1.1a"><mrow id="S2.SS4.2.p2.6.m1.1.2" xref="S2.SS4.2.p2.6.m1.1.2.cmml"><mrow id="S2.SS4.2.p2.6.m1.1.2.2" xref="S2.SS4.2.p2.6.m1.1.2.2.cmml"><mi id="S2.SS4.2.p2.6.m1.1.2.2.2" xref="S2.SS4.2.p2.6.m1.1.2.2.2.cmml">g</mi><mo id="S2.SS4.2.p2.6.m1.1.2.2.1" xref="S2.SS4.2.p2.6.m1.1.2.2.1.cmml">⁢</mo><mrow id="S2.SS4.2.p2.6.m1.1.2.2.3.2" xref="S2.SS4.2.p2.6.m1.1.2.2.cmml"><mo id="S2.SS4.2.p2.6.m1.1.2.2.3.2.1" stretchy="false" xref="S2.SS4.2.p2.6.m1.1.2.2.cmml">(</mo><mn id="S2.SS4.2.p2.6.m1.1.1" xref="S2.SS4.2.p2.6.m1.1.1.cmml">1</mn><mo id="S2.SS4.2.p2.6.m1.1.2.2.3.2.2" stretchy="false" xref="S2.SS4.2.p2.6.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p2.6.m1.1.2.1" xref="S2.SS4.2.p2.6.m1.1.2.1.cmml">=</mo><msub id="S2.SS4.2.p2.6.m1.1.2.3" xref="S2.SS4.2.p2.6.m1.1.2.3.cmml"><mi id="S2.SS4.2.p2.6.m1.1.2.3.2" xref="S2.SS4.2.p2.6.m1.1.2.3.2.cmml">γ</mi><mn id="S2.SS4.2.p2.6.m1.1.2.3.3" xref="S2.SS4.2.p2.6.m1.1.2.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.6.m1.1b"><apply id="S2.SS4.2.p2.6.m1.1.2.cmml" xref="S2.SS4.2.p2.6.m1.1.2"><eq id="S2.SS4.2.p2.6.m1.1.2.1.cmml" xref="S2.SS4.2.p2.6.m1.1.2.1"></eq><apply id="S2.SS4.2.p2.6.m1.1.2.2.cmml" xref="S2.SS4.2.p2.6.m1.1.2.2"><times id="S2.SS4.2.p2.6.m1.1.2.2.1.cmml" xref="S2.SS4.2.p2.6.m1.1.2.2.1"></times><ci id="S2.SS4.2.p2.6.m1.1.2.2.2.cmml" xref="S2.SS4.2.p2.6.m1.1.2.2.2">𝑔</ci><cn id="S2.SS4.2.p2.6.m1.1.1.cmml" type="integer" xref="S2.SS4.2.p2.6.m1.1.1">1</cn></apply><apply id="S2.SS4.2.p2.6.m1.1.2.3.cmml" xref="S2.SS4.2.p2.6.m1.1.2.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.6.m1.1.2.3.1.cmml" xref="S2.SS4.2.p2.6.m1.1.2.3">subscript</csymbol><ci id="S2.SS4.2.p2.6.m1.1.2.3.2.cmml" xref="S2.SS4.2.p2.6.m1.1.2.3.2">𝛾</ci><cn id="S2.SS4.2.p2.6.m1.1.2.3.3.cmml" type="integer" xref="S2.SS4.2.p2.6.m1.1.2.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.6.m1.1c">g(1)=\gamma_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.6.m1.1d">italic_g ( 1 ) = italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and since <math alttext="\gamma_{1}>-1" class="ltx_Math" display="inline" id="S2.SS4.2.p2.7.m2.1"><semantics id="S2.SS4.2.p2.7.m2.1a"><mrow id="S2.SS4.2.p2.7.m2.1.1" xref="S2.SS4.2.p2.7.m2.1.1.cmml"><msub id="S2.SS4.2.p2.7.m2.1.1.2" xref="S2.SS4.2.p2.7.m2.1.1.2.cmml"><mi id="S2.SS4.2.p2.7.m2.1.1.2.2" xref="S2.SS4.2.p2.7.m2.1.1.2.2.cmml">γ</mi><mn id="S2.SS4.2.p2.7.m2.1.1.2.3" xref="S2.SS4.2.p2.7.m2.1.1.2.3.cmml">1</mn></msub><mo id="S2.SS4.2.p2.7.m2.1.1.1" xref="S2.SS4.2.p2.7.m2.1.1.1.cmml">></mo><mrow id="S2.SS4.2.p2.7.m2.1.1.3" xref="S2.SS4.2.p2.7.m2.1.1.3.cmml"><mo id="S2.SS4.2.p2.7.m2.1.1.3a" xref="S2.SS4.2.p2.7.m2.1.1.3.cmml">−</mo><mn id="S2.SS4.2.p2.7.m2.1.1.3.2" xref="S2.SS4.2.p2.7.m2.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.7.m2.1b"><apply id="S2.SS4.2.p2.7.m2.1.1.cmml" xref="S2.SS4.2.p2.7.m2.1.1"><gt id="S2.SS4.2.p2.7.m2.1.1.1.cmml" xref="S2.SS4.2.p2.7.m2.1.1.1"></gt><apply id="S2.SS4.2.p2.7.m2.1.1.2.cmml" xref="S2.SS4.2.p2.7.m2.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.2.p2.7.m2.1.1.2.1.cmml" xref="S2.SS4.2.p2.7.m2.1.1.2">subscript</csymbol><ci id="S2.SS4.2.p2.7.m2.1.1.2.2.cmml" xref="S2.SS4.2.p2.7.m2.1.1.2.2">𝛾</ci><cn id="S2.SS4.2.p2.7.m2.1.1.2.3.cmml" type="integer" xref="S2.SS4.2.p2.7.m2.1.1.2.3">1</cn></apply><apply id="S2.SS4.2.p2.7.m2.1.1.3.cmml" xref="S2.SS4.2.p2.7.m2.1.1.3"><minus id="S2.SS4.2.p2.7.m2.1.1.3.1.cmml" xref="S2.SS4.2.p2.7.m2.1.1.3"></minus><cn id="S2.SS4.2.p2.7.m2.1.1.3.2.cmml" type="integer" xref="S2.SS4.2.p2.7.m2.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.7.m2.1c">\gamma_{1}>-1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.7.m2.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > - 1</annotation></semantics></math> there exists a <math alttext="\theta\in(\theta_{0},1)" class="ltx_Math" display="inline" id="S2.SS4.2.p2.8.m3.2"><semantics id="S2.SS4.2.p2.8.m3.2a"><mrow id="S2.SS4.2.p2.8.m3.2.2" xref="S2.SS4.2.p2.8.m3.2.2.cmml"><mi id="S2.SS4.2.p2.8.m3.2.2.3" xref="S2.SS4.2.p2.8.m3.2.2.3.cmml">θ</mi><mo id="S2.SS4.2.p2.8.m3.2.2.2" xref="S2.SS4.2.p2.8.m3.2.2.2.cmml">∈</mo><mrow id="S2.SS4.2.p2.8.m3.2.2.1.1" xref="S2.SS4.2.p2.8.m3.2.2.1.2.cmml"><mo id="S2.SS4.2.p2.8.m3.2.2.1.1.2" stretchy="false" xref="S2.SS4.2.p2.8.m3.2.2.1.2.cmml">(</mo><msub id="S2.SS4.2.p2.8.m3.2.2.1.1.1" xref="S2.SS4.2.p2.8.m3.2.2.1.1.1.cmml"><mi id="S2.SS4.2.p2.8.m3.2.2.1.1.1.2" xref="S2.SS4.2.p2.8.m3.2.2.1.1.1.2.cmml">θ</mi><mn id="S2.SS4.2.p2.8.m3.2.2.1.1.1.3" xref="S2.SS4.2.p2.8.m3.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS4.2.p2.8.m3.2.2.1.1.3" xref="S2.SS4.2.p2.8.m3.2.2.1.2.cmml">,</mo><mn id="S2.SS4.2.p2.8.m3.1.1" xref="S2.SS4.2.p2.8.m3.1.1.cmml">1</mn><mo id="S2.SS4.2.p2.8.m3.2.2.1.1.4" stretchy="false" xref="S2.SS4.2.p2.8.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.8.m3.2b"><apply id="S2.SS4.2.p2.8.m3.2.2.cmml" xref="S2.SS4.2.p2.8.m3.2.2"><in id="S2.SS4.2.p2.8.m3.2.2.2.cmml" xref="S2.SS4.2.p2.8.m3.2.2.2"></in><ci id="S2.SS4.2.p2.8.m3.2.2.3.cmml" xref="S2.SS4.2.p2.8.m3.2.2.3">𝜃</ci><interval closure="open" id="S2.SS4.2.p2.8.m3.2.2.1.2.cmml" xref="S2.SS4.2.p2.8.m3.2.2.1.1"><apply id="S2.SS4.2.p2.8.m3.2.2.1.1.1.cmml" xref="S2.SS4.2.p2.8.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p2.8.m3.2.2.1.1.1.1.cmml" xref="S2.SS4.2.p2.8.m3.2.2.1.1.1">subscript</csymbol><ci id="S2.SS4.2.p2.8.m3.2.2.1.1.1.2.cmml" xref="S2.SS4.2.p2.8.m3.2.2.1.1.1.2">𝜃</ci><cn id="S2.SS4.2.p2.8.m3.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS4.2.p2.8.m3.2.2.1.1.1.3">0</cn></apply><cn id="S2.SS4.2.p2.8.m3.1.1.cmml" type="integer" xref="S2.SS4.2.p2.8.m3.1.1">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.8.m3.2c">\theta\in(\theta_{0},1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.8.m3.2d">italic_θ ∈ ( italic_θ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , 1 )</annotation></semantics></math> such that <math alttext="\gamma:=g(\theta)>-1" class="ltx_Math" display="inline" id="S2.SS4.2.p2.9.m4.1"><semantics id="S2.SS4.2.p2.9.m4.1a"><mrow id="S2.SS4.2.p2.9.m4.1.2" xref="S2.SS4.2.p2.9.m4.1.2.cmml"><mi id="S2.SS4.2.p2.9.m4.1.2.2" xref="S2.SS4.2.p2.9.m4.1.2.2.cmml">γ</mi><mo id="S2.SS4.2.p2.9.m4.1.2.3" lspace="0.278em" rspace="0.278em" xref="S2.SS4.2.p2.9.m4.1.2.3.cmml">:=</mo><mrow id="S2.SS4.2.p2.9.m4.1.2.4" xref="S2.SS4.2.p2.9.m4.1.2.4.cmml"><mi id="S2.SS4.2.p2.9.m4.1.2.4.2" xref="S2.SS4.2.p2.9.m4.1.2.4.2.cmml">g</mi><mo id="S2.SS4.2.p2.9.m4.1.2.4.1" xref="S2.SS4.2.p2.9.m4.1.2.4.1.cmml">⁢</mo><mrow id="S2.SS4.2.p2.9.m4.1.2.4.3.2" xref="S2.SS4.2.p2.9.m4.1.2.4.cmml"><mo id="S2.SS4.2.p2.9.m4.1.2.4.3.2.1" stretchy="false" xref="S2.SS4.2.p2.9.m4.1.2.4.cmml">(</mo><mi id="S2.SS4.2.p2.9.m4.1.1" xref="S2.SS4.2.p2.9.m4.1.1.cmml">θ</mi><mo id="S2.SS4.2.p2.9.m4.1.2.4.3.2.2" stretchy="false" xref="S2.SS4.2.p2.9.m4.1.2.4.cmml">)</mo></mrow></mrow><mo id="S2.SS4.2.p2.9.m4.1.2.5" xref="S2.SS4.2.p2.9.m4.1.2.5.cmml">></mo><mrow id="S2.SS4.2.p2.9.m4.1.2.6" xref="S2.SS4.2.p2.9.m4.1.2.6.cmml"><mo id="S2.SS4.2.p2.9.m4.1.2.6a" xref="S2.SS4.2.p2.9.m4.1.2.6.cmml">−</mo><mn id="S2.SS4.2.p2.9.m4.1.2.6.2" xref="S2.SS4.2.p2.9.m4.1.2.6.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.9.m4.1b"><apply id="S2.SS4.2.p2.9.m4.1.2.cmml" xref="S2.SS4.2.p2.9.m4.1.2"><and id="S2.SS4.2.p2.9.m4.1.2a.cmml" xref="S2.SS4.2.p2.9.m4.1.2"></and><apply id="S2.SS4.2.p2.9.m4.1.2b.cmml" xref="S2.SS4.2.p2.9.m4.1.2"><csymbol cd="latexml" id="S2.SS4.2.p2.9.m4.1.2.3.cmml" xref="S2.SS4.2.p2.9.m4.1.2.3">assign</csymbol><ci id="S2.SS4.2.p2.9.m4.1.2.2.cmml" xref="S2.SS4.2.p2.9.m4.1.2.2">𝛾</ci><apply id="S2.SS4.2.p2.9.m4.1.2.4.cmml" xref="S2.SS4.2.p2.9.m4.1.2.4"><times id="S2.SS4.2.p2.9.m4.1.2.4.1.cmml" xref="S2.SS4.2.p2.9.m4.1.2.4.1"></times><ci id="S2.SS4.2.p2.9.m4.1.2.4.2.cmml" xref="S2.SS4.2.p2.9.m4.1.2.4.2">𝑔</ci><ci id="S2.SS4.2.p2.9.m4.1.1.cmml" xref="S2.SS4.2.p2.9.m4.1.1">𝜃</ci></apply></apply><apply id="S2.SS4.2.p2.9.m4.1.2c.cmml" xref="S2.SS4.2.p2.9.m4.1.2"><gt id="S2.SS4.2.p2.9.m4.1.2.5.cmml" xref="S2.SS4.2.p2.9.m4.1.2.5"></gt><share href="https://arxiv.org/html/2503.14636v1#S2.SS4.2.p2.9.m4.1.2.4.cmml" id="S2.SS4.2.p2.9.m4.1.2d.cmml" xref="S2.SS4.2.p2.9.m4.1.2"></share><apply id="S2.SS4.2.p2.9.m4.1.2.6.cmml" xref="S2.SS4.2.p2.9.m4.1.2.6"><minus id="S2.SS4.2.p2.9.m4.1.2.6.1.cmml" xref="S2.SS4.2.p2.9.m4.1.2.6"></minus><cn id="S2.SS4.2.p2.9.m4.1.2.6.2.cmml" type="integer" xref="S2.SS4.2.p2.9.m4.1.2.6.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.9.m4.1c">\gamma:=g(\theta)>-1</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.9.m4.1d">italic_γ := italic_g ( italic_θ ) > - 1</annotation></semantics></math>. Define <math alttext="r" class="ltx_Math" display="inline" id="S2.SS4.2.p2.10.m5.1"><semantics id="S2.SS4.2.p2.10.m5.1a"><mi id="S2.SS4.2.p2.10.m5.1.1" xref="S2.SS4.2.p2.10.m5.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.10.m5.1b"><ci id="S2.SS4.2.p2.10.m5.1.1.cmml" xref="S2.SS4.2.p2.10.m5.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.10.m5.1c">r</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.10.m5.1d">italic_r</annotation></semantics></math> and <math alttext="t" class="ltx_Math" display="inline" id="S2.SS4.2.p2.11.m6.1"><semantics id="S2.SS4.2.p2.11.m6.1a"><mi id="S2.SS4.2.p2.11.m6.1.1" xref="S2.SS4.2.p2.11.m6.1.1.cmml">t</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.11.m6.1b"><ci id="S2.SS4.2.p2.11.m6.1.1.cmml" xref="S2.SS4.2.p2.11.m6.1.1">𝑡</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.11.m6.1c">t</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.11.m6.1d">italic_t</annotation></semantics></math> by</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex4a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\frac{1}{p_{1}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{r}\quad\text{ and }\quad t% -\frac{d+\gamma}{r}=s_{1}-\frac{d+\gamma_{1}}{p_{1}}." class="ltx_Math" display="block" id="S2.Ex4a.m1.2"><semantics id="S2.Ex4a.m1.2a"><mrow id="S2.Ex4a.m1.2.2.1"><mrow id="S2.Ex4a.m1.2.2.1.1.2" xref="S2.Ex4a.m1.2.2.1.1.3.cmml"><mrow id="S2.Ex4a.m1.2.2.1.1.1.1" xref="S2.Ex4a.m1.2.2.1.1.1.1.cmml"><mfrac id="S2.Ex4a.m1.2.2.1.1.1.1.3" xref="S2.Ex4a.m1.2.2.1.1.1.1.3.cmml"><mn id="S2.Ex4a.m1.2.2.1.1.1.1.3.2" xref="S2.Ex4a.m1.2.2.1.1.1.1.3.2.cmml">1</mn><msub id="S2.Ex4a.m1.2.2.1.1.1.1.3.3" xref="S2.Ex4a.m1.2.2.1.1.1.1.3.3.cmml"><mi id="S2.Ex4a.m1.2.2.1.1.1.1.3.3.2" 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id="S2.Ex4a.m1.2c">\frac{1}{p_{1}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{r}\quad\text{ and }\quad t% -\frac{d+\gamma}{r}=s_{1}-\frac{d+\gamma_{1}}{p_{1}}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex4a.m1.2d">divide start_ARG 1 end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG = divide start_ARG 1 - italic_θ end_ARG start_ARG italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG + divide start_ARG italic_θ end_ARG start_ARG italic_r end_ARG and italic_t - divide start_ARG italic_d + italic_γ end_ARG start_ARG italic_r end_ARG = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - divide start_ARG italic_d + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.2.p2.15">From the conditions on the parameters it follows that <math alttext="r\in[p_{1},\infty)" class="ltx_Math" display="inline" id="S2.SS4.2.p2.12.m1.2"><semantics id="S2.SS4.2.p2.12.m1.2a"><mrow id="S2.SS4.2.p2.12.m1.2.2" xref="S2.SS4.2.p2.12.m1.2.2.cmml"><mi id="S2.SS4.2.p2.12.m1.2.2.3" xref="S2.SS4.2.p2.12.m1.2.2.3.cmml">r</mi><mo id="S2.SS4.2.p2.12.m1.2.2.2" xref="S2.SS4.2.p2.12.m1.2.2.2.cmml">∈</mo><mrow id="S2.SS4.2.p2.12.m1.2.2.1.1" xref="S2.SS4.2.p2.12.m1.2.2.1.2.cmml"><mo id="S2.SS4.2.p2.12.m1.2.2.1.1.2" stretchy="false" xref="S2.SS4.2.p2.12.m1.2.2.1.2.cmml">[</mo><msub id="S2.SS4.2.p2.12.m1.2.2.1.1.1" xref="S2.SS4.2.p2.12.m1.2.2.1.1.1.cmml"><mi id="S2.SS4.2.p2.12.m1.2.2.1.1.1.2" xref="S2.SS4.2.p2.12.m1.2.2.1.1.1.2.cmml">p</mi><mn id="S2.SS4.2.p2.12.m1.2.2.1.1.1.3" xref="S2.SS4.2.p2.12.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS4.2.p2.12.m1.2.2.1.1.3" xref="S2.SS4.2.p2.12.m1.2.2.1.2.cmml">,</mo><mi id="S2.SS4.2.p2.12.m1.1.1" mathvariant="normal" xref="S2.SS4.2.p2.12.m1.1.1.cmml">∞</mi><mo id="S2.SS4.2.p2.12.m1.2.2.1.1.4" stretchy="false" xref="S2.SS4.2.p2.12.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.12.m1.2b"><apply id="S2.SS4.2.p2.12.m1.2.2.cmml" xref="S2.SS4.2.p2.12.m1.2.2"><in id="S2.SS4.2.p2.12.m1.2.2.2.cmml" xref="S2.SS4.2.p2.12.m1.2.2.2"></in><ci id="S2.SS4.2.p2.12.m1.2.2.3.cmml" xref="S2.SS4.2.p2.12.m1.2.2.3">𝑟</ci><interval closure="closed-open" id="S2.SS4.2.p2.12.m1.2.2.1.2.cmml" xref="S2.SS4.2.p2.12.m1.2.2.1.1"><apply id="S2.SS4.2.p2.12.m1.2.2.1.1.1.cmml" xref="S2.SS4.2.p2.12.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.2.p2.12.m1.2.2.1.1.1.1.cmml" xref="S2.SS4.2.p2.12.m1.2.2.1.1.1">subscript</csymbol><ci id="S2.SS4.2.p2.12.m1.2.2.1.1.1.2.cmml" xref="S2.SS4.2.p2.12.m1.2.2.1.1.1.2">𝑝</ci><cn id="S2.SS4.2.p2.12.m1.2.2.1.1.1.3.cmml" type="integer" xref="S2.SS4.2.p2.12.m1.2.2.1.1.1.3">1</cn></apply><infinity id="S2.SS4.2.p2.12.m1.1.1.cmml" xref="S2.SS4.2.p2.12.m1.1.1"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.12.m1.2c">r\in[p_{1},\infty)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.12.m1.2d">italic_r ∈ [ italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ∞ )</annotation></semantics></math>, and <math alttext="t<s_{0}" class="ltx_Math" display="inline" id="S2.SS4.2.p2.13.m2.1"><semantics id="S2.SS4.2.p2.13.m2.1a"><mrow id="S2.SS4.2.p2.13.m2.1.1" xref="S2.SS4.2.p2.13.m2.1.1.cmml"><mi id="S2.SS4.2.p2.13.m2.1.1.2" xref="S2.SS4.2.p2.13.m2.1.1.2.cmml">t</mi><mo id="S2.SS4.2.p2.13.m2.1.1.1" xref="S2.SS4.2.p2.13.m2.1.1.1.cmml"><</mo><msub id="S2.SS4.2.p2.13.m2.1.1.3" xref="S2.SS4.2.p2.13.m2.1.1.3.cmml"><mi id="S2.SS4.2.p2.13.m2.1.1.3.2" xref="S2.SS4.2.p2.13.m2.1.1.3.2.cmml">s</mi><mn id="S2.SS4.2.p2.13.m2.1.1.3.3" xref="S2.SS4.2.p2.13.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.13.m2.1b"><apply id="S2.SS4.2.p2.13.m2.1.1.cmml" xref="S2.SS4.2.p2.13.m2.1.1"><lt id="S2.SS4.2.p2.13.m2.1.1.1.cmml" xref="S2.SS4.2.p2.13.m2.1.1.1"></lt><ci id="S2.SS4.2.p2.13.m2.1.1.2.cmml" xref="S2.SS4.2.p2.13.m2.1.1.2">𝑡</ci><apply id="S2.SS4.2.p2.13.m2.1.1.3.cmml" xref="S2.SS4.2.p2.13.m2.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.13.m2.1.1.3.1.cmml" xref="S2.SS4.2.p2.13.m2.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p2.13.m2.1.1.3.2.cmml" xref="S2.SS4.2.p2.13.m2.1.1.3.2">𝑠</ci><cn id="S2.SS4.2.p2.13.m2.1.1.3.3.cmml" type="integer" xref="S2.SS4.2.p2.13.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.13.m2.1c">t<s_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.13.m2.1d">italic_t < italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> satisfies <math alttext="s_{1}=\theta t+(1-\theta)s_{0}" class="ltx_Math" display="inline" id="S2.SS4.2.p2.14.m3.1"><semantics id="S2.SS4.2.p2.14.m3.1a"><mrow id="S2.SS4.2.p2.14.m3.1.1" xref="S2.SS4.2.p2.14.m3.1.1.cmml"><msub id="S2.SS4.2.p2.14.m3.1.1.3" xref="S2.SS4.2.p2.14.m3.1.1.3.cmml"><mi id="S2.SS4.2.p2.14.m3.1.1.3.2" xref="S2.SS4.2.p2.14.m3.1.1.3.2.cmml">s</mi><mn id="S2.SS4.2.p2.14.m3.1.1.3.3" xref="S2.SS4.2.p2.14.m3.1.1.3.3.cmml">1</mn></msub><mo id="S2.SS4.2.p2.14.m3.1.1.2" xref="S2.SS4.2.p2.14.m3.1.1.2.cmml">=</mo><mrow id="S2.SS4.2.p2.14.m3.1.1.1" xref="S2.SS4.2.p2.14.m3.1.1.1.cmml"><mrow id="S2.SS4.2.p2.14.m3.1.1.1.3" xref="S2.SS4.2.p2.14.m3.1.1.1.3.cmml"><mi id="S2.SS4.2.p2.14.m3.1.1.1.3.2" xref="S2.SS4.2.p2.14.m3.1.1.1.3.2.cmml">θ</mi><mo id="S2.SS4.2.p2.14.m3.1.1.1.3.1" xref="S2.SS4.2.p2.14.m3.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS4.2.p2.14.m3.1.1.1.3.3" xref="S2.SS4.2.p2.14.m3.1.1.1.3.3.cmml">t</mi></mrow><mo id="S2.SS4.2.p2.14.m3.1.1.1.2" xref="S2.SS4.2.p2.14.m3.1.1.1.2.cmml">+</mo><mrow id="S2.SS4.2.p2.14.m3.1.1.1.1" xref="S2.SS4.2.p2.14.m3.1.1.1.1.cmml"><mrow id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.cmml"><mo id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.cmml"><mn id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.2" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.1" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.3" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.3.cmml">θ</mi></mrow><mo id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS4.2.p2.14.m3.1.1.1.1.2" xref="S2.SS4.2.p2.14.m3.1.1.1.1.2.cmml">⁢</mo><msub id="S2.SS4.2.p2.14.m3.1.1.1.1.3" xref="S2.SS4.2.p2.14.m3.1.1.1.1.3.cmml"><mi id="S2.SS4.2.p2.14.m3.1.1.1.1.3.2" xref="S2.SS4.2.p2.14.m3.1.1.1.1.3.2.cmml">s</mi><mn id="S2.SS4.2.p2.14.m3.1.1.1.1.3.3" xref="S2.SS4.2.p2.14.m3.1.1.1.1.3.3.cmml">0</mn></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.14.m3.1b"><apply id="S2.SS4.2.p2.14.m3.1.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1"><eq id="S2.SS4.2.p2.14.m3.1.1.2.cmml" xref="S2.SS4.2.p2.14.m3.1.1.2"></eq><apply id="S2.SS4.2.p2.14.m3.1.1.3.cmml" xref="S2.SS4.2.p2.14.m3.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.14.m3.1.1.3.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p2.14.m3.1.1.3.2.cmml" xref="S2.SS4.2.p2.14.m3.1.1.3.2">𝑠</ci><cn id="S2.SS4.2.p2.14.m3.1.1.3.3.cmml" type="integer" xref="S2.SS4.2.p2.14.m3.1.1.3.3">1</cn></apply><apply id="S2.SS4.2.p2.14.m3.1.1.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1"><plus id="S2.SS4.2.p2.14.m3.1.1.1.2.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.2"></plus><apply id="S2.SS4.2.p2.14.m3.1.1.1.3.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.3"><times id="S2.SS4.2.p2.14.m3.1.1.1.3.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.3.1"></times><ci id="S2.SS4.2.p2.14.m3.1.1.1.3.2.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.3.2">𝜃</ci><ci id="S2.SS4.2.p2.14.m3.1.1.1.3.3.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.3.3">𝑡</ci></apply><apply id="S2.SS4.2.p2.14.m3.1.1.1.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1"><times id="S2.SS4.2.p2.14.m3.1.1.1.1.2.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1.2"></times><apply id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1"><minus id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.1"></minus><cn id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.2">1</cn><ci id="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1.1.1.1.3">𝜃</ci></apply><apply id="S2.SS4.2.p2.14.m3.1.1.1.1.3.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.14.m3.1.1.1.1.3.1.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p2.14.m3.1.1.1.1.3.2.cmml" xref="S2.SS4.2.p2.14.m3.1.1.1.1.3.2">𝑠</ci><cn id="S2.SS4.2.p2.14.m3.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS4.2.p2.14.m3.1.1.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.14.m3.1c">s_{1}=\theta t+(1-\theta)s_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.14.m3.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_θ italic_t + ( 1 - italic_θ ) italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, it holds that <math alttext="\theta\gamma p_{1}/r+(1-\theta)\gamma_{0}p_{1}/p_{0}=\gamma_{1}" class="ltx_Math" display="inline" id="S2.SS4.2.p2.15.m4.1"><semantics id="S2.SS4.2.p2.15.m4.1a"><mrow id="S2.SS4.2.p2.15.m4.1.1" xref="S2.SS4.2.p2.15.m4.1.1.cmml"><mrow id="S2.SS4.2.p2.15.m4.1.1.1" xref="S2.SS4.2.p2.15.m4.1.1.1.cmml"><mrow id="S2.SS4.2.p2.15.m4.1.1.1.3" xref="S2.SS4.2.p2.15.m4.1.1.1.3.cmml"><mrow id="S2.SS4.2.p2.15.m4.1.1.1.3.2" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.cmml"><mi id="S2.SS4.2.p2.15.m4.1.1.1.3.2.2" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.2.cmml">θ</mi><mo id="S2.SS4.2.p2.15.m4.1.1.1.3.2.1" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.1.cmml">⁢</mo><mi id="S2.SS4.2.p2.15.m4.1.1.1.3.2.3" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.3.cmml">γ</mi><mo id="S2.SS4.2.p2.15.m4.1.1.1.3.2.1a" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.1.cmml">⁢</mo><msub id="S2.SS4.2.p2.15.m4.1.1.1.3.2.4" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.cmml"><mi id="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.2" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.2.cmml">p</mi><mn id="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.3" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.3.cmml">1</mn></msub></mrow><mo id="S2.SS4.2.p2.15.m4.1.1.1.3.1" xref="S2.SS4.2.p2.15.m4.1.1.1.3.1.cmml">/</mo><mi id="S2.SS4.2.p2.15.m4.1.1.1.3.3" xref="S2.SS4.2.p2.15.m4.1.1.1.3.3.cmml">r</mi></mrow><mo id="S2.SS4.2.p2.15.m4.1.1.1.2" xref="S2.SS4.2.p2.15.m4.1.1.1.2.cmml">+</mo><mrow id="S2.SS4.2.p2.15.m4.1.1.1.1" xref="S2.SS4.2.p2.15.m4.1.1.1.1.cmml"><mrow id="S2.SS4.2.p2.15.m4.1.1.1.1.1" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.cmml"><mrow id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.cmml"><mo id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.cmml"><mn id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.2" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.1" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.3" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.3.cmml">θ</mi></mrow><mo id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S2.SS4.2.p2.15.m4.1.1.1.1.1.2" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.SS4.2.p2.15.m4.1.1.1.1.1.3" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.cmml"><mi id="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.2" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.2.cmml">γ</mi><mn id="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.3" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.3.cmml">0</mn></msub><mo id="S2.SS4.2.p2.15.m4.1.1.1.1.1.2a" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.2.cmml">⁢</mo><msub id="S2.SS4.2.p2.15.m4.1.1.1.1.1.4" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.cmml"><mi id="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.2" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.2.cmml">p</mi><mn id="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.3" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.3.cmml">1</mn></msub></mrow><mo id="S2.SS4.2.p2.15.m4.1.1.1.1.2" xref="S2.SS4.2.p2.15.m4.1.1.1.1.2.cmml">/</mo><msub id="S2.SS4.2.p2.15.m4.1.1.1.1.3" xref="S2.SS4.2.p2.15.m4.1.1.1.1.3.cmml"><mi id="S2.SS4.2.p2.15.m4.1.1.1.1.3.2" xref="S2.SS4.2.p2.15.m4.1.1.1.1.3.2.cmml">p</mi><mn id="S2.SS4.2.p2.15.m4.1.1.1.1.3.3" xref="S2.SS4.2.p2.15.m4.1.1.1.1.3.3.cmml">0</mn></msub></mrow></mrow><mo id="S2.SS4.2.p2.15.m4.1.1.2" xref="S2.SS4.2.p2.15.m4.1.1.2.cmml">=</mo><msub id="S2.SS4.2.p2.15.m4.1.1.3" xref="S2.SS4.2.p2.15.m4.1.1.3.cmml"><mi id="S2.SS4.2.p2.15.m4.1.1.3.2" xref="S2.SS4.2.p2.15.m4.1.1.3.2.cmml">γ</mi><mn id="S2.SS4.2.p2.15.m4.1.1.3.3" xref="S2.SS4.2.p2.15.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.2.p2.15.m4.1b"><apply id="S2.SS4.2.p2.15.m4.1.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1"><eq id="S2.SS4.2.p2.15.m4.1.1.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.2"></eq><apply id="S2.SS4.2.p2.15.m4.1.1.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1"><plus id="S2.SS4.2.p2.15.m4.1.1.1.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.2"></plus><apply id="S2.SS4.2.p2.15.m4.1.1.1.3.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3"><divide id="S2.SS4.2.p2.15.m4.1.1.1.3.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.1"></divide><apply id="S2.SS4.2.p2.15.m4.1.1.1.3.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2"><times id="S2.SS4.2.p2.15.m4.1.1.1.3.2.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.1"></times><ci id="S2.SS4.2.p2.15.m4.1.1.1.3.2.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.2">𝜃</ci><ci id="S2.SS4.2.p2.15.m4.1.1.1.3.2.3.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.3">𝛾</ci><apply id="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.4"><csymbol cd="ambiguous" id="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.4">subscript</csymbol><ci id="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.2">𝑝</ci><cn id="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.3.cmml" type="integer" xref="S2.SS4.2.p2.15.m4.1.1.1.3.2.4.3">1</cn></apply></apply><ci id="S2.SS4.2.p2.15.m4.1.1.1.3.3.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.3.3">𝑟</ci></apply><apply id="S2.SS4.2.p2.15.m4.1.1.1.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1"><divide id="S2.SS4.2.p2.15.m4.1.1.1.1.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.2"></divide><apply id="S2.SS4.2.p2.15.m4.1.1.1.1.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1"><times id="S2.SS4.2.p2.15.m4.1.1.1.1.1.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.2"></times><apply id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1"><minus id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.1"></minus><cn id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.2">1</cn><ci id="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.1.1.1.3">𝜃</ci></apply><apply id="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.2">𝛾</ci><cn id="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.3.3">0</cn></apply><apply id="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.4"><csymbol cd="ambiguous" id="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.4">subscript</csymbol><ci id="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.2">𝑝</ci><cn id="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.3.cmml" type="integer" xref="S2.SS4.2.p2.15.m4.1.1.1.1.1.4.3">1</cn></apply></apply><apply id="S2.SS4.2.p2.15.m4.1.1.1.1.3.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.15.m4.1.1.1.1.3.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p2.15.m4.1.1.1.1.3.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.1.1.3.2">𝑝</ci><cn id="S2.SS4.2.p2.15.m4.1.1.1.1.3.3.cmml" type="integer" xref="S2.SS4.2.p2.15.m4.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S2.SS4.2.p2.15.m4.1.1.3.cmml" xref="S2.SS4.2.p2.15.m4.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.2.p2.15.m4.1.1.3.1.cmml" xref="S2.SS4.2.p2.15.m4.1.1.3">subscript</csymbol><ci id="S2.SS4.2.p2.15.m4.1.1.3.2.cmml" xref="S2.SS4.2.p2.15.m4.1.1.3.2">𝛾</ci><cn id="S2.SS4.2.p2.15.m4.1.1.3.3.cmml" type="integer" xref="S2.SS4.2.p2.15.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.2.p2.15.m4.1c">\theta\gamma p_{1}/r+(1-\theta)\gamma_{0}p_{1}/p_{0}=\gamma_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.2.p2.15.m4.1d">italic_θ italic_γ italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_r + ( 1 - italic_θ ) italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, the Gagliardo-Nirenberg type inequality from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 5.1]</cite> gives</p> <table class="ltx_equation ltx_eqn_table" id="S2.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f\|_{F^{s_{1}}_{p_{1},1}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f\|^{1-% \theta}_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\|f\|^{% \theta}_{F^{t}_{r,r}(\mathbb{R}^{d},w_{\gamma};X)}." class="ltx_Math" display="block" id="S2.E8.m1.19"><semantics id="S2.E8.m1.19a"><mrow id="S2.E8.m1.19.19.1" xref="S2.E8.m1.19.19.1.1.cmml"><mrow id="S2.E8.m1.19.19.1.1" xref="S2.E8.m1.19.19.1.1.cmml"><msub id="S2.E8.m1.19.19.1.1.2" xref="S2.E8.m1.19.19.1.1.2.cmml"><mrow id="S2.E8.m1.19.19.1.1.2.2.2" xref="S2.E8.m1.19.19.1.1.2.2.1.cmml"><mo 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xref="S2.E8.m1.15.15.5.5.2.2"><csymbol cd="ambiguous" id="S2.E8.m1.15.15.5.5.2.2.1.cmml" xref="S2.E8.m1.15.15.5.5.2.2">subscript</csymbol><ci id="S2.E8.m1.15.15.5.5.2.2.2.cmml" xref="S2.E8.m1.15.15.5.5.2.2.2">𝑤</ci><ci id="S2.E8.m1.15.15.5.5.2.2.3.cmml" xref="S2.E8.m1.15.15.5.5.2.2.3">𝛾</ci></apply><ci id="S2.E8.m1.13.13.3.3.cmml" xref="S2.E8.m1.13.13.3.3">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.E8.m1.19c">\|f\|_{F^{s_{1}}_{p_{1},1}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f\|^{1-% \theta}_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\|f\|^{% \theta}_{F^{t}_{r,r}(\mathbb{R}^{d},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S2.E8.m1.19d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUPERSCRIPT 1 - italic_θ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ∥ italic_f ∥ start_POSTSUPERSCRIPT italic_θ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r , italic_r end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.2.p2.17">Moreover, by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E5" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.5</span></a>), <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.I2.i2" title="item ii ‣ Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E6" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.6</span></a>), we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S2.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f\|_{F^{t}_{r,r}(\mathbb{R}^{d},w_{\gamma};X)}\eqsim\|f\|_{B^{t}_{r,r}(% \mathbb{R}^{d},w_{\gamma};X)}\leq C\|f\|_{B^{s_{1}}_{p_{1},p_{1}}(\mathbb{R}^{% d},w_{\gamma_{1}};X)}\leq C\|f\|_{F^{s_{1}}_{p_{1},1}(w_{\gamma_{1}};X)}." class="ltx_Math" display="block" id="S2.E9.m1.24"><semantics id="S2.E9.m1.24a"><mrow id="S2.E9.m1.24.24.1" xref="S2.E9.m1.24.24.1.1.cmml"><mrow id="S2.E9.m1.24.24.1.1" xref="S2.E9.m1.24.24.1.1.cmml"><msub id="S2.E9.m1.24.24.1.1.2" xref="S2.E9.m1.24.24.1.1.2.cmml"><mrow id="S2.E9.m1.24.24.1.1.2.2.2" xref="S2.E9.m1.24.24.1.1.2.2.1.cmml"><mo id="S2.E9.m1.24.24.1.1.2.2.2.1" 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id="S2.E9.m1.24c">\|f\|_{F^{t}_{r,r}(\mathbb{R}^{d},w_{\gamma};X)}\eqsim\|f\|_{B^{t}_{r,r}(% \mathbb{R}^{d},w_{\gamma};X)}\leq C\|f\|_{B^{s_{1}}_{p_{1},p_{1}}(\mathbb{R}^{% d},w_{\gamma_{1}};X)}\leq C\|f\|_{F^{s_{1}}_{p_{1},1}(w_{\gamma_{1}};X)}.</annotation><annotation encoding="application/x-llamapun" id="S2.E9.m1.24d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r , italic_r end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≂ ∥ italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_r , italic_r end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , 1 end_POSTSUBSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.2.p2.18">Appealing to (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E6" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.6</span></a>) and substituting (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E9" title="In Proof. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.9</span></a>) in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E8" title="In Proof. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.8</span></a>), yields</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex5a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f\|_{F^{s_{1}}_{p_{1},q_{0}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f\|_{F% ^{s_{1}}_{p_{1},1}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f\|_{F^{s_{0}}_{p_% {0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}" class="ltx_Math" display="block" id="S2.Ex5a.m1.18"><semantics id="S2.Ex5a.m1.18a"><mrow id="S2.Ex5a.m1.18.19" xref="S2.Ex5a.m1.18.19.cmml"><msub id="S2.Ex5a.m1.18.19.2" xref="S2.Ex5a.m1.18.19.2.cmml"><mrow id="S2.Ex5a.m1.18.19.2.2.2" xref="S2.Ex5a.m1.18.19.2.2.1.cmml"><mo id="S2.Ex5a.m1.18.19.2.2.2.1" stretchy="false" xref="S2.Ex5a.m1.18.19.2.2.1.1.cmml">‖</mo><mi id="S2.Ex5a.m1.16.16" xref="S2.Ex5a.m1.16.16.cmml">f</mi><mo id="S2.Ex5a.m1.18.19.2.2.2.2" stretchy="false" xref="S2.Ex5a.m1.18.19.2.2.1.1.cmml">‖</mo></mrow><mrow id="S2.Ex5a.m1.5.5.5" xref="S2.Ex5a.m1.5.5.5.cmml"><msubsup id="S2.Ex5a.m1.5.5.5.7" xref="S2.Ex5a.m1.5.5.5.7.cmml"><mi id="S2.Ex5a.m1.5.5.5.7.2.2" xref="S2.Ex5a.m1.5.5.5.7.2.2.cmml">F</mi><mrow id="S2.Ex5a.m1.2.2.2.2.2.2" xref="S2.Ex5a.m1.2.2.2.2.2.3.cmml"><msub id="S2.Ex5a.m1.1.1.1.1.1.1.1" xref="S2.Ex5a.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex5a.m1.1.1.1.1.1.1.1.2" xref="S2.Ex5a.m1.1.1.1.1.1.1.1.2.cmml">p</mi><mn id="S2.Ex5a.m1.1.1.1.1.1.1.1.3" xref="S2.Ex5a.m1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex5a.m1.2.2.2.2.2.2.3" xref="S2.Ex5a.m1.2.2.2.2.2.3.cmml">,</mo><msub id="S2.Ex5a.m1.2.2.2.2.2.2.2" xref="S2.Ex5a.m1.2.2.2.2.2.2.2.cmml"><mi id="S2.Ex5a.m1.2.2.2.2.2.2.2.2" xref="S2.Ex5a.m1.2.2.2.2.2.2.2.2.cmml">q</mi><mn id="S2.Ex5a.m1.2.2.2.2.2.2.2.3" xref="S2.Ex5a.m1.2.2.2.2.2.2.2.3.cmml">0</mn></msub></mrow><msub id="S2.Ex5a.m1.5.5.5.7.2.3" 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xref="S2.Ex5a.m1.5.5.5.5.2.2.3.2.cmml">γ</mi><mn id="S2.Ex5a.m1.5.5.5.5.2.2.3.3" xref="S2.Ex5a.m1.5.5.5.5.2.2.3.3.cmml">1</mn></msub></msub><mo id="S2.Ex5a.m1.5.5.5.5.2.5" xref="S2.Ex5a.m1.5.5.5.5.3.cmml">;</mo><mi id="S2.Ex5a.m1.3.3.3.3" xref="S2.Ex5a.m1.3.3.3.3.cmml">X</mi><mo id="S2.Ex5a.m1.5.5.5.5.2.6" stretchy="false" xref="S2.Ex5a.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex5a.m1.18.19.3" xref="S2.Ex5a.m1.18.19.3.cmml">≤</mo><mrow id="S2.Ex5a.m1.18.19.4" xref="S2.Ex5a.m1.18.19.4.cmml"><mi id="S2.Ex5a.m1.18.19.4.2" xref="S2.Ex5a.m1.18.19.4.2.cmml">C</mi><mo id="S2.Ex5a.m1.18.19.4.1" xref="S2.Ex5a.m1.18.19.4.1.cmml">⁢</mo><msub id="S2.Ex5a.m1.18.19.4.3" xref="S2.Ex5a.m1.18.19.4.3.cmml"><mrow id="S2.Ex5a.m1.18.19.4.3.2.2" xref="S2.Ex5a.m1.18.19.4.3.2.1.cmml"><mo id="S2.Ex5a.m1.18.19.4.3.2.2.1" stretchy="false" xref="S2.Ex5a.m1.18.19.4.3.2.1.1.cmml">‖</mo><mi id="S2.Ex5a.m1.17.17" xref="S2.Ex5a.m1.17.17.cmml">f</mi><mo id="S2.Ex5a.m1.18.19.4.3.2.2.2" stretchy="false" 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xref="S2.Ex5a.m1.15.15.5.5.2.2.3.2">𝛾</ci><cn id="S2.Ex5a.m1.15.15.5.5.2.2.3.3.cmml" type="integer" xref="S2.Ex5a.m1.15.15.5.5.2.2.3.3">0</cn></apply></apply><ci id="S2.Ex5a.m1.13.13.3.3.cmml" xref="S2.Ex5a.m1.13.13.3.3">𝑋</ci></vector></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex5a.m1.18c">\|f\|_{F^{s_{1}}_{p_{1},q_{0}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f\|_{F% ^{s_{1}}_{p_{1},1}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f\|_{F^{s_{0}}_{p_% {0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex5a.m1.18d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; 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start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) ∩ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S2.SS4.3.p3"> <p class="ltx_p" id="S2.SS4.3.p3.8">The estimate for <math alttext="f\in F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)" class="ltx_Math" display="inline" id="S2.SS4.3.p3.1.m1.5"><semantics 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xref="S2.SS4.3.p3.1.m1.2.2.2.2.2.2.cmml">q</mi><mn id="S2.SS4.3.p3.1.m1.2.2.2.2.2.3" xref="S2.SS4.3.p3.1.m1.2.2.2.2.2.3.cmml">0</mn></msub></mrow><msub id="S2.SS4.3.p3.1.m1.5.5.2.4.2.3" xref="S2.SS4.3.p3.1.m1.5.5.2.4.2.3.cmml"><mi id="S2.SS4.3.p3.1.m1.5.5.2.4.2.3.2" xref="S2.SS4.3.p3.1.m1.5.5.2.4.2.3.2.cmml">s</mi><mn id="S2.SS4.3.p3.1.m1.5.5.2.4.2.3.3" xref="S2.SS4.3.p3.1.m1.5.5.2.4.2.3.3.cmml">0</mn></msub></msubsup><mo id="S2.SS4.3.p3.1.m1.5.5.2.3" xref="S2.SS4.3.p3.1.m1.5.5.2.3.cmml">⁢</mo><mrow id="S2.SS4.3.p3.1.m1.5.5.2.2.2" xref="S2.SS4.3.p3.1.m1.5.5.2.2.3.cmml"><mo id="S2.SS4.3.p3.1.m1.5.5.2.2.2.3" stretchy="false" xref="S2.SS4.3.p3.1.m1.5.5.2.2.3.cmml">(</mo><msup id="S2.SS4.3.p3.1.m1.4.4.1.1.1.1" xref="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.cmml"><mi id="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.2" xref="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.3" xref="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS4.3.p3.1.m1.5.5.2.2.2.4" xref="S2.SS4.3.p3.1.m1.5.5.2.2.3.cmml">,</mo><msub id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.cmml"><mi id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.2" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.2.cmml">w</mi><msub id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.cmml"><mi id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.2" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.2.cmml">γ</mi><mn id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.3" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.3.cmml">0</mn></msub></msub><mo id="S2.SS4.3.p3.1.m1.5.5.2.2.2.5" xref="S2.SS4.3.p3.1.m1.5.5.2.2.3.cmml">;</mo><mi id="S2.SS4.3.p3.1.m1.3.3" xref="S2.SS4.3.p3.1.m1.3.3.cmml">X</mi><mo id="S2.SS4.3.p3.1.m1.5.5.2.2.2.6" stretchy="false" xref="S2.SS4.3.p3.1.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.3.p3.1.m1.5b"><apply id="S2.SS4.3.p3.1.m1.5.5.cmml" xref="S2.SS4.3.p3.1.m1.5.5"><in id="S2.SS4.3.p3.1.m1.5.5.3.cmml" xref="S2.SS4.3.p3.1.m1.5.5.3"></in><ci 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id="S2.SS4.3.p3.1.m1.5.5.2.4.2.3.3.cmml" type="integer" xref="S2.SS4.3.p3.1.m1.5.5.2.4.2.3.3">0</cn></apply></apply><list id="S2.SS4.3.p3.1.m1.2.2.2.3.cmml" xref="S2.SS4.3.p3.1.m1.2.2.2.2"><apply id="S2.SS4.3.p3.1.m1.1.1.1.1.1.cmml" xref="S2.SS4.3.p3.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.3.p3.1.m1.1.1.1.1.1.1.cmml" xref="S2.SS4.3.p3.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS4.3.p3.1.m1.1.1.1.1.1.2.cmml" xref="S2.SS4.3.p3.1.m1.1.1.1.1.1.2">𝑝</ci><cn id="S2.SS4.3.p3.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS4.3.p3.1.m1.1.1.1.1.1.3">0</cn></apply><apply id="S2.SS4.3.p3.1.m1.2.2.2.2.2.cmml" xref="S2.SS4.3.p3.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.3.p3.1.m1.2.2.2.2.2.1.cmml" xref="S2.SS4.3.p3.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S2.SS4.3.p3.1.m1.2.2.2.2.2.2.cmml" xref="S2.SS4.3.p3.1.m1.2.2.2.2.2.2">𝑞</ci><cn id="S2.SS4.3.p3.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS4.3.p3.1.m1.2.2.2.2.2.3">0</cn></apply></list></apply><vector id="S2.SS4.3.p3.1.m1.5.5.2.2.3.cmml" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2"><apply id="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.cmml" xref="S2.SS4.3.p3.1.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.1.cmml" xref="S2.SS4.3.p3.1.m1.4.4.1.1.1.1">superscript</csymbol><ci id="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.2.cmml" xref="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.2">ℝ</ci><ci id="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.3.cmml" xref="S2.SS4.3.p3.1.m1.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.cmml" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.1.cmml" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2">subscript</csymbol><ci id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.2.cmml" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.2">𝑤</ci><apply id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.cmml" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3"><csymbol cd="ambiguous" id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.1.cmml" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3">subscript</csymbol><ci id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.2.cmml" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.2">𝛾</ci><cn id="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.3.cmml" type="integer" xref="S2.SS4.3.p3.1.m1.5.5.2.2.2.2.3.3">0</cn></apply></apply><ci id="S2.SS4.3.p3.1.m1.3.3.cmml" xref="S2.SS4.3.p3.1.m1.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.3.p3.1.m1.5c">f\in F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.3.p3.1.m1.5d">italic_f ∈ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> follows from the Fatou property of Triebel-Lizorkin spaces. We provide the argument for the convenience of the reader. 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id="S2.SS4.3.p3.3.m3.1.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1"><eq id="S2.SS4.3.p3.3.m3.1.1.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.1"></eq><apply id="S2.SS4.3.p3.3.m3.1.1.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.2"><csymbol cd="ambiguous" id="S2.SS4.3.p3.3.m3.1.1.2.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.2">subscript</csymbol><ci id="S2.SS4.3.p3.3.m3.1.1.2.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.2.2">𝑓</ci><ci id="S2.SS4.3.p3.3.m3.1.1.2.3.cmml" xref="S2.SS4.3.p3.3.m3.1.1.2.3">𝑁</ci></apply><apply id="S2.SS4.3.p3.3.m3.1.1.3.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3"><apply id="S2.SS4.3.p3.3.m3.1.1.3.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS4.3.p3.3.m3.1.1.3.1.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1">superscript</csymbol><apply id="S2.SS4.3.p3.3.m3.1.1.3.1.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S2.SS4.3.p3.3.m3.1.1.3.1.2.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1">subscript</csymbol><sum id="S2.SS4.3.p3.3.m3.1.1.3.1.2.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1.2.2"></sum><apply id="S2.SS4.3.p3.3.m3.1.1.3.1.2.3.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1.2.3"><eq id="S2.SS4.3.p3.3.m3.1.1.3.1.2.3.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1.2.3.1"></eq><ci id="S2.SS4.3.p3.3.m3.1.1.3.1.2.3.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1.2.3.2">𝑛</ci><cn id="S2.SS4.3.p3.3.m3.1.1.3.1.2.3.3.cmml" type="integer" xref="S2.SS4.3.p3.3.m3.1.1.3.1.2.3.3">0</cn></apply></apply><ci id="S2.SS4.3.p3.3.m3.1.1.3.1.3.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.1.3">𝑁</ci></apply><apply id="S2.SS4.3.p3.3.m3.1.1.3.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.2"><times id="S2.SS4.3.p3.3.m3.1.1.3.2.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.2.1"></times><apply id="S2.SS4.3.p3.3.m3.1.1.3.2.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.2.2"><csymbol cd="ambiguous" id="S2.SS4.3.p3.3.m3.1.1.3.2.2.1.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.2.2">subscript</csymbol><ci id="S2.SS4.3.p3.3.m3.1.1.3.2.2.2.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.2.2.2">𝑆</ci><ci id="S2.SS4.3.p3.3.m3.1.1.3.2.2.3.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.2.2.3">𝑛</ci></apply><ci id="S2.SS4.3.p3.3.m3.1.1.3.2.3.cmml" xref="S2.SS4.3.p3.3.m3.1.1.3.2.3">𝑓</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.3.p3.3.m3.1c">f_{N}=\sum_{n=0}^{N}S_{n}f</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.3.p3.3.m3.1d">italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_N end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f</annotation></semantics></math> converges to <math alttext="f" class="ltx_Math" display="inline" id="S2.SS4.3.p3.4.m4.1"><semantics id="S2.SS4.3.p3.4.m4.1a"><mi id="S2.SS4.3.p3.4.m4.1.1" xref="S2.SS4.3.p3.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS4.3.p3.4.m4.1b"><ci id="S2.SS4.3.p3.4.m4.1.1.cmml" xref="S2.SS4.3.p3.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.3.p3.4.m4.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.3.p3.4.m4.1d">italic_f</annotation></semantics></math> in <math alttext="\SS^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S2.SS4.3.p3.5.m5.2"><semantics id="S2.SS4.3.p3.5.m5.2a"><mrow id="S2.SS4.3.p3.5.m5.2.2" xref="S2.SS4.3.p3.5.m5.2.2.cmml"><msup id="S2.SS4.3.p3.5.m5.2.2.3" xref="S2.SS4.3.p3.5.m5.2.2.3.cmml"><mi id="S2.SS4.3.p3.5.m5.2.2.3.2" xref="S2.SS4.3.p3.5.m5.2.2.3.2.cmml">SS</mi><mo id="S2.SS4.3.p3.5.m5.2.2.3.3" xref="S2.SS4.3.p3.5.m5.2.2.3.3.cmml">′</mo></msup><mo id="S2.SS4.3.p3.5.m5.2.2.2" xref="S2.SS4.3.p3.5.m5.2.2.2.cmml">⁢</mo><mrow id="S2.SS4.3.p3.5.m5.2.2.1.1" xref="S2.SS4.3.p3.5.m5.2.2.1.2.cmml"><mo id="S2.SS4.3.p3.5.m5.2.2.1.1.2" stretchy="false" xref="S2.SS4.3.p3.5.m5.2.2.1.2.cmml">(</mo><msup id="S2.SS4.3.p3.5.m5.2.2.1.1.1" xref="S2.SS4.3.p3.5.m5.2.2.1.1.1.cmml"><mi id="S2.SS4.3.p3.5.m5.2.2.1.1.1.2" xref="S2.SS4.3.p3.5.m5.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S2.SS4.3.p3.5.m5.2.2.1.1.1.3" xref="S2.SS4.3.p3.5.m5.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S2.SS4.3.p3.5.m5.2.2.1.1.3" xref="S2.SS4.3.p3.5.m5.2.2.1.2.cmml">;</mo><mi id="S2.SS4.3.p3.5.m5.1.1" xref="S2.SS4.3.p3.5.m5.1.1.cmml">X</mi><mo id="S2.SS4.3.p3.5.m5.2.2.1.1.4" stretchy="false" xref="S2.SS4.3.p3.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.3.p3.5.m5.2b"><apply id="S2.SS4.3.p3.5.m5.2.2.cmml" xref="S2.SS4.3.p3.5.m5.2.2"><times id="S2.SS4.3.p3.5.m5.2.2.2.cmml" xref="S2.SS4.3.p3.5.m5.2.2.2"></times><apply id="S2.SS4.3.p3.5.m5.2.2.3.cmml" xref="S2.SS4.3.p3.5.m5.2.2.3"><csymbol cd="ambiguous" id="S2.SS4.3.p3.5.m5.2.2.3.1.cmml" xref="S2.SS4.3.p3.5.m5.2.2.3">superscript</csymbol><ci id="S2.SS4.3.p3.5.m5.2.2.3.2.cmml" xref="S2.SS4.3.p3.5.m5.2.2.3.2">SS</ci><ci id="S2.SS4.3.p3.5.m5.2.2.3.3.cmml" xref="S2.SS4.3.p3.5.m5.2.2.3.3">′</ci></apply><list id="S2.SS4.3.p3.5.m5.2.2.1.2.cmml" xref="S2.SS4.3.p3.5.m5.2.2.1.1"><apply id="S2.SS4.3.p3.5.m5.2.2.1.1.1.cmml" xref="S2.SS4.3.p3.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS4.3.p3.5.m5.2.2.1.1.1.1.cmml" xref="S2.SS4.3.p3.5.m5.2.2.1.1.1">superscript</csymbol><ci id="S2.SS4.3.p3.5.m5.2.2.1.1.1.2.cmml" xref="S2.SS4.3.p3.5.m5.2.2.1.1.1.2">ℝ</ci><ci id="S2.SS4.3.p3.5.m5.2.2.1.1.1.3.cmml" xref="S2.SS4.3.p3.5.m5.2.2.1.1.1.3">𝑑</ci></apply><ci id="S2.SS4.3.p3.5.m5.1.1.cmml" xref="S2.SS4.3.p3.5.m5.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.3.p3.5.m5.2c">\SS^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.3.p3.5.m5.2d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> as <math alttext="N\to\infty" class="ltx_Math" display="inline" id="S2.SS4.3.p3.6.m6.1"><semantics id="S2.SS4.3.p3.6.m6.1a"><mrow id="S2.SS4.3.p3.6.m6.1.1" xref="S2.SS4.3.p3.6.m6.1.1.cmml"><mi id="S2.SS4.3.p3.6.m6.1.1.2" xref="S2.SS4.3.p3.6.m6.1.1.2.cmml">N</mi><mo id="S2.SS4.3.p3.6.m6.1.1.1" stretchy="false" xref="S2.SS4.3.p3.6.m6.1.1.1.cmml">→</mo><mi id="S2.SS4.3.p3.6.m6.1.1.3" mathvariant="normal" xref="S2.SS4.3.p3.6.m6.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.3.p3.6.m6.1b"><apply id="S2.SS4.3.p3.6.m6.1.1.cmml" xref="S2.SS4.3.p3.6.m6.1.1"><ci id="S2.SS4.3.p3.6.m6.1.1.1.cmml" xref="S2.SS4.3.p3.6.m6.1.1.1">→</ci><ci id="S2.SS4.3.p3.6.m6.1.1.2.cmml" xref="S2.SS4.3.p3.6.m6.1.1.2">𝑁</ci><infinity id="S2.SS4.3.p3.6.m6.1.1.3.cmml" xref="S2.SS4.3.p3.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.3.p3.6.m6.1c">N\to\infty</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.3.p3.6.m6.1d">italic_N → ∞</annotation></semantics></math>. Note that for all <math alttext="N\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S2.SS4.3.p3.7.m7.1"><semantics id="S2.SS4.3.p3.7.m7.1a"><mrow id="S2.SS4.3.p3.7.m7.1.1" xref="S2.SS4.3.p3.7.m7.1.1.cmml"><mi id="S2.SS4.3.p3.7.m7.1.1.2" xref="S2.SS4.3.p3.7.m7.1.1.2.cmml">N</mi><mo id="S2.SS4.3.p3.7.m7.1.1.1" xref="S2.SS4.3.p3.7.m7.1.1.1.cmml">∈</mo><msub id="S2.SS4.3.p3.7.m7.1.1.3" xref="S2.SS4.3.p3.7.m7.1.1.3.cmml"><mi id="S2.SS4.3.p3.7.m7.1.1.3.2" xref="S2.SS4.3.p3.7.m7.1.1.3.2.cmml">ℕ</mi><mn id="S2.SS4.3.p3.7.m7.1.1.3.3" xref="S2.SS4.3.p3.7.m7.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS4.3.p3.7.m7.1b"><apply id="S2.SS4.3.p3.7.m7.1.1.cmml" xref="S2.SS4.3.p3.7.m7.1.1"><in id="S2.SS4.3.p3.7.m7.1.1.1.cmml" xref="S2.SS4.3.p3.7.m7.1.1.1"></in><ci id="S2.SS4.3.p3.7.m7.1.1.2.cmml" xref="S2.SS4.3.p3.7.m7.1.1.2">𝑁</ci><apply id="S2.SS4.3.p3.7.m7.1.1.3.cmml" xref="S2.SS4.3.p3.7.m7.1.1.3"><csymbol cd="ambiguous" id="S2.SS4.3.p3.7.m7.1.1.3.1.cmml" xref="S2.SS4.3.p3.7.m7.1.1.3">subscript</csymbol><ci id="S2.SS4.3.p3.7.m7.1.1.3.2.cmml" xref="S2.SS4.3.p3.7.m7.1.1.3.2">ℕ</ci><cn id="S2.SS4.3.p3.7.m7.1.1.3.3.cmml" type="integer" xref="S2.SS4.3.p3.7.m7.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.3.p3.7.m7.1c">N\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.3.p3.7.m7.1d">italic_N ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> we have <math alttext="f_{N}\in F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)\cap F^{s_{1}% }_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)" class="ltx_Math" display="inline" id="S2.SS4.3.p3.8.m8.10"><semantics id="S2.SS4.3.p3.8.m8.10a"><mrow id="S2.SS4.3.p3.8.m8.10.10" xref="S2.SS4.3.p3.8.m8.10.10.cmml"><msub id="S2.SS4.3.p3.8.m8.10.10.6" xref="S2.SS4.3.p3.8.m8.10.10.6.cmml"><mi id="S2.SS4.3.p3.8.m8.10.10.6.2" xref="S2.SS4.3.p3.8.m8.10.10.6.2.cmml">f</mi><mi id="S2.SS4.3.p3.8.m8.10.10.6.3" xref="S2.SS4.3.p3.8.m8.10.10.6.3.cmml">N</mi></msub><mo id="S2.SS4.3.p3.8.m8.10.10.5" xref="S2.SS4.3.p3.8.m8.10.10.5.cmml">∈</mo><mrow id="S2.SS4.3.p3.8.m8.10.10.4" xref="S2.SS4.3.p3.8.m8.10.10.4.cmml"><mrow id="S2.SS4.3.p3.8.m8.8.8.2.2" xref="S2.SS4.3.p3.8.m8.8.8.2.2.cmml"><msubsup id="S2.SS4.3.p3.8.m8.8.8.2.2.4" xref="S2.SS4.3.p3.8.m8.8.8.2.2.4.cmml"><mi id="S2.SS4.3.p3.8.m8.8.8.2.2.4.2.2" xref="S2.SS4.3.p3.8.m8.8.8.2.2.4.2.2.cmml">F</mi><mrow id="S2.SS4.3.p3.8.m8.2.2.2.2" xref="S2.SS4.3.p3.8.m8.2.2.2.3.cmml"><msub id="S2.SS4.3.p3.8.m8.1.1.1.1.1" xref="S2.SS4.3.p3.8.m8.1.1.1.1.1.cmml"><mi id="S2.SS4.3.p3.8.m8.1.1.1.1.1.2" xref="S2.SS4.3.p3.8.m8.1.1.1.1.1.2.cmml">p</mi><mn id="S2.SS4.3.p3.8.m8.1.1.1.1.1.3" xref="S2.SS4.3.p3.8.m8.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS4.3.p3.8.m8.2.2.2.2.3" xref="S2.SS4.3.p3.8.m8.2.2.2.3.cmml">,</mo><msub id="S2.SS4.3.p3.8.m8.2.2.2.2.2" 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xref="S2.SS4.3.p3.8.m8.10.10.4.4.2.2.2.3.3">1</cn></apply></apply><ci id="S2.SS4.3.p3.8.m8.6.6.cmml" xref="S2.SS4.3.p3.8.m8.6.6">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS4.3.p3.8.m8.10c">f_{N}\in F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)\cap F^{s_{1}% }_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)</annotation><annotation encoding="application/x-llamapun" id="S2.SS4.3.p3.8.m8.10d">italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ∈ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) ∩ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E7" title="In Proof. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.7</span></a>). From <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 2.4]</cite> we have that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex6a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f_{N}\|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\leq C\|f% \|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\quad\text{ for % all }N\in\mathbb{N}_{0}," class="ltx_Math" display="block" id="S2.Ex6a.m1.12"><semantics id="S2.Ex6a.m1.12a"><mrow id="S2.Ex6a.m1.12.12.1"><mrow id="S2.Ex6a.m1.12.12.1.1.2" xref="S2.Ex6a.m1.12.12.1.1.3.cmml"><mrow id="S2.Ex6a.m1.12.12.1.1.1.1" xref="S2.Ex6a.m1.12.12.1.1.1.1.cmml"><msub id="S2.Ex6a.m1.12.12.1.1.1.1.1" xref="S2.Ex6a.m1.12.12.1.1.1.1.1.cmml"><mrow id="S2.Ex6a.m1.12.12.1.1.1.1.1.1.1" 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id="S2.Ex6a.m1.12c">\|f_{N}\|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\leq C\|f% \|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\quad\text{ for % all }N\in\mathbb{N}_{0},</annotation><annotation encoding="application/x-llamapun" id="S2.Ex6a.m1.12d">∥ italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT for all italic_N ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.3.p3.10">from which it follows that</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex7a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f_{N}\|_{F^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f% \|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\quad\text{ for % all }N\in\mathbb{N}_{0}." class="ltx_Math" display="block" 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xref="S2.Ex7a.m1.12.12.1.1.2.2.3.2">ℕ</ci><cn id="S2.Ex7a.m1.12.12.1.1.2.2.3.3.cmml" type="integer" xref="S2.Ex7a.m1.12.12.1.1.2.2.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex7a.m1.12c">\|f_{N}\|_{F^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq C\|f% \|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}\quad\text{ for % all }N\in\mathbb{N}_{0}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex7a.m1.12d">∥ italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT for all italic_N ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.3.p3.9">Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.Thmtheorem2" title="Proposition 2.2. ‣ 2.3. Properties of Banach spaces ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.2</span></a> yields <math alttext="f\in F^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)" class="ltx_Math" display="inline" id="S2.SS4.3.p3.9.m1.5"><semantics id="S2.SS4.3.p3.9.m1.5a"><mrow id="S2.SS4.3.p3.9.m1.5.5" xref="S2.SS4.3.p3.9.m1.5.5.cmml"><mi id="S2.SS4.3.p3.9.m1.5.5.4" xref="S2.SS4.3.p3.9.m1.5.5.4.cmml">f</mi><mo id="S2.SS4.3.p3.9.m1.5.5.3" xref="S2.SS4.3.p3.9.m1.5.5.3.cmml">∈</mo><mrow id="S2.SS4.3.p3.9.m1.5.5.2" xref="S2.SS4.3.p3.9.m1.5.5.2.cmml"><msubsup id="S2.SS4.3.p3.9.m1.5.5.2.4" xref="S2.SS4.3.p3.9.m1.5.5.2.4.cmml"><mi id="S2.SS4.3.p3.9.m1.5.5.2.4.2.2" xref="S2.SS4.3.p3.9.m1.5.5.2.4.2.2.cmml">F</mi><mrow id="S2.SS4.3.p3.9.m1.2.2.2.2" xref="S2.SS4.3.p3.9.m1.2.2.2.3.cmml"><msub id="S2.SS4.3.p3.9.m1.1.1.1.1.1" xref="S2.SS4.3.p3.9.m1.1.1.1.1.1.cmml"><mi id="S2.SS4.3.p3.9.m1.1.1.1.1.1.2" xref="S2.SS4.3.p3.9.m1.1.1.1.1.1.2.cmml">p</mi><mn id="S2.SS4.3.p3.9.m1.1.1.1.1.1.3" xref="S2.SS4.3.p3.9.m1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.SS4.3.p3.9.m1.2.2.2.2.3" xref="S2.SS4.3.p3.9.m1.2.2.2.3.cmml">,</mo><msub id="S2.SS4.3.p3.9.m1.2.2.2.2.2" xref="S2.SS4.3.p3.9.m1.2.2.2.2.2.cmml"><mi id="S2.SS4.3.p3.9.m1.2.2.2.2.2.2" xref="S2.SS4.3.p3.9.m1.2.2.2.2.2.2.cmml">q</mi><mn id="S2.SS4.3.p3.9.m1.2.2.2.2.2.3" xref="S2.SS4.3.p3.9.m1.2.2.2.2.2.3.cmml">1</mn></msub></mrow><msub id="S2.SS4.3.p3.9.m1.5.5.2.4.2.3" xref="S2.SS4.3.p3.9.m1.5.5.2.4.2.3.cmml"><mi id="S2.SS4.3.p3.9.m1.5.5.2.4.2.3.2" xref="S2.SS4.3.p3.9.m1.5.5.2.4.2.3.2.cmml">s</mi><mn id="S2.SS4.3.p3.9.m1.5.5.2.4.2.3.3" xref="S2.SS4.3.p3.9.m1.5.5.2.4.2.3.3.cmml">1</mn></msub></msubsup><mo id="S2.SS4.3.p3.9.m1.5.5.2.3" xref="S2.SS4.3.p3.9.m1.5.5.2.3.cmml">⁢</mo><mrow id="S2.SS4.3.p3.9.m1.5.5.2.2.2" xref="S2.SS4.3.p3.9.m1.5.5.2.2.3.cmml"><mo id="S2.SS4.3.p3.9.m1.5.5.2.2.2.3" stretchy="false" xref="S2.SS4.3.p3.9.m1.5.5.2.2.3.cmml">(</mo><msup id="S2.SS4.3.p3.9.m1.4.4.1.1.1.1" 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end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex8a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f\|_{F^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq\liminf_{N% \to\infty}\|f_{N}\|_{F^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}% \leq C\|f\|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}." class="ltx_Math" display="block" id="S2.Ex8a.m1.18"><semantics id="S2.Ex8a.m1.18a"><mrow id="S2.Ex8a.m1.18.18.1" xref="S2.Ex8a.m1.18.18.1.1.cmml"><mrow id="S2.Ex8a.m1.18.18.1.1" xref="S2.Ex8a.m1.18.18.1.1.cmml"><msub id="S2.Ex8a.m1.18.18.1.1.3" xref="S2.Ex8a.m1.18.18.1.1.3.cmml"><mrow id="S2.Ex8a.m1.18.18.1.1.3.2.2" xref="S2.Ex8a.m1.18.18.1.1.3.2.1.cmml"><mo id="S2.Ex8a.m1.18.18.1.1.3.2.2.1" stretchy="false" xref="S2.Ex8a.m1.18.18.1.1.3.2.1.1.cmml">‖</mo><mi id="S2.Ex8a.m1.16.16" xref="S2.Ex8a.m1.16.16.cmml">f</mi><mo id="S2.Ex8a.m1.18.18.1.1.3.2.2.2" stretchy="false" xref="S2.Ex8a.m1.18.18.1.1.3.2.1.1.cmml">‖</mo></mrow><mrow id="S2.Ex8a.m1.5.5.5" xref="S2.Ex8a.m1.5.5.5.cmml"><msubsup id="S2.Ex8a.m1.5.5.5.7" xref="S2.Ex8a.m1.5.5.5.7.cmml"><mi id="S2.Ex8a.m1.5.5.5.7.2.2" xref="S2.Ex8a.m1.5.5.5.7.2.2.cmml">F</mi><mrow id="S2.Ex8a.m1.2.2.2.2.2.2" xref="S2.Ex8a.m1.2.2.2.2.2.3.cmml"><msub id="S2.Ex8a.m1.1.1.1.1.1.1.1" xref="S2.Ex8a.m1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex8a.m1.1.1.1.1.1.1.1.2" xref="S2.Ex8a.m1.1.1.1.1.1.1.1.2.cmml">p</mi><mn id="S2.Ex8a.m1.1.1.1.1.1.1.1.3" xref="S2.Ex8a.m1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S2.Ex8a.m1.2.2.2.2.2.2.3" xref="S2.Ex8a.m1.2.2.2.2.2.3.cmml">,</mo><msub 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xref="S2.Ex8a.m1.5.5.5.5.3.cmml">,</mo><msub id="S2.Ex8a.m1.5.5.5.5.2.2" xref="S2.Ex8a.m1.5.5.5.5.2.2.cmml"><mi id="S2.Ex8a.m1.5.5.5.5.2.2.2" xref="S2.Ex8a.m1.5.5.5.5.2.2.2.cmml">w</mi><msub id="S2.Ex8a.m1.5.5.5.5.2.2.3" xref="S2.Ex8a.m1.5.5.5.5.2.2.3.cmml"><mi id="S2.Ex8a.m1.5.5.5.5.2.2.3.2" xref="S2.Ex8a.m1.5.5.5.5.2.2.3.2.cmml">γ</mi><mn id="S2.Ex8a.m1.5.5.5.5.2.2.3.3" xref="S2.Ex8a.m1.5.5.5.5.2.2.3.3.cmml">1</mn></msub></msub><mo id="S2.Ex8a.m1.5.5.5.5.2.5" xref="S2.Ex8a.m1.5.5.5.5.3.cmml">;</mo><mi id="S2.Ex8a.m1.3.3.3.3" xref="S2.Ex8a.m1.3.3.3.3.cmml">X</mi><mo id="S2.Ex8a.m1.5.5.5.5.2.6" stretchy="false" xref="S2.Ex8a.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex8a.m1.18.18.1.1.4" rspace="0.1389em" xref="S2.Ex8a.m1.18.18.1.1.4.cmml">≤</mo><mrow id="S2.Ex8a.m1.18.18.1.1.1" xref="S2.Ex8a.m1.18.18.1.1.1.cmml"><munder id="S2.Ex8a.m1.18.18.1.1.1.2" xref="S2.Ex8a.m1.18.18.1.1.1.2.cmml"><mo id="S2.Ex8a.m1.18.18.1.1.1.2.2" lspace="0.1389em" movablelimits="false" rspace="0em" xref="S2.Ex8a.m1.18.18.1.1.1.2.2.cmml">lim inf</mo><mrow id="S2.Ex8a.m1.18.18.1.1.1.2.3" xref="S2.Ex8a.m1.18.18.1.1.1.2.3.cmml"><mi id="S2.Ex8a.m1.18.18.1.1.1.2.3.2" xref="S2.Ex8a.m1.18.18.1.1.1.2.3.2.cmml">N</mi><mo id="S2.Ex8a.m1.18.18.1.1.1.2.3.1" stretchy="false" xref="S2.Ex8a.m1.18.18.1.1.1.2.3.1.cmml">→</mo><mi id="S2.Ex8a.m1.18.18.1.1.1.2.3.3" mathvariant="normal" xref="S2.Ex8a.m1.18.18.1.1.1.2.3.3.cmml">∞</mi></mrow></munder><msub id="S2.Ex8a.m1.18.18.1.1.1.1" xref="S2.Ex8a.m1.18.18.1.1.1.1.cmml"><mrow id="S2.Ex8a.m1.18.18.1.1.1.1.1.1" xref="S2.Ex8a.m1.18.18.1.1.1.1.1.2.cmml"><mo id="S2.Ex8a.m1.18.18.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex8a.m1.18.18.1.1.1.1.1.2.1.cmml">‖</mo><msub id="S2.Ex8a.m1.18.18.1.1.1.1.1.1.1" xref="S2.Ex8a.m1.18.18.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex8a.m1.18.18.1.1.1.1.1.1.1.2" xref="S2.Ex8a.m1.18.18.1.1.1.1.1.1.1.2.cmml">f</mi><mi id="S2.Ex8a.m1.18.18.1.1.1.1.1.1.1.3" xref="S2.Ex8a.m1.18.18.1.1.1.1.1.1.1.3.cmml">N</mi></msub><mo 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xref="S2.Ex8a.m1.10.10.5.7.2.3.cmml"><mi id="S2.Ex8a.m1.10.10.5.7.2.3.2" xref="S2.Ex8a.m1.10.10.5.7.2.3.2.cmml">s</mi><mn id="S2.Ex8a.m1.10.10.5.7.2.3.3" xref="S2.Ex8a.m1.10.10.5.7.2.3.3.cmml">1</mn></msub></msubsup><mo id="S2.Ex8a.m1.10.10.5.6" xref="S2.Ex8a.m1.10.10.5.6.cmml">⁢</mo><mrow id="S2.Ex8a.m1.10.10.5.5.2" xref="S2.Ex8a.m1.10.10.5.5.3.cmml"><mo id="S2.Ex8a.m1.10.10.5.5.2.3" stretchy="false" xref="S2.Ex8a.m1.10.10.5.5.3.cmml">(</mo><msup id="S2.Ex8a.m1.9.9.4.4.1.1" xref="S2.Ex8a.m1.9.9.4.4.1.1.cmml"><mi id="S2.Ex8a.m1.9.9.4.4.1.1.2" xref="S2.Ex8a.m1.9.9.4.4.1.1.2.cmml">ℝ</mi><mi id="S2.Ex8a.m1.9.9.4.4.1.1.3" xref="S2.Ex8a.m1.9.9.4.4.1.1.3.cmml">d</mi></msup><mo id="S2.Ex8a.m1.10.10.5.5.2.4" xref="S2.Ex8a.m1.10.10.5.5.3.cmml">,</mo><msub id="S2.Ex8a.m1.10.10.5.5.2.2" xref="S2.Ex8a.m1.10.10.5.5.2.2.cmml"><mi id="S2.Ex8a.m1.10.10.5.5.2.2.2" xref="S2.Ex8a.m1.10.10.5.5.2.2.2.cmml">w</mi><msub id="S2.Ex8a.m1.10.10.5.5.2.2.3" xref="S2.Ex8a.m1.10.10.5.5.2.2.3.cmml"><mi 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xref="S2.Ex8a.m1.13.13.3.3">𝑋</ci></vector></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex8a.m1.18c">\|f\|_{F^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}\leq\liminf_{N% \to\infty}\|f_{N}\|_{F^{s_{1}}_{p_{1},q_{1}}(\mathbb{R}^{d},w_{\gamma_{1}};X)}% \leq C\|f\|_{F^{s_{0}}_{p_{0},q_{0}}(\mathbb{R}^{d},w_{\gamma_{0}};X)}.</annotation><annotation encoding="application/x-llamapun" id="S2.Ex8a.m1.18d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ lim inf start_POSTSUBSCRIPT italic_N → ∞ end_POSTSUBSCRIPT ∥ italic_f start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_q start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS4.3.p3.11">This completes the proof. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S2.SS5"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">2.5. </span>The complex interpolation method</h3> <div class="ltx_para" id="S2.SS5.p1"> <p class="ltx_p" id="S2.SS5.p1.1">We collect some basic properties of the complex interpolation method. For a detailed study of interpolation methods, we refer to, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib9" title="">9</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>]</cite>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S2.SS5.p2"> <p class="ltx_p" id="S2.SS5.p2.15">Let <math alttext="A\subseteq\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS5.p2.1.m1.1"><semantics id="S2.SS5.p2.1.m1.1a"><mrow id="S2.SS5.p2.1.m1.1.1" xref="S2.SS5.p2.1.m1.1.1.cmml"><mi id="S2.SS5.p2.1.m1.1.1.2" xref="S2.SS5.p2.1.m1.1.1.2.cmml">A</mi><mo id="S2.SS5.p2.1.m1.1.1.1" xref="S2.SS5.p2.1.m1.1.1.1.cmml">⊆</mo><mi id="S2.SS5.p2.1.m1.1.1.3" xref="S2.SS5.p2.1.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.1.m1.1b"><apply id="S2.SS5.p2.1.m1.1.1.cmml" xref="S2.SS5.p2.1.m1.1.1"><subset id="S2.SS5.p2.1.m1.1.1.1.cmml" xref="S2.SS5.p2.1.m1.1.1.1"></subset><ci id="S2.SS5.p2.1.m1.1.1.2.cmml" xref="S2.SS5.p2.1.m1.1.1.2">𝐴</ci><ci id="S2.SS5.p2.1.m1.1.1.3.cmml" xref="S2.SS5.p2.1.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.1.m1.1c">A\subseteq\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.1.m1.1d">italic_A ⊆ blackboard_R</annotation></semantics></math> be an interval. Let <math alttext="{\mathbb{S}}_{A}:=\{z\in\mathbb{C}:\operatorname{Re}z\in A\}" class="ltx_Math" display="inline" id="S2.SS5.p2.2.m2.2"><semantics id="S2.SS5.p2.2.m2.2a"><mrow id="S2.SS5.p2.2.m2.2.2" xref="S2.SS5.p2.2.m2.2.2.cmml"><msub id="S2.SS5.p2.2.m2.2.2.4" xref="S2.SS5.p2.2.m2.2.2.4.cmml"><mi id="S2.SS5.p2.2.m2.2.2.4.2" xref="S2.SS5.p2.2.m2.2.2.4.2.cmml">𝕊</mi><mi id="S2.SS5.p2.2.m2.2.2.4.3" xref="S2.SS5.p2.2.m2.2.2.4.3.cmml">A</mi></msub><mo id="S2.SS5.p2.2.m2.2.2.3" lspace="0.278em" rspace="0.278em" xref="S2.SS5.p2.2.m2.2.2.3.cmml">:=</mo><mrow id="S2.SS5.p2.2.m2.2.2.2.2" xref="S2.SS5.p2.2.m2.2.2.2.3.cmml"><mo id="S2.SS5.p2.2.m2.2.2.2.2.3" stretchy="false" xref="S2.SS5.p2.2.m2.2.2.2.3.1.cmml">{</mo><mrow id="S2.SS5.p2.2.m2.1.1.1.1.1" xref="S2.SS5.p2.2.m2.1.1.1.1.1.cmml"><mi id="S2.SS5.p2.2.m2.1.1.1.1.1.2" xref="S2.SS5.p2.2.m2.1.1.1.1.1.2.cmml">z</mi><mo id="S2.SS5.p2.2.m2.1.1.1.1.1.1" xref="S2.SS5.p2.2.m2.1.1.1.1.1.1.cmml">∈</mo><mi id="S2.SS5.p2.2.m2.1.1.1.1.1.3" xref="S2.SS5.p2.2.m2.1.1.1.1.1.3.cmml">ℂ</mi></mrow><mo id="S2.SS5.p2.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S2.SS5.p2.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S2.SS5.p2.2.m2.2.2.2.2.2" xref="S2.SS5.p2.2.m2.2.2.2.2.2.cmml"><mrow id="S2.SS5.p2.2.m2.2.2.2.2.2.2" xref="S2.SS5.p2.2.m2.2.2.2.2.2.2.cmml"><mi id="S2.SS5.p2.2.m2.2.2.2.2.2.2.1" xref="S2.SS5.p2.2.m2.2.2.2.2.2.2.1.cmml">Re</mi><mo id="S2.SS5.p2.2.m2.2.2.2.2.2.2a" lspace="0.167em" xref="S2.SS5.p2.2.m2.2.2.2.2.2.2.cmml">⁡</mo><mi id="S2.SS5.p2.2.m2.2.2.2.2.2.2.2" xref="S2.SS5.p2.2.m2.2.2.2.2.2.2.2.cmml">z</mi></mrow><mo id="S2.SS5.p2.2.m2.2.2.2.2.2.1" xref="S2.SS5.p2.2.m2.2.2.2.2.2.1.cmml">∈</mo><mi id="S2.SS5.p2.2.m2.2.2.2.2.2.3" xref="S2.SS5.p2.2.m2.2.2.2.2.2.3.cmml">A</mi></mrow><mo id="S2.SS5.p2.2.m2.2.2.2.2.5" stretchy="false" xref="S2.SS5.p2.2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.2.m2.2b"><apply id="S2.SS5.p2.2.m2.2.2.cmml" xref="S2.SS5.p2.2.m2.2.2"><csymbol cd="latexml" id="S2.SS5.p2.2.m2.2.2.3.cmml" xref="S2.SS5.p2.2.m2.2.2.3">assign</csymbol><apply id="S2.SS5.p2.2.m2.2.2.4.cmml" xref="S2.SS5.p2.2.m2.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.p2.2.m2.2.2.4.1.cmml" xref="S2.SS5.p2.2.m2.2.2.4">subscript</csymbol><ci id="S2.SS5.p2.2.m2.2.2.4.2.cmml" xref="S2.SS5.p2.2.m2.2.2.4.2">𝕊</ci><ci id="S2.SS5.p2.2.m2.2.2.4.3.cmml" xref="S2.SS5.p2.2.m2.2.2.4.3">𝐴</ci></apply><apply id="S2.SS5.p2.2.m2.2.2.2.3.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2"><csymbol cd="latexml" id="S2.SS5.p2.2.m2.2.2.2.3.1.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2.3">conditional-set</csymbol><apply id="S2.SS5.p2.2.m2.1.1.1.1.1.cmml" xref="S2.SS5.p2.2.m2.1.1.1.1.1"><in id="S2.SS5.p2.2.m2.1.1.1.1.1.1.cmml" xref="S2.SS5.p2.2.m2.1.1.1.1.1.1"></in><ci id="S2.SS5.p2.2.m2.1.1.1.1.1.2.cmml" xref="S2.SS5.p2.2.m2.1.1.1.1.1.2">𝑧</ci><ci id="S2.SS5.p2.2.m2.1.1.1.1.1.3.cmml" xref="S2.SS5.p2.2.m2.1.1.1.1.1.3">ℂ</ci></apply><apply id="S2.SS5.p2.2.m2.2.2.2.2.2.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2.2"><in id="S2.SS5.p2.2.m2.2.2.2.2.2.1.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2.2.1"></in><apply id="S2.SS5.p2.2.m2.2.2.2.2.2.2.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2.2.2"><ci id="S2.SS5.p2.2.m2.2.2.2.2.2.2.1.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2.2.2.1">Re</ci><ci id="S2.SS5.p2.2.m2.2.2.2.2.2.2.2.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2.2.2.2">𝑧</ci></apply><ci id="S2.SS5.p2.2.m2.2.2.2.2.2.3.cmml" xref="S2.SS5.p2.2.m2.2.2.2.2.2.3">𝐴</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.2.m2.2c">{\mathbb{S}}_{A}:=\{z\in\mathbb{C}:\operatorname{Re}z\in A\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.2.m2.2d">blackboard_S start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT := { italic_z ∈ blackboard_C : roman_Re italic_z ∈ italic_A }</annotation></semantics></math> be a strip in the complex plane and we write <math alttext="{\mathbb{S}}" class="ltx_Math" display="inline" id="S2.SS5.p2.3.m3.1"><semantics id="S2.SS5.p2.3.m3.1a"><mi id="S2.SS5.p2.3.m3.1.1" xref="S2.SS5.p2.3.m3.1.1.cmml">𝕊</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.3.m3.1b"><ci id="S2.SS5.p2.3.m3.1.1.cmml" xref="S2.SS5.p2.3.m3.1.1">𝕊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.3.m3.1c">{\mathbb{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.3.m3.1d">blackboard_S</annotation></semantics></math> if <math alttext="A=(0,1)" class="ltx_Math" display="inline" id="S2.SS5.p2.4.m4.2"><semantics id="S2.SS5.p2.4.m4.2a"><mrow id="S2.SS5.p2.4.m4.2.3" xref="S2.SS5.p2.4.m4.2.3.cmml"><mi id="S2.SS5.p2.4.m4.2.3.2" xref="S2.SS5.p2.4.m4.2.3.2.cmml">A</mi><mo id="S2.SS5.p2.4.m4.2.3.1" xref="S2.SS5.p2.4.m4.2.3.1.cmml">=</mo><mrow id="S2.SS5.p2.4.m4.2.3.3.2" xref="S2.SS5.p2.4.m4.2.3.3.1.cmml"><mo id="S2.SS5.p2.4.m4.2.3.3.2.1" stretchy="false" xref="S2.SS5.p2.4.m4.2.3.3.1.cmml">(</mo><mn id="S2.SS5.p2.4.m4.1.1" xref="S2.SS5.p2.4.m4.1.1.cmml">0</mn><mo id="S2.SS5.p2.4.m4.2.3.3.2.2" xref="S2.SS5.p2.4.m4.2.3.3.1.cmml">,</mo><mn id="S2.SS5.p2.4.m4.2.2" xref="S2.SS5.p2.4.m4.2.2.cmml">1</mn><mo id="S2.SS5.p2.4.m4.2.3.3.2.3" stretchy="false" xref="S2.SS5.p2.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.4.m4.2b"><apply id="S2.SS5.p2.4.m4.2.3.cmml" xref="S2.SS5.p2.4.m4.2.3"><eq id="S2.SS5.p2.4.m4.2.3.1.cmml" xref="S2.SS5.p2.4.m4.2.3.1"></eq><ci id="S2.SS5.p2.4.m4.2.3.2.cmml" xref="S2.SS5.p2.4.m4.2.3.2">𝐴</ci><interval closure="open" id="S2.SS5.p2.4.m4.2.3.3.1.cmml" xref="S2.SS5.p2.4.m4.2.3.3.2"><cn id="S2.SS5.p2.4.m4.1.1.cmml" type="integer" xref="S2.SS5.p2.4.m4.1.1">0</cn><cn id="S2.SS5.p2.4.m4.2.2.cmml" type="integer" xref="S2.SS5.p2.4.m4.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.4.m4.2c">A=(0,1)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.4.m4.2d">italic_A = ( 0 , 1 )</annotation></semantics></math>. Let <math alttext="(X_{0},X_{1})" class="ltx_Math" display="inline" id="S2.SS5.p2.5.m5.2"><semantics id="S2.SS5.p2.5.m5.2a"><mrow id="S2.SS5.p2.5.m5.2.2.2" xref="S2.SS5.p2.5.m5.2.2.3.cmml"><mo id="S2.SS5.p2.5.m5.2.2.2.3" stretchy="false" xref="S2.SS5.p2.5.m5.2.2.3.cmml">(</mo><msub id="S2.SS5.p2.5.m5.1.1.1.1" xref="S2.SS5.p2.5.m5.1.1.1.1.cmml"><mi id="S2.SS5.p2.5.m5.1.1.1.1.2" xref="S2.SS5.p2.5.m5.1.1.1.1.2.cmml">X</mi><mn id="S2.SS5.p2.5.m5.1.1.1.1.3" xref="S2.SS5.p2.5.m5.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS5.p2.5.m5.2.2.2.4" xref="S2.SS5.p2.5.m5.2.2.3.cmml">,</mo><msub id="S2.SS5.p2.5.m5.2.2.2.2" xref="S2.SS5.p2.5.m5.2.2.2.2.cmml"><mi id="S2.SS5.p2.5.m5.2.2.2.2.2" xref="S2.SS5.p2.5.m5.2.2.2.2.2.cmml">X</mi><mn id="S2.SS5.p2.5.m5.2.2.2.2.3" xref="S2.SS5.p2.5.m5.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS5.p2.5.m5.2.2.2.5" stretchy="false" xref="S2.SS5.p2.5.m5.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.5.m5.2b"><interval closure="open" id="S2.SS5.p2.5.m5.2.2.3.cmml" xref="S2.SS5.p2.5.m5.2.2.2"><apply id="S2.SS5.p2.5.m5.1.1.1.1.cmml" xref="S2.SS5.p2.5.m5.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p2.5.m5.1.1.1.1.1.cmml" xref="S2.SS5.p2.5.m5.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p2.5.m5.1.1.1.1.2.cmml" xref="S2.SS5.p2.5.m5.1.1.1.1.2">𝑋</ci><cn id="S2.SS5.p2.5.m5.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p2.5.m5.1.1.1.1.3">0</cn></apply><apply id="S2.SS5.p2.5.m5.2.2.2.2.cmml" xref="S2.SS5.p2.5.m5.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.p2.5.m5.2.2.2.2.1.cmml" xref="S2.SS5.p2.5.m5.2.2.2.2">subscript</csymbol><ci id="S2.SS5.p2.5.m5.2.2.2.2.2.cmml" xref="S2.SS5.p2.5.m5.2.2.2.2.2">𝑋</ci><cn id="S2.SS5.p2.5.m5.2.2.2.2.3.cmml" type="integer" xref="S2.SS5.p2.5.m5.2.2.2.2.3">1</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.5.m5.2c">(X_{0},X_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.5.m5.2d">( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> be an interpolation couple of complex Banach spaces, i.e., a pair of Banach spaces which are continuously embedded in a linear Hausdorff space <math alttext="\mathcal{X}" class="ltx_Math" display="inline" id="S2.SS5.p2.6.m6.1"><semantics id="S2.SS5.p2.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S2.SS5.p2.6.m6.1.1" xref="S2.SS5.p2.6.m6.1.1.cmml">𝒳</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.6.m6.1b"><ci id="S2.SS5.p2.6.m6.1.1.cmml" xref="S2.SS5.p2.6.m6.1.1">𝒳</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.6.m6.1c">\mathcal{X}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.6.m6.1d">caligraphic_X</annotation></semantics></math>. We denote by <math alttext="\mathscr{H}(X_{0},X_{1})" class="ltx_Math" display="inline" id="S2.SS5.p2.7.m7.2"><semantics id="S2.SS5.p2.7.m7.2a"><mrow id="S2.SS5.p2.7.m7.2.2" xref="S2.SS5.p2.7.m7.2.2.cmml"><mi class="ltx_font_mathscript" id="S2.SS5.p2.7.m7.2.2.4" xref="S2.SS5.p2.7.m7.2.2.4.cmml">ℋ</mi><mo id="S2.SS5.p2.7.m7.2.2.3" xref="S2.SS5.p2.7.m7.2.2.3.cmml">⁢</mo><mrow id="S2.SS5.p2.7.m7.2.2.2.2" xref="S2.SS5.p2.7.m7.2.2.2.3.cmml"><mo id="S2.SS5.p2.7.m7.2.2.2.2.3" stretchy="false" xref="S2.SS5.p2.7.m7.2.2.2.3.cmml">(</mo><msub id="S2.SS5.p2.7.m7.1.1.1.1.1" xref="S2.SS5.p2.7.m7.1.1.1.1.1.cmml"><mi id="S2.SS5.p2.7.m7.1.1.1.1.1.2" xref="S2.SS5.p2.7.m7.1.1.1.1.1.2.cmml">X</mi><mn id="S2.SS5.p2.7.m7.1.1.1.1.1.3" xref="S2.SS5.p2.7.m7.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS5.p2.7.m7.2.2.2.2.4" xref="S2.SS5.p2.7.m7.2.2.2.3.cmml">,</mo><msub id="S2.SS5.p2.7.m7.2.2.2.2.2" xref="S2.SS5.p2.7.m7.2.2.2.2.2.cmml"><mi id="S2.SS5.p2.7.m7.2.2.2.2.2.2" xref="S2.SS5.p2.7.m7.2.2.2.2.2.2.cmml">X</mi><mn id="S2.SS5.p2.7.m7.2.2.2.2.2.3" xref="S2.SS5.p2.7.m7.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS5.p2.7.m7.2.2.2.2.5" stretchy="false" xref="S2.SS5.p2.7.m7.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.7.m7.2b"><apply id="S2.SS5.p2.7.m7.2.2.cmml" xref="S2.SS5.p2.7.m7.2.2"><times id="S2.SS5.p2.7.m7.2.2.3.cmml" xref="S2.SS5.p2.7.m7.2.2.3"></times><ci id="S2.SS5.p2.7.m7.2.2.4.cmml" xref="S2.SS5.p2.7.m7.2.2.4">ℋ</ci><interval closure="open" id="S2.SS5.p2.7.m7.2.2.2.3.cmml" xref="S2.SS5.p2.7.m7.2.2.2.2"><apply id="S2.SS5.p2.7.m7.1.1.1.1.1.cmml" xref="S2.SS5.p2.7.m7.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p2.7.m7.1.1.1.1.1.1.cmml" xref="S2.SS5.p2.7.m7.1.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p2.7.m7.1.1.1.1.1.2.cmml" xref="S2.SS5.p2.7.m7.1.1.1.1.1.2">𝑋</ci><cn id="S2.SS5.p2.7.m7.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p2.7.m7.1.1.1.1.1.3">0</cn></apply><apply id="S2.SS5.p2.7.m7.2.2.2.2.2.cmml" xref="S2.SS5.p2.7.m7.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.p2.7.m7.2.2.2.2.2.1.cmml" xref="S2.SS5.p2.7.m7.2.2.2.2.2">subscript</csymbol><ci id="S2.SS5.p2.7.m7.2.2.2.2.2.2.cmml" xref="S2.SS5.p2.7.m7.2.2.2.2.2.2">𝑋</ci><cn id="S2.SS5.p2.7.m7.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS5.p2.7.m7.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.7.m7.2c">\mathscr{H}(X_{0},X_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.7.m7.2d">script_H ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> the complex vector space of all continuous functions <math alttext="f:\overline{{\mathbb{S}}}\to X_{0}+X_{1}" class="ltx_Math" display="inline" id="S2.SS5.p2.8.m8.1"><semantics id="S2.SS5.p2.8.m8.1a"><mrow id="S2.SS5.p2.8.m8.1.1" xref="S2.SS5.p2.8.m8.1.1.cmml"><mi id="S2.SS5.p2.8.m8.1.1.2" xref="S2.SS5.p2.8.m8.1.1.2.cmml">f</mi><mo id="S2.SS5.p2.8.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S2.SS5.p2.8.m8.1.1.1.cmml">:</mo><mrow id="S2.SS5.p2.8.m8.1.1.3" xref="S2.SS5.p2.8.m8.1.1.3.cmml"><mover accent="true" id="S2.SS5.p2.8.m8.1.1.3.2" xref="S2.SS5.p2.8.m8.1.1.3.2.cmml"><mi id="S2.SS5.p2.8.m8.1.1.3.2.2" xref="S2.SS5.p2.8.m8.1.1.3.2.2.cmml">𝕊</mi><mo id="S2.SS5.p2.8.m8.1.1.3.2.1" xref="S2.SS5.p2.8.m8.1.1.3.2.1.cmml">¯</mo></mover><mo id="S2.SS5.p2.8.m8.1.1.3.1" stretchy="false" xref="S2.SS5.p2.8.m8.1.1.3.1.cmml">→</mo><mrow id="S2.SS5.p2.8.m8.1.1.3.3" xref="S2.SS5.p2.8.m8.1.1.3.3.cmml"><msub id="S2.SS5.p2.8.m8.1.1.3.3.2" xref="S2.SS5.p2.8.m8.1.1.3.3.2.cmml"><mi id="S2.SS5.p2.8.m8.1.1.3.3.2.2" xref="S2.SS5.p2.8.m8.1.1.3.3.2.2.cmml">X</mi><mn id="S2.SS5.p2.8.m8.1.1.3.3.2.3" xref="S2.SS5.p2.8.m8.1.1.3.3.2.3.cmml">0</mn></msub><mo id="S2.SS5.p2.8.m8.1.1.3.3.1" xref="S2.SS5.p2.8.m8.1.1.3.3.1.cmml">+</mo><msub id="S2.SS5.p2.8.m8.1.1.3.3.3" xref="S2.SS5.p2.8.m8.1.1.3.3.3.cmml"><mi id="S2.SS5.p2.8.m8.1.1.3.3.3.2" xref="S2.SS5.p2.8.m8.1.1.3.3.3.2.cmml">X</mi><mn id="S2.SS5.p2.8.m8.1.1.3.3.3.3" xref="S2.SS5.p2.8.m8.1.1.3.3.3.3.cmml">1</mn></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.8.m8.1b"><apply id="S2.SS5.p2.8.m8.1.1.cmml" xref="S2.SS5.p2.8.m8.1.1"><ci id="S2.SS5.p2.8.m8.1.1.1.cmml" xref="S2.SS5.p2.8.m8.1.1.1">:</ci><ci id="S2.SS5.p2.8.m8.1.1.2.cmml" xref="S2.SS5.p2.8.m8.1.1.2">𝑓</ci><apply id="S2.SS5.p2.8.m8.1.1.3.cmml" xref="S2.SS5.p2.8.m8.1.1.3"><ci id="S2.SS5.p2.8.m8.1.1.3.1.cmml" xref="S2.SS5.p2.8.m8.1.1.3.1">→</ci><apply id="S2.SS5.p2.8.m8.1.1.3.2.cmml" xref="S2.SS5.p2.8.m8.1.1.3.2"><ci id="S2.SS5.p2.8.m8.1.1.3.2.1.cmml" xref="S2.SS5.p2.8.m8.1.1.3.2.1">¯</ci><ci id="S2.SS5.p2.8.m8.1.1.3.2.2.cmml" xref="S2.SS5.p2.8.m8.1.1.3.2.2">𝕊</ci></apply><apply id="S2.SS5.p2.8.m8.1.1.3.3.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3"><plus id="S2.SS5.p2.8.m8.1.1.3.3.1.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3.1"></plus><apply id="S2.SS5.p2.8.m8.1.1.3.3.2.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3.2"><csymbol cd="ambiguous" id="S2.SS5.p2.8.m8.1.1.3.3.2.1.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3.2">subscript</csymbol><ci id="S2.SS5.p2.8.m8.1.1.3.3.2.2.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3.2.2">𝑋</ci><cn id="S2.SS5.p2.8.m8.1.1.3.3.2.3.cmml" type="integer" xref="S2.SS5.p2.8.m8.1.1.3.3.2.3">0</cn></apply><apply id="S2.SS5.p2.8.m8.1.1.3.3.3.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3.3"><csymbol cd="ambiguous" id="S2.SS5.p2.8.m8.1.1.3.3.3.1.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3.3">subscript</csymbol><ci id="S2.SS5.p2.8.m8.1.1.3.3.3.2.cmml" xref="S2.SS5.p2.8.m8.1.1.3.3.3.2">𝑋</ci><cn id="S2.SS5.p2.8.m8.1.1.3.3.3.3.cmml" type="integer" xref="S2.SS5.p2.8.m8.1.1.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.8.m8.1c">f:\overline{{\mathbb{S}}}\to X_{0}+X_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.8.m8.1d">italic_f : over¯ start_ARG blackboard_S end_ARG → italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="f" class="ltx_Math" display="inline" id="S2.SS5.p2.9.m9.1"><semantics id="S2.SS5.p2.9.m9.1a"><mi id="S2.SS5.p2.9.m9.1.1" xref="S2.SS5.p2.9.m9.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.9.m9.1b"><ci id="S2.SS5.p2.9.m9.1.1.cmml" xref="S2.SS5.p2.9.m9.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.9.m9.1c">f</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.9.m9.1d">italic_f</annotation></semantics></math> is holomorphic on <math alttext="{\mathbb{S}}" class="ltx_Math" display="inline" id="S2.SS5.p2.10.m10.1"><semantics id="S2.SS5.p2.10.m10.1a"><mi id="S2.SS5.p2.10.m10.1.1" xref="S2.SS5.p2.10.m10.1.1.cmml">𝕊</mi><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.10.m10.1b"><ci id="S2.SS5.p2.10.m10.1.1.cmml" xref="S2.SS5.p2.10.m10.1.1">𝕊</ci></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.10.m10.1c">{\mathbb{S}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.10.m10.1d">blackboard_S</annotation></semantics></math> with values in <math alttext="X_{0}+X_{1}" class="ltx_Math" display="inline" id="S2.SS5.p2.11.m11.1"><semantics id="S2.SS5.p2.11.m11.1a"><mrow id="S2.SS5.p2.11.m11.1.1" xref="S2.SS5.p2.11.m11.1.1.cmml"><msub id="S2.SS5.p2.11.m11.1.1.2" xref="S2.SS5.p2.11.m11.1.1.2.cmml"><mi id="S2.SS5.p2.11.m11.1.1.2.2" xref="S2.SS5.p2.11.m11.1.1.2.2.cmml">X</mi><mn id="S2.SS5.p2.11.m11.1.1.2.3" xref="S2.SS5.p2.11.m11.1.1.2.3.cmml">0</mn></msub><mo id="S2.SS5.p2.11.m11.1.1.1" xref="S2.SS5.p2.11.m11.1.1.1.cmml">+</mo><msub id="S2.SS5.p2.11.m11.1.1.3" xref="S2.SS5.p2.11.m11.1.1.3.cmml"><mi id="S2.SS5.p2.11.m11.1.1.3.2" xref="S2.SS5.p2.11.m11.1.1.3.2.cmml">X</mi><mn id="S2.SS5.p2.11.m11.1.1.3.3" xref="S2.SS5.p2.11.m11.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.11.m11.1b"><apply id="S2.SS5.p2.11.m11.1.1.cmml" xref="S2.SS5.p2.11.m11.1.1"><plus id="S2.SS5.p2.11.m11.1.1.1.cmml" xref="S2.SS5.p2.11.m11.1.1.1"></plus><apply id="S2.SS5.p2.11.m11.1.1.2.cmml" xref="S2.SS5.p2.11.m11.1.1.2"><csymbol cd="ambiguous" id="S2.SS5.p2.11.m11.1.1.2.1.cmml" xref="S2.SS5.p2.11.m11.1.1.2">subscript</csymbol><ci id="S2.SS5.p2.11.m11.1.1.2.2.cmml" xref="S2.SS5.p2.11.m11.1.1.2.2">𝑋</ci><cn id="S2.SS5.p2.11.m11.1.1.2.3.cmml" type="integer" xref="S2.SS5.p2.11.m11.1.1.2.3">0</cn></apply><apply id="S2.SS5.p2.11.m11.1.1.3.cmml" xref="S2.SS5.p2.11.m11.1.1.3"><csymbol cd="ambiguous" id="S2.SS5.p2.11.m11.1.1.3.1.cmml" xref="S2.SS5.p2.11.m11.1.1.3">subscript</csymbol><ci id="S2.SS5.p2.11.m11.1.1.3.2.cmml" xref="S2.SS5.p2.11.m11.1.1.3.2">𝑋</ci><cn id="S2.SS5.p2.11.m11.1.1.3.3.cmml" type="integer" xref="S2.SS5.p2.11.m11.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.11.m11.1c">X_{0}+X_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.11.m11.1d">italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and the mapping <math alttext="t\mapsto f(j+{\rm i}t)" class="ltx_Math" display="inline" id="S2.SS5.p2.12.m12.1"><semantics id="S2.SS5.p2.12.m12.1a"><mrow id="S2.SS5.p2.12.m12.1.1" xref="S2.SS5.p2.12.m12.1.1.cmml"><mi id="S2.SS5.p2.12.m12.1.1.3" xref="S2.SS5.p2.12.m12.1.1.3.cmml">t</mi><mo id="S2.SS5.p2.12.m12.1.1.2" stretchy="false" xref="S2.SS5.p2.12.m12.1.1.2.cmml">↦</mo><mrow id="S2.SS5.p2.12.m12.1.1.1" xref="S2.SS5.p2.12.m12.1.1.1.cmml"><mi id="S2.SS5.p2.12.m12.1.1.1.3" xref="S2.SS5.p2.12.m12.1.1.1.3.cmml">f</mi><mo id="S2.SS5.p2.12.m12.1.1.1.2" xref="S2.SS5.p2.12.m12.1.1.1.2.cmml">⁢</mo><mrow id="S2.SS5.p2.12.m12.1.1.1.1.1" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.cmml"><mo id="S2.SS5.p2.12.m12.1.1.1.1.1.2" stretchy="false" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.SS5.p2.12.m12.1.1.1.1.1.1" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.cmml"><mi id="S2.SS5.p2.12.m12.1.1.1.1.1.1.2" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.2.cmml">j</mi><mo id="S2.SS5.p2.12.m12.1.1.1.1.1.1.1" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.cmml"><mi id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.1" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.3" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></mrow><mo id="S2.SS5.p2.12.m12.1.1.1.1.1.3" stretchy="false" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.12.m12.1b"><apply id="S2.SS5.p2.12.m12.1.1.cmml" xref="S2.SS5.p2.12.m12.1.1"><csymbol cd="latexml" id="S2.SS5.p2.12.m12.1.1.2.cmml" xref="S2.SS5.p2.12.m12.1.1.2">maps-to</csymbol><ci id="S2.SS5.p2.12.m12.1.1.3.cmml" xref="S2.SS5.p2.12.m12.1.1.3">𝑡</ci><apply id="S2.SS5.p2.12.m12.1.1.1.cmml" xref="S2.SS5.p2.12.m12.1.1.1"><times id="S2.SS5.p2.12.m12.1.1.1.2.cmml" xref="S2.SS5.p2.12.m12.1.1.1.2"></times><ci id="S2.SS5.p2.12.m12.1.1.1.3.cmml" xref="S2.SS5.p2.12.m12.1.1.1.3">𝑓</ci><apply id="S2.SS5.p2.12.m12.1.1.1.1.1.1.cmml" xref="S2.SS5.p2.12.m12.1.1.1.1.1"><plus id="S2.SS5.p2.12.m12.1.1.1.1.1.1.1.cmml" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.1"></plus><ci id="S2.SS5.p2.12.m12.1.1.1.1.1.1.2.cmml" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.2">𝑗</ci><apply id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.cmml" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3"><times id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.1.cmml" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.1"></times><ci id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.2.cmml" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.2">i</ci><ci id="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.3.cmml" xref="S2.SS5.p2.12.m12.1.1.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.12.m12.1c">t\mapsto f(j+{\rm i}t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.12.m12.1d">italic_t ↦ italic_f ( italic_j + roman_i italic_t )</annotation></semantics></math> belongs to <math alttext="C_{{\rm b}}(\mathbb{R};X_{j})" class="ltx_Math" display="inline" id="S2.SS5.p2.13.m13.2"><semantics id="S2.SS5.p2.13.m13.2a"><mrow id="S2.SS5.p2.13.m13.2.2" xref="S2.SS5.p2.13.m13.2.2.cmml"><msub id="S2.SS5.p2.13.m13.2.2.3" xref="S2.SS5.p2.13.m13.2.2.3.cmml"><mi id="S2.SS5.p2.13.m13.2.2.3.2" xref="S2.SS5.p2.13.m13.2.2.3.2.cmml">C</mi><mi id="S2.SS5.p2.13.m13.2.2.3.3" mathvariant="normal" xref="S2.SS5.p2.13.m13.2.2.3.3.cmml">b</mi></msub><mo id="S2.SS5.p2.13.m13.2.2.2" xref="S2.SS5.p2.13.m13.2.2.2.cmml">⁢</mo><mrow id="S2.SS5.p2.13.m13.2.2.1.1" xref="S2.SS5.p2.13.m13.2.2.1.2.cmml"><mo id="S2.SS5.p2.13.m13.2.2.1.1.2" stretchy="false" xref="S2.SS5.p2.13.m13.2.2.1.2.cmml">(</mo><mi id="S2.SS5.p2.13.m13.1.1" xref="S2.SS5.p2.13.m13.1.1.cmml">ℝ</mi><mo id="S2.SS5.p2.13.m13.2.2.1.1.3" xref="S2.SS5.p2.13.m13.2.2.1.2.cmml">;</mo><msub id="S2.SS5.p2.13.m13.2.2.1.1.1" xref="S2.SS5.p2.13.m13.2.2.1.1.1.cmml"><mi id="S2.SS5.p2.13.m13.2.2.1.1.1.2" xref="S2.SS5.p2.13.m13.2.2.1.1.1.2.cmml">X</mi><mi id="S2.SS5.p2.13.m13.2.2.1.1.1.3" xref="S2.SS5.p2.13.m13.2.2.1.1.1.3.cmml">j</mi></msub><mo id="S2.SS5.p2.13.m13.2.2.1.1.4" stretchy="false" xref="S2.SS5.p2.13.m13.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.13.m13.2b"><apply id="S2.SS5.p2.13.m13.2.2.cmml" xref="S2.SS5.p2.13.m13.2.2"><times id="S2.SS5.p2.13.m13.2.2.2.cmml" xref="S2.SS5.p2.13.m13.2.2.2"></times><apply id="S2.SS5.p2.13.m13.2.2.3.cmml" xref="S2.SS5.p2.13.m13.2.2.3"><csymbol cd="ambiguous" id="S2.SS5.p2.13.m13.2.2.3.1.cmml" xref="S2.SS5.p2.13.m13.2.2.3">subscript</csymbol><ci id="S2.SS5.p2.13.m13.2.2.3.2.cmml" xref="S2.SS5.p2.13.m13.2.2.3.2">𝐶</ci><ci id="S2.SS5.p2.13.m13.2.2.3.3.cmml" xref="S2.SS5.p2.13.m13.2.2.3.3">b</ci></apply><list id="S2.SS5.p2.13.m13.2.2.1.2.cmml" xref="S2.SS5.p2.13.m13.2.2.1.1"><ci id="S2.SS5.p2.13.m13.1.1.cmml" xref="S2.SS5.p2.13.m13.1.1">ℝ</ci><apply id="S2.SS5.p2.13.m13.2.2.1.1.1.cmml" xref="S2.SS5.p2.13.m13.2.2.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p2.13.m13.2.2.1.1.1.1.cmml" xref="S2.SS5.p2.13.m13.2.2.1.1.1">subscript</csymbol><ci id="S2.SS5.p2.13.m13.2.2.1.1.1.2.cmml" xref="S2.SS5.p2.13.m13.2.2.1.1.1.2">𝑋</ci><ci id="S2.SS5.p2.13.m13.2.2.1.1.1.3.cmml" xref="S2.SS5.p2.13.m13.2.2.1.1.1.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.13.m13.2c">C_{{\rm b}}(\mathbb{R};X_{j})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.13.m13.2d">italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT ( blackboard_R ; italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="j\in\{0,1\}" class="ltx_Math" display="inline" id="S2.SS5.p2.14.m14.2"><semantics id="S2.SS5.p2.14.m14.2a"><mrow id="S2.SS5.p2.14.m14.2.3" xref="S2.SS5.p2.14.m14.2.3.cmml"><mi id="S2.SS5.p2.14.m14.2.3.2" xref="S2.SS5.p2.14.m14.2.3.2.cmml">j</mi><mo id="S2.SS5.p2.14.m14.2.3.1" xref="S2.SS5.p2.14.m14.2.3.1.cmml">∈</mo><mrow id="S2.SS5.p2.14.m14.2.3.3.2" xref="S2.SS5.p2.14.m14.2.3.3.1.cmml"><mo id="S2.SS5.p2.14.m14.2.3.3.2.1" stretchy="false" xref="S2.SS5.p2.14.m14.2.3.3.1.cmml">{</mo><mn id="S2.SS5.p2.14.m14.1.1" xref="S2.SS5.p2.14.m14.1.1.cmml">0</mn><mo id="S2.SS5.p2.14.m14.2.3.3.2.2" xref="S2.SS5.p2.14.m14.2.3.3.1.cmml">,</mo><mn id="S2.SS5.p2.14.m14.2.2" xref="S2.SS5.p2.14.m14.2.2.cmml">1</mn><mo id="S2.SS5.p2.14.m14.2.3.3.2.3" stretchy="false" xref="S2.SS5.p2.14.m14.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.14.m14.2b"><apply id="S2.SS5.p2.14.m14.2.3.cmml" xref="S2.SS5.p2.14.m14.2.3"><in id="S2.SS5.p2.14.m14.2.3.1.cmml" xref="S2.SS5.p2.14.m14.2.3.1"></in><ci id="S2.SS5.p2.14.m14.2.3.2.cmml" xref="S2.SS5.p2.14.m14.2.3.2">𝑗</ci><set id="S2.SS5.p2.14.m14.2.3.3.1.cmml" xref="S2.SS5.p2.14.m14.2.3.3.2"><cn id="S2.SS5.p2.14.m14.1.1.cmml" type="integer" xref="S2.SS5.p2.14.m14.1.1">0</cn><cn id="S2.SS5.p2.14.m14.2.2.cmml" type="integer" xref="S2.SS5.p2.14.m14.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.14.m14.2c">j\in\{0,1\}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.14.m14.2d">italic_j ∈ { 0 , 1 }</annotation></semantics></math>. The space <math alttext="\mathscr{H}(X_{0},X_{1})" class="ltx_Math" display="inline" id="S2.SS5.p2.15.m15.2"><semantics id="S2.SS5.p2.15.m15.2a"><mrow id="S2.SS5.p2.15.m15.2.2" xref="S2.SS5.p2.15.m15.2.2.cmml"><mi class="ltx_font_mathscript" id="S2.SS5.p2.15.m15.2.2.4" xref="S2.SS5.p2.15.m15.2.2.4.cmml">ℋ</mi><mo id="S2.SS5.p2.15.m15.2.2.3" xref="S2.SS5.p2.15.m15.2.2.3.cmml">⁢</mo><mrow id="S2.SS5.p2.15.m15.2.2.2.2" xref="S2.SS5.p2.15.m15.2.2.2.3.cmml"><mo id="S2.SS5.p2.15.m15.2.2.2.2.3" stretchy="false" xref="S2.SS5.p2.15.m15.2.2.2.3.cmml">(</mo><msub id="S2.SS5.p2.15.m15.1.1.1.1.1" xref="S2.SS5.p2.15.m15.1.1.1.1.1.cmml"><mi id="S2.SS5.p2.15.m15.1.1.1.1.1.2" xref="S2.SS5.p2.15.m15.1.1.1.1.1.2.cmml">X</mi><mn id="S2.SS5.p2.15.m15.1.1.1.1.1.3" xref="S2.SS5.p2.15.m15.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS5.p2.15.m15.2.2.2.2.4" xref="S2.SS5.p2.15.m15.2.2.2.3.cmml">,</mo><msub id="S2.SS5.p2.15.m15.2.2.2.2.2" xref="S2.SS5.p2.15.m15.2.2.2.2.2.cmml"><mi id="S2.SS5.p2.15.m15.2.2.2.2.2.2" xref="S2.SS5.p2.15.m15.2.2.2.2.2.2.cmml">X</mi><mn id="S2.SS5.p2.15.m15.2.2.2.2.2.3" xref="S2.SS5.p2.15.m15.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS5.p2.15.m15.2.2.2.2.5" stretchy="false" xref="S2.SS5.p2.15.m15.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p2.15.m15.2b"><apply id="S2.SS5.p2.15.m15.2.2.cmml" xref="S2.SS5.p2.15.m15.2.2"><times id="S2.SS5.p2.15.m15.2.2.3.cmml" xref="S2.SS5.p2.15.m15.2.2.3"></times><ci id="S2.SS5.p2.15.m15.2.2.4.cmml" xref="S2.SS5.p2.15.m15.2.2.4">ℋ</ci><interval closure="open" id="S2.SS5.p2.15.m15.2.2.2.3.cmml" xref="S2.SS5.p2.15.m15.2.2.2.2"><apply id="S2.SS5.p2.15.m15.1.1.1.1.1.cmml" xref="S2.SS5.p2.15.m15.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p2.15.m15.1.1.1.1.1.1.cmml" xref="S2.SS5.p2.15.m15.1.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p2.15.m15.1.1.1.1.1.2.cmml" xref="S2.SS5.p2.15.m15.1.1.1.1.1.2">𝑋</ci><cn id="S2.SS5.p2.15.m15.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p2.15.m15.1.1.1.1.1.3">0</cn></apply><apply id="S2.SS5.p2.15.m15.2.2.2.2.2.cmml" xref="S2.SS5.p2.15.m15.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.p2.15.m15.2.2.2.2.2.1.cmml" xref="S2.SS5.p2.15.m15.2.2.2.2.2">subscript</csymbol><ci id="S2.SS5.p2.15.m15.2.2.2.2.2.2.cmml" xref="S2.SS5.p2.15.m15.2.2.2.2.2.2">𝑋</ci><cn id="S2.SS5.p2.15.m15.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS5.p2.15.m15.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p2.15.m15.2c">\mathscr{H}(X_{0},X_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p2.15.m15.2d">script_H ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> endowed with the norm</p> <table class="ltx_equation ltx_eqn_table" id="S2.Ex9a"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f\|_{\mathscr{H}(X_{0},X_{1})}:=\max_{j\in\{0,1\}}\sup_{t\in\mathbb{R}}\|f(j% +{\rm i}t)\|_{X_{j}}" class="ltx_Math" display="block" id="S2.Ex9a.m1.6"><semantics id="S2.Ex9a.m1.6a"><mrow id="S2.Ex9a.m1.6.6" xref="S2.Ex9a.m1.6.6.cmml"><msub id="S2.Ex9a.m1.6.6.3" xref="S2.Ex9a.m1.6.6.3.cmml"><mrow id="S2.Ex9a.m1.6.6.3.2.2" xref="S2.Ex9a.m1.6.6.3.2.1.cmml"><mo id="S2.Ex9a.m1.6.6.3.2.2.1" stretchy="false" xref="S2.Ex9a.m1.6.6.3.2.1.1.cmml">‖</mo><mi id="S2.Ex9a.m1.5.5" xref="S2.Ex9a.m1.5.5.cmml">f</mi><mo id="S2.Ex9a.m1.6.6.3.2.2.2" stretchy="false" xref="S2.Ex9a.m1.6.6.3.2.1.1.cmml">‖</mo></mrow><mrow id="S2.Ex9a.m1.2.2.2" xref="S2.Ex9a.m1.2.2.2.cmml"><mi class="ltx_font_mathscript" id="S2.Ex9a.m1.2.2.2.4" xref="S2.Ex9a.m1.2.2.2.4.cmml">ℋ</mi><mo id="S2.Ex9a.m1.2.2.2.3" xref="S2.Ex9a.m1.2.2.2.3.cmml">⁢</mo><mrow id="S2.Ex9a.m1.2.2.2.2.2" xref="S2.Ex9a.m1.2.2.2.2.3.cmml"><mo id="S2.Ex9a.m1.2.2.2.2.2.3" stretchy="false" xref="S2.Ex9a.m1.2.2.2.2.3.cmml">(</mo><msub id="S2.Ex9a.m1.1.1.1.1.1.1" xref="S2.Ex9a.m1.1.1.1.1.1.1.cmml"><mi id="S2.Ex9a.m1.1.1.1.1.1.1.2" xref="S2.Ex9a.m1.1.1.1.1.1.1.2.cmml">X</mi><mn id="S2.Ex9a.m1.1.1.1.1.1.1.3" xref="S2.Ex9a.m1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.Ex9a.m1.2.2.2.2.2.4" xref="S2.Ex9a.m1.2.2.2.2.3.cmml">,</mo><msub id="S2.Ex9a.m1.2.2.2.2.2.2" xref="S2.Ex9a.m1.2.2.2.2.2.2.cmml"><mi id="S2.Ex9a.m1.2.2.2.2.2.2.2" xref="S2.Ex9a.m1.2.2.2.2.2.2.2.cmml">X</mi><mn id="S2.Ex9a.m1.2.2.2.2.2.2.3" xref="S2.Ex9a.m1.2.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.Ex9a.m1.2.2.2.2.2.5" stretchy="false" xref="S2.Ex9a.m1.2.2.2.2.3.cmml">)</mo></mrow></mrow></msub><mo id="S2.Ex9a.m1.6.6.2" lspace="0.278em" rspace="0.278em" xref="S2.Ex9a.m1.6.6.2.cmml">:=</mo><mrow id="S2.Ex9a.m1.6.6.1" xref="S2.Ex9a.m1.6.6.1.cmml"><munder id="S2.Ex9a.m1.6.6.1.3" xref="S2.Ex9a.m1.6.6.1.3.cmml"><mi id="S2.Ex9a.m1.6.6.1.3.2" xref="S2.Ex9a.m1.6.6.1.3.2.cmml">max</mi><mrow id="S2.Ex9a.m1.4.4.2" xref="S2.Ex9a.m1.4.4.2.cmml"><mi id="S2.Ex9a.m1.4.4.2.4" xref="S2.Ex9a.m1.4.4.2.4.cmml">j</mi><mo id="S2.Ex9a.m1.4.4.2.3" xref="S2.Ex9a.m1.4.4.2.3.cmml">∈</mo><mrow id="S2.Ex9a.m1.4.4.2.5.2" xref="S2.Ex9a.m1.4.4.2.5.1.cmml"><mo id="S2.Ex9a.m1.4.4.2.5.2.1" stretchy="false" xref="S2.Ex9a.m1.4.4.2.5.1.cmml">{</mo><mn id="S2.Ex9a.m1.3.3.1.1" xref="S2.Ex9a.m1.3.3.1.1.cmml">0</mn><mo id="S2.Ex9a.m1.4.4.2.5.2.2" xref="S2.Ex9a.m1.4.4.2.5.1.cmml">,</mo><mn id="S2.Ex9a.m1.4.4.2.2" xref="S2.Ex9a.m1.4.4.2.2.cmml">1</mn><mo id="S2.Ex9a.m1.4.4.2.5.2.3" stretchy="false" xref="S2.Ex9a.m1.4.4.2.5.1.cmml">}</mo></mrow></mrow></munder><mo id="S2.Ex9a.m1.6.6.1.2" lspace="0.167em" xref="S2.Ex9a.m1.6.6.1.2.cmml">⁢</mo><mrow id="S2.Ex9a.m1.6.6.1.1" xref="S2.Ex9a.m1.6.6.1.1.cmml"><munder id="S2.Ex9a.m1.6.6.1.1.2" xref="S2.Ex9a.m1.6.6.1.1.2.cmml"><mo id="S2.Ex9a.m1.6.6.1.1.2.2" movablelimits="false" rspace="0em" xref="S2.Ex9a.m1.6.6.1.1.2.2.cmml">sup</mo><mrow id="S2.Ex9a.m1.6.6.1.1.2.3" xref="S2.Ex9a.m1.6.6.1.1.2.3.cmml"><mi id="S2.Ex9a.m1.6.6.1.1.2.3.2" xref="S2.Ex9a.m1.6.6.1.1.2.3.2.cmml">t</mi><mo id="S2.Ex9a.m1.6.6.1.1.2.3.1" xref="S2.Ex9a.m1.6.6.1.1.2.3.1.cmml">∈</mo><mi id="S2.Ex9a.m1.6.6.1.1.2.3.3" xref="S2.Ex9a.m1.6.6.1.1.2.3.3.cmml">ℝ</mi></mrow></munder><msub id="S2.Ex9a.m1.6.6.1.1.1" xref="S2.Ex9a.m1.6.6.1.1.1.cmml"><mrow id="S2.Ex9a.m1.6.6.1.1.1.1.1" xref="S2.Ex9a.m1.6.6.1.1.1.1.2.cmml"><mo id="S2.Ex9a.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S2.Ex9a.m1.6.6.1.1.1.1.2.1.cmml">‖</mo><mrow id="S2.Ex9a.m1.6.6.1.1.1.1.1.1" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.cmml"><mi id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.3" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.3.cmml">f</mi><mo id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.2" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.2" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml">j</mi><mo id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3.1" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3.3" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></mrow><mo id="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.Ex9a.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.Ex9a.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="S2.Ex9a.m1.6.6.1.1.1.1.2.1.cmml">‖</mo></mrow><msub id="S2.Ex9a.m1.6.6.1.1.1.3" xref="S2.Ex9a.m1.6.6.1.1.1.3.cmml"><mi id="S2.Ex9a.m1.6.6.1.1.1.3.2" xref="S2.Ex9a.m1.6.6.1.1.1.3.2.cmml">X</mi><mi id="S2.Ex9a.m1.6.6.1.1.1.3.3" xref="S2.Ex9a.m1.6.6.1.1.1.3.3.cmml">j</mi></msub></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Ex9a.m1.6b"><apply id="S2.Ex9a.m1.6.6.cmml" xref="S2.Ex9a.m1.6.6"><csymbol cd="latexml" id="S2.Ex9a.m1.6.6.2.cmml" xref="S2.Ex9a.m1.6.6.2">assign</csymbol><apply id="S2.Ex9a.m1.6.6.3.cmml" xref="S2.Ex9a.m1.6.6.3"><csymbol cd="ambiguous" id="S2.Ex9a.m1.6.6.3.1.cmml" xref="S2.Ex9a.m1.6.6.3">subscript</csymbol><apply id="S2.Ex9a.m1.6.6.3.2.1.cmml" xref="S2.Ex9a.m1.6.6.3.2.2"><csymbol cd="latexml" id="S2.Ex9a.m1.6.6.3.2.1.1.cmml" xref="S2.Ex9a.m1.6.6.3.2.2.1">norm</csymbol><ci id="S2.Ex9a.m1.5.5.cmml" xref="S2.Ex9a.m1.5.5">𝑓</ci></apply><apply id="S2.Ex9a.m1.2.2.2.cmml" xref="S2.Ex9a.m1.2.2.2"><times id="S2.Ex9a.m1.2.2.2.3.cmml" xref="S2.Ex9a.m1.2.2.2.3"></times><ci id="S2.Ex9a.m1.2.2.2.4.cmml" 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xref="S2.Ex9a.m1.6.6.1.1.1.3.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Ex9a.m1.6c">\|f\|_{\mathscr{H}(X_{0},X_{1})}:=\max_{j\in\{0,1\}}\sup_{t\in\mathbb{R}}\|f(j% +{\rm i}t)\|_{X_{j}}</annotation><annotation encoding="application/x-llamapun" id="S2.Ex9a.m1.6d">∥ italic_f ∥ start_POSTSUBSCRIPT script_H ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT := roman_max start_POSTSUBSCRIPT italic_j ∈ { 0 , 1 } end_POSTSUBSCRIPT roman_sup start_POSTSUBSCRIPT italic_t ∈ blackboard_R end_POSTSUBSCRIPT ∥ italic_f ( italic_j + roman_i italic_t ) ∥ start_POSTSUBSCRIPT italic_X start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S2.SS5.p2.16">is a Banach space.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S2.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.1.1.1">Definition 2.4</span></span><span class="ltx_text ltx_font_bold" id="S2.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S2.Thmtheorem4.p1"> <p class="ltx_p" id="S2.Thmtheorem4.p1.6">Let <math alttext="(X_{0},X_{1})" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.1.m1.2"><semantics id="S2.Thmtheorem4.p1.1.m1.2a"><mrow id="S2.Thmtheorem4.p1.1.m1.2.2.2" xref="S2.Thmtheorem4.p1.1.m1.2.2.3.cmml"><mo id="S2.Thmtheorem4.p1.1.m1.2.2.2.3" stretchy="false" xref="S2.Thmtheorem4.p1.1.m1.2.2.3.cmml">(</mo><msub id="S2.Thmtheorem4.p1.1.m1.1.1.1.1" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.1.m1.1.1.1.1.2" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.1.2.cmml">X</mi><mn id="S2.Thmtheorem4.p1.1.m1.1.1.1.1.3" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.Thmtheorem4.p1.1.m1.2.2.2.4" xref="S2.Thmtheorem4.p1.1.m1.2.2.3.cmml">,</mo><msub id="S2.Thmtheorem4.p1.1.m1.2.2.2.2" xref="S2.Thmtheorem4.p1.1.m1.2.2.2.2.cmml"><mi id="S2.Thmtheorem4.p1.1.m1.2.2.2.2.2" xref="S2.Thmtheorem4.p1.1.m1.2.2.2.2.2.cmml">X</mi><mn id="S2.Thmtheorem4.p1.1.m1.2.2.2.2.3" xref="S2.Thmtheorem4.p1.1.m1.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.Thmtheorem4.p1.1.m1.2.2.2.5" stretchy="false" xref="S2.Thmtheorem4.p1.1.m1.2.2.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.1.m1.2b"><interval closure="open" id="S2.Thmtheorem4.p1.1.m1.2.2.3.cmml" xref="S2.Thmtheorem4.p1.1.m1.2.2.2"><apply id="S2.Thmtheorem4.p1.1.m1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.1.m1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem4.p1.1.m1.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.1.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.1.m1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.1.m1.1.1.1.1.3">0</cn></apply><apply id="S2.Thmtheorem4.p1.1.m1.2.2.2.2.cmml" xref="S2.Thmtheorem4.p1.1.m1.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.1.m1.2.2.2.2.1.cmml" xref="S2.Thmtheorem4.p1.1.m1.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem4.p1.1.m1.2.2.2.2.2.cmml" xref="S2.Thmtheorem4.p1.1.m1.2.2.2.2.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.1.m1.2.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.1.m1.2.2.2.2.3">1</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.1.m1.2c">(X_{0},X_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.1.m1.2d">( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> be an interpolation couple and <math alttext="\theta\in(0,1)" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.2.m2.2"><semantics id="S2.Thmtheorem4.p1.2.m2.2a"><mrow 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id="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.5" stretchy="false" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.2.3.cmml">]</mo></mrow><mi id="S2.Thmtheorem4.p1.3.1.m1.2.2.4" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.4.cmml">θ</mi></msub><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.3.1.m1.2b"><apply id="S2.Thmtheorem4.p1.3.1.m1.2.2.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.1.m1.2.2.3.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.2.2">subscript</csymbol><interval closure="closed" id="S2.Thmtheorem4.p1.3.1.m1.2.2.2.3.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2"><apply id="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.3.1.m1.1.1.1.1.1.3">0</cn></apply><apply id="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.2.2.2.3">1</cn></apply></interval><ci id="S2.Thmtheorem4.p1.3.1.m1.2.2.4.cmml" xref="S2.Thmtheorem4.p1.3.1.m1.2.2.4">𝜃</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.3.1.m1.2c">[X_{0},X_{1}]_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.3.1.m1.2d">[ italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math></em> is the complex vector space of all <math alttext="x\in X_{0}+X_{1}" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.4.m3.1"><semantics id="S2.Thmtheorem4.p1.4.m3.1a"><mrow id="S2.Thmtheorem4.p1.4.m3.1.1" xref="S2.Thmtheorem4.p1.4.m3.1.1.cmml"><mi id="S2.Thmtheorem4.p1.4.m3.1.1.2" xref="S2.Thmtheorem4.p1.4.m3.1.1.2.cmml">x</mi><mo id="S2.Thmtheorem4.p1.4.m3.1.1.1" xref="S2.Thmtheorem4.p1.4.m3.1.1.1.cmml">∈</mo><mrow id="S2.Thmtheorem4.p1.4.m3.1.1.3" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.cmml"><msub id="S2.Thmtheorem4.p1.4.m3.1.1.3.2" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.2.cmml"><mi id="S2.Thmtheorem4.p1.4.m3.1.1.3.2.2" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.2.2.cmml">X</mi><mn id="S2.Thmtheorem4.p1.4.m3.1.1.3.2.3" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.2.3.cmml">0</mn></msub><mo id="S2.Thmtheorem4.p1.4.m3.1.1.3.1" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.1.cmml">+</mo><msub id="S2.Thmtheorem4.p1.4.m3.1.1.3.3" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.3.cmml"><mi id="S2.Thmtheorem4.p1.4.m3.1.1.3.3.2" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.3.2.cmml">X</mi><mn id="S2.Thmtheorem4.p1.4.m3.1.1.3.3.3" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.4.m3.1b"><apply id="S2.Thmtheorem4.p1.4.m3.1.1.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1"><in id="S2.Thmtheorem4.p1.4.m3.1.1.1.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.1"></in><ci id="S2.Thmtheorem4.p1.4.m3.1.1.2.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.2">𝑥</ci><apply id="S2.Thmtheorem4.p1.4.m3.1.1.3.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3"><plus id="S2.Thmtheorem4.p1.4.m3.1.1.3.1.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.1"></plus><apply id="S2.Thmtheorem4.p1.4.m3.1.1.3.2.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.2"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.4.m3.1.1.3.2.1.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.2">subscript</csymbol><ci id="S2.Thmtheorem4.p1.4.m3.1.1.3.2.2.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.2.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.4.m3.1.1.3.2.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.2.3">0</cn></apply><apply id="S2.Thmtheorem4.p1.4.m3.1.1.3.3.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.3"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.4.m3.1.1.3.3.1.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.3">subscript</csymbol><ci id="S2.Thmtheorem4.p1.4.m3.1.1.3.3.2.cmml" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.3.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.4.m3.1.1.3.3.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.4.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.4.m3.1c">x\in X_{0}+X_{1}</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.4.m3.1d">italic_x ∈ italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="f(\theta)=x" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.5.m4.1"><semantics id="S2.Thmtheorem4.p1.5.m4.1a"><mrow id="S2.Thmtheorem4.p1.5.m4.1.2" xref="S2.Thmtheorem4.p1.5.m4.1.2.cmml"><mrow id="S2.Thmtheorem4.p1.5.m4.1.2.2" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.cmml"><mi id="S2.Thmtheorem4.p1.5.m4.1.2.2.2" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.2.cmml">f</mi><mo id="S2.Thmtheorem4.p1.5.m4.1.2.2.1" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.1.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p1.5.m4.1.2.2.3.2" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.cmml"><mo id="S2.Thmtheorem4.p1.5.m4.1.2.2.3.2.1" stretchy="false" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.cmml">(</mo><mi id="S2.Thmtheorem4.p1.5.m4.1.1" xref="S2.Thmtheorem4.p1.5.m4.1.1.cmml">θ</mi><mo id="S2.Thmtheorem4.p1.5.m4.1.2.2.3.2.2" stretchy="false" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S2.Thmtheorem4.p1.5.m4.1.2.1" xref="S2.Thmtheorem4.p1.5.m4.1.2.1.cmml">=</mo><mi id="S2.Thmtheorem4.p1.5.m4.1.2.3" xref="S2.Thmtheorem4.p1.5.m4.1.2.3.cmml">x</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.5.m4.1b"><apply id="S2.Thmtheorem4.p1.5.m4.1.2.cmml" xref="S2.Thmtheorem4.p1.5.m4.1.2"><eq id="S2.Thmtheorem4.p1.5.m4.1.2.1.cmml" xref="S2.Thmtheorem4.p1.5.m4.1.2.1"></eq><apply id="S2.Thmtheorem4.p1.5.m4.1.2.2.cmml" xref="S2.Thmtheorem4.p1.5.m4.1.2.2"><times id="S2.Thmtheorem4.p1.5.m4.1.2.2.1.cmml" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.1"></times><ci id="S2.Thmtheorem4.p1.5.m4.1.2.2.2.cmml" xref="S2.Thmtheorem4.p1.5.m4.1.2.2.2">𝑓</ci><ci id="S2.Thmtheorem4.p1.5.m4.1.1.cmml" xref="S2.Thmtheorem4.p1.5.m4.1.1">𝜃</ci></apply><ci id="S2.Thmtheorem4.p1.5.m4.1.2.3.cmml" xref="S2.Thmtheorem4.p1.5.m4.1.2.3">𝑥</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.5.m4.1c">f(\theta)=x</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.5.m4.1d">italic_f ( italic_θ ) = italic_x</annotation></semantics></math> for some <math alttext="f\in\mathscr{H}(X_{0},X_{1})" class="ltx_Math" display="inline" id="S2.Thmtheorem4.p1.6.m5.2"><semantics id="S2.Thmtheorem4.p1.6.m5.2a"><mrow id="S2.Thmtheorem4.p1.6.m5.2.2" xref="S2.Thmtheorem4.p1.6.m5.2.2.cmml"><mi id="S2.Thmtheorem4.p1.6.m5.2.2.4" xref="S2.Thmtheorem4.p1.6.m5.2.2.4.cmml">f</mi><mo id="S2.Thmtheorem4.p1.6.m5.2.2.3" xref="S2.Thmtheorem4.p1.6.m5.2.2.3.cmml">∈</mo><mrow id="S2.Thmtheorem4.p1.6.m5.2.2.2" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.cmml"><mi class="ltx_font_mathscript" id="S2.Thmtheorem4.p1.6.m5.2.2.2.4" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.4.cmml">ℋ</mi><mo id="S2.Thmtheorem4.p1.6.m5.2.2.2.3" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.3.cmml">⁢</mo><mrow id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.3.cmml"><mo id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.3" stretchy="false" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.3.cmml">(</mo><msub id="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1" xref="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.cmml"><mi id="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.2" xref="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.2.cmml">X</mi><mn id="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.3" xref="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.4" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.3.cmml">,</mo><msub id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.cmml"><mi id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.2" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.2.cmml">X</mi><mn id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.3" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.5" stretchy="false" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.Thmtheorem4.p1.6.m5.2b"><apply id="S2.Thmtheorem4.p1.6.m5.2.2.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2"><in id="S2.Thmtheorem4.p1.6.m5.2.2.3.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.3"></in><ci id="S2.Thmtheorem4.p1.6.m5.2.2.4.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.4">𝑓</ci><apply id="S2.Thmtheorem4.p1.6.m5.2.2.2.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.2"><times id="S2.Thmtheorem4.p1.6.m5.2.2.2.3.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.3"></times><ci id="S2.Thmtheorem4.p1.6.m5.2.2.2.4.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.4">ℋ</ci><interval closure="open" id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.3.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2"><apply id="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.1.cmml" xref="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.2.cmml" xref="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.6.m5.1.1.1.1.1.1.3">0</cn></apply><apply id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.1.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2">subscript</csymbol><ci id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.2.cmml" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.2">𝑋</ci><cn id="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.3.cmml" type="integer" xref="S2.Thmtheorem4.p1.6.m5.2.2.2.2.2.2.3">1</cn></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.Thmtheorem4.p1.6.m5.2c">f\in\mathscr{H}(X_{0},X_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.Thmtheorem4.p1.6.m5.2d">italic_f ∈ script_H ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S2.SS5.p3"> <p class="ltx_p" id="S2.SS5.p3.1">The space <math alttext="[X_{0},X_{1}]_{\theta}" class="ltx_Math" display="inline" id="S2.SS5.p3.1.m1.2"><semantics 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id="S2.SS5.p3.3.m2.2.2.1.3.cmml" xref="S2.SS5.p3.3.m2.2.2.1.3">𝑓</ci><apply id="S2.SS5.p3.3.m2.2.2.1.1.1.1.cmml" xref="S2.SS5.p3.3.m2.2.2.1.1.1"><plus id="S2.SS5.p3.3.m2.2.2.1.1.1.1.1.cmml" xref="S2.SS5.p3.3.m2.2.2.1.1.1.1.1"></plus><ci id="S2.SS5.p3.3.m2.2.2.1.1.1.1.2.cmml" xref="S2.SS5.p3.3.m2.2.2.1.1.1.1.2">𝑧</ci><apply id="S2.SS5.p3.3.m2.2.2.1.1.1.1.3.cmml" xref="S2.SS5.p3.3.m2.2.2.1.1.1.1.3"><times id="S2.SS5.p3.3.m2.2.2.1.1.1.1.3.1.cmml" xref="S2.SS5.p3.3.m2.2.2.1.1.1.1.3.1"></times><ci id="S2.SS5.p3.3.m2.2.2.1.1.1.1.3.2.cmml" xref="S2.SS5.p3.3.m2.2.2.1.1.1.1.3.2">i</ci><ci id="S2.SS5.p3.3.m2.2.2.1.1.1.1.3.3.cmml" xref="S2.SS5.p3.3.m2.2.2.1.1.1.1.3.3">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.3.m2.2c">g_{t}(z):=f(z+{\rm i}t)</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.3.m2.2d">italic_g start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ( italic_z ) := italic_f ( italic_z + roman_i italic_t )</annotation></semantics></math> for <math alttext="z\in\overline{{\mathbb{S}}}" class="ltx_Math" display="inline" id="S2.SS5.p3.4.m3.1"><semantics id="S2.SS5.p3.4.m3.1a"><mrow id="S2.SS5.p3.4.m3.1.1" xref="S2.SS5.p3.4.m3.1.1.cmml"><mi id="S2.SS5.p3.4.m3.1.1.2" xref="S2.SS5.p3.4.m3.1.1.2.cmml">z</mi><mo id="S2.SS5.p3.4.m3.1.1.1" xref="S2.SS5.p3.4.m3.1.1.1.cmml">∈</mo><mover accent="true" id="S2.SS5.p3.4.m3.1.1.3" xref="S2.SS5.p3.4.m3.1.1.3.cmml"><mi id="S2.SS5.p3.4.m3.1.1.3.2" xref="S2.SS5.p3.4.m3.1.1.3.2.cmml">𝕊</mi><mo id="S2.SS5.p3.4.m3.1.1.3.1" xref="S2.SS5.p3.4.m3.1.1.3.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.4.m3.1b"><apply id="S2.SS5.p3.4.m3.1.1.cmml" xref="S2.SS5.p3.4.m3.1.1"><in id="S2.SS5.p3.4.m3.1.1.1.cmml" xref="S2.SS5.p3.4.m3.1.1.1"></in><ci id="S2.SS5.p3.4.m3.1.1.2.cmml" xref="S2.SS5.p3.4.m3.1.1.2">𝑧</ci><apply id="S2.SS5.p3.4.m3.1.1.3.cmml" xref="S2.SS5.p3.4.m3.1.1.3"><ci id="S2.SS5.p3.4.m3.1.1.3.1.cmml" xref="S2.SS5.p3.4.m3.1.1.3.1">¯</ci><ci id="S2.SS5.p3.4.m3.1.1.3.2.cmml" xref="S2.SS5.p3.4.m3.1.1.3.2">𝕊</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.4.m3.1c">z\in\overline{{\mathbb{S}}}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.4.m3.1d">italic_z ∈ over¯ start_ARG blackboard_S end_ARG</annotation></semantics></math> and <math alttext="t\in\mathbb{R}" class="ltx_Math" display="inline" id="S2.SS5.p3.5.m4.1"><semantics id="S2.SS5.p3.5.m4.1a"><mrow id="S2.SS5.p3.5.m4.1.1" xref="S2.SS5.p3.5.m4.1.1.cmml"><mi id="S2.SS5.p3.5.m4.1.1.2" xref="S2.SS5.p3.5.m4.1.1.2.cmml">t</mi><mo id="S2.SS5.p3.5.m4.1.1.1" xref="S2.SS5.p3.5.m4.1.1.1.cmml">∈</mo><mi id="S2.SS5.p3.5.m4.1.1.3" xref="S2.SS5.p3.5.m4.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.5.m4.1b"><apply id="S2.SS5.p3.5.m4.1.1.cmml" xref="S2.SS5.p3.5.m4.1.1"><in id="S2.SS5.p3.5.m4.1.1.1.cmml" xref="S2.SS5.p3.5.m4.1.1.1"></in><ci id="S2.SS5.p3.5.m4.1.1.2.cmml" xref="S2.SS5.p3.5.m4.1.1.2">𝑡</ci><ci id="S2.SS5.p3.5.m4.1.1.3.cmml" xref="S2.SS5.p3.5.m4.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.5.m4.1c">t\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.5.m4.1d">italic_t ∈ blackboard_R</annotation></semantics></math>. Then <math alttext="g_{t}\in\mathscr{H}(X_{0},X_{1})" class="ltx_Math" display="inline" id="S2.SS5.p3.6.m5.2"><semantics id="S2.SS5.p3.6.m5.2a"><mrow id="S2.SS5.p3.6.m5.2.2" xref="S2.SS5.p3.6.m5.2.2.cmml"><msub id="S2.SS5.p3.6.m5.2.2.4" xref="S2.SS5.p3.6.m5.2.2.4.cmml"><mi id="S2.SS5.p3.6.m5.2.2.4.2" xref="S2.SS5.p3.6.m5.2.2.4.2.cmml">g</mi><mi id="S2.SS5.p3.6.m5.2.2.4.3" xref="S2.SS5.p3.6.m5.2.2.4.3.cmml">t</mi></msub><mo id="S2.SS5.p3.6.m5.2.2.3" xref="S2.SS5.p3.6.m5.2.2.3.cmml">∈</mo><mrow id="S2.SS5.p3.6.m5.2.2.2" xref="S2.SS5.p3.6.m5.2.2.2.cmml"><mi class="ltx_font_mathscript" id="S2.SS5.p3.6.m5.2.2.2.4" xref="S2.SS5.p3.6.m5.2.2.2.4.cmml">ℋ</mi><mo id="S2.SS5.p3.6.m5.2.2.2.3" xref="S2.SS5.p3.6.m5.2.2.2.3.cmml">⁢</mo><mrow id="S2.SS5.p3.6.m5.2.2.2.2.2" xref="S2.SS5.p3.6.m5.2.2.2.2.3.cmml"><mo id="S2.SS5.p3.6.m5.2.2.2.2.2.3" stretchy="false" xref="S2.SS5.p3.6.m5.2.2.2.2.3.cmml">(</mo><msub id="S2.SS5.p3.6.m5.1.1.1.1.1.1" xref="S2.SS5.p3.6.m5.1.1.1.1.1.1.cmml"><mi id="S2.SS5.p3.6.m5.1.1.1.1.1.1.2" xref="S2.SS5.p3.6.m5.1.1.1.1.1.1.2.cmml">X</mi><mn id="S2.SS5.p3.6.m5.1.1.1.1.1.1.3" xref="S2.SS5.p3.6.m5.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S2.SS5.p3.6.m5.2.2.2.2.2.4" xref="S2.SS5.p3.6.m5.2.2.2.2.3.cmml">,</mo><msub id="S2.SS5.p3.6.m5.2.2.2.2.2.2" xref="S2.SS5.p3.6.m5.2.2.2.2.2.2.cmml"><mi id="S2.SS5.p3.6.m5.2.2.2.2.2.2.2" xref="S2.SS5.p3.6.m5.2.2.2.2.2.2.2.cmml">X</mi><mn id="S2.SS5.p3.6.m5.2.2.2.2.2.2.3" xref="S2.SS5.p3.6.m5.2.2.2.2.2.2.3.cmml">1</mn></msub><mo id="S2.SS5.p3.6.m5.2.2.2.2.2.5" stretchy="false" xref="S2.SS5.p3.6.m5.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S2.SS5.p3.6.m5.2b"><apply id="S2.SS5.p3.6.m5.2.2.cmml" xref="S2.SS5.p3.6.m5.2.2"><in id="S2.SS5.p3.6.m5.2.2.3.cmml" xref="S2.SS5.p3.6.m5.2.2.3"></in><apply id="S2.SS5.p3.6.m5.2.2.4.cmml" xref="S2.SS5.p3.6.m5.2.2.4"><csymbol cd="ambiguous" id="S2.SS5.p3.6.m5.2.2.4.1.cmml" xref="S2.SS5.p3.6.m5.2.2.4">subscript</csymbol><ci id="S2.SS5.p3.6.m5.2.2.4.2.cmml" xref="S2.SS5.p3.6.m5.2.2.4.2">𝑔</ci><ci id="S2.SS5.p3.6.m5.2.2.4.3.cmml" xref="S2.SS5.p3.6.m5.2.2.4.3">𝑡</ci></apply><apply id="S2.SS5.p3.6.m5.2.2.2.cmml" xref="S2.SS5.p3.6.m5.2.2.2"><times id="S2.SS5.p3.6.m5.2.2.2.3.cmml" xref="S2.SS5.p3.6.m5.2.2.2.3"></times><ci id="S2.SS5.p3.6.m5.2.2.2.4.cmml" xref="S2.SS5.p3.6.m5.2.2.2.4">ℋ</ci><interval closure="open" id="S2.SS5.p3.6.m5.2.2.2.2.3.cmml" xref="S2.SS5.p3.6.m5.2.2.2.2.2"><apply id="S2.SS5.p3.6.m5.1.1.1.1.1.1.cmml" xref="S2.SS5.p3.6.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S2.SS5.p3.6.m5.1.1.1.1.1.1.1.cmml" xref="S2.SS5.p3.6.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S2.SS5.p3.6.m5.1.1.1.1.1.1.2.cmml" xref="S2.SS5.p3.6.m5.1.1.1.1.1.1.2">𝑋</ci><cn id="S2.SS5.p3.6.m5.1.1.1.1.1.1.3.cmml" type="integer" xref="S2.SS5.p3.6.m5.1.1.1.1.1.1.3">0</cn></apply><apply id="S2.SS5.p3.6.m5.2.2.2.2.2.2.cmml" xref="S2.SS5.p3.6.m5.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S2.SS5.p3.6.m5.2.2.2.2.2.2.1.cmml" xref="S2.SS5.p3.6.m5.2.2.2.2.2.2">subscript</csymbol><ci id="S2.SS5.p3.6.m5.2.2.2.2.2.2.2.cmml" xref="S2.SS5.p3.6.m5.2.2.2.2.2.2.2">𝑋</ci><cn id="S2.SS5.p3.6.m5.2.2.2.2.2.2.3.cmml" type="integer" xref="S2.SS5.p3.6.m5.2.2.2.2.2.2.3">1</cn></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S2.SS5.p3.6.m5.2c">g_{t}\in\mathscr{H}(X_{0},X_{1})</annotation><annotation encoding="application/x-llamapun" id="S2.SS5.p3.6.m5.2d">italic_g start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT ∈ script_H ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S2.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f(\theta+{\rm i}t)\|_{[X_{0},X_{1}]_{\theta}}\leq C\|f\|_{\mathscr{H}(X_{0},% X_{1})},\qquad\theta\in(0,1),\,t\in\mathbb{R}." class="ltx_Math" display="block" id="S2.E10.m1.8"><semantics id="S2.E10.m1.8a"><mrow id="S2.E10.m1.8.8.1"><mrow id="S2.E10.m1.8.8.1.1.2" xref="S2.E10.m1.8.8.1.1.3.cmml"><mrow id="S2.E10.m1.8.8.1.1.1.1" xref="S2.E10.m1.8.8.1.1.1.1.cmml"><msub id="S2.E10.m1.8.8.1.1.1.1.1" xref="S2.E10.m1.8.8.1.1.1.1.1.cmml"><mrow id="S2.E10.m1.8.8.1.1.1.1.1.1.1" xref="S2.E10.m1.8.8.1.1.1.1.1.1.2.cmml"><mo id="S2.E10.m1.8.8.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E10.m1.8.8.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.3" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.3.cmml">f</mi><mo id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.2" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.2" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.2.cmml">θ</mi><mo id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.1" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.1.cmml">+</mo><mrow id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3.2" mathvariant="normal" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3.2.cmml">i</mi><mo id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3.1" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3.3" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.3.3.cmml">t</mi></mrow></mrow><mo id="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S2.E10.m1.8.8.1.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S2.E10.m1.8.8.1.1.1.1.1.1.1.3" stretchy="false" 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encoding="application/x-llamapun" id="S2.E10.m1.8d">∥ italic_f ( italic_θ + roman_i italic_t ) ∥ start_POSTSUBSCRIPT [ italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT script_H ( italic_X start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_X start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT , italic_θ ∈ ( 0 , 1 ) , italic_t ∈ blackboard_R .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2.10)</span></td> </tr></tbody> </table> </div> </section> </section> <section class="ltx_section" id="S3"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">3. </span>Trace spaces of Besov and Triebel-Lizorkin spaces</h2> <div class="ltx_para" id="S3.p1"> <p class="ltx_p" id="S3.p1.1">In this section, we characterise the higher-order trace spaces of weighted Besov and Triebel-Lizorkin spaces on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.p1.1.m1.1"><semantics id="S3.p1.1.m1.1a"><msup id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><mi id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml">ℝ</mi><mi id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.p1.1.m1.1b"><apply id="S3.p1.1.m1.1.1.cmml" xref="S3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.1.cmml" xref="S3.p1.1.m1.1.1">superscript</csymbol><ci id="S3.p1.1.m1.1.1.2.cmml" xref="S3.p1.1.m1.1.1.2">ℝ</ci><ci id="S3.p1.1.m1.1.1.3.cmml" xref="S3.p1.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.p1.1.m1.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.SS1" title="3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.1</span></a> we first study the zeroth-order trace operator and in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.SS2" title="3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.2</span></a> we extend the results to higher-order trace operators.</p> </div> <section class="ltx_subsection" id="S3.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.1. </span>The zeroth-order trace operator</h3> <div class="ltx_para" id="S3.SS1.p1"> <p class="ltx_p" id="S3.SS1.p1.1">To define the trace operator we pursue as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>]</cite> and use a concept due to Nikol’skij <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib60" title="">60</a>]</cite>. There exist several more definitions for the trace operator. We refer to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Remark 4.8]</cite> for an overview. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S3.SS1.p2"> <p class="ltx_p" id="S3.SS1.p2.8">Let <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.p2.1.m1.1"><semantics id="S3.SS1.p2.1.m1.1a"><mi id="S3.SS1.p2.1.m1.1.1" xref="S3.SS1.p2.1.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.1.m1.1b"><ci id="S3.SS1.p2.1.m1.1.1.cmml" xref="S3.SS1.p2.1.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.1.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.1.m1.1d">italic_X</annotation></semantics></math> be a Banach space and let <math alttext="f:\mathbb{R}^{d}\to X" class="ltx_Math" display="inline" id="S3.SS1.p2.2.m2.1"><semantics id="S3.SS1.p2.2.m2.1a"><mrow id="S3.SS1.p2.2.m2.1.1" xref="S3.SS1.p2.2.m2.1.1.cmml"><mi id="S3.SS1.p2.2.m2.1.1.2" xref="S3.SS1.p2.2.m2.1.1.2.cmml">f</mi><mo id="S3.SS1.p2.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p2.2.m2.1.1.1.cmml">:</mo><mrow id="S3.SS1.p2.2.m2.1.1.3" xref="S3.SS1.p2.2.m2.1.1.3.cmml"><msup id="S3.SS1.p2.2.m2.1.1.3.2" xref="S3.SS1.p2.2.m2.1.1.3.2.cmml"><mi id="S3.SS1.p2.2.m2.1.1.3.2.2" xref="S3.SS1.p2.2.m2.1.1.3.2.2.cmml">ℝ</mi><mi id="S3.SS1.p2.2.m2.1.1.3.2.3" xref="S3.SS1.p2.2.m2.1.1.3.2.3.cmml">d</mi></msup><mo id="S3.SS1.p2.2.m2.1.1.3.1" stretchy="false" xref="S3.SS1.p2.2.m2.1.1.3.1.cmml">→</mo><mi id="S3.SS1.p2.2.m2.1.1.3.3" xref="S3.SS1.p2.2.m2.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.2.m2.1b"><apply id="S3.SS1.p2.2.m2.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1"><ci id="S3.SS1.p2.2.m2.1.1.1.cmml" xref="S3.SS1.p2.2.m2.1.1.1">:</ci><ci id="S3.SS1.p2.2.m2.1.1.2.cmml" xref="S3.SS1.p2.2.m2.1.1.2">𝑓</ci><apply id="S3.SS1.p2.2.m2.1.1.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3"><ci id="S3.SS1.p2.2.m2.1.1.3.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.1">→</ci><apply id="S3.SS1.p2.2.m2.1.1.3.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.2.m2.1.1.3.2.1.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2">superscript</csymbol><ci id="S3.SS1.p2.2.m2.1.1.3.2.2.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2.2">ℝ</ci><ci id="S3.SS1.p2.2.m2.1.1.3.2.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.2.3">𝑑</ci></apply><ci id="S3.SS1.p2.2.m2.1.1.3.3.cmml" xref="S3.SS1.p2.2.m2.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.2.m2.1c">f:\mathbb{R}^{d}\to X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.2.m2.1d">italic_f : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → italic_X</annotation></semantics></math> be strongly measurable. A function <math alttext="g:\mathbb{R}^{d-1}\to X" class="ltx_Math" display="inline" id="S3.SS1.p2.3.m3.1"><semantics id="S3.SS1.p2.3.m3.1a"><mrow id="S3.SS1.p2.3.m3.1.1" xref="S3.SS1.p2.3.m3.1.1.cmml"><mi id="S3.SS1.p2.3.m3.1.1.2" xref="S3.SS1.p2.3.m3.1.1.2.cmml">g</mi><mo id="S3.SS1.p2.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.p2.3.m3.1.1.1.cmml">:</mo><mrow id="S3.SS1.p2.3.m3.1.1.3" xref="S3.SS1.p2.3.m3.1.1.3.cmml"><msup id="S3.SS1.p2.3.m3.1.1.3.2" xref="S3.SS1.p2.3.m3.1.1.3.2.cmml"><mi id="S3.SS1.p2.3.m3.1.1.3.2.2" xref="S3.SS1.p2.3.m3.1.1.3.2.2.cmml">ℝ</mi><mrow id="S3.SS1.p2.3.m3.1.1.3.2.3" xref="S3.SS1.p2.3.m3.1.1.3.2.3.cmml"><mi id="S3.SS1.p2.3.m3.1.1.3.2.3.2" xref="S3.SS1.p2.3.m3.1.1.3.2.3.2.cmml">d</mi><mo id="S3.SS1.p2.3.m3.1.1.3.2.3.1" xref="S3.SS1.p2.3.m3.1.1.3.2.3.1.cmml">−</mo><mn id="S3.SS1.p2.3.m3.1.1.3.2.3.3" xref="S3.SS1.p2.3.m3.1.1.3.2.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.p2.3.m3.1.1.3.1" stretchy="false" xref="S3.SS1.p2.3.m3.1.1.3.1.cmml">→</mo><mi id="S3.SS1.p2.3.m3.1.1.3.3" xref="S3.SS1.p2.3.m3.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.3.m3.1b"><apply id="S3.SS1.p2.3.m3.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1"><ci id="S3.SS1.p2.3.m3.1.1.1.cmml" xref="S3.SS1.p2.3.m3.1.1.1">:</ci><ci id="S3.SS1.p2.3.m3.1.1.2.cmml" xref="S3.SS1.p2.3.m3.1.1.2">𝑔</ci><apply id="S3.SS1.p2.3.m3.1.1.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3"><ci id="S3.SS1.p2.3.m3.1.1.3.1.cmml" xref="S3.SS1.p2.3.m3.1.1.3.1">→</ci><apply id="S3.SS1.p2.3.m3.1.1.3.2.cmml" xref="S3.SS1.p2.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.3.m3.1.1.3.2.1.cmml" xref="S3.SS1.p2.3.m3.1.1.3.2">superscript</csymbol><ci id="S3.SS1.p2.3.m3.1.1.3.2.2.cmml" xref="S3.SS1.p2.3.m3.1.1.3.2.2">ℝ</ci><apply id="S3.SS1.p2.3.m3.1.1.3.2.3.cmml" xref="S3.SS1.p2.3.m3.1.1.3.2.3"><minus id="S3.SS1.p2.3.m3.1.1.3.2.3.1.cmml" xref="S3.SS1.p2.3.m3.1.1.3.2.3.1"></minus><ci id="S3.SS1.p2.3.m3.1.1.3.2.3.2.cmml" 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encoding="application/x-llamapun" id="S3.SS1.p2.4.m4.1d">italic_f</annotation></semantics></math> on <math alttext="\{0\}\times\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S3.SS1.p2.5.m5.1"><semantics id="S3.SS1.p2.5.m5.1a"><mrow id="S3.SS1.p2.5.m5.1.2" xref="S3.SS1.p2.5.m5.1.2.cmml"><mrow id="S3.SS1.p2.5.m5.1.2.2.2" xref="S3.SS1.p2.5.m5.1.2.2.1.cmml"><mo id="S3.SS1.p2.5.m5.1.2.2.2.1" stretchy="false" xref="S3.SS1.p2.5.m5.1.2.2.1.cmml">{</mo><mn id="S3.SS1.p2.5.m5.1.1" xref="S3.SS1.p2.5.m5.1.1.cmml">0</mn><mo id="S3.SS1.p2.5.m5.1.2.2.2.2" rspace="0.055em" stretchy="false" xref="S3.SS1.p2.5.m5.1.2.2.1.cmml">}</mo></mrow><mo id="S3.SS1.p2.5.m5.1.2.1" rspace="0.222em" xref="S3.SS1.p2.5.m5.1.2.1.cmml">×</mo><msup id="S3.SS1.p2.5.m5.1.2.3" xref="S3.SS1.p2.5.m5.1.2.3.cmml"><mi id="S3.SS1.p2.5.m5.1.2.3.2" xref="S3.SS1.p2.5.m5.1.2.3.2.cmml">ℝ</mi><mrow id="S3.SS1.p2.5.m5.1.2.3.3" xref="S3.SS1.p2.5.m5.1.2.3.3.cmml"><mi id="S3.SS1.p2.5.m5.1.2.3.3.2" xref="S3.SS1.p2.5.m5.1.2.3.3.2.cmml">d</mi><mo id="S3.SS1.p2.5.m5.1.2.3.3.1" xref="S3.SS1.p2.5.m5.1.2.3.3.1.cmml">−</mo><mn id="S3.SS1.p2.5.m5.1.2.3.3.3" xref="S3.SS1.p2.5.m5.1.2.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.5.m5.1b"><apply id="S3.SS1.p2.5.m5.1.2.cmml" xref="S3.SS1.p2.5.m5.1.2"><times id="S3.SS1.p2.5.m5.1.2.1.cmml" xref="S3.SS1.p2.5.m5.1.2.1"></times><set id="S3.SS1.p2.5.m5.1.2.2.1.cmml" xref="S3.SS1.p2.5.m5.1.2.2.2"><cn id="S3.SS1.p2.5.m5.1.1.cmml" type="integer" xref="S3.SS1.p2.5.m5.1.1">0</cn></set><apply id="S3.SS1.p2.5.m5.1.2.3.cmml" xref="S3.SS1.p2.5.m5.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.p2.5.m5.1.2.3.1.cmml" xref="S3.SS1.p2.5.m5.1.2.3">superscript</csymbol><ci id="S3.SS1.p2.5.m5.1.2.3.2.cmml" xref="S3.SS1.p2.5.m5.1.2.3.2">ℝ</ci><apply id="S3.SS1.p2.5.m5.1.2.3.3.cmml" xref="S3.SS1.p2.5.m5.1.2.3.3"><minus id="S3.SS1.p2.5.m5.1.2.3.3.1.cmml" xref="S3.SS1.p2.5.m5.1.2.3.3.1"></minus><ci 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xref="S3.SS1.p2.6.m6.1.1.1.cmml">:</mo><mrow id="S3.SS1.p2.6.m6.1.1.3" xref="S3.SS1.p2.6.m6.1.1.3.cmml"><msup id="S3.SS1.p2.6.m6.1.1.3.2" xref="S3.SS1.p2.6.m6.1.1.3.2.cmml"><mi id="S3.SS1.p2.6.m6.1.1.3.2.2" xref="S3.SS1.p2.6.m6.1.1.3.2.2.cmml">ℝ</mi><mi id="S3.SS1.p2.6.m6.1.1.3.2.3" xref="S3.SS1.p2.6.m6.1.1.3.2.3.cmml">d</mi></msup><mo id="S3.SS1.p2.6.m6.1.1.3.1" stretchy="false" xref="S3.SS1.p2.6.m6.1.1.3.1.cmml">→</mo><mi id="S3.SS1.p2.6.m6.1.1.3.3" xref="S3.SS1.p2.6.m6.1.1.3.3.cmml">X</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.6.m6.1b"><apply id="S3.SS1.p2.6.m6.1.1.cmml" xref="S3.SS1.p2.6.m6.1.1"><ci id="S3.SS1.p2.6.m6.1.1.1.cmml" xref="S3.SS1.p2.6.m6.1.1.1">:</ci><apply id="S3.SS1.p2.6.m6.1.1.2.cmml" xref="S3.SS1.p2.6.m6.1.1.2"><ci id="S3.SS1.p2.6.m6.1.1.2.1.cmml" xref="S3.SS1.p2.6.m6.1.1.2.1">~</ci><ci id="S3.SS1.p2.6.m6.1.1.2.2.cmml" xref="S3.SS1.p2.6.m6.1.1.2.2">𝑓</ci></apply><apply id="S3.SS1.p2.6.m6.1.1.3.cmml" xref="S3.SS1.p2.6.m6.1.1.3"><ci id="S3.SS1.p2.6.m6.1.1.3.1.cmml" xref="S3.SS1.p2.6.m6.1.1.3.1">→</ci><apply id="S3.SS1.p2.6.m6.1.1.3.2.cmml" xref="S3.SS1.p2.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.p2.6.m6.1.1.3.2.1.cmml" xref="S3.SS1.p2.6.m6.1.1.3.2">superscript</csymbol><ci id="S3.SS1.p2.6.m6.1.1.3.2.2.cmml" xref="S3.SS1.p2.6.m6.1.1.3.2.2">ℝ</ci><ci id="S3.SS1.p2.6.m6.1.1.3.2.3.cmml" xref="S3.SS1.p2.6.m6.1.1.3.2.3">𝑑</ci></apply><ci id="S3.SS1.p2.6.m6.1.1.3.3.cmml" xref="S3.SS1.p2.6.m6.1.1.3.3">𝑋</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.6.m6.1c">\widetilde{f}:\mathbb{R}^{d}\to X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.6.m6.1d">over~ start_ARG italic_f end_ARG : blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT → italic_X</annotation></semantics></math>, <math alttext="p\in[1,\infty]" class="ltx_Math" display="inline" id="S3.SS1.p2.7.m7.2"><semantics id="S3.SS1.p2.7.m7.2a"><mrow id="S3.SS1.p2.7.m7.2.3" xref="S3.SS1.p2.7.m7.2.3.cmml"><mi id="S3.SS1.p2.7.m7.2.3.2" xref="S3.SS1.p2.7.m7.2.3.2.cmml">p</mi><mo id="S3.SS1.p2.7.m7.2.3.1" xref="S3.SS1.p2.7.m7.2.3.1.cmml">∈</mo><mrow id="S3.SS1.p2.7.m7.2.3.3.2" xref="S3.SS1.p2.7.m7.2.3.3.1.cmml"><mo id="S3.SS1.p2.7.m7.2.3.3.2.1" stretchy="false" xref="S3.SS1.p2.7.m7.2.3.3.1.cmml">[</mo><mn id="S3.SS1.p2.7.m7.1.1" xref="S3.SS1.p2.7.m7.1.1.cmml">1</mn><mo id="S3.SS1.p2.7.m7.2.3.3.2.2" xref="S3.SS1.p2.7.m7.2.3.3.1.cmml">,</mo><mi id="S3.SS1.p2.7.m7.2.2" mathvariant="normal" xref="S3.SS1.p2.7.m7.2.2.cmml">∞</mi><mo id="S3.SS1.p2.7.m7.2.3.3.2.3" stretchy="false" xref="S3.SS1.p2.7.m7.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.7.m7.2b"><apply id="S3.SS1.p2.7.m7.2.3.cmml" xref="S3.SS1.p2.7.m7.2.3"><in id="S3.SS1.p2.7.m7.2.3.1.cmml" xref="S3.SS1.p2.7.m7.2.3.1"></in><ci id="S3.SS1.p2.7.m7.2.3.2.cmml" xref="S3.SS1.p2.7.m7.2.3.2">𝑝</ci><interval closure="closed" id="S3.SS1.p2.7.m7.2.3.3.1.cmml" xref="S3.SS1.p2.7.m7.2.3.3.2"><cn id="S3.SS1.p2.7.m7.1.1.cmml" type="integer" xref="S3.SS1.p2.7.m7.1.1">1</cn><infinity id="S3.SS1.p2.7.m7.2.2.cmml" xref="S3.SS1.p2.7.m7.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.7.m7.2c">p\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.7.m7.2d">italic_p ∈ [ 1 , ∞ ]</annotation></semantics></math> and <math alttext="\delta>0" class="ltx_Math" display="inline" id="S3.SS1.p2.8.m8.1"><semantics id="S3.SS1.p2.8.m8.1a"><mrow id="S3.SS1.p2.8.m8.1.1" xref="S3.SS1.p2.8.m8.1.1.cmml"><mi id="S3.SS1.p2.8.m8.1.1.2" xref="S3.SS1.p2.8.m8.1.1.2.cmml">δ</mi><mo id="S3.SS1.p2.8.m8.1.1.1" xref="S3.SS1.p2.8.m8.1.1.1.cmml">></mo><mn id="S3.SS1.p2.8.m8.1.1.3" xref="S3.SS1.p2.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.8.m8.1b"><apply id="S3.SS1.p2.8.m8.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1"><gt id="S3.SS1.p2.8.m8.1.1.1.cmml" xref="S3.SS1.p2.8.m8.1.1.1"></gt><ci id="S3.SS1.p2.8.m8.1.1.2.cmml" xref="S3.SS1.p2.8.m8.1.1.2">𝛿</ci><cn id="S3.SS1.p2.8.m8.1.1.3.cmml" type="integer" xref="S3.SS1.p2.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.8.m8.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.8.m8.1d">italic_δ > 0</annotation></semantics></math> such that</p> <ol class="ltx_enumerate" id="S3.I1"> <li class="ltx_item" id="S3.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S3.I1.i1.p1"> <p class="ltx_p" id="S3.I1.i1.p1.2"><math alttext="\widetilde{f}=f" class="ltx_Math" display="inline" id="S3.I1.i1.p1.1.m1.1"><semantics id="S3.I1.i1.p1.1.m1.1a"><mrow id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><mover accent="true" id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml"><mi id="S3.I1.i1.p1.1.m1.1.1.2.2" xref="S3.I1.i1.p1.1.m1.1.1.2.2.cmml">f</mi><mo id="S3.I1.i1.p1.1.m1.1.1.2.1" xref="S3.I1.i1.p1.1.m1.1.1.2.1.cmml">~</mo></mover><mo id="S3.I1.i1.p1.1.m1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.cmml">=</mo><mi id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><eq id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1"></eq><apply id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2"><ci id="S3.I1.i1.p1.1.m1.1.1.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2.1">~</ci><ci id="S3.I1.i1.p1.1.m1.1.1.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2.2">𝑓</ci></apply><ci id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">\widetilde{f}=f</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.1.m1.1d">over~ start_ARG italic_f end_ARG = italic_f</annotation></semantics></math> almost everywhere with respect to the Lebesgue measure on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.I1.i1.p1.2.m2.1"><semantics id="S3.I1.i1.p1.2.m2.1a"><msup id="S3.I1.i1.p1.2.m2.1.1" xref="S3.I1.i1.p1.2.m2.1.1.cmml"><mi id="S3.I1.i1.p1.2.m2.1.1.2" xref="S3.I1.i1.p1.2.m2.1.1.2.cmml">ℝ</mi><mi id="S3.I1.i1.p1.2.m2.1.1.3" xref="S3.I1.i1.p1.2.m2.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.2.m2.1b"><apply id="S3.I1.i1.p1.2.m2.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i1.p1.2.m2.1.1.1.cmml" xref="S3.I1.i1.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I1.i1.p1.2.m2.1.1.2.cmml" xref="S3.I1.i1.p1.2.m2.1.1.2">ℝ</ci><ci id="S3.I1.i1.p1.2.m2.1.1.3.cmml" xref="S3.I1.i1.p1.2.m2.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.2.m2.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i1.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S3.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S3.I1.i2.p1"> <p class="ltx_p" id="S3.I1.i2.p1.2"><math alttext="\widetilde{f}(x_{1},\cdot)\in L^{p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.I1.i2.p1.1.m1.4"><semantics id="S3.I1.i2.p1.1.m1.4a"><mrow id="S3.I1.i2.p1.1.m1.4.4" xref="S3.I1.i2.p1.1.m1.4.4.cmml"><mrow id="S3.I1.i2.p1.1.m1.3.3.1" xref="S3.I1.i2.p1.1.m1.3.3.1.cmml"><mover accent="true" id="S3.I1.i2.p1.1.m1.3.3.1.3" xref="S3.I1.i2.p1.1.m1.3.3.1.3.cmml"><mi id="S3.I1.i2.p1.1.m1.3.3.1.3.2" xref="S3.I1.i2.p1.1.m1.3.3.1.3.2.cmml">f</mi><mo id="S3.I1.i2.p1.1.m1.3.3.1.3.1" xref="S3.I1.i2.p1.1.m1.3.3.1.3.1.cmml">~</mo></mover><mo id="S3.I1.i2.p1.1.m1.3.3.1.2" xref="S3.I1.i2.p1.1.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.I1.i2.p1.1.m1.3.3.1.1.1" xref="S3.I1.i2.p1.1.m1.3.3.1.1.2.cmml"><mo id="S3.I1.i2.p1.1.m1.3.3.1.1.1.2" stretchy="false" xref="S3.I1.i2.p1.1.m1.3.3.1.1.2.cmml">(</mo><msub id="S3.I1.i2.p1.1.m1.3.3.1.1.1.1" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.cmml"><mi id="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.2" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.2.cmml">x</mi><mn id="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.3" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.I1.i2.p1.1.m1.3.3.1.1.1.3" rspace="0em" xref="S3.I1.i2.p1.1.m1.3.3.1.1.2.cmml">,</mo><mo id="S3.I1.i2.p1.1.m1.1.1" lspace="0em" rspace="0em" xref="S3.I1.i2.p1.1.m1.1.1.cmml">⋅</mo><mo id="S3.I1.i2.p1.1.m1.3.3.1.1.1.4" stretchy="false" xref="S3.I1.i2.p1.1.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.I1.i2.p1.1.m1.4.4.3" xref="S3.I1.i2.p1.1.m1.4.4.3.cmml">∈</mo><mrow id="S3.I1.i2.p1.1.m1.4.4.2" xref="S3.I1.i2.p1.1.m1.4.4.2.cmml"><msup id="S3.I1.i2.p1.1.m1.4.4.2.3" 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xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.I1.i2.p1.1.m1.4.4.2.1.1.3" xref="S3.I1.i2.p1.1.m1.4.4.2.1.2.cmml">;</mo><mi id="S3.I1.i2.p1.1.m1.2.2" xref="S3.I1.i2.p1.1.m1.2.2.cmml">X</mi><mo id="S3.I1.i2.p1.1.m1.4.4.2.1.1.4" stretchy="false" xref="S3.I1.i2.p1.1.m1.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.4b"><apply id="S3.I1.i2.p1.1.m1.4.4.cmml" xref="S3.I1.i2.p1.1.m1.4.4"><in id="S3.I1.i2.p1.1.m1.4.4.3.cmml" xref="S3.I1.i2.p1.1.m1.4.4.3"></in><apply id="S3.I1.i2.p1.1.m1.3.3.1.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1"><times id="S3.I1.i2.p1.1.m1.3.3.1.2.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.2"></times><apply id="S3.I1.i2.p1.1.m1.3.3.1.3.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.3"><ci id="S3.I1.i2.p1.1.m1.3.3.1.3.1.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.3.1">~</ci><ci id="S3.I1.i2.p1.1.m1.3.3.1.3.2.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.3.2">𝑓</ci></apply><interval closure="open" id="S3.I1.i2.p1.1.m1.3.3.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1"><apply id="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.2">𝑥</ci><cn id="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.3.3.1.1.1.1.3">1</cn></apply><ci id="S3.I1.i2.p1.1.m1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1">⋅</ci></interval></apply><apply id="S3.I1.i2.p1.1.m1.4.4.2.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2"><times id="S3.I1.i2.p1.1.m1.4.4.2.2.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.2"></times><apply id="S3.I1.i2.p1.1.m1.4.4.2.3.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.3"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.4.4.2.3.1.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.3">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.4.4.2.3.2.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.3.2">𝐿</ci><ci id="S3.I1.i2.p1.1.m1.4.4.2.3.3.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.3.3">𝑝</ci></apply><list id="S3.I1.i2.p1.1.m1.4.4.2.1.2.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1"><apply id="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1">superscript</csymbol><ci id="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.2">ℝ</ci><apply id="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3"><minus id="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.1.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.1"></minus><ci id="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.2.cmml" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.2">𝑑</ci><cn id="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.3.cmml" type="integer" xref="S3.I1.i2.p1.1.m1.4.4.2.1.1.1.3.3">1</cn></apply></apply><ci id="S3.I1.i2.p1.1.m1.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.4c">\widetilde{f}(x_{1},\cdot)\in L^{p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.1.m1.4d">over~ start_ARG italic_f end_ARG ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) ∈ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> for <math alttext="|x_{1}|<\delta" class="ltx_Math" display="inline" id="S3.I1.i2.p1.2.m2.1"><semantics id="S3.I1.i2.p1.2.m2.1a"><mrow id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml"><mrow id="S3.I1.i2.p1.2.m2.1.1.1.1" xref="S3.I1.i2.p1.2.m2.1.1.1.2.cmml"><mo id="S3.I1.i2.p1.2.m2.1.1.1.1.2" stretchy="false" xref="S3.I1.i2.p1.2.m2.1.1.1.2.1.cmml">|</mo><msub id="S3.I1.i2.p1.2.m2.1.1.1.1.1" xref="S3.I1.i2.p1.2.m2.1.1.1.1.1.cmml"><mi id="S3.I1.i2.p1.2.m2.1.1.1.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.1.1.1.2.cmml">x</mi><mn id="S3.I1.i2.p1.2.m2.1.1.1.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.I1.i2.p1.2.m2.1.1.1.1.3" stretchy="false" xref="S3.I1.i2.p1.2.m2.1.1.1.2.1.cmml">|</mo></mrow><mo id="S3.I1.i2.p1.2.m2.1.1.2" xref="S3.I1.i2.p1.2.m2.1.1.2.cmml"><</mo><mi id="S3.I1.i2.p1.2.m2.1.1.3" xref="S3.I1.i2.p1.2.m2.1.1.3.cmml">δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><apply id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1"><lt id="S3.I1.i2.p1.2.m2.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.2"></lt><apply id="S3.I1.i2.p1.2.m2.1.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1.1"><abs id="S3.I1.i2.p1.2.m2.1.1.1.2.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1.1.2"></abs><apply id="S3.I1.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S3.I1.i2.p1.2.m2.1.1.1.1.1.2">𝑥</ci><cn id="S3.I1.i2.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S3.I1.i2.p1.2.m2.1.1.1.1.1.3">1</cn></apply></apply><ci id="S3.I1.i2.p1.2.m2.1.1.3.cmml" xref="S3.I1.i2.p1.2.m2.1.1.3">𝛿</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">|x_{1}|<\delta</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i2.p1.2.m2.1d">| italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | < italic_δ</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S3.I1.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S3.I1.i3.p1"> <p class="ltx_p" id="S3.I1.i3.p1.2"><math alttext="\widetilde{f}(0,\cdot)=g" class="ltx_Math" display="inline" id="S3.I1.i3.p1.1.m1.2"><semantics id="S3.I1.i3.p1.1.m1.2a"><mrow id="S3.I1.i3.p1.1.m1.2.3" xref="S3.I1.i3.p1.1.m1.2.3.cmml"><mrow id="S3.I1.i3.p1.1.m1.2.3.2" xref="S3.I1.i3.p1.1.m1.2.3.2.cmml"><mover accent="true" id="S3.I1.i3.p1.1.m1.2.3.2.2" xref="S3.I1.i3.p1.1.m1.2.3.2.2.cmml"><mi id="S3.I1.i3.p1.1.m1.2.3.2.2.2" xref="S3.I1.i3.p1.1.m1.2.3.2.2.2.cmml">f</mi><mo id="S3.I1.i3.p1.1.m1.2.3.2.2.1" xref="S3.I1.i3.p1.1.m1.2.3.2.2.1.cmml">~</mo></mover><mo id="S3.I1.i3.p1.1.m1.2.3.2.1" xref="S3.I1.i3.p1.1.m1.2.3.2.1.cmml">⁢</mo><mrow id="S3.I1.i3.p1.1.m1.2.3.2.3.2" xref="S3.I1.i3.p1.1.m1.2.3.2.3.1.cmml"><mo id="S3.I1.i3.p1.1.m1.2.3.2.3.2.1" stretchy="false" xref="S3.I1.i3.p1.1.m1.2.3.2.3.1.cmml">(</mo><mn id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml">0</mn><mo id="S3.I1.i3.p1.1.m1.2.3.2.3.2.2" rspace="0em" xref="S3.I1.i3.p1.1.m1.2.3.2.3.1.cmml">,</mo><mo id="S3.I1.i3.p1.1.m1.2.2" lspace="0em" rspace="0em" xref="S3.I1.i3.p1.1.m1.2.2.cmml">⋅</mo><mo id="S3.I1.i3.p1.1.m1.2.3.2.3.2.3" stretchy="false" xref="S3.I1.i3.p1.1.m1.2.3.2.3.1.cmml">)</mo></mrow></mrow><mo id="S3.I1.i3.p1.1.m1.2.3.1" xref="S3.I1.i3.p1.1.m1.2.3.1.cmml">=</mo><mi id="S3.I1.i3.p1.1.m1.2.3.3" xref="S3.I1.i3.p1.1.m1.2.3.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.2b"><apply id="S3.I1.i3.p1.1.m1.2.3.cmml" xref="S3.I1.i3.p1.1.m1.2.3"><eq id="S3.I1.i3.p1.1.m1.2.3.1.cmml" xref="S3.I1.i3.p1.1.m1.2.3.1"></eq><apply id="S3.I1.i3.p1.1.m1.2.3.2.cmml" xref="S3.I1.i3.p1.1.m1.2.3.2"><times id="S3.I1.i3.p1.1.m1.2.3.2.1.cmml" xref="S3.I1.i3.p1.1.m1.2.3.2.1"></times><apply id="S3.I1.i3.p1.1.m1.2.3.2.2.cmml" xref="S3.I1.i3.p1.1.m1.2.3.2.2"><ci id="S3.I1.i3.p1.1.m1.2.3.2.2.1.cmml" xref="S3.I1.i3.p1.1.m1.2.3.2.2.1">~</ci><ci id="S3.I1.i3.p1.1.m1.2.3.2.2.2.cmml" xref="S3.I1.i3.p1.1.m1.2.3.2.2.2">𝑓</ci></apply><interval closure="open" id="S3.I1.i3.p1.1.m1.2.3.2.3.1.cmml" xref="S3.I1.i3.p1.1.m1.2.3.2.3.2"><cn id="S3.I1.i3.p1.1.m1.1.1.cmml" type="integer" xref="S3.I1.i3.p1.1.m1.1.1">0</cn><ci id="S3.I1.i3.p1.1.m1.2.2.cmml" xref="S3.I1.i3.p1.1.m1.2.2">⋅</ci></interval></apply><ci id="S3.I1.i3.p1.1.m1.2.3.3.cmml" xref="S3.I1.i3.p1.1.m1.2.3.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.2c">\widetilde{f}(0,\cdot)=g</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.1.m1.2d">over~ start_ARG italic_f end_ARG ( 0 , ⋅ ) = italic_g</annotation></semantics></math> almost everywhere with respect to the Lebesgue measure on <math alttext="\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S3.I1.i3.p1.2.m2.1"><semantics id="S3.I1.i3.p1.2.m2.1a"><msup id="S3.I1.i3.p1.2.m2.1.1" xref="S3.I1.i3.p1.2.m2.1.1.cmml"><mi id="S3.I1.i3.p1.2.m2.1.1.2" xref="S3.I1.i3.p1.2.m2.1.1.2.cmml">ℝ</mi><mrow id="S3.I1.i3.p1.2.m2.1.1.3" xref="S3.I1.i3.p1.2.m2.1.1.3.cmml"><mi id="S3.I1.i3.p1.2.m2.1.1.3.2" xref="S3.I1.i3.p1.2.m2.1.1.3.2.cmml">d</mi><mo id="S3.I1.i3.p1.2.m2.1.1.3.1" xref="S3.I1.i3.p1.2.m2.1.1.3.1.cmml">−</mo><mn id="S3.I1.i3.p1.2.m2.1.1.3.3" xref="S3.I1.i3.p1.2.m2.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.2.m2.1b"><apply id="S3.I1.i3.p1.2.m2.1.1.cmml" xref="S3.I1.i3.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.1.1.1.cmml" xref="S3.I1.i3.p1.2.m2.1.1">superscript</csymbol><ci id="S3.I1.i3.p1.2.m2.1.1.2.cmml" xref="S3.I1.i3.p1.2.m2.1.1.2">ℝ</ci><apply id="S3.I1.i3.p1.2.m2.1.1.3.cmml" xref="S3.I1.i3.p1.2.m2.1.1.3"><minus id="S3.I1.i3.p1.2.m2.1.1.3.1.cmml" xref="S3.I1.i3.p1.2.m2.1.1.3.1"></minus><ci id="S3.I1.i3.p1.2.m2.1.1.3.2.cmml" xref="S3.I1.i3.p1.2.m2.1.1.3.2">𝑑</ci><cn id="S3.I1.i3.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S3.I1.i3.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.2.m2.1c">\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i3.p1.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S3.I1.i4" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iv)</span> <div class="ltx_para" id="S3.I1.i4.p1"> <p class="ltx_p" 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xref="S3.I1.i4.p1.1.m1.2.2.2.2.1.1.3.2">𝑑</ci><cn id="S3.I1.i4.p1.1.m1.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S3.I1.i4.p1.1.m1.2.2.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.I1.i4.p1.1.m1.1.1.1.1.cmml" xref="S3.I1.i4.p1.1.m1.1.1.1.1">𝑋</ci></list></apply></apply></apply><cn id="S3.I1.i4.p1.1.m1.4.4.3.cmml" type="integer" xref="S3.I1.i4.p1.1.m1.4.4.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.1.m1.4c">\lim_{x_{1}\to 0}\|\widetilde{f}(x_{1},\cdot)-g\|_{L^{p}(\mathbb{R}^{d-1};X)}=0</annotation><annotation encoding="application/x-llamapun" id="S3.I1.i4.p1.1.m1.4d">roman_lim start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → 0 end_POSTSUBSCRIPT ∥ over~ start_ARG italic_f end_ARG ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) - italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S3.SS1.p2.16">By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Remark 4.2 & 4.3]</cite> this definition is independent of <math alttext="\widetilde{f},p" class="ltx_Math" display="inline" id="S3.SS1.p2.9.m1.2"><semantics id="S3.SS1.p2.9.m1.2a"><mrow id="S3.SS1.p2.9.m1.2.3.2" xref="S3.SS1.p2.9.m1.2.3.1.cmml"><mover accent="true" id="S3.SS1.p2.9.m1.1.1" xref="S3.SS1.p2.9.m1.1.1.cmml"><mi id="S3.SS1.p2.9.m1.1.1.2" xref="S3.SS1.p2.9.m1.1.1.2.cmml">f</mi><mo id="S3.SS1.p2.9.m1.1.1.1" xref="S3.SS1.p2.9.m1.1.1.1.cmml">~</mo></mover><mo id="S3.SS1.p2.9.m1.2.3.2.1" xref="S3.SS1.p2.9.m1.2.3.1.cmml">,</mo><mi id="S3.SS1.p2.9.m1.2.2" xref="S3.SS1.p2.9.m1.2.2.cmml">p</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.9.m1.2b"><list id="S3.SS1.p2.9.m1.2.3.1.cmml" xref="S3.SS1.p2.9.m1.2.3.2"><apply id="S3.SS1.p2.9.m1.1.1.cmml" xref="S3.SS1.p2.9.m1.1.1"><ci id="S3.SS1.p2.9.m1.1.1.1.cmml" xref="S3.SS1.p2.9.m1.1.1.1">~</ci><ci id="S3.SS1.p2.9.m1.1.1.2.cmml" xref="S3.SS1.p2.9.m1.1.1.2">𝑓</ci></apply><ci id="S3.SS1.p2.9.m1.2.2.cmml" xref="S3.SS1.p2.9.m1.2.2">𝑝</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.9.m1.2c">\widetilde{f},p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.9.m1.2d">over~ start_ARG italic_f end_ARG , italic_p</annotation></semantics></math> and <math alttext="\delta" class="ltx_Math" display="inline" id="S3.SS1.p2.10.m2.1"><semantics id="S3.SS1.p2.10.m2.1a"><mi id="S3.SS1.p2.10.m2.1.1" xref="S3.SS1.p2.10.m2.1.1.cmml">δ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.10.m2.1b"><ci id="S3.SS1.p2.10.m2.1.1.cmml" xref="S3.SS1.p2.10.m2.1.1">𝛿</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.10.m2.1c">\delta</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.10.m2.1d">italic_δ</annotation></semantics></math> and it coincides with the restriction of <math alttext="f" class="ltx_Math" display="inline" id="S3.SS1.p2.11.m3.1"><semantics id="S3.SS1.p2.11.m3.1a"><mi id="S3.SS1.p2.11.m3.1.1" xref="S3.SS1.p2.11.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.11.m3.1b"><ci id="S3.SS1.p2.11.m3.1.1.cmml" xref="S3.SS1.p2.11.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.11.m3.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.11.m3.1d">italic_f</annotation></semantics></math> to <math alttext="\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}" class="ltx_Math" display="inline" id="S3.SS1.p2.12.m4.4"><semantics id="S3.SS1.p2.12.m4.4a"><mrow id="S3.SS1.p2.12.m4.4.4.2" xref="S3.SS1.p2.12.m4.4.4.3.cmml"><mo id="S3.SS1.p2.12.m4.4.4.2.3" stretchy="false" xref="S3.SS1.p2.12.m4.4.4.3.1.cmml">{</mo><mrow id="S3.SS1.p2.12.m4.3.3.1.1.2" 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id="S3.SS1.p2.12.m4.4.4.2.2.2.1" xref="S3.SS1.p2.12.m4.4.4.2.2.2.1.cmml">~</mo></mover><mo id="S3.SS1.p2.12.m4.4.4.2.2.1" xref="S3.SS1.p2.12.m4.4.4.2.2.1.cmml">∈</mo><msup id="S3.SS1.p2.12.m4.4.4.2.2.3" xref="S3.SS1.p2.12.m4.4.4.2.2.3.cmml"><mi id="S3.SS1.p2.12.m4.4.4.2.2.3.2" xref="S3.SS1.p2.12.m4.4.4.2.2.3.2.cmml">ℝ</mi><mrow id="S3.SS1.p2.12.m4.4.4.2.2.3.3" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3.cmml"><mi id="S3.SS1.p2.12.m4.4.4.2.2.3.3.2" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3.2.cmml">d</mi><mo id="S3.SS1.p2.12.m4.4.4.2.2.3.3.1" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3.1.cmml">−</mo><mn id="S3.SS1.p2.12.m4.4.4.2.2.3.3.3" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3.3.cmml">1</mn></mrow></msup></mrow><mo id="S3.SS1.p2.12.m4.4.4.2.5" stretchy="false" xref="S3.SS1.p2.12.m4.4.4.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.12.m4.4b"><apply id="S3.SS1.p2.12.m4.4.4.3.cmml" xref="S3.SS1.p2.12.m4.4.4.2"><csymbol cd="latexml" id="S3.SS1.p2.12.m4.4.4.3.1.cmml" xref="S3.SS1.p2.12.m4.4.4.2.3">conditional-set</csymbol><interval closure="open" id="S3.SS1.p2.12.m4.3.3.1.1.1.cmml" xref="S3.SS1.p2.12.m4.3.3.1.1.2"><cn id="S3.SS1.p2.12.m4.1.1.cmml" type="integer" xref="S3.SS1.p2.12.m4.1.1">0</cn><apply id="S3.SS1.p2.12.m4.2.2.cmml" xref="S3.SS1.p2.12.m4.2.2"><ci id="S3.SS1.p2.12.m4.2.2.1.cmml" xref="S3.SS1.p2.12.m4.2.2.1">~</ci><ci id="S3.SS1.p2.12.m4.2.2.2.cmml" xref="S3.SS1.p2.12.m4.2.2.2">𝑥</ci></apply></interval><apply id="S3.SS1.p2.12.m4.4.4.2.2.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2"><in id="S3.SS1.p2.12.m4.4.4.2.2.1.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.1"></in><apply id="S3.SS1.p2.12.m4.4.4.2.2.2.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.2"><ci id="S3.SS1.p2.12.m4.4.4.2.2.2.1.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.2.1">~</ci><ci id="S3.SS1.p2.12.m4.4.4.2.2.2.2.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.2.2">𝑥</ci></apply><apply id="S3.SS1.p2.12.m4.4.4.2.2.3.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p2.12.m4.4.4.2.2.3.1.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.3">superscript</csymbol><ci id="S3.SS1.p2.12.m4.4.4.2.2.3.2.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.3.2">ℝ</ci><apply id="S3.SS1.p2.12.m4.4.4.2.2.3.3.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3"><minus id="S3.SS1.p2.12.m4.4.4.2.2.3.3.1.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3.1"></minus><ci id="S3.SS1.p2.12.m4.4.4.2.2.3.3.2.cmml" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3.2">𝑑</ci><cn id="S3.SS1.p2.12.m4.4.4.2.2.3.3.3.cmml" type="integer" xref="S3.SS1.p2.12.m4.4.4.2.2.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.12.m4.4c">\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.12.m4.4d">{ ( 0 , over~ start_ARG italic_x end_ARG ) : over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math> if <math alttext="f" class="ltx_Math" display="inline" id="S3.SS1.p2.13.m5.1"><semantics id="S3.SS1.p2.13.m5.1a"><mi id="S3.SS1.p2.13.m5.1.1" xref="S3.SS1.p2.13.m5.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.13.m5.1b"><ci id="S3.SS1.p2.13.m5.1.1.cmml" xref="S3.SS1.p2.13.m5.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.13.m5.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.13.m5.1d">italic_f</annotation></semantics></math> is continuous on <math alttext="(-\delta,\delta)\times\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S3.SS1.p2.14.m6.2"><semantics id="S3.SS1.p2.14.m6.2a"><mrow id="S3.SS1.p2.14.m6.2.2" xref="S3.SS1.p2.14.m6.2.2.cmml"><mrow id="S3.SS1.p2.14.m6.2.2.1.1" xref="S3.SS1.p2.14.m6.2.2.1.2.cmml"><mo id="S3.SS1.p2.14.m6.2.2.1.1.2" stretchy="false" xref="S3.SS1.p2.14.m6.2.2.1.2.cmml">(</mo><mrow id="S3.SS1.p2.14.m6.2.2.1.1.1" xref="S3.SS1.p2.14.m6.2.2.1.1.1.cmml"><mo id="S3.SS1.p2.14.m6.2.2.1.1.1a" xref="S3.SS1.p2.14.m6.2.2.1.1.1.cmml">−</mo><mi id="S3.SS1.p2.14.m6.2.2.1.1.1.2" xref="S3.SS1.p2.14.m6.2.2.1.1.1.2.cmml">δ</mi></mrow><mo id="S3.SS1.p2.14.m6.2.2.1.1.3" xref="S3.SS1.p2.14.m6.2.2.1.2.cmml">,</mo><mi id="S3.SS1.p2.14.m6.1.1" xref="S3.SS1.p2.14.m6.1.1.cmml">δ</mi><mo id="S3.SS1.p2.14.m6.2.2.1.1.4" rspace="0.055em" stretchy="false" xref="S3.SS1.p2.14.m6.2.2.1.2.cmml">)</mo></mrow><mo id="S3.SS1.p2.14.m6.2.2.2" rspace="0.222em" xref="S3.SS1.p2.14.m6.2.2.2.cmml">×</mo><msup id="S3.SS1.p2.14.m6.2.2.3" xref="S3.SS1.p2.14.m6.2.2.3.cmml"><mi id="S3.SS1.p2.14.m6.2.2.3.2" xref="S3.SS1.p2.14.m6.2.2.3.2.cmml">ℝ</mi><mrow id="S3.SS1.p2.14.m6.2.2.3.3" xref="S3.SS1.p2.14.m6.2.2.3.3.cmml"><mi id="S3.SS1.p2.14.m6.2.2.3.3.2" xref="S3.SS1.p2.14.m6.2.2.3.3.2.cmml">d</mi><mo id="S3.SS1.p2.14.m6.2.2.3.3.1" xref="S3.SS1.p2.14.m6.2.2.3.3.1.cmml">−</mo><mn id="S3.SS1.p2.14.m6.2.2.3.3.3" xref="S3.SS1.p2.14.m6.2.2.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.14.m6.2b"><apply id="S3.SS1.p2.14.m6.2.2.cmml" xref="S3.SS1.p2.14.m6.2.2"><times id="S3.SS1.p2.14.m6.2.2.2.cmml" xref="S3.SS1.p2.14.m6.2.2.2"></times><interval closure="open" id="S3.SS1.p2.14.m6.2.2.1.2.cmml" xref="S3.SS1.p2.14.m6.2.2.1.1"><apply id="S3.SS1.p2.14.m6.2.2.1.1.1.cmml" xref="S3.SS1.p2.14.m6.2.2.1.1.1"><minus id="S3.SS1.p2.14.m6.2.2.1.1.1.1.cmml" xref="S3.SS1.p2.14.m6.2.2.1.1.1"></minus><ci id="S3.SS1.p2.14.m6.2.2.1.1.1.2.cmml" xref="S3.SS1.p2.14.m6.2.2.1.1.1.2">𝛿</ci></apply><ci id="S3.SS1.p2.14.m6.1.1.cmml" xref="S3.SS1.p2.14.m6.1.1">𝛿</ci></interval><apply id="S3.SS1.p2.14.m6.2.2.3.cmml" xref="S3.SS1.p2.14.m6.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.p2.14.m6.2.2.3.1.cmml" xref="S3.SS1.p2.14.m6.2.2.3">superscript</csymbol><ci id="S3.SS1.p2.14.m6.2.2.3.2.cmml" xref="S3.SS1.p2.14.m6.2.2.3.2">ℝ</ci><apply id="S3.SS1.p2.14.m6.2.2.3.3.cmml" xref="S3.SS1.p2.14.m6.2.2.3.3"><minus id="S3.SS1.p2.14.m6.2.2.3.3.1.cmml" xref="S3.SS1.p2.14.m6.2.2.3.3.1"></minus><ci id="S3.SS1.p2.14.m6.2.2.3.3.2.cmml" xref="S3.SS1.p2.14.m6.2.2.3.3.2">𝑑</ci><cn id="S3.SS1.p2.14.m6.2.2.3.3.3.cmml" type="integer" xref="S3.SS1.p2.14.m6.2.2.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.14.m6.2c">(-\delta,\delta)\times\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.14.m6.2d">( - italic_δ , italic_δ ) × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. If the trace exists according to the above definition, then we write <math alttext="\operatorname{Tr}f=g" class="ltx_Math" display="inline" id="S3.SS1.p2.15.m7.1"><semantics id="S3.SS1.p2.15.m7.1a"><mrow id="S3.SS1.p2.15.m7.1.1" xref="S3.SS1.p2.15.m7.1.1.cmml"><mrow id="S3.SS1.p2.15.m7.1.1.2" xref="S3.SS1.p2.15.m7.1.1.2.cmml"><mi id="S3.SS1.p2.15.m7.1.1.2.1" xref="S3.SS1.p2.15.m7.1.1.2.1.cmml">Tr</mi><mo id="S3.SS1.p2.15.m7.1.1.2a" lspace="0.167em" xref="S3.SS1.p2.15.m7.1.1.2.cmml">⁡</mo><mi id="S3.SS1.p2.15.m7.1.1.2.2" xref="S3.SS1.p2.15.m7.1.1.2.2.cmml">f</mi></mrow><mo id="S3.SS1.p2.15.m7.1.1.1" xref="S3.SS1.p2.15.m7.1.1.1.cmml">=</mo><mi id="S3.SS1.p2.15.m7.1.1.3" xref="S3.SS1.p2.15.m7.1.1.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.15.m7.1b"><apply id="S3.SS1.p2.15.m7.1.1.cmml" xref="S3.SS1.p2.15.m7.1.1"><eq id="S3.SS1.p2.15.m7.1.1.1.cmml" xref="S3.SS1.p2.15.m7.1.1.1"></eq><apply id="S3.SS1.p2.15.m7.1.1.2.cmml" xref="S3.SS1.p2.15.m7.1.1.2"><ci id="S3.SS1.p2.15.m7.1.1.2.1.cmml" xref="S3.SS1.p2.15.m7.1.1.2.1">Tr</ci><ci id="S3.SS1.p2.15.m7.1.1.2.2.cmml" xref="S3.SS1.p2.15.m7.1.1.2.2">𝑓</ci></apply><ci id="S3.SS1.p2.15.m7.1.1.3.cmml" xref="S3.SS1.p2.15.m7.1.1.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.15.m7.1c">\operatorname{Tr}f=g</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.15.m7.1d">roman_Tr italic_f = italic_g</annotation></semantics></math>, so that we obtain a linear operator <math alttext="\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.SS1.p2.16.m8.1"><semantics id="S3.SS1.p2.16.m8.1a"><mi id="S3.SS1.p2.16.m8.1.1" xref="S3.SS1.p2.16.m8.1.1.cmml">Tr</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.p2.16.m8.1b"><ci id="S3.SS1.p2.16.m8.1.1.cmml" xref="S3.SS1.p2.16.m8.1.1">Tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p2.16.m8.1c">\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p2.16.m8.1d">roman_Tr</annotation></semantics></math>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S3.SS1.p3"> <p class="ltx_p" id="S3.SS1.p3.2">The following results are extensions of the arguments in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>]</cite> to the case with weights <math alttext="w_{\gamma}(x)=|x_{1}|^{\gamma}" class="ltx_Math" display="inline" id="S3.SS1.p3.1.m1.2"><semantics id="S3.SS1.p3.1.m1.2a"><mrow id="S3.SS1.p3.1.m1.2.2" xref="S3.SS1.p3.1.m1.2.2.cmml"><mrow id="S3.SS1.p3.1.m1.2.2.3" xref="S3.SS1.p3.1.m1.2.2.3.cmml"><msub id="S3.SS1.p3.1.m1.2.2.3.2" xref="S3.SS1.p3.1.m1.2.2.3.2.cmml"><mi id="S3.SS1.p3.1.m1.2.2.3.2.2" xref="S3.SS1.p3.1.m1.2.2.3.2.2.cmml">w</mi><mi id="S3.SS1.p3.1.m1.2.2.3.2.3" xref="S3.SS1.p3.1.m1.2.2.3.2.3.cmml">γ</mi></msub><mo id="S3.SS1.p3.1.m1.2.2.3.1" xref="S3.SS1.p3.1.m1.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.p3.1.m1.2.2.3.3.2" xref="S3.SS1.p3.1.m1.2.2.3.cmml"><mo id="S3.SS1.p3.1.m1.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.p3.1.m1.2.2.3.cmml">(</mo><mi id="S3.SS1.p3.1.m1.1.1" xref="S3.SS1.p3.1.m1.1.1.cmml">x</mi><mo id="S3.SS1.p3.1.m1.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.p3.1.m1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS1.p3.1.m1.2.2.2" xref="S3.SS1.p3.1.m1.2.2.2.cmml">=</mo><msup id="S3.SS1.p3.1.m1.2.2.1" xref="S3.SS1.p3.1.m1.2.2.1.cmml"><mrow id="S3.SS1.p3.1.m1.2.2.1.1.1" xref="S3.SS1.p3.1.m1.2.2.1.1.2.cmml"><mo id="S3.SS1.p3.1.m1.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.p3.1.m1.2.2.1.1.2.1.cmml">|</mo><msub id="S3.SS1.p3.1.m1.2.2.1.1.1.1" xref="S3.SS1.p3.1.m1.2.2.1.1.1.1.cmml"><mi id="S3.SS1.p3.1.m1.2.2.1.1.1.1.2" xref="S3.SS1.p3.1.m1.2.2.1.1.1.1.2.cmml">x</mi><mn id="S3.SS1.p3.1.m1.2.2.1.1.1.1.3" xref="S3.SS1.p3.1.m1.2.2.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.p3.1.m1.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.p3.1.m1.2.2.1.1.2.1.cmml">|</mo></mrow><mi id="S3.SS1.p3.1.m1.2.2.1.3" xref="S3.SS1.p3.1.m1.2.2.1.3.cmml">γ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.1.m1.2b"><apply id="S3.SS1.p3.1.m1.2.2.cmml" xref="S3.SS1.p3.1.m1.2.2"><eq id="S3.SS1.p3.1.m1.2.2.2.cmml" xref="S3.SS1.p3.1.m1.2.2.2"></eq><apply id="S3.SS1.p3.1.m1.2.2.3.cmml" xref="S3.SS1.p3.1.m1.2.2.3"><times id="S3.SS1.p3.1.m1.2.2.3.1.cmml" xref="S3.SS1.p3.1.m1.2.2.3.1"></times><apply id="S3.SS1.p3.1.m1.2.2.3.2.cmml" xref="S3.SS1.p3.1.m1.2.2.3.2"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.2.2.3.2.1.cmml" xref="S3.SS1.p3.1.m1.2.2.3.2">subscript</csymbol><ci id="S3.SS1.p3.1.m1.2.2.3.2.2.cmml" xref="S3.SS1.p3.1.m1.2.2.3.2.2">𝑤</ci><ci id="S3.SS1.p3.1.m1.2.2.3.2.3.cmml" xref="S3.SS1.p3.1.m1.2.2.3.2.3">𝛾</ci></apply><ci id="S3.SS1.p3.1.m1.1.1.cmml" xref="S3.SS1.p3.1.m1.1.1">𝑥</ci></apply><apply id="S3.SS1.p3.1.m1.2.2.1.cmml" xref="S3.SS1.p3.1.m1.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.2.2.1.2.cmml" xref="S3.SS1.p3.1.m1.2.2.1">superscript</csymbol><apply id="S3.SS1.p3.1.m1.2.2.1.1.2.cmml" xref="S3.SS1.p3.1.m1.2.2.1.1.1"><abs id="S3.SS1.p3.1.m1.2.2.1.1.2.1.cmml" xref="S3.SS1.p3.1.m1.2.2.1.1.1.2"></abs><apply id="S3.SS1.p3.1.m1.2.2.1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.p3.1.m1.2.2.1.1.1.1.1.cmml" xref="S3.SS1.p3.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S3.SS1.p3.1.m1.2.2.1.1.1.1.2.cmml" xref="S3.SS1.p3.1.m1.2.2.1.1.1.1.2">𝑥</ci><cn id="S3.SS1.p3.1.m1.2.2.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.p3.1.m1.2.2.1.1.1.1.3">1</cn></apply></apply><ci id="S3.SS1.p3.1.m1.2.2.1.3.cmml" xref="S3.SS1.p3.1.m1.2.2.1.3">𝛾</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.1.m1.2c">w_{\gamma}(x)=|x_{1}|^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.1.m1.2d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ( italic_x ) = | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math>, which can also be found in version 1 of the arXiv preprint of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib59" title="">59</a>]</cite>. We note that radial weights <math alttext="|x|^{\gamma}" class="ltx_Math" display="inline" id="S3.SS1.p3.2.m2.1"><semantics id="S3.SS1.p3.2.m2.1a"><msup id="S3.SS1.p3.2.m2.1.2" xref="S3.SS1.p3.2.m2.1.2.cmml"><mrow id="S3.SS1.p3.2.m2.1.2.2.2" xref="S3.SS1.p3.2.m2.1.2.2.1.cmml"><mo id="S3.SS1.p3.2.m2.1.2.2.2.1" stretchy="false" xref="S3.SS1.p3.2.m2.1.2.2.1.1.cmml">|</mo><mi id="S3.SS1.p3.2.m2.1.1" xref="S3.SS1.p3.2.m2.1.1.cmml">x</mi><mo id="S3.SS1.p3.2.m2.1.2.2.2.2" stretchy="false" xref="S3.SS1.p3.2.m2.1.2.2.1.1.cmml">|</mo></mrow><mi id="S3.SS1.p3.2.m2.1.2.3" xref="S3.SS1.p3.2.m2.1.2.3.cmml">γ</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.p3.2.m2.1b"><apply id="S3.SS1.p3.2.m2.1.2.cmml" xref="S3.SS1.p3.2.m2.1.2"><csymbol cd="ambiguous" id="S3.SS1.p3.2.m2.1.2.1.cmml" xref="S3.SS1.p3.2.m2.1.2">superscript</csymbol><apply id="S3.SS1.p3.2.m2.1.2.2.1.cmml" xref="S3.SS1.p3.2.m2.1.2.2.2"><abs id="S3.SS1.p3.2.m2.1.2.2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.2.2.2.1"></abs><ci id="S3.SS1.p3.2.m2.1.1.cmml" xref="S3.SS1.p3.2.m2.1.1">𝑥</ci></apply><ci id="S3.SS1.p3.2.m2.1.2.3.cmml" xref="S3.SS1.p3.2.m2.1.2.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p3.2.m2.1c">|x|^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p3.2.m2.1d">| italic_x | start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> are considered in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib1" title="">1</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib29" title="">29</a>]</cite>. Anisotropic versions of trace theorems have appeared in, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib34" title="">34</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib46" title="">46</a>, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib47" title="">47</a>]</cite> and see the references therein.</p> </div> <div class="ltx_para" id="S3.SS1.p4"> <p class="ltx_p" id="S3.SS1.p4.1">We start with a lemma analogous to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Lemma 4.5]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.1.1.1">Lemma 3.1</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem1.p1"> <p class="ltx_p" id="S3.Thmtheorem1.p1.10"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem1.p1.10.10">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.1.1.m1.2"><semantics id="S3.Thmtheorem1.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem1.p1.1.1.m1.2.3" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem1.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem1.p1.1.1.m1.1.1" xref="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem1.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem1.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.1.1.m1.2b"><apply id="S3.Thmtheorem1.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3"><in id="S3.Thmtheorem1.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.1"></in><ci id="S3.Thmtheorem1.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.3.3.2"><cn id="S3.Thmtheorem1.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem1.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem1.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.2.2.m2.1"><semantics id="S3.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem1.p1.2.2.m2.1.1" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml"><mo id="S3.Thmtheorem1.p1.2.2.m2.1.1.3a" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.2.2.m2.1b"><apply id="S3.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1"><gt id="S3.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.1"></gt><ci id="S3.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.2">𝛾</ci><apply id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3"><minus id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3"></minus><cn id="S3.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml" type="integer" xref="S3.Thmtheorem1.p1.2.2.m2.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.2.2.m2.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.2.2.m2.1d">italic_γ > - 1</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.3.3.m3.1"><semantics id="S3.Thmtheorem1.p1.3.3.m3.1a"><mi id="S3.Thmtheorem1.p1.3.3.m3.1.1" xref="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.3.3.m3.1b"><ci id="S3.Thmtheorem1.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem1.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.3.3.m3.1d">italic_X</annotation></semantics></math> be a Banach space. Let <math alttext="f\in L^{p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.4.4.m4.3"><semantics id="S3.Thmtheorem1.p1.4.4.m4.3a"><mrow id="S3.Thmtheorem1.p1.4.4.m4.3.3" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.3.3.4" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.4.cmml">f</mi><mo id="S3.Thmtheorem1.p1.4.4.m4.3.3.3" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.3.cmml">∈</mo><mrow id="S3.Thmtheorem1.p1.4.4.m4.3.3.2" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.cmml"><msup id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.2" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.2.cmml">L</mi><mi id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.3" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.3.cmml">p</mi></msup><mo id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.3" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.3.cmml">⁢</mo><mrow id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.3.cmml"><mo id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.3.cmml">(</mo><msup id="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.2" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.3" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.4" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.3.cmml">,</mo><msub id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.cmml"><mi id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.2" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.2.cmml">w</mi><mi id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.3" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.5" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.3.cmml">;</mo><mi id="S3.Thmtheorem1.p1.4.4.m4.1.1" xref="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml">X</mi><mo id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.6" stretchy="false" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.4.4.m4.3b"><apply id="S3.Thmtheorem1.p1.4.4.m4.3.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3"><in id="S3.Thmtheorem1.p1.4.4.m4.3.3.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.3"></in><ci id="S3.Thmtheorem1.p1.4.4.m4.3.3.4.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.4">𝑓</ci><apply id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2"><times id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.3"></times><apply id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4">superscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.2">𝐿</ci><ci id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.4.3">𝑝</ci></apply><vector id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2"><apply id="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.2.2.1.1.1.1.3">𝑑</ci></apply><apply id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.2.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.2">𝑤</ci><ci id="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.3.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.3.3.2.2.2.2.3">𝛾</ci></apply><ci id="S3.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem1.p1.4.4.m4.1.1">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.4.4.m4.3c">f\in L^{p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.4.4.m4.3d">italic_f ∈ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\text{\rm supp\,}\widehat{f}\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq R\}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.5.5.m5.3"><semantics id="S3.Thmtheorem1.p1.5.5.m5.3a"><mrow id="S3.Thmtheorem1.p1.5.5.m5.3.3" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.cmml"><mrow id="S3.Thmtheorem1.p1.5.5.m5.3.3.4" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.cmml"><mtext id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.2" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.2a.cmml">supp </mtext><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.1" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.1.cmml">⁢</mo><mover accent="true" id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.cmml"><mi id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.2" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.2.cmml">f</mi><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.1" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.1.cmml">^</mo></mover></mrow><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.3" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.3.cmml">⊆</mo><mrow id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml"><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.3" stretchy="false" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.3.1.cmml">{</mo><mrow id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.cmml"><mi id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2.cmml">ξ</mi><mo id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.1" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.1.cmml">∈</mo><msup id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.cmml"><mi id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.2" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.2.cmml">ℝ</mi><mi id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.3" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.3.cmml">d</mi></msup></mrow><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.3.1.cmml">:</mo><mrow id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.cmml"><mrow id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.2" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.1.cmml"><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.2.1" stretchy="false" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.1.1.cmml">|</mo><mi id="S3.Thmtheorem1.p1.5.5.m5.1.1" xref="S3.Thmtheorem1.p1.5.5.m5.1.1.cmml">ξ</mi><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.2.2" stretchy="false" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.1.1.cmml">|</mo></mrow><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.1" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.1.cmml">≤</mo><mi id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.cmml">R</mi></mrow><mo id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.5" stretchy="false" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.5.5.m5.3b"><apply id="S3.Thmtheorem1.p1.5.5.m5.3.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3"><subset id="S3.Thmtheorem1.p1.5.5.m5.3.3.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.3"></subset><apply id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4"><times id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.1"></times><ci id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.2a.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.2"><mtext id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.2">supp </mtext></ci><apply id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3"><ci id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.1">^</ci><ci id="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.4.3.2">𝑓</ci></apply></apply><apply id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2"><csymbol cd="latexml" id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.3.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.3">conditional-set</csymbol><apply id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1"><in id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.1"></in><ci id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.2">𝜉</ci><apply id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3">superscript</csymbol><ci id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.2">ℝ</ci><ci id="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.2.2.1.1.1.3.3">𝑑</ci></apply></apply><apply id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2"><leq id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.1"></leq><apply id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.2"><abs id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.2.2.1"></abs><ci id="S3.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.1.1">𝜉</ci></apply><ci id="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3.cmml" xref="S3.Thmtheorem1.p1.5.5.m5.3.3.2.2.2.3">𝑅</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.5.5.m5.3c">\text{\rm supp\,}\widehat{f}\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq R\}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.5.5.m5.3d">supp over^ start_ARG italic_f end_ARG ⊆ { italic_ξ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : | italic_ξ | ≤ italic_R }</annotation></semantics></math> for some <math alttext="R>0" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.6.6.m6.1"><semantics id="S3.Thmtheorem1.p1.6.6.m6.1a"><mrow id="S3.Thmtheorem1.p1.6.6.m6.1.1" xref="S3.Thmtheorem1.p1.6.6.m6.1.1.cmml"><mi id="S3.Thmtheorem1.p1.6.6.m6.1.1.2" xref="S3.Thmtheorem1.p1.6.6.m6.1.1.2.cmml">R</mi><mo id="S3.Thmtheorem1.p1.6.6.m6.1.1.1" xref="S3.Thmtheorem1.p1.6.6.m6.1.1.1.cmml">></mo><mn id="S3.Thmtheorem1.p1.6.6.m6.1.1.3" xref="S3.Thmtheorem1.p1.6.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.6.6.m6.1b"><apply id="S3.Thmtheorem1.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem1.p1.6.6.m6.1.1"><gt id="S3.Thmtheorem1.p1.6.6.m6.1.1.1.cmml" xref="S3.Thmtheorem1.p1.6.6.m6.1.1.1"></gt><ci id="S3.Thmtheorem1.p1.6.6.m6.1.1.2.cmml" xref="S3.Thmtheorem1.p1.6.6.m6.1.1.2">𝑅</ci><cn id="S3.Thmtheorem1.p1.6.6.m6.1.1.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.6.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.6.6.m6.1c">R>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.6.6.m6.1d">italic_R > 0</annotation></semantics></math>. Then there exists a <math alttext="C>0" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.7.7.m7.1"><semantics id="S3.Thmtheorem1.p1.7.7.m7.1a"><mrow id="S3.Thmtheorem1.p1.7.7.m7.1.1" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.cmml"><mi id="S3.Thmtheorem1.p1.7.7.m7.1.1.2" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2.cmml">C</mi><mo id="S3.Thmtheorem1.p1.7.7.m7.1.1.1" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.1.cmml">></mo><mn id="S3.Thmtheorem1.p1.7.7.m7.1.1.3" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.7.7.m7.1b"><apply id="S3.Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1"><gt id="S3.Thmtheorem1.p1.7.7.m7.1.1.1.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.1"></gt><ci id="S3.Thmtheorem1.p1.7.7.m7.1.1.2.cmml" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.2">𝐶</ci><cn id="S3.Thmtheorem1.p1.7.7.m7.1.1.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.7.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.7.7.m7.1c">C>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.7.7.m7.1d">italic_C > 0</annotation></semantics></math>, independent of <math alttext="f" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.8.8.m8.1"><semantics id="S3.Thmtheorem1.p1.8.8.m8.1a"><mi id="S3.Thmtheorem1.p1.8.8.m8.1.1" xref="S3.Thmtheorem1.p1.8.8.m8.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.8.8.m8.1b"><ci id="S3.Thmtheorem1.p1.8.8.m8.1.1.cmml" xref="S3.Thmtheorem1.p1.8.8.m8.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.8.8.m8.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.8.8.m8.1d">italic_f</annotation></semantics></math> and <math alttext="R" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.9.9.m9.1"><semantics id="S3.Thmtheorem1.p1.9.9.m9.1a"><mi id="S3.Thmtheorem1.p1.9.9.m9.1.1" xref="S3.Thmtheorem1.p1.9.9.m9.1.1.cmml">R</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.9.9.m9.1b"><ci id="S3.Thmtheorem1.p1.9.9.m9.1.1.cmml" xref="S3.Thmtheorem1.p1.9.9.m9.1.1">𝑅</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.9.9.m9.1c">R</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.9.9.m9.1d">italic_R</annotation></semantics></math>, such that for all <math alttext="x_{1}\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.Thmtheorem1.p1.10.10.m10.1"><semantics id="S3.Thmtheorem1.p1.10.10.m10.1a"><mrow id="S3.Thmtheorem1.p1.10.10.m10.1.1" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.cmml"><msub id="S3.Thmtheorem1.p1.10.10.m10.1.1.2" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.2.cmml"><mi id="S3.Thmtheorem1.p1.10.10.m10.1.1.2.2" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.2.2.cmml">x</mi><mn id="S3.Thmtheorem1.p1.10.10.m10.1.1.2.3" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.2.3.cmml">1</mn></msub><mo id="S3.Thmtheorem1.p1.10.10.m10.1.1.1" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.1.cmml">∈</mo><mi id="S3.Thmtheorem1.p1.10.10.m10.1.1.3" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem1.p1.10.10.m10.1b"><apply id="S3.Thmtheorem1.p1.10.10.m10.1.1.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.1.1"><in id="S3.Thmtheorem1.p1.10.10.m10.1.1.1.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.1"></in><apply id="S3.Thmtheorem1.p1.10.10.m10.1.1.2.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.2"><csymbol cd="ambiguous" id="S3.Thmtheorem1.p1.10.10.m10.1.1.2.1.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.2">subscript</csymbol><ci id="S3.Thmtheorem1.p1.10.10.m10.1.1.2.2.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.2.2">𝑥</ci><cn id="S3.Thmtheorem1.p1.10.10.m10.1.1.2.3.cmml" type="integer" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.2.3">1</cn></apply><ci id="S3.Thmtheorem1.p1.10.10.m10.1.1.3.cmml" xref="S3.Thmtheorem1.p1.10.10.m10.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem1.p1.10.10.m10.1c">x_{1}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem1.p1.10.10.m10.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math>, we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f(x_{1},\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq CR^{\frac{\gamma+1}{p}}\|f% \|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}." class="ltx_Math" display="block" id="S3.Ex1.m1.8"><semantics id="S3.Ex1.m1.8a"><mrow id="S3.Ex1.m1.8.8.1" xref="S3.Ex1.m1.8.8.1.1.cmml"><mrow id="S3.Ex1.m1.8.8.1.1" xref="S3.Ex1.m1.8.8.1.1.cmml"><msub id="S3.Ex1.m1.8.8.1.1.1" xref="S3.Ex1.m1.8.8.1.1.1.cmml"><mrow id="S3.Ex1.m1.8.8.1.1.1.1.1" xref="S3.Ex1.m1.8.8.1.1.1.1.2.cmml"><mo id="S3.Ex1.m1.8.8.1.1.1.1.1.2" stretchy="false" xref="S3.Ex1.m1.8.8.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex1.m1.8.8.1.1.1.1.1.1" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.cmml"><mi id="S3.Ex1.m1.8.8.1.1.1.1.1.1.3" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.3.cmml">f</mi><mo id="S3.Ex1.m1.8.8.1.1.1.1.1.1.2" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.2.cmml">(</mo><msub id="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.1" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mn id="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.3" rspace="0em" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.2.cmml">,</mo><mo id="S3.Ex1.m1.6.6" lspace="0em" rspace="0em" xref="S3.Ex1.m1.6.6.cmml">⋅</mo><mo id="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.1.4" stretchy="false" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex1.m1.8.8.1.1.1.1.1.3" stretchy="false" xref="S3.Ex1.m1.8.8.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex1.m1.2.2.2" xref="S3.Ex1.m1.2.2.2.cmml"><msup id="S3.Ex1.m1.2.2.2.4" xref="S3.Ex1.m1.2.2.2.4.cmml"><mi id="S3.Ex1.m1.2.2.2.4.2" xref="S3.Ex1.m1.2.2.2.4.2.cmml">L</mi><mi id="S3.Ex1.m1.2.2.2.4.3" xref="S3.Ex1.m1.2.2.2.4.3.cmml">p</mi></msup><mo id="S3.Ex1.m1.2.2.2.3" xref="S3.Ex1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.Ex1.m1.2.2.2.2.1" xref="S3.Ex1.m1.2.2.2.2.2.cmml"><mo id="S3.Ex1.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex1.m1.2.2.2.2.2.cmml">(</mo><msup id="S3.Ex1.m1.2.2.2.2.1.1" xref="S3.Ex1.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex1.m1.2.2.2.2.1.1.2" xref="S3.Ex1.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mrow id="S3.Ex1.m1.2.2.2.2.1.1.3" xref="S3.Ex1.m1.2.2.2.2.1.1.3.cmml"><mi id="S3.Ex1.m1.2.2.2.2.1.1.3.2" xref="S3.Ex1.m1.2.2.2.2.1.1.3.2.cmml">d</mi><mo id="S3.Ex1.m1.2.2.2.2.1.1.3.1" xref="S3.Ex1.m1.2.2.2.2.1.1.3.1.cmml">−</mo><mn id="S3.Ex1.m1.2.2.2.2.1.1.3.3" xref="S3.Ex1.m1.2.2.2.2.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Ex1.m1.2.2.2.2.1.3" xref="S3.Ex1.m1.2.2.2.2.2.cmml">;</mo><mi id="S3.Ex1.m1.1.1.1.1" xref="S3.Ex1.m1.1.1.1.1.cmml">X</mi><mo id="S3.Ex1.m1.2.2.2.2.1.4" stretchy="false" xref="S3.Ex1.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></msub><mo id="S3.Ex1.m1.8.8.1.1.2" xref="S3.Ex1.m1.8.8.1.1.2.cmml">≤</mo><mrow id="S3.Ex1.m1.8.8.1.1.3" xref="S3.Ex1.m1.8.8.1.1.3.cmml"><mi id="S3.Ex1.m1.8.8.1.1.3.2" xref="S3.Ex1.m1.8.8.1.1.3.2.cmml">C</mi><mo id="S3.Ex1.m1.8.8.1.1.3.1" xref="S3.Ex1.m1.8.8.1.1.3.1.cmml">⁢</mo><msup id="S3.Ex1.m1.8.8.1.1.3.3" xref="S3.Ex1.m1.8.8.1.1.3.3.cmml"><mi id="S3.Ex1.m1.8.8.1.1.3.3.2" xref="S3.Ex1.m1.8.8.1.1.3.3.2.cmml">R</mi><mfrac id="S3.Ex1.m1.8.8.1.1.3.3.3" xref="S3.Ex1.m1.8.8.1.1.3.3.3.cmml"><mrow id="S3.Ex1.m1.8.8.1.1.3.3.3.2" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2.cmml"><mi id="S3.Ex1.m1.8.8.1.1.3.3.3.2.2" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2.2.cmml">γ</mi><mo id="S3.Ex1.m1.8.8.1.1.3.3.3.2.1" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2.1.cmml">+</mo><mn id="S3.Ex1.m1.8.8.1.1.3.3.3.2.3" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2.3.cmml">1</mn></mrow><mi id="S3.Ex1.m1.8.8.1.1.3.3.3.3" xref="S3.Ex1.m1.8.8.1.1.3.3.3.3.cmml">p</mi></mfrac></msup><mo id="S3.Ex1.m1.8.8.1.1.3.1a" xref="S3.Ex1.m1.8.8.1.1.3.1.cmml">⁢</mo><msub id="S3.Ex1.m1.8.8.1.1.3.4" xref="S3.Ex1.m1.8.8.1.1.3.4.cmml"><mrow id="S3.Ex1.m1.8.8.1.1.3.4.2.2" xref="S3.Ex1.m1.8.8.1.1.3.4.2.1.cmml"><mo id="S3.Ex1.m1.8.8.1.1.3.4.2.2.1" stretchy="false" xref="S3.Ex1.m1.8.8.1.1.3.4.2.1.1.cmml">‖</mo><mi id="S3.Ex1.m1.7.7" xref="S3.Ex1.m1.7.7.cmml">f</mi><mo id="S3.Ex1.m1.8.8.1.1.3.4.2.2.2" stretchy="false" xref="S3.Ex1.m1.8.8.1.1.3.4.2.1.1.cmml">‖</mo></mrow><mrow id="S3.Ex1.m1.5.5.3" xref="S3.Ex1.m1.5.5.3.cmml"><msup id="S3.Ex1.m1.5.5.3.5" xref="S3.Ex1.m1.5.5.3.5.cmml"><mi id="S3.Ex1.m1.5.5.3.5.2" xref="S3.Ex1.m1.5.5.3.5.2.cmml">L</mi><mi id="S3.Ex1.m1.5.5.3.5.3" xref="S3.Ex1.m1.5.5.3.5.3.cmml">p</mi></msup><mo id="S3.Ex1.m1.5.5.3.4" xref="S3.Ex1.m1.5.5.3.4.cmml">⁢</mo><mrow id="S3.Ex1.m1.5.5.3.3.2" xref="S3.Ex1.m1.5.5.3.3.3.cmml"><mo id="S3.Ex1.m1.5.5.3.3.2.3" stretchy="false" xref="S3.Ex1.m1.5.5.3.3.3.cmml">(</mo><msup id="S3.Ex1.m1.4.4.2.2.1.1" xref="S3.Ex1.m1.4.4.2.2.1.1.cmml"><mi id="S3.Ex1.m1.4.4.2.2.1.1.2" xref="S3.Ex1.m1.4.4.2.2.1.1.2.cmml">ℝ</mi><mi id="S3.Ex1.m1.4.4.2.2.1.1.3" xref="S3.Ex1.m1.4.4.2.2.1.1.3.cmml">d</mi></msup><mo id="S3.Ex1.m1.5.5.3.3.2.4" xref="S3.Ex1.m1.5.5.3.3.3.cmml">,</mo><msub id="S3.Ex1.m1.5.5.3.3.2.2" xref="S3.Ex1.m1.5.5.3.3.2.2.cmml"><mi id="S3.Ex1.m1.5.5.3.3.2.2.2" xref="S3.Ex1.m1.5.5.3.3.2.2.2.cmml">w</mi><mi id="S3.Ex1.m1.5.5.3.3.2.2.3" xref="S3.Ex1.m1.5.5.3.3.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex1.m1.5.5.3.3.2.5" xref="S3.Ex1.m1.5.5.3.3.3.cmml">;</mo><mi id="S3.Ex1.m1.3.3.1.1" xref="S3.Ex1.m1.3.3.1.1.cmml">X</mi><mo id="S3.Ex1.m1.5.5.3.3.2.6" stretchy="false" xref="S3.Ex1.m1.5.5.3.3.3.cmml">)</mo></mrow></mrow></msub></mrow></mrow><mo id="S3.Ex1.m1.8.8.1.2" lspace="0em" xref="S3.Ex1.m1.8.8.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex1.m1.8b"><apply id="S3.Ex1.m1.8.8.1.1.cmml" xref="S3.Ex1.m1.8.8.1"><leq id="S3.Ex1.m1.8.8.1.1.2.cmml" xref="S3.Ex1.m1.8.8.1.1.2"></leq><apply id="S3.Ex1.m1.8.8.1.1.1.cmml" xref="S3.Ex1.m1.8.8.1.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.8.8.1.1.1.2.cmml" xref="S3.Ex1.m1.8.8.1.1.1">subscript</csymbol><apply id="S3.Ex1.m1.8.8.1.1.1.1.2.cmml" xref="S3.Ex1.m1.8.8.1.1.1.1.1"><csymbol cd="latexml" id="S3.Ex1.m1.8.8.1.1.1.1.2.1.cmml" xref="S3.Ex1.m1.8.8.1.1.1.1.1.2">norm</csymbol><apply id="S3.Ex1.m1.8.8.1.1.1.1.1.1.cmml" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1"><times id="S3.Ex1.m1.8.8.1.1.1.1.1.1.2.cmml" xref="S3.Ex1.m1.8.8.1.1.1.1.1.1.2"></times><ci id="S3.Ex1.m1.8.8.1.1.1.1.1.1.3.cmml" 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xref="S3.Ex1.m1.2.2.2.4.2">𝐿</ci><ci id="S3.Ex1.m1.2.2.2.4.3.cmml" xref="S3.Ex1.m1.2.2.2.4.3">𝑝</ci></apply><list id="S3.Ex1.m1.2.2.2.2.2.cmml" xref="S3.Ex1.m1.2.2.2.2.1"><apply id="S3.Ex1.m1.2.2.2.2.1.1.cmml" xref="S3.Ex1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.2.2.2.2.1.1.1.cmml" xref="S3.Ex1.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S3.Ex1.m1.2.2.2.2.1.1.2.cmml" xref="S3.Ex1.m1.2.2.2.2.1.1.2">ℝ</ci><apply id="S3.Ex1.m1.2.2.2.2.1.1.3.cmml" xref="S3.Ex1.m1.2.2.2.2.1.1.3"><minus id="S3.Ex1.m1.2.2.2.2.1.1.3.1.cmml" xref="S3.Ex1.m1.2.2.2.2.1.1.3.1"></minus><ci id="S3.Ex1.m1.2.2.2.2.1.1.3.2.cmml" xref="S3.Ex1.m1.2.2.2.2.1.1.3.2">𝑑</ci><cn id="S3.Ex1.m1.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S3.Ex1.m1.2.2.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex1.m1.1.1.1.1.cmml" xref="S3.Ex1.m1.1.1.1.1">𝑋</ci></list></apply></apply><apply id="S3.Ex1.m1.8.8.1.1.3.cmml" xref="S3.Ex1.m1.8.8.1.1.3"><times id="S3.Ex1.m1.8.8.1.1.3.1.cmml" xref="S3.Ex1.m1.8.8.1.1.3.1"></times><ci id="S3.Ex1.m1.8.8.1.1.3.2.cmml" xref="S3.Ex1.m1.8.8.1.1.3.2">𝐶</ci><apply id="S3.Ex1.m1.8.8.1.1.3.3.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3"><csymbol cd="ambiguous" id="S3.Ex1.m1.8.8.1.1.3.3.1.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3">superscript</csymbol><ci id="S3.Ex1.m1.8.8.1.1.3.3.2.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3.2">𝑅</ci><apply id="S3.Ex1.m1.8.8.1.1.3.3.3.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3.3"><divide id="S3.Ex1.m1.8.8.1.1.3.3.3.1.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3.3"></divide><apply id="S3.Ex1.m1.8.8.1.1.3.3.3.2.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2"><plus id="S3.Ex1.m1.8.8.1.1.3.3.3.2.1.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2.1"></plus><ci id="S3.Ex1.m1.8.8.1.1.3.3.3.2.2.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2.2">𝛾</ci><cn id="S3.Ex1.m1.8.8.1.1.3.3.3.2.3.cmml" type="integer" xref="S3.Ex1.m1.8.8.1.1.3.3.3.2.3">1</cn></apply><ci id="S3.Ex1.m1.8.8.1.1.3.3.3.3.cmml" xref="S3.Ex1.m1.8.8.1.1.3.3.3.3">𝑝</ci></apply></apply><apply id="S3.Ex1.m1.8.8.1.1.3.4.cmml" xref="S3.Ex1.m1.8.8.1.1.3.4"><csymbol cd="ambiguous" id="S3.Ex1.m1.8.8.1.1.3.4.1.cmml" xref="S3.Ex1.m1.8.8.1.1.3.4">subscript</csymbol><apply id="S3.Ex1.m1.8.8.1.1.3.4.2.1.cmml" xref="S3.Ex1.m1.8.8.1.1.3.4.2.2"><csymbol cd="latexml" id="S3.Ex1.m1.8.8.1.1.3.4.2.1.1.cmml" xref="S3.Ex1.m1.8.8.1.1.3.4.2.2.1">norm</csymbol><ci id="S3.Ex1.m1.7.7.cmml" xref="S3.Ex1.m1.7.7">𝑓</ci></apply><apply id="S3.Ex1.m1.5.5.3.cmml" xref="S3.Ex1.m1.5.5.3"><times id="S3.Ex1.m1.5.5.3.4.cmml" xref="S3.Ex1.m1.5.5.3.4"></times><apply id="S3.Ex1.m1.5.5.3.5.cmml" xref="S3.Ex1.m1.5.5.3.5"><csymbol cd="ambiguous" id="S3.Ex1.m1.5.5.3.5.1.cmml" xref="S3.Ex1.m1.5.5.3.5">superscript</csymbol><ci id="S3.Ex1.m1.5.5.3.5.2.cmml" xref="S3.Ex1.m1.5.5.3.5.2">𝐿</ci><ci id="S3.Ex1.m1.5.5.3.5.3.cmml" xref="S3.Ex1.m1.5.5.3.5.3">𝑝</ci></apply><vector id="S3.Ex1.m1.5.5.3.3.3.cmml" xref="S3.Ex1.m1.5.5.3.3.2"><apply id="S3.Ex1.m1.4.4.2.2.1.1.cmml" xref="S3.Ex1.m1.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex1.m1.4.4.2.2.1.1.1.cmml" xref="S3.Ex1.m1.4.4.2.2.1.1">superscript</csymbol><ci id="S3.Ex1.m1.4.4.2.2.1.1.2.cmml" xref="S3.Ex1.m1.4.4.2.2.1.1.2">ℝ</ci><ci id="S3.Ex1.m1.4.4.2.2.1.1.3.cmml" xref="S3.Ex1.m1.4.4.2.2.1.1.3">𝑑</ci></apply><apply id="S3.Ex1.m1.5.5.3.3.2.2.cmml" xref="S3.Ex1.m1.5.5.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex1.m1.5.5.3.3.2.2.1.cmml" xref="S3.Ex1.m1.5.5.3.3.2.2">subscript</csymbol><ci id="S3.Ex1.m1.5.5.3.3.2.2.2.cmml" xref="S3.Ex1.m1.5.5.3.3.2.2.2">𝑤</ci><ci id="S3.Ex1.m1.5.5.3.3.2.2.3.cmml" xref="S3.Ex1.m1.5.5.3.3.2.2.3">𝛾</ci></apply><ci id="S3.Ex1.m1.3.3.1.1.cmml" xref="S3.Ex1.m1.3.3.1.1">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex1.m1.8c">\|f(x_{1},\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq CR^{\frac{\gamma+1}{p}}\|f% \|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex1.m1.8d">∥ italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C italic_R start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S3.SS1.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.1.p1"> <p class="ltx_p" id="S3.SS1.1.p1.1">First of all, note that <math alttext="f" class="ltx_Math" display="inline" id="S3.SS1.1.p1.1.m1.1"><semantics id="S3.SS1.1.p1.1.m1.1a"><mi id="S3.SS1.1.p1.1.m1.1.1" xref="S3.SS1.1.p1.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.1.p1.1.m1.1b"><ci id="S3.SS1.1.p1.1.m1.1.1.cmml" xref="S3.SS1.1.p1.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.1.p1.1.m1.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.1.p1.1.m1.1d">italic_f</annotation></semantics></math> is pointwise defined by the support condition on its Fourier transform, see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Lemma 14.2.9]</cite>.</p> </div> <div class="ltx_para" id="S3.SS1.2.p2"> <p class="ltx_p" id="S3.SS1.2.p2.7"><span class="ltx_text ltx_font_italic" id="S3.SS1.2.p2.1.1">Step 1: the case <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S3.SS1.2.p2.1.1.m1.2"><semantics id="S3.SS1.2.p2.1.1.m1.2a"><mrow id="S3.SS1.2.p2.1.1.m1.2.2" xref="S3.SS1.2.p2.1.1.m1.2.2.cmml"><mi id="S3.SS1.2.p2.1.1.m1.2.2.4" xref="S3.SS1.2.p2.1.1.m1.2.2.4.cmml">γ</mi><mo id="S3.SS1.2.p2.1.1.m1.2.2.3" xref="S3.SS1.2.p2.1.1.m1.2.2.3.cmml">∈</mo><mrow id="S3.SS1.2.p2.1.1.m1.2.2.2.2" xref="S3.SS1.2.p2.1.1.m1.2.2.2.3.cmml"><mo id="S3.SS1.2.p2.1.1.m1.2.2.2.2.3" stretchy="false" xref="S3.SS1.2.p2.1.1.m1.2.2.2.3.cmml">(</mo><mrow id="S3.SS1.2.p2.1.1.m1.1.1.1.1.1" xref="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.cmml"><mo id="S3.SS1.2.p2.1.1.m1.1.1.1.1.1a" xref="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.cmml">−</mo><mn id="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.2" xref="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S3.SS1.2.p2.1.1.m1.2.2.2.2.4" xref="S3.SS1.2.p2.1.1.m1.2.2.2.3.cmml">,</mo><mrow id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.cmml"><mi id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.2" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.2.cmml">p</mi><mo id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.1" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.3" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S3.SS1.2.p2.1.1.m1.2.2.2.2.5" stretchy="false" xref="S3.SS1.2.p2.1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.1.1.m1.2b"><apply id="S3.SS1.2.p2.1.1.m1.2.2.cmml" xref="S3.SS1.2.p2.1.1.m1.2.2"><in id="S3.SS1.2.p2.1.1.m1.2.2.3.cmml" xref="S3.SS1.2.p2.1.1.m1.2.2.3"></in><ci id="S3.SS1.2.p2.1.1.m1.2.2.4.cmml" xref="S3.SS1.2.p2.1.1.m1.2.2.4">𝛾</ci><interval closure="open" id="S3.SS1.2.p2.1.1.m1.2.2.2.3.cmml" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2"><apply id="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.cmml" xref="S3.SS1.2.p2.1.1.m1.1.1.1.1.1"><minus id="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.1.cmml" xref="S3.SS1.2.p2.1.1.m1.1.1.1.1.1"></minus><cn id="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S3.SS1.2.p2.1.1.m1.1.1.1.1.1.2">1</cn></apply><apply id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.cmml" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2"><minus id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.1.cmml" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.1"></minus><ci id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.2.cmml" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.2">𝑝</ci><cn id="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S3.SS1.2.p2.1.1.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.1.1.m1.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.1.1.m1.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>.</span> By scaling and translating it suffices to consider <math alttext="x_{1}=0" class="ltx_Math" display="inline" id="S3.SS1.2.p2.2.m1.1"><semantics id="S3.SS1.2.p2.2.m1.1a"><mrow id="S3.SS1.2.p2.2.m1.1.1" xref="S3.SS1.2.p2.2.m1.1.1.cmml"><msub id="S3.SS1.2.p2.2.m1.1.1.2" xref="S3.SS1.2.p2.2.m1.1.1.2.cmml"><mi id="S3.SS1.2.p2.2.m1.1.1.2.2" xref="S3.SS1.2.p2.2.m1.1.1.2.2.cmml">x</mi><mn id="S3.SS1.2.p2.2.m1.1.1.2.3" xref="S3.SS1.2.p2.2.m1.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.2.p2.2.m1.1.1.1" xref="S3.SS1.2.p2.2.m1.1.1.1.cmml">=</mo><mn id="S3.SS1.2.p2.2.m1.1.1.3" xref="S3.SS1.2.p2.2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.2.m1.1b"><apply id="S3.SS1.2.p2.2.m1.1.1.cmml" xref="S3.SS1.2.p2.2.m1.1.1"><eq id="S3.SS1.2.p2.2.m1.1.1.1.cmml" xref="S3.SS1.2.p2.2.m1.1.1.1"></eq><apply id="S3.SS1.2.p2.2.m1.1.1.2.cmml" xref="S3.SS1.2.p2.2.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.2.p2.2.m1.1.1.2.1.cmml" xref="S3.SS1.2.p2.2.m1.1.1.2">subscript</csymbol><ci id="S3.SS1.2.p2.2.m1.1.1.2.2.cmml" xref="S3.SS1.2.p2.2.m1.1.1.2.2">𝑥</ci><cn id="S3.SS1.2.p2.2.m1.1.1.2.3.cmml" type="integer" xref="S3.SS1.2.p2.2.m1.1.1.2.3">1</cn></apply><cn id="S3.SS1.2.p2.2.m1.1.1.3.cmml" type="integer" xref="S3.SS1.2.p2.2.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.2.m1.1c">x_{1}=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.2.m1.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = 0</annotation></semantics></math> and <math alttext="R=1" class="ltx_Math" display="inline" id="S3.SS1.2.p2.3.m2.1"><semantics id="S3.SS1.2.p2.3.m2.1a"><mrow id="S3.SS1.2.p2.3.m2.1.1" xref="S3.SS1.2.p2.3.m2.1.1.cmml"><mi id="S3.SS1.2.p2.3.m2.1.1.2" xref="S3.SS1.2.p2.3.m2.1.1.2.cmml">R</mi><mo id="S3.SS1.2.p2.3.m2.1.1.1" xref="S3.SS1.2.p2.3.m2.1.1.1.cmml">=</mo><mn id="S3.SS1.2.p2.3.m2.1.1.3" xref="S3.SS1.2.p2.3.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.3.m2.1b"><apply id="S3.SS1.2.p2.3.m2.1.1.cmml" xref="S3.SS1.2.p2.3.m2.1.1"><eq id="S3.SS1.2.p2.3.m2.1.1.1.cmml" xref="S3.SS1.2.p2.3.m2.1.1.1"></eq><ci id="S3.SS1.2.p2.3.m2.1.1.2.cmml" xref="S3.SS1.2.p2.3.m2.1.1.2">𝑅</ci><cn id="S3.SS1.2.p2.3.m2.1.1.3.cmml" type="integer" xref="S3.SS1.2.p2.3.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.3.m2.1c">R=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.3.m2.1d">italic_R = 1</annotation></semantics></math>. Let <math alttext="\widehat{\varphi}\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S3.SS1.2.p2.4.m3.1"><semantics id="S3.SS1.2.p2.4.m3.1a"><mrow id="S3.SS1.2.p2.4.m3.1.1" xref="S3.SS1.2.p2.4.m3.1.1.cmml"><mover accent="true" id="S3.SS1.2.p2.4.m3.1.1.3" xref="S3.SS1.2.p2.4.m3.1.1.3.cmml"><mi id="S3.SS1.2.p2.4.m3.1.1.3.2" xref="S3.SS1.2.p2.4.m3.1.1.3.2.cmml">φ</mi><mo id="S3.SS1.2.p2.4.m3.1.1.3.1" xref="S3.SS1.2.p2.4.m3.1.1.3.1.cmml">^</mo></mover><mo id="S3.SS1.2.p2.4.m3.1.1.2" xref="S3.SS1.2.p2.4.m3.1.1.2.cmml">∈</mo><mrow id="S3.SS1.2.p2.4.m3.1.1.1" xref="S3.SS1.2.p2.4.m3.1.1.1.cmml"><msubsup id="S3.SS1.2.p2.4.m3.1.1.1.3" xref="S3.SS1.2.p2.4.m3.1.1.1.3.cmml"><mi id="S3.SS1.2.p2.4.m3.1.1.1.3.2.2" xref="S3.SS1.2.p2.4.m3.1.1.1.3.2.2.cmml">C</mi><mi id="S3.SS1.2.p2.4.m3.1.1.1.3.2.3" mathvariant="normal" xref="S3.SS1.2.p2.4.m3.1.1.1.3.2.3.cmml">c</mi><mi id="S3.SS1.2.p2.4.m3.1.1.1.3.3" mathvariant="normal" xref="S3.SS1.2.p2.4.m3.1.1.1.3.3.cmml">∞</mi></msubsup><mo id="S3.SS1.2.p2.4.m3.1.1.1.2" xref="S3.SS1.2.p2.4.m3.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.2.p2.4.m3.1.1.1.1.1" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.cmml"><mo id="S3.SS1.2.p2.4.m3.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.cmml">(</mo><msup id="S3.SS1.2.p2.4.m3.1.1.1.1.1.1" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.cmml"><mi id="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.2" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.3" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.2.p2.4.m3.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.4.m3.1b"><apply id="S3.SS1.2.p2.4.m3.1.1.cmml" xref="S3.SS1.2.p2.4.m3.1.1"><in id="S3.SS1.2.p2.4.m3.1.1.2.cmml" xref="S3.SS1.2.p2.4.m3.1.1.2"></in><apply id="S3.SS1.2.p2.4.m3.1.1.3.cmml" xref="S3.SS1.2.p2.4.m3.1.1.3"><ci id="S3.SS1.2.p2.4.m3.1.1.3.1.cmml" xref="S3.SS1.2.p2.4.m3.1.1.3.1">^</ci><ci id="S3.SS1.2.p2.4.m3.1.1.3.2.cmml" xref="S3.SS1.2.p2.4.m3.1.1.3.2">𝜑</ci></apply><apply id="S3.SS1.2.p2.4.m3.1.1.1.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1"><times id="S3.SS1.2.p2.4.m3.1.1.1.2.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.2"></times><apply id="S3.SS1.2.p2.4.m3.1.1.1.3.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.2.p2.4.m3.1.1.1.3.1.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.3">superscript</csymbol><apply id="S3.SS1.2.p2.4.m3.1.1.1.3.2.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.2.p2.4.m3.1.1.1.3.2.1.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.3">subscript</csymbol><ci id="S3.SS1.2.p2.4.m3.1.1.1.3.2.2.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.3.2.2">𝐶</ci><ci id="S3.SS1.2.p2.4.m3.1.1.1.3.2.3.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.3.2.3">c</ci></apply><infinity id="S3.SS1.2.p2.4.m3.1.1.1.3.3.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.3.3"></infinity></apply><apply id="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.1.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.2.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.2">ℝ</ci><ci id="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.3.cmml" xref="S3.SS1.2.p2.4.m3.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.4.m3.1c">\widehat{\varphi}\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.4.m3.1d">over^ start_ARG italic_φ end_ARG ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> such that <math alttext="\widehat{\varphi}=1" class="ltx_Math" display="inline" id="S3.SS1.2.p2.5.m4.1"><semantics id="S3.SS1.2.p2.5.m4.1a"><mrow id="S3.SS1.2.p2.5.m4.1.1" xref="S3.SS1.2.p2.5.m4.1.1.cmml"><mover accent="true" id="S3.SS1.2.p2.5.m4.1.1.2" xref="S3.SS1.2.p2.5.m4.1.1.2.cmml"><mi id="S3.SS1.2.p2.5.m4.1.1.2.2" xref="S3.SS1.2.p2.5.m4.1.1.2.2.cmml">φ</mi><mo id="S3.SS1.2.p2.5.m4.1.1.2.1" xref="S3.SS1.2.p2.5.m4.1.1.2.1.cmml">^</mo></mover><mo id="S3.SS1.2.p2.5.m4.1.1.1" xref="S3.SS1.2.p2.5.m4.1.1.1.cmml">=</mo><mn id="S3.SS1.2.p2.5.m4.1.1.3" xref="S3.SS1.2.p2.5.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.5.m4.1b"><apply id="S3.SS1.2.p2.5.m4.1.1.cmml" xref="S3.SS1.2.p2.5.m4.1.1"><eq id="S3.SS1.2.p2.5.m4.1.1.1.cmml" xref="S3.SS1.2.p2.5.m4.1.1.1"></eq><apply id="S3.SS1.2.p2.5.m4.1.1.2.cmml" xref="S3.SS1.2.p2.5.m4.1.1.2"><ci id="S3.SS1.2.p2.5.m4.1.1.2.1.cmml" xref="S3.SS1.2.p2.5.m4.1.1.2.1">^</ci><ci id="S3.SS1.2.p2.5.m4.1.1.2.2.cmml" xref="S3.SS1.2.p2.5.m4.1.1.2.2">𝜑</ci></apply><cn id="S3.SS1.2.p2.5.m4.1.1.3.cmml" type="integer" xref="S3.SS1.2.p2.5.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.5.m4.1c">\widehat{\varphi}=1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.5.m4.1d">over^ start_ARG italic_φ end_ARG = 1</annotation></semantics></math> on <math alttext="B(0,1)" class="ltx_Math" display="inline" id="S3.SS1.2.p2.6.m5.2"><semantics id="S3.SS1.2.p2.6.m5.2a"><mrow id="S3.SS1.2.p2.6.m5.2.3" xref="S3.SS1.2.p2.6.m5.2.3.cmml"><mi id="S3.SS1.2.p2.6.m5.2.3.2" xref="S3.SS1.2.p2.6.m5.2.3.2.cmml">B</mi><mo id="S3.SS1.2.p2.6.m5.2.3.1" xref="S3.SS1.2.p2.6.m5.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.2.p2.6.m5.2.3.3.2" xref="S3.SS1.2.p2.6.m5.2.3.3.1.cmml"><mo id="S3.SS1.2.p2.6.m5.2.3.3.2.1" stretchy="false" xref="S3.SS1.2.p2.6.m5.2.3.3.1.cmml">(</mo><mn id="S3.SS1.2.p2.6.m5.1.1" xref="S3.SS1.2.p2.6.m5.1.1.cmml">0</mn><mo id="S3.SS1.2.p2.6.m5.2.3.3.2.2" xref="S3.SS1.2.p2.6.m5.2.3.3.1.cmml">,</mo><mn id="S3.SS1.2.p2.6.m5.2.2" xref="S3.SS1.2.p2.6.m5.2.2.cmml">1</mn><mo id="S3.SS1.2.p2.6.m5.2.3.3.2.3" stretchy="false" xref="S3.SS1.2.p2.6.m5.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.6.m5.2b"><apply id="S3.SS1.2.p2.6.m5.2.3.cmml" xref="S3.SS1.2.p2.6.m5.2.3"><times id="S3.SS1.2.p2.6.m5.2.3.1.cmml" xref="S3.SS1.2.p2.6.m5.2.3.1"></times><ci id="S3.SS1.2.p2.6.m5.2.3.2.cmml" xref="S3.SS1.2.p2.6.m5.2.3.2">𝐵</ci><interval closure="open" id="S3.SS1.2.p2.6.m5.2.3.3.1.cmml" xref="S3.SS1.2.p2.6.m5.2.3.3.2"><cn id="S3.SS1.2.p2.6.m5.1.1.cmml" type="integer" xref="S3.SS1.2.p2.6.m5.1.1">0</cn><cn id="S3.SS1.2.p2.6.m5.2.2.cmml" type="integer" xref="S3.SS1.2.p2.6.m5.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.6.m5.2c">B(0,1)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.6.m5.2d">italic_B ( 0 , 1 )</annotation></semantics></math>. Then for every <math alttext="\widetilde{x}\in\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S3.SS1.2.p2.7.m6.1"><semantics id="S3.SS1.2.p2.7.m6.1a"><mrow id="S3.SS1.2.p2.7.m6.1.1" xref="S3.SS1.2.p2.7.m6.1.1.cmml"><mover accent="true" id="S3.SS1.2.p2.7.m6.1.1.2" xref="S3.SS1.2.p2.7.m6.1.1.2.cmml"><mi id="S3.SS1.2.p2.7.m6.1.1.2.2" xref="S3.SS1.2.p2.7.m6.1.1.2.2.cmml">x</mi><mo id="S3.SS1.2.p2.7.m6.1.1.2.1" xref="S3.SS1.2.p2.7.m6.1.1.2.1.cmml">~</mo></mover><mo id="S3.SS1.2.p2.7.m6.1.1.1" xref="S3.SS1.2.p2.7.m6.1.1.1.cmml">∈</mo><msup id="S3.SS1.2.p2.7.m6.1.1.3" xref="S3.SS1.2.p2.7.m6.1.1.3.cmml"><mi id="S3.SS1.2.p2.7.m6.1.1.3.2" xref="S3.SS1.2.p2.7.m6.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS1.2.p2.7.m6.1.1.3.3" xref="S3.SS1.2.p2.7.m6.1.1.3.3.cmml"><mi id="S3.SS1.2.p2.7.m6.1.1.3.3.2" xref="S3.SS1.2.p2.7.m6.1.1.3.3.2.cmml">d</mi><mo id="S3.SS1.2.p2.7.m6.1.1.3.3.1" xref="S3.SS1.2.p2.7.m6.1.1.3.3.1.cmml">−</mo><mn id="S3.SS1.2.p2.7.m6.1.1.3.3.3" xref="S3.SS1.2.p2.7.m6.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.7.m6.1b"><apply id="S3.SS1.2.p2.7.m6.1.1.cmml" xref="S3.SS1.2.p2.7.m6.1.1"><in id="S3.SS1.2.p2.7.m6.1.1.1.cmml" xref="S3.SS1.2.p2.7.m6.1.1.1"></in><apply id="S3.SS1.2.p2.7.m6.1.1.2.cmml" xref="S3.SS1.2.p2.7.m6.1.1.2"><ci id="S3.SS1.2.p2.7.m6.1.1.2.1.cmml" xref="S3.SS1.2.p2.7.m6.1.1.2.1">~</ci><ci id="S3.SS1.2.p2.7.m6.1.1.2.2.cmml" xref="S3.SS1.2.p2.7.m6.1.1.2.2">𝑥</ci></apply><apply id="S3.SS1.2.p2.7.m6.1.1.3.cmml" xref="S3.SS1.2.p2.7.m6.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.2.p2.7.m6.1.1.3.1.cmml" xref="S3.SS1.2.p2.7.m6.1.1.3">superscript</csymbol><ci id="S3.SS1.2.p2.7.m6.1.1.3.2.cmml" xref="S3.SS1.2.p2.7.m6.1.1.3.2">ℝ</ci><apply id="S3.SS1.2.p2.7.m6.1.1.3.3.cmml" xref="S3.SS1.2.p2.7.m6.1.1.3.3"><minus id="S3.SS1.2.p2.7.m6.1.1.3.3.1.cmml" xref="S3.SS1.2.p2.7.m6.1.1.3.3.1"></minus><ci id="S3.SS1.2.p2.7.m6.1.1.3.3.2.cmml" xref="S3.SS1.2.p2.7.m6.1.1.3.3.2">𝑑</ci><cn id="S3.SS1.2.p2.7.m6.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.2.p2.7.m6.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.7.m6.1c">\widetilde{x}\in\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.7.m6.1d">over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> we have</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f(0,\widetilde{x})=\big{(}\mathcal{F}^{-1}(\widehat{\varphi}\widehat{f}\,)\big% {)}(0,\widetilde{x})=\int_{\mathbb{R}^{d-1}}\int_{\mathbb{R}}f(-y_{1},% \widetilde{x}-\widetilde{y})|y_{1}|^{\frac{\gamma}{p}}\varphi(y_{1},\widetilde% {y})|y_{1}|^{-\frac{\gamma}{p}}\hskip 2.0pt\mathrm{d}y_{1}\hskip 2.0pt\mathrm{% d}\widetilde{y}," class="ltx_Math" display="block" id="S3.Ex2.m1.6"><semantics id="S3.Ex2.m1.6a"><mrow id="S3.Ex2.m1.6.6.1" xref="S3.Ex2.m1.6.6.1.1.cmml"><mrow id="S3.Ex2.m1.6.6.1.1" xref="S3.Ex2.m1.6.6.1.1.cmml"><mrow id="S3.Ex2.m1.6.6.1.1.8" xref="S3.Ex2.m1.6.6.1.1.8.cmml"><mi id="S3.Ex2.m1.6.6.1.1.8.2" xref="S3.Ex2.m1.6.6.1.1.8.2.cmml">f</mi><mo id="S3.Ex2.m1.6.6.1.1.8.1" xref="S3.Ex2.m1.6.6.1.1.8.1.cmml">⁢</mo><mrow id="S3.Ex2.m1.6.6.1.1.8.3.2" xref="S3.Ex2.m1.6.6.1.1.8.3.1.cmml"><mo id="S3.Ex2.m1.6.6.1.1.8.3.2.1" stretchy="false" xref="S3.Ex2.m1.6.6.1.1.8.3.1.cmml">(</mo><mn id="S3.Ex2.m1.1.1" xref="S3.Ex2.m1.1.1.cmml">0</mn><mo id="S3.Ex2.m1.6.6.1.1.8.3.2.2" xref="S3.Ex2.m1.6.6.1.1.8.3.1.cmml">,</mo><mover accent="true" id="S3.Ex2.m1.2.2" xref="S3.Ex2.m1.2.2.cmml"><mi id="S3.Ex2.m1.2.2.2" xref="S3.Ex2.m1.2.2.2.cmml">x</mi><mo id="S3.Ex2.m1.2.2.1" xref="S3.Ex2.m1.2.2.1.cmml">~</mo></mover><mo id="S3.Ex2.m1.6.6.1.1.8.3.2.3" stretchy="false" xref="S3.Ex2.m1.6.6.1.1.8.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex2.m1.6.6.1.1.9" xref="S3.Ex2.m1.6.6.1.1.9.cmml">=</mo><mrow id="S3.Ex2.m1.6.6.1.1.1" xref="S3.Ex2.m1.6.6.1.1.1.cmml"><mrow id="S3.Ex2.m1.6.6.1.1.1.1.1" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.cmml"><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex2.m1.6.6.1.1.1.1.1.1" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.cmml"><msup id="S3.Ex2.m1.6.6.1.1.1.1.1.1.3" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.2" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.2.cmml">ℱ</mi><mrow id="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.3" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.3.cmml"><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.3a" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.3.cmml">−</mo><mn id="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.3.2" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msup><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.1.2" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.cmml"><mover accent="true" id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.2.2" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.2.2.cmml">φ</mi><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.2.1" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.2.1.cmml">^</mo></mover><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.3.2.cmml">f</mi><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.3.1" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.3.1.cmml">^</mo></mover></mrow><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex2.m1.6.6.1.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.Ex2.m1.6.6.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex2.m1.6.6.1.1.1.2" xref="S3.Ex2.m1.6.6.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex2.m1.6.6.1.1.1.3.2" xref="S3.Ex2.m1.6.6.1.1.1.3.1.cmml"><mo id="S3.Ex2.m1.6.6.1.1.1.3.2.1" stretchy="false" xref="S3.Ex2.m1.6.6.1.1.1.3.1.cmml">(</mo><mn id="S3.Ex2.m1.3.3" xref="S3.Ex2.m1.3.3.cmml">0</mn><mo id="S3.Ex2.m1.6.6.1.1.1.3.2.2" xref="S3.Ex2.m1.6.6.1.1.1.3.1.cmml">,</mo><mover accent="true" id="S3.Ex2.m1.4.4" xref="S3.Ex2.m1.4.4.cmml"><mi id="S3.Ex2.m1.4.4.2" xref="S3.Ex2.m1.4.4.2.cmml">x</mi><mo id="S3.Ex2.m1.4.4.1" xref="S3.Ex2.m1.4.4.1.cmml">~</mo></mover><mo id="S3.Ex2.m1.6.6.1.1.1.3.2.3" stretchy="false" xref="S3.Ex2.m1.6.6.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex2.m1.6.6.1.1.10" rspace="0.111em" xref="S3.Ex2.m1.6.6.1.1.10.cmml">=</mo><mrow id="S3.Ex2.m1.6.6.1.1.6" xref="S3.Ex2.m1.6.6.1.1.6.cmml"><msub id="S3.Ex2.m1.6.6.1.1.6.6" xref="S3.Ex2.m1.6.6.1.1.6.6.cmml"><mo id="S3.Ex2.m1.6.6.1.1.6.6.2" rspace="0em" xref="S3.Ex2.m1.6.6.1.1.6.6.2.cmml">∫</mo><msup id="S3.Ex2.m1.6.6.1.1.6.6.3" xref="S3.Ex2.m1.6.6.1.1.6.6.3.cmml"><mi id="S3.Ex2.m1.6.6.1.1.6.6.3.2" xref="S3.Ex2.m1.6.6.1.1.6.6.3.2.cmml">ℝ</mi><mrow id="S3.Ex2.m1.6.6.1.1.6.6.3.3" xref="S3.Ex2.m1.6.6.1.1.6.6.3.3.cmml"><mi id="S3.Ex2.m1.6.6.1.1.6.6.3.3.2" xref="S3.Ex2.m1.6.6.1.1.6.6.3.3.2.cmml">d</mi><mo id="S3.Ex2.m1.6.6.1.1.6.6.3.3.1" xref="S3.Ex2.m1.6.6.1.1.6.6.3.3.1.cmml">−</mo><mn id="S3.Ex2.m1.6.6.1.1.6.6.3.3.3" xref="S3.Ex2.m1.6.6.1.1.6.6.3.3.3.cmml">1</mn></mrow></msup></msub><mrow id="S3.Ex2.m1.6.6.1.1.6.5" xref="S3.Ex2.m1.6.6.1.1.6.5.cmml"><msub id="S3.Ex2.m1.6.6.1.1.6.5.6" xref="S3.Ex2.m1.6.6.1.1.6.5.6.cmml"><mo 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0 , over~ start_ARG italic_x end_ARG ) = ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over^ start_ARG italic_φ end_ARG over^ start_ARG italic_f end_ARG ) ) ( 0 , over~ start_ARG italic_x end_ARG ) = ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT italic_f ( - italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG - over~ start_ARG italic_y end_ARG ) | italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT divide start_ARG italic_γ end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT italic_φ ( italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_y end_ARG ) | italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT - divide start_ARG italic_γ end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT roman_d italic_y start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_y end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.2.p2.11">so that Hölder’s inequality implies</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f(0,\widetilde{x})\|_{X}\leq\int_{\mathbb{R}^{d-1}}\|f(\cdot,\widetilde{x}-% \widetilde{y})\|_{L^{p}(\mathbb{R},w_{\gamma};X)}\|\varphi(\cdot,\widetilde{y}% )\|_{L^{p^{\prime}}(\mathbb{R},w_{\gamma^{\prime}})}\hskip 2.0pt\mathrm{d}% \widetilde{y}," class="ltx_Math" display="block" id="S3.Ex3.m1.11"><semantics id="S3.Ex3.m1.11a"><mrow id="S3.Ex3.m1.11.11.1" xref="S3.Ex3.m1.11.11.1.1.cmml"><mrow id="S3.Ex3.m1.11.11.1.1" xref="S3.Ex3.m1.11.11.1.1.cmml"><msub id="S3.Ex3.m1.11.11.1.1.1" xref="S3.Ex3.m1.11.11.1.1.1.cmml"><mrow 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xref="S3.Ex3.m1.11.11.1.1.3.2.4.2.1">~</ci><ci id="S3.Ex3.m1.11.11.1.1.3.2.4.2.2.cmml" xref="S3.Ex3.m1.11.11.1.1.3.2.4.2.2">𝑦</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex3.m1.11c">\|f(0,\widetilde{x})\|_{X}\leq\int_{\mathbb{R}^{d-1}}\|f(\cdot,\widetilde{x}-% \widetilde{y})\|_{L^{p}(\mathbb{R},w_{\gamma};X)}\|\varphi(\cdot,\widetilde{y}% )\|_{L^{p^{\prime}}(\mathbb{R},w_{\gamma^{\prime}})}\hskip 2.0pt\mathrm{d}% \widetilde{y},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex3.m1.11d">∥ italic_f ( 0 , over~ start_ARG italic_x end_ARG ) ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ≤ ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ italic_f ( ⋅ , over~ start_ARG italic_x end_ARG - over~ start_ARG italic_y end_ARG ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ∥ italic_φ ( ⋅ , over~ start_ARG italic_y end_ARG ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT roman_d over~ start_ARG italic_y end_ARG ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.2.p2.9">where <math alttext="p^{\prime}=\frac{p}{p-1}" class="ltx_Math" display="inline" id="S3.SS1.2.p2.8.m1.1"><semantics id="S3.SS1.2.p2.8.m1.1a"><mrow id="S3.SS1.2.p2.8.m1.1.1" xref="S3.SS1.2.p2.8.m1.1.1.cmml"><msup id="S3.SS1.2.p2.8.m1.1.1.2" xref="S3.SS1.2.p2.8.m1.1.1.2.cmml"><mi id="S3.SS1.2.p2.8.m1.1.1.2.2" xref="S3.SS1.2.p2.8.m1.1.1.2.2.cmml">p</mi><mo id="S3.SS1.2.p2.8.m1.1.1.2.3" xref="S3.SS1.2.p2.8.m1.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.2.p2.8.m1.1.1.1" xref="S3.SS1.2.p2.8.m1.1.1.1.cmml">=</mo><mfrac id="S3.SS1.2.p2.8.m1.1.1.3" xref="S3.SS1.2.p2.8.m1.1.1.3.cmml"><mi id="S3.SS1.2.p2.8.m1.1.1.3.2" xref="S3.SS1.2.p2.8.m1.1.1.3.2.cmml">p</mi><mrow id="S3.SS1.2.p2.8.m1.1.1.3.3" xref="S3.SS1.2.p2.8.m1.1.1.3.3.cmml"><mi id="S3.SS1.2.p2.8.m1.1.1.3.3.2" xref="S3.SS1.2.p2.8.m1.1.1.3.3.2.cmml">p</mi><mo id="S3.SS1.2.p2.8.m1.1.1.3.3.1" xref="S3.SS1.2.p2.8.m1.1.1.3.3.1.cmml">−</mo><mn id="S3.SS1.2.p2.8.m1.1.1.3.3.3" xref="S3.SS1.2.p2.8.m1.1.1.3.3.3.cmml">1</mn></mrow></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.8.m1.1b"><apply id="S3.SS1.2.p2.8.m1.1.1.cmml" xref="S3.SS1.2.p2.8.m1.1.1"><eq id="S3.SS1.2.p2.8.m1.1.1.1.cmml" xref="S3.SS1.2.p2.8.m1.1.1.1"></eq><apply id="S3.SS1.2.p2.8.m1.1.1.2.cmml" xref="S3.SS1.2.p2.8.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.2.p2.8.m1.1.1.2.1.cmml" xref="S3.SS1.2.p2.8.m1.1.1.2">superscript</csymbol><ci 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start_ARG italic_p end_ARG start_ARG italic_p - 1 end_ARG</annotation></semantics></math> and <math alttext="\gamma^{\prime}=-\frac{\gamma}{p-1}" class="ltx_Math" display="inline" id="S3.SS1.2.p2.9.m2.1"><semantics id="S3.SS1.2.p2.9.m2.1a"><mrow id="S3.SS1.2.p2.9.m2.1.1" xref="S3.SS1.2.p2.9.m2.1.1.cmml"><msup id="S3.SS1.2.p2.9.m2.1.1.2" xref="S3.SS1.2.p2.9.m2.1.1.2.cmml"><mi id="S3.SS1.2.p2.9.m2.1.1.2.2" xref="S3.SS1.2.p2.9.m2.1.1.2.2.cmml">γ</mi><mo id="S3.SS1.2.p2.9.m2.1.1.2.3" xref="S3.SS1.2.p2.9.m2.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.2.p2.9.m2.1.1.1" xref="S3.SS1.2.p2.9.m2.1.1.1.cmml">=</mo><mrow id="S3.SS1.2.p2.9.m2.1.1.3" xref="S3.SS1.2.p2.9.m2.1.1.3.cmml"><mo id="S3.SS1.2.p2.9.m2.1.1.3a" xref="S3.SS1.2.p2.9.m2.1.1.3.cmml">−</mo><mfrac id="S3.SS1.2.p2.9.m2.1.1.3.2" xref="S3.SS1.2.p2.9.m2.1.1.3.2.cmml"><mi id="S3.SS1.2.p2.9.m2.1.1.3.2.2" xref="S3.SS1.2.p2.9.m2.1.1.3.2.2.cmml">γ</mi><mrow id="S3.SS1.2.p2.9.m2.1.1.3.2.3" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3.cmml"><mi id="S3.SS1.2.p2.9.m2.1.1.3.2.3.2" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3.2.cmml">p</mi><mo id="S3.SS1.2.p2.9.m2.1.1.3.2.3.1" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3.1.cmml">−</mo><mn id="S3.SS1.2.p2.9.m2.1.1.3.2.3.3" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3.3.cmml">1</mn></mrow></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.9.m2.1b"><apply id="S3.SS1.2.p2.9.m2.1.1.cmml" xref="S3.SS1.2.p2.9.m2.1.1"><eq id="S3.SS1.2.p2.9.m2.1.1.1.cmml" xref="S3.SS1.2.p2.9.m2.1.1.1"></eq><apply id="S3.SS1.2.p2.9.m2.1.1.2.cmml" xref="S3.SS1.2.p2.9.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.2.p2.9.m2.1.1.2.1.cmml" xref="S3.SS1.2.p2.9.m2.1.1.2">superscript</csymbol><ci id="S3.SS1.2.p2.9.m2.1.1.2.2.cmml" xref="S3.SS1.2.p2.9.m2.1.1.2.2">𝛾</ci><ci id="S3.SS1.2.p2.9.m2.1.1.2.3.cmml" xref="S3.SS1.2.p2.9.m2.1.1.2.3">′</ci></apply><apply id="S3.SS1.2.p2.9.m2.1.1.3.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3"><minus id="S3.SS1.2.p2.9.m2.1.1.3.1.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3"></minus><apply id="S3.SS1.2.p2.9.m2.1.1.3.2.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3.2"><divide id="S3.SS1.2.p2.9.m2.1.1.3.2.1.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3.2"></divide><ci id="S3.SS1.2.p2.9.m2.1.1.3.2.2.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3.2.2">𝛾</ci><apply id="S3.SS1.2.p2.9.m2.1.1.3.2.3.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3"><minus id="S3.SS1.2.p2.9.m2.1.1.3.2.3.1.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3.1"></minus><ci id="S3.SS1.2.p2.9.m2.1.1.3.2.3.2.cmml" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3.2">𝑝</ci><cn id="S3.SS1.2.p2.9.m2.1.1.3.2.3.3.cmml" type="integer" xref="S3.SS1.2.p2.9.m2.1.1.3.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.9.m2.1c">\gamma^{\prime}=-\frac{\gamma}{p-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.9.m2.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT = - divide start_ARG italic_γ end_ARG start_ARG italic_p - 1 end_ARG</annotation></semantics></math>. By Minkowski’s inequality we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f(0,\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq\|f\|_{L^{p}(\mathbb{R}^{d},w_{% \gamma};X)}\int_{\mathbb{R}^{d-1}}\|\varphi(\cdot,\widetilde{y})\|_{L^{p^{% \prime}}(\mathbb{R},w_{\gamma^{\prime}})}\hskip 2.0pt\mathrm{d}\widetilde{y}=C% \|f\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}," class="ltx_Math" display="block" id="S3.Ex4.m1.17"><semantics id="S3.Ex4.m1.17a"><mrow id="S3.Ex4.m1.17.17.1" xref="S3.Ex4.m1.17.17.1.1.cmml"><mrow id="S3.Ex4.m1.17.17.1.1" xref="S3.Ex4.m1.17.17.1.1.cmml"><msub id="S3.Ex4.m1.17.17.1.1.1" xref="S3.Ex4.m1.17.17.1.1.1.cmml"><mrow id="S3.Ex4.m1.17.17.1.1.1.1.1" xref="S3.Ex4.m1.17.17.1.1.1.1.2.cmml"><mo id="S3.Ex4.m1.17.17.1.1.1.1.1.2" stretchy="false" xref="S3.Ex4.m1.17.17.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex4.m1.17.17.1.1.1.1.1.1" xref="S3.Ex4.m1.17.17.1.1.1.1.1.1.cmml"><mi id="S3.Ex4.m1.17.17.1.1.1.1.1.1.2" xref="S3.Ex4.m1.17.17.1.1.1.1.1.1.2.cmml">f</mi><mo id="S3.Ex4.m1.17.17.1.1.1.1.1.1.1" xref="S3.Ex4.m1.17.17.1.1.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.2" xref="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.1.cmml"><mo id="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.2.1" stretchy="false" xref="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.1.cmml">(</mo><mn id="S3.Ex4.m1.11.11" xref="S3.Ex4.m1.11.11.cmml">0</mn><mo id="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.2.2" rspace="0em" xref="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.1.cmml">,</mo><mo id="S3.Ex4.m1.12.12" lspace="0em" rspace="0em" xref="S3.Ex4.m1.12.12.cmml">⋅</mo><mo id="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.2.3" stretchy="false" xref="S3.Ex4.m1.17.17.1.1.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex4.m1.17.17.1.1.1.1.1.3" stretchy="false" xref="S3.Ex4.m1.17.17.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex4.m1.2.2.2" xref="S3.Ex4.m1.2.2.2.cmml"><msup id="S3.Ex4.m1.2.2.2.4" xref="S3.Ex4.m1.2.2.2.4.cmml"><mi id="S3.Ex4.m1.2.2.2.4.2" xref="S3.Ex4.m1.2.2.2.4.2.cmml">L</mi><mi id="S3.Ex4.m1.2.2.2.4.3" xref="S3.Ex4.m1.2.2.2.4.3.cmml">p</mi></msup><mo id="S3.Ex4.m1.2.2.2.3" xref="S3.Ex4.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.Ex4.m1.2.2.2.2.1" xref="S3.Ex4.m1.2.2.2.2.2.cmml"><mo id="S3.Ex4.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex4.m1.2.2.2.2.2.cmml">(</mo><msup id="S3.Ex4.m1.2.2.2.2.1.1" xref="S3.Ex4.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex4.m1.2.2.2.2.1.1.2" xref="S3.Ex4.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mrow id="S3.Ex4.m1.2.2.2.2.1.1.3" xref="S3.Ex4.m1.2.2.2.2.1.1.3.cmml"><mi id="S3.Ex4.m1.2.2.2.2.1.1.3.2" xref="S3.Ex4.m1.2.2.2.2.1.1.3.2.cmml">d</mi><mo id="S3.Ex4.m1.2.2.2.2.1.1.3.1" xref="S3.Ex4.m1.2.2.2.2.1.1.3.1.cmml">−</mo><mn id="S3.Ex4.m1.2.2.2.2.1.1.3.3" xref="S3.Ex4.m1.2.2.2.2.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Ex4.m1.2.2.2.2.1.3" xref="S3.Ex4.m1.2.2.2.2.2.cmml">;</mo><mi id="S3.Ex4.m1.1.1.1.1" xref="S3.Ex4.m1.1.1.1.1.cmml">X</mi><mo id="S3.Ex4.m1.2.2.2.2.1.4" stretchy="false" xref="S3.Ex4.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></msub><mo id="S3.Ex4.m1.17.17.1.1.4" xref="S3.Ex4.m1.17.17.1.1.4.cmml">≤</mo><mrow id="S3.Ex4.m1.17.17.1.1.2" xref="S3.Ex4.m1.17.17.1.1.2.cmml"><msub id="S3.Ex4.m1.17.17.1.1.2.3" xref="S3.Ex4.m1.17.17.1.1.2.3.cmml"><mrow id="S3.Ex4.m1.17.17.1.1.2.3.2.2" xref="S3.Ex4.m1.17.17.1.1.2.3.2.1.cmml"><mo id="S3.Ex4.m1.17.17.1.1.2.3.2.2.1" stretchy="false" xref="S3.Ex4.m1.17.17.1.1.2.3.2.1.1.cmml">‖</mo><mi id="S3.Ex4.m1.13.13" xref="S3.Ex4.m1.13.13.cmml">f</mi><mo id="S3.Ex4.m1.17.17.1.1.2.3.2.2.2" stretchy="false" xref="S3.Ex4.m1.17.17.1.1.2.3.2.1.1.cmml">‖</mo></mrow><mrow id="S3.Ex4.m1.5.5.3" xref="S3.Ex4.m1.5.5.3.cmml"><msup id="S3.Ex4.m1.5.5.3.5" xref="S3.Ex4.m1.5.5.3.5.cmml"><mi id="S3.Ex4.m1.5.5.3.5.2" xref="S3.Ex4.m1.5.5.3.5.2.cmml">L</mi><mi id="S3.Ex4.m1.5.5.3.5.3" xref="S3.Ex4.m1.5.5.3.5.3.cmml">p</mi></msup><mo id="S3.Ex4.m1.5.5.3.4" xref="S3.Ex4.m1.5.5.3.4.cmml">⁢</mo><mrow id="S3.Ex4.m1.5.5.3.3.2" xref="S3.Ex4.m1.5.5.3.3.3.cmml"><mo id="S3.Ex4.m1.5.5.3.3.2.3" stretchy="false" xref="S3.Ex4.m1.5.5.3.3.3.cmml">(</mo><msup id="S3.Ex4.m1.4.4.2.2.1.1" xref="S3.Ex4.m1.4.4.2.2.1.1.cmml"><mi id="S3.Ex4.m1.4.4.2.2.1.1.2" xref="S3.Ex4.m1.4.4.2.2.1.1.2.cmml">ℝ</mi><mi id="S3.Ex4.m1.4.4.2.2.1.1.3" xref="S3.Ex4.m1.4.4.2.2.1.1.3.cmml">d</mi></msup><mo id="S3.Ex4.m1.5.5.3.3.2.4" xref="S3.Ex4.m1.5.5.3.3.3.cmml">,</mo><msub id="S3.Ex4.m1.5.5.3.3.2.2" xref="S3.Ex4.m1.5.5.3.3.2.2.cmml"><mi id="S3.Ex4.m1.5.5.3.3.2.2.2" xref="S3.Ex4.m1.5.5.3.3.2.2.2.cmml">w</mi><mi id="S3.Ex4.m1.5.5.3.3.2.2.3" xref="S3.Ex4.m1.5.5.3.3.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex4.m1.5.5.3.3.2.5" xref="S3.Ex4.m1.5.5.3.3.3.cmml">;</mo><mi id="S3.Ex4.m1.3.3.1.1" xref="S3.Ex4.m1.3.3.1.1.cmml">X</mi><mo id="S3.Ex4.m1.5.5.3.3.2.6" stretchy="false" xref="S3.Ex4.m1.5.5.3.3.3.cmml">)</mo></mrow></mrow></msub><mo id="S3.Ex4.m1.17.17.1.1.2.2" xref="S3.Ex4.m1.17.17.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex4.m1.17.17.1.1.2.1" xref="S3.Ex4.m1.17.17.1.1.2.1.cmml"><msub id="S3.Ex4.m1.17.17.1.1.2.1.2" xref="S3.Ex4.m1.17.17.1.1.2.1.2.cmml"><mo id="S3.Ex4.m1.17.17.1.1.2.1.2.2" rspace="0em" xref="S3.Ex4.m1.17.17.1.1.2.1.2.2.cmml">∫</mo><msup id="S3.Ex4.m1.17.17.1.1.2.1.2.3" xref="S3.Ex4.m1.17.17.1.1.2.1.2.3.cmml"><mi id="S3.Ex4.m1.17.17.1.1.2.1.2.3.2" xref="S3.Ex4.m1.17.17.1.1.2.1.2.3.2.cmml">ℝ</mi><mrow id="S3.Ex4.m1.17.17.1.1.2.1.2.3.3" xref="S3.Ex4.m1.17.17.1.1.2.1.2.3.3.cmml"><mi id="S3.Ex4.m1.17.17.1.1.2.1.2.3.3.2" xref="S3.Ex4.m1.17.17.1.1.2.1.2.3.3.2.cmml">d</mi><mo id="S3.Ex4.m1.17.17.1.1.2.1.2.3.3.1" xref="S3.Ex4.m1.17.17.1.1.2.1.2.3.3.1.cmml">−</mo><mn id="S3.Ex4.m1.17.17.1.1.2.1.2.3.3.3" xref="S3.Ex4.m1.17.17.1.1.2.1.2.3.3.3.cmml">1</mn></mrow></msup></msub><mrow id="S3.Ex4.m1.17.17.1.1.2.1.1" xref="S3.Ex4.m1.17.17.1.1.2.1.1.cmml"><msub id="S3.Ex4.m1.17.17.1.1.2.1.1.1" xref="S3.Ex4.m1.17.17.1.1.2.1.1.1.cmml"><mrow 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xref="S3.Ex4.m1.10.10.3.4"></times><apply id="S3.Ex4.m1.10.10.3.5.cmml" xref="S3.Ex4.m1.10.10.3.5"><csymbol cd="ambiguous" id="S3.Ex4.m1.10.10.3.5.1.cmml" xref="S3.Ex4.m1.10.10.3.5">superscript</csymbol><ci id="S3.Ex4.m1.10.10.3.5.2.cmml" xref="S3.Ex4.m1.10.10.3.5.2">𝐿</ci><ci id="S3.Ex4.m1.10.10.3.5.3.cmml" xref="S3.Ex4.m1.10.10.3.5.3">𝑝</ci></apply><vector id="S3.Ex4.m1.10.10.3.3.3.cmml" xref="S3.Ex4.m1.10.10.3.3.2"><apply id="S3.Ex4.m1.9.9.2.2.1.1.cmml" xref="S3.Ex4.m1.9.9.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex4.m1.9.9.2.2.1.1.1.cmml" xref="S3.Ex4.m1.9.9.2.2.1.1">superscript</csymbol><ci id="S3.Ex4.m1.9.9.2.2.1.1.2.cmml" xref="S3.Ex4.m1.9.9.2.2.1.1.2">ℝ</ci><ci id="S3.Ex4.m1.9.9.2.2.1.1.3.cmml" xref="S3.Ex4.m1.9.9.2.2.1.1.3">𝑑</ci></apply><apply id="S3.Ex4.m1.10.10.3.3.2.2.cmml" xref="S3.Ex4.m1.10.10.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex4.m1.10.10.3.3.2.2.1.cmml" xref="S3.Ex4.m1.10.10.3.3.2.2">subscript</csymbol><ci id="S3.Ex4.m1.10.10.3.3.2.2.2.cmml" xref="S3.Ex4.m1.10.10.3.3.2.2.2">𝑤</ci><ci id="S3.Ex4.m1.10.10.3.3.2.2.3.cmml" xref="S3.Ex4.m1.10.10.3.3.2.2.3">𝛾</ci></apply><ci id="S3.Ex4.m1.8.8.1.1.cmml" xref="S3.Ex4.m1.8.8.1.1">𝑋</ci></vector></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex4.m1.17c">\|f(0,\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq\|f\|_{L^{p}(\mathbb{R}^{d},w_{% \gamma};X)}\int_{\mathbb{R}^{d-1}}\|\varphi(\cdot,\widetilde{y})\|_{L^{p^{% \prime}}(\mathbb{R},w_{\gamma^{\prime}})}\hskip 2.0pt\mathrm{d}\widetilde{y}=C% \|f\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex4.m1.17d">∥ italic_f ( 0 , ⋅ ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∥ italic_φ ( ⋅ , over~ start_ARG italic_y end_ARG ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT roman_d over~ start_ARG italic_y end_ARG = italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.2.p2.10">using that <math alttext="\gamma^{\prime}>-1" class="ltx_Math" display="inline" id="S3.SS1.2.p2.10.m1.1"><semantics id="S3.SS1.2.p2.10.m1.1a"><mrow id="S3.SS1.2.p2.10.m1.1.1" xref="S3.SS1.2.p2.10.m1.1.1.cmml"><msup id="S3.SS1.2.p2.10.m1.1.1.2" xref="S3.SS1.2.p2.10.m1.1.1.2.cmml"><mi id="S3.SS1.2.p2.10.m1.1.1.2.2" xref="S3.SS1.2.p2.10.m1.1.1.2.2.cmml">γ</mi><mo id="S3.SS1.2.p2.10.m1.1.1.2.3" xref="S3.SS1.2.p2.10.m1.1.1.2.3.cmml">′</mo></msup><mo id="S3.SS1.2.p2.10.m1.1.1.1" xref="S3.SS1.2.p2.10.m1.1.1.1.cmml">></mo><mrow id="S3.SS1.2.p2.10.m1.1.1.3" xref="S3.SS1.2.p2.10.m1.1.1.3.cmml"><mo id="S3.SS1.2.p2.10.m1.1.1.3a" xref="S3.SS1.2.p2.10.m1.1.1.3.cmml">−</mo><mn id="S3.SS1.2.p2.10.m1.1.1.3.2" xref="S3.SS1.2.p2.10.m1.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.2.p2.10.m1.1b"><apply id="S3.SS1.2.p2.10.m1.1.1.cmml" xref="S3.SS1.2.p2.10.m1.1.1"><gt id="S3.SS1.2.p2.10.m1.1.1.1.cmml" xref="S3.SS1.2.p2.10.m1.1.1.1"></gt><apply id="S3.SS1.2.p2.10.m1.1.1.2.cmml" xref="S3.SS1.2.p2.10.m1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.2.p2.10.m1.1.1.2.1.cmml" xref="S3.SS1.2.p2.10.m1.1.1.2">superscript</csymbol><ci id="S3.SS1.2.p2.10.m1.1.1.2.2.cmml" xref="S3.SS1.2.p2.10.m1.1.1.2.2">𝛾</ci><ci id="S3.SS1.2.p2.10.m1.1.1.2.3.cmml" xref="S3.SS1.2.p2.10.m1.1.1.2.3">′</ci></apply><apply id="S3.SS1.2.p2.10.m1.1.1.3.cmml" xref="S3.SS1.2.p2.10.m1.1.1.3"><minus id="S3.SS1.2.p2.10.m1.1.1.3.1.cmml" xref="S3.SS1.2.p2.10.m1.1.1.3"></minus><cn id="S3.SS1.2.p2.10.m1.1.1.3.2.cmml" type="integer" xref="S3.SS1.2.p2.10.m1.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.2.p2.10.m1.1c">\gamma^{\prime}>-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.2.p2.10.m1.1d">italic_γ start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT > - 1</annotation></semantics></math> in the last step.</p> </div> <div class="ltx_para" id="S3.SS1.3.p3"> <p class="ltx_p" id="S3.SS1.3.p3.6"><span class="ltx_text ltx_font_italic" id="S3.SS1.3.p3.1.1">Step 2: the case <math alttext="\gamma\geq p-1" class="ltx_Math" display="inline" id="S3.SS1.3.p3.1.1.m1.1"><semantics id="S3.SS1.3.p3.1.1.m1.1a"><mrow id="S3.SS1.3.p3.1.1.m1.1.1" xref="S3.SS1.3.p3.1.1.m1.1.1.cmml"><mi id="S3.SS1.3.p3.1.1.m1.1.1.2" xref="S3.SS1.3.p3.1.1.m1.1.1.2.cmml">γ</mi><mo id="S3.SS1.3.p3.1.1.m1.1.1.1" xref="S3.SS1.3.p3.1.1.m1.1.1.1.cmml">≥</mo><mrow id="S3.SS1.3.p3.1.1.m1.1.1.3" xref="S3.SS1.3.p3.1.1.m1.1.1.3.cmml"><mi id="S3.SS1.3.p3.1.1.m1.1.1.3.2" xref="S3.SS1.3.p3.1.1.m1.1.1.3.2.cmml">p</mi><mo id="S3.SS1.3.p3.1.1.m1.1.1.3.1" xref="S3.SS1.3.p3.1.1.m1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.3.p3.1.1.m1.1.1.3.3" xref="S3.SS1.3.p3.1.1.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.1.1.m1.1b"><apply id="S3.SS1.3.p3.1.1.m1.1.1.cmml" xref="S3.SS1.3.p3.1.1.m1.1.1"><geq id="S3.SS1.3.p3.1.1.m1.1.1.1.cmml" xref="S3.SS1.3.p3.1.1.m1.1.1.1"></geq><ci id="S3.SS1.3.p3.1.1.m1.1.1.2.cmml" xref="S3.SS1.3.p3.1.1.m1.1.1.2">𝛾</ci><apply id="S3.SS1.3.p3.1.1.m1.1.1.3.cmml" xref="S3.SS1.3.p3.1.1.m1.1.1.3"><minus id="S3.SS1.3.p3.1.1.m1.1.1.3.1.cmml" xref="S3.SS1.3.p3.1.1.m1.1.1.3.1"></minus><ci id="S3.SS1.3.p3.1.1.m1.1.1.3.2.cmml" xref="S3.SS1.3.p3.1.1.m1.1.1.3.2">𝑝</ci><cn id="S3.SS1.3.p3.1.1.m1.1.1.3.3.cmml" type="integer" xref="S3.SS1.3.p3.1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.1.1.m1.1c">\gamma\geq p-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.1.1.m1.1d">italic_γ ≥ italic_p - 1</annotation></semantics></math>.</span> Let <math alttext="\mathcal{F}_{x_{1}}" class="ltx_Math" display="inline" id="S3.SS1.3.p3.2.m1.1"><semantics id="S3.SS1.3.p3.2.m1.1a"><msub id="S3.SS1.3.p3.2.m1.1.1" xref="S3.SS1.3.p3.2.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.3.p3.2.m1.1.1.2" xref="S3.SS1.3.p3.2.m1.1.1.2.cmml">ℱ</mi><msub id="S3.SS1.3.p3.2.m1.1.1.3" xref="S3.SS1.3.p3.2.m1.1.1.3.cmml"><mi id="S3.SS1.3.p3.2.m1.1.1.3.2" xref="S3.SS1.3.p3.2.m1.1.1.3.2.cmml">x</mi><mn id="S3.SS1.3.p3.2.m1.1.1.3.3" xref="S3.SS1.3.p3.2.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.2.m1.1b"><apply id="S3.SS1.3.p3.2.m1.1.1.cmml" xref="S3.SS1.3.p3.2.m1.1.1"><csymbol cd="ambiguous" id="S3.SS1.3.p3.2.m1.1.1.1.cmml" xref="S3.SS1.3.p3.2.m1.1.1">subscript</csymbol><ci id="S3.SS1.3.p3.2.m1.1.1.2.cmml" xref="S3.SS1.3.p3.2.m1.1.1.2">ℱ</ci><apply id="S3.SS1.3.p3.2.m1.1.1.3.cmml" xref="S3.SS1.3.p3.2.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.3.p3.2.m1.1.1.3.1.cmml" xref="S3.SS1.3.p3.2.m1.1.1.3">subscript</csymbol><ci id="S3.SS1.3.p3.2.m1.1.1.3.2.cmml" xref="S3.SS1.3.p3.2.m1.1.1.3.2">𝑥</ci><cn id="S3.SS1.3.p3.2.m1.1.1.3.3.cmml" type="integer" xref="S3.SS1.3.p3.2.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.2.m1.1c">\mathcal{F}_{x_{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.2.m1.1d">caligraphic_F start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be the Fourier transform with respect to <math alttext="x_{1}\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS1.3.p3.3.m2.1"><semantics id="S3.SS1.3.p3.3.m2.1a"><mrow id="S3.SS1.3.p3.3.m2.1.1" xref="S3.SS1.3.p3.3.m2.1.1.cmml"><msub id="S3.SS1.3.p3.3.m2.1.1.2" xref="S3.SS1.3.p3.3.m2.1.1.2.cmml"><mi id="S3.SS1.3.p3.3.m2.1.1.2.2" xref="S3.SS1.3.p3.3.m2.1.1.2.2.cmml">x</mi><mn id="S3.SS1.3.p3.3.m2.1.1.2.3" xref="S3.SS1.3.p3.3.m2.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.3.p3.3.m2.1.1.1" xref="S3.SS1.3.p3.3.m2.1.1.1.cmml">∈</mo><mi id="S3.SS1.3.p3.3.m2.1.1.3" xref="S3.SS1.3.p3.3.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.3.m2.1b"><apply id="S3.SS1.3.p3.3.m2.1.1.cmml" xref="S3.SS1.3.p3.3.m2.1.1"><in id="S3.SS1.3.p3.3.m2.1.1.1.cmml" xref="S3.SS1.3.p3.3.m2.1.1.1"></in><apply id="S3.SS1.3.p3.3.m2.1.1.2.cmml" xref="S3.SS1.3.p3.3.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.3.p3.3.m2.1.1.2.1.cmml" xref="S3.SS1.3.p3.3.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.3.p3.3.m2.1.1.2.2.cmml" xref="S3.SS1.3.p3.3.m2.1.1.2.2">𝑥</ci><cn id="S3.SS1.3.p3.3.m2.1.1.2.3.cmml" type="integer" xref="S3.SS1.3.p3.3.m2.1.1.2.3">1</cn></apply><ci id="S3.SS1.3.p3.3.m2.1.1.3.cmml" xref="S3.SS1.3.p3.3.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.3.m2.1c">x_{1}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.3.m2.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math>. Since <math alttext="\text{\rm supp\,}\widehat{f}\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq R\}" class="ltx_Math" display="inline" id="S3.SS1.3.p3.4.m3.3"><semantics id="S3.SS1.3.p3.4.m3.3a"><mrow id="S3.SS1.3.p3.4.m3.3.3" xref="S3.SS1.3.p3.4.m3.3.3.cmml"><mrow id="S3.SS1.3.p3.4.m3.3.3.4" xref="S3.SS1.3.p3.4.m3.3.3.4.cmml"><mtext id="S3.SS1.3.p3.4.m3.3.3.4.2" xref="S3.SS1.3.p3.4.m3.3.3.4.2a.cmml">supp </mtext><mo id="S3.SS1.3.p3.4.m3.3.3.4.1" xref="S3.SS1.3.p3.4.m3.3.3.4.1.cmml">⁢</mo><mover accent="true" id="S3.SS1.3.p3.4.m3.3.3.4.3" xref="S3.SS1.3.p3.4.m3.3.3.4.3.cmml"><mi id="S3.SS1.3.p3.4.m3.3.3.4.3.2" xref="S3.SS1.3.p3.4.m3.3.3.4.3.2.cmml">f</mi><mo id="S3.SS1.3.p3.4.m3.3.3.4.3.1" xref="S3.SS1.3.p3.4.m3.3.3.4.3.1.cmml">^</mo></mover></mrow><mo id="S3.SS1.3.p3.4.m3.3.3.3" xref="S3.SS1.3.p3.4.m3.3.3.3.cmml">⊆</mo><mrow id="S3.SS1.3.p3.4.m3.3.3.2.2" xref="S3.SS1.3.p3.4.m3.3.3.2.3.cmml"><mo id="S3.SS1.3.p3.4.m3.3.3.2.2.3" stretchy="false" xref="S3.SS1.3.p3.4.m3.3.3.2.3.1.cmml">{</mo><mrow id="S3.SS1.3.p3.4.m3.2.2.1.1.1" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.cmml"><mi id="S3.SS1.3.p3.4.m3.2.2.1.1.1.2" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.2.cmml">ξ</mi><mo id="S3.SS1.3.p3.4.m3.2.2.1.1.1.1" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.1.cmml">∈</mo><msup id="S3.SS1.3.p3.4.m3.2.2.1.1.1.3" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.cmml"><mi id="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.2" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.2.cmml">ℝ</mi><mi id="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.3" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.3.cmml">d</mi></msup></mrow><mo id="S3.SS1.3.p3.4.m3.3.3.2.2.4" lspace="0.278em" rspace="0.278em" xref="S3.SS1.3.p3.4.m3.3.3.2.3.1.cmml">:</mo><mrow id="S3.SS1.3.p3.4.m3.3.3.2.2.2" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.cmml"><mrow id="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.2" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.1.cmml"><mo id="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.2.1" stretchy="false" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.1.1.cmml">|</mo><mi id="S3.SS1.3.p3.4.m3.1.1" xref="S3.SS1.3.p3.4.m3.1.1.cmml">ξ</mi><mo id="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.2.2" stretchy="false" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.1.1.cmml">|</mo></mrow><mo id="S3.SS1.3.p3.4.m3.3.3.2.2.2.1" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.1.cmml">≤</mo><mi id="S3.SS1.3.p3.4.m3.3.3.2.2.2.3" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.3.cmml">R</mi></mrow><mo id="S3.SS1.3.p3.4.m3.3.3.2.2.5" stretchy="false" xref="S3.SS1.3.p3.4.m3.3.3.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.4.m3.3b"><apply id="S3.SS1.3.p3.4.m3.3.3.cmml" xref="S3.SS1.3.p3.4.m3.3.3"><subset id="S3.SS1.3.p3.4.m3.3.3.3.cmml" xref="S3.SS1.3.p3.4.m3.3.3.3"></subset><apply id="S3.SS1.3.p3.4.m3.3.3.4.cmml" xref="S3.SS1.3.p3.4.m3.3.3.4"><times id="S3.SS1.3.p3.4.m3.3.3.4.1.cmml" xref="S3.SS1.3.p3.4.m3.3.3.4.1"></times><ci id="S3.SS1.3.p3.4.m3.3.3.4.2a.cmml" xref="S3.SS1.3.p3.4.m3.3.3.4.2"><mtext id="S3.SS1.3.p3.4.m3.3.3.4.2.cmml" xref="S3.SS1.3.p3.4.m3.3.3.4.2">supp </mtext></ci><apply id="S3.SS1.3.p3.4.m3.3.3.4.3.cmml" xref="S3.SS1.3.p3.4.m3.3.3.4.3"><ci id="S3.SS1.3.p3.4.m3.3.3.4.3.1.cmml" xref="S3.SS1.3.p3.4.m3.3.3.4.3.1">^</ci><ci id="S3.SS1.3.p3.4.m3.3.3.4.3.2.cmml" xref="S3.SS1.3.p3.4.m3.3.3.4.3.2">𝑓</ci></apply></apply><apply id="S3.SS1.3.p3.4.m3.3.3.2.3.cmml" xref="S3.SS1.3.p3.4.m3.3.3.2.2"><csymbol cd="latexml" id="S3.SS1.3.p3.4.m3.3.3.2.3.1.cmml" xref="S3.SS1.3.p3.4.m3.3.3.2.2.3">conditional-set</csymbol><apply id="S3.SS1.3.p3.4.m3.2.2.1.1.1.cmml" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1"><in id="S3.SS1.3.p3.4.m3.2.2.1.1.1.1.cmml" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.1"></in><ci id="S3.SS1.3.p3.4.m3.2.2.1.1.1.2.cmml" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.2">𝜉</ci><apply id="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.cmml" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.1.cmml" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.3">superscript</csymbol><ci id="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.2.cmml" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.2">ℝ</ci><ci id="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.3.cmml" xref="S3.SS1.3.p3.4.m3.2.2.1.1.1.3.3">𝑑</ci></apply></apply><apply id="S3.SS1.3.p3.4.m3.3.3.2.2.2.cmml" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2"><leq id="S3.SS1.3.p3.4.m3.3.3.2.2.2.1.cmml" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.1"></leq><apply id="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.1.cmml" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.2"><abs id="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.1.1.cmml" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.2.2.1"></abs><ci id="S3.SS1.3.p3.4.m3.1.1.cmml" xref="S3.SS1.3.p3.4.m3.1.1">𝜉</ci></apply><ci id="S3.SS1.3.p3.4.m3.3.3.2.2.2.3.cmml" xref="S3.SS1.3.p3.4.m3.3.3.2.2.2.3">𝑅</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.4.m3.3c">\text{\rm supp\,}\widehat{f}\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq R\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.4.m3.3d">supp over^ start_ARG italic_f end_ARG ⊆ { italic_ξ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : | italic_ξ | ≤ italic_R }</annotation></semantics></math> we have that <math alttext="\text{\rm supp\,}\mathcal{F}_{x_{1}}(f(\cdot,\widetilde{x}))\subseteq\{\xi_{1}% \in\mathbb{R}:|\xi_{1}|\leq R\}" class="ltx_Math" display="inline" id="S3.SS1.3.p3.5.m4.5"><semantics id="S3.SS1.3.p3.5.m4.5a"><mrow id="S3.SS1.3.p3.5.m4.5.5" xref="S3.SS1.3.p3.5.m4.5.5.cmml"><mrow id="S3.SS1.3.p3.5.m4.3.3.1" xref="S3.SS1.3.p3.5.m4.3.3.1.cmml"><mtext id="S3.SS1.3.p3.5.m4.3.3.1.3" xref="S3.SS1.3.p3.5.m4.3.3.1.3a.cmml">supp </mtext><mo id="S3.SS1.3.p3.5.m4.3.3.1.2" xref="S3.SS1.3.p3.5.m4.3.3.1.2.cmml">⁢</mo><msub id="S3.SS1.3.p3.5.m4.3.3.1.4" xref="S3.SS1.3.p3.5.m4.3.3.1.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.3.p3.5.m4.3.3.1.4.2" xref="S3.SS1.3.p3.5.m4.3.3.1.4.2.cmml">ℱ</mi><msub id="S3.SS1.3.p3.5.m4.3.3.1.4.3" xref="S3.SS1.3.p3.5.m4.3.3.1.4.3.cmml"><mi id="S3.SS1.3.p3.5.m4.3.3.1.4.3.2" xref="S3.SS1.3.p3.5.m4.3.3.1.4.3.2.cmml">x</mi><mn id="S3.SS1.3.p3.5.m4.3.3.1.4.3.3" xref="S3.SS1.3.p3.5.m4.3.3.1.4.3.3.cmml">1</mn></msub></msub><mo id="S3.SS1.3.p3.5.m4.3.3.1.2a" xref="S3.SS1.3.p3.5.m4.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS1.3.p3.5.m4.3.3.1.1.1" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.cmml"><mo id="S3.SS1.3.p3.5.m4.3.3.1.1.1.2" stretchy="false" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.cmml"><mi id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.2" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.2.cmml">f</mi><mo id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.1" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.1.cmml">⁢</mo><mrow id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.2" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.1.cmml"><mo id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.2.1" stretchy="false" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.1.cmml">(</mo><mo id="S3.SS1.3.p3.5.m4.1.1" lspace="0em" rspace="0em" xref="S3.SS1.3.p3.5.m4.1.1.cmml">⋅</mo><mo id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.2.2" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.1.cmml">,</mo><mover accent="true" id="S3.SS1.3.p3.5.m4.2.2" xref="S3.SS1.3.p3.5.m4.2.2.cmml"><mi id="S3.SS1.3.p3.5.m4.2.2.2" xref="S3.SS1.3.p3.5.m4.2.2.2.cmml">x</mi><mo id="S3.SS1.3.p3.5.m4.2.2.1" xref="S3.SS1.3.p3.5.m4.2.2.1.cmml">~</mo></mover><mo id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.2.3" stretchy="false" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.3.p3.5.m4.3.3.1.1.1.3" stretchy="false" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.3.p3.5.m4.5.5.4" xref="S3.SS1.3.p3.5.m4.5.5.4.cmml">⊆</mo><mrow id="S3.SS1.3.p3.5.m4.5.5.3.2" xref="S3.SS1.3.p3.5.m4.5.5.3.3.cmml"><mo id="S3.SS1.3.p3.5.m4.5.5.3.2.3" stretchy="false" xref="S3.SS1.3.p3.5.m4.5.5.3.3.1.cmml">{</mo><mrow id="S3.SS1.3.p3.5.m4.4.4.2.1.1" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.cmml"><msub id="S3.SS1.3.p3.5.m4.4.4.2.1.1.2" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.cmml"><mi id="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.2" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.2.cmml">ξ</mi><mn id="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.3" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.3.p3.5.m4.4.4.2.1.1.1" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.1.cmml">∈</mo><mi id="S3.SS1.3.p3.5.m4.4.4.2.1.1.3" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.3.cmml">ℝ</mi></mrow><mo id="S3.SS1.3.p3.5.m4.5.5.3.2.4" lspace="0.278em" rspace="0.278em" xref="S3.SS1.3.p3.5.m4.5.5.3.3.1.cmml">:</mo><mrow id="S3.SS1.3.p3.5.m4.5.5.3.2.2" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.cmml"><mrow id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.2.cmml"><mo id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.2" stretchy="false" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.2.1.cmml">|</mo><msub id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.cmml"><mi id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.2" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.2.cmml">ξ</mi><mn id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.3" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.3" stretchy="false" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.2.1.cmml">|</mo></mrow><mo id="S3.SS1.3.p3.5.m4.5.5.3.2.2.2" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.2.cmml">≤</mo><mi id="S3.SS1.3.p3.5.m4.5.5.3.2.2.3" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.3.cmml">R</mi></mrow><mo id="S3.SS1.3.p3.5.m4.5.5.3.2.5" stretchy="false" xref="S3.SS1.3.p3.5.m4.5.5.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.5.m4.5b"><apply id="S3.SS1.3.p3.5.m4.5.5.cmml" xref="S3.SS1.3.p3.5.m4.5.5"><subset id="S3.SS1.3.p3.5.m4.5.5.4.cmml" xref="S3.SS1.3.p3.5.m4.5.5.4"></subset><apply id="S3.SS1.3.p3.5.m4.3.3.1.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1"><times id="S3.SS1.3.p3.5.m4.3.3.1.2.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.2"></times><ci id="S3.SS1.3.p3.5.m4.3.3.1.3a.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.3"><mtext id="S3.SS1.3.p3.5.m4.3.3.1.3.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.3">supp </mtext></ci><apply id="S3.SS1.3.p3.5.m4.3.3.1.4.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.4"><csymbol cd="ambiguous" id="S3.SS1.3.p3.5.m4.3.3.1.4.1.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.4">subscript</csymbol><ci id="S3.SS1.3.p3.5.m4.3.3.1.4.2.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.4.2">ℱ</ci><apply id="S3.SS1.3.p3.5.m4.3.3.1.4.3.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.4.3"><csymbol cd="ambiguous" id="S3.SS1.3.p3.5.m4.3.3.1.4.3.1.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.4.3">subscript</csymbol><ci id="S3.SS1.3.p3.5.m4.3.3.1.4.3.2.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.4.3.2">𝑥</ci><cn id="S3.SS1.3.p3.5.m4.3.3.1.4.3.3.cmml" type="integer" xref="S3.SS1.3.p3.5.m4.3.3.1.4.3.3">1</cn></apply></apply><apply id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1"><times id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.1.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.1"></times><ci id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.2.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.2">𝑓</ci><interval closure="open" id="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.1.cmml" xref="S3.SS1.3.p3.5.m4.3.3.1.1.1.1.3.2"><ci id="S3.SS1.3.p3.5.m4.1.1.cmml" xref="S3.SS1.3.p3.5.m4.1.1">⋅</ci><apply id="S3.SS1.3.p3.5.m4.2.2.cmml" xref="S3.SS1.3.p3.5.m4.2.2"><ci id="S3.SS1.3.p3.5.m4.2.2.1.cmml" xref="S3.SS1.3.p3.5.m4.2.2.1">~</ci><ci id="S3.SS1.3.p3.5.m4.2.2.2.cmml" xref="S3.SS1.3.p3.5.m4.2.2.2">𝑥</ci></apply></interval></apply></apply><apply id="S3.SS1.3.p3.5.m4.5.5.3.3.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2"><csymbol cd="latexml" id="S3.SS1.3.p3.5.m4.5.5.3.3.1.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.3">conditional-set</csymbol><apply id="S3.SS1.3.p3.5.m4.4.4.2.1.1.cmml" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1"><in id="S3.SS1.3.p3.5.m4.4.4.2.1.1.1.cmml" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.1"></in><apply id="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.cmml" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.1.cmml" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.2">subscript</csymbol><ci id="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.2.cmml" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.2">𝜉</ci><cn id="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.3.cmml" type="integer" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.2.3">1</cn></apply><ci id="S3.SS1.3.p3.5.m4.4.4.2.1.1.3.cmml" xref="S3.SS1.3.p3.5.m4.4.4.2.1.1.3">ℝ</ci></apply><apply id="S3.SS1.3.p3.5.m4.5.5.3.2.2.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2"><leq id="S3.SS1.3.p3.5.m4.5.5.3.2.2.2.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.2"></leq><apply id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.2.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1"><abs id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.2.1.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.2"></abs><apply id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.1.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.2.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.2">𝜉</ci><cn id="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.3.cmml" type="integer" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.1.1.1.3">1</cn></apply></apply><ci id="S3.SS1.3.p3.5.m4.5.5.3.2.2.3.cmml" xref="S3.SS1.3.p3.5.m4.5.5.3.2.2.3">𝑅</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.5.m4.5c">\text{\rm supp\,}\mathcal{F}_{x_{1}}(f(\cdot,\widetilde{x}))\subseteq\{\xi_{1}% \in\mathbb{R}:|\xi_{1}|\leq R\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.5.m4.5d">supp caligraphic_F start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_f ( ⋅ , over~ start_ARG italic_x end_ARG ) ) ⊆ { italic_ξ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R : | italic_ξ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ italic_R }</annotation></semantics></math> for fixed <math alttext="\widetilde{x}\in\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S3.SS1.3.p3.6.m5.1"><semantics id="S3.SS1.3.p3.6.m5.1a"><mrow id="S3.SS1.3.p3.6.m5.1.1" xref="S3.SS1.3.p3.6.m5.1.1.cmml"><mover accent="true" id="S3.SS1.3.p3.6.m5.1.1.2" xref="S3.SS1.3.p3.6.m5.1.1.2.cmml"><mi id="S3.SS1.3.p3.6.m5.1.1.2.2" xref="S3.SS1.3.p3.6.m5.1.1.2.2.cmml">x</mi><mo id="S3.SS1.3.p3.6.m5.1.1.2.1" xref="S3.SS1.3.p3.6.m5.1.1.2.1.cmml">~</mo></mover><mo id="S3.SS1.3.p3.6.m5.1.1.1" xref="S3.SS1.3.p3.6.m5.1.1.1.cmml">∈</mo><msup id="S3.SS1.3.p3.6.m5.1.1.3" xref="S3.SS1.3.p3.6.m5.1.1.3.cmml"><mi id="S3.SS1.3.p3.6.m5.1.1.3.2" xref="S3.SS1.3.p3.6.m5.1.1.3.2.cmml">ℝ</mi><mrow id="S3.SS1.3.p3.6.m5.1.1.3.3" xref="S3.SS1.3.p3.6.m5.1.1.3.3.cmml"><mi id="S3.SS1.3.p3.6.m5.1.1.3.3.2" xref="S3.SS1.3.p3.6.m5.1.1.3.3.2.cmml">d</mi><mo id="S3.SS1.3.p3.6.m5.1.1.3.3.1" xref="S3.SS1.3.p3.6.m5.1.1.3.3.1.cmml">−</mo><mn id="S3.SS1.3.p3.6.m5.1.1.3.3.3" xref="S3.SS1.3.p3.6.m5.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.6.m5.1b"><apply id="S3.SS1.3.p3.6.m5.1.1.cmml" xref="S3.SS1.3.p3.6.m5.1.1"><in id="S3.SS1.3.p3.6.m5.1.1.1.cmml" xref="S3.SS1.3.p3.6.m5.1.1.1"></in><apply id="S3.SS1.3.p3.6.m5.1.1.2.cmml" xref="S3.SS1.3.p3.6.m5.1.1.2"><ci id="S3.SS1.3.p3.6.m5.1.1.2.1.cmml" xref="S3.SS1.3.p3.6.m5.1.1.2.1">~</ci><ci id="S3.SS1.3.p3.6.m5.1.1.2.2.cmml" xref="S3.SS1.3.p3.6.m5.1.1.2.2">𝑥</ci></apply><apply id="S3.SS1.3.p3.6.m5.1.1.3.cmml" xref="S3.SS1.3.p3.6.m5.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.3.p3.6.m5.1.1.3.1.cmml" xref="S3.SS1.3.p3.6.m5.1.1.3">superscript</csymbol><ci id="S3.SS1.3.p3.6.m5.1.1.3.2.cmml" xref="S3.SS1.3.p3.6.m5.1.1.3.2">ℝ</ci><apply id="S3.SS1.3.p3.6.m5.1.1.3.3.cmml" xref="S3.SS1.3.p3.6.m5.1.1.3.3"><minus id="S3.SS1.3.p3.6.m5.1.1.3.3.1.cmml" xref="S3.SS1.3.p3.6.m5.1.1.3.3.1"></minus><ci id="S3.SS1.3.p3.6.m5.1.1.3.3.2.cmml" xref="S3.SS1.3.p3.6.m5.1.1.3.3.2">𝑑</ci><cn id="S3.SS1.3.p3.6.m5.1.1.3.3.3.cmml" type="integer" xref="S3.SS1.3.p3.6.m5.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.6.m5.1c">\widetilde{x}\in\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.6.m5.1d">over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Theorem 4.9, Step 1 & 2]</cite>. By the Plancherel-Polya-Nikol’skij type inequality from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 4.1]</cite> we obtain</p> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E1"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E1X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|f\|_{L^{p}(\mathbb{R}^{d};X)}" class="ltx_Math" display="inline" id="S3.E1X.2.1.1.m1.3"><semantics id="S3.E1X.2.1.1.m1.3a"><msub id="S3.E1X.2.1.1.m1.3.4" xref="S3.E1X.2.1.1.m1.3.4.cmml"><mrow id="S3.E1X.2.1.1.m1.3.4.2.2" xref="S3.E1X.2.1.1.m1.3.4.2.1.cmml"><mo id="S3.E1X.2.1.1.m1.3.4.2.2.1" stretchy="false" xref="S3.E1X.2.1.1.m1.3.4.2.1.1.cmml">‖</mo><mi id="S3.E1X.2.1.1.m1.3.3" xref="S3.E1X.2.1.1.m1.3.3.cmml">f</mi><mo id="S3.E1X.2.1.1.m1.3.4.2.2.2" stretchy="false" xref="S3.E1X.2.1.1.m1.3.4.2.1.1.cmml">‖</mo></mrow><mrow id="S3.E1X.2.1.1.m1.2.2.2" xref="S3.E1X.2.1.1.m1.2.2.2.cmml"><msup id="S3.E1X.2.1.1.m1.2.2.2.4" xref="S3.E1X.2.1.1.m1.2.2.2.4.cmml"><mi id="S3.E1X.2.1.1.m1.2.2.2.4.2" xref="S3.E1X.2.1.1.m1.2.2.2.4.2.cmml">L</mi><mi id="S3.E1X.2.1.1.m1.2.2.2.4.3" xref="S3.E1X.2.1.1.m1.2.2.2.4.3.cmml">p</mi></msup><mo id="S3.E1X.2.1.1.m1.2.2.2.3" xref="S3.E1X.2.1.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.E1X.2.1.1.m1.2.2.2.2.1" xref="S3.E1X.2.1.1.m1.2.2.2.2.2.cmml"><mo id="S3.E1X.2.1.1.m1.2.2.2.2.1.2" stretchy="false" xref="S3.E1X.2.1.1.m1.2.2.2.2.2.cmml">(</mo><msup id="S3.E1X.2.1.1.m1.2.2.2.2.1.1" xref="S3.E1X.2.1.1.m1.2.2.2.2.1.1.cmml"><mi id="S3.E1X.2.1.1.m1.2.2.2.2.1.1.2" xref="S3.E1X.2.1.1.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mi id="S3.E1X.2.1.1.m1.2.2.2.2.1.1.3" xref="S3.E1X.2.1.1.m1.2.2.2.2.1.1.3.cmml">d</mi></msup><mo id="S3.E1X.2.1.1.m1.2.2.2.2.1.3" xref="S3.E1X.2.1.1.m1.2.2.2.2.2.cmml">;</mo><mi id="S3.E1X.2.1.1.m1.1.1.1.1" 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id="S3.E1X.2.1.1.m1.2.2.2.4.2.cmml" xref="S3.E1X.2.1.1.m1.2.2.2.4.2">𝐿</ci><ci id="S3.E1X.2.1.1.m1.2.2.2.4.3.cmml" xref="S3.E1X.2.1.1.m1.2.2.2.4.3">𝑝</ci></apply><list id="S3.E1X.2.1.1.m1.2.2.2.2.2.cmml" xref="S3.E1X.2.1.1.m1.2.2.2.2.1"><apply id="S3.E1X.2.1.1.m1.2.2.2.2.1.1.cmml" xref="S3.E1X.2.1.1.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.E1X.2.1.1.m1.2.2.2.2.1.1.1.cmml" xref="S3.E1X.2.1.1.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S3.E1X.2.1.1.m1.2.2.2.2.1.1.2.cmml" xref="S3.E1X.2.1.1.m1.2.2.2.2.1.1.2">ℝ</ci><ci id="S3.E1X.2.1.1.m1.2.2.2.2.1.1.3.cmml" xref="S3.E1X.2.1.1.m1.2.2.2.2.1.1.3">𝑑</ci></apply><ci id="S3.E1X.2.1.1.m1.1.1.1.1.cmml" xref="S3.E1X.2.1.1.m1.1.1.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1X.2.1.1.m1.3c">\displaystyle\|f\|_{L^{p}(\mathbb{R}^{d};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.E1X.2.1.1.m1.3d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p 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\widetilde{x}\Big{)}^{\frac{1}{p}}</annotation><annotation encoding="application/x-llamapun" id="S3.E1X.3.2.2.m1.2d">= ( ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT ∥ italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(3.1)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" 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xref="S3.E1Xa.2.1.1.m1.1.1.1.1">𝑋</ci></vector></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E1Xa.2.1.1.m1.6c">\displaystyle\leq CR^{\frac{\gamma}{p}}\Big{(}\int_{\mathbb{R}^{d-1}}\int_{% \mathbb{R}}|x_{1}|^{\gamma}\|f(x_{1},\widetilde{x})\|_{X}^{p}\hskip 2.0pt% \mathrm{d}x_{1}\hskip 2.0pt\mathrm{d}\widetilde{x}\Big{)}^{\frac{1}{p}}=CR^{% \frac{\gamma}{p}}\|f\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S3.E1Xa.2.1.1.m1.6d">≤ italic_C italic_R start_POSTSUPERSCRIPT divide start_ARG italic_γ end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( ∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT blackboard_R end_POSTSUBSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT ∥ italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT = italic_C italic_R start_POSTSUPERSCRIPT divide start_ARG italic_γ end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S3.SS1.3.p3.7">From Step 1 applied to <math alttext="\gamma=0" class="ltx_Math" display="inline" id="S3.SS1.3.p3.7.m1.1"><semantics id="S3.SS1.3.p3.7.m1.1a"><mrow id="S3.SS1.3.p3.7.m1.1.1" xref="S3.SS1.3.p3.7.m1.1.1.cmml"><mi id="S3.SS1.3.p3.7.m1.1.1.2" xref="S3.SS1.3.p3.7.m1.1.1.2.cmml">γ</mi><mo id="S3.SS1.3.p3.7.m1.1.1.1" xref="S3.SS1.3.p3.7.m1.1.1.1.cmml">=</mo><mn id="S3.SS1.3.p3.7.m1.1.1.3" xref="S3.SS1.3.p3.7.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.7.m1.1b"><apply id="S3.SS1.3.p3.7.m1.1.1.cmml" xref="S3.SS1.3.p3.7.m1.1.1"><eq id="S3.SS1.3.p3.7.m1.1.1.1.cmml" xref="S3.SS1.3.p3.7.m1.1.1.1"></eq><ci id="S3.SS1.3.p3.7.m1.1.1.2.cmml" xref="S3.SS1.3.p3.7.m1.1.1.2">𝛾</ci><cn id="S3.SS1.3.p3.7.m1.1.1.3.cmml" type="integer" xref="S3.SS1.3.p3.7.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.7.m1.1c">\gamma=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.7.m1.1d">italic_γ = 0</annotation></semantics></math> and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E1" title="In Proof. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.1</span></a>), we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f(x_{1},\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq CR^{\frac{1}{p}}\|f\|_{L^{p% }(\mathbb{R}^{d};X)}\leq CR^{\frac{\gamma+1}{p}}\|f\|_{L^{p}(\mathbb{R}^{d},w_% {\gamma};X)}." class="ltx_Math" display="block" id="S3.Ex5.m1.11"><semantics id="S3.Ex5.m1.11a"><mrow id="S3.Ex5.m1.11.11.1" xref="S3.Ex5.m1.11.11.1.1.cmml"><mrow id="S3.Ex5.m1.11.11.1.1" xref="S3.Ex5.m1.11.11.1.1.cmml"><msub id="S3.Ex5.m1.11.11.1.1.1" xref="S3.Ex5.m1.11.11.1.1.1.cmml"><mrow id="S3.Ex5.m1.11.11.1.1.1.1.1" xref="S3.Ex5.m1.11.11.1.1.1.1.2.cmml"><mo id="S3.Ex5.m1.11.11.1.1.1.1.1.2" stretchy="false" 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italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C italic_R start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C italic_R start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ∥ italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.3.p3.8">Note that this argument is actually valid for any <math alttext="\gamma>0" class="ltx_Math" display="inline" id="S3.SS1.3.p3.8.m1.1"><semantics id="S3.SS1.3.p3.8.m1.1a"><mrow id="S3.SS1.3.p3.8.m1.1.1" xref="S3.SS1.3.p3.8.m1.1.1.cmml"><mi id="S3.SS1.3.p3.8.m1.1.1.2" xref="S3.SS1.3.p3.8.m1.1.1.2.cmml">γ</mi><mo id="S3.SS1.3.p3.8.m1.1.1.1" xref="S3.SS1.3.p3.8.m1.1.1.1.cmml">></mo><mn id="S3.SS1.3.p3.8.m1.1.1.3" xref="S3.SS1.3.p3.8.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.3.p3.8.m1.1b"><apply id="S3.SS1.3.p3.8.m1.1.1.cmml" xref="S3.SS1.3.p3.8.m1.1.1"><gt id="S3.SS1.3.p3.8.m1.1.1.1.cmml" xref="S3.SS1.3.p3.8.m1.1.1.1"></gt><ci id="S3.SS1.3.p3.8.m1.1.1.2.cmml" xref="S3.SS1.3.p3.8.m1.1.1.2">𝛾</ci><cn id="S3.SS1.3.p3.8.m1.1.1.3.cmml" type="integer" xref="S3.SS1.3.p3.8.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.3.p3.8.m1.1c">\gamma>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.3.p3.8.m1.1d">italic_γ > 0</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS1.p5"> <p class="ltx_p" id="S3.SS1.p5.1">From Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.1</span></a> it follows that the trace operator exists on certain weighted Besov spaces. The following result is a weighted version of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Proposition 4.4]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S3.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.1.1.1">Proposition 3.2</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem2.p1"> <p class="ltx_p" id="S3.Thmtheorem2.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem2.p1.5.5">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.1.1.m1.2"><semantics id="S3.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem2.p1.1.1.m1.2.3" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem2.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem2.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem2.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem2.p1.1.1.m1.1.1" xref="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem2.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem2.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem2.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem2.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.1.1.m1.2b"><apply id="S3.Thmtheorem2.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.2.3"><in id="S3.Thmtheorem2.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.1"></in><ci id="S3.Thmtheorem2.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.2.3.3.2"><cn id="S3.Thmtheorem2.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem2.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem2.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.2.2.m2.1"><semantics id="S3.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S3.Thmtheorem2.p1.2.2.m2.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S3.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><mo id="S3.Thmtheorem2.p1.2.2.m2.1.1.3a" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.2.2.m2.1b"><apply id="S3.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1"><gt id="S3.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.1"></gt><ci id="S3.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.2">𝛾</ci><apply id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3"><minus id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3"></minus><cn id="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" type="integer" xref="S3.Thmtheorem2.p1.2.2.m2.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.2.2.m2.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.2.2.m2.1d">italic_γ > - 1</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.3.3.m3.1"><semantics id="S3.Thmtheorem2.p1.3.3.m3.1a"><mi id="S3.Thmtheorem2.p1.3.3.m3.1.1" xref="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem2.p1.3.3.m3.1b"><ci id="S3.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem2.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.3.3.m3.1d">italic_X</annotation></semantics></math> be a Banach space. Then the trace of a function <math alttext="f\in B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.4.4.m4.5"><semantics id="S3.Thmtheorem2.p1.4.4.m4.5a"><mrow id="S3.Thmtheorem2.p1.4.4.m4.5.5" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.cmml"><mi id="S3.Thmtheorem2.p1.4.4.m4.5.5.4" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.4.cmml">f</mi><mo id="S3.Thmtheorem2.p1.4.4.m4.5.5.3" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.3.cmml">∈</mo><mrow id="S3.Thmtheorem2.p1.4.4.m4.5.5.2" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.2.cmml"><msubsup id="S3.Thmtheorem2.p1.4.4.m4.5.5.2.4" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.2.4.cmml"><mi id="S3.Thmtheorem2.p1.4.4.m4.5.5.2.4.2.2" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.2.4.2.2.cmml">B</mi><mrow id="S3.Thmtheorem2.p1.4.4.m4.2.2.2.4" xref="S3.Thmtheorem2.p1.4.4.m4.2.2.2.3.cmml"><mi id="S3.Thmtheorem2.p1.4.4.m4.1.1.1.1" xref="S3.Thmtheorem2.p1.4.4.m4.1.1.1.1.cmml">p</mi><mo id="S3.Thmtheorem2.p1.4.4.m4.2.2.2.4.1" 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xref="S3.Thmtheorem2.p1.4.4.m4.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.4.4.m4.5.5.2.2.2.2.1.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.2.2.2.2">subscript</csymbol><ci id="S3.Thmtheorem2.p1.4.4.m4.5.5.2.2.2.2.2.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.2.2.2.2.2">𝑤</ci><ci id="S3.Thmtheorem2.p1.4.4.m4.5.5.2.2.2.2.3.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S3.Thmtheorem2.p1.4.4.m4.3.3.cmml" xref="S3.Thmtheorem2.p1.4.4.m4.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.4.4.m4.5c">f\in B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.4.4.m4.5d">italic_f ∈ italic_B start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> exists and for any <math alttext="(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S3.Thmtheorem2.p1.5.5.m5.2"><semantics id="S3.Thmtheorem2.p1.5.5.m5.2a"><mrow id="S3.Thmtheorem2.p1.5.5.m5.2.2" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.cmml"><msub id="S3.Thmtheorem2.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.cmml"><mrow id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2.cmml">φ</mi><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.cmml"><mi id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.2" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.2.cmml">n</mi><mo id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.1" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.1.cmml">≥</mo><mn id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.3" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.3.cmml">0</mn></mrow></msub><mo id="S3.Thmtheorem2.p1.5.5.m5.2.2.3" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.3.cmml">∈</mo><mrow id="S3.Thmtheorem2.p1.5.5.m5.2.2.2" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.cmml"><mi id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.3" mathvariant="normal" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.3.cmml">Φ</mi><mo id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.2" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1.1.cmml"><mo 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xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1">subscript</csymbol><apply id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.2">𝜑</ci><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3"><geq id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.1"></geq><ci id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.2">𝑛</ci><cn id="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem2.p1.5.5.m5.1.1.1.3.3">0</cn></apply></apply><apply id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2"><times id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.2"></times><ci id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.3.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.3">Φ</ci><apply id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1">superscript</csymbol><ci id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem2.p1.5.5.m5.2.2.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem2.p1.5.5.m5.2c">(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem2.p1.5.5.m5.2d">( italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math>, we have</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}f=\sum_{n=0}^{\infty}\varphi_{n}\ast f(0,\cdot)\qquad\text{in% }L^{p}(\mathbb{R}^{d-1};X)." class="ltx_Math" display="block" id="S3.Ex6.m1.4"><semantics id="S3.Ex6.m1.4a"><mrow id="S3.Ex6.m1.4.4.1" xref="S3.Ex6.m1.4.4.1.1.cmml"><mrow id="S3.Ex6.m1.4.4.1.1" xref="S3.Ex6.m1.4.4.1.1.cmml"><mrow id="S3.Ex6.m1.4.4.1.1.4" xref="S3.Ex6.m1.4.4.1.1.4.cmml"><mi id="S3.Ex6.m1.4.4.1.1.4.1" xref="S3.Ex6.m1.4.4.1.1.4.1.cmml">Tr</mi><mo id="S3.Ex6.m1.4.4.1.1.4a" lspace="0.167em" xref="S3.Ex6.m1.4.4.1.1.4.cmml">⁡</mo><mi id="S3.Ex6.m1.4.4.1.1.4.2" xref="S3.Ex6.m1.4.4.1.1.4.2.cmml">f</mi></mrow><mo id="S3.Ex6.m1.4.4.1.1.3" rspace="0.111em" xref="S3.Ex6.m1.4.4.1.1.3.cmml">=</mo><mrow id="S3.Ex6.m1.4.4.1.1.2.2" xref="S3.Ex6.m1.4.4.1.1.2.3.cmml"><mrow id="S3.Ex6.m1.4.4.1.1.1.1.1" xref="S3.Ex6.m1.4.4.1.1.1.1.1.cmml"><munderover id="S3.Ex6.m1.4.4.1.1.1.1.1.1" xref="S3.Ex6.m1.4.4.1.1.1.1.1.1.cmml"><mo id="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.2" movablelimits="false" xref="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3" xref="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3.cmml"><mi id="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3.2" xref="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3.1" xref="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3.1.cmml">=</mo><mn id="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3.3" xref="S3.Ex6.m1.4.4.1.1.1.1.1.1.2.3.3.cmml">0</mn></mrow><mi id="S3.Ex6.m1.4.4.1.1.1.1.1.1.3" mathvariant="normal" xref="S3.Ex6.m1.4.4.1.1.1.1.1.1.3.cmml">∞</mi></munderover><mrow id="S3.Ex6.m1.4.4.1.1.1.1.1.2" 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</mtext></ci><apply id="S3.Ex6.m1.4.4.1.1.2.2.2.4.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.4"><csymbol cd="ambiguous" id="S3.Ex6.m1.4.4.1.1.2.2.2.4.1.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.4">superscript</csymbol><ci id="S3.Ex6.m1.4.4.1.1.2.2.2.4.2.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.4.2">𝐿</ci><ci id="S3.Ex6.m1.4.4.1.1.2.2.2.4.3.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.4.3">𝑝</ci></apply><list id="S3.Ex6.m1.4.4.1.1.2.2.2.1.2.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1"><apply id="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.1.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1">superscript</csymbol><ci id="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.2.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.2">ℝ</ci><apply id="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3"><minus id="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3.1.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3.1"></minus><ci id="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3.2.cmml" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3.2">𝑑</ci><cn id="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.Ex6.m1.4.4.1.1.2.2.2.1.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex6.m1.3.3.cmml" xref="S3.Ex6.m1.3.3">𝑋</ci></list></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex6.m1.4c">\operatorname{Tr}f=\sum_{n=0}^{\infty}\varphi_{n}\ast f(0,\cdot)\qquad\text{in% }L^{p}(\mathbb{R}^{d-1};X).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex6.m1.4d">roman_Tr italic_f = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_f ( 0 , ⋅ ) in italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S3.SS1.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.4.p1"> <p class="ltx_p" id="S3.SS1.4.p1.9">Let <math alttext="f\in B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS1.4.p1.1.m1.5"><semantics id="S3.SS1.4.p1.1.m1.5a"><mrow id="S3.SS1.4.p1.1.m1.5.5" xref="S3.SS1.4.p1.1.m1.5.5.cmml"><mi id="S3.SS1.4.p1.1.m1.5.5.4" xref="S3.SS1.4.p1.1.m1.5.5.4.cmml">f</mi><mo id="S3.SS1.4.p1.1.m1.5.5.3" xref="S3.SS1.4.p1.1.m1.5.5.3.cmml">∈</mo><mrow id="S3.SS1.4.p1.1.m1.5.5.2" xref="S3.SS1.4.p1.1.m1.5.5.2.cmml"><msubsup id="S3.SS1.4.p1.1.m1.5.5.2.4" xref="S3.SS1.4.p1.1.m1.5.5.2.4.cmml"><mi id="S3.SS1.4.p1.1.m1.5.5.2.4.2.2" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.2.cmml">B</mi><mrow id="S3.SS1.4.p1.1.m1.2.2.2.4" xref="S3.SS1.4.p1.1.m1.2.2.2.3.cmml"><mi id="S3.SS1.4.p1.1.m1.1.1.1.1" xref="S3.SS1.4.p1.1.m1.1.1.1.1.cmml">p</mi><mo id="S3.SS1.4.p1.1.m1.2.2.2.4.1" xref="S3.SS1.4.p1.1.m1.2.2.2.3.cmml">,</mo><mn id="S3.SS1.4.p1.1.m1.2.2.2.2" xref="S3.SS1.4.p1.1.m1.2.2.2.2.cmml">1</mn></mrow><mfrac id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.cmml"><mrow id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.cmml"><mi id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.2" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.2.cmml">γ</mi><mo id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.1" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.1.cmml">+</mo><mn id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.3" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.3.cmml">1</mn></mrow><mi id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.3" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.3.cmml">p</mi></mfrac></msubsup><mo id="S3.SS1.4.p1.1.m1.5.5.2.3" xref="S3.SS1.4.p1.1.m1.5.5.2.3.cmml">⁢</mo><mrow id="S3.SS1.4.p1.1.m1.5.5.2.2.2" xref="S3.SS1.4.p1.1.m1.5.5.2.2.3.cmml"><mo id="S3.SS1.4.p1.1.m1.5.5.2.2.2.3" stretchy="false" xref="S3.SS1.4.p1.1.m1.5.5.2.2.3.cmml">(</mo><msup id="S3.SS1.4.p1.1.m1.4.4.1.1.1.1" xref="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.cmml"><mi id="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.2" xref="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.3" xref="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.4.p1.1.m1.5.5.2.2.2.4" xref="S3.SS1.4.p1.1.m1.5.5.2.2.3.cmml">,</mo><msub id="S3.SS1.4.p1.1.m1.5.5.2.2.2.2" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.cmml"><mi id="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.2" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.2.cmml">w</mi><mi id="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.3" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS1.4.p1.1.m1.5.5.2.2.2.5" xref="S3.SS1.4.p1.1.m1.5.5.2.2.3.cmml">;</mo><mi id="S3.SS1.4.p1.1.m1.3.3" xref="S3.SS1.4.p1.1.m1.3.3.cmml">X</mi><mo id="S3.SS1.4.p1.1.m1.5.5.2.2.2.6" stretchy="false" xref="S3.SS1.4.p1.1.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.1.m1.5b"><apply id="S3.SS1.4.p1.1.m1.5.5.cmml" xref="S3.SS1.4.p1.1.m1.5.5"><in id="S3.SS1.4.p1.1.m1.5.5.3.cmml" xref="S3.SS1.4.p1.1.m1.5.5.3"></in><ci id="S3.SS1.4.p1.1.m1.5.5.4.cmml" xref="S3.SS1.4.p1.1.m1.5.5.4">𝑓</ci><apply id="S3.SS1.4.p1.1.m1.5.5.2.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2"><times id="S3.SS1.4.p1.1.m1.5.5.2.3.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.3"></times><apply id="S3.SS1.4.p1.1.m1.5.5.2.4.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4"><csymbol cd="ambiguous" id="S3.SS1.4.p1.1.m1.5.5.2.4.1.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4">subscript</csymbol><apply id="S3.SS1.4.p1.1.m1.5.5.2.4.2.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4"><csymbol cd="ambiguous" id="S3.SS1.4.p1.1.m1.5.5.2.4.2.1.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4">superscript</csymbol><ci id="S3.SS1.4.p1.1.m1.5.5.2.4.2.2.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.2">𝐵</ci><apply id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3"><divide id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.1.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3"></divide><apply id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2"><plus id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.1.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.1"></plus><ci id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.2.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.2">𝛾</ci><cn id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.3.cmml" type="integer" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.2.3">1</cn></apply><ci id="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.3.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.4.2.3.3">𝑝</ci></apply></apply><list id="S3.SS1.4.p1.1.m1.2.2.2.3.cmml" xref="S3.SS1.4.p1.1.m1.2.2.2.4"><ci id="S3.SS1.4.p1.1.m1.1.1.1.1.cmml" xref="S3.SS1.4.p1.1.m1.1.1.1.1">𝑝</ci><cn id="S3.SS1.4.p1.1.m1.2.2.2.2.cmml" type="integer" xref="S3.SS1.4.p1.1.m1.2.2.2.2">1</cn></list></apply><vector id="S3.SS1.4.p1.1.m1.5.5.2.2.3.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2"><apply id="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.cmml" xref="S3.SS1.4.p1.1.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.1.cmml" xref="S3.SS1.4.p1.1.m1.4.4.1.1.1.1">superscript</csymbol><ci id="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.2.cmml" xref="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.2">ℝ</ci><ci id="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.3.cmml" xref="S3.SS1.4.p1.1.m1.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.1.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2.2">subscript</csymbol><ci id="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.2.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.2">𝑤</ci><ci id="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.3.cmml" xref="S3.SS1.4.p1.1.m1.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S3.SS1.4.p1.1.m1.3.3.cmml" xref="S3.SS1.4.p1.1.m1.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.1.m1.5c">f\in B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.1.m1.5d">italic_f ∈ italic_B start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>, <math alttext="(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S3.SS1.4.p1.2.m2.2"><semantics id="S3.SS1.4.p1.2.m2.2a"><mrow id="S3.SS1.4.p1.2.m2.2.2" xref="S3.SS1.4.p1.2.m2.2.2.cmml"><msub id="S3.SS1.4.p1.2.m2.1.1.1" xref="S3.SS1.4.p1.2.m2.1.1.1.cmml"><mrow id="S3.SS1.4.p1.2.m2.1.1.1.1.1" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.cmml"><mo id="S3.SS1.4.p1.2.m2.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.4.p1.2.m2.1.1.1.1.1.1" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.cmml"><mi id="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.2" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.2.cmml">φ</mi><mi id="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.3" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.SS1.4.p1.2.m2.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.4.p1.2.m2.1.1.1.3" xref="S3.SS1.4.p1.2.m2.1.1.1.3.cmml"><mi id="S3.SS1.4.p1.2.m2.1.1.1.3.2" xref="S3.SS1.4.p1.2.m2.1.1.1.3.2.cmml">n</mi><mo id="S3.SS1.4.p1.2.m2.1.1.1.3.1" xref="S3.SS1.4.p1.2.m2.1.1.1.3.1.cmml">≥</mo><mn id="S3.SS1.4.p1.2.m2.1.1.1.3.3" xref="S3.SS1.4.p1.2.m2.1.1.1.3.3.cmml">0</mn></mrow></msub><mo id="S3.SS1.4.p1.2.m2.2.2.3" xref="S3.SS1.4.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S3.SS1.4.p1.2.m2.2.2.2" xref="S3.SS1.4.p1.2.m2.2.2.2.cmml"><mi id="S3.SS1.4.p1.2.m2.2.2.2.3" mathvariant="normal" xref="S3.SS1.4.p1.2.m2.2.2.2.3.cmml">Φ</mi><mo id="S3.SS1.4.p1.2.m2.2.2.2.2" xref="S3.SS1.4.p1.2.m2.2.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.4.p1.2.m2.2.2.2.1.1" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.cmml"><mo id="S3.SS1.4.p1.2.m2.2.2.2.1.1.2" stretchy="false" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.cmml">(</mo><msup id="S3.SS1.4.p1.2.m2.2.2.2.1.1.1" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.cmml"><mi id="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.2" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.3" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.4.p1.2.m2.2.2.2.1.1.3" stretchy="false" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.2.m2.2b"><apply id="S3.SS1.4.p1.2.m2.2.2.cmml" xref="S3.SS1.4.p1.2.m2.2.2"><in id="S3.SS1.4.p1.2.m2.2.2.3.cmml" xref="S3.SS1.4.p1.2.m2.2.2.3"></in><apply id="S3.SS1.4.p1.2.m2.1.1.1.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.2.m2.1.1.1.2.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1">subscript</csymbol><apply id="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.1.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.2.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.2">𝜑</ci><ci id="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.3.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1.1.1.1.3">𝑛</ci></apply><apply id="S3.SS1.4.p1.2.m2.1.1.1.3.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1.3"><geq id="S3.SS1.4.p1.2.m2.1.1.1.3.1.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1.3.1"></geq><ci id="S3.SS1.4.p1.2.m2.1.1.1.3.2.cmml" xref="S3.SS1.4.p1.2.m2.1.1.1.3.2">𝑛</ci><cn id="S3.SS1.4.p1.2.m2.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.4.p1.2.m2.1.1.1.3.3">0</cn></apply></apply><apply id="S3.SS1.4.p1.2.m2.2.2.2.cmml" xref="S3.SS1.4.p1.2.m2.2.2.2"><times id="S3.SS1.4.p1.2.m2.2.2.2.2.cmml" xref="S3.SS1.4.p1.2.m2.2.2.2.2"></times><ci id="S3.SS1.4.p1.2.m2.2.2.2.3.cmml" xref="S3.SS1.4.p1.2.m2.2.2.2.3">Φ</ci><apply id="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.cmml" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.1.cmml" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1">superscript</csymbol><ci id="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.2.cmml" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.2">ℝ</ci><ci id="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.3.cmml" xref="S3.SS1.4.p1.2.m2.2.2.2.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.2.m2.2c">(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.2.m2.2d">( italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> and set <math alttext="\widetilde{f}=\sum_{n=0}^{\infty}S_{n}f" class="ltx_Math" display="inline" id="S3.SS1.4.p1.3.m3.1"><semantics id="S3.SS1.4.p1.3.m3.1a"><mrow id="S3.SS1.4.p1.3.m3.1.1" xref="S3.SS1.4.p1.3.m3.1.1.cmml"><mover accent="true" id="S3.SS1.4.p1.3.m3.1.1.2" xref="S3.SS1.4.p1.3.m3.1.1.2.cmml"><mi id="S3.SS1.4.p1.3.m3.1.1.2.2" xref="S3.SS1.4.p1.3.m3.1.1.2.2.cmml">f</mi><mo id="S3.SS1.4.p1.3.m3.1.1.2.1" xref="S3.SS1.4.p1.3.m3.1.1.2.1.cmml">~</mo></mover><mo id="S3.SS1.4.p1.3.m3.1.1.1" rspace="0.111em" xref="S3.SS1.4.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S3.SS1.4.p1.3.m3.1.1.3" xref="S3.SS1.4.p1.3.m3.1.1.3.cmml"><msubsup id="S3.SS1.4.p1.3.m3.1.1.3.1" xref="S3.SS1.4.p1.3.m3.1.1.3.1.cmml"><mo id="S3.SS1.4.p1.3.m3.1.1.3.1.2.2" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.2.cmml">∑</mo><mrow id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.cmml"><mi id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.2" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.2.cmml">n</mi><mo id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.1" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.1.cmml">=</mo><mn id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.3" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.3.cmml">0</mn></mrow><mi id="S3.SS1.4.p1.3.m3.1.1.3.1.3" mathvariant="normal" xref="S3.SS1.4.p1.3.m3.1.1.3.1.3.cmml">∞</mi></msubsup><mrow id="S3.SS1.4.p1.3.m3.1.1.3.2" xref="S3.SS1.4.p1.3.m3.1.1.3.2.cmml"><msub id="S3.SS1.4.p1.3.m3.1.1.3.2.2" xref="S3.SS1.4.p1.3.m3.1.1.3.2.2.cmml"><mi id="S3.SS1.4.p1.3.m3.1.1.3.2.2.2" xref="S3.SS1.4.p1.3.m3.1.1.3.2.2.2.cmml">S</mi><mi id="S3.SS1.4.p1.3.m3.1.1.3.2.2.3" xref="S3.SS1.4.p1.3.m3.1.1.3.2.2.3.cmml">n</mi></msub><mo id="S3.SS1.4.p1.3.m3.1.1.3.2.1" xref="S3.SS1.4.p1.3.m3.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS1.4.p1.3.m3.1.1.3.2.3" xref="S3.SS1.4.p1.3.m3.1.1.3.2.3.cmml">f</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.3.m3.1b"><apply id="S3.SS1.4.p1.3.m3.1.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1"><eq id="S3.SS1.4.p1.3.m3.1.1.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.1"></eq><apply id="S3.SS1.4.p1.3.m3.1.1.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.2"><ci id="S3.SS1.4.p1.3.m3.1.1.2.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.2.1">~</ci><ci id="S3.SS1.4.p1.3.m3.1.1.2.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.2.2">𝑓</ci></apply><apply id="S3.SS1.4.p1.3.m3.1.1.3.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3"><apply id="S3.SS1.4.p1.3.m3.1.1.3.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.3.m3.1.1.3.1.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1">superscript</csymbol><apply id="S3.SS1.4.p1.3.m3.1.1.3.1.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.3.m3.1.1.3.1.2.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1">subscript</csymbol><sum id="S3.SS1.4.p1.3.m3.1.1.3.1.2.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.2"></sum><apply id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3"><eq id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.1"></eq><ci id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.2">𝑛</ci><cn id="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.3.cmml" type="integer" xref="S3.SS1.4.p1.3.m3.1.1.3.1.2.3.3">0</cn></apply></apply><infinity id="S3.SS1.4.p1.3.m3.1.1.3.1.3.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.1.3"></infinity></apply><apply id="S3.SS1.4.p1.3.m3.1.1.3.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.2"><times id="S3.SS1.4.p1.3.m3.1.1.3.2.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.2.1"></times><apply id="S3.SS1.4.p1.3.m3.1.1.3.2.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.SS1.4.p1.3.m3.1.1.3.2.2.1.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.2.2">subscript</csymbol><ci id="S3.SS1.4.p1.3.m3.1.1.3.2.2.2.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.2.2.2">𝑆</ci><ci id="S3.SS1.4.p1.3.m3.1.1.3.2.2.3.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.2.2.3">𝑛</ci></apply><ci id="S3.SS1.4.p1.3.m3.1.1.3.2.3.cmml" xref="S3.SS1.4.p1.3.m3.1.1.3.2.3">𝑓</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.3.m3.1c">\widetilde{f}=\sum_{n=0}^{\infty}S_{n}f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.3.m3.1d">over~ start_ARG italic_f end_ARG = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f</annotation></semantics></math>. We verify the conditions for the definition of the trace operator. Since <math alttext="f\in B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS1.4.p1.4.m4.5"><semantics id="S3.SS1.4.p1.4.m4.5a"><mrow id="S3.SS1.4.p1.4.m4.5.5" xref="S3.SS1.4.p1.4.m4.5.5.cmml"><mi id="S3.SS1.4.p1.4.m4.5.5.4" xref="S3.SS1.4.p1.4.m4.5.5.4.cmml">f</mi><mo id="S3.SS1.4.p1.4.m4.5.5.3" xref="S3.SS1.4.p1.4.m4.5.5.3.cmml">∈</mo><mrow id="S3.SS1.4.p1.4.m4.5.5.2" xref="S3.SS1.4.p1.4.m4.5.5.2.cmml"><msubsup id="S3.SS1.4.p1.4.m4.5.5.2.4" xref="S3.SS1.4.p1.4.m4.5.5.2.4.cmml"><mi id="S3.SS1.4.p1.4.m4.5.5.2.4.2.2" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.2.cmml">B</mi><mrow id="S3.SS1.4.p1.4.m4.2.2.2.4" xref="S3.SS1.4.p1.4.m4.2.2.2.3.cmml"><mi id="S3.SS1.4.p1.4.m4.1.1.1.1" xref="S3.SS1.4.p1.4.m4.1.1.1.1.cmml">p</mi><mo id="S3.SS1.4.p1.4.m4.2.2.2.4.1" xref="S3.SS1.4.p1.4.m4.2.2.2.3.cmml">,</mo><mn id="S3.SS1.4.p1.4.m4.2.2.2.2" xref="S3.SS1.4.p1.4.m4.2.2.2.2.cmml">1</mn></mrow><mfrac id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.cmml"><mrow id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.cmml"><mi id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.2" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.2.cmml">γ</mi><mo id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.1" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.1.cmml">+</mo><mn id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.3" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.3.cmml">1</mn></mrow><mi id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.3" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.3.cmml">p</mi></mfrac></msubsup><mo id="S3.SS1.4.p1.4.m4.5.5.2.3" xref="S3.SS1.4.p1.4.m4.5.5.2.3.cmml">⁢</mo><mrow id="S3.SS1.4.p1.4.m4.5.5.2.2.2" xref="S3.SS1.4.p1.4.m4.5.5.2.2.3.cmml"><mo id="S3.SS1.4.p1.4.m4.5.5.2.2.2.3" stretchy="false" xref="S3.SS1.4.p1.4.m4.5.5.2.2.3.cmml">(</mo><msup id="S3.SS1.4.p1.4.m4.4.4.1.1.1.1" xref="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.cmml"><mi id="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.2" xref="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.3" xref="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.4.p1.4.m4.5.5.2.2.2.4" xref="S3.SS1.4.p1.4.m4.5.5.2.2.3.cmml">,</mo><msub id="S3.SS1.4.p1.4.m4.5.5.2.2.2.2" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.cmml"><mi id="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.2" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.2.cmml">w</mi><mi id="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.3" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS1.4.p1.4.m4.5.5.2.2.2.5" xref="S3.SS1.4.p1.4.m4.5.5.2.2.3.cmml">;</mo><mi id="S3.SS1.4.p1.4.m4.3.3" xref="S3.SS1.4.p1.4.m4.3.3.cmml">X</mi><mo id="S3.SS1.4.p1.4.m4.5.5.2.2.2.6" stretchy="false" xref="S3.SS1.4.p1.4.m4.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.4.m4.5b"><apply id="S3.SS1.4.p1.4.m4.5.5.cmml" xref="S3.SS1.4.p1.4.m4.5.5"><in id="S3.SS1.4.p1.4.m4.5.5.3.cmml" xref="S3.SS1.4.p1.4.m4.5.5.3"></in><ci id="S3.SS1.4.p1.4.m4.5.5.4.cmml" xref="S3.SS1.4.p1.4.m4.5.5.4">𝑓</ci><apply id="S3.SS1.4.p1.4.m4.5.5.2.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2"><times id="S3.SS1.4.p1.4.m4.5.5.2.3.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.3"></times><apply id="S3.SS1.4.p1.4.m4.5.5.2.4.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4"><csymbol cd="ambiguous" id="S3.SS1.4.p1.4.m4.5.5.2.4.1.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4">subscript</csymbol><apply id="S3.SS1.4.p1.4.m4.5.5.2.4.2.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4"><csymbol cd="ambiguous" id="S3.SS1.4.p1.4.m4.5.5.2.4.2.1.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4">superscript</csymbol><ci id="S3.SS1.4.p1.4.m4.5.5.2.4.2.2.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.2">𝐵</ci><apply id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3"><divide id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.1.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3"></divide><apply id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2"><plus id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.1.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.1"></plus><ci id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.2.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.2">𝛾</ci><cn id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.3.cmml" type="integer" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.2.3">1</cn></apply><ci id="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.3.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.4.2.3.3">𝑝</ci></apply></apply><list id="S3.SS1.4.p1.4.m4.2.2.2.3.cmml" xref="S3.SS1.4.p1.4.m4.2.2.2.4"><ci id="S3.SS1.4.p1.4.m4.1.1.1.1.cmml" xref="S3.SS1.4.p1.4.m4.1.1.1.1">𝑝</ci><cn id="S3.SS1.4.p1.4.m4.2.2.2.2.cmml" type="integer" xref="S3.SS1.4.p1.4.m4.2.2.2.2">1</cn></list></apply><vector id="S3.SS1.4.p1.4.m4.5.5.2.2.3.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2"><apply id="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.cmml" xref="S3.SS1.4.p1.4.m4.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.1.cmml" xref="S3.SS1.4.p1.4.m4.4.4.1.1.1.1">superscript</csymbol><ci id="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.2.cmml" xref="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.2">ℝ</ci><ci id="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.3.cmml" xref="S3.SS1.4.p1.4.m4.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.1.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2.2">subscript</csymbol><ci id="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.2.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.2">𝑤</ci><ci id="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.3.cmml" xref="S3.SS1.4.p1.4.m4.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S3.SS1.4.p1.4.m4.3.3.cmml" xref="S3.SS1.4.p1.4.m4.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.4.m4.5c">f\in B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.4.m4.5d">italic_f ∈ italic_B start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> we have <math alttext="\sum_{n=0}^{\infty}\|S_{n}f\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}<\infty" class="ltx_Math" display="inline" id="S3.SS1.4.p1.5.m5.4"><semantics id="S3.SS1.4.p1.5.m5.4a"><mrow id="S3.SS1.4.p1.5.m5.4.4" xref="S3.SS1.4.p1.5.m5.4.4.cmml"><mrow id="S3.SS1.4.p1.5.m5.4.4.1" xref="S3.SS1.4.p1.5.m5.4.4.1.cmml"><msubsup id="S3.SS1.4.p1.5.m5.4.4.1.2" xref="S3.SS1.4.p1.5.m5.4.4.1.2.cmml"><mo id="S3.SS1.4.p1.5.m5.4.4.1.2.2.2" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.2.cmml">∑</mo><mrow id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.cmml"><mi id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.2" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.2.cmml">n</mi><mo id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.1" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.1.cmml">=</mo><mn id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.3" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.3.cmml">0</mn></mrow><mi id="S3.SS1.4.p1.5.m5.4.4.1.2.3" mathvariant="normal" xref="S3.SS1.4.p1.5.m5.4.4.1.2.3.cmml">∞</mi></msubsup><msub id="S3.SS1.4.p1.5.m5.4.4.1.1" xref="S3.SS1.4.p1.5.m5.4.4.1.1.cmml"><mrow id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.2.cmml"><mo id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.2" lspace="0em" stretchy="false" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.2.1.cmml">‖</mo><mrow id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.cmml"><msub id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.2" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.2.cmml"><mi id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.2.2" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.2.2.cmml">S</mi><mi id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.2.3" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.1" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.3" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.1.3.cmml">f</mi></mrow><mo id="S3.SS1.4.p1.5.m5.4.4.1.1.1.1.3" stretchy="false" xref="S3.SS1.4.p1.5.m5.4.4.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.SS1.4.p1.5.m5.3.3.3" xref="S3.SS1.4.p1.5.m5.3.3.3.cmml"><msup id="S3.SS1.4.p1.5.m5.3.3.3.5" xref="S3.SS1.4.p1.5.m5.3.3.3.5.cmml"><mi id="S3.SS1.4.p1.5.m5.3.3.3.5.2" xref="S3.SS1.4.p1.5.m5.3.3.3.5.2.cmml">L</mi><mi id="S3.SS1.4.p1.5.m5.3.3.3.5.3" xref="S3.SS1.4.p1.5.m5.3.3.3.5.3.cmml">p</mi></msup><mo id="S3.SS1.4.p1.5.m5.3.3.3.4" xref="S3.SS1.4.p1.5.m5.3.3.3.4.cmml">⁢</mo><mrow id="S3.SS1.4.p1.5.m5.3.3.3.3.2" xref="S3.SS1.4.p1.5.m5.3.3.3.3.3.cmml"><mo id="S3.SS1.4.p1.5.m5.3.3.3.3.2.3" stretchy="false" xref="S3.SS1.4.p1.5.m5.3.3.3.3.3.cmml">(</mo><msup id="S3.SS1.4.p1.5.m5.2.2.2.2.1.1" xref="S3.SS1.4.p1.5.m5.2.2.2.2.1.1.cmml"><mi id="S3.SS1.4.p1.5.m5.2.2.2.2.1.1.2" xref="S3.SS1.4.p1.5.m5.2.2.2.2.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.4.p1.5.m5.2.2.2.2.1.1.3" xref="S3.SS1.4.p1.5.m5.2.2.2.2.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.4.p1.5.m5.3.3.3.3.2.4" xref="S3.SS1.4.p1.5.m5.3.3.3.3.3.cmml">,</mo><msub id="S3.SS1.4.p1.5.m5.3.3.3.3.2.2" xref="S3.SS1.4.p1.5.m5.3.3.3.3.2.2.cmml"><mi id="S3.SS1.4.p1.5.m5.3.3.3.3.2.2.2" xref="S3.SS1.4.p1.5.m5.3.3.3.3.2.2.2.cmml">w</mi><mi id="S3.SS1.4.p1.5.m5.3.3.3.3.2.2.3" xref="S3.SS1.4.p1.5.m5.3.3.3.3.2.2.3.cmml">γ</mi></msub><mo id="S3.SS1.4.p1.5.m5.3.3.3.3.2.5" xref="S3.SS1.4.p1.5.m5.3.3.3.3.3.cmml">;</mo><mi id="S3.SS1.4.p1.5.m5.1.1.1.1" xref="S3.SS1.4.p1.5.m5.1.1.1.1.cmml">X</mi><mo id="S3.SS1.4.p1.5.m5.3.3.3.3.2.6" stretchy="false" xref="S3.SS1.4.p1.5.m5.3.3.3.3.3.cmml">)</mo></mrow></mrow></msub></mrow><mo id="S3.SS1.4.p1.5.m5.4.4.2" xref="S3.SS1.4.p1.5.m5.4.4.2.cmml"><</mo><mi id="S3.SS1.4.p1.5.m5.4.4.3" mathvariant="normal" xref="S3.SS1.4.p1.5.m5.4.4.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.5.m5.4b"><apply id="S3.SS1.4.p1.5.m5.4.4.cmml" xref="S3.SS1.4.p1.5.m5.4.4"><lt id="S3.SS1.4.p1.5.m5.4.4.2.cmml" xref="S3.SS1.4.p1.5.m5.4.4.2"></lt><apply id="S3.SS1.4.p1.5.m5.4.4.1.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1"><apply id="S3.SS1.4.p1.5.m5.4.4.1.2.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2"><csymbol cd="ambiguous" id="S3.SS1.4.p1.5.m5.4.4.1.2.1.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2">superscript</csymbol><apply id="S3.SS1.4.p1.5.m5.4.4.1.2.2.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2"><csymbol cd="ambiguous" id="S3.SS1.4.p1.5.m5.4.4.1.2.2.1.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2">subscript</csymbol><sum id="S3.SS1.4.p1.5.m5.4.4.1.2.2.2.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.2"></sum><apply id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3"><eq id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.1.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.1"></eq><ci id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.2.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.2">𝑛</ci><cn id="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.3.cmml" type="integer" xref="S3.SS1.4.p1.5.m5.4.4.1.2.2.3.3">0</cn></apply></apply><infinity id="S3.SS1.4.p1.5.m5.4.4.1.2.3.cmml" xref="S3.SS1.4.p1.5.m5.4.4.1.2.3"></infinity></apply><apply 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end_POSTSUPERSCRIPT ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT < ∞</annotation></semantics></math> and therefore the series converges to <math alttext="f" class="ltx_Math" display="inline" id="S3.SS1.4.p1.6.m6.1"><semantics id="S3.SS1.4.p1.6.m6.1a"><mi id="S3.SS1.4.p1.6.m6.1.1" xref="S3.SS1.4.p1.6.m6.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.6.m6.1b"><ci id="S3.SS1.4.p1.6.m6.1.1.cmml" xref="S3.SS1.4.p1.6.m6.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.6.m6.1c">f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.6.m6.1d">italic_f</annotation></semantics></math> in <math alttext="L^{p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS1.4.p1.7.m7.3"><semantics id="S3.SS1.4.p1.7.m7.3a"><mrow id="S3.SS1.4.p1.7.m7.3.3" xref="S3.SS1.4.p1.7.m7.3.3.cmml"><msup id="S3.SS1.4.p1.7.m7.3.3.4" xref="S3.SS1.4.p1.7.m7.3.3.4.cmml"><mi id="S3.SS1.4.p1.7.m7.3.3.4.2" xref="S3.SS1.4.p1.7.m7.3.3.4.2.cmml">L</mi><mi id="S3.SS1.4.p1.7.m7.3.3.4.3" xref="S3.SS1.4.p1.7.m7.3.3.4.3.cmml">p</mi></msup><mo id="S3.SS1.4.p1.7.m7.3.3.3" xref="S3.SS1.4.p1.7.m7.3.3.3.cmml">⁢</mo><mrow id="S3.SS1.4.p1.7.m7.3.3.2.2" xref="S3.SS1.4.p1.7.m7.3.3.2.3.cmml"><mo id="S3.SS1.4.p1.7.m7.3.3.2.2.3" stretchy="false" xref="S3.SS1.4.p1.7.m7.3.3.2.3.cmml">(</mo><msup id="S3.SS1.4.p1.7.m7.2.2.1.1.1" xref="S3.SS1.4.p1.7.m7.2.2.1.1.1.cmml"><mi id="S3.SS1.4.p1.7.m7.2.2.1.1.1.2" xref="S3.SS1.4.p1.7.m7.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.4.p1.7.m7.2.2.1.1.1.3" xref="S3.SS1.4.p1.7.m7.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.4.p1.7.m7.3.3.2.2.4" xref="S3.SS1.4.p1.7.m7.3.3.2.3.cmml">,</mo><msub id="S3.SS1.4.p1.7.m7.3.3.2.2.2" xref="S3.SS1.4.p1.7.m7.3.3.2.2.2.cmml"><mi id="S3.SS1.4.p1.7.m7.3.3.2.2.2.2" xref="S3.SS1.4.p1.7.m7.3.3.2.2.2.2.cmml">w</mi><mi id="S3.SS1.4.p1.7.m7.3.3.2.2.2.3" xref="S3.SS1.4.p1.7.m7.3.3.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS1.4.p1.7.m7.3.3.2.2.5" xref="S3.SS1.4.p1.7.m7.3.3.2.3.cmml">;</mo><mi id="S3.SS1.4.p1.7.m7.1.1" xref="S3.SS1.4.p1.7.m7.1.1.cmml">X</mi><mo id="S3.SS1.4.p1.7.m7.3.3.2.2.6" stretchy="false" xref="S3.SS1.4.p1.7.m7.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.7.m7.3b"><apply id="S3.SS1.4.p1.7.m7.3.3.cmml" xref="S3.SS1.4.p1.7.m7.3.3"><times id="S3.SS1.4.p1.7.m7.3.3.3.cmml" xref="S3.SS1.4.p1.7.m7.3.3.3"></times><apply id="S3.SS1.4.p1.7.m7.3.3.4.cmml" xref="S3.SS1.4.p1.7.m7.3.3.4"><csymbol cd="ambiguous" id="S3.SS1.4.p1.7.m7.3.3.4.1.cmml" xref="S3.SS1.4.p1.7.m7.3.3.4">superscript</csymbol><ci id="S3.SS1.4.p1.7.m7.3.3.4.2.cmml" xref="S3.SS1.4.p1.7.m7.3.3.4.2">𝐿</ci><ci id="S3.SS1.4.p1.7.m7.3.3.4.3.cmml" xref="S3.SS1.4.p1.7.m7.3.3.4.3">𝑝</ci></apply><vector id="S3.SS1.4.p1.7.m7.3.3.2.3.cmml" xref="S3.SS1.4.p1.7.m7.3.3.2.2"><apply id="S3.SS1.4.p1.7.m7.2.2.1.1.1.cmml" xref="S3.SS1.4.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.7.m7.2.2.1.1.1.1.cmml" xref="S3.SS1.4.p1.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.4.p1.7.m7.2.2.1.1.1.2.cmml" xref="S3.SS1.4.p1.7.m7.2.2.1.1.1.2">ℝ</ci><ci id="S3.SS1.4.p1.7.m7.2.2.1.1.1.3.cmml" xref="S3.SS1.4.p1.7.m7.2.2.1.1.1.3">𝑑</ci></apply><apply id="S3.SS1.4.p1.7.m7.3.3.2.2.2.cmml" xref="S3.SS1.4.p1.7.m7.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.4.p1.7.m7.3.3.2.2.2.1.cmml" xref="S3.SS1.4.p1.7.m7.3.3.2.2.2">subscript</csymbol><ci id="S3.SS1.4.p1.7.m7.3.3.2.2.2.2.cmml" xref="S3.SS1.4.p1.7.m7.3.3.2.2.2.2">𝑤</ci><ci id="S3.SS1.4.p1.7.m7.3.3.2.2.2.3.cmml" xref="S3.SS1.4.p1.7.m7.3.3.2.2.2.3">𝛾</ci></apply><ci id="S3.SS1.4.p1.7.m7.1.1.cmml" xref="S3.SS1.4.p1.7.m7.1.1">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.7.m7.3c">L^{p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.7.m7.3d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> and thus <math alttext="f=\widetilde{f}" class="ltx_Math" display="inline" id="S3.SS1.4.p1.8.m8.1"><semantics id="S3.SS1.4.p1.8.m8.1a"><mrow id="S3.SS1.4.p1.8.m8.1.1" xref="S3.SS1.4.p1.8.m8.1.1.cmml"><mi id="S3.SS1.4.p1.8.m8.1.1.2" xref="S3.SS1.4.p1.8.m8.1.1.2.cmml">f</mi><mo id="S3.SS1.4.p1.8.m8.1.1.1" xref="S3.SS1.4.p1.8.m8.1.1.1.cmml">=</mo><mover accent="true" id="S3.SS1.4.p1.8.m8.1.1.3" xref="S3.SS1.4.p1.8.m8.1.1.3.cmml"><mi id="S3.SS1.4.p1.8.m8.1.1.3.2" xref="S3.SS1.4.p1.8.m8.1.1.3.2.cmml">f</mi><mo id="S3.SS1.4.p1.8.m8.1.1.3.1" xref="S3.SS1.4.p1.8.m8.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.8.m8.1b"><apply id="S3.SS1.4.p1.8.m8.1.1.cmml" xref="S3.SS1.4.p1.8.m8.1.1"><eq id="S3.SS1.4.p1.8.m8.1.1.1.cmml" xref="S3.SS1.4.p1.8.m8.1.1.1"></eq><ci id="S3.SS1.4.p1.8.m8.1.1.2.cmml" xref="S3.SS1.4.p1.8.m8.1.1.2">𝑓</ci><apply id="S3.SS1.4.p1.8.m8.1.1.3.cmml" xref="S3.SS1.4.p1.8.m8.1.1.3"><ci id="S3.SS1.4.p1.8.m8.1.1.3.1.cmml" xref="S3.SS1.4.p1.8.m8.1.1.3.1">~</ci><ci id="S3.SS1.4.p1.8.m8.1.1.3.2.cmml" xref="S3.SS1.4.p1.8.m8.1.1.3.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.8.m8.1c">f=\widetilde{f}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.8.m8.1d">italic_f = over~ start_ARG italic_f end_ARG</annotation></semantics></math> almost everywhere on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S3.SS1.4.p1.9.m9.1"><semantics id="S3.SS1.4.p1.9.m9.1a"><msup id="S3.SS1.4.p1.9.m9.1.1" xref="S3.SS1.4.p1.9.m9.1.1.cmml"><mi id="S3.SS1.4.p1.9.m9.1.1.2" xref="S3.SS1.4.p1.9.m9.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.4.p1.9.m9.1.1.3" xref="S3.SS1.4.p1.9.m9.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S3.SS1.4.p1.9.m9.1b"><apply id="S3.SS1.4.p1.9.m9.1.1.cmml" xref="S3.SS1.4.p1.9.m9.1.1"><csymbol cd="ambiguous" id="S3.SS1.4.p1.9.m9.1.1.1.cmml" xref="S3.SS1.4.p1.9.m9.1.1">superscript</csymbol><ci id="S3.SS1.4.p1.9.m9.1.1.2.cmml" xref="S3.SS1.4.p1.9.m9.1.1.2">ℝ</ci><ci id="S3.SS1.4.p1.9.m9.1.1.3.cmml" xref="S3.SS1.4.p1.9.m9.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.4.p1.9.m9.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.4.p1.9.m9.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.5.p2"> <p class="ltx_p" id="S3.SS1.5.p2.2">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.1</span></a> (using that <math alttext="\text{\rm supp\,}\mathcal{F}(S_{n}f)\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq 3% \cdot 2^{n-1}\}" class="ltx_Math" display="inline" id="S3.SS1.5.p2.1.m1.4"><semantics id="S3.SS1.5.p2.1.m1.4a"><mrow id="S3.SS1.5.p2.1.m1.4.4" xref="S3.SS1.5.p2.1.m1.4.4.cmml"><mrow id="S3.SS1.5.p2.1.m1.2.2.1" xref="S3.SS1.5.p2.1.m1.2.2.1.cmml"><mtext id="S3.SS1.5.p2.1.m1.2.2.1.3" xref="S3.SS1.5.p2.1.m1.2.2.1.3a.cmml">supp </mtext><mo id="S3.SS1.5.p2.1.m1.2.2.1.2" xref="S3.SS1.5.p2.1.m1.2.2.1.2.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.SS1.5.p2.1.m1.2.2.1.4" xref="S3.SS1.5.p2.1.m1.2.2.1.4.cmml">ℱ</mi><mo id="S3.SS1.5.p2.1.m1.2.2.1.2a" xref="S3.SS1.5.p2.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.5.p2.1.m1.2.2.1.1.1" xref="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.cmml"><mo id="S3.SS1.5.p2.1.m1.2.2.1.1.1.2" 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id="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2.cmml" xref="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2.1.cmml" xref="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2">subscript</csymbol><ci id="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2.2.cmml" xref="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2.2">𝑆</ci><ci id="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2.3.cmml" xref="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.2.3">𝑛</ci></apply><ci id="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.3.cmml" xref="S3.SS1.5.p2.1.m1.2.2.1.1.1.1.3">𝑓</ci></apply></apply><apply id="S3.SS1.5.p2.1.m1.4.4.3.3.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2"><csymbol cd="latexml" id="S3.SS1.5.p2.1.m1.4.4.3.3.1.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.3">conditional-set</csymbol><apply id="S3.SS1.5.p2.1.m1.3.3.2.1.1.cmml" xref="S3.SS1.5.p2.1.m1.3.3.2.1.1"><in id="S3.SS1.5.p2.1.m1.3.3.2.1.1.1.cmml" xref="S3.SS1.5.p2.1.m1.3.3.2.1.1.1"></in><ci id="S3.SS1.5.p2.1.m1.3.3.2.1.1.2.cmml" xref="S3.SS1.5.p2.1.m1.3.3.2.1.1.2">𝜉</ci><apply id="S3.SS1.5.p2.1.m1.3.3.2.1.1.3.cmml" xref="S3.SS1.5.p2.1.m1.3.3.2.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.5.p2.1.m1.3.3.2.1.1.3.1.cmml" xref="S3.SS1.5.p2.1.m1.3.3.2.1.1.3">superscript</csymbol><ci id="S3.SS1.5.p2.1.m1.3.3.2.1.1.3.2.cmml" xref="S3.SS1.5.p2.1.m1.3.3.2.1.1.3.2">ℝ</ci><ci id="S3.SS1.5.p2.1.m1.3.3.2.1.1.3.3.cmml" xref="S3.SS1.5.p2.1.m1.3.3.2.1.1.3.3">𝑑</ci></apply></apply><apply id="S3.SS1.5.p2.1.m1.4.4.3.2.2.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2"><leq id="S3.SS1.5.p2.1.m1.4.4.3.2.2.1.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.1"></leq><apply id="S3.SS1.5.p2.1.m1.4.4.3.2.2.2.1.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.2.2"><abs id="S3.SS1.5.p2.1.m1.4.4.3.2.2.2.1.1.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.2.2.1"></abs><ci id="S3.SS1.5.p2.1.m1.1.1.cmml" xref="S3.SS1.5.p2.1.m1.1.1">𝜉</ci></apply><apply id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3"><ci id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.1.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.1">⋅</ci><cn id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.2.cmml" type="integer" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.2">3</cn><apply id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3"><csymbol cd="ambiguous" id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.1.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3">superscript</csymbol><cn id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.2.cmml" type="integer" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.2">2</cn><apply id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3"><minus id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3.1.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3.1"></minus><ci id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3.2.cmml" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3.2">𝑛</ci><cn id="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3.3.cmml" type="integer" xref="S3.SS1.5.p2.1.m1.4.4.3.2.2.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p2.1.m1.4c">\text{\rm supp\,}\mathcal{F}(S_{n}f)\subseteq\{\xi\in\mathbb{R}^{d}:|\xi|\leq 3% \cdot 2^{n-1}\}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p2.1.m1.4d">supp caligraphic_F ( italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ) ⊆ { italic_ξ ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT : | italic_ξ | ≤ 3 ⋅ 2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math>) we obtain for every <math alttext="x_{1}\in\mathbb{R}" class="ltx_Math" display="inline" id="S3.SS1.5.p2.2.m2.1"><semantics id="S3.SS1.5.p2.2.m2.1a"><mrow id="S3.SS1.5.p2.2.m2.1.1" xref="S3.SS1.5.p2.2.m2.1.1.cmml"><msub id="S3.SS1.5.p2.2.m2.1.1.2" xref="S3.SS1.5.p2.2.m2.1.1.2.cmml"><mi id="S3.SS1.5.p2.2.m2.1.1.2.2" xref="S3.SS1.5.p2.2.m2.1.1.2.2.cmml">x</mi><mn id="S3.SS1.5.p2.2.m2.1.1.2.3" xref="S3.SS1.5.p2.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.5.p2.2.m2.1.1.1" xref="S3.SS1.5.p2.2.m2.1.1.1.cmml">∈</mo><mi id="S3.SS1.5.p2.2.m2.1.1.3" xref="S3.SS1.5.p2.2.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.5.p2.2.m2.1b"><apply id="S3.SS1.5.p2.2.m2.1.1.cmml" xref="S3.SS1.5.p2.2.m2.1.1"><in id="S3.SS1.5.p2.2.m2.1.1.1.cmml" xref="S3.SS1.5.p2.2.m2.1.1.1"></in><apply id="S3.SS1.5.p2.2.m2.1.1.2.cmml" xref="S3.SS1.5.p2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.5.p2.2.m2.1.1.2.1.cmml" xref="S3.SS1.5.p2.2.m2.1.1.2">subscript</csymbol><ci id="S3.SS1.5.p2.2.m2.1.1.2.2.cmml" xref="S3.SS1.5.p2.2.m2.1.1.2.2">𝑥</ci><cn id="S3.SS1.5.p2.2.m2.1.1.2.3.cmml" type="integer" xref="S3.SS1.5.p2.2.m2.1.1.2.3">1</cn></apply><ci id="S3.SS1.5.p2.2.m2.1.1.3.cmml" xref="S3.SS1.5.p2.2.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.5.p2.2.m2.1c">x_{1}\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.5.p2.2.m2.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R</annotation></semantics></math> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx5"> <tbody id="S3.Ex7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\widetilde{f}(x_{1},\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}" class="ltx_Math" display="inline" id="S3.Ex7.m1.4"><semantics id="S3.Ex7.m1.4a"><msub id="S3.Ex7.m1.4.4" xref="S3.Ex7.m1.4.4.cmml"><mrow id="S3.Ex7.m1.4.4.1.1" xref="S3.Ex7.m1.4.4.1.2.cmml"><mo id="S3.Ex7.m1.4.4.1.1.2" stretchy="false" xref="S3.Ex7.m1.4.4.1.2.1.cmml">‖</mo><mrow id="S3.Ex7.m1.4.4.1.1.1" xref="S3.Ex7.m1.4.4.1.1.1.cmml"><mover accent="true" id="S3.Ex7.m1.4.4.1.1.1.3" xref="S3.Ex7.m1.4.4.1.1.1.3.cmml"><mi id="S3.Ex7.m1.4.4.1.1.1.3.2" xref="S3.Ex7.m1.4.4.1.1.1.3.2.cmml">f</mi><mo id="S3.Ex7.m1.4.4.1.1.1.3.1" xref="S3.Ex7.m1.4.4.1.1.1.3.1.cmml">~</mo></mover><mo id="S3.Ex7.m1.4.4.1.1.1.2" xref="S3.Ex7.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex7.m1.4.4.1.1.1.1.1" xref="S3.Ex7.m1.4.4.1.1.1.1.2.cmml"><mo id="S3.Ex7.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.Ex7.m1.4.4.1.1.1.1.2.cmml">(</mo><msub id="S3.Ex7.m1.4.4.1.1.1.1.1.1" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S3.Ex7.m1.4.4.1.1.1.1.1.1.2" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.2.cmml">x</mi><mn id="S3.Ex7.m1.4.4.1.1.1.1.1.1.3" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.Ex7.m1.4.4.1.1.1.1.1.3" rspace="0em" xref="S3.Ex7.m1.4.4.1.1.1.1.2.cmml">,</mo><mo id="S3.Ex7.m1.3.3" lspace="0em" rspace="0em" xref="S3.Ex7.m1.3.3.cmml">⋅</mo><mo id="S3.Ex7.m1.4.4.1.1.1.1.1.4" stretchy="false" xref="S3.Ex7.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex7.m1.4.4.1.1.3" stretchy="false" xref="S3.Ex7.m1.4.4.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex7.m1.2.2.2" xref="S3.Ex7.m1.2.2.2.cmml"><msup id="S3.Ex7.m1.2.2.2.4" xref="S3.Ex7.m1.2.2.2.4.cmml"><mi id="S3.Ex7.m1.2.2.2.4.2" xref="S3.Ex7.m1.2.2.2.4.2.cmml">L</mi><mi id="S3.Ex7.m1.2.2.2.4.3" xref="S3.Ex7.m1.2.2.2.4.3.cmml">p</mi></msup><mo id="S3.Ex7.m1.2.2.2.3" xref="S3.Ex7.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.Ex7.m1.2.2.2.2.1" xref="S3.Ex7.m1.2.2.2.2.2.cmml"><mo id="S3.Ex7.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex7.m1.2.2.2.2.2.cmml">(</mo><msup id="S3.Ex7.m1.2.2.2.2.1.1" xref="S3.Ex7.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex7.m1.2.2.2.2.1.1.2" xref="S3.Ex7.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mrow id="S3.Ex7.m1.2.2.2.2.1.1.3" xref="S3.Ex7.m1.2.2.2.2.1.1.3.cmml"><mi id="S3.Ex7.m1.2.2.2.2.1.1.3.2" xref="S3.Ex7.m1.2.2.2.2.1.1.3.2.cmml">d</mi><mo id="S3.Ex7.m1.2.2.2.2.1.1.3.1" xref="S3.Ex7.m1.2.2.2.2.1.1.3.1.cmml">−</mo><mn id="S3.Ex7.m1.2.2.2.2.1.1.3.3" xref="S3.Ex7.m1.2.2.2.2.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Ex7.m1.2.2.2.2.1.3" xref="S3.Ex7.m1.2.2.2.2.2.cmml">;</mo><mi id="S3.Ex7.m1.1.1.1.1" xref="S3.Ex7.m1.1.1.1.1.cmml">X</mi><mo id="S3.Ex7.m1.2.2.2.2.1.4" stretchy="false" xref="S3.Ex7.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Ex7.m1.4b"><apply id="S3.Ex7.m1.4.4.cmml" xref="S3.Ex7.m1.4.4"><csymbol cd="ambiguous" id="S3.Ex7.m1.4.4.2.cmml" xref="S3.Ex7.m1.4.4">subscript</csymbol><apply id="S3.Ex7.m1.4.4.1.2.cmml" xref="S3.Ex7.m1.4.4.1.1"><csymbol cd="latexml" id="S3.Ex7.m1.4.4.1.2.1.cmml" xref="S3.Ex7.m1.4.4.1.1.2">norm</csymbol><apply id="S3.Ex7.m1.4.4.1.1.1.cmml" xref="S3.Ex7.m1.4.4.1.1.1"><times id="S3.Ex7.m1.4.4.1.1.1.2.cmml" xref="S3.Ex7.m1.4.4.1.1.1.2"></times><apply id="S3.Ex7.m1.4.4.1.1.1.3.cmml" xref="S3.Ex7.m1.4.4.1.1.1.3"><ci id="S3.Ex7.m1.4.4.1.1.1.3.1.cmml" xref="S3.Ex7.m1.4.4.1.1.1.3.1">~</ci><ci id="S3.Ex7.m1.4.4.1.1.1.3.2.cmml" xref="S3.Ex7.m1.4.4.1.1.1.3.2">𝑓</ci></apply><interval closure="open" id="S3.Ex7.m1.4.4.1.1.1.1.2.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1"><apply id="S3.Ex7.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex7.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1">subscript</csymbol><ci id="S3.Ex7.m1.4.4.1.1.1.1.1.1.2.cmml" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.2">𝑥</ci><cn id="S3.Ex7.m1.4.4.1.1.1.1.1.1.3.cmml" type="integer" xref="S3.Ex7.m1.4.4.1.1.1.1.1.1.3">1</cn></apply><ci id="S3.Ex7.m1.3.3.cmml" xref="S3.Ex7.m1.3.3">⋅</ci></interval></apply></apply><apply id="S3.Ex7.m1.2.2.2.cmml" xref="S3.Ex7.m1.2.2.2"><times id="S3.Ex7.m1.2.2.2.3.cmml" xref="S3.Ex7.m1.2.2.2.3"></times><apply id="S3.Ex7.m1.2.2.2.4.cmml" xref="S3.Ex7.m1.2.2.2.4"><csymbol cd="ambiguous" id="S3.Ex7.m1.2.2.2.4.1.cmml" xref="S3.Ex7.m1.2.2.2.4">superscript</csymbol><ci id="S3.Ex7.m1.2.2.2.4.2.cmml" xref="S3.Ex7.m1.2.2.2.4.2">𝐿</ci><ci id="S3.Ex7.m1.2.2.2.4.3.cmml" xref="S3.Ex7.m1.2.2.2.4.3">𝑝</ci></apply><list id="S3.Ex7.m1.2.2.2.2.2.cmml" xref="S3.Ex7.m1.2.2.2.2.1"><apply id="S3.Ex7.m1.2.2.2.2.1.1.cmml" xref="S3.Ex7.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex7.m1.2.2.2.2.1.1.1.cmml" xref="S3.Ex7.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S3.Ex7.m1.2.2.2.2.1.1.2.cmml" xref="S3.Ex7.m1.2.2.2.2.1.1.2">ℝ</ci><apply id="S3.Ex7.m1.2.2.2.2.1.1.3.cmml" xref="S3.Ex7.m1.2.2.2.2.1.1.3"><minus id="S3.Ex7.m1.2.2.2.2.1.1.3.1.cmml" xref="S3.Ex7.m1.2.2.2.2.1.1.3.1"></minus><ci id="S3.Ex7.m1.2.2.2.2.1.1.3.2.cmml" xref="S3.Ex7.m1.2.2.2.2.1.1.3.2">𝑑</ci><cn id="S3.Ex7.m1.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S3.Ex7.m1.2.2.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex7.m1.1.1.1.1.cmml" xref="S3.Ex7.m1.1.1.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m1.4c">\displaystyle\|\widetilde{f}(x_{1},\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m1.4d">∥ over~ start_ARG italic_f end_ARG ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="S3.Ex7.m2.4.4.1.1.1.1.3.3">0</cn></apply></apply></apply><apply id="S3.Ex7.m2.4.4.1.3.cmml" xref="S3.Ex7.m2.4.4.1.3"><csymbol cd="ambiguous" id="S3.Ex7.m2.4.4.1.3.1.cmml" xref="S3.Ex7.m2.4.4.1.3">superscript</csymbol><ci id="S3.Ex7.m2.4.4.1.3.2.cmml" xref="S3.Ex7.m2.4.4.1.3.2">ℓ</ci><cn id="S3.Ex7.m2.4.4.1.3.3.cmml" type="integer" xref="S3.Ex7.m2.4.4.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex7.m2.4c">\displaystyle\leq\big{\|}(\|S_{n}f(x_{1},\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)})% _{n\geq 0}\big{\|}_{\ell^{1}}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex7.m2.4d">≤ ∥ ( ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex8"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\big{\|}(2^{\frac{\gamma+1}{p}n}\|S_{n}f\|_{L^{p}(\mathbb{R% }^{d},w_{\gamma};X)})_{n\geq 0}\big{\|}_{\ell^{1}}=C\|f\|_{B^{\frac{\gamma+1}{% p}}_{p,1}(\mathbb{R}^{d-1};X)}." class="ltx_Math" display="inline" id="S3.Ex8.m1.9"><semantics id="S3.Ex8.m1.9a"><mrow id="S3.Ex8.m1.9.9.1" xref="S3.Ex8.m1.9.9.1.1.cmml"><mrow id="S3.Ex8.m1.9.9.1.1" xref="S3.Ex8.m1.9.9.1.1.cmml"><mi id="S3.Ex8.m1.9.9.1.1.3" xref="S3.Ex8.m1.9.9.1.1.3.cmml"></mi><mo id="S3.Ex8.m1.9.9.1.1.4" xref="S3.Ex8.m1.9.9.1.1.4.cmml">≤</mo><mrow 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encoding="application/x-llamapun" id="S3.Ex8.m1.9d">≤ italic_C ∥ ( 2 start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG italic_n end_POSTSUPERSCRIPT ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS1.6.p3"> <p class="ltx_p" id="S3.SS1.6.p3.8">Finally, from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.1</span></a> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Lemma 2.3]</cite>, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|S_{n}f(x_{1},\cdot)-S_{n}f(0,\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq C\,2^{% \frac{\gamma+1}{p}n}|x_{1}|2^{n}\|\mathcal{M}(\|S_{n}f\|^{r})\|^{\frac{1}{r}}_% {L^{p}(\mathbb{R}^{d},w_{\gamma};X)}," class="ltx_Math" display="block" id="S3.Ex9.m1.9"><semantics id="S3.Ex9.m1.9a"><mrow id="S3.Ex9.m1.9.9.1" xref="S3.Ex9.m1.9.9.1.1.cmml"><mrow id="S3.Ex9.m1.9.9.1.1" xref="S3.Ex9.m1.9.9.1.1.cmml"><msub id="S3.Ex9.m1.9.9.1.1.1" xref="S3.Ex9.m1.9.9.1.1.1.cmml"><mrow id="S3.Ex9.m1.9.9.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.1.1.2.cmml"><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex9.m1.9.9.1.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.cmml"><mrow id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.cmml"><msub id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.3" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.3.2" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.3.2.cmml">S</mi><mi id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.3.3" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.3.3.cmml">n</mi></msub><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.2" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.4" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.4.cmml">f</mi><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.2a" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.2.cmml">(</mo><msub id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.cmml">x</mi><mn id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.3" rspace="0em" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.2.cmml">,</mo><mo id="S3.Ex9.m1.6.6" lspace="0em" rspace="0em" xref="S3.Ex9.m1.6.6.cmml">⋅</mo><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.1.4" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.2" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.2.cmml">−</mo><mrow id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.cmml"><msub id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.2" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.2.cmml"><mi id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.2.2" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.2.2.cmml">S</mi><mi id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.2.3" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.1" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.3" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.3.cmml">f</mi><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.1a" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.2" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.1.cmml"><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.2.1" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.1.cmml">(</mo><mn id="S3.Ex9.m1.7.7" xref="S3.Ex9.m1.7.7.cmml">0</mn><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.2.2" rspace="0em" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.1.cmml">,</mo><mo id="S3.Ex9.m1.8.8" lspace="0em" rspace="0em" xref="S3.Ex9.m1.8.8.cmml">⋅</mo><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.2.3" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.1.1.1.1.3.4.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex9.m1.9.9.1.1.1.1.1.3" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex9.m1.2.2.2" xref="S3.Ex9.m1.2.2.2.cmml"><msup id="S3.Ex9.m1.2.2.2.4" xref="S3.Ex9.m1.2.2.2.4.cmml"><mi id="S3.Ex9.m1.2.2.2.4.2" xref="S3.Ex9.m1.2.2.2.4.2.cmml">L</mi><mi id="S3.Ex9.m1.2.2.2.4.3" xref="S3.Ex9.m1.2.2.2.4.3.cmml">p</mi></msup><mo id="S3.Ex9.m1.2.2.2.3" xref="S3.Ex9.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.Ex9.m1.2.2.2.2.1" xref="S3.Ex9.m1.2.2.2.2.2.cmml"><mo id="S3.Ex9.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex9.m1.2.2.2.2.2.cmml">(</mo><msup id="S3.Ex9.m1.2.2.2.2.1.1" xref="S3.Ex9.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex9.m1.2.2.2.2.1.1.2" xref="S3.Ex9.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mrow id="S3.Ex9.m1.2.2.2.2.1.1.3" xref="S3.Ex9.m1.2.2.2.2.1.1.3.cmml"><mi id="S3.Ex9.m1.2.2.2.2.1.1.3.2" xref="S3.Ex9.m1.2.2.2.2.1.1.3.2.cmml">d</mi><mo id="S3.Ex9.m1.2.2.2.2.1.1.3.1" xref="S3.Ex9.m1.2.2.2.2.1.1.3.1.cmml">−</mo><mn id="S3.Ex9.m1.2.2.2.2.1.1.3.3" xref="S3.Ex9.m1.2.2.2.2.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Ex9.m1.2.2.2.2.1.3" xref="S3.Ex9.m1.2.2.2.2.2.cmml">;</mo><mi id="S3.Ex9.m1.1.1.1.1" xref="S3.Ex9.m1.1.1.1.1.cmml">X</mi><mo id="S3.Ex9.m1.2.2.2.2.1.4" stretchy="false" xref="S3.Ex9.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></msub><mo id="S3.Ex9.m1.9.9.1.1.3" xref="S3.Ex9.m1.9.9.1.1.3.cmml">≤</mo><mrow id="S3.Ex9.m1.9.9.1.1.2" xref="S3.Ex9.m1.9.9.1.1.2.cmml"><mi id="S3.Ex9.m1.9.9.1.1.2.3" xref="S3.Ex9.m1.9.9.1.1.2.3.cmml">C</mi><mo id="S3.Ex9.m1.9.9.1.1.2.2" xref="S3.Ex9.m1.9.9.1.1.2.2.cmml">⁢</mo><msup id="S3.Ex9.m1.9.9.1.1.2.4" xref="S3.Ex9.m1.9.9.1.1.2.4.cmml"><mn id="S3.Ex9.m1.9.9.1.1.2.4.2" xref="S3.Ex9.m1.9.9.1.1.2.4.2.cmml"> 2</mn><mrow id="S3.Ex9.m1.9.9.1.1.2.4.3" xref="S3.Ex9.m1.9.9.1.1.2.4.3.cmml"><mfrac id="S3.Ex9.m1.9.9.1.1.2.4.3.2" xref="S3.Ex9.m1.9.9.1.1.2.4.3.2.cmml"><mrow id="S3.Ex9.m1.9.9.1.1.2.4.3.2.2" xref="S3.Ex9.m1.9.9.1.1.2.4.3.2.2.cmml"><mi id="S3.Ex9.m1.9.9.1.1.2.4.3.2.2.2" xref="S3.Ex9.m1.9.9.1.1.2.4.3.2.2.2.cmml">γ</mi><mo id="S3.Ex9.m1.9.9.1.1.2.4.3.2.2.1" xref="S3.Ex9.m1.9.9.1.1.2.4.3.2.2.1.cmml">+</mo><mn id="S3.Ex9.m1.9.9.1.1.2.4.3.2.2.3" xref="S3.Ex9.m1.9.9.1.1.2.4.3.2.2.3.cmml">1</mn></mrow><mi id="S3.Ex9.m1.9.9.1.1.2.4.3.2.3" xref="S3.Ex9.m1.9.9.1.1.2.4.3.2.3.cmml">p</mi></mfrac><mo id="S3.Ex9.m1.9.9.1.1.2.4.3.1" xref="S3.Ex9.m1.9.9.1.1.2.4.3.1.cmml">⁢</mo><mi id="S3.Ex9.m1.9.9.1.1.2.4.3.3" xref="S3.Ex9.m1.9.9.1.1.2.4.3.3.cmml">n</mi></mrow></msup><mo id="S3.Ex9.m1.9.9.1.1.2.2a" xref="S3.Ex9.m1.9.9.1.1.2.2.cmml">⁢</mo><msubsup id="S3.Ex9.m1.9.9.1.1.2.1" xref="S3.Ex9.m1.9.9.1.1.2.1.cmml"><mrow id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.2.cmml"><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.2.1.cmml">|</mo><mrow id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.cmml"><msub id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.4" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.4.cmml"><mi id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.4.2" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.4.2.cmml">x</mi><mn id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.4.3" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.4.3.cmml">1</mn></msub><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.3" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.3.cmml">⁢</mo><mrow id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.2.1.cmml">|</mo><msup id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.1.cmml"><mn id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.1.2" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.1.2.cmml">2</mn><mi id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.1.3" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.1.3.cmml">n</mi></msup><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.3a" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.3.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.5" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.5.cmml">ℳ</mi><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.3b" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.3.cmml">⁢</mo><mrow id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.cmml"><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.2" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.cmml">(</mo><msup id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.cmml"><mrow id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.1.2.cmml"><mo id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.1.1.2" stretchy="false" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.1.1.1" xref="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.1.1.1.cmml"><msub id="S3.Ex9.m1.9.9.1.1.2.1.1.1.1.1.2.1.1.1.1.1.2" 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id="S3.Ex9.m1.5.5.3.4.cmml" xref="S3.Ex9.m1.5.5.3.4"></times><apply id="S3.Ex9.m1.5.5.3.5.cmml" xref="S3.Ex9.m1.5.5.3.5"><csymbol cd="ambiguous" id="S3.Ex9.m1.5.5.3.5.1.cmml" xref="S3.Ex9.m1.5.5.3.5">superscript</csymbol><ci id="S3.Ex9.m1.5.5.3.5.2.cmml" xref="S3.Ex9.m1.5.5.3.5.2">𝐿</ci><ci id="S3.Ex9.m1.5.5.3.5.3.cmml" xref="S3.Ex9.m1.5.5.3.5.3">𝑝</ci></apply><vector id="S3.Ex9.m1.5.5.3.3.3.cmml" xref="S3.Ex9.m1.5.5.3.3.2"><apply id="S3.Ex9.m1.4.4.2.2.1.1.cmml" xref="S3.Ex9.m1.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex9.m1.4.4.2.2.1.1.1.cmml" xref="S3.Ex9.m1.4.4.2.2.1.1">superscript</csymbol><ci id="S3.Ex9.m1.4.4.2.2.1.1.2.cmml" xref="S3.Ex9.m1.4.4.2.2.1.1.2">ℝ</ci><ci id="S3.Ex9.m1.4.4.2.2.1.1.3.cmml" xref="S3.Ex9.m1.4.4.2.2.1.1.3">𝑑</ci></apply><apply id="S3.Ex9.m1.5.5.3.3.2.2.cmml" xref="S3.Ex9.m1.5.5.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex9.m1.5.5.3.3.2.2.1.cmml" xref="S3.Ex9.m1.5.5.3.3.2.2">subscript</csymbol><ci id="S3.Ex9.m1.5.5.3.3.2.2.2.cmml" xref="S3.Ex9.m1.5.5.3.3.2.2.2">𝑤</ci><ci id="S3.Ex9.m1.5.5.3.3.2.2.3.cmml" xref="S3.Ex9.m1.5.5.3.3.2.2.3">𝛾</ci></apply><ci id="S3.Ex9.m1.3.3.1.1.cmml" xref="S3.Ex9.m1.3.3.1.1">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex9.m1.9c">\|S_{n}f(x_{1},\cdot)-S_{n}f(0,\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq C\,2^{% \frac{\gamma+1}{p}n}|x_{1}|2^{n}\|\mathcal{M}(\|S_{n}f\|^{r})\|^{\frac{1}{r}}_% {L^{p}(\mathbb{R}^{d},w_{\gamma};X)},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex9.m1.9d">∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) - italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( 0 , ⋅ ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C 2 start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG italic_n end_POSTSUPERSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ caligraphic_M ( ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ∥ start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT ) ∥ start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_r end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.6.p3.5">for some <math alttext="r\in(0,p)" class="ltx_Math" display="inline" id="S3.SS1.6.p3.1.m1.2"><semantics id="S3.SS1.6.p3.1.m1.2a"><mrow id="S3.SS1.6.p3.1.m1.2.3" xref="S3.SS1.6.p3.1.m1.2.3.cmml"><mi id="S3.SS1.6.p3.1.m1.2.3.2" xref="S3.SS1.6.p3.1.m1.2.3.2.cmml">r</mi><mo id="S3.SS1.6.p3.1.m1.2.3.1" xref="S3.SS1.6.p3.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.SS1.6.p3.1.m1.2.3.3.2" xref="S3.SS1.6.p3.1.m1.2.3.3.1.cmml"><mo id="S3.SS1.6.p3.1.m1.2.3.3.2.1" stretchy="false" xref="S3.SS1.6.p3.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.SS1.6.p3.1.m1.1.1" xref="S3.SS1.6.p3.1.m1.1.1.cmml">0</mn><mo id="S3.SS1.6.p3.1.m1.2.3.3.2.2" xref="S3.SS1.6.p3.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.SS1.6.p3.1.m1.2.2" xref="S3.SS1.6.p3.1.m1.2.2.cmml">p</mi><mo id="S3.SS1.6.p3.1.m1.2.3.3.2.3" stretchy="false" xref="S3.SS1.6.p3.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.6.p3.1.m1.2b"><apply id="S3.SS1.6.p3.1.m1.2.3.cmml" xref="S3.SS1.6.p3.1.m1.2.3"><in id="S3.SS1.6.p3.1.m1.2.3.1.cmml" xref="S3.SS1.6.p3.1.m1.2.3.1"></in><ci id="S3.SS1.6.p3.1.m1.2.3.2.cmml" xref="S3.SS1.6.p3.1.m1.2.3.2">𝑟</ci><interval closure="open" id="S3.SS1.6.p3.1.m1.2.3.3.1.cmml" xref="S3.SS1.6.p3.1.m1.2.3.3.2"><cn id="S3.SS1.6.p3.1.m1.1.1.cmml" type="integer" xref="S3.SS1.6.p3.1.m1.1.1">0</cn><ci id="S3.SS1.6.p3.1.m1.2.2.cmml" xref="S3.SS1.6.p3.1.m1.2.2">𝑝</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.6.p3.1.m1.2c">r\in(0,p)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.6.p3.1.m1.2d">italic_r ∈ ( 0 , italic_p )</annotation></semantics></math>, where <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.SS1.6.p3.2.m2.1"><semantics id="S3.SS1.6.p3.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.6.p3.2.m2.1.1" xref="S3.SS1.6.p3.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.6.p3.2.m2.1b"><ci id="S3.SS1.6.p3.2.m2.1.1.cmml" xref="S3.SS1.6.p3.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.6.p3.2.m2.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.6.p3.2.m2.1d">caligraphic_M</annotation></semantics></math> denotes the Hardy-Littlewood maximal operator. Pick <math alttext="r" class="ltx_Math" display="inline" id="S3.SS1.6.p3.3.m3.1"><semantics id="S3.SS1.6.p3.3.m3.1a"><mi id="S3.SS1.6.p3.3.m3.1.1" xref="S3.SS1.6.p3.3.m3.1.1.cmml">r</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.6.p3.3.m3.1b"><ci id="S3.SS1.6.p3.3.m3.1.1.cmml" xref="S3.SS1.6.p3.3.m3.1.1">𝑟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.6.p3.3.m3.1c">r</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.6.p3.3.m3.1d">italic_r</annotation></semantics></math> such that <math alttext="w_{\gamma}\in A_{\frac{p}{r}}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S3.SS1.6.p3.4.m4.1"><semantics id="S3.SS1.6.p3.4.m4.1a"><mrow id="S3.SS1.6.p3.4.m4.1.1" xref="S3.SS1.6.p3.4.m4.1.1.cmml"><msub id="S3.SS1.6.p3.4.m4.1.1.3" xref="S3.SS1.6.p3.4.m4.1.1.3.cmml"><mi id="S3.SS1.6.p3.4.m4.1.1.3.2" xref="S3.SS1.6.p3.4.m4.1.1.3.2.cmml">w</mi><mi id="S3.SS1.6.p3.4.m4.1.1.3.3" xref="S3.SS1.6.p3.4.m4.1.1.3.3.cmml">γ</mi></msub><mo 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cd="ambiguous" id="S3.SS1.6.p3.4.m4.1.1.1.3.1.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.3">subscript</csymbol><ci id="S3.SS1.6.p3.4.m4.1.1.1.3.2.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.3.2">𝐴</ci><apply id="S3.SS1.6.p3.4.m4.1.1.1.3.3.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.3.3"><divide id="S3.SS1.6.p3.4.m4.1.1.1.3.3.1.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.3.3"></divide><ci id="S3.SS1.6.p3.4.m4.1.1.1.3.3.2.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.3.3.2">𝑝</ci><ci id="S3.SS1.6.p3.4.m4.1.1.1.3.3.3.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.3.3.3">𝑟</ci></apply></apply><apply id="S3.SS1.6.p3.4.m4.1.1.1.1.1.1.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.6.p3.4.m4.1.1.1.1.1.1.1.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S3.SS1.6.p3.4.m4.1.1.1.1.1.1.2.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.1.1.1.2">ℝ</ci><ci id="S3.SS1.6.p3.4.m4.1.1.1.1.1.1.3.cmml" xref="S3.SS1.6.p3.4.m4.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.6.p3.4.m4.1c">w_{\gamma}\in A_{\frac{p}{r}}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.6.p3.4.m4.1d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ∈ italic_A start_POSTSUBSCRIPT divide start_ARG italic_p end_ARG start_ARG italic_r end_ARG end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> and using the boundedness of <math alttext="\mathcal{M}" class="ltx_Math" display="inline" id="S3.SS1.6.p3.5.m5.1"><semantics id="S3.SS1.6.p3.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS1.6.p3.5.m5.1.1" xref="S3.SS1.6.p3.5.m5.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.6.p3.5.m5.1b"><ci id="S3.SS1.6.p3.5.m5.1.1.cmml" xref="S3.SS1.6.p3.5.m5.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.6.p3.5.m5.1c">\mathcal{M}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.6.p3.5.m5.1d">caligraphic_M</annotation></semantics></math> gives</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|S_{n}f(x_{1},\cdot)-S_{n}f(0,\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq C\,2^{% \frac{\gamma+1}{p}n}|x_{1}|2^{n}\|S_{n}f\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)% },\qquad n\in\mathbb{N}_{0}." class="ltx_Math" display="block" id="S3.Ex10.m1.9"><semantics id="S3.Ex10.m1.9a"><mrow id="S3.Ex10.m1.9.9.1"><mrow id="S3.Ex10.m1.9.9.1.1.2" xref="S3.Ex10.m1.9.9.1.1.3.cmml"><mrow id="S3.Ex10.m1.9.9.1.1.1.1" xref="S3.Ex10.m1.9.9.1.1.1.1.cmml"><msub id="S3.Ex10.m1.9.9.1.1.1.1.1" xref="S3.Ex10.m1.9.9.1.1.1.1.1.cmml"><mrow id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.cmml"><msub id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.3.2.cmml">S</mi><mi id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.3.3" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.3.3.cmml">n</mi></msub><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.4" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.4.cmml">f</mi><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.2a" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.cmml"><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.1.1.1.2" stretchy="false" 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id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.2.2" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.2.2.cmml">S</mi><mi id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.2.3" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.1" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.3" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.3.cmml">f</mi><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.1a" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mrow id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.2" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.1.cmml"><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.2.1" stretchy="false" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.1.cmml">(</mo><mn id="S3.Ex10.m1.7.7" xref="S3.Ex10.m1.7.7.cmml">0</mn><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.2.2" rspace="0em" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.1.cmml">,</mo><mo id="S3.Ex10.m1.8.8" lspace="0em" rspace="0em" xref="S3.Ex10.m1.8.8.cmml">⋅</mo><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.2.3" stretchy="false" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.1.3.4.1.cmml">)</mo></mrow></mrow></mrow><mo id="S3.Ex10.m1.9.9.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex10.m1.9.9.1.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex10.m1.2.2.2" xref="S3.Ex10.m1.2.2.2.cmml"><msup id="S3.Ex10.m1.2.2.2.4" xref="S3.Ex10.m1.2.2.2.4.cmml"><mi id="S3.Ex10.m1.2.2.2.4.2" xref="S3.Ex10.m1.2.2.2.4.2.cmml">L</mi><mi id="S3.Ex10.m1.2.2.2.4.3" xref="S3.Ex10.m1.2.2.2.4.3.cmml">p</mi></msup><mo id="S3.Ex10.m1.2.2.2.3" xref="S3.Ex10.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.Ex10.m1.2.2.2.2.1" xref="S3.Ex10.m1.2.2.2.2.2.cmml"><mo id="S3.Ex10.m1.2.2.2.2.1.2" stretchy="false" xref="S3.Ex10.m1.2.2.2.2.2.cmml">(</mo><msup id="S3.Ex10.m1.2.2.2.2.1.1" xref="S3.Ex10.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex10.m1.2.2.2.2.1.1.2" xref="S3.Ex10.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mrow id="S3.Ex10.m1.2.2.2.2.1.1.3" xref="S3.Ex10.m1.2.2.2.2.1.1.3.cmml"><mi id="S3.Ex10.m1.2.2.2.2.1.1.3.2" 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id="S3.Ex10.m1.9.9.1.1.2.2.3.3.cmml" type="integer" xref="S3.Ex10.m1.9.9.1.1.2.2.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex10.m1.9c">\|S_{n}f(x_{1},\cdot)-S_{n}f(0,\cdot)\|_{L^{p}(\mathbb{R}^{d-1};X)}\leq C\,2^{% \frac{\gamma+1}{p}n}|x_{1}|2^{n}\|S_{n}f\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)% },\qquad n\in\mathbb{N}_{0}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex10.m1.9d">∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) - italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ( 0 , ⋅ ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C 2 start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG italic_n end_POSTSUPERSCRIPT | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT , italic_n ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.6.p3.7">Continuing as in the proof of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Proposition 4.4]</cite> shows that <math alttext="\lim_{x_{1}\to 0}\widetilde{f}(x_{1},\cdot)=\widetilde{f}(0,\cdot)" class="ltx_Math" display="inline" id="S3.SS1.6.p3.6.m1.4"><semantics id="S3.SS1.6.p3.6.m1.4a"><mrow 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xref="S3.SS1.6.p3.6.m1.4.4.1.2.3.2">subscript</csymbol><ci id="S3.SS1.6.p3.6.m1.4.4.1.2.3.2.2.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.2.3.2.2">𝑥</ci><cn id="S3.SS1.6.p3.6.m1.4.4.1.2.3.2.3.cmml" type="integer" xref="S3.SS1.6.p3.6.m1.4.4.1.2.3.2.3">1</cn></apply><cn id="S3.SS1.6.p3.6.m1.4.4.1.2.3.3.cmml" type="integer" xref="S3.SS1.6.p3.6.m1.4.4.1.2.3.3">0</cn></apply></apply><apply id="S3.SS1.6.p3.6.m1.4.4.1.1.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1"><times id="S3.SS1.6.p3.6.m1.4.4.1.1.2.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.2"></times><apply id="S3.SS1.6.p3.6.m1.4.4.1.1.3.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.3"><ci id="S3.SS1.6.p3.6.m1.4.4.1.1.3.1.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.3.1">~</ci><ci id="S3.SS1.6.p3.6.m1.4.4.1.1.3.2.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.3.2">𝑓</ci></apply><interval closure="open" id="S3.SS1.6.p3.6.m1.4.4.1.1.1.2.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.1.1"><apply id="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1.1.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1">subscript</csymbol><ci id="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1.2.cmml" xref="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1.2">𝑥</ci><cn id="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1.3.cmml" type="integer" xref="S3.SS1.6.p3.6.m1.4.4.1.1.1.1.1.3">1</cn></apply><ci id="S3.SS1.6.p3.6.m1.1.1.cmml" xref="S3.SS1.6.p3.6.m1.1.1">⋅</ci></interval></apply></apply><apply id="S3.SS1.6.p3.6.m1.4.4.3.cmml" xref="S3.SS1.6.p3.6.m1.4.4.3"><times id="S3.SS1.6.p3.6.m1.4.4.3.1.cmml" xref="S3.SS1.6.p3.6.m1.4.4.3.1"></times><apply id="S3.SS1.6.p3.6.m1.4.4.3.2.cmml" xref="S3.SS1.6.p3.6.m1.4.4.3.2"><ci id="S3.SS1.6.p3.6.m1.4.4.3.2.1.cmml" xref="S3.SS1.6.p3.6.m1.4.4.3.2.1">~</ci><ci id="S3.SS1.6.p3.6.m1.4.4.3.2.2.cmml" xref="S3.SS1.6.p3.6.m1.4.4.3.2.2">𝑓</ci></apply><interval closure="open" id="S3.SS1.6.p3.6.m1.4.4.3.3.1.cmml" xref="S3.SS1.6.p3.6.m1.4.4.3.3.2"><cn id="S3.SS1.6.p3.6.m1.2.2.cmml" type="integer" xref="S3.SS1.6.p3.6.m1.2.2">0</cn><ci id="S3.SS1.6.p3.6.m1.3.3.cmml" xref="S3.SS1.6.p3.6.m1.3.3">⋅</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.6.p3.6.m1.4c">\lim_{x_{1}\to 0}\widetilde{f}(x_{1},\cdot)=\widetilde{f}(0,\cdot)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.6.p3.6.m1.4d">roman_lim start_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT → 0 end_POSTSUBSCRIPT over~ start_ARG italic_f end_ARG ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , ⋅ ) = over~ start_ARG italic_f end_ARG ( 0 , ⋅ )</annotation></semantics></math> in <math alttext="L^{p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS1.6.p3.7.m2.2"><semantics id="S3.SS1.6.p3.7.m2.2a"><mrow id="S3.SS1.6.p3.7.m2.2.2" xref="S3.SS1.6.p3.7.m2.2.2.cmml"><msup id="S3.SS1.6.p3.7.m2.2.2.3" xref="S3.SS1.6.p3.7.m2.2.2.3.cmml"><mi id="S3.SS1.6.p3.7.m2.2.2.3.2" xref="S3.SS1.6.p3.7.m2.2.2.3.2.cmml">L</mi><mi id="S3.SS1.6.p3.7.m2.2.2.3.3" xref="S3.SS1.6.p3.7.m2.2.2.3.3.cmml">p</mi></msup><mo id="S3.SS1.6.p3.7.m2.2.2.2" xref="S3.SS1.6.p3.7.m2.2.2.2.cmml">⁢</mo><mrow id="S3.SS1.6.p3.7.m2.2.2.1.1" xref="S3.SS1.6.p3.7.m2.2.2.1.2.cmml"><mo id="S3.SS1.6.p3.7.m2.2.2.1.1.2" stretchy="false" xref="S3.SS1.6.p3.7.m2.2.2.1.2.cmml">(</mo><msup id="S3.SS1.6.p3.7.m2.2.2.1.1.1" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.cmml"><mi id="S3.SS1.6.p3.7.m2.2.2.1.1.1.2" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.cmml"><mi id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.2" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.2.cmml">d</mi><mo id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.1" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.3" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.6.p3.7.m2.2.2.1.1.3" xref="S3.SS1.6.p3.7.m2.2.2.1.2.cmml">;</mo><mi id="S3.SS1.6.p3.7.m2.1.1" xref="S3.SS1.6.p3.7.m2.1.1.cmml">X</mi><mo id="S3.SS1.6.p3.7.m2.2.2.1.1.4" stretchy="false" xref="S3.SS1.6.p3.7.m2.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.6.p3.7.m2.2b"><apply id="S3.SS1.6.p3.7.m2.2.2.cmml" xref="S3.SS1.6.p3.7.m2.2.2"><times id="S3.SS1.6.p3.7.m2.2.2.2.cmml" xref="S3.SS1.6.p3.7.m2.2.2.2"></times><apply id="S3.SS1.6.p3.7.m2.2.2.3.cmml" xref="S3.SS1.6.p3.7.m2.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.6.p3.7.m2.2.2.3.1.cmml" xref="S3.SS1.6.p3.7.m2.2.2.3">superscript</csymbol><ci id="S3.SS1.6.p3.7.m2.2.2.3.2.cmml" xref="S3.SS1.6.p3.7.m2.2.2.3.2">𝐿</ci><ci id="S3.SS1.6.p3.7.m2.2.2.3.3.cmml" xref="S3.SS1.6.p3.7.m2.2.2.3.3">𝑝</ci></apply><list id="S3.SS1.6.p3.7.m2.2.2.1.2.cmml" xref="S3.SS1.6.p3.7.m2.2.2.1.1"><apply id="S3.SS1.6.p3.7.m2.2.2.1.1.1.cmml" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.6.p3.7.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.6.p3.7.m2.2.2.1.1.1.2.cmml" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.2">ℝ</ci><apply id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.cmml" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3"><minus id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.1.cmml" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.1"></minus><ci id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.2.cmml" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.2">𝑑</ci><cn id="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.6.p3.7.m2.2.2.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.6.p3.7.m2.1.1.cmml" xref="S3.SS1.6.p3.7.m2.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.6.p3.7.m2.2c">L^{p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.6.p3.7.m2.2d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S3.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.1.1.1">Remark 3.3</span></span><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem3.p1"> <p class="ltx_p" id="S3.Thmtheorem3.p1.4">Note that for <math alttext="\gamma>0" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.1.m1.1"><semantics id="S3.Thmtheorem3.p1.1.m1.1a"><mrow id="S3.Thmtheorem3.p1.1.m1.1.1" xref="S3.Thmtheorem3.p1.1.m1.1.1.cmml"><mi id="S3.Thmtheorem3.p1.1.m1.1.1.2" xref="S3.Thmtheorem3.p1.1.m1.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem3.p1.1.m1.1.1.1" xref="S3.Thmtheorem3.p1.1.m1.1.1.1.cmml">></mo><mn id="S3.Thmtheorem3.p1.1.m1.1.1.3" xref="S3.Thmtheorem3.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.1.m1.1b"><apply id="S3.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.m1.1.1"><gt id="S3.Thmtheorem3.p1.1.m1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.1.m1.1.1.1"></gt><ci id="S3.Thmtheorem3.p1.1.m1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.1.m1.1.1.2">𝛾</ci><cn id="S3.Thmtheorem3.p1.1.m1.1.1.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.1.m1.1c">\gamma>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.1.m1.1d">italic_γ > 0</annotation></semantics></math> we have <math alttext="B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)\hookrightarrow B^{% \frac{1}{p}}_{p,1}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.2.m2.9"><semantics id="S3.Thmtheorem3.p1.2.m2.9a"><mrow id="S3.Thmtheorem3.p1.2.m2.9.9" xref="S3.Thmtheorem3.p1.2.m2.9.9.cmml"><mrow id="S3.Thmtheorem3.p1.2.m2.8.8.2" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.cmml"><msubsup id="S3.Thmtheorem3.p1.2.m2.8.8.2.4" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.4.cmml"><mi id="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.2" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.2.cmml">B</mi><mrow id="S3.Thmtheorem3.p1.2.m2.2.2.2.4" xref="S3.Thmtheorem3.p1.2.m2.2.2.2.3.cmml"><mi id="S3.Thmtheorem3.p1.2.m2.1.1.1.1" xref="S3.Thmtheorem3.p1.2.m2.1.1.1.1.cmml">p</mi><mo id="S3.Thmtheorem3.p1.2.m2.2.2.2.4.1" xref="S3.Thmtheorem3.p1.2.m2.2.2.2.3.cmml">,</mo><mn id="S3.Thmtheorem3.p1.2.m2.2.2.2.2" xref="S3.Thmtheorem3.p1.2.m2.2.2.2.2.cmml">1</mn></mrow><mfrac id="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.cmml"><mrow id="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2.cmml"><mi id="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2.2" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2.2.cmml">γ</mi><mo id="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2.1" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2.1.cmml">+</mo><mn id="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2.3" xref="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.2.3.cmml">1</mn></mrow><mi id="S3.Thmtheorem3.p1.2.m2.8.8.2.4.2.3.3" 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id="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1.cmml" xref="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1.2.cmml" xref="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1.3.cmml" xref="S3.Thmtheorem3.p1.2.m2.9.9.3.1.1.1.3">𝑑</ci></apply><ci id="S3.Thmtheorem3.p1.2.m2.6.6.cmml" xref="S3.Thmtheorem3.p1.2.m2.6.6">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.2.m2.9c">B^{\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)\hookrightarrow B^{% \frac{1}{p}}_{p,1}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.2.m2.9d">italic_B start_POSTSUPERSCRIPT divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ↪ italic_B start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.Thmtheorem3" title="Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>. Thus, Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem2" title="Proposition 3.2. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.2</span></a> for <math alttext="\gamma>0" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.3.m3.1"><semantics id="S3.Thmtheorem3.p1.3.m3.1a"><mrow id="S3.Thmtheorem3.p1.3.m3.1.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.cmml"><mi id="S3.Thmtheorem3.p1.3.m3.1.1.2" xref="S3.Thmtheorem3.p1.3.m3.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem3.p1.3.m3.1.1.1" xref="S3.Thmtheorem3.p1.3.m3.1.1.1.cmml">></mo><mn id="S3.Thmtheorem3.p1.3.m3.1.1.3" xref="S3.Thmtheorem3.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.3.m3.1b"><apply id="S3.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1"><gt id="S3.Thmtheorem3.p1.3.m3.1.1.1.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.1"></gt><ci id="S3.Thmtheorem3.p1.3.m3.1.1.2.cmml" xref="S3.Thmtheorem3.p1.3.m3.1.1.2">𝛾</ci><cn id="S3.Thmtheorem3.p1.3.m3.1.1.3.cmml" type="integer" xref="S3.Thmtheorem3.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.3.m3.1c">\gamma>0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.3.m3.1d">italic_γ > 0</annotation></semantics></math> can also be deduced from the unweighted result. This argument does not apply for <math alttext="\gamma\in(-1,0)" class="ltx_Math" display="inline" id="S3.Thmtheorem3.p1.4.m4.2"><semantics id="S3.Thmtheorem3.p1.4.m4.2a"><mrow id="S3.Thmtheorem3.p1.4.m4.2.2" xref="S3.Thmtheorem3.p1.4.m4.2.2.cmml"><mi id="S3.Thmtheorem3.p1.4.m4.2.2.3" xref="S3.Thmtheorem3.p1.4.m4.2.2.3.cmml">γ</mi><mo id="S3.Thmtheorem3.p1.4.m4.2.2.2" xref="S3.Thmtheorem3.p1.4.m4.2.2.2.cmml">∈</mo><mrow id="S3.Thmtheorem3.p1.4.m4.2.2.1.1" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.2.cmml"><mo id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.2.cmml">(</mo><mrow id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1a" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.cmml">−</mo><mn id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.2" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.2.cmml">1</mn></mrow><mo id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.3" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.2.cmml">,</mo><mn id="S3.Thmtheorem3.p1.4.m4.1.1" xref="S3.Thmtheorem3.p1.4.m4.1.1.cmml">0</mn><mo id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.4" stretchy="false" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem3.p1.4.m4.2b"><apply id="S3.Thmtheorem3.p1.4.m4.2.2.cmml" xref="S3.Thmtheorem3.p1.4.m4.2.2"><in id="S3.Thmtheorem3.p1.4.m4.2.2.2.cmml" xref="S3.Thmtheorem3.p1.4.m4.2.2.2"></in><ci id="S3.Thmtheorem3.p1.4.m4.2.2.3.cmml" xref="S3.Thmtheorem3.p1.4.m4.2.2.3">𝛾</ci><interval closure="open" id="S3.Thmtheorem3.p1.4.m4.2.2.1.2.cmml" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.1"><apply id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.cmml" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1"><minus id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1"></minus><cn id="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem3.p1.4.m4.2.2.1.1.1.2">1</cn></apply><cn id="S3.Thmtheorem3.p1.4.m4.1.1.cmml" type="integer" xref="S3.Thmtheorem3.p1.4.m4.1.1">0</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem3.p1.4.m4.2c">\gamma\in(-1,0)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem3.p1.4.m4.2d">italic_γ ∈ ( - 1 , 0 )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS1.p6"> <p class="ltx_p" id="S3.SS1.p6.2">With the above results, the trace space of a weighted Besov space can be characterised following the arguments from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Theorem 4.9 & Proposition 4.12]</cite> for the unweighted case. For <math alttext="p=q\in(1,\infty)" class="ltx_Math" display="inline" id="S3.SS1.p6.1.m1.2"><semantics id="S3.SS1.p6.1.m1.2a"><mrow id="S3.SS1.p6.1.m1.2.3" xref="S3.SS1.p6.1.m1.2.3.cmml"><mi id="S3.SS1.p6.1.m1.2.3.2" xref="S3.SS1.p6.1.m1.2.3.2.cmml">p</mi><mo id="S3.SS1.p6.1.m1.2.3.3" xref="S3.SS1.p6.1.m1.2.3.3.cmml">=</mo><mi id="S3.SS1.p6.1.m1.2.3.4" xref="S3.SS1.p6.1.m1.2.3.4.cmml">q</mi><mo id="S3.SS1.p6.1.m1.2.3.5" xref="S3.SS1.p6.1.m1.2.3.5.cmml">∈</mo><mrow id="S3.SS1.p6.1.m1.2.3.6.2" xref="S3.SS1.p6.1.m1.2.3.6.1.cmml"><mo id="S3.SS1.p6.1.m1.2.3.6.2.1" stretchy="false" xref="S3.SS1.p6.1.m1.2.3.6.1.cmml">(</mo><mn id="S3.SS1.p6.1.m1.1.1" xref="S3.SS1.p6.1.m1.1.1.cmml">1</mn><mo id="S3.SS1.p6.1.m1.2.3.6.2.2" xref="S3.SS1.p6.1.m1.2.3.6.1.cmml">,</mo><mi id="S3.SS1.p6.1.m1.2.2" mathvariant="normal" xref="S3.SS1.p6.1.m1.2.2.cmml">∞</mi><mo id="S3.SS1.p6.1.m1.2.3.6.2.3" stretchy="false" xref="S3.SS1.p6.1.m1.2.3.6.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.1.m1.2b"><apply id="S3.SS1.p6.1.m1.2.3.cmml" xref="S3.SS1.p6.1.m1.2.3"><and id="S3.SS1.p6.1.m1.2.3a.cmml" xref="S3.SS1.p6.1.m1.2.3"></and><apply id="S3.SS1.p6.1.m1.2.3b.cmml" xref="S3.SS1.p6.1.m1.2.3"><eq id="S3.SS1.p6.1.m1.2.3.3.cmml" xref="S3.SS1.p6.1.m1.2.3.3"></eq><ci id="S3.SS1.p6.1.m1.2.3.2.cmml" xref="S3.SS1.p6.1.m1.2.3.2">𝑝</ci><ci id="S3.SS1.p6.1.m1.2.3.4.cmml" xref="S3.SS1.p6.1.m1.2.3.4">𝑞</ci></apply><apply id="S3.SS1.p6.1.m1.2.3c.cmml" xref="S3.SS1.p6.1.m1.2.3"><in id="S3.SS1.p6.1.m1.2.3.5.cmml" xref="S3.SS1.p6.1.m1.2.3.5"></in><share href="https://arxiv.org/html/2503.14636v1#S3.SS1.p6.1.m1.2.3.4.cmml" id="S3.SS1.p6.1.m1.2.3d.cmml" xref="S3.SS1.p6.1.m1.2.3"></share><interval closure="open" id="S3.SS1.p6.1.m1.2.3.6.1.cmml" xref="S3.SS1.p6.1.m1.2.3.6.2"><cn id="S3.SS1.p6.1.m1.1.1.cmml" type="integer" xref="S3.SS1.p6.1.m1.1.1">1</cn><infinity id="S3.SS1.p6.1.m1.2.2.cmml" xref="S3.SS1.p6.1.m1.2.2"></infinity></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.1.m1.2c">p=q\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.1.m1.2d">italic_p = italic_q ∈ ( 1 , ∞ )</annotation></semantics></math> and <math alttext="A_{p}" class="ltx_Math" display="inline" id="S3.SS1.p6.2.m2.1"><semantics id="S3.SS1.p6.2.m2.1a"><msub id="S3.SS1.p6.2.m2.1.1" xref="S3.SS1.p6.2.m2.1.1.cmml"><mi id="S3.SS1.p6.2.m2.1.1.2" xref="S3.SS1.p6.2.m2.1.1.2.cmml">A</mi><mi id="S3.SS1.p6.2.m2.1.1.3" xref="S3.SS1.p6.2.m2.1.1.3.cmml">p</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS1.p6.2.m2.1b"><apply id="S3.SS1.p6.2.m2.1.1.cmml" xref="S3.SS1.p6.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS1.p6.2.m2.1.1.1.cmml" xref="S3.SS1.p6.2.m2.1.1">subscript</csymbol><ci id="S3.SS1.p6.2.m2.1.1.2.cmml" xref="S3.SS1.p6.2.m2.1.1.2">𝐴</ci><ci id="S3.SS1.p6.2.m2.1.1.3.cmml" xref="S3.SS1.p6.2.m2.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.p6.2.m2.1c">A_{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.p6.2.m2.1d">italic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT</annotation></semantics></math> weights the result was already shown in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib23" title="">23</a>, Théorème 7.1]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.1.1.1">Theorem 3.4</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem4.p1"> <p class="ltx_p" id="S3.Thmtheorem4.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.6.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.1.1.m1.2"><semantics id="S3.Thmtheorem4.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem4.p1.1.1.m1.2.3" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem4.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem4.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem4.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem4.p1.1.1.m1.1.1" xref="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem4.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem4.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem4.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.1.1.m1.2b"><apply id="S3.Thmtheorem4.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.2.3"><in id="S3.Thmtheorem4.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.1"></in><ci id="S3.Thmtheorem4.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.2.3.3.2"><cn id="S3.Thmtheorem4.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem4.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem4.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem4.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.2.2.m2.2"><semantics id="S3.Thmtheorem4.p1.2.2.m2.2a"><mrow id="S3.Thmtheorem4.p1.2.2.m2.2.3" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.cmml"><mi id="S3.Thmtheorem4.p1.2.2.m2.2.3.2" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.2.cmml">q</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.2.3.1" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem4.p1.2.2.m2.2.3.3.2" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml"><mo id="S3.Thmtheorem4.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml">[</mo><mn id="S3.Thmtheorem4.p1.2.2.m2.1.1" xref="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml">1</mn><mo id="S3.Thmtheorem4.p1.2.2.m2.2.3.3.2.2" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.2.2.m2.2.2" mathvariant="normal" xref="S3.Thmtheorem4.p1.2.2.m2.2.2.cmml">∞</mi><mo id="S3.Thmtheorem4.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.2.2.m2.2b"><apply id="S3.Thmtheorem4.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.2.3"><in id="S3.Thmtheorem4.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.1"></in><ci id="S3.Thmtheorem4.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S3.Thmtheorem4.p1.2.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.2.3.3.2"><cn id="S3.Thmtheorem4.p1.2.2.m2.1.1.cmml" type="integer" xref="S3.Thmtheorem4.p1.2.2.m2.1.1">1</cn><infinity id="S3.Thmtheorem4.p1.2.2.m2.2.2.cmml" xref="S3.Thmtheorem4.p1.2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.3.3.m3.1"><semantics id="S3.Thmtheorem4.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem4.p1.3.3.m3.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem4.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem4.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.cmml"><mo id="S3.Thmtheorem4.p1.3.3.m3.1.1.3a" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.3.3.m3.1b"><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1"><gt id="S3.Thmtheorem4.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.1"></gt><ci id="S3.Thmtheorem4.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.2">𝛾</ci><apply id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3"><minus id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3"></minus><cn id="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2.cmml" type="integer" xref="S3.Thmtheorem4.p1.3.3.m3.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.3.3.m3.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.3.3.m3.1d">italic_γ > - 1</annotation></semantics></math>, <math alttext="s>\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.4.4.m4.1"><semantics id="S3.Thmtheorem4.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem4.p1.4.4.m4.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml">s</mi><mo id="S3.Thmtheorem4.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.1.cmml">></mo><mfrac id="S3.Thmtheorem4.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.cmml"><mrow id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.cmml"><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.2" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.2.cmml">γ</mi><mo id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.1" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.cmml">+</mo><mn id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.3" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.3.cmml">1</mn></mrow><mi id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml">p</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.4.4.m4.1b"><apply id="S3.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1"><gt id="S3.Thmtheorem4.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.1"></gt><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.2">𝑠</ci><apply id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3"><divide id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3"></divide><apply id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2"><plus id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.1.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.1"></plus><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.2.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.2">𝛾</ci><cn id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.3.cmml" type="integer" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.2.3">1</cn></apply><ci id="S3.Thmtheorem4.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem4.p1.4.4.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.4.4.m4.1c">s>\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.4.4.m4.1d">italic_s > divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.5.5.m5.1"><semantics id="S3.Thmtheorem4.p1.5.5.m5.1a"><mi id="S3.Thmtheorem4.p1.5.5.m5.1.1" xref="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.5.5.m5.1b"><ci id="S3.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem4.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Let <math alttext="\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.6.6.m6.1"><semantics id="S3.Thmtheorem4.p1.6.6.m6.1a"><mi id="S3.Thmtheorem4.p1.6.6.m6.1.1" xref="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml">Tr</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.6.6.m6.1b"><ci id="S3.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem4.p1.6.6.m6.1.1">Tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.6.6.m6.1c">\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.6.6.m6.1d">roman_Tr</annotation></semantics></math> be as in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem2" title="Proposition 3.2. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.2</span></a>. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}:B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B_{p,q}^{s-\frac% {\gamma+1}{p}}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S3.Ex11.m1.9"><semantics id="S3.Ex11.m1.9a"><mrow id="S3.Ex11.m1.9.9" xref="S3.Ex11.m1.9.9.cmml"><mi id="S3.Ex11.m1.9.9.5" xref="S3.Ex11.m1.9.9.5.cmml">Tr</mi><mo id="S3.Ex11.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S3.Ex11.m1.9.9.4.cmml">:</mo><mrow id="S3.Ex11.m1.9.9.3" xref="S3.Ex11.m1.9.9.3.cmml"><mrow id="S3.Ex11.m1.8.8.2.2" xref="S3.Ex11.m1.8.8.2.2.cmml"><msubsup id="S3.Ex11.m1.8.8.2.2.4" xref="S3.Ex11.m1.8.8.2.2.4.cmml"><mi id="S3.Ex11.m1.8.8.2.2.4.2.2" xref="S3.Ex11.m1.8.8.2.2.4.2.2.cmml">B</mi><mrow id="S3.Ex11.m1.2.2.2.4" xref="S3.Ex11.m1.2.2.2.3.cmml"><mi id="S3.Ex11.m1.1.1.1.1" xref="S3.Ex11.m1.1.1.1.1.cmml">p</mi><mo id="S3.Ex11.m1.2.2.2.4.1" xref="S3.Ex11.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex11.m1.2.2.2.2" xref="S3.Ex11.m1.2.2.2.2.cmml">q</mi></mrow><mi id="S3.Ex11.m1.8.8.2.2.4.2.3" xref="S3.Ex11.m1.8.8.2.2.4.2.3.cmml">s</mi></msubsup><mo id="S3.Ex11.m1.8.8.2.2.3" xref="S3.Ex11.m1.8.8.2.2.3.cmml">⁢</mo><mrow id="S3.Ex11.m1.8.8.2.2.2.2" xref="S3.Ex11.m1.8.8.2.2.2.3.cmml"><mo id="S3.Ex11.m1.8.8.2.2.2.2.3" stretchy="false" xref="S3.Ex11.m1.8.8.2.2.2.3.cmml">(</mo><msup id="S3.Ex11.m1.7.7.1.1.1.1.1" xref="S3.Ex11.m1.7.7.1.1.1.1.1.cmml"><mi id="S3.Ex11.m1.7.7.1.1.1.1.1.2" xref="S3.Ex11.m1.7.7.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.Ex11.m1.7.7.1.1.1.1.1.3" xref="S3.Ex11.m1.7.7.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.Ex11.m1.8.8.2.2.2.2.4" xref="S3.Ex11.m1.8.8.2.2.2.3.cmml">,</mo><msub id="S3.Ex11.m1.8.8.2.2.2.2.2" xref="S3.Ex11.m1.8.8.2.2.2.2.2.cmml"><mi id="S3.Ex11.m1.8.8.2.2.2.2.2.2" xref="S3.Ex11.m1.8.8.2.2.2.2.2.2.cmml">w</mi><mi id="S3.Ex11.m1.8.8.2.2.2.2.2.3" xref="S3.Ex11.m1.8.8.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex11.m1.8.8.2.2.2.2.5" xref="S3.Ex11.m1.8.8.2.2.2.3.cmml">;</mo><mi id="S3.Ex11.m1.5.5" xref="S3.Ex11.m1.5.5.cmml">X</mi><mo id="S3.Ex11.m1.8.8.2.2.2.2.6" stretchy="false" xref="S3.Ex11.m1.8.8.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex11.m1.9.9.3.4" stretchy="false" xref="S3.Ex11.m1.9.9.3.4.cmml">→</mo><mrow id="S3.Ex11.m1.9.9.3.3" xref="S3.Ex11.m1.9.9.3.3.cmml"><msubsup id="S3.Ex11.m1.9.9.3.3.3" xref="S3.Ex11.m1.9.9.3.3.3.cmml"><mi id="S3.Ex11.m1.9.9.3.3.3.2.2" xref="S3.Ex11.m1.9.9.3.3.3.2.2.cmml">B</mi><mrow id="S3.Ex11.m1.4.4.2.4" xref="S3.Ex11.m1.4.4.2.3.cmml"><mi id="S3.Ex11.m1.3.3.1.1" xref="S3.Ex11.m1.3.3.1.1.cmml">p</mi><mo id="S3.Ex11.m1.4.4.2.4.1" xref="S3.Ex11.m1.4.4.2.3.cmml">,</mo><mi id="S3.Ex11.m1.4.4.2.2" xref="S3.Ex11.m1.4.4.2.2.cmml">q</mi></mrow><mrow id="S3.Ex11.m1.9.9.3.3.3.3" xref="S3.Ex11.m1.9.9.3.3.3.3.cmml"><mi id="S3.Ex11.m1.9.9.3.3.3.3.2" xref="S3.Ex11.m1.9.9.3.3.3.3.2.cmml">s</mi><mo id="S3.Ex11.m1.9.9.3.3.3.3.1" xref="S3.Ex11.m1.9.9.3.3.3.3.1.cmml">−</mo><mfrac id="S3.Ex11.m1.9.9.3.3.3.3.3" xref="S3.Ex11.m1.9.9.3.3.3.3.3.cmml"><mrow id="S3.Ex11.m1.9.9.3.3.3.3.3.2" xref="S3.Ex11.m1.9.9.3.3.3.3.3.2.cmml"><mi id="S3.Ex11.m1.9.9.3.3.3.3.3.2.2" xref="S3.Ex11.m1.9.9.3.3.3.3.3.2.2.cmml">γ</mi><mo id="S3.Ex11.m1.9.9.3.3.3.3.3.2.1" xref="S3.Ex11.m1.9.9.3.3.3.3.3.2.1.cmml">+</mo><mn id="S3.Ex11.m1.9.9.3.3.3.3.3.2.3" xref="S3.Ex11.m1.9.9.3.3.3.3.3.2.3.cmml">1</mn></mrow><mi id="S3.Ex11.m1.9.9.3.3.3.3.3.3" xref="S3.Ex11.m1.9.9.3.3.3.3.3.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.Ex11.m1.9.9.3.3.2" xref="S3.Ex11.m1.9.9.3.3.2.cmml">⁢</mo><mrow id="S3.Ex11.m1.9.9.3.3.1.1" xref="S3.Ex11.m1.9.9.3.3.1.2.cmml"><mo id="S3.Ex11.m1.9.9.3.3.1.1.2" stretchy="false" xref="S3.Ex11.m1.9.9.3.3.1.2.cmml">(</mo><msup id="S3.Ex11.m1.9.9.3.3.1.1.1" xref="S3.Ex11.m1.9.9.3.3.1.1.1.cmml"><mi id="S3.Ex11.m1.9.9.3.3.1.1.1.2" xref="S3.Ex11.m1.9.9.3.3.1.1.1.2.cmml">ℝ</mi><mrow id="S3.Ex11.m1.9.9.3.3.1.1.1.3" xref="S3.Ex11.m1.9.9.3.3.1.1.1.3.cmml"><mi id="S3.Ex11.m1.9.9.3.3.1.1.1.3.2" xref="S3.Ex11.m1.9.9.3.3.1.1.1.3.2.cmml">d</mi><mo id="S3.Ex11.m1.9.9.3.3.1.1.1.3.1" xref="S3.Ex11.m1.9.9.3.3.1.1.1.3.1.cmml">−</mo><mn id="S3.Ex11.m1.9.9.3.3.1.1.1.3.3" xref="S3.Ex11.m1.9.9.3.3.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Ex11.m1.9.9.3.3.1.1.3" xref="S3.Ex11.m1.9.9.3.3.1.2.cmml">;</mo><mi id="S3.Ex11.m1.6.6" xref="S3.Ex11.m1.6.6.cmml">X</mi><mo id="S3.Ex11.m1.9.9.3.3.1.1.4" stretchy="false" xref="S3.Ex11.m1.9.9.3.3.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex11.m1.9b"><apply id="S3.Ex11.m1.9.9.cmml" xref="S3.Ex11.m1.9.9"><ci id="S3.Ex11.m1.9.9.4.cmml" xref="S3.Ex11.m1.9.9.4">:</ci><ci id="S3.Ex11.m1.9.9.5.cmml" xref="S3.Ex11.m1.9.9.5">Tr</ci><apply id="S3.Ex11.m1.9.9.3.cmml" xref="S3.Ex11.m1.9.9.3"><ci id="S3.Ex11.m1.9.9.3.4.cmml" xref="S3.Ex11.m1.9.9.3.4">→</ci><apply id="S3.Ex11.m1.8.8.2.2.cmml" xref="S3.Ex11.m1.8.8.2.2"><times id="S3.Ex11.m1.8.8.2.2.3.cmml" xref="S3.Ex11.m1.8.8.2.2.3"></times><apply id="S3.Ex11.m1.8.8.2.2.4.cmml" xref="S3.Ex11.m1.8.8.2.2.4"><csymbol cd="ambiguous" id="S3.Ex11.m1.8.8.2.2.4.1.cmml" xref="S3.Ex11.m1.8.8.2.2.4">subscript</csymbol><apply id="S3.Ex11.m1.8.8.2.2.4.2.cmml" xref="S3.Ex11.m1.8.8.2.2.4"><csymbol cd="ambiguous" id="S3.Ex11.m1.8.8.2.2.4.2.1.cmml" xref="S3.Ex11.m1.8.8.2.2.4">superscript</csymbol><ci id="S3.Ex11.m1.8.8.2.2.4.2.2.cmml" xref="S3.Ex11.m1.8.8.2.2.4.2.2">𝐵</ci><ci id="S3.Ex11.m1.8.8.2.2.4.2.3.cmml" xref="S3.Ex11.m1.8.8.2.2.4.2.3">𝑠</ci></apply><list id="S3.Ex11.m1.2.2.2.3.cmml" xref="S3.Ex11.m1.2.2.2.4"><ci id="S3.Ex11.m1.1.1.1.1.cmml" xref="S3.Ex11.m1.1.1.1.1">𝑝</ci><ci id="S3.Ex11.m1.2.2.2.2.cmml" xref="S3.Ex11.m1.2.2.2.2">𝑞</ci></list></apply><vector id="S3.Ex11.m1.8.8.2.2.2.3.cmml" xref="S3.Ex11.m1.8.8.2.2.2.2"><apply id="S3.Ex11.m1.7.7.1.1.1.1.1.cmml" 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xref="S3.Ex11.m1.9.9.3.3.1.1.1.3.2">𝑑</ci><cn id="S3.Ex11.m1.9.9.3.3.1.1.1.3.3.cmml" type="integer" xref="S3.Ex11.m1.9.9.3.3.1.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex11.m1.6.6.cmml" xref="S3.Ex11.m1.6.6">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex11.m1.9c">\operatorname{Tr}:B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B_{p,q}^{s-\frac% {\gamma+1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex11.m1.9d">roman_Tr : italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem4.p1.10"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem4.p1.10.4">is a continuous and surjective operator. Moreover, there exists a continuous right inverse <math alttext="\operatorname{ext}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.7.1.m1.1"><semantics id="S3.Thmtheorem4.p1.7.1.m1.1a"><mi id="S3.Thmtheorem4.p1.7.1.m1.1.1" xref="S3.Thmtheorem4.p1.7.1.m1.1.1.cmml">ext</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.7.1.m1.1b"><ci id="S3.Thmtheorem4.p1.7.1.m1.1.1.cmml" xref="S3.Thmtheorem4.p1.7.1.m1.1.1">ext</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.7.1.m1.1c">\operatorname{ext}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.7.1.m1.1d">roman_ext</annotation></semantics></math> of <math alttext="\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.8.2.m2.1"><semantics id="S3.Thmtheorem4.p1.8.2.m2.1a"><mi id="S3.Thmtheorem4.p1.8.2.m2.1.1" xref="S3.Thmtheorem4.p1.8.2.m2.1.1.cmml">Tr</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.8.2.m2.1b"><ci id="S3.Thmtheorem4.p1.8.2.m2.1.1.cmml" xref="S3.Thmtheorem4.p1.8.2.m2.1.1">Tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.8.2.m2.1c">\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.8.2.m2.1d">roman_Tr</annotation></semantics></math> which is independent of <math alttext="s,p,q,\gamma" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.9.3.m3.4"><semantics id="S3.Thmtheorem4.p1.9.3.m3.4a"><mrow id="S3.Thmtheorem4.p1.9.3.m3.4.5.2" xref="S3.Thmtheorem4.p1.9.3.m3.4.5.1.cmml"><mi id="S3.Thmtheorem4.p1.9.3.m3.1.1" xref="S3.Thmtheorem4.p1.9.3.m3.1.1.cmml">s</mi><mo id="S3.Thmtheorem4.p1.9.3.m3.4.5.2.1" xref="S3.Thmtheorem4.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.9.3.m3.2.2" xref="S3.Thmtheorem4.p1.9.3.m3.2.2.cmml">p</mi><mo id="S3.Thmtheorem4.p1.9.3.m3.4.5.2.2" xref="S3.Thmtheorem4.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.9.3.m3.3.3" xref="S3.Thmtheorem4.p1.9.3.m3.3.3.cmml">q</mi><mo id="S3.Thmtheorem4.p1.9.3.m3.4.5.2.3" xref="S3.Thmtheorem4.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem4.p1.9.3.m3.4.4" xref="S3.Thmtheorem4.p1.9.3.m3.4.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.9.3.m3.4b"><list id="S3.Thmtheorem4.p1.9.3.m3.4.5.1.cmml" xref="S3.Thmtheorem4.p1.9.3.m3.4.5.2"><ci id="S3.Thmtheorem4.p1.9.3.m3.1.1.cmml" xref="S3.Thmtheorem4.p1.9.3.m3.1.1">𝑠</ci><ci id="S3.Thmtheorem4.p1.9.3.m3.2.2.cmml" xref="S3.Thmtheorem4.p1.9.3.m3.2.2">𝑝</ci><ci id="S3.Thmtheorem4.p1.9.3.m3.3.3.cmml" xref="S3.Thmtheorem4.p1.9.3.m3.3.3">𝑞</ci><ci id="S3.Thmtheorem4.p1.9.3.m3.4.4.cmml" xref="S3.Thmtheorem4.p1.9.3.m3.4.4">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.9.3.m3.4c">s,p,q,\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.9.3.m3.4d">italic_s , italic_p , italic_q , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem4.p1.10.4.m4.1"><semantics id="S3.Thmtheorem4.p1.10.4.m4.1a"><mi id="S3.Thmtheorem4.p1.10.4.m4.1.1" xref="S3.Thmtheorem4.p1.10.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem4.p1.10.4.m4.1b"><ci id="S3.Thmtheorem4.p1.10.4.m4.1.1.cmml" xref="S3.Thmtheorem4.p1.10.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem4.p1.10.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem4.p1.10.4.m4.1d">italic_X</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.10"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.7.p1"> <p class="ltx_p" id="S3.SS1.7.p1.7">Let <math alttext="(\varphi_{n})_{n\geq 0}\in\Phi(\mathbb{R}^{d})" class="ltx_Math" display="inline" 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xref="S3.SS1.7.p1.4.m4.1.1.cmml">X</mi><mo id="S3.SS1.7.p1.4.m4.2.2.1.1.1.4" stretchy="false" xref="S3.SS1.7.p1.4.m4.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.7.p1.4.m4.2b"><apply id="S3.SS1.7.p1.4.m4.2.2.cmml" xref="S3.SS1.7.p1.4.m4.2.2"><in id="S3.SS1.7.p1.4.m4.2.2.2.cmml" xref="S3.SS1.7.p1.4.m4.2.2.2"></in><ci id="S3.SS1.7.p1.4.m4.2.2.3.cmml" xref="S3.SS1.7.p1.4.m4.2.2.3">𝑔</ci><apply id="S3.SS1.7.p1.4.m4.2.2.1.cmml" xref="S3.SS1.7.p1.4.m4.2.2.1"><times id="S3.SS1.7.p1.4.m4.2.2.1.2.cmml" xref="S3.SS1.7.p1.4.m4.2.2.1.2"></times><apply id="S3.SS1.7.p1.4.m4.2.2.1.3.cmml" xref="S3.SS1.7.p1.4.m4.2.2.1.3"><csymbol cd="ambiguous" id="S3.SS1.7.p1.4.m4.2.2.1.3.1.cmml" xref="S3.SS1.7.p1.4.m4.2.2.1.3">superscript</csymbol><ci id="S3.SS1.7.p1.4.m4.2.2.1.3.2.cmml" xref="S3.SS1.7.p1.4.m4.2.2.1.3.2">SS</ci><ci id="S3.SS1.7.p1.4.m4.2.2.1.3.3.cmml" xref="S3.SS1.7.p1.4.m4.2.2.1.3.3">′</ci></apply><list id="S3.SS1.7.p1.4.m4.2.2.1.1.2.cmml" 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id="S3.SS1.7.p1.4.m4.2d">italic_g ∈ roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> and define <math alttext="S_{n}f:=\varphi_{n}\ast f" class="ltx_Math" display="inline" id="S3.SS1.7.p1.5.m5.1"><semantics id="S3.SS1.7.p1.5.m5.1a"><mrow id="S3.SS1.7.p1.5.m5.1.1" xref="S3.SS1.7.p1.5.m5.1.1.cmml"><mrow id="S3.SS1.7.p1.5.m5.1.1.2" xref="S3.SS1.7.p1.5.m5.1.1.2.cmml"><msub id="S3.SS1.7.p1.5.m5.1.1.2.2" xref="S3.SS1.7.p1.5.m5.1.1.2.2.cmml"><mi id="S3.SS1.7.p1.5.m5.1.1.2.2.2" xref="S3.SS1.7.p1.5.m5.1.1.2.2.2.cmml">S</mi><mi id="S3.SS1.7.p1.5.m5.1.1.2.2.3" xref="S3.SS1.7.p1.5.m5.1.1.2.2.3.cmml">n</mi></msub><mo id="S3.SS1.7.p1.5.m5.1.1.2.1" xref="S3.SS1.7.p1.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.7.p1.5.m5.1.1.2.3" xref="S3.SS1.7.p1.5.m5.1.1.2.3.cmml">f</mi></mrow><mo id="S3.SS1.7.p1.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.7.p1.5.m5.1.1.1.cmml">:=</mo><mrow id="S3.SS1.7.p1.5.m5.1.1.3" xref="S3.SS1.7.p1.5.m5.1.1.3.cmml"><msub id="S3.SS1.7.p1.5.m5.1.1.3.2" xref="S3.SS1.7.p1.5.m5.1.1.3.2.cmml"><mi id="S3.SS1.7.p1.5.m5.1.1.3.2.2" xref="S3.SS1.7.p1.5.m5.1.1.3.2.2.cmml">φ</mi><mi id="S3.SS1.7.p1.5.m5.1.1.3.2.3" xref="S3.SS1.7.p1.5.m5.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.SS1.7.p1.5.m5.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.7.p1.5.m5.1.1.3.1.cmml">∗</mo><mi id="S3.SS1.7.p1.5.m5.1.1.3.3" xref="S3.SS1.7.p1.5.m5.1.1.3.3.cmml">f</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.7.p1.5.m5.1b"><apply id="S3.SS1.7.p1.5.m5.1.1.cmml" xref="S3.SS1.7.p1.5.m5.1.1"><csymbol cd="latexml" id="S3.SS1.7.p1.5.m5.1.1.1.cmml" xref="S3.SS1.7.p1.5.m5.1.1.1">assign</csymbol><apply id="S3.SS1.7.p1.5.m5.1.1.2.cmml" xref="S3.SS1.7.p1.5.m5.1.1.2"><times id="S3.SS1.7.p1.5.m5.1.1.2.1.cmml" xref="S3.SS1.7.p1.5.m5.1.1.2.1"></times><apply id="S3.SS1.7.p1.5.m5.1.1.2.2.cmml" xref="S3.SS1.7.p1.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.7.p1.5.m5.1.1.2.2.1.cmml" xref="S3.SS1.7.p1.5.m5.1.1.2.2">subscript</csymbol><ci id="S3.SS1.7.p1.5.m5.1.1.2.2.2.cmml" xref="S3.SS1.7.p1.5.m5.1.1.2.2.2">𝑆</ci><ci id="S3.SS1.7.p1.5.m5.1.1.2.2.3.cmml" xref="S3.SS1.7.p1.5.m5.1.1.2.2.3">𝑛</ci></apply><ci id="S3.SS1.7.p1.5.m5.1.1.2.3.cmml" xref="S3.SS1.7.p1.5.m5.1.1.2.3">𝑓</ci></apply><apply id="S3.SS1.7.p1.5.m5.1.1.3.cmml" xref="S3.SS1.7.p1.5.m5.1.1.3"><ci id="S3.SS1.7.p1.5.m5.1.1.3.1.cmml" xref="S3.SS1.7.p1.5.m5.1.1.3.1">∗</ci><apply id="S3.SS1.7.p1.5.m5.1.1.3.2.cmml" xref="S3.SS1.7.p1.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.7.p1.5.m5.1.1.3.2.1.cmml" xref="S3.SS1.7.p1.5.m5.1.1.3.2">subscript</csymbol><ci id="S3.SS1.7.p1.5.m5.1.1.3.2.2.cmml" xref="S3.SS1.7.p1.5.m5.1.1.3.2.2">𝜑</ci><ci id="S3.SS1.7.p1.5.m5.1.1.3.2.3.cmml" xref="S3.SS1.7.p1.5.m5.1.1.3.2.3">𝑛</ci></apply><ci id="S3.SS1.7.p1.5.m5.1.1.3.3.cmml" xref="S3.SS1.7.p1.5.m5.1.1.3.3">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.7.p1.5.m5.1c">S_{n}f:=\varphi_{n}\ast f</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.7.p1.5.m5.1d">italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_f := italic_φ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_f</annotation></semantics></math> and <math alttext="T_{n}g:=\phi_{n}\ast g" class="ltx_Math" display="inline" id="S3.SS1.7.p1.6.m6.1"><semantics id="S3.SS1.7.p1.6.m6.1a"><mrow id="S3.SS1.7.p1.6.m6.1.1" xref="S3.SS1.7.p1.6.m6.1.1.cmml"><mrow id="S3.SS1.7.p1.6.m6.1.1.2" xref="S3.SS1.7.p1.6.m6.1.1.2.cmml"><msub id="S3.SS1.7.p1.6.m6.1.1.2.2" xref="S3.SS1.7.p1.6.m6.1.1.2.2.cmml"><mi id="S3.SS1.7.p1.6.m6.1.1.2.2.2" xref="S3.SS1.7.p1.6.m6.1.1.2.2.2.cmml">T</mi><mi id="S3.SS1.7.p1.6.m6.1.1.2.2.3" xref="S3.SS1.7.p1.6.m6.1.1.2.2.3.cmml">n</mi></msub><mo id="S3.SS1.7.p1.6.m6.1.1.2.1" xref="S3.SS1.7.p1.6.m6.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.7.p1.6.m6.1.1.2.3" xref="S3.SS1.7.p1.6.m6.1.1.2.3.cmml">g</mi></mrow><mo id="S3.SS1.7.p1.6.m6.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.7.p1.6.m6.1.1.1.cmml">:=</mo><mrow id="S3.SS1.7.p1.6.m6.1.1.3" xref="S3.SS1.7.p1.6.m6.1.1.3.cmml"><msub id="S3.SS1.7.p1.6.m6.1.1.3.2" xref="S3.SS1.7.p1.6.m6.1.1.3.2.cmml"><mi id="S3.SS1.7.p1.6.m6.1.1.3.2.2" xref="S3.SS1.7.p1.6.m6.1.1.3.2.2.cmml">ϕ</mi><mi id="S3.SS1.7.p1.6.m6.1.1.3.2.3" xref="S3.SS1.7.p1.6.m6.1.1.3.2.3.cmml">n</mi></msub><mo id="S3.SS1.7.p1.6.m6.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS1.7.p1.6.m6.1.1.3.1.cmml">∗</mo><mi id="S3.SS1.7.p1.6.m6.1.1.3.3" xref="S3.SS1.7.p1.6.m6.1.1.3.3.cmml">g</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.7.p1.6.m6.1b"><apply id="S3.SS1.7.p1.6.m6.1.1.cmml" xref="S3.SS1.7.p1.6.m6.1.1"><csymbol cd="latexml" id="S3.SS1.7.p1.6.m6.1.1.1.cmml" xref="S3.SS1.7.p1.6.m6.1.1.1">assign</csymbol><apply id="S3.SS1.7.p1.6.m6.1.1.2.cmml" xref="S3.SS1.7.p1.6.m6.1.1.2"><times id="S3.SS1.7.p1.6.m6.1.1.2.1.cmml" xref="S3.SS1.7.p1.6.m6.1.1.2.1"></times><apply id="S3.SS1.7.p1.6.m6.1.1.2.2.cmml" xref="S3.SS1.7.p1.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.7.p1.6.m6.1.1.2.2.1.cmml" xref="S3.SS1.7.p1.6.m6.1.1.2.2">subscript</csymbol><ci id="S3.SS1.7.p1.6.m6.1.1.2.2.2.cmml" xref="S3.SS1.7.p1.6.m6.1.1.2.2.2">𝑇</ci><ci id="S3.SS1.7.p1.6.m6.1.1.2.2.3.cmml" xref="S3.SS1.7.p1.6.m6.1.1.2.2.3">𝑛</ci></apply><ci id="S3.SS1.7.p1.6.m6.1.1.2.3.cmml" xref="S3.SS1.7.p1.6.m6.1.1.2.3">𝑔</ci></apply><apply id="S3.SS1.7.p1.6.m6.1.1.3.cmml" xref="S3.SS1.7.p1.6.m6.1.1.3"><ci id="S3.SS1.7.p1.6.m6.1.1.3.1.cmml" xref="S3.SS1.7.p1.6.m6.1.1.3.1">∗</ci><apply id="S3.SS1.7.p1.6.m6.1.1.3.2.cmml" xref="S3.SS1.7.p1.6.m6.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS1.7.p1.6.m6.1.1.3.2.1.cmml" xref="S3.SS1.7.p1.6.m6.1.1.3.2">subscript</csymbol><ci id="S3.SS1.7.p1.6.m6.1.1.3.2.2.cmml" xref="S3.SS1.7.p1.6.m6.1.1.3.2.2">italic-ϕ</ci><ci id="S3.SS1.7.p1.6.m6.1.1.3.2.3.cmml" xref="S3.SS1.7.p1.6.m6.1.1.3.2.3">𝑛</ci></apply><ci id="S3.SS1.7.p1.6.m6.1.1.3.3.cmml" xref="S3.SS1.7.p1.6.m6.1.1.3.3">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.7.p1.6.m6.1c">T_{n}g:=\phi_{n}\ast g</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.7.p1.6.m6.1d">italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g := italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_g</annotation></semantics></math>. Moreover, set <math alttext="S_{-1}f=T_{-1}g=0" class="ltx_Math" display="inline" id="S3.SS1.7.p1.7.m7.1"><semantics id="S3.SS1.7.p1.7.m7.1a"><mrow id="S3.SS1.7.p1.7.m7.1.1" xref="S3.SS1.7.p1.7.m7.1.1.cmml"><mrow id="S3.SS1.7.p1.7.m7.1.1.2" xref="S3.SS1.7.p1.7.m7.1.1.2.cmml"><msub id="S3.SS1.7.p1.7.m7.1.1.2.2" xref="S3.SS1.7.p1.7.m7.1.1.2.2.cmml"><mi id="S3.SS1.7.p1.7.m7.1.1.2.2.2" xref="S3.SS1.7.p1.7.m7.1.1.2.2.2.cmml">S</mi><mrow id="S3.SS1.7.p1.7.m7.1.1.2.2.3" xref="S3.SS1.7.p1.7.m7.1.1.2.2.3.cmml"><mo id="S3.SS1.7.p1.7.m7.1.1.2.2.3a" xref="S3.SS1.7.p1.7.m7.1.1.2.2.3.cmml">−</mo><mn id="S3.SS1.7.p1.7.m7.1.1.2.2.3.2" xref="S3.SS1.7.p1.7.m7.1.1.2.2.3.2.cmml">1</mn></mrow></msub><mo id="S3.SS1.7.p1.7.m7.1.1.2.1" xref="S3.SS1.7.p1.7.m7.1.1.2.1.cmml">⁢</mo><mi id="S3.SS1.7.p1.7.m7.1.1.2.3" xref="S3.SS1.7.p1.7.m7.1.1.2.3.cmml">f</mi></mrow><mo id="S3.SS1.7.p1.7.m7.1.1.3" xref="S3.SS1.7.p1.7.m7.1.1.3.cmml">=</mo><mrow id="S3.SS1.7.p1.7.m7.1.1.4" xref="S3.SS1.7.p1.7.m7.1.1.4.cmml"><msub id="S3.SS1.7.p1.7.m7.1.1.4.2" xref="S3.SS1.7.p1.7.m7.1.1.4.2.cmml"><mi id="S3.SS1.7.p1.7.m7.1.1.4.2.2" xref="S3.SS1.7.p1.7.m7.1.1.4.2.2.cmml">T</mi><mrow id="S3.SS1.7.p1.7.m7.1.1.4.2.3" xref="S3.SS1.7.p1.7.m7.1.1.4.2.3.cmml"><mo id="S3.SS1.7.p1.7.m7.1.1.4.2.3a" xref="S3.SS1.7.p1.7.m7.1.1.4.2.3.cmml">−</mo><mn id="S3.SS1.7.p1.7.m7.1.1.4.2.3.2" xref="S3.SS1.7.p1.7.m7.1.1.4.2.3.2.cmml">1</mn></mrow></msub><mo id="S3.SS1.7.p1.7.m7.1.1.4.1" xref="S3.SS1.7.p1.7.m7.1.1.4.1.cmml">⁢</mo><mi id="S3.SS1.7.p1.7.m7.1.1.4.3" xref="S3.SS1.7.p1.7.m7.1.1.4.3.cmml">g</mi></mrow><mo id="S3.SS1.7.p1.7.m7.1.1.5" xref="S3.SS1.7.p1.7.m7.1.1.5.cmml">=</mo><mn id="S3.SS1.7.p1.7.m7.1.1.6" xref="S3.SS1.7.p1.7.m7.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.7.p1.7.m7.1b"><apply id="S3.SS1.7.p1.7.m7.1.1.cmml" xref="S3.SS1.7.p1.7.m7.1.1"><and id="S3.SS1.7.p1.7.m7.1.1a.cmml" xref="S3.SS1.7.p1.7.m7.1.1"></and><apply id="S3.SS1.7.p1.7.m7.1.1b.cmml" xref="S3.SS1.7.p1.7.m7.1.1"><eq id="S3.SS1.7.p1.7.m7.1.1.3.cmml" xref="S3.SS1.7.p1.7.m7.1.1.3"></eq><apply id="S3.SS1.7.p1.7.m7.1.1.2.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2"><times id="S3.SS1.7.p1.7.m7.1.1.2.1.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2.1"></times><apply id="S3.SS1.7.p1.7.m7.1.1.2.2.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.7.p1.7.m7.1.1.2.2.1.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2.2">subscript</csymbol><ci id="S3.SS1.7.p1.7.m7.1.1.2.2.2.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2.2.2">𝑆</ci><apply id="S3.SS1.7.p1.7.m7.1.1.2.2.3.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2.2.3"><minus id="S3.SS1.7.p1.7.m7.1.1.2.2.3.1.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2.2.3"></minus><cn id="S3.SS1.7.p1.7.m7.1.1.2.2.3.2.cmml" type="integer" xref="S3.SS1.7.p1.7.m7.1.1.2.2.3.2">1</cn></apply></apply><ci id="S3.SS1.7.p1.7.m7.1.1.2.3.cmml" xref="S3.SS1.7.p1.7.m7.1.1.2.3">𝑓</ci></apply><apply id="S3.SS1.7.p1.7.m7.1.1.4.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4"><times id="S3.SS1.7.p1.7.m7.1.1.4.1.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4.1"></times><apply id="S3.SS1.7.p1.7.m7.1.1.4.2.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4.2"><csymbol cd="ambiguous" id="S3.SS1.7.p1.7.m7.1.1.4.2.1.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4.2">subscript</csymbol><ci id="S3.SS1.7.p1.7.m7.1.1.4.2.2.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4.2.2">𝑇</ci><apply id="S3.SS1.7.p1.7.m7.1.1.4.2.3.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4.2.3"><minus id="S3.SS1.7.p1.7.m7.1.1.4.2.3.1.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4.2.3"></minus><cn id="S3.SS1.7.p1.7.m7.1.1.4.2.3.2.cmml" type="integer" xref="S3.SS1.7.p1.7.m7.1.1.4.2.3.2">1</cn></apply></apply><ci id="S3.SS1.7.p1.7.m7.1.1.4.3.cmml" xref="S3.SS1.7.p1.7.m7.1.1.4.3">𝑔</ci></apply></apply><apply id="S3.SS1.7.p1.7.m7.1.1c.cmml" xref="S3.SS1.7.p1.7.m7.1.1"><eq id="S3.SS1.7.p1.7.m7.1.1.5.cmml" xref="S3.SS1.7.p1.7.m7.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S3.SS1.7.p1.7.m7.1.1.4.cmml" id="S3.SS1.7.p1.7.m7.1.1d.cmml" xref="S3.SS1.7.p1.7.m7.1.1"></share><cn id="S3.SS1.7.p1.7.m7.1.1.6.cmml" type="integer" xref="S3.SS1.7.p1.7.m7.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.7.p1.7.m7.1c">S_{-1}f=T_{-1}g=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.7.p1.7.m7.1d">italic_S start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_f = italic_T start_POSTSUBSCRIPT - 1 end_POSTSUBSCRIPT italic_g = 0</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.8.p2"> <p class="ltx_p" id="S3.SS1.8.p2.1"><span class="ltx_text ltx_font_italic" id="S3.SS1.8.p2.1.1">Step 1: trace operator.</span> The continuity of the trace operator can be shown similarly as in Step 4 of the proof of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Theorem 4.9]</cite> if we apply Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem1" title="Lemma 3.1. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.1</span></a> instead of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Equation (4.5)]</cite>.</p> </div> <div class="ltx_para" id="S3.SS1.9.p3"> <p class="ltx_p" id="S3.SS1.9.p3.5"><span class="ltx_text ltx_font_italic" id="S3.SS1.9.p3.5.1">Step 2: extension operator.</span> The extension operator is defined similarly as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib68" title="">68</a>, Section 2.7.2]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>]</cite>. Let <math alttext="g\in B^{s-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS1.9.p3.1.m1.4"><semantics id="S3.SS1.9.p3.1.m1.4a"><mrow id="S3.SS1.9.p3.1.m1.4.4" xref="S3.SS1.9.p3.1.m1.4.4.cmml"><mi id="S3.SS1.9.p3.1.m1.4.4.3" xref="S3.SS1.9.p3.1.m1.4.4.3.cmml">g</mi><mo id="S3.SS1.9.p3.1.m1.4.4.2" xref="S3.SS1.9.p3.1.m1.4.4.2.cmml">∈</mo><mrow id="S3.SS1.9.p3.1.m1.4.4.1" xref="S3.SS1.9.p3.1.m1.4.4.1.cmml"><msubsup id="S3.SS1.9.p3.1.m1.4.4.1.3" xref="S3.SS1.9.p3.1.m1.4.4.1.3.cmml"><mi id="S3.SS1.9.p3.1.m1.4.4.1.3.2.2" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.2.cmml">B</mi><mrow id="S3.SS1.9.p3.1.m1.2.2.2.4" xref="S3.SS1.9.p3.1.m1.2.2.2.3.cmml"><mi id="S3.SS1.9.p3.1.m1.1.1.1.1" xref="S3.SS1.9.p3.1.m1.1.1.1.1.cmml">p</mi><mo id="S3.SS1.9.p3.1.m1.2.2.2.4.1" xref="S3.SS1.9.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS1.9.p3.1.m1.2.2.2.2" xref="S3.SS1.9.p3.1.m1.2.2.2.2.cmml">q</mi></mrow><mrow id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.cmml"><mi id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.2" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.2.cmml">s</mi><mo id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.1" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.cmml"><mrow id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.cmml"><mi id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.2" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.2.cmml">γ</mi><mo id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.1" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.1.cmml">+</mo><mn id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.3" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.3.cmml">1</mn></mrow><mi id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.3" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.SS1.9.p3.1.m1.4.4.1.2" xref="S3.SS1.9.p3.1.m1.4.4.1.2.cmml">⁢</mo><mrow id="S3.SS1.9.p3.1.m1.4.4.1.1.1" xref="S3.SS1.9.p3.1.m1.4.4.1.1.2.cmml"><mo id="S3.SS1.9.p3.1.m1.4.4.1.1.1.2" stretchy="false" xref="S3.SS1.9.p3.1.m1.4.4.1.1.2.cmml">(</mo><msup id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.cmml"><mi id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.2" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.cmml"><mi id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.2" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.2.cmml">d</mi><mo id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.1" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.3" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.9.p3.1.m1.4.4.1.1.1.3" xref="S3.SS1.9.p3.1.m1.4.4.1.1.2.cmml">;</mo><mi id="S3.SS1.9.p3.1.m1.3.3" xref="S3.SS1.9.p3.1.m1.3.3.cmml">X</mi><mo id="S3.SS1.9.p3.1.m1.4.4.1.1.1.4" stretchy="false" xref="S3.SS1.9.p3.1.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.1.m1.4b"><apply id="S3.SS1.9.p3.1.m1.4.4.cmml" xref="S3.SS1.9.p3.1.m1.4.4"><in id="S3.SS1.9.p3.1.m1.4.4.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.2"></in><ci id="S3.SS1.9.p3.1.m1.4.4.3.cmml" xref="S3.SS1.9.p3.1.m1.4.4.3">𝑔</ci><apply id="S3.SS1.9.p3.1.m1.4.4.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1"><times id="S3.SS1.9.p3.1.m1.4.4.1.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.2"></times><apply id="S3.SS1.9.p3.1.m1.4.4.1.3.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS1.9.p3.1.m1.4.4.1.3.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3">subscript</csymbol><apply id="S3.SS1.9.p3.1.m1.4.4.1.3.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS1.9.p3.1.m1.4.4.1.3.2.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3">superscript</csymbol><ci id="S3.SS1.9.p3.1.m1.4.4.1.3.2.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.2">𝐵</ci><apply id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3"><minus id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.1"></minus><ci id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.2">𝑠</ci><apply id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3"><divide id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3"></divide><apply id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2"><plus id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.1"></plus><ci id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.2">𝛾</ci><cn id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.3.cmml" type="integer" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.2.3">1</cn></apply><ci id="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.3.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.3.2.3.3.3">𝑝</ci></apply></apply></apply><list id="S3.SS1.9.p3.1.m1.2.2.2.3.cmml" xref="S3.SS1.9.p3.1.m1.2.2.2.4"><ci id="S3.SS1.9.p3.1.m1.1.1.1.1.cmml" xref="S3.SS1.9.p3.1.m1.1.1.1.1">𝑝</ci><ci id="S3.SS1.9.p3.1.m1.2.2.2.2.cmml" xref="S3.SS1.9.p3.1.m1.2.2.2.2">𝑞</ci></list></apply><list id="S3.SS1.9.p3.1.m1.4.4.1.1.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1"><apply id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1">superscript</csymbol><ci id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.2">ℝ</ci><apply id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3"><minus id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.1.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.1"></minus><ci id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.2.cmml" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.9.p3.1.m1.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.9.p3.1.m1.3.3.cmml" xref="S3.SS1.9.p3.1.m1.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.1.m1.4c">g\in B^{s-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.1.m1.4d">italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> and take <math alttext="(\rho_{n})_{n\geq 0}\in\Phi(\mathbb{R})" class="ltx_Math" display="inline" id="S3.SS1.9.p3.2.m2.2"><semantics id="S3.SS1.9.p3.2.m2.2a"><mrow id="S3.SS1.9.p3.2.m2.2.2" xref="S3.SS1.9.p3.2.m2.2.2.cmml"><msub id="S3.SS1.9.p3.2.m2.2.2.1" xref="S3.SS1.9.p3.2.m2.2.2.1.cmml"><mrow id="S3.SS1.9.p3.2.m2.2.2.1.1.1" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.cmml"><mo id="S3.SS1.9.p3.2.m2.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.cmml">(</mo><msub id="S3.SS1.9.p3.2.m2.2.2.1.1.1.1" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.cmml"><mi id="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.2" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.2.cmml">ρ</mi><mi id="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.3" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.3.cmml">n</mi></msub><mo id="S3.SS1.9.p3.2.m2.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS1.9.p3.2.m2.2.2.1.3" xref="S3.SS1.9.p3.2.m2.2.2.1.3.cmml"><mi id="S3.SS1.9.p3.2.m2.2.2.1.3.2" xref="S3.SS1.9.p3.2.m2.2.2.1.3.2.cmml">n</mi><mo id="S3.SS1.9.p3.2.m2.2.2.1.3.1" xref="S3.SS1.9.p3.2.m2.2.2.1.3.1.cmml">≥</mo><mn id="S3.SS1.9.p3.2.m2.2.2.1.3.3" xref="S3.SS1.9.p3.2.m2.2.2.1.3.3.cmml">0</mn></mrow></msub><mo id="S3.SS1.9.p3.2.m2.2.2.2" xref="S3.SS1.9.p3.2.m2.2.2.2.cmml">∈</mo><mrow id="S3.SS1.9.p3.2.m2.2.2.3" xref="S3.SS1.9.p3.2.m2.2.2.3.cmml"><mi id="S3.SS1.9.p3.2.m2.2.2.3.2" mathvariant="normal" xref="S3.SS1.9.p3.2.m2.2.2.3.2.cmml">Φ</mi><mo id="S3.SS1.9.p3.2.m2.2.2.3.1" xref="S3.SS1.9.p3.2.m2.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS1.9.p3.2.m2.2.2.3.3.2" xref="S3.SS1.9.p3.2.m2.2.2.3.cmml"><mo id="S3.SS1.9.p3.2.m2.2.2.3.3.2.1" stretchy="false" xref="S3.SS1.9.p3.2.m2.2.2.3.cmml">(</mo><mi id="S3.SS1.9.p3.2.m2.1.1" xref="S3.SS1.9.p3.2.m2.1.1.cmml">ℝ</mi><mo id="S3.SS1.9.p3.2.m2.2.2.3.3.2.2" stretchy="false" xref="S3.SS1.9.p3.2.m2.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.2.m2.2b"><apply id="S3.SS1.9.p3.2.m2.2.2.cmml" xref="S3.SS1.9.p3.2.m2.2.2"><in id="S3.SS1.9.p3.2.m2.2.2.2.cmml" xref="S3.SS1.9.p3.2.m2.2.2.2"></in><apply id="S3.SS1.9.p3.2.m2.2.2.1.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1"><csymbol cd="ambiguous" id="S3.SS1.9.p3.2.m2.2.2.1.2.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1">subscript</csymbol><apply id="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.1.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1">subscript</csymbol><ci id="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.2.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.2">𝜌</ci><ci id="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.3.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1.1.1.1.3">𝑛</ci></apply><apply id="S3.SS1.9.p3.2.m2.2.2.1.3.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1.3"><geq id="S3.SS1.9.p3.2.m2.2.2.1.3.1.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1.3.1"></geq><ci id="S3.SS1.9.p3.2.m2.2.2.1.3.2.cmml" xref="S3.SS1.9.p3.2.m2.2.2.1.3.2">𝑛</ci><cn id="S3.SS1.9.p3.2.m2.2.2.1.3.3.cmml" type="integer" xref="S3.SS1.9.p3.2.m2.2.2.1.3.3">0</cn></apply></apply><apply id="S3.SS1.9.p3.2.m2.2.2.3.cmml" xref="S3.SS1.9.p3.2.m2.2.2.3"><times id="S3.SS1.9.p3.2.m2.2.2.3.1.cmml" xref="S3.SS1.9.p3.2.m2.2.2.3.1"></times><ci id="S3.SS1.9.p3.2.m2.2.2.3.2.cmml" xref="S3.SS1.9.p3.2.m2.2.2.3.2">Φ</ci><ci id="S3.SS1.9.p3.2.m2.1.1.cmml" xref="S3.SS1.9.p3.2.m2.1.1">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.2.m2.2c">(\rho_{n})_{n\geq 0}\in\Phi(\mathbb{R})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.2.m2.2d">( italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R )</annotation></semantics></math> such that <math alttext="\rho_{1}(0)=2" class="ltx_Math" display="inline" id="S3.SS1.9.p3.3.m3.1"><semantics id="S3.SS1.9.p3.3.m3.1a"><mrow id="S3.SS1.9.p3.3.m3.1.2" xref="S3.SS1.9.p3.3.m3.1.2.cmml"><mrow id="S3.SS1.9.p3.3.m3.1.2.2" xref="S3.SS1.9.p3.3.m3.1.2.2.cmml"><msub id="S3.SS1.9.p3.3.m3.1.2.2.2" xref="S3.SS1.9.p3.3.m3.1.2.2.2.cmml"><mi id="S3.SS1.9.p3.3.m3.1.2.2.2.2" xref="S3.SS1.9.p3.3.m3.1.2.2.2.2.cmml">ρ</mi><mn id="S3.SS1.9.p3.3.m3.1.2.2.2.3" xref="S3.SS1.9.p3.3.m3.1.2.2.2.3.cmml">1</mn></msub><mo id="S3.SS1.9.p3.3.m3.1.2.2.1" xref="S3.SS1.9.p3.3.m3.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.9.p3.3.m3.1.2.2.3.2" xref="S3.SS1.9.p3.3.m3.1.2.2.cmml"><mo id="S3.SS1.9.p3.3.m3.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.9.p3.3.m3.1.2.2.cmml">(</mo><mn id="S3.SS1.9.p3.3.m3.1.1" xref="S3.SS1.9.p3.3.m3.1.1.cmml">0</mn><mo id="S3.SS1.9.p3.3.m3.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.9.p3.3.m3.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.9.p3.3.m3.1.2.1" xref="S3.SS1.9.p3.3.m3.1.2.1.cmml">=</mo><mn id="S3.SS1.9.p3.3.m3.1.2.3" xref="S3.SS1.9.p3.3.m3.1.2.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.3.m3.1b"><apply id="S3.SS1.9.p3.3.m3.1.2.cmml" xref="S3.SS1.9.p3.3.m3.1.2"><eq id="S3.SS1.9.p3.3.m3.1.2.1.cmml" xref="S3.SS1.9.p3.3.m3.1.2.1"></eq><apply id="S3.SS1.9.p3.3.m3.1.2.2.cmml" xref="S3.SS1.9.p3.3.m3.1.2.2"><times id="S3.SS1.9.p3.3.m3.1.2.2.1.cmml" xref="S3.SS1.9.p3.3.m3.1.2.2.1"></times><apply id="S3.SS1.9.p3.3.m3.1.2.2.2.cmml" xref="S3.SS1.9.p3.3.m3.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.9.p3.3.m3.1.2.2.2.1.cmml" xref="S3.SS1.9.p3.3.m3.1.2.2.2">subscript</csymbol><ci id="S3.SS1.9.p3.3.m3.1.2.2.2.2.cmml" xref="S3.SS1.9.p3.3.m3.1.2.2.2.2">𝜌</ci><cn id="S3.SS1.9.p3.3.m3.1.2.2.2.3.cmml" type="integer" xref="S3.SS1.9.p3.3.m3.1.2.2.2.3">1</cn></apply><cn id="S3.SS1.9.p3.3.m3.1.1.cmml" type="integer" xref="S3.SS1.9.p3.3.m3.1.1">0</cn></apply><cn id="S3.SS1.9.p3.3.m3.1.2.3.cmml" type="integer" xref="S3.SS1.9.p3.3.m3.1.2.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.3.m3.1c">\rho_{1}(0)=2</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.3.m3.1d">italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 0 ) = 2</annotation></semantics></math> (and therefore <math alttext="\rho_{n}(0)=2^{n}" class="ltx_Math" display="inline" id="S3.SS1.9.p3.4.m4.1"><semantics id="S3.SS1.9.p3.4.m4.1a"><mrow id="S3.SS1.9.p3.4.m4.1.2" xref="S3.SS1.9.p3.4.m4.1.2.cmml"><mrow id="S3.SS1.9.p3.4.m4.1.2.2" xref="S3.SS1.9.p3.4.m4.1.2.2.cmml"><msub id="S3.SS1.9.p3.4.m4.1.2.2.2" xref="S3.SS1.9.p3.4.m4.1.2.2.2.cmml"><mi id="S3.SS1.9.p3.4.m4.1.2.2.2.2" xref="S3.SS1.9.p3.4.m4.1.2.2.2.2.cmml">ρ</mi><mi id="S3.SS1.9.p3.4.m4.1.2.2.2.3" xref="S3.SS1.9.p3.4.m4.1.2.2.2.3.cmml">n</mi></msub><mo id="S3.SS1.9.p3.4.m4.1.2.2.1" xref="S3.SS1.9.p3.4.m4.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.9.p3.4.m4.1.2.2.3.2" xref="S3.SS1.9.p3.4.m4.1.2.2.cmml"><mo id="S3.SS1.9.p3.4.m4.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.9.p3.4.m4.1.2.2.cmml">(</mo><mn id="S3.SS1.9.p3.4.m4.1.1" xref="S3.SS1.9.p3.4.m4.1.1.cmml">0</mn><mo id="S3.SS1.9.p3.4.m4.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.9.p3.4.m4.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.9.p3.4.m4.1.2.1" xref="S3.SS1.9.p3.4.m4.1.2.1.cmml">=</mo><msup id="S3.SS1.9.p3.4.m4.1.2.3" xref="S3.SS1.9.p3.4.m4.1.2.3.cmml"><mn id="S3.SS1.9.p3.4.m4.1.2.3.2" xref="S3.SS1.9.p3.4.m4.1.2.3.2.cmml">2</mn><mi id="S3.SS1.9.p3.4.m4.1.2.3.3" xref="S3.SS1.9.p3.4.m4.1.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.4.m4.1b"><apply id="S3.SS1.9.p3.4.m4.1.2.cmml" xref="S3.SS1.9.p3.4.m4.1.2"><eq id="S3.SS1.9.p3.4.m4.1.2.1.cmml" xref="S3.SS1.9.p3.4.m4.1.2.1"></eq><apply id="S3.SS1.9.p3.4.m4.1.2.2.cmml" xref="S3.SS1.9.p3.4.m4.1.2.2"><times id="S3.SS1.9.p3.4.m4.1.2.2.1.cmml" xref="S3.SS1.9.p3.4.m4.1.2.2.1"></times><apply id="S3.SS1.9.p3.4.m4.1.2.2.2.cmml" xref="S3.SS1.9.p3.4.m4.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.9.p3.4.m4.1.2.2.2.1.cmml" xref="S3.SS1.9.p3.4.m4.1.2.2.2">subscript</csymbol><ci id="S3.SS1.9.p3.4.m4.1.2.2.2.2.cmml" xref="S3.SS1.9.p3.4.m4.1.2.2.2.2">𝜌</ci><ci id="S3.SS1.9.p3.4.m4.1.2.2.2.3.cmml" xref="S3.SS1.9.p3.4.m4.1.2.2.2.3">𝑛</ci></apply><cn id="S3.SS1.9.p3.4.m4.1.1.cmml" type="integer" xref="S3.SS1.9.p3.4.m4.1.1">0</cn></apply><apply id="S3.SS1.9.p3.4.m4.1.2.3.cmml" xref="S3.SS1.9.p3.4.m4.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.9.p3.4.m4.1.2.3.1.cmml" xref="S3.SS1.9.p3.4.m4.1.2.3">superscript</csymbol><cn id="S3.SS1.9.p3.4.m4.1.2.3.2.cmml" type="integer" xref="S3.SS1.9.p3.4.m4.1.2.3.2">2</cn><ci id="S3.SS1.9.p3.4.m4.1.2.3.3.cmml" xref="S3.SS1.9.p3.4.m4.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.4.m4.1c">\rho_{n}(0)=2^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.4.m4.1d">italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> for all <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S3.SS1.9.p3.5.m5.1"><semantics id="S3.SS1.9.p3.5.m5.1a"><mrow id="S3.SS1.9.p3.5.m5.1.1" xref="S3.SS1.9.p3.5.m5.1.1.cmml"><mi id="S3.SS1.9.p3.5.m5.1.1.2" xref="S3.SS1.9.p3.5.m5.1.1.2.cmml">n</mi><mo id="S3.SS1.9.p3.5.m5.1.1.1" xref="S3.SS1.9.p3.5.m5.1.1.1.cmml">≥</mo><mn id="S3.SS1.9.p3.5.m5.1.1.3" xref="S3.SS1.9.p3.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.5.m5.1b"><apply id="S3.SS1.9.p3.5.m5.1.1.cmml" xref="S3.SS1.9.p3.5.m5.1.1"><geq id="S3.SS1.9.p3.5.m5.1.1.1.cmml" xref="S3.SS1.9.p3.5.m5.1.1.1"></geq><ci id="S3.SS1.9.p3.5.m5.1.1.2.cmml" xref="S3.SS1.9.p3.5.m5.1.1.2">𝑛</ci><cn id="S3.SS1.9.p3.5.m5.1.1.3.cmml" type="integer" xref="S3.SS1.9.p3.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.5.m5.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.5.m5.1d">italic_n ≥ 0</annotation></semantics></math>). We set</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{ext}g(x_{1},\widetilde{x})=\sum_{n=0}^{\infty}2^{-n}\rho_{n}(x_{% 1})T_{n}g(\widetilde{x})\quad\text{ in }\SS^{\prime}(\mathbb{R}^{d};X)." class="ltx_Math" display="block" id="S3.Ex12.m1.4"><semantics id="S3.Ex12.m1.4a"><mrow id="S3.Ex12.m1.4.4.1" xref="S3.Ex12.m1.4.4.1.1.cmml"><mrow id="S3.Ex12.m1.4.4.1.1" xref="S3.Ex12.m1.4.4.1.1.cmml"><mrow id="S3.Ex12.m1.4.4.1.1.1" xref="S3.Ex12.m1.4.4.1.1.1.cmml"><mrow id="S3.Ex12.m1.4.4.1.1.1.3" xref="S3.Ex12.m1.4.4.1.1.1.3.cmml"><mi id="S3.Ex12.m1.4.4.1.1.1.3.1" xref="S3.Ex12.m1.4.4.1.1.1.3.1.cmml">ext</mi><mo id="S3.Ex12.m1.4.4.1.1.1.3a" lspace="0.167em" xref="S3.Ex12.m1.4.4.1.1.1.3.cmml">⁡</mo><mi id="S3.Ex12.m1.4.4.1.1.1.3.2" xref="S3.Ex12.m1.4.4.1.1.1.3.2.cmml">g</mi></mrow><mo id="S3.Ex12.m1.4.4.1.1.1.2" xref="S3.Ex12.m1.4.4.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex12.m1.4.4.1.1.1.1.1" xref="S3.Ex12.m1.4.4.1.1.1.1.2.cmml"><mo id="S3.Ex12.m1.4.4.1.1.1.1.1.2" stretchy="false" xref="S3.Ex12.m1.4.4.1.1.1.1.2.cmml">(</mo><msub id="S3.Ex12.m1.4.4.1.1.1.1.1.1" xref="S3.Ex12.m1.4.4.1.1.1.1.1.1.cmml"><mi id="S3.Ex12.m1.4.4.1.1.1.1.1.1.2" xref="S3.Ex12.m1.4.4.1.1.1.1.1.1.2.cmml">x</mi><mn id="S3.Ex12.m1.4.4.1.1.1.1.1.1.3" xref="S3.Ex12.m1.4.4.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.Ex12.m1.4.4.1.1.1.1.1.3" xref="S3.Ex12.m1.4.4.1.1.1.1.2.cmml">,</mo><mover accent="true" id="S3.Ex12.m1.1.1" xref="S3.Ex12.m1.1.1.cmml"><mi id="S3.Ex12.m1.1.1.2" xref="S3.Ex12.m1.1.1.2.cmml">x</mi><mo id="S3.Ex12.m1.1.1.1" xref="S3.Ex12.m1.1.1.1.cmml">~</mo></mover><mo id="S3.Ex12.m1.4.4.1.1.1.1.1.4" stretchy="false" xref="S3.Ex12.m1.4.4.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex12.m1.4.4.1.1.4" rspace="0.111em" xref="S3.Ex12.m1.4.4.1.1.4.cmml">=</mo><mrow id="S3.Ex12.m1.4.4.1.1.3.2" 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id="S3.Ex12.m1.3.3.cmml" xref="S3.Ex12.m1.3.3">𝑋</ci></list></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex12.m1.4c">\operatorname{ext}g(x_{1},\widetilde{x})=\sum_{n=0}^{\infty}2^{-n}\rho_{n}(x_{% 1})T_{n}g(\widetilde{x})\quad\text{ in }\SS^{\prime}(\mathbb{R}^{d};X).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex12.m1.4d">roman_ext italic_g ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_x end_ARG ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g ( over~ start_ARG italic_x end_ARG ) in roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.9.p3.9">Note that <math alttext="\operatorname{ext}" class="ltx_Math" display="inline" id="S3.SS1.9.p3.6.m1.1"><semantics id="S3.SS1.9.p3.6.m1.1a"><mi id="S3.SS1.9.p3.6.m1.1.1" xref="S3.SS1.9.p3.6.m1.1.1.cmml">ext</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.6.m1.1b"><ci id="S3.SS1.9.p3.6.m1.1.1.cmml" xref="S3.SS1.9.p3.6.m1.1.1">ext</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.6.m1.1c">\operatorname{ext}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.6.m1.1d">roman_ext</annotation></semantics></math> is independent of <math alttext="s,p,q,\gamma" class="ltx_Math" display="inline" id="S3.SS1.9.p3.7.m2.4"><semantics id="S3.SS1.9.p3.7.m2.4a"><mrow id="S3.SS1.9.p3.7.m2.4.5.2" xref="S3.SS1.9.p3.7.m2.4.5.1.cmml"><mi id="S3.SS1.9.p3.7.m2.1.1" xref="S3.SS1.9.p3.7.m2.1.1.cmml">s</mi><mo id="S3.SS1.9.p3.7.m2.4.5.2.1" xref="S3.SS1.9.p3.7.m2.4.5.1.cmml">,</mo><mi id="S3.SS1.9.p3.7.m2.2.2" xref="S3.SS1.9.p3.7.m2.2.2.cmml">p</mi><mo id="S3.SS1.9.p3.7.m2.4.5.2.2" xref="S3.SS1.9.p3.7.m2.4.5.1.cmml">,</mo><mi id="S3.SS1.9.p3.7.m2.3.3" xref="S3.SS1.9.p3.7.m2.3.3.cmml">q</mi><mo id="S3.SS1.9.p3.7.m2.4.5.2.3" xref="S3.SS1.9.p3.7.m2.4.5.1.cmml">,</mo><mi id="S3.SS1.9.p3.7.m2.4.4" xref="S3.SS1.9.p3.7.m2.4.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.7.m2.4b"><list id="S3.SS1.9.p3.7.m2.4.5.1.cmml" xref="S3.SS1.9.p3.7.m2.4.5.2"><ci id="S3.SS1.9.p3.7.m2.1.1.cmml" xref="S3.SS1.9.p3.7.m2.1.1">𝑠</ci><ci id="S3.SS1.9.p3.7.m2.2.2.cmml" xref="S3.SS1.9.p3.7.m2.2.2">𝑝</ci><ci id="S3.SS1.9.p3.7.m2.3.3.cmml" xref="S3.SS1.9.p3.7.m2.3.3">𝑞</ci><ci id="S3.SS1.9.p3.7.m2.4.4.cmml" xref="S3.SS1.9.p3.7.m2.4.4">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.7.m2.4c">s,p,q,\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.7.m2.4d">italic_s , italic_p , italic_q , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S3.SS1.9.p3.8.m3.1"><semantics id="S3.SS1.9.p3.8.m3.1a"><mi id="S3.SS1.9.p3.8.m3.1.1" xref="S3.SS1.9.p3.8.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.8.m3.1b"><ci id="S3.SS1.9.p3.8.m3.1.1.cmml" xref="S3.SS1.9.p3.8.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.8.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.8.m3.1d">italic_X</annotation></semantics></math>. Moreover, since <math alttext="\rho_{n}(x_{1})=2^{n-1}\rho_{1}(2^{n-1}x_{1})" class="ltx_Math" display="inline" id="S3.SS1.9.p3.9.m4.2"><semantics id="S3.SS1.9.p3.9.m4.2a"><mrow id="S3.SS1.9.p3.9.m4.2.2" xref="S3.SS1.9.p3.9.m4.2.2.cmml"><mrow id="S3.SS1.9.p3.9.m4.1.1.1" xref="S3.SS1.9.p3.9.m4.1.1.1.cmml"><msub id="S3.SS1.9.p3.9.m4.1.1.1.3" xref="S3.SS1.9.p3.9.m4.1.1.1.3.cmml"><mi id="S3.SS1.9.p3.9.m4.1.1.1.3.2" xref="S3.SS1.9.p3.9.m4.1.1.1.3.2.cmml">ρ</mi><mi id="S3.SS1.9.p3.9.m4.1.1.1.3.3" xref="S3.SS1.9.p3.9.m4.1.1.1.3.3.cmml">n</mi></msub><mo id="S3.SS1.9.p3.9.m4.1.1.1.2" xref="S3.SS1.9.p3.9.m4.1.1.1.2.cmml">⁢</mo><mrow id="S3.SS1.9.p3.9.m4.1.1.1.1.1" xref="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.cmml"><mo id="S3.SS1.9.p3.9.m4.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.cmml">(</mo><msub id="S3.SS1.9.p3.9.m4.1.1.1.1.1.1" xref="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.cmml"><mi id="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.2" xref="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.2.cmml">x</mi><mn id="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.3" xref="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS1.9.p3.9.m4.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.9.p3.9.m4.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.SS1.9.p3.9.m4.2.2.3" xref="S3.SS1.9.p3.9.m4.2.2.3.cmml">=</mo><mrow id="S3.SS1.9.p3.9.m4.2.2.2" xref="S3.SS1.9.p3.9.m4.2.2.2.cmml"><msup id="S3.SS1.9.p3.9.m4.2.2.2.3" xref="S3.SS1.9.p3.9.m4.2.2.2.3.cmml"><mn id="S3.SS1.9.p3.9.m4.2.2.2.3.2" xref="S3.SS1.9.p3.9.m4.2.2.2.3.2.cmml">2</mn><mrow id="S3.SS1.9.p3.9.m4.2.2.2.3.3" xref="S3.SS1.9.p3.9.m4.2.2.2.3.3.cmml"><mi id="S3.SS1.9.p3.9.m4.2.2.2.3.3.2" xref="S3.SS1.9.p3.9.m4.2.2.2.3.3.2.cmml">n</mi><mo id="S3.SS1.9.p3.9.m4.2.2.2.3.3.1" xref="S3.SS1.9.p3.9.m4.2.2.2.3.3.1.cmml">−</mo><mn id="S3.SS1.9.p3.9.m4.2.2.2.3.3.3" xref="S3.SS1.9.p3.9.m4.2.2.2.3.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.9.p3.9.m4.2.2.2.2" xref="S3.SS1.9.p3.9.m4.2.2.2.2.cmml">⁢</mo><msub id="S3.SS1.9.p3.9.m4.2.2.2.4" 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cd="ambiguous" id="S3.SS1.9.p3.9.m4.2.2.2.1.1.1.3.1.cmml" xref="S3.SS1.9.p3.9.m4.2.2.2.1.1.1.3">subscript</csymbol><ci id="S3.SS1.9.p3.9.m4.2.2.2.1.1.1.3.2.cmml" xref="S3.SS1.9.p3.9.m4.2.2.2.1.1.1.3.2">𝑥</ci><cn id="S3.SS1.9.p3.9.m4.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.9.p3.9.m4.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.9.m4.2c">\rho_{n}(x_{1})=2^{n-1}\rho_{1}(2^{n-1}x_{1})</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.9.m4.2d">italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( 2 start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math> it holds that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E2"> 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xref="S3.E2.m1.7.7.1.1.2.2.1"></in><ci id="S3.E2.m1.7.7.1.1.2.2.2.cmml" xref="S3.E2.m1.7.7.1.1.2.2.2">𝑛</ci><apply id="S3.E2.m1.7.7.1.1.2.2.3.cmml" xref="S3.E2.m1.7.7.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.E2.m1.7.7.1.1.2.2.3.1.cmml" xref="S3.E2.m1.7.7.1.1.2.2.3">subscript</csymbol><ci id="S3.E2.m1.7.7.1.1.2.2.3.2.cmml" xref="S3.E2.m1.7.7.1.1.2.2.3.2">ℕ</ci><cn id="S3.E2.m1.7.7.1.1.2.2.3.3.cmml" type="integer" xref="S3.E2.m1.7.7.1.1.2.2.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E2.m1.7c">\|\rho_{n}\|_{L^{p}(\mathbb{R},w_{\gamma})}=2^{(n-1)(1-\frac{\gamma+1}{p})}\|% \rho_{1}\|_{L^{p}(\mathbb{R},w_{\gamma})},\qquad n\in\mathbb{N}_{0}.</annotation><annotation encoding="application/x-llamapun" id="S3.E2.m1.7d">∥ italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT = 2 start_POSTSUPERSCRIPT ( italic_n - 1 ) ( 1 - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT , italic_n ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.9.p3.11">Therefore, using (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E2" title="In Proof. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.2</span></a>), Young’s inequality, <math alttext="s-\frac{\gamma+1}{p}>0" class="ltx_Math" display="inline" id="S3.SS1.9.p3.10.m1.1"><semantics id="S3.SS1.9.p3.10.m1.1a"><mrow id="S3.SS1.9.p3.10.m1.1.1" xref="S3.SS1.9.p3.10.m1.1.1.cmml"><mrow id="S3.SS1.9.p3.10.m1.1.1.2" xref="S3.SS1.9.p3.10.m1.1.1.2.cmml"><mi id="S3.SS1.9.p3.10.m1.1.1.2.2" xref="S3.SS1.9.p3.10.m1.1.1.2.2.cmml">s</mi><mo id="S3.SS1.9.p3.10.m1.1.1.2.1" xref="S3.SS1.9.p3.10.m1.1.1.2.1.cmml">−</mo><mfrac id="S3.SS1.9.p3.10.m1.1.1.2.3" xref="S3.SS1.9.p3.10.m1.1.1.2.3.cmml"><mrow id="S3.SS1.9.p3.10.m1.1.1.2.3.2" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2.cmml"><mi id="S3.SS1.9.p3.10.m1.1.1.2.3.2.2" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2.2.cmml">γ</mi><mo id="S3.SS1.9.p3.10.m1.1.1.2.3.2.1" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2.1.cmml">+</mo><mn id="S3.SS1.9.p3.10.m1.1.1.2.3.2.3" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S3.SS1.9.p3.10.m1.1.1.2.3.3" xref="S3.SS1.9.p3.10.m1.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S3.SS1.9.p3.10.m1.1.1.1" xref="S3.SS1.9.p3.10.m1.1.1.1.cmml">></mo><mn id="S3.SS1.9.p3.10.m1.1.1.3" xref="S3.SS1.9.p3.10.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.10.m1.1b"><apply id="S3.SS1.9.p3.10.m1.1.1.cmml" xref="S3.SS1.9.p3.10.m1.1.1"><gt id="S3.SS1.9.p3.10.m1.1.1.1.cmml" xref="S3.SS1.9.p3.10.m1.1.1.1"></gt><apply id="S3.SS1.9.p3.10.m1.1.1.2.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2"><minus id="S3.SS1.9.p3.10.m1.1.1.2.1.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.1"></minus><ci id="S3.SS1.9.p3.10.m1.1.1.2.2.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.2">𝑠</ci><apply id="S3.SS1.9.p3.10.m1.1.1.2.3.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.3"><divide id="S3.SS1.9.p3.10.m1.1.1.2.3.1.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.3"></divide><apply id="S3.SS1.9.p3.10.m1.1.1.2.3.2.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2"><plus id="S3.SS1.9.p3.10.m1.1.1.2.3.2.1.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2.1"></plus><ci id="S3.SS1.9.p3.10.m1.1.1.2.3.2.2.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2.2">𝛾</ci><cn id="S3.SS1.9.p3.10.m1.1.1.2.3.2.3.cmml" type="integer" xref="S3.SS1.9.p3.10.m1.1.1.2.3.2.3">1</cn></apply><ci id="S3.SS1.9.p3.10.m1.1.1.2.3.3.cmml" xref="S3.SS1.9.p3.10.m1.1.1.2.3.3">𝑝</ci></apply></apply><cn id="S3.SS1.9.p3.10.m1.1.1.3.cmml" type="integer" xref="S3.SS1.9.p3.10.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.10.m1.1c">s-\frac{\gamma+1}{p}>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.10.m1.1d">italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG > 0</annotation></semantics></math> and <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.SS1.9.p3.11.m2.1"><semantics id="S3.SS1.9.p3.11.m2.1a"><mrow id="S3.SS1.9.p3.11.m2.1.1" xref="S3.SS1.9.p3.11.m2.1.1.cmml"><mi id="S3.SS1.9.p3.11.m2.1.1.2" xref="S3.SS1.9.p3.11.m2.1.1.2.cmml">γ</mi><mo id="S3.SS1.9.p3.11.m2.1.1.1" xref="S3.SS1.9.p3.11.m2.1.1.1.cmml">></mo><mrow id="S3.SS1.9.p3.11.m2.1.1.3" xref="S3.SS1.9.p3.11.m2.1.1.3.cmml"><mo id="S3.SS1.9.p3.11.m2.1.1.3a" xref="S3.SS1.9.p3.11.m2.1.1.3.cmml">−</mo><mn id="S3.SS1.9.p3.11.m2.1.1.3.2" xref="S3.SS1.9.p3.11.m2.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.11.m2.1b"><apply id="S3.SS1.9.p3.11.m2.1.1.cmml" xref="S3.SS1.9.p3.11.m2.1.1"><gt id="S3.SS1.9.p3.11.m2.1.1.1.cmml" xref="S3.SS1.9.p3.11.m2.1.1.1"></gt><ci id="S3.SS1.9.p3.11.m2.1.1.2.cmml" xref="S3.SS1.9.p3.11.m2.1.1.2">𝛾</ci><apply id="S3.SS1.9.p3.11.m2.1.1.3.cmml" xref="S3.SS1.9.p3.11.m2.1.1.3"><minus id="S3.SS1.9.p3.11.m2.1.1.3.1.cmml" xref="S3.SS1.9.p3.11.m2.1.1.3"></minus><cn id="S3.SS1.9.p3.11.m2.1.1.3.2.cmml" type="integer" xref="S3.SS1.9.p3.11.m2.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.11.m2.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.11.m2.1d">italic_γ > - 1</annotation></semantics></math>, we find</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx6"> <tbody id="S3.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{n=0}^{\infty}2^{-n}\|\rho_{n}T_{n}g\|_{L^{p}(\mathbb{R}^{d}% ,w_{\gamma};X)}" class="ltx_Math" display="inline" id="S3.Ex13.m1.4"><semantics id="S3.Ex13.m1.4a"><mrow id="S3.Ex13.m1.4.4" xref="S3.Ex13.m1.4.4.cmml"><mstyle displaystyle="true" id="S3.Ex13.m1.4.4.2" xref="S3.Ex13.m1.4.4.2.cmml"><munderover id="S3.Ex13.m1.4.4.2a" xref="S3.Ex13.m1.4.4.2.cmml"><mo id="S3.Ex13.m1.4.4.2.2.2" movablelimits="false" 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xref="S3.Ex13.m1.2.2.2.2.1.1.2">ℝ</ci><ci id="S3.Ex13.m1.2.2.2.2.1.1.3.cmml" xref="S3.Ex13.m1.2.2.2.2.1.1.3">𝑑</ci></apply><apply id="S3.Ex13.m1.3.3.3.3.2.2.cmml" xref="S3.Ex13.m1.3.3.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex13.m1.3.3.3.3.2.2.1.cmml" xref="S3.Ex13.m1.3.3.3.3.2.2">subscript</csymbol><ci id="S3.Ex13.m1.3.3.3.3.2.2.2.cmml" xref="S3.Ex13.m1.3.3.3.3.2.2.2">𝑤</ci><ci id="S3.Ex13.m1.3.3.3.3.2.2.3.cmml" xref="S3.Ex13.m1.3.3.3.3.2.2.3">𝛾</ci></apply><ci id="S3.Ex13.m1.1.1.1.1.cmml" xref="S3.Ex13.m1.1.1.1.1">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex13.m1.4c">\displaystyle\sum_{n=0}^{\infty}2^{-n}\|\rho_{n}T_{n}g\|_{L^{p}(\mathbb{R}^{d}% ,w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex13.m1.4d">∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ∥ italic_ρ 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xref="S3.Ex13.m2.4.4.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex13.m2.4.4.2.2.1.1.1.cmml" xref="S3.Ex13.m2.4.4.2.2.1.1">superscript</csymbol><ci id="S3.Ex13.m2.4.4.2.2.1.1.2.cmml" xref="S3.Ex13.m2.4.4.2.2.1.1.2">ℝ</ci><apply id="S3.Ex13.m2.4.4.2.2.1.1.3.cmml" xref="S3.Ex13.m2.4.4.2.2.1.1.3"><minus id="S3.Ex13.m2.4.4.2.2.1.1.3.1.cmml" xref="S3.Ex13.m2.4.4.2.2.1.1.3.1"></minus><ci id="S3.Ex13.m2.4.4.2.2.1.1.3.2.cmml" xref="S3.Ex13.m2.4.4.2.2.1.1.3.2">𝑑</ci><cn id="S3.Ex13.m2.4.4.2.2.1.1.3.3.cmml" type="integer" xref="S3.Ex13.m2.4.4.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex13.m2.3.3.1.1.cmml" xref="S3.Ex13.m2.3.3.1.1">𝑋</ci></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex13.m2.6c">\displaystyle=\sum_{n=0}^{\infty}2^{-n}\|\rho_{n}\|_{L^{p}(\mathbb{R},w_{% \gamma})}\|T_{n}g\|_{L^{p}(\mathbb{R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex13.m2.6d">= ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ∥ italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ∥ italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq 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encoding="application/x-llamapun" id="S3.Ex14.m1.9d">≤ italic_C ∥ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - italic_n divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ≤ italic_C ∥ italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.9.p3.16">where the constant <math alttext="C" class="ltx_Math" display="inline" id="S3.SS1.9.p3.12.m1.1"><semantics id="S3.SS1.9.p3.12.m1.1a"><mi id="S3.SS1.9.p3.12.m1.1.1" xref="S3.SS1.9.p3.12.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.12.m1.1b"><ci id="S3.SS1.9.p3.12.m1.1.1.cmml" xref="S3.SS1.9.p3.12.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.12.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.12.m1.1d">italic_C</annotation></semantics></math> is independent of <math alttext="g" class="ltx_Math" display="inline" id="S3.SS1.9.p3.13.m2.1"><semantics id="S3.SS1.9.p3.13.m2.1a"><mi id="S3.SS1.9.p3.13.m2.1.1" xref="S3.SS1.9.p3.13.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.13.m2.1b"><ci id="S3.SS1.9.p3.13.m2.1.1.cmml" xref="S3.SS1.9.p3.13.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.13.m2.1c">g</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.13.m2.1d">italic_g</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.9.p3.14.m3.1"><semantics id="S3.SS1.9.p3.14.m3.1a"><mi id="S3.SS1.9.p3.14.m3.1.1" xref="S3.SS1.9.p3.14.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.14.m3.1b"><ci id="S3.SS1.9.p3.14.m3.1.1.cmml" xref="S3.SS1.9.p3.14.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.14.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.14.m3.1d">italic_n</annotation></semantics></math>. This shows that <math alttext="\operatorname{ext}g" class="ltx_Math" display="inline" id="S3.SS1.9.p3.15.m4.1"><semantics id="S3.SS1.9.p3.15.m4.1a"><mrow id="S3.SS1.9.p3.15.m4.1.1" xref="S3.SS1.9.p3.15.m4.1.1.cmml"><mi id="S3.SS1.9.p3.15.m4.1.1.1" xref="S3.SS1.9.p3.15.m4.1.1.1.cmml">ext</mi><mo id="S3.SS1.9.p3.15.m4.1.1a" lspace="0.167em" xref="S3.SS1.9.p3.15.m4.1.1.cmml">⁡</mo><mi id="S3.SS1.9.p3.15.m4.1.1.2" xref="S3.SS1.9.p3.15.m4.1.1.2.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.15.m4.1b"><apply id="S3.SS1.9.p3.15.m4.1.1.cmml" xref="S3.SS1.9.p3.15.m4.1.1"><ci id="S3.SS1.9.p3.15.m4.1.1.1.cmml" xref="S3.SS1.9.p3.15.m4.1.1.1">ext</ci><ci id="S3.SS1.9.p3.15.m4.1.1.2.cmml" xref="S3.SS1.9.p3.15.m4.1.1.2">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.15.m4.1c">\operatorname{ext}g</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.15.m4.1d">roman_ext italic_g</annotation></semantics></math> is well defined in <math alttext="L^{p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS1.9.p3.16.m5.3"><semantics id="S3.SS1.9.p3.16.m5.3a"><mrow id="S3.SS1.9.p3.16.m5.3.3" xref="S3.SS1.9.p3.16.m5.3.3.cmml"><msup id="S3.SS1.9.p3.16.m5.3.3.4" xref="S3.SS1.9.p3.16.m5.3.3.4.cmml"><mi id="S3.SS1.9.p3.16.m5.3.3.4.2" xref="S3.SS1.9.p3.16.m5.3.3.4.2.cmml">L</mi><mi id="S3.SS1.9.p3.16.m5.3.3.4.3" xref="S3.SS1.9.p3.16.m5.3.3.4.3.cmml">p</mi></msup><mo id="S3.SS1.9.p3.16.m5.3.3.3" xref="S3.SS1.9.p3.16.m5.3.3.3.cmml">⁢</mo><mrow id="S3.SS1.9.p3.16.m5.3.3.2.2" xref="S3.SS1.9.p3.16.m5.3.3.2.3.cmml"><mo id="S3.SS1.9.p3.16.m5.3.3.2.2.3" stretchy="false" xref="S3.SS1.9.p3.16.m5.3.3.2.3.cmml">(</mo><msup id="S3.SS1.9.p3.16.m5.2.2.1.1.1" xref="S3.SS1.9.p3.16.m5.2.2.1.1.1.cmml"><mi id="S3.SS1.9.p3.16.m5.2.2.1.1.1.2" xref="S3.SS1.9.p3.16.m5.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.9.p3.16.m5.2.2.1.1.1.3" xref="S3.SS1.9.p3.16.m5.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.9.p3.16.m5.3.3.2.2.4" xref="S3.SS1.9.p3.16.m5.3.3.2.3.cmml">,</mo><msub id="S3.SS1.9.p3.16.m5.3.3.2.2.2" xref="S3.SS1.9.p3.16.m5.3.3.2.2.2.cmml"><mi id="S3.SS1.9.p3.16.m5.3.3.2.2.2.2" xref="S3.SS1.9.p3.16.m5.3.3.2.2.2.2.cmml">w</mi><mi id="S3.SS1.9.p3.16.m5.3.3.2.2.2.3" xref="S3.SS1.9.p3.16.m5.3.3.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS1.9.p3.16.m5.3.3.2.2.5" xref="S3.SS1.9.p3.16.m5.3.3.2.3.cmml">;</mo><mi id="S3.SS1.9.p3.16.m5.1.1" xref="S3.SS1.9.p3.16.m5.1.1.cmml">X</mi><mo id="S3.SS1.9.p3.16.m5.3.3.2.2.6" stretchy="false" xref="S3.SS1.9.p3.16.m5.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.16.m5.3b"><apply id="S3.SS1.9.p3.16.m5.3.3.cmml" xref="S3.SS1.9.p3.16.m5.3.3"><times id="S3.SS1.9.p3.16.m5.3.3.3.cmml" xref="S3.SS1.9.p3.16.m5.3.3.3"></times><apply id="S3.SS1.9.p3.16.m5.3.3.4.cmml" xref="S3.SS1.9.p3.16.m5.3.3.4"><csymbol cd="ambiguous" id="S3.SS1.9.p3.16.m5.3.3.4.1.cmml" xref="S3.SS1.9.p3.16.m5.3.3.4">superscript</csymbol><ci id="S3.SS1.9.p3.16.m5.3.3.4.2.cmml" xref="S3.SS1.9.p3.16.m5.3.3.4.2">𝐿</ci><ci id="S3.SS1.9.p3.16.m5.3.3.4.3.cmml" xref="S3.SS1.9.p3.16.m5.3.3.4.3">𝑝</ci></apply><vector id="S3.SS1.9.p3.16.m5.3.3.2.3.cmml" xref="S3.SS1.9.p3.16.m5.3.3.2.2"><apply id="S3.SS1.9.p3.16.m5.2.2.1.1.1.cmml" xref="S3.SS1.9.p3.16.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.9.p3.16.m5.2.2.1.1.1.1.cmml" xref="S3.SS1.9.p3.16.m5.2.2.1.1.1">superscript</csymbol><ci id="S3.SS1.9.p3.16.m5.2.2.1.1.1.2.cmml" xref="S3.SS1.9.p3.16.m5.2.2.1.1.1.2">ℝ</ci><ci id="S3.SS1.9.p3.16.m5.2.2.1.1.1.3.cmml" xref="S3.SS1.9.p3.16.m5.2.2.1.1.1.3">𝑑</ci></apply><apply id="S3.SS1.9.p3.16.m5.3.3.2.2.2.cmml" xref="S3.SS1.9.p3.16.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.9.p3.16.m5.3.3.2.2.2.1.cmml" xref="S3.SS1.9.p3.16.m5.3.3.2.2.2">subscript</csymbol><ci id="S3.SS1.9.p3.16.m5.3.3.2.2.2.2.cmml" xref="S3.SS1.9.p3.16.m5.3.3.2.2.2.2">𝑤</ci><ci id="S3.SS1.9.p3.16.m5.3.3.2.2.2.3.cmml" xref="S3.SS1.9.p3.16.m5.3.3.2.2.2.3">𝛾</ci></apply><ci id="S3.SS1.9.p3.16.m5.1.1.cmml" xref="S3.SS1.9.p3.16.m5.1.1">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.16.m5.3c">L^{p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.16.m5.3d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. Now (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E3" title="In 2.2. Weighted function spaces ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E2" title="In Proof. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.2</span></a>) imply that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx7"> <tbody id="S3.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|S_{n}\operatorname{ext}g\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S3.Ex15.m1.4"><semantics id="S3.Ex15.m1.4a"><msub id="S3.Ex15.m1.4.4" xref="S3.Ex15.m1.4.4.cmml"><mrow id="S3.Ex15.m1.4.4.1.1" xref="S3.Ex15.m1.4.4.1.2.cmml"><mo id="S3.Ex15.m1.4.4.1.1.2" stretchy="false" xref="S3.Ex15.m1.4.4.1.2.1.cmml">‖</mo><mrow id="S3.Ex15.m1.4.4.1.1.1" xref="S3.Ex15.m1.4.4.1.1.1.cmml"><msub id="S3.Ex15.m1.4.4.1.1.1.2" xref="S3.Ex15.m1.4.4.1.1.1.2.cmml"><mi id="S3.Ex15.m1.4.4.1.1.1.2.2" xref="S3.Ex15.m1.4.4.1.1.1.2.2.cmml">S</mi><mi id="S3.Ex15.m1.4.4.1.1.1.2.3" xref="S3.Ex15.m1.4.4.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.Ex15.m1.4.4.1.1.1.1" lspace="0.167em" xref="S3.Ex15.m1.4.4.1.1.1.1.cmml">⁢</mo><mrow id="S3.Ex15.m1.4.4.1.1.1.3" xref="S3.Ex15.m1.4.4.1.1.1.3.cmml"><mi id="S3.Ex15.m1.4.4.1.1.1.3.1" xref="S3.Ex15.m1.4.4.1.1.1.3.1.cmml">ext</mi><mo id="S3.Ex15.m1.4.4.1.1.1.3a" lspace="0.167em" xref="S3.Ex15.m1.4.4.1.1.1.3.cmml">⁡</mo><mi id="S3.Ex15.m1.4.4.1.1.1.3.2" xref="S3.Ex15.m1.4.4.1.1.1.3.2.cmml">g</mi></mrow></mrow><mo id="S3.Ex15.m1.4.4.1.1.3" stretchy="false" xref="S3.Ex15.m1.4.4.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex15.m1.3.3.3" xref="S3.Ex15.m1.3.3.3.cmml"><msup id="S3.Ex15.m1.3.3.3.5" xref="S3.Ex15.m1.3.3.3.5.cmml"><mi id="S3.Ex15.m1.3.3.3.5.2" xref="S3.Ex15.m1.3.3.3.5.2.cmml">L</mi><mi id="S3.Ex15.m1.3.3.3.5.3" xref="S3.Ex15.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S3.Ex15.m1.3.3.3.4" xref="S3.Ex15.m1.3.3.3.4.cmml">⁢</mo><mrow id="S3.Ex15.m1.3.3.3.3.2" 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xref="S3.Ex15.m1.4.4.1.1.1.3.2">𝑔</ci></apply></apply></apply><apply id="S3.Ex15.m1.3.3.3.cmml" xref="S3.Ex15.m1.3.3.3"><times id="S3.Ex15.m1.3.3.3.4.cmml" xref="S3.Ex15.m1.3.3.3.4"></times><apply id="S3.Ex15.m1.3.3.3.5.cmml" xref="S3.Ex15.m1.3.3.3.5"><csymbol cd="ambiguous" id="S3.Ex15.m1.3.3.3.5.1.cmml" xref="S3.Ex15.m1.3.3.3.5">superscript</csymbol><ci id="S3.Ex15.m1.3.3.3.5.2.cmml" xref="S3.Ex15.m1.3.3.3.5.2">𝐿</ci><ci id="S3.Ex15.m1.3.3.3.5.3.cmml" xref="S3.Ex15.m1.3.3.3.5.3">𝑝</ci></apply><vector id="S3.Ex15.m1.3.3.3.3.3.cmml" xref="S3.Ex15.m1.3.3.3.3.2"><apply id="S3.Ex15.m1.2.2.2.2.1.1.cmml" xref="S3.Ex15.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Ex15.m1.2.2.2.2.1.1.1.cmml" xref="S3.Ex15.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S3.Ex15.m1.2.2.2.2.1.1.2.cmml" xref="S3.Ex15.m1.2.2.2.2.1.1.2">ℝ</ci><ci id="S3.Ex15.m1.2.2.2.2.1.1.3.cmml" xref="S3.Ex15.m1.2.2.2.2.1.1.3">𝑑</ci></apply><apply id="S3.Ex15.m1.3.3.3.3.2.2.cmml" xref="S3.Ex15.m1.3.3.3.3.2.2"><csymbol cd="ambiguous" id="S3.Ex15.m1.3.3.3.3.2.2.1.cmml" xref="S3.Ex15.m1.3.3.3.3.2.2">subscript</csymbol><ci id="S3.Ex15.m1.3.3.3.3.2.2.2.cmml" xref="S3.Ex15.m1.3.3.3.3.2.2.2">𝑤</ci><ci id="S3.Ex15.m1.3.3.3.3.2.2.3.cmml" xref="S3.Ex15.m1.3.3.3.3.2.2.3">𝛾</ci></apply><ci id="S3.Ex15.m1.1.1.1.1.cmml" xref="S3.Ex15.m1.1.1.1.1">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex15.m1.4c">\displaystyle\|S_{n}\operatorname{ext}g\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex15.m1.4d">∥ italic_S start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT roman_ext italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S3.Ex15.m2.4.4.1.1.cmml" xref="S3.Ex15.m2.4.4.1.1">𝑋</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex15.m2.7c">\displaystyle\leq C\sum_{j=-1}^{1}2^{-(n+j)}\|\rho_{n+j}\|_{L^{p}(\mathbb{R},w% _{\gamma})}\|T_{n+j}g\|_{L^{p}(\mathbb{R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex15.m2.7d">≤ italic_C ∑ start_POSTSUBSCRIPT italic_j = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - ( italic_n + italic_j ) end_POSTSUPERSCRIPT ∥ italic_ρ start_POSTSUBSCRIPT italic_n + italic_j end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ∥ italic_T start_POSTSUBSCRIPT italic_n + italic_j end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\sum_{j=-1}^{1}2^{-(n+j)\frac{\gamma+1}{p}}\|T_{n+j}g\|_{L^% {p}(\mathbb{R}^{d-1};X)}," class="ltx_Math" display="inline" id="S3.Ex16.m1.4"><semantics id="S3.Ex16.m1.4a"><mrow id="S3.Ex16.m1.4.4.1" xref="S3.Ex16.m1.4.4.1.1.cmml"><mrow id="S3.Ex16.m1.4.4.1.1" xref="S3.Ex16.m1.4.4.1.1.cmml"><mi id="S3.Ex16.m1.4.4.1.1.3" xref="S3.Ex16.m1.4.4.1.1.3.cmml"></mi><mo id="S3.Ex16.m1.4.4.1.1.2" xref="S3.Ex16.m1.4.4.1.1.2.cmml">≤</mo><mrow id="S3.Ex16.m1.4.4.1.1.1" xref="S3.Ex16.m1.4.4.1.1.1.cmml"><mi id="S3.Ex16.m1.4.4.1.1.1.3" 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xref="S3.Ex16.m1.3.3.2.2.1.1">superscript</csymbol><ci id="S3.Ex16.m1.3.3.2.2.1.1.2.cmml" xref="S3.Ex16.m1.3.3.2.2.1.1.2">ℝ</ci><apply id="S3.Ex16.m1.3.3.2.2.1.1.3.cmml" xref="S3.Ex16.m1.3.3.2.2.1.1.3"><minus id="S3.Ex16.m1.3.3.2.2.1.1.3.1.cmml" xref="S3.Ex16.m1.3.3.2.2.1.1.3.1"></minus><ci id="S3.Ex16.m1.3.3.2.2.1.1.3.2.cmml" xref="S3.Ex16.m1.3.3.2.2.1.1.3.2">𝑑</ci><cn id="S3.Ex16.m1.3.3.2.2.1.1.3.3.cmml" type="integer" xref="S3.Ex16.m1.3.3.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex16.m1.2.2.1.1.cmml" xref="S3.Ex16.m1.2.2.1.1">𝑋</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex16.m1.4c">\displaystyle\leq C\sum_{j=-1}^{1}2^{-(n+j)\frac{\gamma+1}{p}}\|T_{n+j}g\|_{L^% {p}(\mathbb{R}^{d-1};X)},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex16.m1.4d">≤ italic_C ∑ start_POSTSUBSCRIPT italic_j = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - ( italic_n + italic_j ) divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT italic_n + italic_j end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.9.p3.19">where the constant <math alttext="C" class="ltx_Math" display="inline" id="S3.SS1.9.p3.17.m1.1"><semantics id="S3.SS1.9.p3.17.m1.1a"><mi id="S3.SS1.9.p3.17.m1.1.1" xref="S3.SS1.9.p3.17.m1.1.1.cmml">C</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.17.m1.1b"><ci id="S3.SS1.9.p3.17.m1.1.1.cmml" xref="S3.SS1.9.p3.17.m1.1.1">𝐶</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.17.m1.1c">C</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.17.m1.1d">italic_C</annotation></semantics></math> is independent of <math alttext="g" class="ltx_Math" display="inline" id="S3.SS1.9.p3.18.m2.1"><semantics id="S3.SS1.9.p3.18.m2.1a"><mi id="S3.SS1.9.p3.18.m2.1.1" xref="S3.SS1.9.p3.18.m2.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.18.m2.1b"><ci id="S3.SS1.9.p3.18.m2.1.1.cmml" xref="S3.SS1.9.p3.18.m2.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.18.m2.1c">g</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.18.m2.1d">italic_g</annotation></semantics></math> and <math alttext="n" class="ltx_Math" display="inline" id="S3.SS1.9.p3.19.m3.1"><semantics id="S3.SS1.9.p3.19.m3.1a"><mi id="S3.SS1.9.p3.19.m3.1.1" xref="S3.SS1.9.p3.19.m3.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.19.m3.1b"><ci id="S3.SS1.9.p3.19.m3.1.1.cmml" xref="S3.SS1.9.p3.19.m3.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.19.m3.1c">n</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.19.m3.1d">italic_n</annotation></semantics></math>. This implies that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex17"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|\operatorname{ext}g\|_{B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)}\leq C\sum_{% j=-1}^{1}\big{\|}\big{(}2^{n(s-\frac{\gamma+1}{p})}\|T_{n+j}g\|_{L^{p}(\mathbb% {R}^{d-1};X)}\big{)}_{n\geq 0}\big{\|}_{\ell^{q}}\leq\|g\|_{B^{s-\frac{\gamma+% 1}{p}}_{p,q}(\mathbb{R}^{d-1};X)}." class="ltx_Math" display="block" id="S3.Ex17.m1.14"><semantics id="S3.Ex17.m1.14a"><mrow id="S3.Ex17.m1.14.14.1" xref="S3.Ex17.m1.14.14.1.1.cmml"><mrow id="S3.Ex17.m1.14.14.1.1" xref="S3.Ex17.m1.14.14.1.1.cmml"><msub id="S3.Ex17.m1.14.14.1.1.1" xref="S3.Ex17.m1.14.14.1.1.1.cmml"><mrow id="S3.Ex17.m1.14.14.1.1.1.1.1" xref="S3.Ex17.m1.14.14.1.1.1.1.2.cmml"><mo id="S3.Ex17.m1.14.14.1.1.1.1.1.2" stretchy="false" xref="S3.Ex17.m1.14.14.1.1.1.1.2.1.cmml">‖</mo><mrow 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id="S3.Ex17.m1.12.12.4.4.1.1.3.1.cmml" xref="S3.Ex17.m1.12.12.4.4.1.1.3.1"></minus><ci id="S3.Ex17.m1.12.12.4.4.1.1.3.2.cmml" xref="S3.Ex17.m1.12.12.4.4.1.1.3.2">𝑑</ci><cn id="S3.Ex17.m1.12.12.4.4.1.1.3.3.cmml" type="integer" xref="S3.Ex17.m1.12.12.4.4.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex17.m1.11.11.3.3.cmml" xref="S3.Ex17.m1.11.11.3.3">𝑋</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex17.m1.14c">\|\operatorname{ext}g\|_{B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)}\leq C\sum_{% j=-1}^{1}\big{\|}\big{(}2^{n(s-\frac{\gamma+1}{p})}\|T_{n+j}g\|_{L^{p}(\mathbb% {R}^{d-1};X)}\big{)}_{n\geq 0}\big{\|}_{\ell^{q}}\leq\|g\|_{B^{s-\frac{\gamma+% 1}{p}}_{p,q}(\mathbb{R}^{d-1};X)}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex17.m1.14d">∥ roman_ext italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∑ start_POSTSUBSCRIPT italic_j = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∥ ( 2 start_POSTSUPERSCRIPT italic_n ( italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT italic_n + italic_j end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT end_POSTSUBSCRIPT ≤ ∥ italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.9.p3.20">This proves the continuity of <math alttext="\operatorname{ext}" class="ltx_Math" display="inline" id="S3.SS1.9.p3.20.m1.1"><semantics id="S3.SS1.9.p3.20.m1.1a"><mi id="S3.SS1.9.p3.20.m1.1.1" xref="S3.SS1.9.p3.20.m1.1.1.cmml">ext</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.9.p3.20.m1.1b"><ci id="S3.SS1.9.p3.20.m1.1.1.cmml" xref="S3.SS1.9.p3.20.m1.1.1">ext</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.9.p3.20.m1.1c">\operatorname{ext}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.9.p3.20.m1.1d">roman_ext</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS1.10.p4"> <p class="ltx_p" id="S3.SS1.10.p4.12">Since <math alttext="\rho_{n}(0)=2^{n}" class="ltx_Math" display="inline" id="S3.SS1.10.p4.1.m1.1"><semantics id="S3.SS1.10.p4.1.m1.1a"><mrow id="S3.SS1.10.p4.1.m1.1.2" xref="S3.SS1.10.p4.1.m1.1.2.cmml"><mrow id="S3.SS1.10.p4.1.m1.1.2.2" xref="S3.SS1.10.p4.1.m1.1.2.2.cmml"><msub id="S3.SS1.10.p4.1.m1.1.2.2.2" xref="S3.SS1.10.p4.1.m1.1.2.2.2.cmml"><mi id="S3.SS1.10.p4.1.m1.1.2.2.2.2" xref="S3.SS1.10.p4.1.m1.1.2.2.2.2.cmml">ρ</mi><mi id="S3.SS1.10.p4.1.m1.1.2.2.2.3" xref="S3.SS1.10.p4.1.m1.1.2.2.2.3.cmml">n</mi></msub><mo id="S3.SS1.10.p4.1.m1.1.2.2.1" xref="S3.SS1.10.p4.1.m1.1.2.2.1.cmml">⁢</mo><mrow id="S3.SS1.10.p4.1.m1.1.2.2.3.2" xref="S3.SS1.10.p4.1.m1.1.2.2.cmml"><mo id="S3.SS1.10.p4.1.m1.1.2.2.3.2.1" stretchy="false" xref="S3.SS1.10.p4.1.m1.1.2.2.cmml">(</mo><mn id="S3.SS1.10.p4.1.m1.1.1" xref="S3.SS1.10.p4.1.m1.1.1.cmml">0</mn><mo id="S3.SS1.10.p4.1.m1.1.2.2.3.2.2" stretchy="false" xref="S3.SS1.10.p4.1.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.10.p4.1.m1.1.2.1" xref="S3.SS1.10.p4.1.m1.1.2.1.cmml">=</mo><msup id="S3.SS1.10.p4.1.m1.1.2.3" xref="S3.SS1.10.p4.1.m1.1.2.3.cmml"><mn id="S3.SS1.10.p4.1.m1.1.2.3.2" xref="S3.SS1.10.p4.1.m1.1.2.3.2.cmml">2</mn><mi id="S3.SS1.10.p4.1.m1.1.2.3.3" xref="S3.SS1.10.p4.1.m1.1.2.3.3.cmml">n</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.1.m1.1b"><apply id="S3.SS1.10.p4.1.m1.1.2.cmml" xref="S3.SS1.10.p4.1.m1.1.2"><eq id="S3.SS1.10.p4.1.m1.1.2.1.cmml" xref="S3.SS1.10.p4.1.m1.1.2.1"></eq><apply id="S3.SS1.10.p4.1.m1.1.2.2.cmml" xref="S3.SS1.10.p4.1.m1.1.2.2"><times id="S3.SS1.10.p4.1.m1.1.2.2.1.cmml" xref="S3.SS1.10.p4.1.m1.1.2.2.1"></times><apply id="S3.SS1.10.p4.1.m1.1.2.2.2.cmml" xref="S3.SS1.10.p4.1.m1.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.10.p4.1.m1.1.2.2.2.1.cmml" xref="S3.SS1.10.p4.1.m1.1.2.2.2">subscript</csymbol><ci id="S3.SS1.10.p4.1.m1.1.2.2.2.2.cmml" xref="S3.SS1.10.p4.1.m1.1.2.2.2.2">𝜌</ci><ci id="S3.SS1.10.p4.1.m1.1.2.2.2.3.cmml" xref="S3.SS1.10.p4.1.m1.1.2.2.2.3">𝑛</ci></apply><cn id="S3.SS1.10.p4.1.m1.1.1.cmml" type="integer" xref="S3.SS1.10.p4.1.m1.1.1">0</cn></apply><apply id="S3.SS1.10.p4.1.m1.1.2.3.cmml" xref="S3.SS1.10.p4.1.m1.1.2.3"><csymbol cd="ambiguous" id="S3.SS1.10.p4.1.m1.1.2.3.1.cmml" xref="S3.SS1.10.p4.1.m1.1.2.3">superscript</csymbol><cn id="S3.SS1.10.p4.1.m1.1.2.3.2.cmml" type="integer" xref="S3.SS1.10.p4.1.m1.1.2.3.2">2</cn><ci id="S3.SS1.10.p4.1.m1.1.2.3.3.cmml" xref="S3.SS1.10.p4.1.m1.1.2.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.1.m1.1c">\rho_{n}(0)=2^{n}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.1.m1.1d">italic_ρ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( 0 ) = 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> it follows that <math alttext="\operatorname{Tr}(\operatorname{ext}g)=g" class="ltx_Math" display="inline" id="S3.SS1.10.p4.2.m2.2"><semantics id="S3.SS1.10.p4.2.m2.2a"><mrow id="S3.SS1.10.p4.2.m2.2.2" xref="S3.SS1.10.p4.2.m2.2.2.cmml"><mrow id="S3.SS1.10.p4.2.m2.2.2.1.1" xref="S3.SS1.10.p4.2.m2.2.2.1.2.cmml"><mi id="S3.SS1.10.p4.2.m2.1.1" xref="S3.SS1.10.p4.2.m2.1.1.cmml">Tr</mi><mo id="S3.SS1.10.p4.2.m2.2.2.1.1a" xref="S3.SS1.10.p4.2.m2.2.2.1.2.cmml">⁡</mo><mrow id="S3.SS1.10.p4.2.m2.2.2.1.1.1" xref="S3.SS1.10.p4.2.m2.2.2.1.2.cmml"><mo id="S3.SS1.10.p4.2.m2.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.10.p4.2.m2.2.2.1.2.cmml">(</mo><mrow id="S3.SS1.10.p4.2.m2.2.2.1.1.1.1" xref="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.cmml"><mi id="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.1" xref="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.1.cmml">ext</mi><mo id="S3.SS1.10.p4.2.m2.2.2.1.1.1.1a" lspace="0.167em" xref="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.cmml">⁡</mo><mi id="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.2" xref="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.2.cmml">g</mi></mrow><mo id="S3.SS1.10.p4.2.m2.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.10.p4.2.m2.2.2.1.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.10.p4.2.m2.2.2.2" xref="S3.SS1.10.p4.2.m2.2.2.2.cmml">=</mo><mi id="S3.SS1.10.p4.2.m2.2.2.3" xref="S3.SS1.10.p4.2.m2.2.2.3.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.2.m2.2b"><apply id="S3.SS1.10.p4.2.m2.2.2.cmml" xref="S3.SS1.10.p4.2.m2.2.2"><eq id="S3.SS1.10.p4.2.m2.2.2.2.cmml" xref="S3.SS1.10.p4.2.m2.2.2.2"></eq><apply id="S3.SS1.10.p4.2.m2.2.2.1.2.cmml" xref="S3.SS1.10.p4.2.m2.2.2.1.1"><ci id="S3.SS1.10.p4.2.m2.1.1.cmml" xref="S3.SS1.10.p4.2.m2.1.1">Tr</ci><apply id="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.cmml" xref="S3.SS1.10.p4.2.m2.2.2.1.1.1.1"><ci id="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.1.cmml" xref="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.1">ext</ci><ci id="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.2.cmml" xref="S3.SS1.10.p4.2.m2.2.2.1.1.1.1.2">𝑔</ci></apply></apply><ci id="S3.SS1.10.p4.2.m2.2.2.3.cmml" xref="S3.SS1.10.p4.2.m2.2.2.3">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.2.m2.2c">\operatorname{Tr}(\operatorname{ext}g)=g</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.2.m2.2d">roman_Tr ( roman_ext italic_g ) = italic_g</annotation></semantics></math> for all <math alttext="g\in\SS(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS1.10.p4.3.m3.2"><semantics id="S3.SS1.10.p4.3.m3.2a"><mrow id="S3.SS1.10.p4.3.m3.2.2" xref="S3.SS1.10.p4.3.m3.2.2.cmml"><mi id="S3.SS1.10.p4.3.m3.2.2.3" xref="S3.SS1.10.p4.3.m3.2.2.3.cmml">g</mi><mo id="S3.SS1.10.p4.3.m3.2.2.2" xref="S3.SS1.10.p4.3.m3.2.2.2.cmml">∈</mo><mrow id="S3.SS1.10.p4.3.m3.2.2.1" xref="S3.SS1.10.p4.3.m3.2.2.1.cmml"><mi id="S3.SS1.10.p4.3.m3.2.2.1.3" xref="S3.SS1.10.p4.3.m3.2.2.1.3.cmml">SS</mi><mo id="S3.SS1.10.p4.3.m3.2.2.1.2" xref="S3.SS1.10.p4.3.m3.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS1.10.p4.3.m3.2.2.1.1.1" xref="S3.SS1.10.p4.3.m3.2.2.1.1.2.cmml"><mo id="S3.SS1.10.p4.3.m3.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.10.p4.3.m3.2.2.1.1.2.cmml">(</mo><msup id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.cmml"><mi id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.2" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.cmml"><mi id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.2" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.2.cmml">d</mi><mo id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.1" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.3" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.10.p4.3.m3.2.2.1.1.1.3" xref="S3.SS1.10.p4.3.m3.2.2.1.1.2.cmml">;</mo><mi id="S3.SS1.10.p4.3.m3.1.1" xref="S3.SS1.10.p4.3.m3.1.1.cmml">X</mi><mo id="S3.SS1.10.p4.3.m3.2.2.1.1.1.4" stretchy="false" xref="S3.SS1.10.p4.3.m3.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.3.m3.2b"><apply id="S3.SS1.10.p4.3.m3.2.2.cmml" xref="S3.SS1.10.p4.3.m3.2.2"><in id="S3.SS1.10.p4.3.m3.2.2.2.cmml" xref="S3.SS1.10.p4.3.m3.2.2.2"></in><ci id="S3.SS1.10.p4.3.m3.2.2.3.cmml" xref="S3.SS1.10.p4.3.m3.2.2.3">𝑔</ci><apply id="S3.SS1.10.p4.3.m3.2.2.1.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1"><times id="S3.SS1.10.p4.3.m3.2.2.1.2.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.2"></times><ci id="S3.SS1.10.p4.3.m3.2.2.1.3.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.3">SS</ci><list id="S3.SS1.10.p4.3.m3.2.2.1.1.2.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1"><apply id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.1.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1">superscript</csymbol><ci id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.2.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.2">ℝ</ci><apply id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3"><minus id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.1.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.1"></minus><ci id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.2.cmml" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.10.p4.3.m3.2.2.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.10.p4.3.m3.1.1.cmml" xref="S3.SS1.10.p4.3.m3.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.3.m3.2c">g\in\SS(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.3.m3.2d">italic_g ∈ roman_SS ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>. By density (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Lemma 3.8]</cite>) this extends to all <math alttext="g\in B^{s-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS1.10.p4.4.m4.4"><semantics id="S3.SS1.10.p4.4.m4.4a"><mrow id="S3.SS1.10.p4.4.m4.4.4" xref="S3.SS1.10.p4.4.m4.4.4.cmml"><mi id="S3.SS1.10.p4.4.m4.4.4.3" xref="S3.SS1.10.p4.4.m4.4.4.3.cmml">g</mi><mo id="S3.SS1.10.p4.4.m4.4.4.2" xref="S3.SS1.10.p4.4.m4.4.4.2.cmml">∈</mo><mrow id="S3.SS1.10.p4.4.m4.4.4.1" xref="S3.SS1.10.p4.4.m4.4.4.1.cmml"><msubsup id="S3.SS1.10.p4.4.m4.4.4.1.3" xref="S3.SS1.10.p4.4.m4.4.4.1.3.cmml"><mi id="S3.SS1.10.p4.4.m4.4.4.1.3.2.2" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.2.cmml">B</mi><mrow id="S3.SS1.10.p4.4.m4.2.2.2.4" xref="S3.SS1.10.p4.4.m4.2.2.2.3.cmml"><mi id="S3.SS1.10.p4.4.m4.1.1.1.1" xref="S3.SS1.10.p4.4.m4.1.1.1.1.cmml">p</mi><mo id="S3.SS1.10.p4.4.m4.2.2.2.4.1" xref="S3.SS1.10.p4.4.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS1.10.p4.4.m4.2.2.2.2" xref="S3.SS1.10.p4.4.m4.2.2.2.2.cmml">q</mi></mrow><mrow id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.cmml"><mi id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.2" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.2.cmml">s</mi><mo id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.1" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.cmml"><mrow id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.cmml"><mi id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.2" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.2.cmml">γ</mi><mo id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.1" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.1.cmml">+</mo><mn id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.3" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.3.cmml">1</mn></mrow><mi id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.3" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.SS1.10.p4.4.m4.4.4.1.2" xref="S3.SS1.10.p4.4.m4.4.4.1.2.cmml">⁢</mo><mrow id="S3.SS1.10.p4.4.m4.4.4.1.1.1" xref="S3.SS1.10.p4.4.m4.4.4.1.1.2.cmml"><mo id="S3.SS1.10.p4.4.m4.4.4.1.1.1.2" stretchy="false" xref="S3.SS1.10.p4.4.m4.4.4.1.1.2.cmml">(</mo><msup id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.cmml"><mi id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.2" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.cmml"><mi id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.2" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.2.cmml">d</mi><mo id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.1" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.3" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.10.p4.4.m4.4.4.1.1.1.3" xref="S3.SS1.10.p4.4.m4.4.4.1.1.2.cmml">;</mo><mi id="S3.SS1.10.p4.4.m4.3.3" xref="S3.SS1.10.p4.4.m4.3.3.cmml">X</mi><mo id="S3.SS1.10.p4.4.m4.4.4.1.1.1.4" stretchy="false" xref="S3.SS1.10.p4.4.m4.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.4.m4.4b"><apply id="S3.SS1.10.p4.4.m4.4.4.cmml" xref="S3.SS1.10.p4.4.m4.4.4"><in id="S3.SS1.10.p4.4.m4.4.4.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.2"></in><ci id="S3.SS1.10.p4.4.m4.4.4.3.cmml" xref="S3.SS1.10.p4.4.m4.4.4.3">𝑔</ci><apply id="S3.SS1.10.p4.4.m4.4.4.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1"><times id="S3.SS1.10.p4.4.m4.4.4.1.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.2"></times><apply id="S3.SS1.10.p4.4.m4.4.4.1.3.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS1.10.p4.4.m4.4.4.1.3.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3">subscript</csymbol><apply id="S3.SS1.10.p4.4.m4.4.4.1.3.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS1.10.p4.4.m4.4.4.1.3.2.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3">superscript</csymbol><ci id="S3.SS1.10.p4.4.m4.4.4.1.3.2.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.2">𝐵</ci><apply id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3"><minus id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.1"></minus><ci id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.2">𝑠</ci><apply id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3"><divide id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3"></divide><apply id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2"><plus id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.1"></plus><ci id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.2">𝛾</ci><cn id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.3.cmml" type="integer" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.2.3">1</cn></apply><ci id="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.3.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.3.2.3.3.3">𝑝</ci></apply></apply></apply><list id="S3.SS1.10.p4.4.m4.2.2.2.3.cmml" xref="S3.SS1.10.p4.4.m4.2.2.2.4"><ci id="S3.SS1.10.p4.4.m4.1.1.1.1.cmml" xref="S3.SS1.10.p4.4.m4.1.1.1.1">𝑝</ci><ci id="S3.SS1.10.p4.4.m4.2.2.2.2.cmml" xref="S3.SS1.10.p4.4.m4.2.2.2.2">𝑞</ci></list></apply><list id="S3.SS1.10.p4.4.m4.4.4.1.1.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1"><apply id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1">superscript</csymbol><ci id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.2">ℝ</ci><apply id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3"><minus id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.1.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.1"></minus><ci id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.2.cmml" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.10.p4.4.m4.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.10.p4.4.m4.3.3.cmml" xref="S3.SS1.10.p4.4.m4.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.4.m4.4c">g\in B^{s-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.4.m4.4d">italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> with <math alttext="q<\infty" class="ltx_Math" display="inline" id="S3.SS1.10.p4.5.m5.1"><semantics id="S3.SS1.10.p4.5.m5.1a"><mrow id="S3.SS1.10.p4.5.m5.1.1" xref="S3.SS1.10.p4.5.m5.1.1.cmml"><mi id="S3.SS1.10.p4.5.m5.1.1.2" xref="S3.SS1.10.p4.5.m5.1.1.2.cmml">q</mi><mo id="S3.SS1.10.p4.5.m5.1.1.1" xref="S3.SS1.10.p4.5.m5.1.1.1.cmml"><</mo><mi id="S3.SS1.10.p4.5.m5.1.1.3" mathvariant="normal" xref="S3.SS1.10.p4.5.m5.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.5.m5.1b"><apply id="S3.SS1.10.p4.5.m5.1.1.cmml" xref="S3.SS1.10.p4.5.m5.1.1"><lt id="S3.SS1.10.p4.5.m5.1.1.1.cmml" xref="S3.SS1.10.p4.5.m5.1.1.1"></lt><ci id="S3.SS1.10.p4.5.m5.1.1.2.cmml" xref="S3.SS1.10.p4.5.m5.1.1.2">𝑞</ci><infinity id="S3.SS1.10.p4.5.m5.1.1.3.cmml" xref="S3.SS1.10.p4.5.m5.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.5.m5.1c">q<\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.5.m5.1d">italic_q < ∞</annotation></semantics></math>. If <math alttext="q=\infty" class="ltx_Math" display="inline" id="S3.SS1.10.p4.6.m6.1"><semantics id="S3.SS1.10.p4.6.m6.1a"><mrow id="S3.SS1.10.p4.6.m6.1.1" xref="S3.SS1.10.p4.6.m6.1.1.cmml"><mi id="S3.SS1.10.p4.6.m6.1.1.2" xref="S3.SS1.10.p4.6.m6.1.1.2.cmml">q</mi><mo id="S3.SS1.10.p4.6.m6.1.1.1" xref="S3.SS1.10.p4.6.m6.1.1.1.cmml">=</mo><mi id="S3.SS1.10.p4.6.m6.1.1.3" mathvariant="normal" xref="S3.SS1.10.p4.6.m6.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.6.m6.1b"><apply id="S3.SS1.10.p4.6.m6.1.1.cmml" xref="S3.SS1.10.p4.6.m6.1.1"><eq id="S3.SS1.10.p4.6.m6.1.1.1.cmml" xref="S3.SS1.10.p4.6.m6.1.1.1"></eq><ci id="S3.SS1.10.p4.6.m6.1.1.2.cmml" xref="S3.SS1.10.p4.6.m6.1.1.2">𝑞</ci><infinity id="S3.SS1.10.p4.6.m6.1.1.3.cmml" xref="S3.SS1.10.p4.6.m6.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.6.m6.1c">q=\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.6.m6.1d">italic_q = ∞</annotation></semantics></math>, then we have <math alttext="B^{s-\frac{\gamma+1}{p}}_{p,\infty}(\mathbb{R}^{d-1};X)\hookrightarrow B^{s-% \frac{\gamma+1}{p}-\varepsilon}_{p,1}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS1.10.p4.7.m7.8"><semantics id="S3.SS1.10.p4.7.m7.8a"><mrow id="S3.SS1.10.p4.7.m7.8.8" xref="S3.SS1.10.p4.7.m7.8.8.cmml"><mrow id="S3.SS1.10.p4.7.m7.7.7.1" xref="S3.SS1.10.p4.7.m7.7.7.1.cmml"><msubsup id="S3.SS1.10.p4.7.m7.7.7.1.3" xref="S3.SS1.10.p4.7.m7.7.7.1.3.cmml"><mi id="S3.SS1.10.p4.7.m7.7.7.1.3.2.2" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.2.cmml">B</mi><mrow id="S3.SS1.10.p4.7.m7.2.2.2.4" xref="S3.SS1.10.p4.7.m7.2.2.2.3.cmml"><mi id="S3.SS1.10.p4.7.m7.1.1.1.1" xref="S3.SS1.10.p4.7.m7.1.1.1.1.cmml">p</mi><mo id="S3.SS1.10.p4.7.m7.2.2.2.4.1" xref="S3.SS1.10.p4.7.m7.2.2.2.3.cmml">,</mo><mi id="S3.SS1.10.p4.7.m7.2.2.2.2" mathvariant="normal" xref="S3.SS1.10.p4.7.m7.2.2.2.2.cmml">∞</mi></mrow><mrow id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.cmml"><mi id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.2" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.2.cmml">s</mi><mo id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.1" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.cmml"><mrow id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2.cmml"><mi id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2.2" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2.2.cmml">γ</mi><mo id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2.1" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2.1.cmml">+</mo><mn id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2.3" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.2.3.cmml">1</mn></mrow><mi id="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.3" xref="S3.SS1.10.p4.7.m7.7.7.1.3.2.3.3.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.SS1.10.p4.7.m7.7.7.1.2" xref="S3.SS1.10.p4.7.m7.7.7.1.2.cmml">⁢</mo><mrow id="S3.SS1.10.p4.7.m7.7.7.1.1.1" xref="S3.SS1.10.p4.7.m7.7.7.1.1.2.cmml"><mo id="S3.SS1.10.p4.7.m7.7.7.1.1.1.2" stretchy="false" xref="S3.SS1.10.p4.7.m7.7.7.1.1.2.cmml">(</mo><msup id="S3.SS1.10.p4.7.m7.7.7.1.1.1.1" xref="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.cmml"><mi id="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.2" xref="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3" xref="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3.cmml"><mi id="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3.2" xref="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3.2.cmml">d</mi><mo id="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3.1" xref="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3.3" xref="S3.SS1.10.p4.7.m7.7.7.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.10.p4.7.m7.7.7.1.1.1.3" xref="S3.SS1.10.p4.7.m7.7.7.1.1.2.cmml">;</mo><mi id="S3.SS1.10.p4.7.m7.5.5" xref="S3.SS1.10.p4.7.m7.5.5.cmml">X</mi><mo id="S3.SS1.10.p4.7.m7.7.7.1.1.1.4" stretchy="false" xref="S3.SS1.10.p4.7.m7.7.7.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.SS1.10.p4.7.m7.8.8.3" stretchy="false" xref="S3.SS1.10.p4.7.m7.8.8.3.cmml">↪</mo><mrow id="S3.SS1.10.p4.7.m7.8.8.2" xref="S3.SS1.10.p4.7.m7.8.8.2.cmml"><msubsup id="S3.SS1.10.p4.7.m7.8.8.2.3" xref="S3.SS1.10.p4.7.m7.8.8.2.3.cmml"><mi id="S3.SS1.10.p4.7.m7.8.8.2.3.2.2" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.2.cmml">B</mi><mrow id="S3.SS1.10.p4.7.m7.4.4.2.4" xref="S3.SS1.10.p4.7.m7.4.4.2.3.cmml"><mi id="S3.SS1.10.p4.7.m7.3.3.1.1" xref="S3.SS1.10.p4.7.m7.3.3.1.1.cmml">p</mi><mo id="S3.SS1.10.p4.7.m7.4.4.2.4.1" xref="S3.SS1.10.p4.7.m7.4.4.2.3.cmml">,</mo><mn id="S3.SS1.10.p4.7.m7.4.4.2.2" xref="S3.SS1.10.p4.7.m7.4.4.2.2.cmml">1</mn></mrow><mrow id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.cmml"><mi id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.2" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.2.cmml">s</mi><mo id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.1" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.1.cmml">−</mo><mfrac id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.cmml"><mrow id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.cmml"><mi id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.2" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.2.cmml">γ</mi><mo id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.1" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.1.cmml">+</mo><mn id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.3" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.3.cmml">1</mn></mrow><mi id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.3" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.3.cmml">p</mi></mfrac><mo id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.1a" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.1.cmml">−</mo><mi id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.4" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.4.cmml">ε</mi></mrow></msubsup><mo id="S3.SS1.10.p4.7.m7.8.8.2.2" xref="S3.SS1.10.p4.7.m7.8.8.2.2.cmml">⁢</mo><mrow id="S3.SS1.10.p4.7.m7.8.8.2.1.1" xref="S3.SS1.10.p4.7.m7.8.8.2.1.2.cmml"><mo id="S3.SS1.10.p4.7.m7.8.8.2.1.1.2" stretchy="false" xref="S3.SS1.10.p4.7.m7.8.8.2.1.2.cmml">(</mo><msup id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.cmml"><mi id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.2" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.cmml"><mi id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.2" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.2.cmml">d</mi><mo id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.1" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.1.cmml">−</mo><mn id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.3" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS1.10.p4.7.m7.8.8.2.1.1.3" xref="S3.SS1.10.p4.7.m7.8.8.2.1.2.cmml">;</mo><mi id="S3.SS1.10.p4.7.m7.6.6" xref="S3.SS1.10.p4.7.m7.6.6.cmml">X</mi><mo id="S3.SS1.10.p4.7.m7.8.8.2.1.1.4" stretchy="false" xref="S3.SS1.10.p4.7.m7.8.8.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.7.m7.8b"><apply id="S3.SS1.10.p4.7.m7.8.8.cmml" 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xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3"></divide><apply id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2"><plus id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.1.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.1"></plus><ci id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.2.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.2">𝛾</ci><cn id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.3.cmml" type="integer" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.2.3">1</cn></apply><ci id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.3.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.3.3">𝑝</ci></apply><ci id="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.4.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.3.2.3.4">𝜀</ci></apply></apply><list id="S3.SS1.10.p4.7.m7.4.4.2.3.cmml" xref="S3.SS1.10.p4.7.m7.4.4.2.4"><ci id="S3.SS1.10.p4.7.m7.3.3.1.1.cmml" xref="S3.SS1.10.p4.7.m7.3.3.1.1">𝑝</ci><cn id="S3.SS1.10.p4.7.m7.4.4.2.2.cmml" type="integer" xref="S3.SS1.10.p4.7.m7.4.4.2.2">1</cn></list></apply><list id="S3.SS1.10.p4.7.m7.8.8.2.1.2.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1"><apply id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.1.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1">superscript</csymbol><ci id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.2.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.2">ℝ</ci><apply id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3"><minus id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.1.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.1"></minus><ci id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.2.cmml" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.2">𝑑</ci><cn id="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.10.p4.7.m7.8.8.2.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.10.p4.7.m7.6.6.cmml" xref="S3.SS1.10.p4.7.m7.6.6">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.7.m7.8c">B^{s-\frac{\gamma+1}{p}}_{p,\infty}(\mathbb{R}^{d-1};X)\hookrightarrow B^{s-% \frac{\gamma+1}{p}-\varepsilon}_{p,1}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.7.m7.8d">italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ↪ italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG - italic_ε end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.SS1.10.p4.8.m8.1"><semantics id="S3.SS1.10.p4.8.m8.1a"><mrow id="S3.SS1.10.p4.8.m8.1.1" xref="S3.SS1.10.p4.8.m8.1.1.cmml"><mi id="S3.SS1.10.p4.8.m8.1.1.2" xref="S3.SS1.10.p4.8.m8.1.1.2.cmml">ε</mi><mo id="S3.SS1.10.p4.8.m8.1.1.1" xref="S3.SS1.10.p4.8.m8.1.1.1.cmml">></mo><mn id="S3.SS1.10.p4.8.m8.1.1.3" xref="S3.SS1.10.p4.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.8.m8.1b"><apply id="S3.SS1.10.p4.8.m8.1.1.cmml" xref="S3.SS1.10.p4.8.m8.1.1"><gt id="S3.SS1.10.p4.8.m8.1.1.1.cmml" xref="S3.SS1.10.p4.8.m8.1.1.1"></gt><ci id="S3.SS1.10.p4.8.m8.1.1.2.cmml" xref="S3.SS1.10.p4.8.m8.1.1.2">𝜀</ci><cn id="S3.SS1.10.p4.8.m8.1.1.3.cmml" type="integer" xref="S3.SS1.10.p4.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.8.m8.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.8.m8.1d">italic_ε > 0</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Theorem 14.4.19]</cite>). This shows that <math alttext="\operatorname{ext}" class="ltx_Math" display="inline" id="S3.SS1.10.p4.9.m9.1"><semantics id="S3.SS1.10.p4.9.m9.1a"><mi id="S3.SS1.10.p4.9.m9.1.1" xref="S3.SS1.10.p4.9.m9.1.1.cmml">ext</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.9.m9.1b"><ci id="S3.SS1.10.p4.9.m9.1.1.cmml" xref="S3.SS1.10.p4.9.m9.1.1">ext</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.9.m9.1c">\operatorname{ext}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.9.m9.1d">roman_ext</annotation></semantics></math> is also the right inverse for <math alttext="\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.SS1.10.p4.10.m10.1"><semantics id="S3.SS1.10.p4.10.m10.1a"><mi id="S3.SS1.10.p4.10.m10.1.1" xref="S3.SS1.10.p4.10.m10.1.1.cmml">Tr</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.10.m10.1b"><ci id="S3.SS1.10.p4.10.m10.1.1.cmml" xref="S3.SS1.10.p4.10.m10.1.1">Tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.10.m10.1c">\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.10.m10.1d">roman_Tr</annotation></semantics></math> in this case. Finally, the existence of a right inverse of <math alttext="\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.SS1.10.p4.11.m11.1"><semantics id="S3.SS1.10.p4.11.m11.1a"><mi id="S3.SS1.10.p4.11.m11.1.1" xref="S3.SS1.10.p4.11.m11.1.1.cmml">Tr</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.11.m11.1b"><ci id="S3.SS1.10.p4.11.m11.1.1.cmml" xref="S3.SS1.10.p4.11.m11.1.1">Tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.11.m11.1c">\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.11.m11.1d">roman_Tr</annotation></semantics></math> implies that <math alttext="\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.SS1.10.p4.12.m12.1"><semantics id="S3.SS1.10.p4.12.m12.1a"><mi id="S3.SS1.10.p4.12.m12.1.1" xref="S3.SS1.10.p4.12.m12.1.1.cmml">Tr</mi><annotation-xml encoding="MathML-Content" id="S3.SS1.10.p4.12.m12.1b"><ci id="S3.SS1.10.p4.12.m12.1.1.cmml" xref="S3.SS1.10.p4.12.m12.1.1">Tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.10.p4.12.m12.1c">\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.10.p4.12.m12.1d">roman_Tr</annotation></semantics></math> is surjective. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS1.p7"> <p class="ltx_p" id="S3.SS1.p7.1">Using Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem4" title="Theorem 3.4. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.4</span></a> and the Sobolev embeddings for Triebel-Lizorkin spaces from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.Thmtheorem3" title="Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>, we characterise the trace spaces for Triebel-Lizorkin spaces. In contrast to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib62" title="">62</a>, Lemma 4.15]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib29" title="">29</a>, Theorem 3.6]</cite>, the following proof is Fourier analytic and does not use an atomic approach. In the unweighted scalar-valued setting, other methods are used in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib68" title="">68</a>, Theorem 2.7.2]</cite> to obtain the trace space for Triebel-Lizorkin spaces.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.1.1.1">Theorem 3.5</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem5.p1"> <p class="ltx_p" id="S3.Thmtheorem5.p1.5"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem5.p1.5.5">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.1.1.m1.2"><semantics id="S3.Thmtheorem5.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem5.p1.1.1.m1.2.3" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem5.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem5.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem5.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem5.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem5.p1.1.1.m1.1.1" xref="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem5.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem5.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem5.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.1.1.m1.2b"><apply id="S3.Thmtheorem5.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.2.3"><in id="S3.Thmtheorem5.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.1"></in><ci id="S3.Thmtheorem5.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.2.3.3.2"><cn id="S3.Thmtheorem5.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem5.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem5.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.2.2.m2.2"><semantics id="S3.Thmtheorem5.p1.2.2.m2.2a"><mrow id="S3.Thmtheorem5.p1.2.2.m2.2.3" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.cmml"><mi id="S3.Thmtheorem5.p1.2.2.m2.2.3.2" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.2.cmml">q</mi><mo id="S3.Thmtheorem5.p1.2.2.m2.2.3.1" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem5.p1.2.2.m2.2.3.3.2" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.3.1.cmml"><mo id="S3.Thmtheorem5.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.3.1.cmml">[</mo><mn id="S3.Thmtheorem5.p1.2.2.m2.1.1" xref="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml">1</mn><mo id="S3.Thmtheorem5.p1.2.2.m2.2.3.3.2.2" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.2.2.m2.2.2" mathvariant="normal" xref="S3.Thmtheorem5.p1.2.2.m2.2.2.cmml">∞</mi><mo id="S3.Thmtheorem5.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.2.2.m2.2b"><apply id="S3.Thmtheorem5.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.2.3"><in id="S3.Thmtheorem5.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.1"></in><ci id="S3.Thmtheorem5.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S3.Thmtheorem5.p1.2.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.2.3.3.2"><cn id="S3.Thmtheorem5.p1.2.2.m2.1.1.cmml" type="integer" xref="S3.Thmtheorem5.p1.2.2.m2.1.1">1</cn><infinity id="S3.Thmtheorem5.p1.2.2.m2.2.2.cmml" xref="S3.Thmtheorem5.p1.2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.3.3.m3.1"><semantics id="S3.Thmtheorem5.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem5.p1.3.3.m3.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem5.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem5.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem5.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.cmml"><mo id="S3.Thmtheorem5.p1.3.3.m3.1.1.3a" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.3.3.m3.1b"><apply id="S3.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1"><gt id="S3.Thmtheorem5.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.1"></gt><ci id="S3.Thmtheorem5.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.2">𝛾</ci><apply id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3"><minus id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3"></minus><cn id="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2.cmml" type="integer" xref="S3.Thmtheorem5.p1.3.3.m3.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.3.3.m3.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.3.3.m3.1d">italic_γ > - 1</annotation></semantics></math>, <math alttext="s>\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.4.4.m4.1"><semantics id="S3.Thmtheorem5.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem5.p1.4.4.m4.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.2.cmml">s</mi><mo id="S3.Thmtheorem5.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1.cmml">></mo><mfrac id="S3.Thmtheorem5.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.cmml"><mrow id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.cmml"><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2.cmml">γ</mi><mo id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.1" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.1.cmml">+</mo><mn id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3.cmml">1</mn></mrow><mi id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.cmml">p</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.4.4.m4.1b"><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1"><gt id="S3.Thmtheorem5.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.1"></gt><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.2">𝑠</ci><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3"><divide id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3"></divide><apply id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2"><plus id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.1.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.1"></plus><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.2">𝛾</ci><cn id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3.cmml" type="integer" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.2.3">1</cn></apply><ci id="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3.cmml" xref="S3.Thmtheorem5.p1.4.4.m4.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.4.4.m4.1c">s>\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.4.4.m4.1d">italic_s > divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.5.5.m5.1"><semantics id="S3.Thmtheorem5.p1.5.5.m5.1a"><mi id="S3.Thmtheorem5.p1.5.5.m5.1.1" xref="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.5.5.m5.1b"><ci id="S3.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem5.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}:F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B_{p,p}^{s-\frac% {\gamma+1}{p}}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S3.Ex18.m1.9"><semantics id="S3.Ex18.m1.9a"><mrow id="S3.Ex18.m1.9.9" xref="S3.Ex18.m1.9.9.cmml"><mi id="S3.Ex18.m1.9.9.5" xref="S3.Ex18.m1.9.9.5.cmml">Tr</mi><mo id="S3.Ex18.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S3.Ex18.m1.9.9.4.cmml">:</mo><mrow id="S3.Ex18.m1.9.9.3" xref="S3.Ex18.m1.9.9.3.cmml"><mrow id="S3.Ex18.m1.8.8.2.2" xref="S3.Ex18.m1.8.8.2.2.cmml"><msubsup id="S3.Ex18.m1.8.8.2.2.4" xref="S3.Ex18.m1.8.8.2.2.4.cmml"><mi id="S3.Ex18.m1.8.8.2.2.4.2.2" xref="S3.Ex18.m1.8.8.2.2.4.2.2.cmml">F</mi><mrow id="S3.Ex18.m1.2.2.2.4" xref="S3.Ex18.m1.2.2.2.3.cmml"><mi id="S3.Ex18.m1.1.1.1.1" xref="S3.Ex18.m1.1.1.1.1.cmml">p</mi><mo id="S3.Ex18.m1.2.2.2.4.1" xref="S3.Ex18.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex18.m1.2.2.2.2" xref="S3.Ex18.m1.2.2.2.2.cmml">q</mi></mrow><mi id="S3.Ex18.m1.8.8.2.2.4.2.3" xref="S3.Ex18.m1.8.8.2.2.4.2.3.cmml">s</mi></msubsup><mo id="S3.Ex18.m1.8.8.2.2.3" xref="S3.Ex18.m1.8.8.2.2.3.cmml">⁢</mo><mrow id="S3.Ex18.m1.8.8.2.2.2.2" xref="S3.Ex18.m1.8.8.2.2.2.3.cmml"><mo id="S3.Ex18.m1.8.8.2.2.2.2.3" stretchy="false" xref="S3.Ex18.m1.8.8.2.2.2.3.cmml">(</mo><msup id="S3.Ex18.m1.7.7.1.1.1.1.1" xref="S3.Ex18.m1.7.7.1.1.1.1.1.cmml"><mi id="S3.Ex18.m1.7.7.1.1.1.1.1.2" xref="S3.Ex18.m1.7.7.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.Ex18.m1.7.7.1.1.1.1.1.3" xref="S3.Ex18.m1.7.7.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.Ex18.m1.8.8.2.2.2.2.4" xref="S3.Ex18.m1.8.8.2.2.2.3.cmml">,</mo><msub id="S3.Ex18.m1.8.8.2.2.2.2.2" xref="S3.Ex18.m1.8.8.2.2.2.2.2.cmml"><mi id="S3.Ex18.m1.8.8.2.2.2.2.2.2" xref="S3.Ex18.m1.8.8.2.2.2.2.2.2.cmml">w</mi><mi id="S3.Ex18.m1.8.8.2.2.2.2.2.3" xref="S3.Ex18.m1.8.8.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex18.m1.8.8.2.2.2.2.5" xref="S3.Ex18.m1.8.8.2.2.2.3.cmml">;</mo><mi id="S3.Ex18.m1.5.5" xref="S3.Ex18.m1.5.5.cmml">X</mi><mo id="S3.Ex18.m1.8.8.2.2.2.2.6" stretchy="false" xref="S3.Ex18.m1.8.8.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex18.m1.9.9.3.4" stretchy="false" xref="S3.Ex18.m1.9.9.3.4.cmml">→</mo><mrow id="S3.Ex18.m1.9.9.3.3" xref="S3.Ex18.m1.9.9.3.3.cmml"><msubsup id="S3.Ex18.m1.9.9.3.3.3" xref="S3.Ex18.m1.9.9.3.3.3.cmml"><mi id="S3.Ex18.m1.9.9.3.3.3.2.2" xref="S3.Ex18.m1.9.9.3.3.3.2.2.cmml">B</mi><mrow id="S3.Ex18.m1.4.4.2.4" xref="S3.Ex18.m1.4.4.2.3.cmml"><mi id="S3.Ex18.m1.3.3.1.1" xref="S3.Ex18.m1.3.3.1.1.cmml">p</mi><mo id="S3.Ex18.m1.4.4.2.4.1" xref="S3.Ex18.m1.4.4.2.3.cmml">,</mo><mi id="S3.Ex18.m1.4.4.2.2" xref="S3.Ex18.m1.4.4.2.2.cmml">p</mi></mrow><mrow id="S3.Ex18.m1.9.9.3.3.3.3" xref="S3.Ex18.m1.9.9.3.3.3.3.cmml"><mi id="S3.Ex18.m1.9.9.3.3.3.3.2" xref="S3.Ex18.m1.9.9.3.3.3.3.2.cmml">s</mi><mo id="S3.Ex18.m1.9.9.3.3.3.3.1" xref="S3.Ex18.m1.9.9.3.3.3.3.1.cmml">−</mo><mfrac id="S3.Ex18.m1.9.9.3.3.3.3.3" xref="S3.Ex18.m1.9.9.3.3.3.3.3.cmml"><mrow id="S3.Ex18.m1.9.9.3.3.3.3.3.2" xref="S3.Ex18.m1.9.9.3.3.3.3.3.2.cmml"><mi id="S3.Ex18.m1.9.9.3.3.3.3.3.2.2" xref="S3.Ex18.m1.9.9.3.3.3.3.3.2.2.cmml">γ</mi><mo id="S3.Ex18.m1.9.9.3.3.3.3.3.2.1" xref="S3.Ex18.m1.9.9.3.3.3.3.3.2.1.cmml">+</mo><mn id="S3.Ex18.m1.9.9.3.3.3.3.3.2.3" xref="S3.Ex18.m1.9.9.3.3.3.3.3.2.3.cmml">1</mn></mrow><mi id="S3.Ex18.m1.9.9.3.3.3.3.3.3" xref="S3.Ex18.m1.9.9.3.3.3.3.3.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.Ex18.m1.9.9.3.3.2" xref="S3.Ex18.m1.9.9.3.3.2.cmml">⁢</mo><mrow id="S3.Ex18.m1.9.9.3.3.1.1" xref="S3.Ex18.m1.9.9.3.3.1.2.cmml"><mo id="S3.Ex18.m1.9.9.3.3.1.1.2" stretchy="false" xref="S3.Ex18.m1.9.9.3.3.1.2.cmml">(</mo><msup id="S3.Ex18.m1.9.9.3.3.1.1.1" xref="S3.Ex18.m1.9.9.3.3.1.1.1.cmml"><mi id="S3.Ex18.m1.9.9.3.3.1.1.1.2" 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xref="S3.Ex18.m1.9.9.3.3.1.1.1.3.2">𝑑</ci><cn id="S3.Ex18.m1.9.9.3.3.1.1.1.3.3.cmml" type="integer" xref="S3.Ex18.m1.9.9.3.3.1.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex18.m1.6.6.cmml" xref="S3.Ex18.m1.6.6">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex18.m1.9c">\operatorname{Tr}:F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B_{p,p}^{s-\frac% {\gamma+1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex18.m1.9d">roman_Tr : italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem5.p1.9"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem5.p1.9.4">is a continuous and surjective operator. Moreover, there exists a continuous right inverse <math alttext="\operatorname{ext}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.6.1.m1.1"><semantics id="S3.Thmtheorem5.p1.6.1.m1.1a"><mi id="S3.Thmtheorem5.p1.6.1.m1.1.1" xref="S3.Thmtheorem5.p1.6.1.m1.1.1.cmml">ext</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.6.1.m1.1b"><ci id="S3.Thmtheorem5.p1.6.1.m1.1.1.cmml" xref="S3.Thmtheorem5.p1.6.1.m1.1.1">ext</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.6.1.m1.1c">\operatorname{ext}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.6.1.m1.1d">roman_ext</annotation></semantics></math> of <math alttext="\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.7.2.m2.1"><semantics id="S3.Thmtheorem5.p1.7.2.m2.1a"><mi id="S3.Thmtheorem5.p1.7.2.m2.1.1" xref="S3.Thmtheorem5.p1.7.2.m2.1.1.cmml">Tr</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.7.2.m2.1b"><ci id="S3.Thmtheorem5.p1.7.2.m2.1.1.cmml" xref="S3.Thmtheorem5.p1.7.2.m2.1.1">Tr</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.7.2.m2.1c">\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.7.2.m2.1d">roman_Tr</annotation></semantics></math> which is independent of <math alttext="s,p,q,\gamma" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.8.3.m3.4"><semantics id="S3.Thmtheorem5.p1.8.3.m3.4a"><mrow id="S3.Thmtheorem5.p1.8.3.m3.4.5.2" xref="S3.Thmtheorem5.p1.8.3.m3.4.5.1.cmml"><mi id="S3.Thmtheorem5.p1.8.3.m3.1.1" xref="S3.Thmtheorem5.p1.8.3.m3.1.1.cmml">s</mi><mo id="S3.Thmtheorem5.p1.8.3.m3.4.5.2.1" xref="S3.Thmtheorem5.p1.8.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.8.3.m3.2.2" xref="S3.Thmtheorem5.p1.8.3.m3.2.2.cmml">p</mi><mo id="S3.Thmtheorem5.p1.8.3.m3.4.5.2.2" xref="S3.Thmtheorem5.p1.8.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.8.3.m3.3.3" xref="S3.Thmtheorem5.p1.8.3.m3.3.3.cmml">q</mi><mo id="S3.Thmtheorem5.p1.8.3.m3.4.5.2.3" xref="S3.Thmtheorem5.p1.8.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem5.p1.8.3.m3.4.4" xref="S3.Thmtheorem5.p1.8.3.m3.4.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.8.3.m3.4b"><list id="S3.Thmtheorem5.p1.8.3.m3.4.5.1.cmml" xref="S3.Thmtheorem5.p1.8.3.m3.4.5.2"><ci id="S3.Thmtheorem5.p1.8.3.m3.1.1.cmml" xref="S3.Thmtheorem5.p1.8.3.m3.1.1">𝑠</ci><ci id="S3.Thmtheorem5.p1.8.3.m3.2.2.cmml" xref="S3.Thmtheorem5.p1.8.3.m3.2.2">𝑝</ci><ci id="S3.Thmtheorem5.p1.8.3.m3.3.3.cmml" xref="S3.Thmtheorem5.p1.8.3.m3.3.3">𝑞</ci><ci id="S3.Thmtheorem5.p1.8.3.m3.4.4.cmml" xref="S3.Thmtheorem5.p1.8.3.m3.4.4">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.8.3.m3.4c">s,p,q,\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.8.3.m3.4d">italic_s , italic_p , italic_q , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem5.p1.9.4.m4.1"><semantics id="S3.Thmtheorem5.p1.9.4.m4.1a"><mi id="S3.Thmtheorem5.p1.9.4.m4.1.1" xref="S3.Thmtheorem5.p1.9.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem5.p1.9.4.m4.1b"><ci id="S3.Thmtheorem5.p1.9.4.m4.1.1.cmml" xref="S3.Thmtheorem5.p1.9.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem5.p1.9.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem5.p1.9.4.m4.1d">italic_X</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS1.12"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS1.11.p1"> <p class="ltx_p" id="S3.SS1.11.p1.6"><span class="ltx_text ltx_font_italic" id="S3.SS1.11.p1.6.1">Step 1: trace operator.</span> Let <math alttext="f\in 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xref="S3.SS1.11.p1.1.m1.5.5.2.4.3.cmml">s</mi></msubsup><mo id="S3.SS1.11.p1.1.m1.5.5.2.3" xref="S3.SS1.11.p1.1.m1.5.5.2.3.cmml">⁢</mo><mrow id="S3.SS1.11.p1.1.m1.5.5.2.2.2" xref="S3.SS1.11.p1.1.m1.5.5.2.2.3.cmml"><mo id="S3.SS1.11.p1.1.m1.5.5.2.2.2.3" stretchy="false" xref="S3.SS1.11.p1.1.m1.5.5.2.2.3.cmml">(</mo><msup id="S3.SS1.11.p1.1.m1.4.4.1.1.1.1" xref="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.cmml"><mi id="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.2" xref="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.3" xref="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS1.11.p1.1.m1.5.5.2.2.2.4" xref="S3.SS1.11.p1.1.m1.5.5.2.2.3.cmml">,</mo><msub id="S3.SS1.11.p1.1.m1.5.5.2.2.2.2" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.cmml"><mi id="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.2" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.2.cmml">w</mi><mi id="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.3" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS1.11.p1.1.m1.5.5.2.2.2.5" xref="S3.SS1.11.p1.1.m1.5.5.2.2.3.cmml">;</mo><mi id="S3.SS1.11.p1.1.m1.3.3" xref="S3.SS1.11.p1.1.m1.3.3.cmml">X</mi><mo id="S3.SS1.11.p1.1.m1.5.5.2.2.2.6" stretchy="false" xref="S3.SS1.11.p1.1.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.11.p1.1.m1.5b"><apply id="S3.SS1.11.p1.1.m1.5.5.cmml" xref="S3.SS1.11.p1.1.m1.5.5"><in id="S3.SS1.11.p1.1.m1.5.5.3.cmml" xref="S3.SS1.11.p1.1.m1.5.5.3"></in><ci id="S3.SS1.11.p1.1.m1.5.5.4.cmml" xref="S3.SS1.11.p1.1.m1.5.5.4">𝑓</ci><apply id="S3.SS1.11.p1.1.m1.5.5.2.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2"><times id="S3.SS1.11.p1.1.m1.5.5.2.3.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.3"></times><apply id="S3.SS1.11.p1.1.m1.5.5.2.4.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.4"><csymbol cd="ambiguous" id="S3.SS1.11.p1.1.m1.5.5.2.4.1.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.4">superscript</csymbol><apply id="S3.SS1.11.p1.1.m1.5.5.2.4.2.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.4"><csymbol cd="ambiguous" id="S3.SS1.11.p1.1.m1.5.5.2.4.2.1.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.4">subscript</csymbol><ci id="S3.SS1.11.p1.1.m1.5.5.2.4.2.2.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.4.2.2">𝐹</ci><list id="S3.SS1.11.p1.1.m1.2.2.2.3.cmml" xref="S3.SS1.11.p1.1.m1.2.2.2.4"><ci id="S3.SS1.11.p1.1.m1.1.1.1.1.cmml" xref="S3.SS1.11.p1.1.m1.1.1.1.1">𝑝</ci><ci id="S3.SS1.11.p1.1.m1.2.2.2.2.cmml" xref="S3.SS1.11.p1.1.m1.2.2.2.2">𝑞</ci></list></apply><ci id="S3.SS1.11.p1.1.m1.5.5.2.4.3.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.4.3">𝑠</ci></apply><vector id="S3.SS1.11.p1.1.m1.5.5.2.2.3.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2"><apply id="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.cmml" xref="S3.SS1.11.p1.1.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.1.cmml" xref="S3.SS1.11.p1.1.m1.4.4.1.1.1.1">superscript</csymbol><ci id="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.2.cmml" xref="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.2">ℝ</ci><ci id="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.3.cmml" xref="S3.SS1.11.p1.1.m1.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.1.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2.2">subscript</csymbol><ci id="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.2.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.2">𝑤</ci><ci id="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.3.cmml" xref="S3.SS1.11.p1.1.m1.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S3.SS1.11.p1.1.m1.3.3.cmml" xref="S3.SS1.11.p1.1.m1.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.11.p1.1.m1.5c">f\in F_{p,q}^{s}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.11.p1.1.m1.5d">italic_f ∈ italic_F start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> and for <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.SS1.11.p1.2.m2.1"><semantics id="S3.SS1.11.p1.2.m2.1a"><mrow id="S3.SS1.11.p1.2.m2.1.1" xref="S3.SS1.11.p1.2.m2.1.1.cmml"><mi id="S3.SS1.11.p1.2.m2.1.1.2" xref="S3.SS1.11.p1.2.m2.1.1.2.cmml">ε</mi><mo id="S3.SS1.11.p1.2.m2.1.1.1" xref="S3.SS1.11.p1.2.m2.1.1.1.cmml">></mo><mn id="S3.SS1.11.p1.2.m2.1.1.3" xref="S3.SS1.11.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.11.p1.2.m2.1b"><apply id="S3.SS1.11.p1.2.m2.1.1.cmml" xref="S3.SS1.11.p1.2.m2.1.1"><gt id="S3.SS1.11.p1.2.m2.1.1.1.cmml" xref="S3.SS1.11.p1.2.m2.1.1.1"></gt><ci id="S3.SS1.11.p1.2.m2.1.1.2.cmml" xref="S3.SS1.11.p1.2.m2.1.1.2">𝜀</ci><cn id="S3.SS1.11.p1.2.m2.1.1.3.cmml" type="integer" xref="S3.SS1.11.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.11.p1.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.11.p1.2.m2.1d">italic_ε > 0</annotation></semantics></math> small take <math alttext="s_{1}:=s-\frac{\varepsilon}{p}" class="ltx_Math" display="inline" id="S3.SS1.11.p1.3.m3.1"><semantics id="S3.SS1.11.p1.3.m3.1a"><mrow id="S3.SS1.11.p1.3.m3.1.1" xref="S3.SS1.11.p1.3.m3.1.1.cmml"><msub id="S3.SS1.11.p1.3.m3.1.1.2" xref="S3.SS1.11.p1.3.m3.1.1.2.cmml"><mi id="S3.SS1.11.p1.3.m3.1.1.2.2" xref="S3.SS1.11.p1.3.m3.1.1.2.2.cmml">s</mi><mn id="S3.SS1.11.p1.3.m3.1.1.2.3" xref="S3.SS1.11.p1.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.11.p1.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.11.p1.3.m3.1.1.1.cmml">:=</mo><mrow id="S3.SS1.11.p1.3.m3.1.1.3" xref="S3.SS1.11.p1.3.m3.1.1.3.cmml"><mi id="S3.SS1.11.p1.3.m3.1.1.3.2" xref="S3.SS1.11.p1.3.m3.1.1.3.2.cmml">s</mi><mo id="S3.SS1.11.p1.3.m3.1.1.3.1" xref="S3.SS1.11.p1.3.m3.1.1.3.1.cmml">−</mo><mfrac id="S3.SS1.11.p1.3.m3.1.1.3.3" xref="S3.SS1.11.p1.3.m3.1.1.3.3.cmml"><mi id="S3.SS1.11.p1.3.m3.1.1.3.3.2" xref="S3.SS1.11.p1.3.m3.1.1.3.3.2.cmml">ε</mi><mi id="S3.SS1.11.p1.3.m3.1.1.3.3.3" xref="S3.SS1.11.p1.3.m3.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.11.p1.3.m3.1b"><apply id="S3.SS1.11.p1.3.m3.1.1.cmml" xref="S3.SS1.11.p1.3.m3.1.1"><csymbol cd="latexml" id="S3.SS1.11.p1.3.m3.1.1.1.cmml" xref="S3.SS1.11.p1.3.m3.1.1.1">assign</csymbol><apply id="S3.SS1.11.p1.3.m3.1.1.2.cmml" xref="S3.SS1.11.p1.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.11.p1.3.m3.1.1.2.1.cmml" xref="S3.SS1.11.p1.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS1.11.p1.3.m3.1.1.2.2.cmml" xref="S3.SS1.11.p1.3.m3.1.1.2.2">𝑠</ci><cn id="S3.SS1.11.p1.3.m3.1.1.2.3.cmml" type="integer" xref="S3.SS1.11.p1.3.m3.1.1.2.3">1</cn></apply><apply id="S3.SS1.11.p1.3.m3.1.1.3.cmml" xref="S3.SS1.11.p1.3.m3.1.1.3"><minus id="S3.SS1.11.p1.3.m3.1.1.3.1.cmml" xref="S3.SS1.11.p1.3.m3.1.1.3.1"></minus><ci id="S3.SS1.11.p1.3.m3.1.1.3.2.cmml" xref="S3.SS1.11.p1.3.m3.1.1.3.2">𝑠</ci><apply id="S3.SS1.11.p1.3.m3.1.1.3.3.cmml" xref="S3.SS1.11.p1.3.m3.1.1.3.3"><divide id="S3.SS1.11.p1.3.m3.1.1.3.3.1.cmml" xref="S3.SS1.11.p1.3.m3.1.1.3.3"></divide><ci id="S3.SS1.11.p1.3.m3.1.1.3.3.2.cmml" xref="S3.SS1.11.p1.3.m3.1.1.3.3.2">𝜀</ci><ci id="S3.SS1.11.p1.3.m3.1.1.3.3.3.cmml" xref="S3.SS1.11.p1.3.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.11.p1.3.m3.1c">s_{1}:=s-\frac{\varepsilon}{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.11.p1.3.m3.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := italic_s - divide start_ARG italic_ε end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and <math alttext="\gamma_{1}:=\gamma-\varepsilon" class="ltx_Math" display="inline" id="S3.SS1.11.p1.4.m4.1"><semantics id="S3.SS1.11.p1.4.m4.1a"><mrow id="S3.SS1.11.p1.4.m4.1.1" xref="S3.SS1.11.p1.4.m4.1.1.cmml"><msub id="S3.SS1.11.p1.4.m4.1.1.2" xref="S3.SS1.11.p1.4.m4.1.1.2.cmml"><mi id="S3.SS1.11.p1.4.m4.1.1.2.2" xref="S3.SS1.11.p1.4.m4.1.1.2.2.cmml">γ</mi><mn id="S3.SS1.11.p1.4.m4.1.1.2.3" xref="S3.SS1.11.p1.4.m4.1.1.2.3.cmml">1</mn></msub><mo id="S3.SS1.11.p1.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS1.11.p1.4.m4.1.1.1.cmml">:=</mo><mrow id="S3.SS1.11.p1.4.m4.1.1.3" xref="S3.SS1.11.p1.4.m4.1.1.3.cmml"><mi id="S3.SS1.11.p1.4.m4.1.1.3.2" xref="S3.SS1.11.p1.4.m4.1.1.3.2.cmml">γ</mi><mo id="S3.SS1.11.p1.4.m4.1.1.3.1" xref="S3.SS1.11.p1.4.m4.1.1.3.1.cmml">−</mo><mi id="S3.SS1.11.p1.4.m4.1.1.3.3" xref="S3.SS1.11.p1.4.m4.1.1.3.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.11.p1.4.m4.1b"><apply id="S3.SS1.11.p1.4.m4.1.1.cmml" xref="S3.SS1.11.p1.4.m4.1.1"><csymbol cd="latexml" id="S3.SS1.11.p1.4.m4.1.1.1.cmml" xref="S3.SS1.11.p1.4.m4.1.1.1">assign</csymbol><apply id="S3.SS1.11.p1.4.m4.1.1.2.cmml" xref="S3.SS1.11.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.11.p1.4.m4.1.1.2.1.cmml" xref="S3.SS1.11.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS1.11.p1.4.m4.1.1.2.2.cmml" xref="S3.SS1.11.p1.4.m4.1.1.2.2">𝛾</ci><cn id="S3.SS1.11.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S3.SS1.11.p1.4.m4.1.1.2.3">1</cn></apply><apply id="S3.SS1.11.p1.4.m4.1.1.3.cmml" xref="S3.SS1.11.p1.4.m4.1.1.3"><minus id="S3.SS1.11.p1.4.m4.1.1.3.1.cmml" xref="S3.SS1.11.p1.4.m4.1.1.3.1"></minus><ci id="S3.SS1.11.p1.4.m4.1.1.3.2.cmml" xref="S3.SS1.11.p1.4.m4.1.1.3.2">𝛾</ci><ci id="S3.SS1.11.p1.4.m4.1.1.3.3.cmml" xref="S3.SS1.11.p1.4.m4.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.11.p1.4.m4.1c">\gamma_{1}:=\gamma-\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.11.p1.4.m4.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := italic_γ - italic_ε</annotation></semantics></math>. Then <math alttext="\gamma_{1}/p<\gamma/p" class="ltx_Math" display="inline" id="S3.SS1.11.p1.5.m5.1"><semantics id="S3.SS1.11.p1.5.m5.1a"><mrow id="S3.SS1.11.p1.5.m5.1.1" xref="S3.SS1.11.p1.5.m5.1.1.cmml"><mrow id="S3.SS1.11.p1.5.m5.1.1.2" xref="S3.SS1.11.p1.5.m5.1.1.2.cmml"><msub id="S3.SS1.11.p1.5.m5.1.1.2.2" xref="S3.SS1.11.p1.5.m5.1.1.2.2.cmml"><mi id="S3.SS1.11.p1.5.m5.1.1.2.2.2" xref="S3.SS1.11.p1.5.m5.1.1.2.2.2.cmml">γ</mi><mn id="S3.SS1.11.p1.5.m5.1.1.2.2.3" xref="S3.SS1.11.p1.5.m5.1.1.2.2.3.cmml">1</mn></msub><mo id="S3.SS1.11.p1.5.m5.1.1.2.1" xref="S3.SS1.11.p1.5.m5.1.1.2.1.cmml">/</mo><mi id="S3.SS1.11.p1.5.m5.1.1.2.3" xref="S3.SS1.11.p1.5.m5.1.1.2.3.cmml">p</mi></mrow><mo id="S3.SS1.11.p1.5.m5.1.1.1" xref="S3.SS1.11.p1.5.m5.1.1.1.cmml"><</mo><mrow id="S3.SS1.11.p1.5.m5.1.1.3" xref="S3.SS1.11.p1.5.m5.1.1.3.cmml"><mi id="S3.SS1.11.p1.5.m5.1.1.3.2" xref="S3.SS1.11.p1.5.m5.1.1.3.2.cmml">γ</mi><mo id="S3.SS1.11.p1.5.m5.1.1.3.1" xref="S3.SS1.11.p1.5.m5.1.1.3.1.cmml">/</mo><mi id="S3.SS1.11.p1.5.m5.1.1.3.3" xref="S3.SS1.11.p1.5.m5.1.1.3.3.cmml">p</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.11.p1.5.m5.1b"><apply id="S3.SS1.11.p1.5.m5.1.1.cmml" xref="S3.SS1.11.p1.5.m5.1.1"><lt id="S3.SS1.11.p1.5.m5.1.1.1.cmml" xref="S3.SS1.11.p1.5.m5.1.1.1"></lt><apply id="S3.SS1.11.p1.5.m5.1.1.2.cmml" xref="S3.SS1.11.p1.5.m5.1.1.2"><divide id="S3.SS1.11.p1.5.m5.1.1.2.1.cmml" xref="S3.SS1.11.p1.5.m5.1.1.2.1"></divide><apply id="S3.SS1.11.p1.5.m5.1.1.2.2.cmml" xref="S3.SS1.11.p1.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS1.11.p1.5.m5.1.1.2.2.1.cmml" xref="S3.SS1.11.p1.5.m5.1.1.2.2">subscript</csymbol><ci id="S3.SS1.11.p1.5.m5.1.1.2.2.2.cmml" xref="S3.SS1.11.p1.5.m5.1.1.2.2.2">𝛾</ci><cn id="S3.SS1.11.p1.5.m5.1.1.2.2.3.cmml" type="integer" xref="S3.SS1.11.p1.5.m5.1.1.2.2.3">1</cn></apply><ci id="S3.SS1.11.p1.5.m5.1.1.2.3.cmml" xref="S3.SS1.11.p1.5.m5.1.1.2.3">𝑝</ci></apply><apply id="S3.SS1.11.p1.5.m5.1.1.3.cmml" xref="S3.SS1.11.p1.5.m5.1.1.3"><divide id="S3.SS1.11.p1.5.m5.1.1.3.1.cmml" xref="S3.SS1.11.p1.5.m5.1.1.3.1"></divide><ci id="S3.SS1.11.p1.5.m5.1.1.3.2.cmml" xref="S3.SS1.11.p1.5.m5.1.1.3.2">𝛾</ci><ci id="S3.SS1.11.p1.5.m5.1.1.3.3.cmml" xref="S3.SS1.11.p1.5.m5.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.11.p1.5.m5.1c">\gamma_{1}/p<\gamma/p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.11.p1.5.m5.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT / italic_p < italic_γ / italic_p</annotation></semantics></math> and <math alttext="s-(d+\gamma)/p=s_{1}-(d+\gamma_{1})/p" class="ltx_Math" display="inline" id="S3.SS1.11.p1.6.m6.2"><semantics id="S3.SS1.11.p1.6.m6.2a"><mrow id="S3.SS1.11.p1.6.m6.2.2" xref="S3.SS1.11.p1.6.m6.2.2.cmml"><mrow id="S3.SS1.11.p1.6.m6.1.1.1" xref="S3.SS1.11.p1.6.m6.1.1.1.cmml"><mi id="S3.SS1.11.p1.6.m6.1.1.1.3" xref="S3.SS1.11.p1.6.m6.1.1.1.3.cmml">s</mi><mo id="S3.SS1.11.p1.6.m6.1.1.1.2" xref="S3.SS1.11.p1.6.m6.1.1.1.2.cmml">−</mo><mrow id="S3.SS1.11.p1.6.m6.1.1.1.1" xref="S3.SS1.11.p1.6.m6.1.1.1.1.cmml"><mrow id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.cmml"><mo id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.cmml"><mi id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.2" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.2.cmml">d</mi><mo id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.1" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.3" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.3.cmml">γ</mi></mrow><mo id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.11.p1.6.m6.1.1.1.1.2" xref="S3.SS1.11.p1.6.m6.1.1.1.1.2.cmml">/</mo><mi id="S3.SS1.11.p1.6.m6.1.1.1.1.3" xref="S3.SS1.11.p1.6.m6.1.1.1.1.3.cmml">p</mi></mrow></mrow><mo id="S3.SS1.11.p1.6.m6.2.2.3" xref="S3.SS1.11.p1.6.m6.2.2.3.cmml">=</mo><mrow id="S3.SS1.11.p1.6.m6.2.2.2" xref="S3.SS1.11.p1.6.m6.2.2.2.cmml"><msub id="S3.SS1.11.p1.6.m6.2.2.2.3" xref="S3.SS1.11.p1.6.m6.2.2.2.3.cmml"><mi id="S3.SS1.11.p1.6.m6.2.2.2.3.2" xref="S3.SS1.11.p1.6.m6.2.2.2.3.2.cmml">s</mi><mn id="S3.SS1.11.p1.6.m6.2.2.2.3.3" xref="S3.SS1.11.p1.6.m6.2.2.2.3.3.cmml">1</mn></msub><mo id="S3.SS1.11.p1.6.m6.2.2.2.2" xref="S3.SS1.11.p1.6.m6.2.2.2.2.cmml">−</mo><mrow id="S3.SS1.11.p1.6.m6.2.2.2.1" xref="S3.SS1.11.p1.6.m6.2.2.2.1.cmml"><mrow id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.cmml"><mo id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.2" stretchy="false" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.cmml"><mi id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.2" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.2.cmml">d</mi><mo id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.1" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.1.cmml">+</mo><msub id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.cmml"><mi id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.2" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.2.cmml">γ</mi><mn id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.3" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.3" stretchy="false" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS1.11.p1.6.m6.2.2.2.1.2" xref="S3.SS1.11.p1.6.m6.2.2.2.1.2.cmml">/</mo><mi id="S3.SS1.11.p1.6.m6.2.2.2.1.3" xref="S3.SS1.11.p1.6.m6.2.2.2.1.3.cmml">p</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.11.p1.6.m6.2b"><apply id="S3.SS1.11.p1.6.m6.2.2.cmml" xref="S3.SS1.11.p1.6.m6.2.2"><eq id="S3.SS1.11.p1.6.m6.2.2.3.cmml" xref="S3.SS1.11.p1.6.m6.2.2.3"></eq><apply id="S3.SS1.11.p1.6.m6.1.1.1.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1"><minus id="S3.SS1.11.p1.6.m6.1.1.1.2.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.2"></minus><ci id="S3.SS1.11.p1.6.m6.1.1.1.3.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.3">𝑠</ci><apply id="S3.SS1.11.p1.6.m6.1.1.1.1.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.1"><divide id="S3.SS1.11.p1.6.m6.1.1.1.1.2.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.1.2"></divide><apply id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1"><plus id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.1.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.1"></plus><ci id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.2.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.2">𝑑</ci><ci id="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.3.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.1.1.1.1.3">𝛾</ci></apply><ci id="S3.SS1.11.p1.6.m6.1.1.1.1.3.cmml" xref="S3.SS1.11.p1.6.m6.1.1.1.1.3">𝑝</ci></apply></apply><apply id="S3.SS1.11.p1.6.m6.2.2.2.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2"><minus id="S3.SS1.11.p1.6.m6.2.2.2.2.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.2"></minus><apply id="S3.SS1.11.p1.6.m6.2.2.2.3.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.3"><csymbol cd="ambiguous" id="S3.SS1.11.p1.6.m6.2.2.2.3.1.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.3">subscript</csymbol><ci id="S3.SS1.11.p1.6.m6.2.2.2.3.2.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.3.2">𝑠</ci><cn id="S3.SS1.11.p1.6.m6.2.2.2.3.3.cmml" type="integer" xref="S3.SS1.11.p1.6.m6.2.2.2.3.3">1</cn></apply><apply id="S3.SS1.11.p1.6.m6.2.2.2.1.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1"><divide id="S3.SS1.11.p1.6.m6.2.2.2.1.2.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.2"></divide><apply id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1"><plus id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.1.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.1"></plus><ci id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.2.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.2">𝑑</ci><apply id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.1.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3">subscript</csymbol><ci id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.2.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.2">𝛾</ci><cn id="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.11.p1.6.m6.2.2.2.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.11.p1.6.m6.2.2.2.1.3.cmml" xref="S3.SS1.11.p1.6.m6.2.2.2.1.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.11.p1.6.m6.2c">s-(d+\gamma)/p=s_{1}-(d+\gamma_{1})/p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.11.p1.6.m6.2d">italic_s - ( italic_d + italic_γ ) / italic_p = italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - ( italic_d + italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) / italic_p</annotation></semantics></math>. So by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem4" title="Theorem 3.4. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.4</span></a>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E5" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.5</span></a>) and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.Thmtheorem3" title="Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx8"> <tbody id="S3.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\operatorname{Tr}f\|_{B^{s-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}% ^{d-1};X)}" class="ltx_Math" display="inline" id="S3.Ex19.m1.5"><semantics id="S3.Ex19.m1.5a"><msub id="S3.Ex19.m1.5.5" xref="S3.Ex19.m1.5.5.cmml"><mrow id="S3.Ex19.m1.5.5.1.1" xref="S3.Ex19.m1.5.5.1.2.cmml"><mo id="S3.Ex19.m1.5.5.1.1.2" stretchy="false" xref="S3.Ex19.m1.5.5.1.2.1.cmml">‖</mo><mrow id="S3.Ex19.m1.5.5.1.1.1" xref="S3.Ex19.m1.5.5.1.1.1.cmml"><mi id="S3.Ex19.m1.5.5.1.1.1.1" xref="S3.Ex19.m1.5.5.1.1.1.1.cmml">Tr</mi><mo id="S3.Ex19.m1.5.5.1.1.1a" lspace="0.167em" xref="S3.Ex19.m1.5.5.1.1.1.cmml">⁡</mo><mi id="S3.Ex19.m1.5.5.1.1.1.2" xref="S3.Ex19.m1.5.5.1.1.1.2.cmml">f</mi></mrow><mo id="S3.Ex19.m1.5.5.1.1.3" stretchy="false" xref="S3.Ex19.m1.5.5.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex19.m1.4.4.4" xref="S3.Ex19.m1.4.4.4.cmml"><msubsup id="S3.Ex19.m1.4.4.4.6" xref="S3.Ex19.m1.4.4.4.6.cmml"><mi id="S3.Ex19.m1.4.4.4.6.2.2" xref="S3.Ex19.m1.4.4.4.6.2.2.cmml">B</mi><mrow id="S3.Ex19.m1.2.2.2.2.2.4" xref="S3.Ex19.m1.2.2.2.2.2.3.cmml"><mi id="S3.Ex19.m1.1.1.1.1.1.1" xref="S3.Ex19.m1.1.1.1.1.1.1.cmml">p</mi><mo id="S3.Ex19.m1.2.2.2.2.2.4.1" xref="S3.Ex19.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S3.Ex19.m1.2.2.2.2.2.2" xref="S3.Ex19.m1.2.2.2.2.2.2.cmml">p</mi></mrow><mrow id="S3.Ex19.m1.4.4.4.6.2.3" xref="S3.Ex19.m1.4.4.4.6.2.3.cmml"><mi id="S3.Ex19.m1.4.4.4.6.2.3.2" xref="S3.Ex19.m1.4.4.4.6.2.3.2.cmml">s</mi><mo id="S3.Ex19.m1.4.4.4.6.2.3.1" xref="S3.Ex19.m1.4.4.4.6.2.3.1.cmml">−</mo><mfrac id="S3.Ex19.m1.4.4.4.6.2.3.3" 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xref="S3.Ex19.m1.4.4.4.4.1.1.3.2.cmml">d</mi><mo id="S3.Ex19.m1.4.4.4.4.1.1.3.1" xref="S3.Ex19.m1.4.4.4.4.1.1.3.1.cmml">−</mo><mn id="S3.Ex19.m1.4.4.4.4.1.1.3.3" xref="S3.Ex19.m1.4.4.4.4.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Ex19.m1.4.4.4.4.1.3" xref="S3.Ex19.m1.4.4.4.4.2.cmml">;</mo><mi id="S3.Ex19.m1.3.3.3.3" xref="S3.Ex19.m1.3.3.3.3.cmml">X</mi><mo id="S3.Ex19.m1.4.4.4.4.1.4" stretchy="false" xref="S3.Ex19.m1.4.4.4.4.2.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Ex19.m1.5b"><apply id="S3.Ex19.m1.5.5.cmml" xref="S3.Ex19.m1.5.5"><csymbol cd="ambiguous" id="S3.Ex19.m1.5.5.2.cmml" xref="S3.Ex19.m1.5.5">subscript</csymbol><apply id="S3.Ex19.m1.5.5.1.2.cmml" xref="S3.Ex19.m1.5.5.1.1"><csymbol cd="latexml" id="S3.Ex19.m1.5.5.1.2.1.cmml" xref="S3.Ex19.m1.5.5.1.1.2">norm</csymbol><apply id="S3.Ex19.m1.5.5.1.1.1.cmml" xref="S3.Ex19.m1.5.5.1.1.1"><ci id="S3.Ex19.m1.5.5.1.1.1.1.cmml" xref="S3.Ex19.m1.5.5.1.1.1.1">Tr</ci><ci id="S3.Ex19.m1.5.5.1.1.1.2.cmml" xref="S3.Ex19.m1.5.5.1.1.1.2">𝑓</ci></apply></apply><apply id="S3.Ex19.m1.4.4.4.cmml" xref="S3.Ex19.m1.4.4.4"><times id="S3.Ex19.m1.4.4.4.5.cmml" xref="S3.Ex19.m1.4.4.4.5"></times><apply id="S3.Ex19.m1.4.4.4.6.cmml" xref="S3.Ex19.m1.4.4.4.6"><csymbol cd="ambiguous" id="S3.Ex19.m1.4.4.4.6.1.cmml" xref="S3.Ex19.m1.4.4.4.6">subscript</csymbol><apply id="S3.Ex19.m1.4.4.4.6.2.cmml" xref="S3.Ex19.m1.4.4.4.6"><csymbol cd="ambiguous" id="S3.Ex19.m1.4.4.4.6.2.1.cmml" xref="S3.Ex19.m1.4.4.4.6">superscript</csymbol><ci id="S3.Ex19.m1.4.4.4.6.2.2.cmml" xref="S3.Ex19.m1.4.4.4.6.2.2">𝐵</ci><apply id="S3.Ex19.m1.4.4.4.6.2.3.cmml" xref="S3.Ex19.m1.4.4.4.6.2.3"><minus id="S3.Ex19.m1.4.4.4.6.2.3.1.cmml" xref="S3.Ex19.m1.4.4.4.6.2.3.1"></minus><ci id="S3.Ex19.m1.4.4.4.6.2.3.2.cmml" xref="S3.Ex19.m1.4.4.4.6.2.3.2">𝑠</ci><apply id="S3.Ex19.m1.4.4.4.6.2.3.3.cmml" xref="S3.Ex19.m1.4.4.4.6.2.3.3"><divide id="S3.Ex19.m1.4.4.4.6.2.3.3.1.cmml" 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id="S3.Ex19.m1.4.4.4.4.1.1.2.cmml" xref="S3.Ex19.m1.4.4.4.4.1.1.2">ℝ</ci><apply id="S3.Ex19.m1.4.4.4.4.1.1.3.cmml" xref="S3.Ex19.m1.4.4.4.4.1.1.3"><minus id="S3.Ex19.m1.4.4.4.4.1.1.3.1.cmml" xref="S3.Ex19.m1.4.4.4.4.1.1.3.1"></minus><ci id="S3.Ex19.m1.4.4.4.4.1.1.3.2.cmml" xref="S3.Ex19.m1.4.4.4.4.1.1.3.2">𝑑</ci><cn id="S3.Ex19.m1.4.4.4.4.1.1.3.3.cmml" type="integer" xref="S3.Ex19.m1.4.4.4.4.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex19.m1.3.3.3.3.cmml" xref="S3.Ex19.m1.3.3.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex19.m1.5c">\displaystyle\|\operatorname{Tr}f\|_{B^{s-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}% ^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex19.m1.5d">∥ roman_Tr italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\|\operatorname{Tr}f\|_{B_{p,p}^{s_{1}-\frac{\gamma_{1}+1}{p}}(% \mathbb{R}^{d-1};X)}\leq C\|f\|_{B^{s_{1}}_{p,p}(\mathbb{R}^{d},w_{\gamma_{1}}% ;X)}" class="ltx_Math" display="inline" id="S3.Ex19.m2.11"><semantics id="S3.Ex19.m2.11a"><mrow id="S3.Ex19.m2.11.11" xref="S3.Ex19.m2.11.11.cmml"><mi id="S3.Ex19.m2.11.11.3" xref="S3.Ex19.m2.11.11.3.cmml"></mi><mo id="S3.Ex19.m2.11.11.4" xref="S3.Ex19.m2.11.11.4.cmml">=</mo><msub id="S3.Ex19.m2.11.11.1" xref="S3.Ex19.m2.11.11.1.cmml"><mrow id="S3.Ex19.m2.11.11.1.1.1" xref="S3.Ex19.m2.11.11.1.1.2.cmml"><mo id="S3.Ex19.m2.11.11.1.1.1.2" stretchy="false" xref="S3.Ex19.m2.11.11.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex19.m2.11.11.1.1.1.1" xref="S3.Ex19.m2.11.11.1.1.1.1.cmml"><mi id="S3.Ex19.m2.11.11.1.1.1.1.1" 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start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell 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id="S3.Ex20.m1.13c">\displaystyle\leq C\|f\|_{F^{s_{1}}_{p,p}(\mathbb{R}^{d},w_{\gamma_{1}};X)}% \leq C\|f\|_{F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex20.m1.13d">≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td 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xref="S3.SS1.12.p2.1.m1.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS1.12.p2.1.m1.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS1.12.p2.1.m1.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS1.12.p2.1.m1.3.3.cmml" xref="S3.SS1.12.p2.1.m1.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.12.p2.1.m1.4c">g\in B_{p,p}^{s-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.12.p2.1.m1.4d">italic_g ∈ italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> and for <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.SS1.12.p2.2.m2.1"><semantics id="S3.SS1.12.p2.2.m2.1a"><mrow id="S3.SS1.12.p2.2.m2.1.1" xref="S3.SS1.12.p2.2.m2.1.1.cmml"><mi id="S3.SS1.12.p2.2.m2.1.1.2" xref="S3.SS1.12.p2.2.m2.1.1.2.cmml">ε</mi><mo id="S3.SS1.12.p2.2.m2.1.1.1" xref="S3.SS1.12.p2.2.m2.1.1.1.cmml">></mo><mn id="S3.SS1.12.p2.2.m2.1.1.3" xref="S3.SS1.12.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.12.p2.2.m2.1b"><apply id="S3.SS1.12.p2.2.m2.1.1.cmml" xref="S3.SS1.12.p2.2.m2.1.1"><gt id="S3.SS1.12.p2.2.m2.1.1.1.cmml" xref="S3.SS1.12.p2.2.m2.1.1.1"></gt><ci id="S3.SS1.12.p2.2.m2.1.1.2.cmml" xref="S3.SS1.12.p2.2.m2.1.1.2">𝜀</ci><cn id="S3.SS1.12.p2.2.m2.1.1.3.cmml" type="integer" xref="S3.SS1.12.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.12.p2.2.m2.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.12.p2.2.m2.1d">italic_ε > 0</annotation></semantics></math> small take <math alttext="s_{0}=s+\varepsilon/p" class="ltx_Math" display="inline" id="S3.SS1.12.p2.3.m3.1"><semantics id="S3.SS1.12.p2.3.m3.1a"><mrow id="S3.SS1.12.p2.3.m3.1.1" xref="S3.SS1.12.p2.3.m3.1.1.cmml"><msub id="S3.SS1.12.p2.3.m3.1.1.2" xref="S3.SS1.12.p2.3.m3.1.1.2.cmml"><mi id="S3.SS1.12.p2.3.m3.1.1.2.2" xref="S3.SS1.12.p2.3.m3.1.1.2.2.cmml">s</mi><mn id="S3.SS1.12.p2.3.m3.1.1.2.3" xref="S3.SS1.12.p2.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS1.12.p2.3.m3.1.1.1" xref="S3.SS1.12.p2.3.m3.1.1.1.cmml">=</mo><mrow id="S3.SS1.12.p2.3.m3.1.1.3" xref="S3.SS1.12.p2.3.m3.1.1.3.cmml"><mi id="S3.SS1.12.p2.3.m3.1.1.3.2" xref="S3.SS1.12.p2.3.m3.1.1.3.2.cmml">s</mi><mo id="S3.SS1.12.p2.3.m3.1.1.3.1" xref="S3.SS1.12.p2.3.m3.1.1.3.1.cmml">+</mo><mrow id="S3.SS1.12.p2.3.m3.1.1.3.3" xref="S3.SS1.12.p2.3.m3.1.1.3.3.cmml"><mi id="S3.SS1.12.p2.3.m3.1.1.3.3.2" xref="S3.SS1.12.p2.3.m3.1.1.3.3.2.cmml">ε</mi><mo id="S3.SS1.12.p2.3.m3.1.1.3.3.1" xref="S3.SS1.12.p2.3.m3.1.1.3.3.1.cmml">/</mo><mi id="S3.SS1.12.p2.3.m3.1.1.3.3.3" xref="S3.SS1.12.p2.3.m3.1.1.3.3.3.cmml">p</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.12.p2.3.m3.1b"><apply id="S3.SS1.12.p2.3.m3.1.1.cmml" xref="S3.SS1.12.p2.3.m3.1.1"><eq id="S3.SS1.12.p2.3.m3.1.1.1.cmml" xref="S3.SS1.12.p2.3.m3.1.1.1"></eq><apply id="S3.SS1.12.p2.3.m3.1.1.2.cmml" xref="S3.SS1.12.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.12.p2.3.m3.1.1.2.1.cmml" xref="S3.SS1.12.p2.3.m3.1.1.2">subscript</csymbol><ci id="S3.SS1.12.p2.3.m3.1.1.2.2.cmml" xref="S3.SS1.12.p2.3.m3.1.1.2.2">𝑠</ci><cn id="S3.SS1.12.p2.3.m3.1.1.2.3.cmml" type="integer" xref="S3.SS1.12.p2.3.m3.1.1.2.3">0</cn></apply><apply id="S3.SS1.12.p2.3.m3.1.1.3.cmml" xref="S3.SS1.12.p2.3.m3.1.1.3"><plus id="S3.SS1.12.p2.3.m3.1.1.3.1.cmml" xref="S3.SS1.12.p2.3.m3.1.1.3.1"></plus><ci id="S3.SS1.12.p2.3.m3.1.1.3.2.cmml" xref="S3.SS1.12.p2.3.m3.1.1.3.2">𝑠</ci><apply id="S3.SS1.12.p2.3.m3.1.1.3.3.cmml" xref="S3.SS1.12.p2.3.m3.1.1.3.3"><divide id="S3.SS1.12.p2.3.m3.1.1.3.3.1.cmml" xref="S3.SS1.12.p2.3.m3.1.1.3.3.1"></divide><ci id="S3.SS1.12.p2.3.m3.1.1.3.3.2.cmml" xref="S3.SS1.12.p2.3.m3.1.1.3.3.2">𝜀</ci><ci id="S3.SS1.12.p2.3.m3.1.1.3.3.3.cmml" xref="S3.SS1.12.p2.3.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.12.p2.3.m3.1c">s_{0}=s+\varepsilon/p</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.12.p2.3.m3.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_s + italic_ε / italic_p</annotation></semantics></math> and <math alttext="\gamma_{0}=\gamma+\varepsilon" class="ltx_Math" display="inline" id="S3.SS1.12.p2.4.m4.1"><semantics id="S3.SS1.12.p2.4.m4.1a"><mrow id="S3.SS1.12.p2.4.m4.1.1" xref="S3.SS1.12.p2.4.m4.1.1.cmml"><msub id="S3.SS1.12.p2.4.m4.1.1.2" xref="S3.SS1.12.p2.4.m4.1.1.2.cmml"><mi id="S3.SS1.12.p2.4.m4.1.1.2.2" xref="S3.SS1.12.p2.4.m4.1.1.2.2.cmml">γ</mi><mn id="S3.SS1.12.p2.4.m4.1.1.2.3" xref="S3.SS1.12.p2.4.m4.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS1.12.p2.4.m4.1.1.1" xref="S3.SS1.12.p2.4.m4.1.1.1.cmml">=</mo><mrow id="S3.SS1.12.p2.4.m4.1.1.3" xref="S3.SS1.12.p2.4.m4.1.1.3.cmml"><mi id="S3.SS1.12.p2.4.m4.1.1.3.2" xref="S3.SS1.12.p2.4.m4.1.1.3.2.cmml">γ</mi><mo id="S3.SS1.12.p2.4.m4.1.1.3.1" xref="S3.SS1.12.p2.4.m4.1.1.3.1.cmml">+</mo><mi id="S3.SS1.12.p2.4.m4.1.1.3.3" xref="S3.SS1.12.p2.4.m4.1.1.3.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.12.p2.4.m4.1b"><apply id="S3.SS1.12.p2.4.m4.1.1.cmml" xref="S3.SS1.12.p2.4.m4.1.1"><eq id="S3.SS1.12.p2.4.m4.1.1.1.cmml" xref="S3.SS1.12.p2.4.m4.1.1.1"></eq><apply id="S3.SS1.12.p2.4.m4.1.1.2.cmml" xref="S3.SS1.12.p2.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS1.12.p2.4.m4.1.1.2.1.cmml" xref="S3.SS1.12.p2.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS1.12.p2.4.m4.1.1.2.2.cmml" xref="S3.SS1.12.p2.4.m4.1.1.2.2">𝛾</ci><cn id="S3.SS1.12.p2.4.m4.1.1.2.3.cmml" type="integer" xref="S3.SS1.12.p2.4.m4.1.1.2.3">0</cn></apply><apply id="S3.SS1.12.p2.4.m4.1.1.3.cmml" xref="S3.SS1.12.p2.4.m4.1.1.3"><plus id="S3.SS1.12.p2.4.m4.1.1.3.1.cmml" xref="S3.SS1.12.p2.4.m4.1.1.3.1"></plus><ci id="S3.SS1.12.p2.4.m4.1.1.3.2.cmml" xref="S3.SS1.12.p2.4.m4.1.1.3.2">𝛾</ci><ci id="S3.SS1.12.p2.4.m4.1.1.3.3.cmml" xref="S3.SS1.12.p2.4.m4.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.12.p2.4.m4.1c">\gamma_{0}=\gamma+\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.12.p2.4.m4.1d">italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_γ + italic_ε</annotation></semantics></math>. Then by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.Thmtheorem3" title="Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E5" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.5</span></a>) and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem4" title="Theorem 3.4. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.4</span></a>, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx9"> <tbody id="S3.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\operatorname{ext}g\|_{F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S3.Ex21.m1.6"><semantics id="S3.Ex21.m1.6a"><msub id="S3.Ex21.m1.6.6" xref="S3.Ex21.m1.6.6.cmml"><mrow id="S3.Ex21.m1.6.6.1.1" xref="S3.Ex21.m1.6.6.1.2.cmml"><mo id="S3.Ex21.m1.6.6.1.1.2" stretchy="false" xref="S3.Ex21.m1.6.6.1.2.1.cmml">‖</mo><mrow id="S3.Ex21.m1.6.6.1.1.1" xref="S3.Ex21.m1.6.6.1.1.1.cmml"><mi id="S3.Ex21.m1.6.6.1.1.1.1" xref="S3.Ex21.m1.6.6.1.1.1.1.cmml">ext</mi><mo id="S3.Ex21.m1.6.6.1.1.1a" lspace="0.167em" xref="S3.Ex21.m1.6.6.1.1.1.cmml">⁡</mo><mi id="S3.Ex21.m1.6.6.1.1.1.2" xref="S3.Ex21.m1.6.6.1.1.1.2.cmml">g</mi></mrow><mo id="S3.Ex21.m1.6.6.1.1.3" stretchy="false" xref="S3.Ex21.m1.6.6.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex21.m1.5.5.5" xref="S3.Ex21.m1.5.5.5.cmml"><msubsup id="S3.Ex21.m1.5.5.5.7" xref="S3.Ex21.m1.5.5.5.7.cmml"><mi id="S3.Ex21.m1.5.5.5.7.2.2" xref="S3.Ex21.m1.5.5.5.7.2.2.cmml">F</mi><mrow id="S3.Ex21.m1.2.2.2.2.2.4" xref="S3.Ex21.m1.2.2.2.2.2.3.cmml"><mi id="S3.Ex21.m1.1.1.1.1.1.1" xref="S3.Ex21.m1.1.1.1.1.1.1.cmml">p</mi><mo id="S3.Ex21.m1.2.2.2.2.2.4.1" xref="S3.Ex21.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S3.Ex21.m1.2.2.2.2.2.2" xref="S3.Ex21.m1.2.2.2.2.2.2.cmml">q</mi></mrow><mi id="S3.Ex21.m1.5.5.5.7.2.3" xref="S3.Ex21.m1.5.5.5.7.2.3.cmml">s</mi></msubsup><mo id="S3.Ex21.m1.5.5.5.6" xref="S3.Ex21.m1.5.5.5.6.cmml">⁢</mo><mrow id="S3.Ex21.m1.5.5.5.5.2" xref="S3.Ex21.m1.5.5.5.5.3.cmml"><mo id="S3.Ex21.m1.5.5.5.5.2.3" stretchy="false" xref="S3.Ex21.m1.5.5.5.5.3.cmml">(</mo><msup id="S3.Ex21.m1.4.4.4.4.1.1" xref="S3.Ex21.m1.4.4.4.4.1.1.cmml"><mi id="S3.Ex21.m1.4.4.4.4.1.1.2" xref="S3.Ex21.m1.4.4.4.4.1.1.2.cmml">ℝ</mi><mi id="S3.Ex21.m1.4.4.4.4.1.1.3" xref="S3.Ex21.m1.4.4.4.4.1.1.3.cmml">d</mi></msup><mo id="S3.Ex21.m1.5.5.5.5.2.4" xref="S3.Ex21.m1.5.5.5.5.3.cmml">,</mo><msub id="S3.Ex21.m1.5.5.5.5.2.2" xref="S3.Ex21.m1.5.5.5.5.2.2.cmml"><mi id="S3.Ex21.m1.5.5.5.5.2.2.2" xref="S3.Ex21.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S3.Ex21.m1.5.5.5.5.2.2.3" xref="S3.Ex21.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex21.m1.5.5.5.5.2.5" xref="S3.Ex21.m1.5.5.5.5.3.cmml">;</mo><mi id="S3.Ex21.m1.3.3.3.3" xref="S3.Ex21.m1.3.3.3.3.cmml">X</mi><mo id="S3.Ex21.m1.5.5.5.5.2.6" stretchy="false" xref="S3.Ex21.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Ex21.m1.6b"><apply id="S3.Ex21.m1.6.6.cmml" xref="S3.Ex21.m1.6.6"><csymbol cd="ambiguous" id="S3.Ex21.m1.6.6.2.cmml" xref="S3.Ex21.m1.6.6">subscript</csymbol><apply id="S3.Ex21.m1.6.6.1.2.cmml" xref="S3.Ex21.m1.6.6.1.1"><csymbol cd="latexml" id="S3.Ex21.m1.6.6.1.2.1.cmml" xref="S3.Ex21.m1.6.6.1.1.2">norm</csymbol><apply id="S3.Ex21.m1.6.6.1.1.1.cmml" xref="S3.Ex21.m1.6.6.1.1.1"><ci id="S3.Ex21.m1.6.6.1.1.1.1.cmml" xref="S3.Ex21.m1.6.6.1.1.1.1">ext</ci><ci id="S3.Ex21.m1.6.6.1.1.1.2.cmml" xref="S3.Ex21.m1.6.6.1.1.1.2">𝑔</ci></apply></apply><apply id="S3.Ex21.m1.5.5.5.cmml" xref="S3.Ex21.m1.5.5.5"><times id="S3.Ex21.m1.5.5.5.6.cmml" xref="S3.Ex21.m1.5.5.5.6"></times><apply id="S3.Ex21.m1.5.5.5.7.cmml" xref="S3.Ex21.m1.5.5.5.7"><csymbol cd="ambiguous" id="S3.Ex21.m1.5.5.5.7.1.cmml" xref="S3.Ex21.m1.5.5.5.7">subscript</csymbol><apply id="S3.Ex21.m1.5.5.5.7.2.cmml" xref="S3.Ex21.m1.5.5.5.7"><csymbol cd="ambiguous" id="S3.Ex21.m1.5.5.5.7.2.1.cmml" xref="S3.Ex21.m1.5.5.5.7">superscript</csymbol><ci id="S3.Ex21.m1.5.5.5.7.2.2.cmml" xref="S3.Ex21.m1.5.5.5.7.2.2">𝐹</ci><ci id="S3.Ex21.m1.5.5.5.7.2.3.cmml" xref="S3.Ex21.m1.5.5.5.7.2.3">𝑠</ci></apply><list id="S3.Ex21.m1.2.2.2.2.2.3.cmml" xref="S3.Ex21.m1.2.2.2.2.2.4"><ci id="S3.Ex21.m1.1.1.1.1.1.1.cmml" xref="S3.Ex21.m1.1.1.1.1.1.1">𝑝</ci><ci id="S3.Ex21.m1.2.2.2.2.2.2.cmml" xref="S3.Ex21.m1.2.2.2.2.2.2">𝑞</ci></list></apply><vector id="S3.Ex21.m1.5.5.5.5.3.cmml" xref="S3.Ex21.m1.5.5.5.5.2"><apply id="S3.Ex21.m1.4.4.4.4.1.1.cmml" xref="S3.Ex21.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S3.Ex21.m1.4.4.4.4.1.1.1.cmml" xref="S3.Ex21.m1.4.4.4.4.1.1">superscript</csymbol><ci id="S3.Ex21.m1.4.4.4.4.1.1.2.cmml" xref="S3.Ex21.m1.4.4.4.4.1.1.2">ℝ</ci><ci id="S3.Ex21.m1.4.4.4.4.1.1.3.cmml" xref="S3.Ex21.m1.4.4.4.4.1.1.3">𝑑</ci></apply><apply id="S3.Ex21.m1.5.5.5.5.2.2.cmml" xref="S3.Ex21.m1.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S3.Ex21.m1.5.5.5.5.2.2.1.cmml" xref="S3.Ex21.m1.5.5.5.5.2.2">subscript</csymbol><ci id="S3.Ex21.m1.5.5.5.5.2.2.2.cmml" xref="S3.Ex21.m1.5.5.5.5.2.2.2">𝑤</ci><ci id="S3.Ex21.m1.5.5.5.5.2.2.3.cmml" xref="S3.Ex21.m1.5.5.5.5.2.2.3">𝛾</ci></apply><ci id="S3.Ex21.m1.3.3.3.3.cmml" xref="S3.Ex21.m1.3.3.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex21.m1.6c">\displaystyle\|\operatorname{ext}g\|_{F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex21.m1.6d">∥ roman_ext italic_g ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|\operatorname{ext}g\|_{F^{s_{0}}_{p,p}(\mathbb{R}^{d},w_{% \gamma_{0}};X)}\leq C\|\operatorname{ext}g\|_{B^{s_{0}}_{p,p}(\mathbb{R}^{d},w% _{\gamma_{0}};X)}" class="ltx_Math" display="inline" id="S3.Ex21.m2.12"><semantics id="S3.Ex21.m2.12a"><mrow id="S3.Ex21.m2.12.12" xref="S3.Ex21.m2.12.12.cmml"><mi id="S3.Ex21.m2.12.12.4" xref="S3.Ex21.m2.12.12.4.cmml"></mi><mo id="S3.Ex21.m2.12.12.5" xref="S3.Ex21.m2.12.12.5.cmml">≤</mo><mrow id="S3.Ex21.m2.11.11.1" xref="S3.Ex21.m2.11.11.1.cmml"><mi id="S3.Ex21.m2.11.11.1.3" xref="S3.Ex21.m2.11.11.1.3.cmml">C</mi><mo id="S3.Ex21.m2.11.11.1.2" xref="S3.Ex21.m2.11.11.1.2.cmml">⁢</mo><msub id="S3.Ex21.m2.11.11.1.1" xref="S3.Ex21.m2.11.11.1.1.cmml"><mrow id="S3.Ex21.m2.11.11.1.1.1.1" xref="S3.Ex21.m2.11.11.1.1.1.2.cmml"><mo id="S3.Ex21.m2.11.11.1.1.1.1.2" stretchy="false" xref="S3.Ex21.m2.11.11.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex21.m2.11.11.1.1.1.1.1" xref="S3.Ex21.m2.11.11.1.1.1.1.1.cmml"><mi id="S3.Ex21.m2.11.11.1.1.1.1.1.1" xref="S3.Ex21.m2.11.11.1.1.1.1.1.1.cmml">ext</mi><mo id="S3.Ex21.m2.11.11.1.1.1.1.1a" lspace="0.167em" xref="S3.Ex21.m2.11.11.1.1.1.1.1.cmml">⁡</mo><mi id="S3.Ex21.m2.11.11.1.1.1.1.1.2" xref="S3.Ex21.m2.11.11.1.1.1.1.1.2.cmml">g</mi></mrow><mo id="S3.Ex21.m2.11.11.1.1.1.1.3" stretchy="false" xref="S3.Ex21.m2.11.11.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex21.m2.5.5.5" xref="S3.Ex21.m2.5.5.5.cmml"><msubsup id="S3.Ex21.m2.5.5.5.7" xref="S3.Ex21.m2.5.5.5.7.cmml"><mi id="S3.Ex21.m2.5.5.5.7.2.2" xref="S3.Ex21.m2.5.5.5.7.2.2.cmml">F</mi><mrow id="S3.Ex21.m2.2.2.2.2.2.4" xref="S3.Ex21.m2.2.2.2.2.2.3.cmml"><mi id="S3.Ex21.m2.1.1.1.1.1.1" xref="S3.Ex21.m2.1.1.1.1.1.1.cmml">p</mi><mo id="S3.Ex21.m2.2.2.2.2.2.4.1" xref="S3.Ex21.m2.2.2.2.2.2.3.cmml">,</mo><mi id="S3.Ex21.m2.2.2.2.2.2.2" xref="S3.Ex21.m2.2.2.2.2.2.2.cmml">p</mi></mrow><msub id="S3.Ex21.m2.5.5.5.7.2.3" xref="S3.Ex21.m2.5.5.5.7.2.3.cmml"><mi id="S3.Ex21.m2.5.5.5.7.2.3.2" xref="S3.Ex21.m2.5.5.5.7.2.3.2.cmml">s</mi><mn id="S3.Ex21.m2.5.5.5.7.2.3.3" xref="S3.Ex21.m2.5.5.5.7.2.3.3.cmml">0</mn></msub></msubsup><mo id="S3.Ex21.m2.5.5.5.6" xref="S3.Ex21.m2.5.5.5.6.cmml">⁢</mo><mrow id="S3.Ex21.m2.5.5.5.5.2" xref="S3.Ex21.m2.5.5.5.5.3.cmml"><mo id="S3.Ex21.m2.5.5.5.5.2.3" stretchy="false" xref="S3.Ex21.m2.5.5.5.5.3.cmml">(</mo><msup id="S3.Ex21.m2.4.4.4.4.1.1" xref="S3.Ex21.m2.4.4.4.4.1.1.cmml"><mi id="S3.Ex21.m2.4.4.4.4.1.1.2" xref="S3.Ex21.m2.4.4.4.4.1.1.2.cmml">ℝ</mi><mi id="S3.Ex21.m2.4.4.4.4.1.1.3" xref="S3.Ex21.m2.4.4.4.4.1.1.3.cmml">d</mi></msup><mo id="S3.Ex21.m2.5.5.5.5.2.4" xref="S3.Ex21.m2.5.5.5.5.3.cmml">,</mo><msub id="S3.Ex21.m2.5.5.5.5.2.2" xref="S3.Ex21.m2.5.5.5.5.2.2.cmml"><mi id="S3.Ex21.m2.5.5.5.5.2.2.2" xref="S3.Ex21.m2.5.5.5.5.2.2.2.cmml">w</mi><msub id="S3.Ex21.m2.5.5.5.5.2.2.3" xref="S3.Ex21.m2.5.5.5.5.2.2.3.cmml"><mi id="S3.Ex21.m2.5.5.5.5.2.2.3.2" xref="S3.Ex21.m2.5.5.5.5.2.2.3.2.cmml">γ</mi><mn id="S3.Ex21.m2.5.5.5.5.2.2.3.3" xref="S3.Ex21.m2.5.5.5.5.2.2.3.3.cmml">0</mn></msub></msub><mo id="S3.Ex21.m2.5.5.5.5.2.5" xref="S3.Ex21.m2.5.5.5.5.3.cmml">;</mo><mi id="S3.Ex21.m2.3.3.3.3" xref="S3.Ex21.m2.3.3.3.3.cmml">X</mi><mo id="S3.Ex21.m2.5.5.5.5.2.6" stretchy="false" xref="S3.Ex21.m2.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub></mrow><mo id="S3.Ex21.m2.12.12.6" xref="S3.Ex21.m2.12.12.6.cmml">≤</mo><mrow id="S3.Ex21.m2.12.12.2" xref="S3.Ex21.m2.12.12.2.cmml"><mi id="S3.Ex21.m2.12.12.2.3" xref="S3.Ex21.m2.12.12.2.3.cmml">C</mi><mo id="S3.Ex21.m2.12.12.2.2" xref="S3.Ex21.m2.12.12.2.2.cmml">⁢</mo><msub id="S3.Ex21.m2.12.12.2.1" xref="S3.Ex21.m2.12.12.2.1.cmml"><mrow id="S3.Ex21.m2.12.12.2.1.1.1" xref="S3.Ex21.m2.12.12.2.1.1.2.cmml"><mo id="S3.Ex21.m2.12.12.2.1.1.1.2" stretchy="false" xref="S3.Ex21.m2.12.12.2.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex21.m2.12.12.2.1.1.1.1" xref="S3.Ex21.m2.12.12.2.1.1.1.1.cmml"><mi id="S3.Ex21.m2.12.12.2.1.1.1.1.1" xref="S3.Ex21.m2.12.12.2.1.1.1.1.1.cmml">ext</mi><mo id="S3.Ex21.m2.12.12.2.1.1.1.1a" lspace="0.167em" xref="S3.Ex21.m2.12.12.2.1.1.1.1.cmml">⁡</mo><mi id="S3.Ex21.m2.12.12.2.1.1.1.1.2" xref="S3.Ex21.m2.12.12.2.1.1.1.1.2.cmml">g</mi></mrow><mo id="S3.Ex21.m2.12.12.2.1.1.1.3" stretchy="false" 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xref="S3.Ex21.m2.10.10.5.5.2.2.3.3">0</cn></apply></apply><ci id="S3.Ex21.m2.8.8.3.3.cmml" xref="S3.Ex21.m2.8.8.3.3">𝑋</ci></vector></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex21.m2.12c">\displaystyle\leq C\|\operatorname{ext}g\|_{F^{s_{0}}_{p,p}(\mathbb{R}^{d},w_{% \gamma_{0}};X)}\leq C\|\operatorname{ext}g\|_{B^{s_{0}}_{p,p}(\mathbb{R}^{d},w% _{\gamma_{0}};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex21.m2.12d">≤ italic_C ∥ roman_ext italic_g ∥ start_POSTSUBSCRIPT italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ roman_ext italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|g\|_{B^{s_{0}-\frac{\gamma_{0}+1}{p}}_{p,p}(\mathbb{R}^{d% -1};X)}=\|g\|_{B^{s-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)}." class="ltx_Math" display="inline" id="S3.Ex22.m1.11"><semantics id="S3.Ex22.m1.11a"><mrow id="S3.Ex22.m1.11.11.1" xref="S3.Ex22.m1.11.11.1.1.cmml"><mrow id="S3.Ex22.m1.11.11.1.1" 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end_POSTSUBSCRIPT - divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT = ∥ italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS1.12.p2.5">The identity <math alttext="\operatorname{Tr}\mathop{\circ}\nolimits\operatorname{ext}=\operatorname{id}" class="ltx_Math" display="inline" id="S3.SS1.12.p2.5.m1.1"><semantics id="S3.SS1.12.p2.5.m1.1a"><mrow id="S3.SS1.12.p2.5.m1.1.1" xref="S3.SS1.12.p2.5.m1.1.1.cmml"><mrow id="S3.SS1.12.p2.5.m1.1.1.2" xref="S3.SS1.12.p2.5.m1.1.1.2.cmml"><mi id="S3.SS1.12.p2.5.m1.1.1.2.2" xref="S3.SS1.12.p2.5.m1.1.1.2.2.cmml">Tr</mi><mo id="S3.SS1.12.p2.5.m1.1.1.2.1" lspace="0.167em" xref="S3.SS1.12.p2.5.m1.1.1.2.1.cmml">⁢</mo><mrow id="S3.SS1.12.p2.5.m1.1.1.2.3" xref="S3.SS1.12.p2.5.m1.1.1.2.3.cmml"><mo id="S3.SS1.12.p2.5.m1.1.1.2.3.1" rspace="0.167em" xref="S3.SS1.12.p2.5.m1.1.1.2.3.1.cmml">∘</mo><mi id="S3.SS1.12.p2.5.m1.1.1.2.3.2" xref="S3.SS1.12.p2.5.m1.1.1.2.3.2.cmml">ext</mi></mrow></mrow><mo id="S3.SS1.12.p2.5.m1.1.1.1" xref="S3.SS1.12.p2.5.m1.1.1.1.cmml">=</mo><mi id="S3.SS1.12.p2.5.m1.1.1.3" xref="S3.SS1.12.p2.5.m1.1.1.3.cmml">id</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS1.12.p2.5.m1.1b"><apply id="S3.SS1.12.p2.5.m1.1.1.cmml" xref="S3.SS1.12.p2.5.m1.1.1"><eq id="S3.SS1.12.p2.5.m1.1.1.1.cmml" xref="S3.SS1.12.p2.5.m1.1.1.1"></eq><apply id="S3.SS1.12.p2.5.m1.1.1.2.cmml" xref="S3.SS1.12.p2.5.m1.1.1.2"><times id="S3.SS1.12.p2.5.m1.1.1.2.1.cmml" xref="S3.SS1.12.p2.5.m1.1.1.2.1"></times><ci id="S3.SS1.12.p2.5.m1.1.1.2.2.cmml" xref="S3.SS1.12.p2.5.m1.1.1.2.2">Tr</ci><apply id="S3.SS1.12.p2.5.m1.1.1.2.3.cmml" xref="S3.SS1.12.p2.5.m1.1.1.2.3"><compose id="S3.SS1.12.p2.5.m1.1.1.2.3.1.cmml" xref="S3.SS1.12.p2.5.m1.1.1.2.3.1"></compose><ci id="S3.SS1.12.p2.5.m1.1.1.2.3.2.cmml" xref="S3.SS1.12.p2.5.m1.1.1.2.3.2">ext</ci></apply></apply><ci id="S3.SS1.12.p2.5.m1.1.1.3.cmml" xref="S3.SS1.12.p2.5.m1.1.1.3">id</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS1.12.p2.5.m1.1c">\operatorname{Tr}\mathop{\circ}\nolimits\operatorname{ext}=\operatorname{id}</annotation><annotation encoding="application/x-llamapun" id="S3.SS1.12.p2.5.m1.1d">roman_Tr ∘ roman_ext = roman_id</annotation></semantics></math> follows from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem4" title="Theorem 3.4. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.4</span></a>. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S3.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">3.2. </span>The higher-order trace operator</h3> <div class="ltx_para" id="S3.SS2.p1"> <p class="ltx_p" id="S3.SS2.p1.4">In this section, we study the <math alttext="m" class="ltx_Math" display="inline" id="S3.SS2.p1.1.m1.1"><semantics id="S3.SS2.p1.1.m1.1a"><mi id="S3.SS2.p1.1.m1.1.1" xref="S3.SS2.p1.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.1.m1.1b"><ci id="S3.SS2.p1.1.m1.1.1.cmml" xref="S3.SS2.p1.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.1.m1.1d">italic_m</annotation></semantics></math>-th order trace operator <math alttext="\operatorname{Tr}_{m}=\operatorname{Tr}\mathop{\circ}\nolimits\partial_{1}^{m}" class="ltx_Math" display="inline" id="S3.SS2.p1.2.m2.1"><semantics id="S3.SS2.p1.2.m2.1a"><mrow id="S3.SS2.p1.2.m2.1.1" xref="S3.SS2.p1.2.m2.1.1.cmml"><msub id="S3.SS2.p1.2.m2.1.1.2" xref="S3.SS2.p1.2.m2.1.1.2.cmml"><mi id="S3.SS2.p1.2.m2.1.1.2.2" xref="S3.SS2.p1.2.m2.1.1.2.2.cmml">Tr</mi><mi id="S3.SS2.p1.2.m2.1.1.2.3" xref="S3.SS2.p1.2.m2.1.1.2.3.cmml">m</mi></msub><mo id="S3.SS2.p1.2.m2.1.1.1" xref="S3.SS2.p1.2.m2.1.1.1.cmml">=</mo><mrow id="S3.SS2.p1.2.m2.1.1.3" xref="S3.SS2.p1.2.m2.1.1.3.cmml"><mi id="S3.SS2.p1.2.m2.1.1.3.2" xref="S3.SS2.p1.2.m2.1.1.3.2.cmml">Tr</mi><mo id="S3.SS2.p1.2.m2.1.1.3.1" lspace="0.167em" xref="S3.SS2.p1.2.m2.1.1.3.1.cmml">⁢</mo><mrow id="S3.SS2.p1.2.m2.1.1.3.3" xref="S3.SS2.p1.2.m2.1.1.3.3.cmml"><mo id="S3.SS2.p1.2.m2.1.1.3.3.1" rspace="0.0835em" xref="S3.SS2.p1.2.m2.1.1.3.3.1.cmml">∘</mo><msubsup id="S3.SS2.p1.2.m2.1.1.3.3.2" 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xref="S3.SS2.p1.2.m2.1.1.3.1"></times><ci id="S3.SS2.p1.2.m2.1.1.3.2.cmml" xref="S3.SS2.p1.2.m2.1.1.3.2">Tr</ci><apply id="S3.SS2.p1.2.m2.1.1.3.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3"><compose id="S3.SS2.p1.2.m2.1.1.3.3.1.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3.1"></compose><apply id="S3.SS2.p1.2.m2.1.1.3.3.2.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.1.1.3.3.2.1.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3.2">superscript</csymbol><apply id="S3.SS2.p1.2.m2.1.1.3.3.2.2.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.p1.2.m2.1.1.3.3.2.2.1.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3.2">subscript</csymbol><partialdiff id="S3.SS2.p1.2.m2.1.1.3.3.2.2.2.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3.2.2.2"></partialdiff><cn id="S3.SS2.p1.2.m2.1.1.3.3.2.2.3.cmml" type="integer" xref="S3.SS2.p1.2.m2.1.1.3.3.2.2.3">1</cn></apply><ci id="S3.SS2.p1.2.m2.1.1.3.3.2.3.cmml" xref="S3.SS2.p1.2.m2.1.1.3.3.2.3">𝑚</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.2.m2.1c">\operatorname{Tr}_{m}=\operatorname{Tr}\mathop{\circ}\nolimits\partial_{1}^{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = roman_Tr ∘ ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="m\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S3.SS2.p1.3.m3.1"><semantics id="S3.SS2.p1.3.m3.1a"><mrow id="S3.SS2.p1.3.m3.1.1" xref="S3.SS2.p1.3.m3.1.1.cmml"><mi id="S3.SS2.p1.3.m3.1.1.2" xref="S3.SS2.p1.3.m3.1.1.2.cmml">m</mi><mo id="S3.SS2.p1.3.m3.1.1.1" xref="S3.SS2.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S3.SS2.p1.3.m3.1.1.3" xref="S3.SS2.p1.3.m3.1.1.3.cmml"><mi id="S3.SS2.p1.3.m3.1.1.3.2" xref="S3.SS2.p1.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S3.SS2.p1.3.m3.1.1.3.3" xref="S3.SS2.p1.3.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.3.m3.1b"><apply id="S3.SS2.p1.3.m3.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1"><in id="S3.SS2.p1.3.m3.1.1.1.cmml" xref="S3.SS2.p1.3.m3.1.1.1"></in><ci id="S3.SS2.p1.3.m3.1.1.2.cmml" xref="S3.SS2.p1.3.m3.1.1.2">𝑚</ci><apply id="S3.SS2.p1.3.m3.1.1.3.cmml" xref="S3.SS2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.p1.3.m3.1.1.3.1.cmml" xref="S3.SS2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS2.p1.3.m3.1.1.3.2.cmml" xref="S3.SS2.p1.3.m3.1.1.3.2">ℕ</ci><cn id="S3.SS2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S3.SS2.p1.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.3.m3.1c">m\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.3.m3.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Furthermore, we will write <math alttext="\operatorname{Tr}_{0}=\operatorname{Tr}" class="ltx_Math" display="inline" id="S3.SS2.p1.4.m4.1"><semantics id="S3.SS2.p1.4.m4.1a"><mrow id="S3.SS2.p1.4.m4.1.1" xref="S3.SS2.p1.4.m4.1.1.cmml"><msub id="S3.SS2.p1.4.m4.1.1.2" xref="S3.SS2.p1.4.m4.1.1.2.cmml"><mi id="S3.SS2.p1.4.m4.1.1.2.2" xref="S3.SS2.p1.4.m4.1.1.2.2.cmml">Tr</mi><mn id="S3.SS2.p1.4.m4.1.1.2.3" xref="S3.SS2.p1.4.m4.1.1.2.3.cmml">0</mn></msub><mo id="S3.SS2.p1.4.m4.1.1.1" xref="S3.SS2.p1.4.m4.1.1.1.cmml">=</mo><mi id="S3.SS2.p1.4.m4.1.1.3" xref="S3.SS2.p1.4.m4.1.1.3.cmml">Tr</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.p1.4.m4.1b"><apply id="S3.SS2.p1.4.m4.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1"><eq id="S3.SS2.p1.4.m4.1.1.1.cmml" xref="S3.SS2.p1.4.m4.1.1.1"></eq><apply id="S3.SS2.p1.4.m4.1.1.2.cmml" xref="S3.SS2.p1.4.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.p1.4.m4.1.1.2.1.cmml" xref="S3.SS2.p1.4.m4.1.1.2">subscript</csymbol><ci id="S3.SS2.p1.4.m4.1.1.2.2.cmml" xref="S3.SS2.p1.4.m4.1.1.2.2">Tr</ci><cn id="S3.SS2.p1.4.m4.1.1.2.3.cmml" type="integer" xref="S3.SS2.p1.4.m4.1.1.2.3">0</cn></apply><ci id="S3.SS2.p1.4.m4.1.1.3.cmml" xref="S3.SS2.p1.4.m4.1.1.3">Tr</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p1.4.m4.1c">\operatorname{Tr}_{0}=\operatorname{Tr}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p1.4.m4.1d">roman_Tr start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = roman_Tr</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.p2"> <p class="ltx_p" id="S3.SS2.p2.1">We start with higher-order trace spaces for Besov spaces. To this end, we first define the <math alttext="m" class="ltx_Math" display="inline" id="S3.SS2.p2.1.m1.1"><semantics id="S3.SS2.p2.1.m1.1a"><mi id="S3.SS2.p2.1.m1.1.1" xref="S3.SS2.p2.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.p2.1.m1.1b"><ci id="S3.SS2.p2.1.m1.1.1.cmml" xref="S3.SS2.p2.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p2.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p2.1.m1.1d">italic_m</annotation></semantics></math>-th order extension operator.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S3.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.1.1.1">Definition 3.6</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem6.p1"> <p class="ltx_p" id="S3.Thmtheorem6.p1.2">Let <math alttext="\eta_{0}\in C_{\mathrm{c}}^{\infty}((-1,1))" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.1.m1.2"><semantics id="S3.Thmtheorem6.p1.1.m1.2a"><mrow id="S3.Thmtheorem6.p1.1.m1.2.2" xref="S3.Thmtheorem6.p1.1.m1.2.2.cmml"><msub id="S3.Thmtheorem6.p1.1.m1.2.2.3" xref="S3.Thmtheorem6.p1.1.m1.2.2.3.cmml"><mi id="S3.Thmtheorem6.p1.1.m1.2.2.3.2" xref="S3.Thmtheorem6.p1.1.m1.2.2.3.2.cmml">η</mi><mn id="S3.Thmtheorem6.p1.1.m1.2.2.3.3" xref="S3.Thmtheorem6.p1.1.m1.2.2.3.3.cmml">0</mn></msub><mo id="S3.Thmtheorem6.p1.1.m1.2.2.2" xref="S3.Thmtheorem6.p1.1.m1.2.2.2.cmml">∈</mo><mrow id="S3.Thmtheorem6.p1.1.m1.2.2.1" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.cmml"><msubsup id="S3.Thmtheorem6.p1.1.m1.2.2.1.3" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3.cmml"><mi id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.2" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.2.cmml">C</mi><mi id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.3.cmml">c</mi><mi id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3.3.cmml">∞</mi></msubsup><mo id="S3.Thmtheorem6.p1.1.m1.2.2.1.2" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.cmml"><mo id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.cmml">(</mo><mrow id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.2.cmml"><mo id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.2.cmml">(</mo><mrow id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1a" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.cmml">−</mo><mn id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.2" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.3" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.2.cmml">,</mo><mn id="S3.Thmtheorem6.p1.1.m1.1.1" xref="S3.Thmtheorem6.p1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.4" stretchy="false" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.2.cmml">)</mo></mrow><mo id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.1.m1.2b"><apply id="S3.Thmtheorem6.p1.1.m1.2.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2"><in id="S3.Thmtheorem6.p1.1.m1.2.2.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.2"></in><apply id="S3.Thmtheorem6.p1.1.m1.2.2.3.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.1.m1.2.2.3.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.3">subscript</csymbol><ci id="S3.Thmtheorem6.p1.1.m1.2.2.3.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.3.2">𝜂</ci><cn id="S3.Thmtheorem6.p1.1.m1.2.2.3.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.1.m1.2.2.3.3">0</cn></apply><apply id="S3.Thmtheorem6.p1.1.m1.2.2.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1"><times id="S3.Thmtheorem6.p1.1.m1.2.2.1.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.2"></times><apply id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3">superscript</csymbol><apply id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3">subscript</csymbol><ci id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.2">𝐶</ci><ci id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.3.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3.2.3">c</ci></apply><infinity id="S3.Thmtheorem6.p1.1.m1.2.2.1.3.3.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.3.3"></infinity></apply><interval closure="open" id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1"><apply id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1"><minus id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1"></minus><cn id="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml" type="integer" xref="S3.Thmtheorem6.p1.1.m1.2.2.1.1.1.1.1.1.2">1</cn></apply><cn id="S3.Thmtheorem6.p1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem6.p1.1.m1.1.1">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.1.m1.2c">\eta_{0}\in C_{\mathrm{c}}^{\infty}((-1,1))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.1.m1.2d">italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( ( - 1 , 1 ) )</annotation></semantics></math> and <math alttext="\eta\in C_{\mathrm{c}}^{\infty}((1,\frac{3}{2}))" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.2.m2.3"><semantics id="S3.Thmtheorem6.p1.2.m2.3a"><mrow id="S3.Thmtheorem6.p1.2.m2.3.3" xref="S3.Thmtheorem6.p1.2.m2.3.3.cmml"><mi id="S3.Thmtheorem6.p1.2.m2.3.3.3" xref="S3.Thmtheorem6.p1.2.m2.3.3.3.cmml">η</mi><mo id="S3.Thmtheorem6.p1.2.m2.3.3.2" xref="S3.Thmtheorem6.p1.2.m2.3.3.2.cmml">∈</mo><mrow id="S3.Thmtheorem6.p1.2.m2.3.3.1" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.cmml"><msubsup id="S3.Thmtheorem6.p1.2.m2.3.3.1.3" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3.cmml"><mi id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.2" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.2.cmml">C</mi><mi id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.3.cmml">c</mi><mi id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3.3.cmml">∞</mi></msubsup><mo id="S3.Thmtheorem6.p1.2.m2.3.3.1.2" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.cmml"><mo id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.cmml">(</mo><mrow id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.2" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.2.1" stretchy="false" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.1.cmml">(</mo><mn id="S3.Thmtheorem6.p1.2.m2.1.1" xref="S3.Thmtheorem6.p1.2.m2.1.1.cmml">1</mn><mo id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.2.2" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.1.cmml">,</mo><mfrac id="S3.Thmtheorem6.p1.2.m2.2.2" xref="S3.Thmtheorem6.p1.2.m2.2.2.cmml"><mn id="S3.Thmtheorem6.p1.2.m2.2.2.2" xref="S3.Thmtheorem6.p1.2.m2.2.2.2.cmml">3</mn><mn id="S3.Thmtheorem6.p1.2.m2.2.2.3" xref="S3.Thmtheorem6.p1.2.m2.2.2.3.cmml">2</mn></mfrac><mo id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.2.3" stretchy="false" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.2.m2.3b"><apply id="S3.Thmtheorem6.p1.2.m2.3.3.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3"><in id="S3.Thmtheorem6.p1.2.m2.3.3.2.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.2"></in><ci id="S3.Thmtheorem6.p1.2.m2.3.3.3.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.3">𝜂</ci><apply id="S3.Thmtheorem6.p1.2.m2.3.3.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1"><times id="S3.Thmtheorem6.p1.2.m2.3.3.1.2.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.2"></times><apply id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3">superscript</csymbol><apply id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3">subscript</csymbol><ci id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.2.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.2">𝐶</ci><ci id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.3.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3.2.3">c</ci></apply><infinity id="S3.Thmtheorem6.p1.2.m2.3.3.1.3.3.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.3.3"></infinity></apply><interval closure="open" id="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.3.3.1.1.1.1.2"><cn id="S3.Thmtheorem6.p1.2.m2.1.1.cmml" type="integer" xref="S3.Thmtheorem6.p1.2.m2.1.1">1</cn><apply id="S3.Thmtheorem6.p1.2.m2.2.2.cmml" xref="S3.Thmtheorem6.p1.2.m2.2.2"><divide id="S3.Thmtheorem6.p1.2.m2.2.2.1.cmml" xref="S3.Thmtheorem6.p1.2.m2.2.2"></divide><cn id="S3.Thmtheorem6.p1.2.m2.2.2.2.cmml" type="integer" xref="S3.Thmtheorem6.p1.2.m2.2.2.2">3</cn><cn id="S3.Thmtheorem6.p1.2.m2.2.2.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.2.m2.2.2.3">2</cn></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.2.m2.3c">\eta\in C_{\mathrm{c}}^{\infty}((1,\frac{3}{2}))</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.2.m2.3d">italic_η ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( ( 1 , divide start_ARG 3 end_ARG start_ARG 2 end_ARG ) )</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{F}^{-1}\eta_{0}(0)=\mathcal{F}^{-1}\eta(0)=1," class="ltx_Math" display="block" id="S3.E3.m1.3"><semantics id="S3.E3.m1.3a"><mrow id="S3.E3.m1.3.3.1" xref="S3.E3.m1.3.3.1.1.cmml"><mrow id="S3.E3.m1.3.3.1.1" xref="S3.E3.m1.3.3.1.1.cmml"><mrow id="S3.E3.m1.3.3.1.1.2" xref="S3.E3.m1.3.3.1.1.2.cmml"><msup id="S3.E3.m1.3.3.1.1.2.2" xref="S3.E3.m1.3.3.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E3.m1.3.3.1.1.2.2.2" xref="S3.E3.m1.3.3.1.1.2.2.2.cmml">ℱ</mi><mrow id="S3.E3.m1.3.3.1.1.2.2.3" xref="S3.E3.m1.3.3.1.1.2.2.3.cmml"><mo id="S3.E3.m1.3.3.1.1.2.2.3a" xref="S3.E3.m1.3.3.1.1.2.2.3.cmml">−</mo><mn id="S3.E3.m1.3.3.1.1.2.2.3.2" xref="S3.E3.m1.3.3.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.E3.m1.3.3.1.1.2.1" xref="S3.E3.m1.3.3.1.1.2.1.cmml">⁢</mo><msub id="S3.E3.m1.3.3.1.1.2.3" xref="S3.E3.m1.3.3.1.1.2.3.cmml"><mi id="S3.E3.m1.3.3.1.1.2.3.2" xref="S3.E3.m1.3.3.1.1.2.3.2.cmml">η</mi><mn id="S3.E3.m1.3.3.1.1.2.3.3" xref="S3.E3.m1.3.3.1.1.2.3.3.cmml">0</mn></msub><mo id="S3.E3.m1.3.3.1.1.2.1a" xref="S3.E3.m1.3.3.1.1.2.1.cmml">⁢</mo><mrow id="S3.E3.m1.3.3.1.1.2.4.2" xref="S3.E3.m1.3.3.1.1.2.cmml"><mo id="S3.E3.m1.3.3.1.1.2.4.2.1" stretchy="false" xref="S3.E3.m1.3.3.1.1.2.cmml">(</mo><mn id="S3.E3.m1.1.1" xref="S3.E3.m1.1.1.cmml">0</mn><mo id="S3.E3.m1.3.3.1.1.2.4.2.2" stretchy="false" xref="S3.E3.m1.3.3.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E3.m1.3.3.1.1.3" xref="S3.E3.m1.3.3.1.1.3.cmml">=</mo><mrow id="S3.E3.m1.3.3.1.1.4" xref="S3.E3.m1.3.3.1.1.4.cmml"><msup id="S3.E3.m1.3.3.1.1.4.2" xref="S3.E3.m1.3.3.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E3.m1.3.3.1.1.4.2.2" xref="S3.E3.m1.3.3.1.1.4.2.2.cmml">ℱ</mi><mrow id="S3.E3.m1.3.3.1.1.4.2.3" xref="S3.E3.m1.3.3.1.1.4.2.3.cmml"><mo id="S3.E3.m1.3.3.1.1.4.2.3a" xref="S3.E3.m1.3.3.1.1.4.2.3.cmml">−</mo><mn id="S3.E3.m1.3.3.1.1.4.2.3.2" xref="S3.E3.m1.3.3.1.1.4.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.E3.m1.3.3.1.1.4.1" xref="S3.E3.m1.3.3.1.1.4.1.cmml">⁢</mo><mi id="S3.E3.m1.3.3.1.1.4.3" xref="S3.E3.m1.3.3.1.1.4.3.cmml">η</mi><mo id="S3.E3.m1.3.3.1.1.4.1a" xref="S3.E3.m1.3.3.1.1.4.1.cmml">⁢</mo><mrow id="S3.E3.m1.3.3.1.1.4.4.2" xref="S3.E3.m1.3.3.1.1.4.cmml"><mo id="S3.E3.m1.3.3.1.1.4.4.2.1" stretchy="false" xref="S3.E3.m1.3.3.1.1.4.cmml">(</mo><mn id="S3.E3.m1.2.2" xref="S3.E3.m1.2.2.cmml">0</mn><mo id="S3.E3.m1.3.3.1.1.4.4.2.2" stretchy="false" xref="S3.E3.m1.3.3.1.1.4.cmml">)</mo></mrow></mrow><mo id="S3.E3.m1.3.3.1.1.5" xref="S3.E3.m1.3.3.1.1.5.cmml">=</mo><mn id="S3.E3.m1.3.3.1.1.6" xref="S3.E3.m1.3.3.1.1.6.cmml">1</mn></mrow><mo id="S3.E3.m1.3.3.1.2" xref="S3.E3.m1.3.3.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E3.m1.3b"><apply id="S3.E3.m1.3.3.1.1.cmml" xref="S3.E3.m1.3.3.1"><and id="S3.E3.m1.3.3.1.1a.cmml" xref="S3.E3.m1.3.3.1"></and><apply id="S3.E3.m1.3.3.1.1b.cmml" xref="S3.E3.m1.3.3.1"><eq id="S3.E3.m1.3.3.1.1.3.cmml" xref="S3.E3.m1.3.3.1.1.3"></eq><apply id="S3.E3.m1.3.3.1.1.2.cmml" xref="S3.E3.m1.3.3.1.1.2"><times id="S3.E3.m1.3.3.1.1.2.1.cmml" xref="S3.E3.m1.3.3.1.1.2.1"></times><apply id="S3.E3.m1.3.3.1.1.2.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.2.2.1.cmml" xref="S3.E3.m1.3.3.1.1.2.2">superscript</csymbol><ci id="S3.E3.m1.3.3.1.1.2.2.2.cmml" xref="S3.E3.m1.3.3.1.1.2.2.2">ℱ</ci><apply id="S3.E3.m1.3.3.1.1.2.2.3.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3"><minus id="S3.E3.m1.3.3.1.1.2.2.3.1.cmml" xref="S3.E3.m1.3.3.1.1.2.2.3"></minus><cn id="S3.E3.m1.3.3.1.1.2.2.3.2.cmml" type="integer" xref="S3.E3.m1.3.3.1.1.2.2.3.2">1</cn></apply></apply><apply id="S3.E3.m1.3.3.1.1.2.3.cmml" xref="S3.E3.m1.3.3.1.1.2.3"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.2.3.1.cmml" xref="S3.E3.m1.3.3.1.1.2.3">subscript</csymbol><ci id="S3.E3.m1.3.3.1.1.2.3.2.cmml" xref="S3.E3.m1.3.3.1.1.2.3.2">𝜂</ci><cn id="S3.E3.m1.3.3.1.1.2.3.3.cmml" type="integer" xref="S3.E3.m1.3.3.1.1.2.3.3">0</cn></apply><cn id="S3.E3.m1.1.1.cmml" type="integer" xref="S3.E3.m1.1.1">0</cn></apply><apply id="S3.E3.m1.3.3.1.1.4.cmml" xref="S3.E3.m1.3.3.1.1.4"><times id="S3.E3.m1.3.3.1.1.4.1.cmml" xref="S3.E3.m1.3.3.1.1.4.1"></times><apply id="S3.E3.m1.3.3.1.1.4.2.cmml" xref="S3.E3.m1.3.3.1.1.4.2"><csymbol cd="ambiguous" id="S3.E3.m1.3.3.1.1.4.2.1.cmml" xref="S3.E3.m1.3.3.1.1.4.2">superscript</csymbol><ci id="S3.E3.m1.3.3.1.1.4.2.2.cmml" xref="S3.E3.m1.3.3.1.1.4.2.2">ℱ</ci><apply id="S3.E3.m1.3.3.1.1.4.2.3.cmml" xref="S3.E3.m1.3.3.1.1.4.2.3"><minus id="S3.E3.m1.3.3.1.1.4.2.3.1.cmml" xref="S3.E3.m1.3.3.1.1.4.2.3"></minus><cn id="S3.E3.m1.3.3.1.1.4.2.3.2.cmml" type="integer" xref="S3.E3.m1.3.3.1.1.4.2.3.2">1</cn></apply></apply><ci id="S3.E3.m1.3.3.1.1.4.3.cmml" xref="S3.E3.m1.3.3.1.1.4.3">𝜂</ci><cn id="S3.E3.m1.2.2.cmml" type="integer" xref="S3.E3.m1.2.2">0</cn></apply></apply><apply id="S3.E3.m1.3.3.1.1c.cmml" xref="S3.E3.m1.3.3.1"><eq id="S3.E3.m1.3.3.1.1.5.cmml" xref="S3.E3.m1.3.3.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S3.E3.m1.3.3.1.1.4.cmml" id="S3.E3.m1.3.3.1.1d.cmml" xref="S3.E3.m1.3.3.1"></share><cn id="S3.E3.m1.3.3.1.1.6.cmml" type="integer" xref="S3.E3.m1.3.3.1.1.6">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E3.m1.3c">\mathcal{F}^{-1}\eta_{0}(0)=\mathcal{F}^{-1}\eta(0)=1,</annotation><annotation encoding="application/x-llamapun" id="S3.E3.m1.3d">caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 0 ) = caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η ( 0 ) = 1 ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem6.p1.5">where <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.3.m1.1"><semantics id="S3.Thmtheorem6.p1.3.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.Thmtheorem6.p1.3.m1.1.1" xref="S3.Thmtheorem6.p1.3.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.3.m1.1b"><ci id="S3.Thmtheorem6.p1.3.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.3.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.3.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.3.m1.1d">caligraphic_F</annotation></semantics></math> is the one-dimensional Fourier transform. For <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.4.m2.1"><semantics id="S3.Thmtheorem6.p1.4.m2.1a"><mrow id="S3.Thmtheorem6.p1.4.m2.1.1" xref="S3.Thmtheorem6.p1.4.m2.1.1.cmml"><mi id="S3.Thmtheorem6.p1.4.m2.1.1.2" xref="S3.Thmtheorem6.p1.4.m2.1.1.2.cmml">m</mi><mo id="S3.Thmtheorem6.p1.4.m2.1.1.1" xref="S3.Thmtheorem6.p1.4.m2.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem6.p1.4.m2.1.1.3" xref="S3.Thmtheorem6.p1.4.m2.1.1.3.cmml"><mi id="S3.Thmtheorem6.p1.4.m2.1.1.3.2" xref="S3.Thmtheorem6.p1.4.m2.1.1.3.2.cmml">ℕ</mi><mn id="S3.Thmtheorem6.p1.4.m2.1.1.3.3" xref="S3.Thmtheorem6.p1.4.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.4.m2.1b"><apply id="S3.Thmtheorem6.p1.4.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m2.1.1"><in id="S3.Thmtheorem6.p1.4.m2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.4.m2.1.1.1"></in><ci id="S3.Thmtheorem6.p1.4.m2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.4.m2.1.1.2">𝑚</ci><apply id="S3.Thmtheorem6.p1.4.m2.1.1.3.cmml" xref="S3.Thmtheorem6.p1.4.m2.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.4.m2.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.4.m2.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem6.p1.4.m2.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.4.m2.1.1.3.2">ℕ</ci><cn id="S3.Thmtheorem6.p1.4.m2.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.4.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.4.m2.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.4.m2.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="D:=-{\rm i}\partial_{1}" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.5.m3.1"><semantics id="S3.Thmtheorem6.p1.5.m3.1a"><mrow id="S3.Thmtheorem6.p1.5.m3.1.1" xref="S3.Thmtheorem6.p1.5.m3.1.1.cmml"><mi id="S3.Thmtheorem6.p1.5.m3.1.1.2" xref="S3.Thmtheorem6.p1.5.m3.1.1.2.cmml">D</mi><mo id="S3.Thmtheorem6.p1.5.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.Thmtheorem6.p1.5.m3.1.1.1.cmml">:=</mo><mrow id="S3.Thmtheorem6.p1.5.m3.1.1.3" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.cmml"><mo id="S3.Thmtheorem6.p1.5.m3.1.1.3a" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.cmml">−</mo><mrow id="S3.Thmtheorem6.p1.5.m3.1.1.3.2" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.cmml"><mi id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.2" mathvariant="normal" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.2.cmml">i</mi><mo id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.1" lspace="0em" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.1.cmml">⁢</mo><msub id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.cmml"><mo id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.2" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.2.cmml">∂</mo><mn id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.3" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.3.cmml">1</mn></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.5.m3.1b"><apply id="S3.Thmtheorem6.p1.5.m3.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1"><csymbol cd="latexml" id="S3.Thmtheorem6.p1.5.m3.1.1.1.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.1">assign</csymbol><ci id="S3.Thmtheorem6.p1.5.m3.1.1.2.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.2">𝐷</ci><apply id="S3.Thmtheorem6.p1.5.m3.1.1.3.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3"><minus id="S3.Thmtheorem6.p1.5.m3.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3"></minus><apply id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2"><times id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.1.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.1"></times><ci id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.2.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.2">i</ci><apply id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.1.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3">subscript</csymbol><partialdiff id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.2.cmml" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.2"></partialdiff><cn id="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.5.m3.1.1.3.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.5.m3.1c">D:=-{\rm i}\partial_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.5.m3.1d">italic_D := - roman_i ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we define</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\eta_{0}^{m}:=(-D)^{m}\eta_{0}/m!\quad\text{ and }\quad\eta_{j}^{m}:=(-D)^{m}% \eta(2^{-j}\cdot)/m!,\quad j\in\mathbb{N}_{1}." class="ltx_math_unparsed" display="block" id="S3.Ex23.m1.1"><semantics id="S3.Ex23.m1.1a"><mrow id="S3.Ex23.m1.1b"><msubsup id="S3.Ex23.m1.1.2"><mi id="S3.Ex23.m1.1.2.2.2">η</mi><mn id="S3.Ex23.m1.1.2.2.3">0</mn><mi id="S3.Ex23.m1.1.2.3">m</mi></msubsup><mo id="S3.Ex23.m1.1.3" lspace="0.278em" rspace="0.278em">:=</mo><msup id="S3.Ex23.m1.1.4"><mrow id="S3.Ex23.m1.1.4.2"><mo id="S3.Ex23.m1.1.4.2.1" stretchy="false">(</mo><mo id="S3.Ex23.m1.1.4.2.2" lspace="0em">−</mo><mi id="S3.Ex23.m1.1.4.2.3">D</mi><mo id="S3.Ex23.m1.1.4.2.4" stretchy="false">)</mo></mrow><mi id="S3.Ex23.m1.1.4.3">m</mi></msup><msub id="S3.Ex23.m1.1.5"><mi id="S3.Ex23.m1.1.5.2">η</mi><mn id="S3.Ex23.m1.1.5.3">0</mn></msub><mo id="S3.Ex23.m1.1.6">/</mo><mi id="S3.Ex23.m1.1.7">m</mi><mo id="S3.Ex23.m1.1.8">!</mo><mspace id="S3.Ex23.m1.1.9" width="1em"></mspace><mtext id="S3.Ex23.m1.1.1"> and </mtext><mspace id="S3.Ex23.m1.1.10" width="1em"></mspace><msubsup id="S3.Ex23.m1.1.11"><mi id="S3.Ex23.m1.1.11.2.2">η</mi><mi id="S3.Ex23.m1.1.11.2.3">j</mi><mi id="S3.Ex23.m1.1.11.3">m</mi></msubsup><mo id="S3.Ex23.m1.1.12" lspace="0.278em" rspace="0.278em">:=</mo><msup id="S3.Ex23.m1.1.13"><mrow id="S3.Ex23.m1.1.13.2"><mo id="S3.Ex23.m1.1.13.2.1" stretchy="false">(</mo><mo id="S3.Ex23.m1.1.13.2.2" lspace="0em">−</mo><mi id="S3.Ex23.m1.1.13.2.3">D</mi><mo id="S3.Ex23.m1.1.13.2.4" stretchy="false">)</mo></mrow><mi id="S3.Ex23.m1.1.13.3">m</mi></msup><mi id="S3.Ex23.m1.1.14">η</mi><mrow id="S3.Ex23.m1.1.15"><mo id="S3.Ex23.m1.1.15.1" stretchy="false">(</mo><msup id="S3.Ex23.m1.1.15.2"><mn id="S3.Ex23.m1.1.15.2.2">2</mn><mrow id="S3.Ex23.m1.1.15.2.3"><mo id="S3.Ex23.m1.1.15.2.3a">−</mo><mi id="S3.Ex23.m1.1.15.2.3.2">j</mi></mrow></msup><mo id="S3.Ex23.m1.1.15.3" lspace="0.222em" rspace="0em">⋅</mo><mo id="S3.Ex23.m1.1.15.4" stretchy="false">)</mo></mrow><mo id="S3.Ex23.m1.1.16">/</mo><mi id="S3.Ex23.m1.1.17">m</mi><mo id="S3.Ex23.m1.1.18">!</mo><mo id="S3.Ex23.m1.1.19" rspace="1.167em">,</mo><mi id="S3.Ex23.m1.1.20">j</mi><mo id="S3.Ex23.m1.1.21">∈</mo><msub id="S3.Ex23.m1.1.22"><mi id="S3.Ex23.m1.1.22.2">ℕ</mi><mn id="S3.Ex23.m1.1.22.3">1</mn></msub><mo id="S3.Ex23.m1.1.23" lspace="0em">.</mo></mrow><annotation encoding="application/x-tex" id="S3.Ex23.m1.1c">\eta_{0}^{m}:=(-D)^{m}\eta_{0}/m!\quad\text{ and }\quad\eta_{j}^{m}:=(-D)^{m}% \eta(2^{-j}\cdot)/m!,\quad j\in\mathbb{N}_{1}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex23.m1.1d">italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT := ( - italic_D ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT / italic_m ! and italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT := ( - italic_D ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_η ( 2 start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT ⋅ ) / italic_m ! , italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem6.p1.7">Let <math alttext="(\phi_{j})_{j\geq 0}\in\Phi(\mathbb{R}^{d-1})" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.6.m1.2"><semantics id="S3.Thmtheorem6.p1.6.m1.2a"><mrow id="S3.Thmtheorem6.p1.6.m1.2.2" xref="S3.Thmtheorem6.p1.6.m1.2.2.cmml"><msub id="S3.Thmtheorem6.p1.6.m1.1.1.1" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.cmml"><mrow id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.cmml"><mo id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.cmml"><mi id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.2" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.3" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.Thmtheorem6.p1.6.m1.1.1.1.3" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3.cmml"><mi id="S3.Thmtheorem6.p1.6.m1.1.1.1.3.2" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3.2.cmml">j</mi><mo id="S3.Thmtheorem6.p1.6.m1.1.1.1.3.1" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3.1.cmml">≥</mo><mn id="S3.Thmtheorem6.p1.6.m1.1.1.1.3.3" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3.3.cmml">0</mn></mrow></msub><mo id="S3.Thmtheorem6.p1.6.m1.2.2.3" xref="S3.Thmtheorem6.p1.6.m1.2.2.3.cmml">∈</mo><mrow id="S3.Thmtheorem6.p1.6.m1.2.2.2" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.cmml"><mi id="S3.Thmtheorem6.p1.6.m1.2.2.2.3" mathvariant="normal" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.3.cmml">Φ</mi><mo id="S3.Thmtheorem6.p1.6.m1.2.2.2.2" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.2.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.cmml"><mo id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.cmml">(</mo><msup id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.cmml"><mi id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.2" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.2.cmml">ℝ</mi><mrow id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.cmml"><mi id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.2" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.2.cmml">d</mi><mo id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.1" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.1.cmml">−</mo><mn id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.3" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.3" stretchy="false" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.6.m1.2b"><apply id="S3.Thmtheorem6.p1.6.m1.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2"><in id="S3.Thmtheorem6.p1.6.m1.2.2.3.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.3"></in><apply id="S3.Thmtheorem6.p1.6.m1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.6.m1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1">subscript</csymbol><apply id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.2">italic-ϕ</ci><ci id="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.1.1.1.3">𝑗</ci></apply><apply id="S3.Thmtheorem6.p1.6.m1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3"><geq id="S3.Thmtheorem6.p1.6.m1.1.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3.1"></geq><ci id="S3.Thmtheorem6.p1.6.m1.1.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3.2">𝑗</ci><cn id="S3.Thmtheorem6.p1.6.m1.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.6.m1.1.1.1.3.3">0</cn></apply></apply><apply id="S3.Thmtheorem6.p1.6.m1.2.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2"><times id="S3.Thmtheorem6.p1.6.m1.2.2.2.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.2"></times><ci id="S3.Thmtheorem6.p1.6.m1.2.2.2.3.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.3">Φ</ci><apply id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1">superscript</csymbol><ci id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.2">ℝ</ci><apply id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3"><minus id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.1"></minus><ci id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.2">𝑑</ci><cn id="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.6.m1.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.6.m1.2c">(\phi_{j})_{j\geq 0}\in\Phi(\mathbb{R}^{d-1})</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.6.m1.2d">( italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT )</annotation></semantics></math>. For any <math alttext="g\in\SS(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.Thmtheorem6.p1.7.m2.2"><semantics id="S3.Thmtheorem6.p1.7.m2.2a"><mrow id="S3.Thmtheorem6.p1.7.m2.2.2" xref="S3.Thmtheorem6.p1.7.m2.2.2.cmml"><mi id="S3.Thmtheorem6.p1.7.m2.2.2.3" xref="S3.Thmtheorem6.p1.7.m2.2.2.3.cmml">g</mi><mo id="S3.Thmtheorem6.p1.7.m2.2.2.2" xref="S3.Thmtheorem6.p1.7.m2.2.2.2.cmml">∈</mo><mrow id="S3.Thmtheorem6.p1.7.m2.2.2.1" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.cmml"><mi id="S3.Thmtheorem6.p1.7.m2.2.2.1.3" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.3.cmml">SS</mi><mo id="S3.Thmtheorem6.p1.7.m2.2.2.1.2" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.2.cmml">⁢</mo><mrow id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.2.cmml"><mo id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.2" stretchy="false" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.2.cmml">(</mo><msup id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.cmml"><mi id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.2" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.cmml"><mi id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.2" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.2.cmml">d</mi><mo id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.1" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.1.cmml">−</mo><mn id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.3" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.3" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.2.cmml">;</mo><mi id="S3.Thmtheorem6.p1.7.m2.1.1" xref="S3.Thmtheorem6.p1.7.m2.1.1.cmml">X</mi><mo id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.4" stretchy="false" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem6.p1.7.m2.2b"><apply id="S3.Thmtheorem6.p1.7.m2.2.2.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2"><in id="S3.Thmtheorem6.p1.7.m2.2.2.2.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.2"></in><ci id="S3.Thmtheorem6.p1.7.m2.2.2.3.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.3">𝑔</ci><apply id="S3.Thmtheorem6.p1.7.m2.2.2.1.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1"><times id="S3.Thmtheorem6.p1.7.m2.2.2.1.2.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.2"></times><ci id="S3.Thmtheorem6.p1.7.m2.2.2.1.3.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.3">SS</ci><list id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.2.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1"><apply id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.2">ℝ</ci><apply id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3"><minus id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.1.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.1"></minus><ci id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.2.cmml" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.2">𝑑</ci><cn id="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem6.p1.7.m2.2.2.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.Thmtheorem6.p1.7.m2.1.1.cmml" xref="S3.Thmtheorem6.p1.7.m2.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.7.m2.2c">g\in\SS(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.7.m2.2d">italic_g ∈ roman_SS ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> we define</p> <table class="ltx_equation ltx_eqn_table" id="S3.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{ext}_{m}g(x_{1},\widetilde{x}):=\sum_{j=0}^{\infty}2^{-j}(% \mathcal{F}^{-1}\eta_{j}^{m})(x_{1})(\phi_{j}\ast g)(\widetilde{x})," class="ltx_Math" display="block" id="S3.E4.m1.3"><semantics id="S3.E4.m1.3a"><mrow id="S3.E4.m1.3.3.1" xref="S3.E4.m1.3.3.1.1.cmml"><mrow id="S3.E4.m1.3.3.1.1" xref="S3.E4.m1.3.3.1.1.cmml"><mrow id="S3.E4.m1.3.3.1.1.1" xref="S3.E4.m1.3.3.1.1.1.cmml"><mrow id="S3.E4.m1.3.3.1.1.1.3" xref="S3.E4.m1.3.3.1.1.1.3.cmml"><msub id="S3.E4.m1.3.3.1.1.1.3.1" xref="S3.E4.m1.3.3.1.1.1.3.1.cmml"><mi id="S3.E4.m1.3.3.1.1.1.3.1.2" xref="S3.E4.m1.3.3.1.1.1.3.1.2.cmml">ext</mi><mi id="S3.E4.m1.3.3.1.1.1.3.1.3" xref="S3.E4.m1.3.3.1.1.1.3.1.3.cmml">m</mi></msub><mo id="S3.E4.m1.3.3.1.1.1.3a" lspace="0.167em" xref="S3.E4.m1.3.3.1.1.1.3.cmml">⁡</mo><mi id="S3.E4.m1.3.3.1.1.1.3.2" xref="S3.E4.m1.3.3.1.1.1.3.2.cmml">g</mi></mrow><mo id="S3.E4.m1.3.3.1.1.1.2" xref="S3.E4.m1.3.3.1.1.1.2.cmml">⁢</mo><mrow id="S3.E4.m1.3.3.1.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.2.cmml"><mo id="S3.E4.m1.3.3.1.1.1.1.1.2" stretchy="false" xref="S3.E4.m1.3.3.1.1.1.1.2.cmml">(</mo><msub id="S3.E4.m1.3.3.1.1.1.1.1.1" xref="S3.E4.m1.3.3.1.1.1.1.1.1.cmml"><mi id="S3.E4.m1.3.3.1.1.1.1.1.1.2" xref="S3.E4.m1.3.3.1.1.1.1.1.1.2.cmml">x</mi><mn id="S3.E4.m1.3.3.1.1.1.1.1.1.3" xref="S3.E4.m1.3.3.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.E4.m1.3.3.1.1.1.1.1.3" xref="S3.E4.m1.3.3.1.1.1.1.2.cmml">,</mo><mover accent="true" id="S3.E4.m1.1.1" xref="S3.E4.m1.1.1.cmml"><mi id="S3.E4.m1.1.1.2" xref="S3.E4.m1.1.1.2.cmml">x</mi><mo id="S3.E4.m1.1.1.1" xref="S3.E4.m1.1.1.1.cmml">~</mo></mover><mo id="S3.E4.m1.3.3.1.1.1.1.1.4" rspace="0.278em" stretchy="false" xref="S3.E4.m1.3.3.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.E4.m1.3.3.1.1.5" rspace="0.111em" xref="S3.E4.m1.3.3.1.1.5.cmml">:=</mo><mrow id="S3.E4.m1.3.3.1.1.4" xref="S3.E4.m1.3.3.1.1.4.cmml"><munderover id="S3.E4.m1.3.3.1.1.4.4" xref="S3.E4.m1.3.3.1.1.4.4.cmml"><mo id="S3.E4.m1.3.3.1.1.4.4.2.2" movablelimits="false" xref="S3.E4.m1.3.3.1.1.4.4.2.2.cmml">∑</mo><mrow id="S3.E4.m1.3.3.1.1.4.4.2.3" xref="S3.E4.m1.3.3.1.1.4.4.2.3.cmml"><mi id="S3.E4.m1.3.3.1.1.4.4.2.3.2" xref="S3.E4.m1.3.3.1.1.4.4.2.3.2.cmml">j</mi><mo id="S3.E4.m1.3.3.1.1.4.4.2.3.1" xref="S3.E4.m1.3.3.1.1.4.4.2.3.1.cmml">=</mo><mn id="S3.E4.m1.3.3.1.1.4.4.2.3.3" xref="S3.E4.m1.3.3.1.1.4.4.2.3.3.cmml">0</mn></mrow><mi id="S3.E4.m1.3.3.1.1.4.4.3" mathvariant="normal" xref="S3.E4.m1.3.3.1.1.4.4.3.cmml">∞</mi></munderover><mrow id="S3.E4.m1.3.3.1.1.4.3" xref="S3.E4.m1.3.3.1.1.4.3.cmml"><msup id="S3.E4.m1.3.3.1.1.4.3.5" xref="S3.E4.m1.3.3.1.1.4.3.5.cmml"><mn id="S3.E4.m1.3.3.1.1.4.3.5.2" xref="S3.E4.m1.3.3.1.1.4.3.5.2.cmml">2</mn><mrow id="S3.E4.m1.3.3.1.1.4.3.5.3" xref="S3.E4.m1.3.3.1.1.4.3.5.3.cmml"><mo id="S3.E4.m1.3.3.1.1.4.3.5.3a" xref="S3.E4.m1.3.3.1.1.4.3.5.3.cmml">−</mo><mi id="S3.E4.m1.3.3.1.1.4.3.5.3.2" xref="S3.E4.m1.3.3.1.1.4.3.5.3.2.cmml">j</mi></mrow></msup><mo id="S3.E4.m1.3.3.1.1.4.3.4" xref="S3.E4.m1.3.3.1.1.4.3.4.cmml">⁢</mo><mrow id="S3.E4.m1.3.3.1.1.2.1.1.1" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.cmml"><mo id="S3.E4.m1.3.3.1.1.2.1.1.1.2" stretchy="false" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.cmml">(</mo><mrow id="S3.E4.m1.3.3.1.1.2.1.1.1.1" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.cmml"><msup id="S3.E4.m1.3.3.1.1.2.1.1.1.1.2" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.2" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.2.cmml">ℱ</mi><mrow id="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.3" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.3.cmml"><mo id="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.3a" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.3.cmml">−</mo><mn id="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.3.2" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S3.E4.m1.3.3.1.1.2.1.1.1.1.1" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.1.cmml">⁢</mo><msubsup id="S3.E4.m1.3.3.1.1.2.1.1.1.1.3" xref="S3.E4.m1.3.3.1.1.2.1.1.1.1.3.cmml"><mi 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xref="S3.Thmtheorem6.p1.8.m1.2.2.1.1.1">superscript</csymbol><ci id="S3.Thmtheorem6.p1.8.m1.2.2.1.1.1.2.cmml" xref="S3.Thmtheorem6.p1.8.m1.2.2.1.1.1.2">ℝ</ci><ci id="S3.Thmtheorem6.p1.8.m1.2.2.1.1.1.3.cmml" xref="S3.Thmtheorem6.p1.8.m1.2.2.1.1.1.3">𝑑</ci></apply><ci id="S3.Thmtheorem6.p1.8.m1.1.1.cmml" xref="S3.Thmtheorem6.p1.8.m1.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem6.p1.8.m1.2c">\SS^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem6.p1.8.m1.2d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S3.SS2.p3"> <p class="ltx_p" id="S3.SS2.p3.1">The following lemma is analogous to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Lemma VIII.1.2.7]</cite> and shows the continuity of the extension operator <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S3.SS2.p3.1.m1.1"><semantics id="S3.SS2.p3.1.m1.1a"><msub id="S3.SS2.p3.1.m1.1.1" xref="S3.SS2.p3.1.m1.1.1.cmml"><mi id="S3.SS2.p3.1.m1.1.1.2" xref="S3.SS2.p3.1.m1.1.1.2.cmml">ext</mi><mi id="S3.SS2.p3.1.m1.1.1.3" xref="S3.SS2.p3.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.p3.1.m1.1b"><apply id="S3.SS2.p3.1.m1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.p3.1.m1.1.1.1.cmml" xref="S3.SS2.p3.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.p3.1.m1.1.1.2.cmml" xref="S3.SS2.p3.1.m1.1.1.2">ext</ci><ci id="S3.SS2.p3.1.m1.1.1.3.cmml" xref="S3.SS2.p3.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.p3.1.m1.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.p3.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> on Besov spaces.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S3.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.1.1.1">Lemma 3.7</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem7.p1"> <p class="ltx_p" id="S3.Thmtheorem7.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.6.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.1.1.m1.2"><semantics id="S3.Thmtheorem7.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem7.p1.1.1.m1.2.3" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem7.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem7.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem7.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem7.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem7.p1.1.1.m1.1.1" xref="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem7.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem7.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem7.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem7.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.1.1.m1.2b"><apply id="S3.Thmtheorem7.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.2.3"><in id="S3.Thmtheorem7.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.1"></in><ci id="S3.Thmtheorem7.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.2.3.3.2"><cn id="S3.Thmtheorem7.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem7.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem7.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem7.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.2.2.m2.2"><semantics id="S3.Thmtheorem7.p1.2.2.m2.2a"><mrow id="S3.Thmtheorem7.p1.2.2.m2.2.3" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.cmml"><mi id="S3.Thmtheorem7.p1.2.2.m2.2.3.2" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.2.cmml">q</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.2.3.1" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem7.p1.2.2.m2.2.3.3.2" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.3.1.cmml"><mo id="S3.Thmtheorem7.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.3.1.cmml">[</mo><mn id="S3.Thmtheorem7.p1.2.2.m2.1.1" xref="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml">1</mn><mo id="S3.Thmtheorem7.p1.2.2.m2.2.3.3.2.2" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem7.p1.2.2.m2.2.2" mathvariant="normal" xref="S3.Thmtheorem7.p1.2.2.m2.2.2.cmml">∞</mi><mo id="S3.Thmtheorem7.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.2.2.m2.2b"><apply id="S3.Thmtheorem7.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3"><in id="S3.Thmtheorem7.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.1"></in><ci id="S3.Thmtheorem7.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S3.Thmtheorem7.p1.2.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.3.3.2"><cn id="S3.Thmtheorem7.p1.2.2.m2.1.1.cmml" type="integer" xref="S3.Thmtheorem7.p1.2.2.m2.1.1">1</cn><infinity id="S3.Thmtheorem7.p1.2.2.m2.2.2.cmml" xref="S3.Thmtheorem7.p1.2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.3.3.m3.1"><semantics id="S3.Thmtheorem7.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem7.p1.3.3.m3.1.1" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem7.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.2.cmml">m</mi><mo id="S3.Thmtheorem7.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem7.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem7.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S3.Thmtheorem7.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.3.3.m3.1b"><apply id="S3.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1"><in id="S3.Thmtheorem7.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem7.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.2">𝑚</ci><apply id="S3.Thmtheorem7.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem7.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem7.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.3.2">ℕ</ci><cn id="S3.Thmtheorem7.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem7.p1.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.3.3.m3.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.3.3.m3.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.4.4.m4.1"><semantics id="S3.Thmtheorem7.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem7.p1.4.4.m4.1.1" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem7.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem7.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem7.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.3.cmml"><mo id="S3.Thmtheorem7.p1.4.4.m4.1.1.3a" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem7.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.4.4.m4.1b"><apply id="S3.Thmtheorem7.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1"><gt id="S3.Thmtheorem7.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.1"></gt><ci id="S3.Thmtheorem7.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.2">𝛾</ci><apply id="S3.Thmtheorem7.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.3"><minus id="S3.Thmtheorem7.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.3"></minus><cn id="S3.Thmtheorem7.p1.4.4.m4.1.1.3.2.cmml" type="integer" xref="S3.Thmtheorem7.p1.4.4.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.4.4.m4.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.4.4.m4.1d">italic_γ > - 1</annotation></semantics></math>, <math alttext="s>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.5.5.m5.1"><semantics id="S3.Thmtheorem7.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem7.p1.5.5.m5.1.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.2.cmml">s</mi><mo id="S3.Thmtheorem7.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem7.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml">m</mi><mo id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.1.cmml">+</mo><mfrac id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.cmml"><mrow id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.cmml"><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.2" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.1" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.5.5.m5.1b"><apply id="S3.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1"><gt id="S3.Thmtheorem7.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.1"></gt><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.2">𝑠</ci><apply id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3"><plus id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.1"></plus><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.2">𝑚</ci><apply id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3"><divide id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3"></divide><apply id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2"><plus id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.1"></plus><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.2">𝛾</ci><cn id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.2.3">1</cn></apply><ci id="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.3.cmml" xref="S3.Thmtheorem7.p1.5.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.5.5.m5.1c">s>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.5.5.m5.1d">italic_s > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem7.p1.6.6.m6.1"><semantics id="S3.Thmtheorem7.p1.6.6.m6.1a"><mi id="S3.Thmtheorem7.p1.6.6.m6.1.1" xref="S3.Thmtheorem7.p1.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem7.p1.6.6.m6.1b"><ci id="S3.Thmtheorem7.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem7.p1.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem7.p1.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem7.p1.6.6.m6.1d">italic_X</annotation></semantics></math> be a Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{ext}_{m}:B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)\to B% ^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="block" id="S3.Ex24.m1.9"><semantics id="S3.Ex24.m1.9a"><mrow id="S3.Ex24.m1.9.9" xref="S3.Ex24.m1.9.9.cmml"><msub id="S3.Ex24.m1.9.9.5" xref="S3.Ex24.m1.9.9.5.cmml"><mi id="S3.Ex24.m1.9.9.5.2" xref="S3.Ex24.m1.9.9.5.2.cmml">ext</mi><mi id="S3.Ex24.m1.9.9.5.3" xref="S3.Ex24.m1.9.9.5.3.cmml">m</mi></msub><mo id="S3.Ex24.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S3.Ex24.m1.9.9.4.cmml">:</mo><mrow id="S3.Ex24.m1.9.9.3" xref="S3.Ex24.m1.9.9.3.cmml"><mrow id="S3.Ex24.m1.7.7.1.1" xref="S3.Ex24.m1.7.7.1.1.cmml"><msubsup id="S3.Ex24.m1.7.7.1.1.3" xref="S3.Ex24.m1.7.7.1.1.3.cmml"><mi id="S3.Ex24.m1.7.7.1.1.3.2.2" xref="S3.Ex24.m1.7.7.1.1.3.2.2.cmml">B</mi><mrow id="S3.Ex24.m1.2.2.2.4" xref="S3.Ex24.m1.2.2.2.3.cmml"><mi id="S3.Ex24.m1.1.1.1.1" xref="S3.Ex24.m1.1.1.1.1.cmml">p</mi><mo id="S3.Ex24.m1.2.2.2.4.1" xref="S3.Ex24.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex24.m1.2.2.2.2" xref="S3.Ex24.m1.2.2.2.2.cmml">q</mi></mrow><mrow id="S3.Ex24.m1.7.7.1.1.3.2.3" xref="S3.Ex24.m1.7.7.1.1.3.2.3.cmml"><mi id="S3.Ex24.m1.7.7.1.1.3.2.3.2" xref="S3.Ex24.m1.7.7.1.1.3.2.3.2.cmml">s</mi><mo id="S3.Ex24.m1.7.7.1.1.3.2.3.1" xref="S3.Ex24.m1.7.7.1.1.3.2.3.1.cmml">−</mo><mi id="S3.Ex24.m1.7.7.1.1.3.2.3.3" xref="S3.Ex24.m1.7.7.1.1.3.2.3.3.cmml">m</mi><mo id="S3.Ex24.m1.7.7.1.1.3.2.3.1a" xref="S3.Ex24.m1.7.7.1.1.3.2.3.1.cmml">−</mo><mfrac id="S3.Ex24.m1.7.7.1.1.3.2.3.4" xref="S3.Ex24.m1.7.7.1.1.3.2.3.4.cmml"><mrow id="S3.Ex24.m1.7.7.1.1.3.2.3.4.2" xref="S3.Ex24.m1.7.7.1.1.3.2.3.4.2.cmml"><mi id="S3.Ex24.m1.7.7.1.1.3.2.3.4.2.2" xref="S3.Ex24.m1.7.7.1.1.3.2.3.4.2.2.cmml">γ</mi><mo id="S3.Ex24.m1.7.7.1.1.3.2.3.4.2.1" xref="S3.Ex24.m1.7.7.1.1.3.2.3.4.2.1.cmml">+</mo><mn id="S3.Ex24.m1.7.7.1.1.3.2.3.4.2.3" xref="S3.Ex24.m1.7.7.1.1.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S3.Ex24.m1.7.7.1.1.3.2.3.4.3" xref="S3.Ex24.m1.7.7.1.1.3.2.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.Ex24.m1.7.7.1.1.2" xref="S3.Ex24.m1.7.7.1.1.2.cmml">⁢</mo><mrow id="S3.Ex24.m1.7.7.1.1.1.1" xref="S3.Ex24.m1.7.7.1.1.1.2.cmml"><mo id="S3.Ex24.m1.7.7.1.1.1.1.2" stretchy="false" xref="S3.Ex24.m1.7.7.1.1.1.2.cmml">(</mo><msup id="S3.Ex24.m1.7.7.1.1.1.1.1" xref="S3.Ex24.m1.7.7.1.1.1.1.1.cmml"><mi id="S3.Ex24.m1.7.7.1.1.1.1.1.2" xref="S3.Ex24.m1.7.7.1.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.Ex24.m1.7.7.1.1.1.1.1.3" xref="S3.Ex24.m1.7.7.1.1.1.1.1.3.cmml"><mi id="S3.Ex24.m1.7.7.1.1.1.1.1.3.2" xref="S3.Ex24.m1.7.7.1.1.1.1.1.3.2.cmml">d</mi><mo id="S3.Ex24.m1.7.7.1.1.1.1.1.3.1" xref="S3.Ex24.m1.7.7.1.1.1.1.1.3.1.cmml">−</mo><mn id="S3.Ex24.m1.7.7.1.1.1.1.1.3.3" xref="S3.Ex24.m1.7.7.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.Ex24.m1.7.7.1.1.1.1.3" xref="S3.Ex24.m1.7.7.1.1.1.2.cmml">;</mo><mi id="S3.Ex24.m1.5.5" xref="S3.Ex24.m1.5.5.cmml">X</mi><mo id="S3.Ex24.m1.7.7.1.1.1.1.4" stretchy="false" xref="S3.Ex24.m1.7.7.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S3.Ex24.m1.9.9.3.4" stretchy="false" xref="S3.Ex24.m1.9.9.3.4.cmml">→</mo><mrow id="S3.Ex24.m1.9.9.3.3" xref="S3.Ex24.m1.9.9.3.3.cmml"><msubsup id="S3.Ex24.m1.9.9.3.3.4" xref="S3.Ex24.m1.9.9.3.3.4.cmml"><mi id="S3.Ex24.m1.9.9.3.3.4.2.2" xref="S3.Ex24.m1.9.9.3.3.4.2.2.cmml">B</mi><mrow id="S3.Ex24.m1.4.4.2.4" xref="S3.Ex24.m1.4.4.2.3.cmml"><mi id="S3.Ex24.m1.3.3.1.1" xref="S3.Ex24.m1.3.3.1.1.cmml">p</mi><mo id="S3.Ex24.m1.4.4.2.4.1" xref="S3.Ex24.m1.4.4.2.3.cmml">,</mo><mi id="S3.Ex24.m1.4.4.2.2" xref="S3.Ex24.m1.4.4.2.2.cmml">q</mi></mrow><mi id="S3.Ex24.m1.9.9.3.3.4.2.3" xref="S3.Ex24.m1.9.9.3.3.4.2.3.cmml">s</mi></msubsup><mo id="S3.Ex24.m1.9.9.3.3.3" xref="S3.Ex24.m1.9.9.3.3.3.cmml">⁢</mo><mrow id="S3.Ex24.m1.9.9.3.3.2.2" xref="S3.Ex24.m1.9.9.3.3.2.3.cmml"><mo id="S3.Ex24.m1.9.9.3.3.2.2.3" stretchy="false" xref="S3.Ex24.m1.9.9.3.3.2.3.cmml">(</mo><msup id="S3.Ex24.m1.8.8.2.2.1.1.1" xref="S3.Ex24.m1.8.8.2.2.1.1.1.cmml"><mi id="S3.Ex24.m1.8.8.2.2.1.1.1.2" xref="S3.Ex24.m1.8.8.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.Ex24.m1.8.8.2.2.1.1.1.3" xref="S3.Ex24.m1.8.8.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.Ex24.m1.9.9.3.3.2.2.4" xref="S3.Ex24.m1.9.9.3.3.2.3.cmml">,</mo><msub id="S3.Ex24.m1.9.9.3.3.2.2.2" xref="S3.Ex24.m1.9.9.3.3.2.2.2.cmml"><mi id="S3.Ex24.m1.9.9.3.3.2.2.2.2" xref="S3.Ex24.m1.9.9.3.3.2.2.2.2.cmml">w</mi><mi id="S3.Ex24.m1.9.9.3.3.2.2.2.3" xref="S3.Ex24.m1.9.9.3.3.2.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex24.m1.9.9.3.3.2.2.5" xref="S3.Ex24.m1.9.9.3.3.2.3.cmml">;</mo><mi id="S3.Ex24.m1.6.6" xref="S3.Ex24.m1.6.6.cmml">X</mi><mo id="S3.Ex24.m1.9.9.3.3.2.2.6" stretchy="false" xref="S3.Ex24.m1.9.9.3.3.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex24.m1.9b"><apply id="S3.Ex24.m1.9.9.cmml" xref="S3.Ex24.m1.9.9"><ci id="S3.Ex24.m1.9.9.4.cmml" xref="S3.Ex24.m1.9.9.4">:</ci><apply id="S3.Ex24.m1.9.9.5.cmml" xref="S3.Ex24.m1.9.9.5"><csymbol cd="ambiguous" id="S3.Ex24.m1.9.9.5.1.cmml" xref="S3.Ex24.m1.9.9.5">subscript</csymbol><ci id="S3.Ex24.m1.9.9.5.2.cmml" xref="S3.Ex24.m1.9.9.5.2">ext</ci><ci id="S3.Ex24.m1.9.9.5.3.cmml" xref="S3.Ex24.m1.9.9.5.3">𝑚</ci></apply><apply id="S3.Ex24.m1.9.9.3.cmml" xref="S3.Ex24.m1.9.9.3"><ci id="S3.Ex24.m1.9.9.3.4.cmml" xref="S3.Ex24.m1.9.9.3.4">→</ci><apply id="S3.Ex24.m1.7.7.1.1.cmml" xref="S3.Ex24.m1.7.7.1.1"><times id="S3.Ex24.m1.7.7.1.1.2.cmml" xref="S3.Ex24.m1.7.7.1.1.2"></times><apply id="S3.Ex24.m1.7.7.1.1.3.cmml" xref="S3.Ex24.m1.7.7.1.1.3"><csymbol cd="ambiguous" id="S3.Ex24.m1.7.7.1.1.3.1.cmml" xref="S3.Ex24.m1.7.7.1.1.3">subscript</csymbol><apply 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xref="S3.Ex24.m1.6.6">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex24.m1.9c">\operatorname{ext}_{m}:B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)\to B% ^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex24.m1.9d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem7.p1.7"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem7.p1.7.1">is continuous.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS2.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS2.1.p1"> <p class="ltx_p" id="S3.SS2.1.p1.9"><span class="ltx_text ltx_font_italic" id="S3.SS2.1.p1.9.1">Step 1: preparations.</span> For <math alttext="m,j\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.1.m1.2"><semantics id="S3.SS2.1.p1.1.m1.2a"><mrow id="S3.SS2.1.p1.1.m1.2.3" xref="S3.SS2.1.p1.1.m1.2.3.cmml"><mrow id="S3.SS2.1.p1.1.m1.2.3.2.2" xref="S3.SS2.1.p1.1.m1.2.3.2.1.cmml"><mi id="S3.SS2.1.p1.1.m1.1.1" xref="S3.SS2.1.p1.1.m1.1.1.cmml">m</mi><mo id="S3.SS2.1.p1.1.m1.2.3.2.2.1" xref="S3.SS2.1.p1.1.m1.2.3.2.1.cmml">,</mo><mi id="S3.SS2.1.p1.1.m1.2.2" xref="S3.SS2.1.p1.1.m1.2.2.cmml">j</mi></mrow><mo 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id="S3.SS2.1.p1.2.m2.2.2.2.2.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.2"></times><ci id="S3.SS2.1.p1.2.m2.2.2.2.3.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.3">Φ</ci><apply id="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.1.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.1.1">superscript</csymbol><ci id="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.2.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.2">ℝ</ci><apply id="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3"><minus id="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3.1.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3.1"></minus><ci id="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3.2.cmml" xref="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3.2">𝑑</ci><cn id="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.1.p1.2.m2.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.2.m2.2c">(\phi_{j})_{j\geq 0}\in\Phi(\mathbb{R}^{d-1})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.2.m2.2d">( italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT )</annotation></semantics></math> let <math alttext="\eta_{j}^{m}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.3.m3.1"><semantics id="S3.SS2.1.p1.3.m3.1a"><msubsup id="S3.SS2.1.p1.3.m3.1.1" xref="S3.SS2.1.p1.3.m3.1.1.cmml"><mi id="S3.SS2.1.p1.3.m3.1.1.2.2" xref="S3.SS2.1.p1.3.m3.1.1.2.2.cmml">η</mi><mi id="S3.SS2.1.p1.3.m3.1.1.2.3" xref="S3.SS2.1.p1.3.m3.1.1.2.3.cmml">j</mi><mi id="S3.SS2.1.p1.3.m3.1.1.3" xref="S3.SS2.1.p1.3.m3.1.1.3.cmml">m</mi></msubsup><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.3.m3.1b"><apply id="S3.SS2.1.p1.3.m3.1.1.cmml" xref="S3.SS2.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.3.m3.1.1.1.cmml" xref="S3.SS2.1.p1.3.m3.1.1">superscript</csymbol><apply id="S3.SS2.1.p1.3.m3.1.1.2.cmml" xref="S3.SS2.1.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.3.m3.1.1.2.1.cmml" xref="S3.SS2.1.p1.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.3.m3.1.1.2.2.cmml" xref="S3.SS2.1.p1.3.m3.1.1.2.2">𝜂</ci><ci id="S3.SS2.1.p1.3.m3.1.1.2.3.cmml" xref="S3.SS2.1.p1.3.m3.1.1.2.3">𝑗</ci></apply><ci id="S3.SS2.1.p1.3.m3.1.1.3.cmml" xref="S3.SS2.1.p1.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.3.m3.1c">\eta_{j}^{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.3.m3.1d">italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.4.m4.1"><semantics id="S3.SS2.1.p1.4.m4.1a"><msub id="S3.SS2.1.p1.4.m4.1.1" xref="S3.SS2.1.p1.4.m4.1.1.cmml"><mi id="S3.SS2.1.p1.4.m4.1.1.2" xref="S3.SS2.1.p1.4.m4.1.1.2.cmml">ext</mi><mi id="S3.SS2.1.p1.4.m4.1.1.3" xref="S3.SS2.1.p1.4.m4.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.4.m4.1b"><apply id="S3.SS2.1.p1.4.m4.1.1.cmml" xref="S3.SS2.1.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.4.m4.1.1.1.cmml" xref="S3.SS2.1.p1.4.m4.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.4.m4.1.1.2.cmml" xref="S3.SS2.1.p1.4.m4.1.1.2">ext</ci><ci id="S3.SS2.1.p1.4.m4.1.1.3.cmml" xref="S3.SS2.1.p1.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.4.m4.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.4.m4.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> be as defined in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem6" title="Definition 3.6. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.6</span></a>. Let <math alttext="\mathcal{F}_{d}" class="ltx_Math" display="inline" id="S3.SS2.1.p1.5.m5.1"><semantics id="S3.SS2.1.p1.5.m5.1a"><msub id="S3.SS2.1.p1.5.m5.1.1" xref="S3.SS2.1.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.5.m5.1.1.2" xref="S3.SS2.1.p1.5.m5.1.1.2.cmml">ℱ</mi><mi id="S3.SS2.1.p1.5.m5.1.1.3" xref="S3.SS2.1.p1.5.m5.1.1.3.cmml">d</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.5.m5.1b"><apply id="S3.SS2.1.p1.5.m5.1.1.cmml" xref="S3.SS2.1.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.5.m5.1.1.1.cmml" xref="S3.SS2.1.p1.5.m5.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.5.m5.1.1.2.cmml" xref="S3.SS2.1.p1.5.m5.1.1.2">ℱ</ci><ci id="S3.SS2.1.p1.5.m5.1.1.3.cmml" xref="S3.SS2.1.p1.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.5.m5.1c">\mathcal{F}_{d}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.5.m5.1d">caligraphic_F start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT</annotation></semantics></math> be the <math alttext="d" class="ltx_Math" display="inline" id="S3.SS2.1.p1.6.m6.1"><semantics id="S3.SS2.1.p1.6.m6.1a"><mi id="S3.SS2.1.p1.6.m6.1.1" xref="S3.SS2.1.p1.6.m6.1.1.cmml">d</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.6.m6.1b"><ci id="S3.SS2.1.p1.6.m6.1.1.cmml" xref="S3.SS2.1.p1.6.m6.1.1">𝑑</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.6.m6.1c">d</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.6.m6.1d">italic_d</annotation></semantics></math>-dimensional Fourier transform. Moreover, let <math alttext="(\rho_{j})_{j\geq 0}\in\Phi(\mathbb{R})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.7.m7.2"><semantics id="S3.SS2.1.p1.7.m7.2a"><mrow id="S3.SS2.1.p1.7.m7.2.2" xref="S3.SS2.1.p1.7.m7.2.2.cmml"><msub id="S3.SS2.1.p1.7.m7.2.2.1" xref="S3.SS2.1.p1.7.m7.2.2.1.cmml"><mrow id="S3.SS2.1.p1.7.m7.2.2.1.1.1" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.cmml"><mo id="S3.SS2.1.p1.7.m7.2.2.1.1.1.2" stretchy="false" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.cmml">(</mo><msub id="S3.SS2.1.p1.7.m7.2.2.1.1.1.1" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.cmml"><mi id="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.2" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.2.cmml">ρ</mi><mi id="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.3" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.SS2.1.p1.7.m7.2.2.1.1.1.3" stretchy="false" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS2.1.p1.7.m7.2.2.1.3" xref="S3.SS2.1.p1.7.m7.2.2.1.3.cmml"><mi id="S3.SS2.1.p1.7.m7.2.2.1.3.2" xref="S3.SS2.1.p1.7.m7.2.2.1.3.2.cmml">j</mi><mo id="S3.SS2.1.p1.7.m7.2.2.1.3.1" xref="S3.SS2.1.p1.7.m7.2.2.1.3.1.cmml">≥</mo><mn id="S3.SS2.1.p1.7.m7.2.2.1.3.3" xref="S3.SS2.1.p1.7.m7.2.2.1.3.3.cmml">0</mn></mrow></msub><mo id="S3.SS2.1.p1.7.m7.2.2.2" xref="S3.SS2.1.p1.7.m7.2.2.2.cmml">∈</mo><mrow id="S3.SS2.1.p1.7.m7.2.2.3" xref="S3.SS2.1.p1.7.m7.2.2.3.cmml"><mi id="S3.SS2.1.p1.7.m7.2.2.3.2" mathvariant="normal" xref="S3.SS2.1.p1.7.m7.2.2.3.2.cmml">Φ</mi><mo id="S3.SS2.1.p1.7.m7.2.2.3.1" xref="S3.SS2.1.p1.7.m7.2.2.3.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.7.m7.2.2.3.3.2" xref="S3.SS2.1.p1.7.m7.2.2.3.cmml"><mo id="S3.SS2.1.p1.7.m7.2.2.3.3.2.1" stretchy="false" xref="S3.SS2.1.p1.7.m7.2.2.3.cmml">(</mo><mi id="S3.SS2.1.p1.7.m7.1.1" xref="S3.SS2.1.p1.7.m7.1.1.cmml">ℝ</mi><mo id="S3.SS2.1.p1.7.m7.2.2.3.3.2.2" stretchy="false" xref="S3.SS2.1.p1.7.m7.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.7.m7.2b"><apply id="S3.SS2.1.p1.7.m7.2.2.cmml" xref="S3.SS2.1.p1.7.m7.2.2"><in id="S3.SS2.1.p1.7.m7.2.2.2.cmml" xref="S3.SS2.1.p1.7.m7.2.2.2"></in><apply id="S3.SS2.1.p1.7.m7.2.2.1.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.7.m7.2.2.1.2.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1">subscript</csymbol><apply id="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.2.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.2">𝜌</ci><ci id="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.3.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1.1.1.1.3">𝑗</ci></apply><apply id="S3.SS2.1.p1.7.m7.2.2.1.3.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1.3"><geq id="S3.SS2.1.p1.7.m7.2.2.1.3.1.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1.3.1"></geq><ci id="S3.SS2.1.p1.7.m7.2.2.1.3.2.cmml" xref="S3.SS2.1.p1.7.m7.2.2.1.3.2">𝑗</ci><cn id="S3.SS2.1.p1.7.m7.2.2.1.3.3.cmml" type="integer" xref="S3.SS2.1.p1.7.m7.2.2.1.3.3">0</cn></apply></apply><apply id="S3.SS2.1.p1.7.m7.2.2.3.cmml" xref="S3.SS2.1.p1.7.m7.2.2.3"><times id="S3.SS2.1.p1.7.m7.2.2.3.1.cmml" xref="S3.SS2.1.p1.7.m7.2.2.3.1"></times><ci id="S3.SS2.1.p1.7.m7.2.2.3.2.cmml" xref="S3.SS2.1.p1.7.m7.2.2.3.2">Φ</ci><ci id="S3.SS2.1.p1.7.m7.1.1.cmml" xref="S3.SS2.1.p1.7.m7.1.1">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.7.m7.2c">(\rho_{j})_{j\geq 0}\in\Phi(\mathbb{R})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.7.m7.2d">( italic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R )</annotation></semantics></math> and set <math alttext="\mathcal{F}_{d}\varphi_{j}(x):=(\mathcal{F}_{1}\rho_{j})(x_{1})(\mathcal{F}_{d% -1}\phi_{j})(\widetilde{x})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.8.m8.5"><semantics id="S3.SS2.1.p1.8.m8.5a"><mrow id="S3.SS2.1.p1.8.m8.5.5" xref="S3.SS2.1.p1.8.m8.5.5.cmml"><mrow id="S3.SS2.1.p1.8.m8.5.5.5" xref="S3.SS2.1.p1.8.m8.5.5.5.cmml"><msub id="S3.SS2.1.p1.8.m8.5.5.5.2" xref="S3.SS2.1.p1.8.m8.5.5.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.1.p1.8.m8.5.5.5.2.2" xref="S3.SS2.1.p1.8.m8.5.5.5.2.2.cmml">ℱ</mi><mi id="S3.SS2.1.p1.8.m8.5.5.5.2.3" xref="S3.SS2.1.p1.8.m8.5.5.5.2.3.cmml">d</mi></msub><mo id="S3.SS2.1.p1.8.m8.5.5.5.1" xref="S3.SS2.1.p1.8.m8.5.5.5.1.cmml">⁢</mo><msub id="S3.SS2.1.p1.8.m8.5.5.5.3" xref="S3.SS2.1.p1.8.m8.5.5.5.3.cmml"><mi id="S3.SS2.1.p1.8.m8.5.5.5.3.2" xref="S3.SS2.1.p1.8.m8.5.5.5.3.2.cmml">φ</mi><mi id="S3.SS2.1.p1.8.m8.5.5.5.3.3" xref="S3.SS2.1.p1.8.m8.5.5.5.3.3.cmml">j</mi></msub><mo id="S3.SS2.1.p1.8.m8.5.5.5.1a" xref="S3.SS2.1.p1.8.m8.5.5.5.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.8.m8.5.5.5.4.2" xref="S3.SS2.1.p1.8.m8.5.5.5.cmml"><mo id="S3.SS2.1.p1.8.m8.5.5.5.4.2.1" stretchy="false" xref="S3.SS2.1.p1.8.m8.5.5.5.cmml">(</mo><mi id="S3.SS2.1.p1.8.m8.1.1" 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xref="S3.SS2.1.p1.8.m8.2.2.1.cmml">~</mo></mover><mo id="S3.SS2.1.p1.8.m8.5.5.3.5.2.2" stretchy="false" xref="S3.SS2.1.p1.8.m8.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.8.m8.5b"><apply id="S3.SS2.1.p1.8.m8.5.5.cmml" xref="S3.SS2.1.p1.8.m8.5.5"><csymbol cd="latexml" id="S3.SS2.1.p1.8.m8.5.5.4.cmml" xref="S3.SS2.1.p1.8.m8.5.5.4">assign</csymbol><apply id="S3.SS2.1.p1.8.m8.5.5.5.cmml" xref="S3.SS2.1.p1.8.m8.5.5.5"><times id="S3.SS2.1.p1.8.m8.5.5.5.1.cmml" xref="S3.SS2.1.p1.8.m8.5.5.5.1"></times><apply id="S3.SS2.1.p1.8.m8.5.5.5.2.cmml" xref="S3.SS2.1.p1.8.m8.5.5.5.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.8.m8.5.5.5.2.1.cmml" xref="S3.SS2.1.p1.8.m8.5.5.5.2">subscript</csymbol><ci id="S3.SS2.1.p1.8.m8.5.5.5.2.2.cmml" xref="S3.SS2.1.p1.8.m8.5.5.5.2.2">ℱ</ci><ci id="S3.SS2.1.p1.8.m8.5.5.5.2.3.cmml" xref="S3.SS2.1.p1.8.m8.5.5.5.2.3">𝑑</ci></apply><apply id="S3.SS2.1.p1.8.m8.5.5.5.3.cmml" xref="S3.SS2.1.p1.8.m8.5.5.5.3"><csymbol 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xref="S3.SS2.1.p1.8.m8.5.5.3.3.1"><times id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.1.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.1"></times><apply id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.1.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2">subscript</csymbol><ci id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.2.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.2">ℱ</ci><apply id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3"><minus id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3.1.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3.1"></minus><ci id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3.2.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3.2">𝑑</ci><cn id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3.3.cmml" type="integer" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.2.3.3">1</cn></apply></apply><apply id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3.1.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3">subscript</csymbol><ci id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3.2.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3.2">italic-ϕ</ci><ci id="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3.3.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.3.1.1.3.3">𝑗</ci></apply></apply><apply id="S3.SS2.1.p1.8.m8.2.2.cmml" xref="S3.SS2.1.p1.8.m8.5.5.3.5.2"><ci id="S3.SS2.1.p1.8.m8.2.2.1.cmml" xref="S3.SS2.1.p1.8.m8.2.2.1">~</ci><ci id="S3.SS2.1.p1.8.m8.2.2.2.cmml" xref="S3.SS2.1.p1.8.m8.2.2.2">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.8.m8.5c">\mathcal{F}_{d}\varphi_{j}(x):=(\mathcal{F}_{1}\rho_{j})(x_{1})(\mathcal{F}_{d% -1}\phi_{j})(\widetilde{x})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.8.m8.5d">caligraphic_F start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x ) := ( caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ( caligraphic_F start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) ( over~ start_ARG italic_x end_ARG )</annotation></semantics></math>. For notational convenience we write <math alttext="\widehat{\varphi}_{j}(x)=\widehat{\rho}_{j}(x_{1})\widehat{\phi}_{j}(% \widetilde{x})" class="ltx_Math" display="inline" id="S3.SS2.1.p1.9.m9.3"><semantics id="S3.SS2.1.p1.9.m9.3a"><mrow id="S3.SS2.1.p1.9.m9.3.3" xref="S3.SS2.1.p1.9.m9.3.3.cmml"><mrow id="S3.SS2.1.p1.9.m9.3.3.3" xref="S3.SS2.1.p1.9.m9.3.3.3.cmml"><msub id="S3.SS2.1.p1.9.m9.3.3.3.2" xref="S3.SS2.1.p1.9.m9.3.3.3.2.cmml"><mover accent="true" id="S3.SS2.1.p1.9.m9.3.3.3.2.2" xref="S3.SS2.1.p1.9.m9.3.3.3.2.2.cmml"><mi id="S3.SS2.1.p1.9.m9.3.3.3.2.2.2" xref="S3.SS2.1.p1.9.m9.3.3.3.2.2.2.cmml">φ</mi><mo id="S3.SS2.1.p1.9.m9.3.3.3.2.2.1" xref="S3.SS2.1.p1.9.m9.3.3.3.2.2.1.cmml">^</mo></mover><mi id="S3.SS2.1.p1.9.m9.3.3.3.2.3" xref="S3.SS2.1.p1.9.m9.3.3.3.2.3.cmml">j</mi></msub><mo id="S3.SS2.1.p1.9.m9.3.3.3.1" xref="S3.SS2.1.p1.9.m9.3.3.3.1.cmml">⁢</mo><mrow id="S3.SS2.1.p1.9.m9.3.3.3.3.2" xref="S3.SS2.1.p1.9.m9.3.3.3.cmml"><mo id="S3.SS2.1.p1.9.m9.3.3.3.3.2.1" stretchy="false" xref="S3.SS2.1.p1.9.m9.3.3.3.cmml">(</mo><mi id="S3.SS2.1.p1.9.m9.1.1" xref="S3.SS2.1.p1.9.m9.1.1.cmml">x</mi><mo id="S3.SS2.1.p1.9.m9.3.3.3.3.2.2" stretchy="false" xref="S3.SS2.1.p1.9.m9.3.3.3.cmml">)</mo></mrow></mrow><mo id="S3.SS2.1.p1.9.m9.3.3.2" xref="S3.SS2.1.p1.9.m9.3.3.2.cmml">=</mo><mrow id="S3.SS2.1.p1.9.m9.3.3.1" xref="S3.SS2.1.p1.9.m9.3.3.1.cmml"><msub id="S3.SS2.1.p1.9.m9.3.3.1.3" xref="S3.SS2.1.p1.9.m9.3.3.1.3.cmml"><mover accent="true" id="S3.SS2.1.p1.9.m9.3.3.1.3.2" xref="S3.SS2.1.p1.9.m9.3.3.1.3.2.cmml"><mi id="S3.SS2.1.p1.9.m9.3.3.1.3.2.2" xref="S3.SS2.1.p1.9.m9.3.3.1.3.2.2.cmml">ρ</mi><mo id="S3.SS2.1.p1.9.m9.3.3.1.3.2.1" xref="S3.SS2.1.p1.9.m9.3.3.1.3.2.1.cmml">^</mo></mover><mi id="S3.SS2.1.p1.9.m9.3.3.1.3.3" xref="S3.SS2.1.p1.9.m9.3.3.1.3.3.cmml">j</mi></msub><mo id="S3.SS2.1.p1.9.m9.3.3.1.2" xref="S3.SS2.1.p1.9.m9.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS2.1.p1.9.m9.3.3.1.1.1" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.cmml"><mo id="S3.SS2.1.p1.9.m9.3.3.1.1.1.2" stretchy="false" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.cmml">(</mo><msub id="S3.SS2.1.p1.9.m9.3.3.1.1.1.1" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.cmml"><mi id="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.2" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.2.cmml">x</mi><mn id="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.3" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S3.SS2.1.p1.9.m9.3.3.1.1.1.3" stretchy="false" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S3.SS2.1.p1.9.m9.3.3.1.2a" xref="S3.SS2.1.p1.9.m9.3.3.1.2.cmml">⁢</mo><msub id="S3.SS2.1.p1.9.m9.3.3.1.4" xref="S3.SS2.1.p1.9.m9.3.3.1.4.cmml"><mover accent="true" id="S3.SS2.1.p1.9.m9.3.3.1.4.2" xref="S3.SS2.1.p1.9.m9.3.3.1.4.2.cmml"><mi id="S3.SS2.1.p1.9.m9.3.3.1.4.2.2" xref="S3.SS2.1.p1.9.m9.3.3.1.4.2.2.cmml">ϕ</mi><mo id="S3.SS2.1.p1.9.m9.3.3.1.4.2.1" xref="S3.SS2.1.p1.9.m9.3.3.1.4.2.1.cmml">^</mo></mover><mi id="S3.SS2.1.p1.9.m9.3.3.1.4.3" xref="S3.SS2.1.p1.9.m9.3.3.1.4.3.cmml">j</mi></msub><mo id="S3.SS2.1.p1.9.m9.3.3.1.2b" xref="S3.SS2.1.p1.9.m9.3.3.1.2.cmml">⁢</mo><mrow id="S3.SS2.1.p1.9.m9.3.3.1.5.2" xref="S3.SS2.1.p1.9.m9.2.2.cmml"><mo id="S3.SS2.1.p1.9.m9.3.3.1.5.2.1" stretchy="false" xref="S3.SS2.1.p1.9.m9.2.2.cmml">(</mo><mover accent="true" id="S3.SS2.1.p1.9.m9.2.2" xref="S3.SS2.1.p1.9.m9.2.2.cmml"><mi id="S3.SS2.1.p1.9.m9.2.2.2" xref="S3.SS2.1.p1.9.m9.2.2.2.cmml">x</mi><mo id="S3.SS2.1.p1.9.m9.2.2.1" xref="S3.SS2.1.p1.9.m9.2.2.1.cmml">~</mo></mover><mo id="S3.SS2.1.p1.9.m9.3.3.1.5.2.2" stretchy="false" xref="S3.SS2.1.p1.9.m9.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.1.p1.9.m9.3b"><apply id="S3.SS2.1.p1.9.m9.3.3.cmml" xref="S3.SS2.1.p1.9.m9.3.3"><eq id="S3.SS2.1.p1.9.m9.3.3.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.2"></eq><apply id="S3.SS2.1.p1.9.m9.3.3.3.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3"><times id="S3.SS2.1.p1.9.m9.3.3.3.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3.1"></times><apply id="S3.SS2.1.p1.9.m9.3.3.3.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.1.p1.9.m9.3.3.3.2.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3.2">subscript</csymbol><apply id="S3.SS2.1.p1.9.m9.3.3.3.2.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3.2.2"><ci id="S3.SS2.1.p1.9.m9.3.3.3.2.2.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3.2.2.1">^</ci><ci id="S3.SS2.1.p1.9.m9.3.3.3.2.2.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3.2.2.2">𝜑</ci></apply><ci id="S3.SS2.1.p1.9.m9.3.3.3.2.3.cmml" xref="S3.SS2.1.p1.9.m9.3.3.3.2.3">𝑗</ci></apply><ci id="S3.SS2.1.p1.9.m9.1.1.cmml" xref="S3.SS2.1.p1.9.m9.1.1">𝑥</ci></apply><apply id="S3.SS2.1.p1.9.m9.3.3.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1"><times id="S3.SS2.1.p1.9.m9.3.3.1.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.2"></times><apply id="S3.SS2.1.p1.9.m9.3.3.1.3.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.3"><csymbol cd="ambiguous" id="S3.SS2.1.p1.9.m9.3.3.1.3.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.3">subscript</csymbol><apply id="S3.SS2.1.p1.9.m9.3.3.1.3.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.3.2"><ci id="S3.SS2.1.p1.9.m9.3.3.1.3.2.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.3.2.1">^</ci><ci id="S3.SS2.1.p1.9.m9.3.3.1.3.2.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.3.2.2">𝜌</ci></apply><ci id="S3.SS2.1.p1.9.m9.3.3.1.3.3.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.3.3">𝑗</ci></apply><apply id="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1">subscript</csymbol><ci id="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.2">𝑥</ci><cn id="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.3.cmml" type="integer" xref="S3.SS2.1.p1.9.m9.3.3.1.1.1.1.3">1</cn></apply><apply id="S3.SS2.1.p1.9.m9.3.3.1.4.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.4"><csymbol cd="ambiguous" id="S3.SS2.1.p1.9.m9.3.3.1.4.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.4">subscript</csymbol><apply id="S3.SS2.1.p1.9.m9.3.3.1.4.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.4.2"><ci id="S3.SS2.1.p1.9.m9.3.3.1.4.2.1.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.4.2.1">^</ci><ci id="S3.SS2.1.p1.9.m9.3.3.1.4.2.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.4.2.2">italic-ϕ</ci></apply><ci id="S3.SS2.1.p1.9.m9.3.3.1.4.3.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.4.3">𝑗</ci></apply><apply id="S3.SS2.1.p1.9.m9.2.2.cmml" xref="S3.SS2.1.p1.9.m9.3.3.1.5.2"><ci id="S3.SS2.1.p1.9.m9.2.2.1.cmml" xref="S3.SS2.1.p1.9.m9.2.2.1">~</ci><ci id="S3.SS2.1.p1.9.m9.2.2.2.cmml" xref="S3.SS2.1.p1.9.m9.2.2.2">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.1.p1.9.m9.3c">\widehat{\varphi}_{j}(x)=\widehat{\rho}_{j}(x_{1})\widehat{\phi}_{j}(% \widetilde{x})</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.1.p1.9.m9.3d">over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x ) = over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( over~ start_ARG italic_x end_ARG )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S3.SS2.2.p2"> <p class="ltx_p" id="S3.SS2.2.p2.1">Let <math alttext="g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS2.2.p2.1.m1.4"><semantics id="S3.SS2.2.p2.1.m1.4a"><mrow id="S3.SS2.2.p2.1.m1.4.4" xref="S3.SS2.2.p2.1.m1.4.4.cmml"><mi id="S3.SS2.2.p2.1.m1.4.4.3" xref="S3.SS2.2.p2.1.m1.4.4.3.cmml">g</mi><mo id="S3.SS2.2.p2.1.m1.4.4.2" xref="S3.SS2.2.p2.1.m1.4.4.2.cmml">∈</mo><mrow id="S3.SS2.2.p2.1.m1.4.4.1" xref="S3.SS2.2.p2.1.m1.4.4.1.cmml"><msubsup id="S3.SS2.2.p2.1.m1.4.4.1.3" xref="S3.SS2.2.p2.1.m1.4.4.1.3.cmml"><mi id="S3.SS2.2.p2.1.m1.4.4.1.3.2.2" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.2.cmml">B</mi><mrow id="S3.SS2.2.p2.1.m1.2.2.2.4" xref="S3.SS2.2.p2.1.m1.2.2.2.3.cmml"><mi id="S3.SS2.2.p2.1.m1.1.1.1.1" xref="S3.SS2.2.p2.1.m1.1.1.1.1.cmml">p</mi><mo id="S3.SS2.2.p2.1.m1.2.2.2.4.1" xref="S3.SS2.2.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS2.2.p2.1.m1.2.2.2.2" xref="S3.SS2.2.p2.1.m1.2.2.2.2.cmml">q</mi></mrow><mrow id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.cmml"><mi id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.2" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.2.cmml">s</mi><mo id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.1" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.1.cmml">−</mo><mi id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.3" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.3.cmml">m</mi><mo id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.1a" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4.cmml"><mrow id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4.2" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4.2.cmml"><mi id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4.2.2" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4.2.2.cmml">γ</mi><mo id="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4.2.1" xref="S3.SS2.2.p2.1.m1.4.4.1.3.2.3.4.2.1.cmml">+</mo><mn 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notational convenience we introduce</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widehat{g}:=\mathcal{F}_{d-1}g\quad\text{ and }\quad g_{j}:=(\mathcal{F}^{-1}% _{1}\eta_{j}^{m})(\phi_{j}\ast g)\quad\text{ for }j\in\mathbb{N}_{0}." class="ltx_Math" display="block" id="S3.Ex25.m1.2"><semantics id="S3.Ex25.m1.2a"><mrow id="S3.Ex25.m1.2.2.1"><mrow id="S3.Ex25.m1.2.2.1.1.2" xref="S3.Ex25.m1.2.2.1.1.3.cmml"><mrow id="S3.Ex25.m1.2.2.1.1.1.1" xref="S3.Ex25.m1.2.2.1.1.1.1.cmml"><mover accent="true" id="S3.Ex25.m1.2.2.1.1.1.1.3" xref="S3.Ex25.m1.2.2.1.1.1.1.3.cmml"><mi id="S3.Ex25.m1.2.2.1.1.1.1.3.2" xref="S3.Ex25.m1.2.2.1.1.1.1.3.2.cmml">g</mi><mo id="S3.Ex25.m1.2.2.1.1.1.1.3.1" xref="S3.Ex25.m1.2.2.1.1.1.1.3.1.cmml">^</mo></mover><mo id="S3.Ex25.m1.2.2.1.1.1.1.2" lspace="0.278em" rspace="0.278em" 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end_ARG := caligraphic_F start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT italic_g and italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT := ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) ( italic_ϕ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∗ italic_g ) for italic_j ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.2.p2.2">Note that for <math alttext="k,j\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.2.m1.2"><semantics id="S3.SS2.2.p2.2.m1.2a"><mrow id="S3.SS2.2.p2.2.m1.2.3" xref="S3.SS2.2.p2.2.m1.2.3.cmml"><mrow id="S3.SS2.2.p2.2.m1.2.3.2.2" xref="S3.SS2.2.p2.2.m1.2.3.2.1.cmml"><mi id="S3.SS2.2.p2.2.m1.1.1" xref="S3.SS2.2.p2.2.m1.1.1.cmml">k</mi><mo id="S3.SS2.2.p2.2.m1.2.3.2.2.1" xref="S3.SS2.2.p2.2.m1.2.3.2.1.cmml">,</mo><mi id="S3.SS2.2.p2.2.m1.2.2" xref="S3.SS2.2.p2.2.m1.2.2.cmml">j</mi></mrow><mo id="S3.SS2.2.p2.2.m1.2.3.1" xref="S3.SS2.2.p2.2.m1.2.3.1.cmml">∈</mo><msub id="S3.SS2.2.p2.2.m1.2.3.3" xref="S3.SS2.2.p2.2.m1.2.3.3.cmml"><mi id="S3.SS2.2.p2.2.m1.2.3.3.2" xref="S3.SS2.2.p2.2.m1.2.3.3.2.cmml">ℕ</mi><mn id="S3.SS2.2.p2.2.m1.2.3.3.3" xref="S3.SS2.2.p2.2.m1.2.3.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.2.m1.2b"><apply id="S3.SS2.2.p2.2.m1.2.3.cmml" xref="S3.SS2.2.p2.2.m1.2.3"><in id="S3.SS2.2.p2.2.m1.2.3.1.cmml" xref="S3.SS2.2.p2.2.m1.2.3.1"></in><list id="S3.SS2.2.p2.2.m1.2.3.2.1.cmml" xref="S3.SS2.2.p2.2.m1.2.3.2.2"><ci id="S3.SS2.2.p2.2.m1.1.1.cmml" xref="S3.SS2.2.p2.2.m1.1.1">𝑘</ci><ci id="S3.SS2.2.p2.2.m1.2.2.cmml" xref="S3.SS2.2.p2.2.m1.2.2">𝑗</ci></list><apply id="S3.SS2.2.p2.2.m1.2.3.3.cmml" xref="S3.SS2.2.p2.2.m1.2.3.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.2.m1.2.3.3.1.cmml" xref="S3.SS2.2.p2.2.m1.2.3.3">subscript</csymbol><ci id="S3.SS2.2.p2.2.m1.2.3.3.2.cmml" xref="S3.SS2.2.p2.2.m1.2.3.3.2">ℕ</ci><cn id="S3.SS2.2.p2.2.m1.2.3.3.3.cmml" type="integer" xref="S3.SS2.2.p2.2.m1.2.3.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.2.m1.2c">k,j\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.2.m1.2d">italic_k , italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx10"> <tbody id="S3.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\varphi_{k}\ast g_{j}=\mathcal{F}_{d}^{-1}(\widehat{\varphi}_{k}% \mathcal{F}_{d}g_{j})=\mathcal{F}^{-1}_{d}\big{[}\widehat{\rho}_{k}\widehat{% \phi}_{k}\mathcal{F}_{d}\big{(}(\mathcal{F}_{1}^{-1}\eta_{j}^{m})(\mathcal{F}^% {-1}_{d-1}\widehat{\phi}_{j}\widehat{g})\big{)}\big{]}=\mathcal{F}^{-1}_{d}% \big{[}(\widehat{\rho}_{k}\eta_{j}^{m})(\widehat{\phi}_{k}\widehat{\phi}_{j}% \widehat{g})\big{]}," class="ltx_Math" display="inline" id="S3.Ex26.m1.1"><semantics id="S3.Ex26.m1.1a"><mrow id="S3.Ex26.m1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.cmml"><mrow id="S3.Ex26.m1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.cmml"><mrow id="S3.Ex26.m1.1.1.1.1.5" xref="S3.Ex26.m1.1.1.1.1.5.cmml"><msub id="S3.Ex26.m1.1.1.1.1.5.2" xref="S3.Ex26.m1.1.1.1.1.5.2.cmml"><mi id="S3.Ex26.m1.1.1.1.1.5.2.2" xref="S3.Ex26.m1.1.1.1.1.5.2.2.cmml">φ</mi><mi id="S3.Ex26.m1.1.1.1.1.5.2.3" xref="S3.Ex26.m1.1.1.1.1.5.2.3.cmml">k</mi></msub><mo id="S3.Ex26.m1.1.1.1.1.5.1" lspace="0.222em" rspace="0.222em" xref="S3.Ex26.m1.1.1.1.1.5.1.cmml">∗</mo><msub id="S3.Ex26.m1.1.1.1.1.5.3" xref="S3.Ex26.m1.1.1.1.1.5.3.cmml"><mi id="S3.Ex26.m1.1.1.1.1.5.3.2" xref="S3.Ex26.m1.1.1.1.1.5.3.2.cmml">g</mi><mi id="S3.Ex26.m1.1.1.1.1.5.3.3" xref="S3.Ex26.m1.1.1.1.1.5.3.3.cmml">j</mi></msub></mrow><mo id="S3.Ex26.m1.1.1.1.1.6" xref="S3.Ex26.m1.1.1.1.1.6.cmml">=</mo><mrow id="S3.Ex26.m1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.1.cmml"><msubsup id="S3.Ex26.m1.1.1.1.1.1.3" xref="S3.Ex26.m1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex26.m1.1.1.1.1.1.3.2.2" xref="S3.Ex26.m1.1.1.1.1.1.3.2.2.cmml">ℱ</mi><mi id="S3.Ex26.m1.1.1.1.1.1.3.2.3" xref="S3.Ex26.m1.1.1.1.1.1.3.2.3.cmml">d</mi><mrow id="S3.Ex26.m1.1.1.1.1.1.3.3" xref="S3.Ex26.m1.1.1.1.1.1.3.3.cmml"><mo id="S3.Ex26.m1.1.1.1.1.1.3.3a" xref="S3.Ex26.m1.1.1.1.1.1.3.3.cmml">−</mo><mn id="S3.Ex26.m1.1.1.1.1.1.3.3.2" xref="S3.Ex26.m1.1.1.1.1.1.3.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.Ex26.m1.1.1.1.1.1.2" xref="S3.Ex26.m1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex26.m1.1.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex26.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex26.m1.1.1.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.cmml"><msub id="S3.Ex26.m1.1.1.1.1.1.1.1.1.2" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.cmml"><mover accent="true" id="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.2" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.2.2" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.2.2.cmml">φ</mi><mo id="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.2.1" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.2.1.cmml">^</mo></mover><mi id="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.3" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S3.Ex26.m1.1.1.1.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S3.Ex26.m1.1.1.1.1.1.1.1.1.3" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex26.m1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.3.2.cmml">ℱ</mi><mi id="S3.Ex26.m1.1.1.1.1.1.1.1.1.3.3" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.3.3.cmml">d</mi></msub><mo id="S3.Ex26.m1.1.1.1.1.1.1.1.1.1a" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><msub id="S3.Ex26.m1.1.1.1.1.1.1.1.1.4" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.4.cmml"><mi id="S3.Ex26.m1.1.1.1.1.1.1.1.1.4.2" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.4.2.cmml">g</mi><mi id="S3.Ex26.m1.1.1.1.1.1.1.1.1.4.3" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.4.3.cmml">j</mi></msub></mrow><mo id="S3.Ex26.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex26.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex26.m1.1.1.1.1.7" xref="S3.Ex26.m1.1.1.1.1.7.cmml">=</mo><mrow id="S3.Ex26.m1.1.1.1.1.2" xref="S3.Ex26.m1.1.1.1.1.2.cmml"><msubsup id="S3.Ex26.m1.1.1.1.1.2.3" xref="S3.Ex26.m1.1.1.1.1.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex26.m1.1.1.1.1.2.3.2.2" xref="S3.Ex26.m1.1.1.1.1.2.3.2.2.cmml">ℱ</mi><mi id="S3.Ex26.m1.1.1.1.1.2.3.3" xref="S3.Ex26.m1.1.1.1.1.2.3.3.cmml">d</mi><mrow id="S3.Ex26.m1.1.1.1.1.2.3.2.3" xref="S3.Ex26.m1.1.1.1.1.2.3.2.3.cmml"><mo id="S3.Ex26.m1.1.1.1.1.2.3.2.3a" xref="S3.Ex26.m1.1.1.1.1.2.3.2.3.cmml">−</mo><mn id="S3.Ex26.m1.1.1.1.1.2.3.2.3.2" xref="S3.Ex26.m1.1.1.1.1.2.3.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.Ex26.m1.1.1.1.1.2.2" xref="S3.Ex26.m1.1.1.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.2.cmml"><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.2" maxsize="120%" minsize="120%" xref="S3.Ex26.m1.1.1.1.1.2.1.2.1.cmml">[</mo><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.cmml"><msub id="S3.Ex26.m1.1.1.1.1.2.1.1.1.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.cmml"><mover accent="true" id="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.2.cmml"><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.2.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.2.2.cmml">ρ</mi><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.2.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.2.1.cmml">^</mo></mover><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.3.3.cmml">k</mi></msub><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.2.cmml">⁢</mo><msub id="S3.Ex26.m1.1.1.1.1.2.1.1.1.4" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.cmml"><mover accent="true" id="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.2.cmml"><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.2.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.2.2.cmml">ϕ</mi><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.2.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.2.1.cmml">^</mo></mover><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.4.3.cmml">k</mi></msub><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.2a" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.2.cmml">⁢</mo><msub id="S3.Ex26.m1.1.1.1.1.2.1.1.1.5" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex26.m1.1.1.1.1.2.1.1.1.5.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.5.2.cmml">ℱ</mi><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.5.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.5.3.cmml">d</mi></msub><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.2b" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.2.cmml">⁢</mo><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.cmml"><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.cmml"><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.cmml"><msubsup id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.2.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.2.2.cmml">ℱ</mi><mn id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.2.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.2.3.cmml">1</mn><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.3.cmml"><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.3a" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.3.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.1.cmml">⁢</mo><msubsup id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3.2.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3.2.2.cmml">η</mi><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3.2.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3.2.3.cmml">j</mi><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.3.3.cmml">m</mi></msubsup></mrow><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.3.cmml">⁢</mo><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.cmml"><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.2" stretchy="false" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.cmml">(</mo><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.cmml"><msubsup id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.2.cmml">ℱ</mi><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3.cmml"><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3.2.cmml">d</mi><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3.1.cmml">−</mo><mn id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.3.3.cmml">1</mn></mrow><mrow id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.3.cmml"><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.3a" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.3.cmml">−</mo><mn id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.3.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.2.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.1.cmml">⁢</mo><msub id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.cmml"><mover accent="true" id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.2.cmml"><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.2.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.2.2.cmml">ϕ</mi><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.2.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.2.1.cmml">^</mo></mover><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.3" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.3.3.cmml">j</mi></msub><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.1a" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.1.cmml">⁢</mo><mover accent="true" id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.4" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.4.cmml"><mi id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.4.2" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.4.2.cmml">g</mi><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.4.1" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.4.1.cmml">^</mo></mover></mrow><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.3" stretchy="false" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.2.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.3" maxsize="120%" minsize="120%" xref="S3.Ex26.m1.1.1.1.1.2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex26.m1.1.1.1.1.2.1.1.3" maxsize="120%" minsize="120%" xref="S3.Ex26.m1.1.1.1.1.2.1.2.1.cmml">]</mo></mrow></mrow><mo id="S3.Ex26.m1.1.1.1.1.8" xref="S3.Ex26.m1.1.1.1.1.8.cmml">=</mo><mrow id="S3.Ex26.m1.1.1.1.1.3" xref="S3.Ex26.m1.1.1.1.1.3.cmml"><msubsup id="S3.Ex26.m1.1.1.1.1.3.3" xref="S3.Ex26.m1.1.1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.Ex26.m1.1.1.1.1.3.3.2.2" xref="S3.Ex26.m1.1.1.1.1.3.3.2.2.cmml">ℱ</mi><mi id="S3.Ex26.m1.1.1.1.1.3.3.3" xref="S3.Ex26.m1.1.1.1.1.3.3.3.cmml">d</mi><mrow id="S3.Ex26.m1.1.1.1.1.3.3.2.3" xref="S3.Ex26.m1.1.1.1.1.3.3.2.3.cmml"><mo id="S3.Ex26.m1.1.1.1.1.3.3.2.3a" xref="S3.Ex26.m1.1.1.1.1.3.3.2.3.cmml">−</mo><mn id="S3.Ex26.m1.1.1.1.1.3.3.2.3.2" xref="S3.Ex26.m1.1.1.1.1.3.3.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.Ex26.m1.1.1.1.1.3.2" xref="S3.Ex26.m1.1.1.1.1.3.2.cmml">⁢</mo><mrow id="S3.Ex26.m1.1.1.1.1.3.1.1" xref="S3.Ex26.m1.1.1.1.1.3.1.2.cmml"><mo id="S3.Ex26.m1.1.1.1.1.3.1.1.2" maxsize="120%" minsize="120%" xref="S3.Ex26.m1.1.1.1.1.3.1.2.1.cmml">[</mo><mrow id="S3.Ex26.m1.1.1.1.1.3.1.1.1" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.cmml"><mrow id="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.cmml"><mo id="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.2" stretchy="false" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.cmml"><msub id="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.2" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.2.cmml"><mover accent="true" id="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.2.2" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.2.2.cmml"><mi id="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.2.2.2" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.2.2.2.cmml">ρ</mi><mo id="S3.Ex26.m1.1.1.1.1.3.1.1.1.1.1.1.2.2.1" 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id="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.3.2.1.cmml" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.3.2.1">^</ci><ci id="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.3.2.2.cmml" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.3.2.2">italic-ϕ</ci></apply><ci id="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.3.3.cmml" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.3.3">𝑗</ci></apply><apply id="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.4.cmml" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.4"><ci id="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.4.1.cmml" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.4.1">^</ci><ci id="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.4.2.cmml" xref="S3.Ex26.m1.1.1.1.1.3.1.1.1.2.1.1.4.2">𝑔</ci></apply></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex26.m1.1c">\displaystyle\varphi_{k}\ast g_{j}=\mathcal{F}_{d}^{-1}(\widehat{\varphi}_{k}% \mathcal{F}_{d}g_{j})=\mathcal{F}^{-1}_{d}\big{[}\widehat{\rho}_{k}\widehat{% \phi}_{k}\mathcal{F}_{d}\big{(}(\mathcal{F}_{1}^{-1}\eta_{j}^{m})(\mathcal{F}^% {-1}_{d-1}\widehat{\phi}_{j}\widehat{g})\big{)}\big{]}=\mathcal{F}^{-1}_{d}% \big{[}(\widehat{\rho}_{k}\eta_{j}^{m})(\widehat{\phi}_{k}\widehat{\phi}_{j}% \widehat{g})\big{]},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex26.m1.1d">italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = caligraphic_F start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( over^ start_ARG italic_φ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT caligraphic_F start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT [ over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT caligraphic_F start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ( ( caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d - 1 end_POSTSUBSCRIPT over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT over^ start_ARG italic_g end_ARG ) ) ] = caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT [ ( over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) ( over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT over^ start_ARG italic_ϕ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT over^ start_ARG italic_g end_ARG ) ] 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href="https://arxiv.org/html/2503.14636v1#S3.SS2.2.p2.3.m1.2.2.2.cmml" id="S3.SS2.2.p2.3.m1.2.2d.cmml" xref="S3.SS2.2.p2.3.m1.2.2"></share><apply id="S3.SS2.2.p2.3.m1.2.2.7.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7"><times id="S3.SS2.2.p2.3.m1.2.2.7.1.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.1"></times><ci id="S3.SS2.2.p2.3.m1.2.2.7.2a.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.2"><mtext id="S3.SS2.2.p2.3.m1.2.2.7.2.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.2">supp </mtext></ci><apply id="S3.SS2.2.p2.3.m1.2.2.7.3.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.3.m1.2.2.7.3.1.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.3">subscript</csymbol><apply id="S3.SS2.2.p2.3.m1.2.2.7.3.2.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.3.2"><ci id="S3.SS2.2.p2.3.m1.2.2.7.3.2.1.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.3.2.1">^</ci><ci id="S3.SS2.2.p2.3.m1.2.2.7.3.2.2.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.3.2.2">𝜌</ci></apply><ci id="S3.SS2.2.p2.3.m1.2.2.7.3.3.cmml" xref="S3.SS2.2.p2.3.m1.2.2.7.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.3.m1.2c">\text{\rm supp\,}\eta_{j}^{m}\subseteq\{\xi_{1}\in\mathbb{R}:2^{j}<|\xi_{1}|<3% \cdot 2^{j-1}\}\subseteq\text{\rm supp\,}\widehat{\rho}_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.3.m1.2d">supp italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ⊆ { italic_ξ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R : 2 start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT < | italic_ξ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | < 3 ⋅ 2 start_POSTSUPERSCRIPT italic_j - 1 end_POSTSUPERSCRIPT } ⊆ supp over^ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S3.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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id="S3.E5.m1.1.1.1.1.1.1.2.3.3" xref="S3.E5.m1.1.1.1.1.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="S3.E5.m1.1.1.1.1.1.1.1" xref="S3.E5.m1.1.1.1.1.1.1.1.cmml">=</mo><mn id="S3.E5.m1.1.1.1.1.1.1.3" xref="S3.E5.m1.1.1.1.1.1.1.3.cmml">0</mn></mrow><mspace id="S3.E5.m1.1.1.1.1.2.3" width="1em" xref="S3.E5.m1.1.1.1.1.3a.cmml"></mspace><mrow id="S3.E5.m1.1.1.1.1.2.2" xref="S3.E5.m1.1.1.1.1.2.2.cmml"><mrow id="S3.E5.m1.1.1.1.1.2.2.1" xref="S3.E5.m1.1.1.1.1.2.2.1.cmml"><mtext id="S3.E5.m1.1.1.1.1.2.2.1.3" xref="S3.E5.m1.1.1.1.1.2.2.1.3a.cmml">if </mtext><mo id="S3.E5.m1.1.1.1.1.2.2.1.2" lspace="0.170em" xref="S3.E5.m1.1.1.1.1.2.2.1.2.cmml">⁢</mo><mrow id="S3.E5.m1.1.1.1.1.2.2.1.1.1" xref="S3.E5.m1.1.1.1.1.2.2.1.1.2.cmml"><mo id="S3.E5.m1.1.1.1.1.2.2.1.1.1.2" stretchy="false" xref="S3.E5.m1.1.1.1.1.2.2.1.1.2.1.cmml">|</mo><mrow id="S3.E5.m1.1.1.1.1.2.2.1.1.1.1" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.cmml"><mi id="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.2" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.2.cmml">k</mi><mo 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id="S3.E5.m1.1.1.1.1.2.2.1.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1"><times id="S3.E5.m1.1.1.1.1.2.2.1.2.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.2"></times><ci id="S3.E5.m1.1.1.1.1.2.2.1.3a.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.3"><mtext id="S3.E5.m1.1.1.1.1.2.2.1.3.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.3">if </mtext></ci><apply id="S3.E5.m1.1.1.1.1.2.2.1.1.2.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1"><abs id="S3.E5.m1.1.1.1.1.2.2.1.1.2.1.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1.2"></abs><apply id="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1.1"><minus id="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.1.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.1"></minus><ci id="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.2.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.2">𝑘</ci><ci id="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.3.cmml" xref="S3.E5.m1.1.1.1.1.2.2.1.1.1.1.3">𝑗</ci></apply></apply></apply><cn id="S3.E5.m1.1.1.1.1.2.2.3.cmml" type="integer" xref="S3.E5.m1.1.1.1.1.2.2.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E5.m1.1c">\varphi_{k}\ast g_{j}=0\quad\text{if }\,|k-j|\geq 2.</annotation><annotation encoding="application/x-llamapun" id="S3.E5.m1.1d">italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0 if | italic_k - italic_j | ≥ 2 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.2.p2.4">Moreover, by properties of the Fourier transform, we obtain for <math alttext="j\geq 1" class="ltx_Math" display="inline" id="S3.SS2.2.p2.4.m1.1"><semantics id="S3.SS2.2.p2.4.m1.1a"><mrow id="S3.SS2.2.p2.4.m1.1.1" xref="S3.SS2.2.p2.4.m1.1.1.cmml"><mi id="S3.SS2.2.p2.4.m1.1.1.2" xref="S3.SS2.2.p2.4.m1.1.1.2.cmml">j</mi><mo id="S3.SS2.2.p2.4.m1.1.1.1" xref="S3.SS2.2.p2.4.m1.1.1.1.cmml">≥</mo><mn id="S3.SS2.2.p2.4.m1.1.1.3" xref="S3.SS2.2.p2.4.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.4.m1.1b"><apply id="S3.SS2.2.p2.4.m1.1.1.cmml" xref="S3.SS2.2.p2.4.m1.1.1"><geq id="S3.SS2.2.p2.4.m1.1.1.1.cmml" xref="S3.SS2.2.p2.4.m1.1.1.1"></geq><ci id="S3.SS2.2.p2.4.m1.1.1.2.cmml" xref="S3.SS2.2.p2.4.m1.1.1.2">𝑗</ci><cn id="S3.SS2.2.p2.4.m1.1.1.3.cmml" type="integer" xref="S3.SS2.2.p2.4.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.4.m1.1c">j\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.4.m1.1d">italic_j ≥ 1</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx11"> <tbody id="S3.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{F}_{1}^{-1}\eta^{m}_{j}=\frac{2^{-jm}}{m!}\mathcal{F}^{-% 1}_{1}\big{(}((-D)^{m}\eta)(2^{j}\cdot)\big{)}=\frac{2^{-j(m-1)}}{m!}\big{(}% \mathcal{F}_{1}^{-1}((-D)^{m}\eta)\big{)}(2^{j}\cdot)," class="ltx_math_unparsed" display="inline" id="S3.Ex27.m1.1"><semantics id="S3.Ex27.m1.1a"><mrow id="S3.Ex27.m1.1b"><msubsup id="S3.Ex27.m1.1.2"><mi class="ltx_font_mathcaligraphic" id="S3.Ex27.m1.1.2.2.2">ℱ</mi><mn id="S3.Ex27.m1.1.2.2.3">1</mn><mrow id="S3.Ex27.m1.1.2.3"><mo id="S3.Ex27.m1.1.2.3a">−</mo><mn id="S3.Ex27.m1.1.2.3.2">1</mn></mrow></msubsup><msubsup id="S3.Ex27.m1.1.3"><mi id="S3.Ex27.m1.1.3.2.2">η</mi><mi id="S3.Ex27.m1.1.3.3">j</mi><mi id="S3.Ex27.m1.1.3.2.3">m</mi></msubsup><mo id="S3.Ex27.m1.1.4">=</mo><mstyle displaystyle="true" id="S3.Ex27.m1.1.5"><mfrac id="S3.Ex27.m1.1.5a"><msup id="S3.Ex27.m1.1.5.2"><mn id="S3.Ex27.m1.1.5.2.2">2</mn><mrow id="S3.Ex27.m1.1.5.2.3"><mo id="S3.Ex27.m1.1.5.2.3a">−</mo><mrow id="S3.Ex27.m1.1.5.2.3.2"><mi id="S3.Ex27.m1.1.5.2.3.2.2">j</mi><mo id="S3.Ex27.m1.1.5.2.3.2.1">⁢</mo><mi id="S3.Ex27.m1.1.5.2.3.2.3">m</mi></mrow></mrow></msup><mrow id="S3.Ex27.m1.1.5.3"><mi id="S3.Ex27.m1.1.5.3.2">m</mi><mo id="S3.Ex27.m1.1.5.3.1">!</mo></mrow></mfrac></mstyle><msubsup id="S3.Ex27.m1.1.6"><mi class="ltx_font_mathcaligraphic" id="S3.Ex27.m1.1.6.2.2">ℱ</mi><mn id="S3.Ex27.m1.1.6.3">1</mn><mrow id="S3.Ex27.m1.1.6.2.3"><mo id="S3.Ex27.m1.1.6.2.3a">−</mo><mn id="S3.Ex27.m1.1.6.2.3.2">1</mn></mrow></msubsup><mrow id="S3.Ex27.m1.1.7"><mo id="S3.Ex27.m1.1.7.1" maxsize="120%" minsize="120%">(</mo><mrow id="S3.Ex27.m1.1.7.2"><mo id="S3.Ex27.m1.1.7.2.1" stretchy="false">(</mo><msup id="S3.Ex27.m1.1.7.2.2"><mrow id="S3.Ex27.m1.1.7.2.2.2"><mo id="S3.Ex27.m1.1.7.2.2.2.1" stretchy="false">(</mo><mo id="S3.Ex27.m1.1.7.2.2.2.2" lspace="0em">−</mo><mi id="S3.Ex27.m1.1.7.2.2.2.3">D</mi><mo id="S3.Ex27.m1.1.7.2.2.2.4" stretchy="false">)</mo></mrow><mi id="S3.Ex27.m1.1.7.2.2.3">m</mi></msup><mi id="S3.Ex27.m1.1.7.2.3">η</mi><mo id="S3.Ex27.m1.1.7.2.4" stretchy="false">)</mo></mrow><mrow id="S3.Ex27.m1.1.7.3"><mo id="S3.Ex27.m1.1.7.3.1" stretchy="false">(</mo><msup id="S3.Ex27.m1.1.7.3.2"><mn id="S3.Ex27.m1.1.7.3.2.2">2</mn><mi id="S3.Ex27.m1.1.7.3.2.3">j</mi></msup><mo id="S3.Ex27.m1.1.7.3.3" lspace="0.222em" rspace="0em">⋅</mo><mo id="S3.Ex27.m1.1.7.3.4" stretchy="false">)</mo></mrow><mo id="S3.Ex27.m1.1.7.4" maxsize="120%" minsize="120%">)</mo></mrow><mo id="S3.Ex27.m1.1.8">=</mo><mstyle displaystyle="true" id="S3.Ex27.m1.1.1"><mfrac id="S3.Ex27.m1.1.1a"><msup id="S3.Ex27.m1.1.1.1"><mn id="S3.Ex27.m1.1.1.1.3">2</mn><mrow id="S3.Ex27.m1.1.1.1.1.1"><mo id="S3.Ex27.m1.1.1.1.1.1a">−</mo><mrow id="S3.Ex27.m1.1.1.1.1.1.1"><mi id="S3.Ex27.m1.1.1.1.1.1.1.3">j</mi><mo id="S3.Ex27.m1.1.1.1.1.1.1.2">⁢</mo><mrow id="S3.Ex27.m1.1.1.1.1.1.1.1.1"><mo id="S3.Ex27.m1.1.1.1.1.1.1.1.1.2" stretchy="false">(</mo><mrow id="S3.Ex27.m1.1.1.1.1.1.1.1.1.1"><mi id="S3.Ex27.m1.1.1.1.1.1.1.1.1.1.2">m</mi><mo id="S3.Ex27.m1.1.1.1.1.1.1.1.1.1.1">−</mo><mn id="S3.Ex27.m1.1.1.1.1.1.1.1.1.1.3">1</mn></mrow><mo id="S3.Ex27.m1.1.1.1.1.1.1.1.1.3" stretchy="false">)</mo></mrow></mrow></mrow></msup><mrow id="S3.Ex27.m1.1.1.3"><mi id="S3.Ex27.m1.1.1.3.2">m</mi><mo id="S3.Ex27.m1.1.1.3.1">!</mo></mrow></mfrac></mstyle><mrow id="S3.Ex27.m1.1.9"><mo id="S3.Ex27.m1.1.9.1" maxsize="120%" minsize="120%">(</mo><msubsup id="S3.Ex27.m1.1.9.2"><mi class="ltx_font_mathcaligraphic" id="S3.Ex27.m1.1.9.2.2.2">ℱ</mi><mn id="S3.Ex27.m1.1.9.2.2.3">1</mn><mrow id="S3.Ex27.m1.1.9.2.3"><mo id="S3.Ex27.m1.1.9.2.3a">−</mo><mn id="S3.Ex27.m1.1.9.2.3.2">1</mn></mrow></msubsup><mrow id="S3.Ex27.m1.1.9.3"><mo id="S3.Ex27.m1.1.9.3.1" stretchy="false">(</mo><msup id="S3.Ex27.m1.1.9.3.2"><mrow id="S3.Ex27.m1.1.9.3.2.2"><mo id="S3.Ex27.m1.1.9.3.2.2.1" stretchy="false">(</mo><mo id="S3.Ex27.m1.1.9.3.2.2.2" lspace="0em">−</mo><mi id="S3.Ex27.m1.1.9.3.2.2.3">D</mi><mo id="S3.Ex27.m1.1.9.3.2.2.4" stretchy="false">)</mo></mrow><mi id="S3.Ex27.m1.1.9.3.2.3">m</mi></msup><mi id="S3.Ex27.m1.1.9.3.3">η</mi><mo id="S3.Ex27.m1.1.9.3.4" stretchy="false">)</mo></mrow><mo id="S3.Ex27.m1.1.9.4" maxsize="120%" minsize="120%">)</mo></mrow><mrow id="S3.Ex27.m1.1.10"><mo id="S3.Ex27.m1.1.10.1" stretchy="false">(</mo><msup id="S3.Ex27.m1.1.10.2"><mn id="S3.Ex27.m1.1.10.2.2">2</mn><mi id="S3.Ex27.m1.1.10.2.3">j</mi></msup><mo id="S3.Ex27.m1.1.10.3" lspace="0.222em" rspace="0em">⋅</mo><mo id="S3.Ex27.m1.1.10.4" stretchy="false">)</mo></mrow><mo id="S3.Ex27.m1.1.11">,</mo></mrow><annotation encoding="application/x-tex" id="S3.Ex27.m1.1c">\displaystyle\mathcal{F}_{1}^{-1}\eta^{m}_{j}=\frac{2^{-jm}}{m!}\mathcal{F}^{-% 1}_{1}\big{(}((-D)^{m}\eta)(2^{j}\cdot)\big{)}=\frac{2^{-j(m-1)}}{m!}\big{(}% \mathcal{F}_{1}^{-1}((-D)^{m}\eta)\big{)}(2^{j}\cdot),</annotation><annotation encoding="application/x-llamapun" id="S3.Ex27.m1.1d">caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = divide start_ARG 2 start_POSTSUPERSCRIPT - italic_j italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_m ! end_ARG caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ( ( ( - italic_D ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_η ) ( 2 start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ⋅ ) ) = divide start_ARG 2 start_POSTSUPERSCRIPT - italic_j ( italic_m - 1 ) end_POSTSUPERSCRIPT end_ARG start_ARG italic_m ! end_ARG ( caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT ( ( - italic_D ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_η ) ) ( 2 start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ⋅ ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.2.p2.5">and with a substitution <math alttext="x\mapsto 2^{-j}x" class="ltx_Math" display="inline" id="S3.SS2.2.p2.5.m1.1"><semantics id="S3.SS2.2.p2.5.m1.1a"><mrow id="S3.SS2.2.p2.5.m1.1.1" xref="S3.SS2.2.p2.5.m1.1.1.cmml"><mi id="S3.SS2.2.p2.5.m1.1.1.2" xref="S3.SS2.2.p2.5.m1.1.1.2.cmml">x</mi><mo id="S3.SS2.2.p2.5.m1.1.1.1" stretchy="false" xref="S3.SS2.2.p2.5.m1.1.1.1.cmml">↦</mo><mrow id="S3.SS2.2.p2.5.m1.1.1.3" xref="S3.SS2.2.p2.5.m1.1.1.3.cmml"><msup id="S3.SS2.2.p2.5.m1.1.1.3.2" xref="S3.SS2.2.p2.5.m1.1.1.3.2.cmml"><mn id="S3.SS2.2.p2.5.m1.1.1.3.2.2" xref="S3.SS2.2.p2.5.m1.1.1.3.2.2.cmml">2</mn><mrow id="S3.SS2.2.p2.5.m1.1.1.3.2.3" xref="S3.SS2.2.p2.5.m1.1.1.3.2.3.cmml"><mo id="S3.SS2.2.p2.5.m1.1.1.3.2.3a" xref="S3.SS2.2.p2.5.m1.1.1.3.2.3.cmml">−</mo><mi id="S3.SS2.2.p2.5.m1.1.1.3.2.3.2" xref="S3.SS2.2.p2.5.m1.1.1.3.2.3.2.cmml">j</mi></mrow></msup><mo id="S3.SS2.2.p2.5.m1.1.1.3.1" xref="S3.SS2.2.p2.5.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.2.p2.5.m1.1.1.3.3" xref="S3.SS2.2.p2.5.m1.1.1.3.3.cmml">x</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.5.m1.1b"><apply id="S3.SS2.2.p2.5.m1.1.1.cmml" xref="S3.SS2.2.p2.5.m1.1.1"><csymbol cd="latexml" id="S3.SS2.2.p2.5.m1.1.1.1.cmml" xref="S3.SS2.2.p2.5.m1.1.1.1">maps-to</csymbol><ci id="S3.SS2.2.p2.5.m1.1.1.2.cmml" xref="S3.SS2.2.p2.5.m1.1.1.2">𝑥</ci><apply id="S3.SS2.2.p2.5.m1.1.1.3.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3"><times id="S3.SS2.2.p2.5.m1.1.1.3.1.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3.1"></times><apply id="S3.SS2.2.p2.5.m1.1.1.3.2.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.5.m1.1.1.3.2.1.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3.2">superscript</csymbol><cn id="S3.SS2.2.p2.5.m1.1.1.3.2.2.cmml" type="integer" xref="S3.SS2.2.p2.5.m1.1.1.3.2.2">2</cn><apply id="S3.SS2.2.p2.5.m1.1.1.3.2.3.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3.2.3"><minus id="S3.SS2.2.p2.5.m1.1.1.3.2.3.1.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3.2.3"></minus><ci id="S3.SS2.2.p2.5.m1.1.1.3.2.3.2.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3.2.3.2">𝑗</ci></apply></apply><ci id="S3.SS2.2.p2.5.m1.1.1.3.3.cmml" xref="S3.SS2.2.p2.5.m1.1.1.3.3">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.5.m1.1c">x\mapsto 2^{-j}x</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.5.m1.1d">italic_x ↦ 2 start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_x</annotation></semantics></math> we obtain that</p> <table class="ltx_equation ltx_eqn_table" id="S3.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|\mathcal{F}_{1}^{-1}\eta_{j}^{m}\|_{L^{p}(\mathbb{R},w_{\gamma})}=2^{j(1-m-% \frac{\gamma+1}{p})}\|\mathcal{F}_{1}^{-1}\eta^{m}\|_{L^{p}(\mathbb{R},w_{% \gamma})}," class="ltx_Math" display="block" id="S3.E6.m1.6"><semantics id="S3.E6.m1.6a"><mrow id="S3.E6.m1.6.6.1" xref="S3.E6.m1.6.6.1.1.cmml"><mrow id="S3.E6.m1.6.6.1.1" xref="S3.E6.m1.6.6.1.1.cmml"><msub id="S3.E6.m1.6.6.1.1.1" xref="S3.E6.m1.6.6.1.1.1.cmml"><mrow id="S3.E6.m1.6.6.1.1.1.1.1" xref="S3.E6.m1.6.6.1.1.1.1.2.cmml"><mo id="S3.E6.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S3.E6.m1.6.6.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.E6.m1.6.6.1.1.1.1.1.1" xref="S3.E6.m1.6.6.1.1.1.1.1.1.cmml"><msubsup id="S3.E6.m1.6.6.1.1.1.1.1.1.2" xref="S3.E6.m1.6.6.1.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E6.m1.6.6.1.1.1.1.1.1.2.2.2" xref="S3.E6.m1.6.6.1.1.1.1.1.1.2.2.2.cmml">ℱ</mi><mn id="S3.E6.m1.6.6.1.1.1.1.1.1.2.2.3" xref="S3.E6.m1.6.6.1.1.1.1.1.1.2.2.3.cmml">1</mn><mrow id="S3.E6.m1.6.6.1.1.1.1.1.1.2.3" xref="S3.E6.m1.6.6.1.1.1.1.1.1.2.3.cmml"><mo id="S3.E6.m1.6.6.1.1.1.1.1.1.2.3a" xref="S3.E6.m1.6.6.1.1.1.1.1.1.2.3.cmml">−</mo><mn id="S3.E6.m1.6.6.1.1.1.1.1.1.2.3.2" xref="S3.E6.m1.6.6.1.1.1.1.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.E6.m1.6.6.1.1.1.1.1.1.1" xref="S3.E6.m1.6.6.1.1.1.1.1.1.1.cmml">⁢</mo><msubsup id="S3.E6.m1.6.6.1.1.1.1.1.1.3" xref="S3.E6.m1.6.6.1.1.1.1.1.1.3.cmml"><mi id="S3.E6.m1.6.6.1.1.1.1.1.1.3.2.2" xref="S3.E6.m1.6.6.1.1.1.1.1.1.3.2.2.cmml">η</mi><mi id="S3.E6.m1.6.6.1.1.1.1.1.1.3.2.3" xref="S3.E6.m1.6.6.1.1.1.1.1.1.3.2.3.cmml">j</mi><mi id="S3.E6.m1.6.6.1.1.1.1.1.1.3.3" xref="S3.E6.m1.6.6.1.1.1.1.1.1.3.3.cmml">m</mi></msubsup></mrow><mo id="S3.E6.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="S3.E6.m1.6.6.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.E6.m1.2.2.2" xref="S3.E6.m1.2.2.2.cmml"><msup id="S3.E6.m1.2.2.2.4" xref="S3.E6.m1.2.2.2.4.cmml"><mi id="S3.E6.m1.2.2.2.4.2" xref="S3.E6.m1.2.2.2.4.2.cmml">L</mi><mi id="S3.E6.m1.2.2.2.4.3" xref="S3.E6.m1.2.2.2.4.3.cmml">p</mi></msup><mo id="S3.E6.m1.2.2.2.3" xref="S3.E6.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.E6.m1.2.2.2.2.1" xref="S3.E6.m1.2.2.2.2.2.cmml"><mo id="S3.E6.m1.2.2.2.2.1.2" stretchy="false" xref="S3.E6.m1.2.2.2.2.2.cmml">(</mo><mi id="S3.E6.m1.1.1.1.1" xref="S3.E6.m1.1.1.1.1.cmml">ℝ</mi><mo id="S3.E6.m1.2.2.2.2.1.3" xref="S3.E6.m1.2.2.2.2.2.cmml">,</mo><msub id="S3.E6.m1.2.2.2.2.1.1" xref="S3.E6.m1.2.2.2.2.1.1.cmml"><mi id="S3.E6.m1.2.2.2.2.1.1.2" xref="S3.E6.m1.2.2.2.2.1.1.2.cmml">w</mi><mi id="S3.E6.m1.2.2.2.2.1.1.3" xref="S3.E6.m1.2.2.2.2.1.1.3.cmml">γ</mi></msub><mo id="S3.E6.m1.2.2.2.2.1.4" stretchy="false" xref="S3.E6.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></msub><mo id="S3.E6.m1.6.6.1.1.3" xref="S3.E6.m1.6.6.1.1.3.cmml">=</mo><mrow id="S3.E6.m1.6.6.1.1.2" xref="S3.E6.m1.6.6.1.1.2.cmml"><msup id="S3.E6.m1.6.6.1.1.2.3" xref="S3.E6.m1.6.6.1.1.2.3.cmml"><mn id="S3.E6.m1.6.6.1.1.2.3.2" xref="S3.E6.m1.6.6.1.1.2.3.2.cmml">2</mn><mrow id="S3.E6.m1.3.3.1" xref="S3.E6.m1.3.3.1.cmml"><mi id="S3.E6.m1.3.3.1.3" xref="S3.E6.m1.3.3.1.3.cmml">j</mi><mo id="S3.E6.m1.3.3.1.2" xref="S3.E6.m1.3.3.1.2.cmml">⁢</mo><mrow id="S3.E6.m1.3.3.1.1.1" xref="S3.E6.m1.3.3.1.1.1.1.cmml"><mo id="S3.E6.m1.3.3.1.1.1.2" stretchy="false" xref="S3.E6.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="S3.E6.m1.3.3.1.1.1.1" xref="S3.E6.m1.3.3.1.1.1.1.cmml"><mn id="S3.E6.m1.3.3.1.1.1.1.2" xref="S3.E6.m1.3.3.1.1.1.1.2.cmml">1</mn><mo id="S3.E6.m1.3.3.1.1.1.1.1" xref="S3.E6.m1.3.3.1.1.1.1.1.cmml">−</mo><mi id="S3.E6.m1.3.3.1.1.1.1.3" xref="S3.E6.m1.3.3.1.1.1.1.3.cmml">m</mi><mo id="S3.E6.m1.3.3.1.1.1.1.1a" xref="S3.E6.m1.3.3.1.1.1.1.1.cmml">−</mo><mfrac id="S3.E6.m1.3.3.1.1.1.1.4" xref="S3.E6.m1.3.3.1.1.1.1.4.cmml"><mrow id="S3.E6.m1.3.3.1.1.1.1.4.2" xref="S3.E6.m1.3.3.1.1.1.1.4.2.cmml"><mi id="S3.E6.m1.3.3.1.1.1.1.4.2.2" xref="S3.E6.m1.3.3.1.1.1.1.4.2.2.cmml">γ</mi><mo id="S3.E6.m1.3.3.1.1.1.1.4.2.1" xref="S3.E6.m1.3.3.1.1.1.1.4.2.1.cmml">+</mo><mn id="S3.E6.m1.3.3.1.1.1.1.4.2.3" xref="S3.E6.m1.3.3.1.1.1.1.4.2.3.cmml">1</mn></mrow><mi id="S3.E6.m1.3.3.1.1.1.1.4.3" xref="S3.E6.m1.3.3.1.1.1.1.4.3.cmml">p</mi></mfrac></mrow><mo id="S3.E6.m1.3.3.1.1.1.3" stretchy="false" xref="S3.E6.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></msup><mo id="S3.E6.m1.6.6.1.1.2.2" xref="S3.E6.m1.6.6.1.1.2.2.cmml">⁢</mo><msub id="S3.E6.m1.6.6.1.1.2.1" xref="S3.E6.m1.6.6.1.1.2.1.cmml"><mrow id="S3.E6.m1.6.6.1.1.2.1.1.1" xref="S3.E6.m1.6.6.1.1.2.1.1.2.cmml"><mo id="S3.E6.m1.6.6.1.1.2.1.1.1.2" stretchy="false" xref="S3.E6.m1.6.6.1.1.2.1.1.2.1.cmml">‖</mo><mrow id="S3.E6.m1.6.6.1.1.2.1.1.1.1" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.cmml"><msubsup id="S3.E6.m1.6.6.1.1.2.1.1.1.1.2" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.2.2" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.2.2.cmml">ℱ</mi><mn id="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.2.3" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.2.3.cmml">1</mn><mrow id="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.3" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.3.cmml"><mo id="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.3a" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.3.cmml">−</mo><mn id="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.3.2" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.2.3.2.cmml">1</mn></mrow></msubsup><mo id="S3.E6.m1.6.6.1.1.2.1.1.1.1.1" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.1.cmml">⁢</mo><msup id="S3.E6.m1.6.6.1.1.2.1.1.1.1.3" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.3.cmml"><mi id="S3.E6.m1.6.6.1.1.2.1.1.1.1.3.2" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.3.2.cmml">η</mi><mi id="S3.E6.m1.6.6.1.1.2.1.1.1.1.3.3" xref="S3.E6.m1.6.6.1.1.2.1.1.1.1.3.3.cmml">m</mi></msup></mrow><mo id="S3.E6.m1.6.6.1.1.2.1.1.1.3" stretchy="false" xref="S3.E6.m1.6.6.1.1.2.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.E6.m1.5.5.2" xref="S3.E6.m1.5.5.2.cmml"><msup id="S3.E6.m1.5.5.2.4" xref="S3.E6.m1.5.5.2.4.cmml"><mi id="S3.E6.m1.5.5.2.4.2" xref="S3.E6.m1.5.5.2.4.2.cmml">L</mi><mi id="S3.E6.m1.5.5.2.4.3" xref="S3.E6.m1.5.5.2.4.3.cmml">p</mi></msup><mo id="S3.E6.m1.5.5.2.3" xref="S3.E6.m1.5.5.2.3.cmml">⁢</mo><mrow id="S3.E6.m1.5.5.2.2.1" xref="S3.E6.m1.5.5.2.2.2.cmml"><mo id="S3.E6.m1.5.5.2.2.1.2" stretchy="false" xref="S3.E6.m1.5.5.2.2.2.cmml">(</mo><mi id="S3.E6.m1.4.4.1.1" xref="S3.E6.m1.4.4.1.1.cmml">ℝ</mi><mo 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xref="S3.E6.m1.5.5.2.2.1.1.3">𝛾</ci></apply></interval></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E6.m1.6c">\|\mathcal{F}_{1}^{-1}\eta_{j}^{m}\|_{L^{p}(\mathbb{R},w_{\gamma})}=2^{j(1-m-% \frac{\gamma+1}{p})}\|\mathcal{F}_{1}^{-1}\eta^{m}\|_{L^{p}(\mathbb{R},w_{% \gamma})},</annotation><annotation encoding="application/x-llamapun" id="S3.E6.m1.6d">∥ caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT = 2 start_POSTSUPERSCRIPT italic_j ( 1 - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ caligraphic_F start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.2.p2.9">where <math alttext="\eta^{m}:=(-D)^{m}\eta/m!" class="ltx_Math" display="inline" id="S3.SS2.2.p2.6.m1.1"><semantics id="S3.SS2.2.p2.6.m1.1a"><mrow id="S3.SS2.2.p2.6.m1.1.1" xref="S3.SS2.2.p2.6.m1.1.1.cmml"><msup id="S3.SS2.2.p2.6.m1.1.1.3" xref="S3.SS2.2.p2.6.m1.1.1.3.cmml"><mi id="S3.SS2.2.p2.6.m1.1.1.3.2" xref="S3.SS2.2.p2.6.m1.1.1.3.2.cmml">η</mi><mi id="S3.SS2.2.p2.6.m1.1.1.3.3" xref="S3.SS2.2.p2.6.m1.1.1.3.3.cmml">m</mi></msup><mo id="S3.SS2.2.p2.6.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S3.SS2.2.p2.6.m1.1.1.2.cmml">:=</mo><mrow id="S3.SS2.2.p2.6.m1.1.1.1" xref="S3.SS2.2.p2.6.m1.1.1.1.cmml"><mrow id="S3.SS2.2.p2.6.m1.1.1.1.1" xref="S3.SS2.2.p2.6.m1.1.1.1.1.cmml"><msup id="S3.SS2.2.p2.6.m1.1.1.1.1.1" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.cmml"><mrow id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1a" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.2" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.2.cmml">D</mi></mrow><mo id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mi id="S3.SS2.2.p2.6.m1.1.1.1.1.1.3" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.3.cmml">m</mi></msup><mo id="S3.SS2.2.p2.6.m1.1.1.1.1.2" xref="S3.SS2.2.p2.6.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S3.SS2.2.p2.6.m1.1.1.1.1.3" xref="S3.SS2.2.p2.6.m1.1.1.1.1.3.cmml">η</mi></mrow><mo id="S3.SS2.2.p2.6.m1.1.1.1.2" xref="S3.SS2.2.p2.6.m1.1.1.1.2.cmml">/</mo><mrow id="S3.SS2.2.p2.6.m1.1.1.1.3" xref="S3.SS2.2.p2.6.m1.1.1.1.3.cmml"><mi id="S3.SS2.2.p2.6.m1.1.1.1.3.2" xref="S3.SS2.2.p2.6.m1.1.1.1.3.2.cmml">m</mi><mo id="S3.SS2.2.p2.6.m1.1.1.1.3.1" xref="S3.SS2.2.p2.6.m1.1.1.1.3.1.cmml">!</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.6.m1.1b"><apply id="S3.SS2.2.p2.6.m1.1.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1"><csymbol cd="latexml" id="S3.SS2.2.p2.6.m1.1.1.2.cmml" xref="S3.SS2.2.p2.6.m1.1.1.2">assign</csymbol><apply id="S3.SS2.2.p2.6.m1.1.1.3.cmml" xref="S3.SS2.2.p2.6.m1.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.6.m1.1.1.3.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1.3">superscript</csymbol><ci id="S3.SS2.2.p2.6.m1.1.1.3.2.cmml" xref="S3.SS2.2.p2.6.m1.1.1.3.2">𝜂</ci><ci id="S3.SS2.2.p2.6.m1.1.1.3.3.cmml" xref="S3.SS2.2.p2.6.m1.1.1.3.3">𝑚</ci></apply><apply id="S3.SS2.2.p2.6.m1.1.1.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1"><divide id="S3.SS2.2.p2.6.m1.1.1.1.2.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.2"></divide><apply id="S3.SS2.2.p2.6.m1.1.1.1.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1"><times id="S3.SS2.2.p2.6.m1.1.1.1.1.2.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.2"></times><apply id="S3.SS2.2.p2.6.m1.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.2.p2.6.m1.1.1.1.1.1.2.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1">superscript</csymbol><apply id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1"><minus id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1"></minus><ci id="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.2.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.1.1.1.2">𝐷</ci></apply><ci id="S3.SS2.2.p2.6.m1.1.1.1.1.1.3.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.1.3">𝑚</ci></apply><ci id="S3.SS2.2.p2.6.m1.1.1.1.1.3.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.1.3">𝜂</ci></apply><apply id="S3.SS2.2.p2.6.m1.1.1.1.3.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.3"><factorial id="S3.SS2.2.p2.6.m1.1.1.1.3.1.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.3.1"></factorial><ci id="S3.SS2.2.p2.6.m1.1.1.1.3.2.cmml" xref="S3.SS2.2.p2.6.m1.1.1.1.3.2">𝑚</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.6.m1.1c">\eta^{m}:=(-D)^{m}\eta/m!</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.6.m1.1d">italic_η start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT := ( - italic_D ) start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_η / italic_m !</annotation></semantics></math>. Define <math alttext="s_{n}g:=\sum_{j=0}^{n}2^{-j}g_{j}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.7.m2.1"><semantics id="S3.SS2.2.p2.7.m2.1a"><mrow id="S3.SS2.2.p2.7.m2.1.1" xref="S3.SS2.2.p2.7.m2.1.1.cmml"><mrow id="S3.SS2.2.p2.7.m2.1.1.2" xref="S3.SS2.2.p2.7.m2.1.1.2.cmml"><msub id="S3.SS2.2.p2.7.m2.1.1.2.2" xref="S3.SS2.2.p2.7.m2.1.1.2.2.cmml"><mi id="S3.SS2.2.p2.7.m2.1.1.2.2.2" xref="S3.SS2.2.p2.7.m2.1.1.2.2.2.cmml">s</mi><mi id="S3.SS2.2.p2.7.m2.1.1.2.2.3" xref="S3.SS2.2.p2.7.m2.1.1.2.2.3.cmml">n</mi></msub><mo id="S3.SS2.2.p2.7.m2.1.1.2.1" xref="S3.SS2.2.p2.7.m2.1.1.2.1.cmml">⁢</mo><mi id="S3.SS2.2.p2.7.m2.1.1.2.3" xref="S3.SS2.2.p2.7.m2.1.1.2.3.cmml">g</mi></mrow><mo id="S3.SS2.2.p2.7.m2.1.1.1" lspace="0.278em" rspace="0.111em" xref="S3.SS2.2.p2.7.m2.1.1.1.cmml">:=</mo><mrow id="S3.SS2.2.p2.7.m2.1.1.3" xref="S3.SS2.2.p2.7.m2.1.1.3.cmml"><msubsup id="S3.SS2.2.p2.7.m2.1.1.3.1" xref="S3.SS2.2.p2.7.m2.1.1.3.1.cmml"><mo id="S3.SS2.2.p2.7.m2.1.1.3.1.2.2" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.2.cmml">∑</mo><mrow id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.cmml"><mi id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.2" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.2.cmml">j</mi><mo id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.1" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.1.cmml">=</mo><mn id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.3" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.3.cmml">0</mn></mrow><mi id="S3.SS2.2.p2.7.m2.1.1.3.1.3" xref="S3.SS2.2.p2.7.m2.1.1.3.1.3.cmml">n</mi></msubsup><mrow id="S3.SS2.2.p2.7.m2.1.1.3.2" xref="S3.SS2.2.p2.7.m2.1.1.3.2.cmml"><msup id="S3.SS2.2.p2.7.m2.1.1.3.2.2" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.cmml"><mn id="S3.SS2.2.p2.7.m2.1.1.3.2.2.2" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.2.cmml">2</mn><mrow id="S3.SS2.2.p2.7.m2.1.1.3.2.2.3" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.cmml"><mo id="S3.SS2.2.p2.7.m2.1.1.3.2.2.3a" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.cmml">−</mo><mi id="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.2" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.2.cmml">j</mi></mrow></msup><mo id="S3.SS2.2.p2.7.m2.1.1.3.2.1" xref="S3.SS2.2.p2.7.m2.1.1.3.2.1.cmml">⁢</mo><msub id="S3.SS2.2.p2.7.m2.1.1.3.2.3" xref="S3.SS2.2.p2.7.m2.1.1.3.2.3.cmml"><mi id="S3.SS2.2.p2.7.m2.1.1.3.2.3.2" xref="S3.SS2.2.p2.7.m2.1.1.3.2.3.2.cmml">g</mi><mi id="S3.SS2.2.p2.7.m2.1.1.3.2.3.3" xref="S3.SS2.2.p2.7.m2.1.1.3.2.3.3.cmml">j</mi></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.7.m2.1b"><apply id="S3.SS2.2.p2.7.m2.1.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1"><csymbol cd="latexml" id="S3.SS2.2.p2.7.m2.1.1.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.1">assign</csymbol><apply id="S3.SS2.2.p2.7.m2.1.1.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.2"><times id="S3.SS2.2.p2.7.m2.1.1.2.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.2.1"></times><apply id="S3.SS2.2.p2.7.m2.1.1.2.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.7.m2.1.1.2.2.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.2.2">subscript</csymbol><ci id="S3.SS2.2.p2.7.m2.1.1.2.2.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.2.2.2">𝑠</ci><ci id="S3.SS2.2.p2.7.m2.1.1.2.2.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.2.2.3">𝑛</ci></apply><ci id="S3.SS2.2.p2.7.m2.1.1.2.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.2.3">𝑔</ci></apply><apply id="S3.SS2.2.p2.7.m2.1.1.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3"><apply id="S3.SS2.2.p2.7.m2.1.1.3.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS2.2.p2.7.m2.1.1.3.1.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1">superscript</csymbol><apply id="S3.SS2.2.p2.7.m2.1.1.3.1.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS2.2.p2.7.m2.1.1.3.1.2.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1">subscript</csymbol><sum id="S3.SS2.2.p2.7.m2.1.1.3.1.2.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.2"></sum><apply id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3"><eq id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.1"></eq><ci id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.2">𝑗</ci><cn id="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.3.cmml" type="integer" xref="S3.SS2.2.p2.7.m2.1.1.3.1.2.3.3">0</cn></apply></apply><ci id="S3.SS2.2.p2.7.m2.1.1.3.1.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.1.3">𝑛</ci></apply><apply id="S3.SS2.2.p2.7.m2.1.1.3.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2"><times id="S3.SS2.2.p2.7.m2.1.1.3.2.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.1"></times><apply id="S3.SS2.2.p2.7.m2.1.1.3.2.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2"><csymbol cd="ambiguous" id="S3.SS2.2.p2.7.m2.1.1.3.2.2.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2">superscript</csymbol><cn id="S3.SS2.2.p2.7.m2.1.1.3.2.2.2.cmml" type="integer" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.2">2</cn><apply id="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.3"><minus id="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.3"></minus><ci id="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.2.3.2">𝑗</ci></apply></apply><apply id="S3.SS2.2.p2.7.m2.1.1.3.2.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.7.m2.1.1.3.2.3.1.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.3">subscript</csymbol><ci id="S3.SS2.2.p2.7.m2.1.1.3.2.3.2.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.3.2">𝑔</ci><ci id="S3.SS2.2.p2.7.m2.1.1.3.2.3.3.cmml" xref="S3.SS2.2.p2.7.m2.1.1.3.2.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.7.m2.1c">s_{n}g:=\sum_{j=0}^{n}2^{-j}g_{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.7.m2.1d">italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g := ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="n\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.SS2.2.p2.8.m3.1"><semantics id="S3.SS2.2.p2.8.m3.1a"><mrow id="S3.SS2.2.p2.8.m3.1.1" xref="S3.SS2.2.p2.8.m3.1.1.cmml"><mi id="S3.SS2.2.p2.8.m3.1.1.2" xref="S3.SS2.2.p2.8.m3.1.1.2.cmml">n</mi><mo id="S3.SS2.2.p2.8.m3.1.1.1" xref="S3.SS2.2.p2.8.m3.1.1.1.cmml">∈</mo><msub id="S3.SS2.2.p2.8.m3.1.1.3" xref="S3.SS2.2.p2.8.m3.1.1.3.cmml"><mi id="S3.SS2.2.p2.8.m3.1.1.3.2" xref="S3.SS2.2.p2.8.m3.1.1.3.2.cmml">ℕ</mi><mn id="S3.SS2.2.p2.8.m3.1.1.3.3" xref="S3.SS2.2.p2.8.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.8.m3.1b"><apply id="S3.SS2.2.p2.8.m3.1.1.cmml" xref="S3.SS2.2.p2.8.m3.1.1"><in id="S3.SS2.2.p2.8.m3.1.1.1.cmml" xref="S3.SS2.2.p2.8.m3.1.1.1"></in><ci id="S3.SS2.2.p2.8.m3.1.1.2.cmml" xref="S3.SS2.2.p2.8.m3.1.1.2">𝑛</ci><apply id="S3.SS2.2.p2.8.m3.1.1.3.cmml" xref="S3.SS2.2.p2.8.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.2.p2.8.m3.1.1.3.1.cmml" xref="S3.SS2.2.p2.8.m3.1.1.3">subscript</csymbol><ci id="S3.SS2.2.p2.8.m3.1.1.3.2.cmml" xref="S3.SS2.2.p2.8.m3.1.1.3.2">ℕ</ci><cn id="S3.SS2.2.p2.8.m3.1.1.3.3.cmml" type="integer" xref="S3.SS2.2.p2.8.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.8.m3.1c">n\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.8.m3.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, for <math alttext="0\leq k\leq n+1" class="ltx_Math" display="inline" id="S3.SS2.2.p2.9.m4.1"><semantics id="S3.SS2.2.p2.9.m4.1a"><mrow id="S3.SS2.2.p2.9.m4.1.1" xref="S3.SS2.2.p2.9.m4.1.1.cmml"><mn id="S3.SS2.2.p2.9.m4.1.1.2" xref="S3.SS2.2.p2.9.m4.1.1.2.cmml">0</mn><mo id="S3.SS2.2.p2.9.m4.1.1.3" xref="S3.SS2.2.p2.9.m4.1.1.3.cmml">≤</mo><mi id="S3.SS2.2.p2.9.m4.1.1.4" xref="S3.SS2.2.p2.9.m4.1.1.4.cmml">k</mi><mo id="S3.SS2.2.p2.9.m4.1.1.5" xref="S3.SS2.2.p2.9.m4.1.1.5.cmml">≤</mo><mrow id="S3.SS2.2.p2.9.m4.1.1.6" xref="S3.SS2.2.p2.9.m4.1.1.6.cmml"><mi id="S3.SS2.2.p2.9.m4.1.1.6.2" xref="S3.SS2.2.p2.9.m4.1.1.6.2.cmml">n</mi><mo id="S3.SS2.2.p2.9.m4.1.1.6.1" xref="S3.SS2.2.p2.9.m4.1.1.6.1.cmml">+</mo><mn id="S3.SS2.2.p2.9.m4.1.1.6.3" xref="S3.SS2.2.p2.9.m4.1.1.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.9.m4.1b"><apply id="S3.SS2.2.p2.9.m4.1.1.cmml" xref="S3.SS2.2.p2.9.m4.1.1"><and id="S3.SS2.2.p2.9.m4.1.1a.cmml" xref="S3.SS2.2.p2.9.m4.1.1"></and><apply id="S3.SS2.2.p2.9.m4.1.1b.cmml" xref="S3.SS2.2.p2.9.m4.1.1"><leq id="S3.SS2.2.p2.9.m4.1.1.3.cmml" xref="S3.SS2.2.p2.9.m4.1.1.3"></leq><cn id="S3.SS2.2.p2.9.m4.1.1.2.cmml" type="integer" xref="S3.SS2.2.p2.9.m4.1.1.2">0</cn><ci id="S3.SS2.2.p2.9.m4.1.1.4.cmml" xref="S3.SS2.2.p2.9.m4.1.1.4">𝑘</ci></apply><apply id="S3.SS2.2.p2.9.m4.1.1c.cmml" xref="S3.SS2.2.p2.9.m4.1.1"><leq id="S3.SS2.2.p2.9.m4.1.1.5.cmml" xref="S3.SS2.2.p2.9.m4.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S3.SS2.2.p2.9.m4.1.1.4.cmml" id="S3.SS2.2.p2.9.m4.1.1d.cmml" xref="S3.SS2.2.p2.9.m4.1.1"></share><apply id="S3.SS2.2.p2.9.m4.1.1.6.cmml" xref="S3.SS2.2.p2.9.m4.1.1.6"><plus id="S3.SS2.2.p2.9.m4.1.1.6.1.cmml" xref="S3.SS2.2.p2.9.m4.1.1.6.1"></plus><ci id="S3.SS2.2.p2.9.m4.1.1.6.2.cmml" xref="S3.SS2.2.p2.9.m4.1.1.6.2">𝑛</ci><cn id="S3.SS2.2.p2.9.m4.1.1.6.3.cmml" type="integer" xref="S3.SS2.2.p2.9.m4.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.9.m4.1c">0\leq k\leq n+1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.9.m4.1d">0 ≤ italic_k ≤ italic_n + 1</annotation></semantics></math> we obtain from (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E5" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.5</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E3" title="In 2.2. Weighted function spaces ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E6" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.6</span></a>) that</p> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E7"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E7X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 2^{ks}\|\varphi_{k}\ast" class="ltx_math_unparsed" display="inline" id="S3.E7X.2.1.1.m1.1"><semantics id="S3.E7X.2.1.1.m1.1a"><mrow id="S3.E7X.2.1.1.m1.1b"><msup id="S3.E7X.2.1.1.m1.1.1"><mn id="S3.E7X.2.1.1.m1.1.1.2">2</mn><mrow id="S3.E7X.2.1.1.m1.1.1.3"><mi id="S3.E7X.2.1.1.m1.1.1.3.2">k</mi><mo id="S3.E7X.2.1.1.m1.1.1.3.1">⁢</mo><mi id="S3.E7X.2.1.1.m1.1.1.3.3">s</mi></mrow></msup><mo id="S3.E7X.2.1.1.m1.1.2" lspace="0em" rspace="0.167em">∥</mo><msub id="S3.E7X.2.1.1.m1.1.3"><mi id="S3.E7X.2.1.1.m1.1.3.2">φ</mi><mi id="S3.E7X.2.1.1.m1.1.3.3">k</mi></msub><mo id="S3.E7X.2.1.1.m1.1.4" lspace="0.222em">∗</mo></mrow><annotation encoding="application/x-tex" id="S3.E7X.2.1.1.m1.1c">\displaystyle 2^{ks}\|\varphi_{k}\ast</annotation><annotation encoding="application/x-llamapun" id="S3.E7X.2.1.1.m1.1d">2 start_POSTSUPERSCRIPT italic_k italic_s end_POSTSUPERSCRIPT ∥ italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle s_{n}g\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S3.E7X.3.2.2.m1.4"><semantics id="S3.E7X.3.2.2.m1.4a"><msub id="S3.E7X.3.2.2.m1.4.4.1" xref="S3.E7X.3.2.2.m1.4.4.2.cmml"><mrow id="S3.E7X.3.2.2.m1.4.4.1.1" xref="S3.E7X.3.2.2.m1.4.4.2.cmml"><mrow id="S3.E7X.3.2.2.m1.4.4.1.1.1" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.cmml"><msub id="S3.E7X.3.2.2.m1.4.4.1.1.1.2" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.2.cmml"><mi id="S3.E7X.3.2.2.m1.4.4.1.1.1.2.2" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.2.2.cmml">s</mi><mi id="S3.E7X.3.2.2.m1.4.4.1.1.1.2.3" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.E7X.3.2.2.m1.4.4.1.1.1.1" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.1.cmml">⁢</mo><mi id="S3.E7X.3.2.2.m1.4.4.1.1.1.3" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.3.cmml">g</mi></mrow><mo fence="true" id="S3.E7X.3.2.2.m1.4.4.1.1.2" lspace="0em" xref="S3.E7X.3.2.2.m1.4.4.2.1.cmml">∥</mo></mrow><mrow id="S3.E7X.3.2.2.m1.3.3.3" xref="S3.E7X.3.2.2.m1.3.3.3.cmml"><msup id="S3.E7X.3.2.2.m1.3.3.3.5" xref="S3.E7X.3.2.2.m1.3.3.3.5.cmml"><mi id="S3.E7X.3.2.2.m1.3.3.3.5.2" xref="S3.E7X.3.2.2.m1.3.3.3.5.2.cmml">L</mi><mi id="S3.E7X.3.2.2.m1.3.3.3.5.3" xref="S3.E7X.3.2.2.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S3.E7X.3.2.2.m1.3.3.3.4" xref="S3.E7X.3.2.2.m1.3.3.3.4.cmml">⁢</mo><mrow id="S3.E7X.3.2.2.m1.3.3.3.3.2" xref="S3.E7X.3.2.2.m1.3.3.3.3.3.cmml"><mo id="S3.E7X.3.2.2.m1.3.3.3.3.2.3" stretchy="false" xref="S3.E7X.3.2.2.m1.3.3.3.3.3.cmml">(</mo><msup id="S3.E7X.3.2.2.m1.2.2.2.2.1.1" xref="S3.E7X.3.2.2.m1.2.2.2.2.1.1.cmml"><mi id="S3.E7X.3.2.2.m1.2.2.2.2.1.1.2" xref="S3.E7X.3.2.2.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mi id="S3.E7X.3.2.2.m1.2.2.2.2.1.1.3" xref="S3.E7X.3.2.2.m1.2.2.2.2.1.1.3.cmml">d</mi></msup><mo id="S3.E7X.3.2.2.m1.3.3.3.3.2.4" xref="S3.E7X.3.2.2.m1.3.3.3.3.3.cmml">,</mo><msub id="S3.E7X.3.2.2.m1.3.3.3.3.2.2" xref="S3.E7X.3.2.2.m1.3.3.3.3.2.2.cmml"><mi id="S3.E7X.3.2.2.m1.3.3.3.3.2.2.2" xref="S3.E7X.3.2.2.m1.3.3.3.3.2.2.2.cmml">w</mi><mi id="S3.E7X.3.2.2.m1.3.3.3.3.2.2.3" xref="S3.E7X.3.2.2.m1.3.3.3.3.2.2.3.cmml">γ</mi></msub><mo id="S3.E7X.3.2.2.m1.3.3.3.3.2.5" xref="S3.E7X.3.2.2.m1.3.3.3.3.3.cmml">;</mo><mi id="S3.E7X.3.2.2.m1.1.1.1.1" xref="S3.E7X.3.2.2.m1.1.1.1.1.cmml">X</mi><mo id="S3.E7X.3.2.2.m1.3.3.3.3.2.6" stretchy="false" xref="S3.E7X.3.2.2.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.E7X.3.2.2.m1.4b"><apply id="S3.E7X.3.2.2.m1.4.4.2.cmml" xref="S3.E7X.3.2.2.m1.4.4.1"><csymbol cd="latexml" id="S3.E7X.3.2.2.m1.4.4.2.1.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.2">evaluated-at</csymbol><apply id="S3.E7X.3.2.2.m1.4.4.1.1.1.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.1"><times id="S3.E7X.3.2.2.m1.4.4.1.1.1.1.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.1"></times><apply id="S3.E7X.3.2.2.m1.4.4.1.1.1.2.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.2"><csymbol cd="ambiguous" id="S3.E7X.3.2.2.m1.4.4.1.1.1.2.1.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.2">subscript</csymbol><ci id="S3.E7X.3.2.2.m1.4.4.1.1.1.2.2.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.2.2">𝑠</ci><ci id="S3.E7X.3.2.2.m1.4.4.1.1.1.2.3.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.2.3">𝑛</ci></apply><ci id="S3.E7X.3.2.2.m1.4.4.1.1.1.3.cmml" xref="S3.E7X.3.2.2.m1.4.4.1.1.1.3">𝑔</ci></apply><apply id="S3.E7X.3.2.2.m1.3.3.3.cmml" xref="S3.E7X.3.2.2.m1.3.3.3"><times id="S3.E7X.3.2.2.m1.3.3.3.4.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.4"></times><apply id="S3.E7X.3.2.2.m1.3.3.3.5.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.5"><csymbol cd="ambiguous" id="S3.E7X.3.2.2.m1.3.3.3.5.1.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.5">superscript</csymbol><ci id="S3.E7X.3.2.2.m1.3.3.3.5.2.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.5.2">𝐿</ci><ci id="S3.E7X.3.2.2.m1.3.3.3.5.3.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.5.3">𝑝</ci></apply><vector id="S3.E7X.3.2.2.m1.3.3.3.3.3.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.3.2"><apply id="S3.E7X.3.2.2.m1.2.2.2.2.1.1.cmml" xref="S3.E7X.3.2.2.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.E7X.3.2.2.m1.2.2.2.2.1.1.1.cmml" xref="S3.E7X.3.2.2.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S3.E7X.3.2.2.m1.2.2.2.2.1.1.2.cmml" xref="S3.E7X.3.2.2.m1.2.2.2.2.1.1.2">ℝ</ci><ci id="S3.E7X.3.2.2.m1.2.2.2.2.1.1.3.cmml" xref="S3.E7X.3.2.2.m1.2.2.2.2.1.1.3">𝑑</ci></apply><apply id="S3.E7X.3.2.2.m1.3.3.3.3.2.2.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.3.2.2"><csymbol cd="ambiguous" id="S3.E7X.3.2.2.m1.3.3.3.3.2.2.1.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.3.2.2">subscript</csymbol><ci id="S3.E7X.3.2.2.m1.3.3.3.3.2.2.2.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.3.2.2.2">𝑤</ci><ci id="S3.E7X.3.2.2.m1.3.3.3.3.2.2.3.cmml" xref="S3.E7X.3.2.2.m1.3.3.3.3.2.2.3">𝛾</ci></apply><ci id="S3.E7X.3.2.2.m1.1.1.1.1.cmml" xref="S3.E7X.3.2.2.m1.1.1.1.1">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7X.3.2.2.m1.4c">\displaystyle s_{n}g\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.E7X.3.2.2.m1.4d">italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="5"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(3.7)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E7Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=2^{ks}\Big{\|}\sum_{j=k-1}^{k+1}\varphi_{k}\ast 2^{-j}g_{j}\Big{% \|}_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S3.E7Xa.2.1.1.m1.4"><semantics id="S3.E7Xa.2.1.1.m1.4a"><mrow id="S3.E7Xa.2.1.1.m1.4.4" xref="S3.E7Xa.2.1.1.m1.4.4.cmml"><mi id="S3.E7Xa.2.1.1.m1.4.4.3" xref="S3.E7Xa.2.1.1.m1.4.4.3.cmml"></mi><mo id="S3.E7Xa.2.1.1.m1.4.4.2" xref="S3.E7Xa.2.1.1.m1.4.4.2.cmml">=</mo><mrow id="S3.E7Xa.2.1.1.m1.4.4.1" xref="S3.E7Xa.2.1.1.m1.4.4.1.cmml"><msup id="S3.E7Xa.2.1.1.m1.4.4.1.3" xref="S3.E7Xa.2.1.1.m1.4.4.1.3.cmml"><mn id="S3.E7Xa.2.1.1.m1.4.4.1.3.2" xref="S3.E7Xa.2.1.1.m1.4.4.1.3.2.cmml">2</mn><mrow id="S3.E7Xa.2.1.1.m1.4.4.1.3.3" 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + 1 end_POSTSUPERSCRIPT italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ 2 start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E7Xb"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\sum_{j=-1}^{1}2^{(k+j)s-(k+j)}\|g_{k+j}\|_{L^{p}(\mathbb{R% }^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S3.E7Xb.2.1.1.m1.6"><semantics 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id="S3.E7Xc.2.1.1.m1.6.6.2.2.1.1.3.3.cmml" type="integer" xref="S3.E7Xc.2.1.1.m1.6.6.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.E7Xc.2.1.1.m1.5.5.1.1.cmml" xref="S3.E7Xc.2.1.1.m1.5.5.1.1">𝑋</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7Xc.2.1.1.m1.8c">\displaystyle=C\sum_{j=-1}^{1}2^{(k+j)s-(k+j)}\|\mathcal{F}^{-1}_{1}\eta_{k+j}% ^{m}\|_{L^{p}(\mathbb{R},w_{\gamma})}\|\phi_{k+j}\ast g\|_{L^{p}(\mathbb{R}^{d% -1};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.E7Xc.2.1.1.m1.8d">= italic_C ∑ start_POSTSUBSCRIPT italic_j = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT ( italic_k + italic_j ) italic_s - ( italic_k + italic_j ) end_POSTSUPERSCRIPT ∥ caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_η start_POSTSUBSCRIPT italic_k + italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ∥ italic_ϕ start_POSTSUBSCRIPT italic_k + italic_j end_POSTSUBSCRIPT ∗ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E7Xd"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\sum_{j=-1}^{1}2^{(k+j)(s-m-\frac{\gamma+1}{p})}\|\phi_{k+j% }\ast g\|_{L^{p}(\mathbb{R}^{d-1};X)}," class="ltx_Math" display="inline" id="S3.E7Xd.2.1.1.m1.5"><semantics id="S3.E7Xd.2.1.1.m1.5a"><mrow 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xref="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.2">ℝ</ci><apply id="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3.cmml" xref="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3"><minus id="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3.1.cmml" xref="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3.1"></minus><ci id="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3.2.cmml" xref="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3.2">𝑑</ci><cn id="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3.3.cmml" type="integer" xref="S3.E7Xd.2.1.1.m1.4.4.2.2.1.1.3.3">1</cn></apply></apply><ci id="S3.E7Xd.2.1.1.m1.3.3.1.1.cmml" xref="S3.E7Xd.2.1.1.m1.3.3.1.1">𝑋</ci></list></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E7Xd.2.1.1.m1.5c">\displaystyle\leq C\sum_{j=-1}^{1}2^{(k+j)(s-m-\frac{\gamma+1}{p})}\|\phi_{k+j% }\ast g\|_{L^{p}(\mathbb{R}^{d-1};X)},</annotation><annotation encoding="application/x-llamapun" id="S3.E7Xd.2.1.1.m1.5d">≤ italic_C ∑ start_POSTSUBSCRIPT italic_j = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT ( italic_k + italic_j ) ( italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_ϕ start_POSTSUBSCRIPT italic_k + italic_j end_POSTSUBSCRIPT ∗ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S3.SS2.2.p2.10">and if <math alttext="k\geq n+2" class="ltx_Math" display="inline" id="S3.SS2.2.p2.10.m1.1"><semantics id="S3.SS2.2.p2.10.m1.1a"><mrow id="S3.SS2.2.p2.10.m1.1.1" xref="S3.SS2.2.p2.10.m1.1.1.cmml"><mi id="S3.SS2.2.p2.10.m1.1.1.2" xref="S3.SS2.2.p2.10.m1.1.1.2.cmml">k</mi><mo id="S3.SS2.2.p2.10.m1.1.1.1" xref="S3.SS2.2.p2.10.m1.1.1.1.cmml">≥</mo><mrow id="S3.SS2.2.p2.10.m1.1.1.3" xref="S3.SS2.2.p2.10.m1.1.1.3.cmml"><mi id="S3.SS2.2.p2.10.m1.1.1.3.2" xref="S3.SS2.2.p2.10.m1.1.1.3.2.cmml">n</mi><mo id="S3.SS2.2.p2.10.m1.1.1.3.1" xref="S3.SS2.2.p2.10.m1.1.1.3.1.cmml">+</mo><mn id="S3.SS2.2.p2.10.m1.1.1.3.3" xref="S3.SS2.2.p2.10.m1.1.1.3.3.cmml">2</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.2.p2.10.m1.1b"><apply id="S3.SS2.2.p2.10.m1.1.1.cmml" xref="S3.SS2.2.p2.10.m1.1.1"><geq id="S3.SS2.2.p2.10.m1.1.1.1.cmml" xref="S3.SS2.2.p2.10.m1.1.1.1"></geq><ci id="S3.SS2.2.p2.10.m1.1.1.2.cmml" xref="S3.SS2.2.p2.10.m1.1.1.2">𝑘</ci><apply id="S3.SS2.2.p2.10.m1.1.1.3.cmml" xref="S3.SS2.2.p2.10.m1.1.1.3"><plus id="S3.SS2.2.p2.10.m1.1.1.3.1.cmml" xref="S3.SS2.2.p2.10.m1.1.1.3.1"></plus><ci id="S3.SS2.2.p2.10.m1.1.1.3.2.cmml" xref="S3.SS2.2.p2.10.m1.1.1.3.2">𝑛</ci><cn id="S3.SS2.2.p2.10.m1.1.1.3.3.cmml" type="integer" xref="S3.SS2.2.p2.10.m1.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.2.p2.10.m1.1c">k\geq n+2</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.2.p2.10.m1.1d">italic_k ≥ italic_n + 2</annotation></semantics></math> we find</p> <table class="ltx_equation ltx_eqn_table" id="S3.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="2^{ks}\|\varphi_{k}\ast s_{n}g\|_{L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}=0." class="ltx_Math" display="block" id="S3.E8.m1.4"><semantics id="S3.E8.m1.4a"><mrow id="S3.E8.m1.4.4.1" xref="S3.E8.m1.4.4.1.1.cmml"><mrow id="S3.E8.m1.4.4.1.1" xref="S3.E8.m1.4.4.1.1.cmml"><mrow id="S3.E8.m1.4.4.1.1.1" xref="S3.E8.m1.4.4.1.1.1.cmml"><msup id="S3.E8.m1.4.4.1.1.1.3" xref="S3.E8.m1.4.4.1.1.1.3.cmml"><mn id="S3.E8.m1.4.4.1.1.1.3.2" xref="S3.E8.m1.4.4.1.1.1.3.2.cmml">2</mn><mrow id="S3.E8.m1.4.4.1.1.1.3.3" xref="S3.E8.m1.4.4.1.1.1.3.3.cmml"><mi id="S3.E8.m1.4.4.1.1.1.3.3.2" xref="S3.E8.m1.4.4.1.1.1.3.3.2.cmml">k</mi><mo id="S3.E8.m1.4.4.1.1.1.3.3.1" xref="S3.E8.m1.4.4.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.E8.m1.4.4.1.1.1.3.3.3" xref="S3.E8.m1.4.4.1.1.1.3.3.3.cmml">s</mi></mrow></msup><mo id="S3.E8.m1.4.4.1.1.1.2" xref="S3.E8.m1.4.4.1.1.1.2.cmml">⁢</mo><msub id="S3.E8.m1.4.4.1.1.1.1" xref="S3.E8.m1.4.4.1.1.1.1.cmml"><mrow id="S3.E8.m1.4.4.1.1.1.1.1.1" xref="S3.E8.m1.4.4.1.1.1.1.1.2.cmml"><mo id="S3.E8.m1.4.4.1.1.1.1.1.1.2" stretchy="false" xref="S3.E8.m1.4.4.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.E8.m1.4.4.1.1.1.1.1.1.1" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.cmml"><mrow id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.cmml"><msub id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.cmml"><mi id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.2" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.2.cmml">φ</mi><mi id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.3" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.3.cmml">k</mi></msub><mo id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.1.cmml">∗</mo><msub id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.3" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.3.cmml"><mi id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.3.2" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.3.2.cmml">s</mi><mi id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.3.3" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.3.3.cmml">n</mi></msub></mrow><mo id="S3.E8.m1.4.4.1.1.1.1.1.1.1.1" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.E8.m1.4.4.1.1.1.1.1.1.1.3" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.E8.m1.4.4.1.1.1.1.1.1.3" stretchy="false" xref="S3.E8.m1.4.4.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.E8.m1.3.3.3" xref="S3.E8.m1.3.3.3.cmml"><msup id="S3.E8.m1.3.3.3.5" xref="S3.E8.m1.3.3.3.5.cmml"><mi id="S3.E8.m1.3.3.3.5.2" xref="S3.E8.m1.3.3.3.5.2.cmml">L</mi><mi id="S3.E8.m1.3.3.3.5.3" xref="S3.E8.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S3.E8.m1.3.3.3.4" xref="S3.E8.m1.3.3.3.4.cmml">⁢</mo><mrow id="S3.E8.m1.3.3.3.3.2" xref="S3.E8.m1.3.3.3.3.3.cmml"><mo id="S3.E8.m1.3.3.3.3.2.3" stretchy="false" xref="S3.E8.m1.3.3.3.3.3.cmml">(</mo><msubsup id="S3.E8.m1.2.2.2.2.1.1" xref="S3.E8.m1.2.2.2.2.1.1.cmml"><mi id="S3.E8.m1.2.2.2.2.1.1.2.2" xref="S3.E8.m1.2.2.2.2.1.1.2.2.cmml">ℝ</mi><mo id="S3.E8.m1.2.2.2.2.1.1.3" xref="S3.E8.m1.2.2.2.2.1.1.3.cmml">+</mo><mi id="S3.E8.m1.2.2.2.2.1.1.2.3" xref="S3.E8.m1.2.2.2.2.1.1.2.3.cmml">d</mi></msubsup><mo id="S3.E8.m1.3.3.3.3.2.4" xref="S3.E8.m1.3.3.3.3.3.cmml">,</mo><msub id="S3.E8.m1.3.3.3.3.2.2" xref="S3.E8.m1.3.3.3.3.2.2.cmml"><mi id="S3.E8.m1.3.3.3.3.2.2.2" xref="S3.E8.m1.3.3.3.3.2.2.2.cmml">w</mi><mi id="S3.E8.m1.3.3.3.3.2.2.3" xref="S3.E8.m1.3.3.3.3.2.2.3.cmml">γ</mi></msub><mo id="S3.E8.m1.3.3.3.3.2.5" xref="S3.E8.m1.3.3.3.3.3.cmml">;</mo><mi id="S3.E8.m1.1.1.1.1" xref="S3.E8.m1.1.1.1.1.cmml">X</mi><mo id="S3.E8.m1.3.3.3.3.2.6" stretchy="false" xref="S3.E8.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow></msub></mrow><mo id="S3.E8.m1.4.4.1.1.2" xref="S3.E8.m1.4.4.1.1.2.cmml">=</mo><mn id="S3.E8.m1.4.4.1.1.3" xref="S3.E8.m1.4.4.1.1.3.cmml">0</mn></mrow><mo id="S3.E8.m1.4.4.1.2" lspace="0em" xref="S3.E8.m1.4.4.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S3.E8.m1.4b"><apply id="S3.E8.m1.4.4.1.1.cmml" xref="S3.E8.m1.4.4.1"><eq id="S3.E8.m1.4.4.1.1.2.cmml" xref="S3.E8.m1.4.4.1.1.2"></eq><apply id="S3.E8.m1.4.4.1.1.1.cmml" xref="S3.E8.m1.4.4.1.1.1"><times id="S3.E8.m1.4.4.1.1.1.2.cmml" xref="S3.E8.m1.4.4.1.1.1.2"></times><apply id="S3.E8.m1.4.4.1.1.1.3.cmml" xref="S3.E8.m1.4.4.1.1.1.3"><csymbol cd="ambiguous" id="S3.E8.m1.4.4.1.1.1.3.1.cmml" xref="S3.E8.m1.4.4.1.1.1.3">superscript</csymbol><cn id="S3.E8.m1.4.4.1.1.1.3.2.cmml" type="integer" xref="S3.E8.m1.4.4.1.1.1.3.2">2</cn><apply id="S3.E8.m1.4.4.1.1.1.3.3.cmml" xref="S3.E8.m1.4.4.1.1.1.3.3"><times id="S3.E8.m1.4.4.1.1.1.3.3.1.cmml" xref="S3.E8.m1.4.4.1.1.1.3.3.1"></times><ci id="S3.E8.m1.4.4.1.1.1.3.3.2.cmml" xref="S3.E8.m1.4.4.1.1.1.3.3.2">𝑘</ci><ci id="S3.E8.m1.4.4.1.1.1.3.3.3.cmml" xref="S3.E8.m1.4.4.1.1.1.3.3.3">𝑠</ci></apply></apply><apply id="S3.E8.m1.4.4.1.1.1.1.cmml" xref="S3.E8.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.E8.m1.4.4.1.1.1.1.2.cmml" xref="S3.E8.m1.4.4.1.1.1.1">subscript</csymbol><apply id="S3.E8.m1.4.4.1.1.1.1.1.2.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1"><csymbol cd="latexml" id="S3.E8.m1.4.4.1.1.1.1.1.2.1.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1.2">norm</csymbol><apply id="S3.E8.m1.4.4.1.1.1.1.1.1.1.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1"><times id="S3.E8.m1.4.4.1.1.1.1.1.1.1.1.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.1"></times><apply id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2"><ci id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.1.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.1">∗</ci><apply id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S3.E8.m1.4.4.1.1.1.1.1.1.1.2.2.2.cmml" 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xref="S3.E8.m1.3.3.3.5.2">𝐿</ci><ci id="S3.E8.m1.3.3.3.5.3.cmml" xref="S3.E8.m1.3.3.3.5.3">𝑝</ci></apply><vector id="S3.E8.m1.3.3.3.3.3.cmml" xref="S3.E8.m1.3.3.3.3.2"><apply id="S3.E8.m1.2.2.2.2.1.1.cmml" xref="S3.E8.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.2.2.1.1.1.cmml" xref="S3.E8.m1.2.2.2.2.1.1">subscript</csymbol><apply id="S3.E8.m1.2.2.2.2.1.1.2.cmml" xref="S3.E8.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S3.E8.m1.2.2.2.2.1.1.2.1.cmml" xref="S3.E8.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S3.E8.m1.2.2.2.2.1.1.2.2.cmml" xref="S3.E8.m1.2.2.2.2.1.1.2.2">ℝ</ci><ci id="S3.E8.m1.2.2.2.2.1.1.2.3.cmml" xref="S3.E8.m1.2.2.2.2.1.1.2.3">𝑑</ci></apply><plus id="S3.E8.m1.2.2.2.2.1.1.3.cmml" xref="S3.E8.m1.2.2.2.2.1.1.3"></plus></apply><apply id="S3.E8.m1.3.3.3.3.2.2.cmml" xref="S3.E8.m1.3.3.3.3.2.2"><csymbol cd="ambiguous" id="S3.E8.m1.3.3.3.3.2.2.1.cmml" xref="S3.E8.m1.3.3.3.3.2.2">subscript</csymbol><ci id="S3.E8.m1.3.3.3.3.2.2.2.cmml" xref="S3.E8.m1.3.3.3.3.2.2.2">𝑤</ci><ci id="S3.E8.m1.3.3.3.3.2.2.3.cmml" xref="S3.E8.m1.3.3.3.3.2.2.3">𝛾</ci></apply><ci id="S3.E8.m1.1.1.1.1.cmml" xref="S3.E8.m1.1.1.1.1">𝑋</ci></vector></apply></apply></apply><cn id="S3.E8.m1.4.4.1.1.3.cmml" type="integer" xref="S3.E8.m1.4.4.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E8.m1.4c">2^{ks}\|\varphi_{k}\ast s_{n}g\|_{L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}=0.</annotation><annotation encoding="application/x-llamapun" id="S3.E8.m1.4d">2 start_POSTSUPERSCRIPT italic_k italic_s end_POSTSUPERSCRIPT ∥ italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.8)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S3.SS2.3.p3"> <p class="ltx_p" id="S3.SS2.3.p3.1"><span class="ltx_text ltx_font_italic" id="S3.SS2.3.p3.1.1">Step 2: existence.</span> For <math alttext="g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS2.3.p3.1.m1.4"><semantics id="S3.SS2.3.p3.1.m1.4a"><mrow id="S3.SS2.3.p3.1.m1.4.4" xref="S3.SS2.3.p3.1.m1.4.4.cmml"><mi id="S3.SS2.3.p3.1.m1.4.4.3" xref="S3.SS2.3.p3.1.m1.4.4.3.cmml">g</mi><mo id="S3.SS2.3.p3.1.m1.4.4.2" xref="S3.SS2.3.p3.1.m1.4.4.2.cmml">∈</mo><mrow id="S3.SS2.3.p3.1.m1.4.4.1" xref="S3.SS2.3.p3.1.m1.4.4.1.cmml"><msubsup id="S3.SS2.3.p3.1.m1.4.4.1.3" xref="S3.SS2.3.p3.1.m1.4.4.1.3.cmml"><mi id="S3.SS2.3.p3.1.m1.4.4.1.3.2.2" xref="S3.SS2.3.p3.1.m1.4.4.1.3.2.2.cmml">B</mi><mrow id="S3.SS2.3.p3.1.m1.2.2.2.4" xref="S3.SS2.3.p3.1.m1.2.2.2.3.cmml"><mi id="S3.SS2.3.p3.1.m1.1.1.1.1" xref="S3.SS2.3.p3.1.m1.1.1.1.1.cmml">p</mi><mo id="S3.SS2.3.p3.1.m1.2.2.2.4.1" xref="S3.SS2.3.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS2.3.p3.1.m1.2.2.2.2" xref="S3.SS2.3.p3.1.m1.2.2.2.2.cmml">q</mi></mrow><mrow id="S3.SS2.3.p3.1.m1.4.4.1.3.2.3" xref="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.cmml"><mi id="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.2" xref="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.2.cmml">s</mi><mo id="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.1" xref="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.1.cmml">−</mo><mi id="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.3" xref="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.3.cmml">m</mi><mo id="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.1a" xref="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.4" xref="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.4.cmml"><mrow id="S3.SS2.3.p3.1.m1.4.4.1.3.2.3.4.2" 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end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> we prove that</p> <ol class="ltx_enumerate" id="S3.I2"> <li class="ltx_item" id="S3.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S3.I2.i1.p1"> <p class="ltx_p" id="S3.I2.i1.p1.3"><math alttext="\operatorname{ext}_{m}g" class="ltx_Math" display="inline" id="S3.I2.i1.p1.1.m1.1"><semantics id="S3.I2.i1.p1.1.m1.1a"><mrow id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml"><msub id="S3.I2.i1.p1.1.m1.1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.1.cmml"><mi id="S3.I2.i1.p1.1.m1.1.1.1.2" xref="S3.I2.i1.p1.1.m1.1.1.1.2.cmml">ext</mi><mi id="S3.I2.i1.p1.1.m1.1.1.1.3" xref="S3.I2.i1.p1.1.m1.1.1.1.3.cmml">m</mi></msub><mo id="S3.I2.i1.p1.1.m1.1.1a" lspace="0.167em" xref="S3.I2.i1.p1.1.m1.1.1.cmml">⁡</mo><mi id="S3.I2.i1.p1.1.m1.1.1.2" xref="S3.I2.i1.p1.1.m1.1.1.2.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.1b"><apply id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1"><apply id="S3.I2.i1.p1.1.m1.1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.1.1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1.1">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.1.1.1.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.1.2">ext</ci><ci id="S3.I2.i1.p1.1.m1.1.1.1.3.cmml" xref="S3.I2.i1.p1.1.m1.1.1.1.3">𝑚</ci></apply><ci id="S3.I2.i1.p1.1.m1.1.1.2.cmml" xref="S3.I2.i1.p1.1.m1.1.1.2">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.1c">\operatorname{ext}_{m}g</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g</annotation></semantics></math> exists in <math alttext="B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.I2.i1.p1.2.m2.5"><semantics id="S3.I2.i1.p1.2.m2.5a"><mrow id="S3.I2.i1.p1.2.m2.5.5" xref="S3.I2.i1.p1.2.m2.5.5.cmml"><msubsup id="S3.I2.i1.p1.2.m2.5.5.4" xref="S3.I2.i1.p1.2.m2.5.5.4.cmml"><mi id="S3.I2.i1.p1.2.m2.5.5.4.2.2" xref="S3.I2.i1.p1.2.m2.5.5.4.2.2.cmml">B</mi><mrow id="S3.I2.i1.p1.2.m2.2.2.2.4" xref="S3.I2.i1.p1.2.m2.2.2.2.3.cmml"><mi id="S3.I2.i1.p1.2.m2.1.1.1.1" xref="S3.I2.i1.p1.2.m2.1.1.1.1.cmml">p</mi><mo id="S3.I2.i1.p1.2.m2.2.2.2.4.1" xref="S3.I2.i1.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S3.I2.i1.p1.2.m2.2.2.2.2" xref="S3.I2.i1.p1.2.m2.2.2.2.2.cmml">q</mi></mrow><mi id="S3.I2.i1.p1.2.m2.5.5.4.2.3" xref="S3.I2.i1.p1.2.m2.5.5.4.2.3.cmml">s</mi></msubsup><mo id="S3.I2.i1.p1.2.m2.5.5.3" xref="S3.I2.i1.p1.2.m2.5.5.3.cmml">⁢</mo><mrow id="S3.I2.i1.p1.2.m2.5.5.2.2" xref="S3.I2.i1.p1.2.m2.5.5.2.3.cmml"><mo id="S3.I2.i1.p1.2.m2.5.5.2.2.3" stretchy="false" xref="S3.I2.i1.p1.2.m2.5.5.2.3.cmml">(</mo><msup id="S3.I2.i1.p1.2.m2.4.4.1.1.1" xref="S3.I2.i1.p1.2.m2.4.4.1.1.1.cmml"><mi id="S3.I2.i1.p1.2.m2.4.4.1.1.1.2" xref="S3.I2.i1.p1.2.m2.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S3.I2.i1.p1.2.m2.4.4.1.1.1.3" xref="S3.I2.i1.p1.2.m2.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S3.I2.i1.p1.2.m2.5.5.2.2.4" xref="S3.I2.i1.p1.2.m2.5.5.2.3.cmml">,</mo><msub id="S3.I2.i1.p1.2.m2.5.5.2.2.2" xref="S3.I2.i1.p1.2.m2.5.5.2.2.2.cmml"><mi id="S3.I2.i1.p1.2.m2.5.5.2.2.2.2" xref="S3.I2.i1.p1.2.m2.5.5.2.2.2.2.cmml">w</mi><mi id="S3.I2.i1.p1.2.m2.5.5.2.2.2.3" xref="S3.I2.i1.p1.2.m2.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S3.I2.i1.p1.2.m2.5.5.2.2.5" xref="S3.I2.i1.p1.2.m2.5.5.2.3.cmml">;</mo><mi id="S3.I2.i1.p1.2.m2.3.3" xref="S3.I2.i1.p1.2.m2.3.3.cmml">X</mi><mo id="S3.I2.i1.p1.2.m2.5.5.2.2.6" stretchy="false" xref="S3.I2.i1.p1.2.m2.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.2.m2.5b"><apply id="S3.I2.i1.p1.2.m2.5.5.cmml" xref="S3.I2.i1.p1.2.m2.5.5"><times id="S3.I2.i1.p1.2.m2.5.5.3.cmml" xref="S3.I2.i1.p1.2.m2.5.5.3"></times><apply id="S3.I2.i1.p1.2.m2.5.5.4.cmml" xref="S3.I2.i1.p1.2.m2.5.5.4"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.5.5.4.1.cmml" xref="S3.I2.i1.p1.2.m2.5.5.4">subscript</csymbol><apply id="S3.I2.i1.p1.2.m2.5.5.4.2.cmml" xref="S3.I2.i1.p1.2.m2.5.5.4"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.5.5.4.2.1.cmml" xref="S3.I2.i1.p1.2.m2.5.5.4">superscript</csymbol><ci id="S3.I2.i1.p1.2.m2.5.5.4.2.2.cmml" xref="S3.I2.i1.p1.2.m2.5.5.4.2.2">𝐵</ci><ci id="S3.I2.i1.p1.2.m2.5.5.4.2.3.cmml" xref="S3.I2.i1.p1.2.m2.5.5.4.2.3">𝑠</ci></apply><list id="S3.I2.i1.p1.2.m2.2.2.2.3.cmml" xref="S3.I2.i1.p1.2.m2.2.2.2.4"><ci id="S3.I2.i1.p1.2.m2.1.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.1.1.1.1">𝑝</ci><ci id="S3.I2.i1.p1.2.m2.2.2.2.2.cmml" xref="S3.I2.i1.p1.2.m2.2.2.2.2">𝑞</ci></list></apply><vector id="S3.I2.i1.p1.2.m2.5.5.2.3.cmml" xref="S3.I2.i1.p1.2.m2.5.5.2.2"><apply id="S3.I2.i1.p1.2.m2.4.4.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.4.4.1.1.1.1.cmml" xref="S3.I2.i1.p1.2.m2.4.4.1.1.1">superscript</csymbol><ci id="S3.I2.i1.p1.2.m2.4.4.1.1.1.2.cmml" xref="S3.I2.i1.p1.2.m2.4.4.1.1.1.2">ℝ</ci><ci id="S3.I2.i1.p1.2.m2.4.4.1.1.1.3.cmml" xref="S3.I2.i1.p1.2.m2.4.4.1.1.1.3">𝑑</ci></apply><apply id="S3.I2.i1.p1.2.m2.5.5.2.2.2.cmml" xref="S3.I2.i1.p1.2.m2.5.5.2.2.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.2.m2.5.5.2.2.2.1.cmml" xref="S3.I2.i1.p1.2.m2.5.5.2.2.2">subscript</csymbol><ci id="S3.I2.i1.p1.2.m2.5.5.2.2.2.2.cmml" xref="S3.I2.i1.p1.2.m2.5.5.2.2.2.2">𝑤</ci><ci id="S3.I2.i1.p1.2.m2.5.5.2.2.2.3.cmml" xref="S3.I2.i1.p1.2.m2.5.5.2.2.2.3">𝛾</ci></apply><ci id="S3.I2.i1.p1.2.m2.3.3.cmml" xref="S3.I2.i1.p1.2.m2.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.2.m2.5c">B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.2.m2.5d">italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> if <math alttext="q\in[1,\infty)" class="ltx_Math" display="inline" id="S3.I2.i1.p1.3.m3.2"><semantics id="S3.I2.i1.p1.3.m3.2a"><mrow id="S3.I2.i1.p1.3.m3.2.3" xref="S3.I2.i1.p1.3.m3.2.3.cmml"><mi id="S3.I2.i1.p1.3.m3.2.3.2" xref="S3.I2.i1.p1.3.m3.2.3.2.cmml">q</mi><mo id="S3.I2.i1.p1.3.m3.2.3.1" xref="S3.I2.i1.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S3.I2.i1.p1.3.m3.2.3.3.2" xref="S3.I2.i1.p1.3.m3.2.3.3.1.cmml"><mo id="S3.I2.i1.p1.3.m3.2.3.3.2.1" stretchy="false" xref="S3.I2.i1.p1.3.m3.2.3.3.1.cmml">[</mo><mn id="S3.I2.i1.p1.3.m3.1.1" xref="S3.I2.i1.p1.3.m3.1.1.cmml">1</mn><mo id="S3.I2.i1.p1.3.m3.2.3.3.2.2" xref="S3.I2.i1.p1.3.m3.2.3.3.1.cmml">,</mo><mi id="S3.I2.i1.p1.3.m3.2.2" mathvariant="normal" xref="S3.I2.i1.p1.3.m3.2.2.cmml">∞</mi><mo id="S3.I2.i1.p1.3.m3.2.3.3.2.3" stretchy="false" xref="S3.I2.i1.p1.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.3.m3.2b"><apply id="S3.I2.i1.p1.3.m3.2.3.cmml" xref="S3.I2.i1.p1.3.m3.2.3"><in id="S3.I2.i1.p1.3.m3.2.3.1.cmml" xref="S3.I2.i1.p1.3.m3.2.3.1"></in><ci id="S3.I2.i1.p1.3.m3.2.3.2.cmml" xref="S3.I2.i1.p1.3.m3.2.3.2">𝑞</ci><interval closure="closed-open" id="S3.I2.i1.p1.3.m3.2.3.3.1.cmml" xref="S3.I2.i1.p1.3.m3.2.3.3.2"><cn id="S3.I2.i1.p1.3.m3.1.1.cmml" type="integer" xref="S3.I2.i1.p1.3.m3.1.1">1</cn><infinity id="S3.I2.i1.p1.3.m3.2.2.cmml" xref="S3.I2.i1.p1.3.m3.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.3.m3.2c">q\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i1.p1.3.m3.2d">italic_q ∈ [ 1 , ∞ )</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S3.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S3.I2.i2.p1"> <p class="ltx_p" id="S3.I2.i2.p1.4"><math alttext="\operatorname{ext}_{m}g" class="ltx_Math" display="inline" id="S3.I2.i2.p1.1.m1.1"><semantics id="S3.I2.i2.p1.1.m1.1a"><mrow id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml"><msub id="S3.I2.i2.p1.1.m1.1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.1.cmml"><mi id="S3.I2.i2.p1.1.m1.1.1.1.2" xref="S3.I2.i2.p1.1.m1.1.1.1.2.cmml">ext</mi><mi id="S3.I2.i2.p1.1.m1.1.1.1.3" xref="S3.I2.i2.p1.1.m1.1.1.1.3.cmml">m</mi></msub><mo id="S3.I2.i2.p1.1.m1.1.1a" lspace="0.167em" xref="S3.I2.i2.p1.1.m1.1.1.cmml">⁡</mo><mi id="S3.I2.i2.p1.1.m1.1.1.2" xref="S3.I2.i2.p1.1.m1.1.1.2.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.1b"><apply id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1"><apply id="S3.I2.i2.p1.1.m1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1.2">ext</ci><ci id="S3.I2.i2.p1.1.m1.1.1.1.3.cmml" xref="S3.I2.i2.p1.1.m1.1.1.1.3">𝑚</ci></apply><ci id="S3.I2.i2.p1.1.m1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.1.1.2">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.1.m1.1c">\operatorname{ext}_{m}g</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g</annotation></semantics></math> exists in <math alttext="B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.I2.i2.p1.2.m2.5"><semantics id="S3.I2.i2.p1.2.m2.5a"><mrow id="S3.I2.i2.p1.2.m2.5.5" xref="S3.I2.i2.p1.2.m2.5.5.cmml"><msubsup id="S3.I2.i2.p1.2.m2.5.5.4" xref="S3.I2.i2.p1.2.m2.5.5.4.cmml"><mi id="S3.I2.i2.p1.2.m2.5.5.4.2.2" xref="S3.I2.i2.p1.2.m2.5.5.4.2.2.cmml">B</mi><mrow id="S3.I2.i2.p1.2.m2.2.2.2.4" 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xref="S3.I2.i2.p1.2.m2.5.5.2.2.2.cmml"><mi id="S3.I2.i2.p1.2.m2.5.5.2.2.2.2" xref="S3.I2.i2.p1.2.m2.5.5.2.2.2.2.cmml">w</mi><mi id="S3.I2.i2.p1.2.m2.5.5.2.2.2.3" xref="S3.I2.i2.p1.2.m2.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S3.I2.i2.p1.2.m2.5.5.2.2.5" xref="S3.I2.i2.p1.2.m2.5.5.2.3.cmml">;</mo><mi id="S3.I2.i2.p1.2.m2.3.3" xref="S3.I2.i2.p1.2.m2.3.3.cmml">X</mi><mo id="S3.I2.i2.p1.2.m2.5.5.2.2.6" stretchy="false" xref="S3.I2.i2.p1.2.m2.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.2.m2.5b"><apply id="S3.I2.i2.p1.2.m2.5.5.cmml" xref="S3.I2.i2.p1.2.m2.5.5"><times id="S3.I2.i2.p1.2.m2.5.5.3.cmml" xref="S3.I2.i2.p1.2.m2.5.5.3"></times><apply id="S3.I2.i2.p1.2.m2.5.5.4.cmml" xref="S3.I2.i2.p1.2.m2.5.5.4"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.5.5.4.1.cmml" xref="S3.I2.i2.p1.2.m2.5.5.4">subscript</csymbol><apply id="S3.I2.i2.p1.2.m2.5.5.4.2.cmml" xref="S3.I2.i2.p1.2.m2.5.5.4"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.5.5.4.2.1.cmml" 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xref="S3.I2.i2.p1.2.m2.5.5.2.2.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.2.m2.5.5.2.2.2.1.cmml" xref="S3.I2.i2.p1.2.m2.5.5.2.2.2">subscript</csymbol><ci id="S3.I2.i2.p1.2.m2.5.5.2.2.2.2.cmml" xref="S3.I2.i2.p1.2.m2.5.5.2.2.2.2">𝑤</ci><ci id="S3.I2.i2.p1.2.m2.5.5.2.2.2.3.cmml" xref="S3.I2.i2.p1.2.m2.5.5.2.2.2.3">𝛾</ci></apply><ci id="S3.I2.i2.p1.2.m2.3.3.cmml" xref="S3.I2.i2.p1.2.m2.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.2.m2.5c">B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.2.m2.5d">italic_B start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for any <math alttext="t<s" class="ltx_Math" display="inline" id="S3.I2.i2.p1.3.m3.1"><semantics id="S3.I2.i2.p1.3.m3.1a"><mrow id="S3.I2.i2.p1.3.m3.1.1" xref="S3.I2.i2.p1.3.m3.1.1.cmml"><mi id="S3.I2.i2.p1.3.m3.1.1.2" xref="S3.I2.i2.p1.3.m3.1.1.2.cmml">t</mi><mo id="S3.I2.i2.p1.3.m3.1.1.1" xref="S3.I2.i2.p1.3.m3.1.1.1.cmml"><</mo><mi id="S3.I2.i2.p1.3.m3.1.1.3" xref="S3.I2.i2.p1.3.m3.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.3.m3.1b"><apply id="S3.I2.i2.p1.3.m3.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1"><lt id="S3.I2.i2.p1.3.m3.1.1.1.cmml" xref="S3.I2.i2.p1.3.m3.1.1.1"></lt><ci id="S3.I2.i2.p1.3.m3.1.1.2.cmml" xref="S3.I2.i2.p1.3.m3.1.1.2">𝑡</ci><ci id="S3.I2.i2.p1.3.m3.1.1.3.cmml" xref="S3.I2.i2.p1.3.m3.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.3.m3.1c">t<s</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.3.m3.1d">italic_t < italic_s</annotation></semantics></math> if <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.I2.i2.p1.4.m4.2"><semantics id="S3.I2.i2.p1.4.m4.2a"><mrow id="S3.I2.i2.p1.4.m4.2.3" xref="S3.I2.i2.p1.4.m4.2.3.cmml"><mi id="S3.I2.i2.p1.4.m4.2.3.2" xref="S3.I2.i2.p1.4.m4.2.3.2.cmml">q</mi><mo id="S3.I2.i2.p1.4.m4.2.3.1" xref="S3.I2.i2.p1.4.m4.2.3.1.cmml">∈</mo><mrow id="S3.I2.i2.p1.4.m4.2.3.3.2" xref="S3.I2.i2.p1.4.m4.2.3.3.1.cmml"><mo id="S3.I2.i2.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S3.I2.i2.p1.4.m4.2.3.3.1.cmml">[</mo><mn id="S3.I2.i2.p1.4.m4.1.1" xref="S3.I2.i2.p1.4.m4.1.1.cmml">1</mn><mo id="S3.I2.i2.p1.4.m4.2.3.3.2.2" xref="S3.I2.i2.p1.4.m4.2.3.3.1.cmml">,</mo><mi id="S3.I2.i2.p1.4.m4.2.2" mathvariant="normal" xref="S3.I2.i2.p1.4.m4.2.2.cmml">∞</mi><mo id="S3.I2.i2.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S3.I2.i2.p1.4.m4.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.4.m4.2b"><apply id="S3.I2.i2.p1.4.m4.2.3.cmml" xref="S3.I2.i2.p1.4.m4.2.3"><in id="S3.I2.i2.p1.4.m4.2.3.1.cmml" xref="S3.I2.i2.p1.4.m4.2.3.1"></in><ci id="S3.I2.i2.p1.4.m4.2.3.2.cmml" xref="S3.I2.i2.p1.4.m4.2.3.2">𝑞</ci><interval closure="closed" id="S3.I2.i2.p1.4.m4.2.3.3.1.cmml" xref="S3.I2.i2.p1.4.m4.2.3.3.2"><cn id="S3.I2.i2.p1.4.m4.1.1.cmml" type="integer" xref="S3.I2.i2.p1.4.m4.1.1">1</cn><infinity id="S3.I2.i2.p1.4.m4.2.2.cmml" xref="S3.I2.i2.p1.4.m4.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i2.p1.4.m4.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.I2.i2.p1.4.m4.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S3.SS2.3.p3.3">If <math alttext="q\in[1,\infty)" class="ltx_Math" display="inline" id="S3.SS2.3.p3.2.m1.2"><semantics id="S3.SS2.3.p3.2.m1.2a"><mrow id="S3.SS2.3.p3.2.m1.2.3" xref="S3.SS2.3.p3.2.m1.2.3.cmml"><mi id="S3.SS2.3.p3.2.m1.2.3.2" xref="S3.SS2.3.p3.2.m1.2.3.2.cmml">q</mi><mo id="S3.SS2.3.p3.2.m1.2.3.1" xref="S3.SS2.3.p3.2.m1.2.3.1.cmml">∈</mo><mrow id="S3.SS2.3.p3.2.m1.2.3.3.2" xref="S3.SS2.3.p3.2.m1.2.3.3.1.cmml"><mo id="S3.SS2.3.p3.2.m1.2.3.3.2.1" stretchy="false" xref="S3.SS2.3.p3.2.m1.2.3.3.1.cmml">[</mo><mn id="S3.SS2.3.p3.2.m1.1.1" xref="S3.SS2.3.p3.2.m1.1.1.cmml">1</mn><mo id="S3.SS2.3.p3.2.m1.2.3.3.2.2" xref="S3.SS2.3.p3.2.m1.2.3.3.1.cmml">,</mo><mi id="S3.SS2.3.p3.2.m1.2.2" mathvariant="normal" xref="S3.SS2.3.p3.2.m1.2.2.cmml">∞</mi><mo id="S3.SS2.3.p3.2.m1.2.3.3.2.3" stretchy="false" xref="S3.SS2.3.p3.2.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.2.m1.2b"><apply id="S3.SS2.3.p3.2.m1.2.3.cmml" xref="S3.SS2.3.p3.2.m1.2.3"><in id="S3.SS2.3.p3.2.m1.2.3.1.cmml" xref="S3.SS2.3.p3.2.m1.2.3.1"></in><ci id="S3.SS2.3.p3.2.m1.2.3.2.cmml" xref="S3.SS2.3.p3.2.m1.2.3.2">𝑞</ci><interval closure="closed-open" id="S3.SS2.3.p3.2.m1.2.3.3.1.cmml" xref="S3.SS2.3.p3.2.m1.2.3.3.2"><cn id="S3.SS2.3.p3.2.m1.1.1.cmml" type="integer" xref="S3.SS2.3.p3.2.m1.1.1">1</cn><infinity id="S3.SS2.3.p3.2.m1.2.2.cmml" xref="S3.SS2.3.p3.2.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.2.m1.2c">q\in[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.2.m1.2d">italic_q ∈ [ 1 , ∞ )</annotation></semantics></math> and <math alttext="0<\ell<n" class="ltx_Math" display="inline" id="S3.SS2.3.p3.3.m2.1"><semantics id="S3.SS2.3.p3.3.m2.1a"><mrow id="S3.SS2.3.p3.3.m2.1.1" xref="S3.SS2.3.p3.3.m2.1.1.cmml"><mn id="S3.SS2.3.p3.3.m2.1.1.2" xref="S3.SS2.3.p3.3.m2.1.1.2.cmml">0</mn><mo id="S3.SS2.3.p3.3.m2.1.1.3" xref="S3.SS2.3.p3.3.m2.1.1.3.cmml"><</mo><mi id="S3.SS2.3.p3.3.m2.1.1.4" mathvariant="normal" xref="S3.SS2.3.p3.3.m2.1.1.4.cmml">ℓ</mi><mo id="S3.SS2.3.p3.3.m2.1.1.5" xref="S3.SS2.3.p3.3.m2.1.1.5.cmml"><</mo><mi id="S3.SS2.3.p3.3.m2.1.1.6" xref="S3.SS2.3.p3.3.m2.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.3.m2.1b"><apply id="S3.SS2.3.p3.3.m2.1.1.cmml" xref="S3.SS2.3.p3.3.m2.1.1"><and id="S3.SS2.3.p3.3.m2.1.1a.cmml" xref="S3.SS2.3.p3.3.m2.1.1"></and><apply id="S3.SS2.3.p3.3.m2.1.1b.cmml" xref="S3.SS2.3.p3.3.m2.1.1"><lt id="S3.SS2.3.p3.3.m2.1.1.3.cmml" xref="S3.SS2.3.p3.3.m2.1.1.3"></lt><cn id="S3.SS2.3.p3.3.m2.1.1.2.cmml" type="integer" xref="S3.SS2.3.p3.3.m2.1.1.2">0</cn><ci id="S3.SS2.3.p3.3.m2.1.1.4.cmml" xref="S3.SS2.3.p3.3.m2.1.1.4">ℓ</ci></apply><apply id="S3.SS2.3.p3.3.m2.1.1c.cmml" xref="S3.SS2.3.p3.3.m2.1.1"><lt id="S3.SS2.3.p3.3.m2.1.1.5.cmml" xref="S3.SS2.3.p3.3.m2.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S3.SS2.3.p3.3.m2.1.1.4.cmml" id="S3.SS2.3.p3.3.m2.1.1d.cmml" xref="S3.SS2.3.p3.3.m2.1.1"></share><ci id="S3.SS2.3.p3.3.m2.1.1.6.cmml" xref="S3.SS2.3.p3.3.m2.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.3.m2.1c">0<\ell<n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.3.m2.1d">0 < roman_ℓ < italic_n</annotation></semantics></math>, then by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E5" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.5</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E7" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.7</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E8" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.8</span></a>), we have</p> <table class="ltx_equationgroup ltx_eqn_table" id="S3.E9"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E9X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|s_{n}g-s_{\ell}g\|_{B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S3.E9X.2.1.1.m1.6"><semantics id="S3.E9X.2.1.1.m1.6a"><msub id="S3.E9X.2.1.1.m1.6.6" xref="S3.E9X.2.1.1.m1.6.6.cmml"><mrow id="S3.E9X.2.1.1.m1.6.6.1.1" xref="S3.E9X.2.1.1.m1.6.6.1.2.cmml"><mo id="S3.E9X.2.1.1.m1.6.6.1.1.2" stretchy="false" xref="S3.E9X.2.1.1.m1.6.6.1.2.1.cmml">‖</mo><mrow id="S3.E9X.2.1.1.m1.6.6.1.1.1" xref="S3.E9X.2.1.1.m1.6.6.1.1.1.cmml"><mrow id="S3.E9X.2.1.1.m1.6.6.1.1.1.2" 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xref="S3.E9X.2.1.1.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.E9X.2.1.1.m1.6b"><apply id="S3.E9X.2.1.1.m1.6.6.cmml" xref="S3.E9X.2.1.1.m1.6.6"><csymbol cd="ambiguous" id="S3.E9X.2.1.1.m1.6.6.2.cmml" xref="S3.E9X.2.1.1.m1.6.6">subscript</csymbol><apply id="S3.E9X.2.1.1.m1.6.6.1.2.cmml" xref="S3.E9X.2.1.1.m1.6.6.1.1"><csymbol cd="latexml" id="S3.E9X.2.1.1.m1.6.6.1.2.1.cmml" xref="S3.E9X.2.1.1.m1.6.6.1.1.2">norm</csymbol><apply id="S3.E9X.2.1.1.m1.6.6.1.1.1.cmml" xref="S3.E9X.2.1.1.m1.6.6.1.1.1"><minus id="S3.E9X.2.1.1.m1.6.6.1.1.1.1.cmml" xref="S3.E9X.2.1.1.m1.6.6.1.1.1.1"></minus><apply id="S3.E9X.2.1.1.m1.6.6.1.1.1.2.cmml" xref="S3.E9X.2.1.1.m1.6.6.1.1.1.2"><times id="S3.E9X.2.1.1.m1.6.6.1.1.1.2.1.cmml" xref="S3.E9X.2.1.1.m1.6.6.1.1.1.2.1"></times><apply id="S3.E9X.2.1.1.m1.6.6.1.1.1.2.2.cmml" xref="S3.E9X.2.1.1.m1.6.6.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.E9X.2.1.1.m1.6.6.1.1.1.2.2.1.cmml" 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xref="S3.E9X.2.1.1.m1.3.3.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9X.2.1.1.m1.6c">\displaystyle\|s_{n}g-s_{\ell}g\|_{B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S3.E9X.2.1.1.m1.6d">∥ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g - italic_s start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\Big{(}\sum_{k=\ell}^{n+1}\big{(}2^{ks}\|\varphi_{k}\ast(s_{n}g-% s_{\ell}g)\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}\big{)}^{q}\Big{)}^{\frac{1}{% q}}" 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id="S3.E9X.3.2.2.m1.4.4.1.3.cmml" xref="S3.E9X.3.2.2.m1.4.4.1.3"><divide id="S3.E9X.3.2.2.m1.4.4.1.3.1.cmml" xref="S3.E9X.3.2.2.m1.4.4.1.3"></divide><cn id="S3.E9X.3.2.2.m1.4.4.1.3.2.cmml" type="integer" xref="S3.E9X.3.2.2.m1.4.4.1.3.2">1</cn><ci id="S3.E9X.3.2.2.m1.4.4.1.3.3.cmml" xref="S3.E9X.3.2.2.m1.4.4.1.3.3">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9X.3.2.2.m1.4c">\displaystyle=\Big{(}\sum_{k=\ell}^{n+1}\big{(}2^{ks}\|\varphi_{k}\ast(s_{n}g-% s_{\ell}g)\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X)}\big{)}^{q}\Big{)}^{\frac{1}{% q}}</annotation><annotation encoding="application/x-llamapun" id="S3.E9X.3.2.2.m1.4d">= ( ∑ start_POSTSUBSCRIPT italic_k = roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT ( 2 start_POSTSUPERSCRIPT italic_k italic_s end_POSTSUPERSCRIPT ∥ italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ( italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g - italic_s start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_g ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_q end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="3"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(3.9)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E9Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq 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xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.2.3.2.cmml">n</mi><mo id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.2.3.1" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.2.3.1.cmml">+</mo><mn id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.2.3.3" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.2.3.3.cmml">1</mn></mrow></munderover></mstyle><msup id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.cmml"><mrow id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.2" maxsize="120%" minsize="120%" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.2" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.2.cmml"><munderover id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.2a" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.2.cmml"><mo 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xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.2.3.cmml">1</mn></munderover></mstyle><mrow id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.cmml"><msup id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.3" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.cmml"><mn id="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.E9Xa.2.1.1.m1.5.5.1.1.1.1.1.1.1.1.1.1.3.2.cmml">2</mn><mrow id="S3.E9Xa.2.1.1.m1.2.2.2" xref="S3.E9Xa.2.1.1.m1.2.2.2.cmml"><mrow id="S3.E9Xa.2.1.1.m1.1.1.1.1.1" xref="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.cmml"><mo id="S3.E9Xa.2.1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1" xref="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.cmml"><mi id="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.2" xref="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.2.cmml">k</mi><mo id="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.1" xref="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.3" xref="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.3.cmml">j</mi></mrow><mo id="S3.E9Xa.2.1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S3.E9Xa.2.1.1.m1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S3.E9Xa.2.1.1.m1.2.2.2.3" xref="S3.E9Xa.2.1.1.m1.2.2.2.3.cmml">⁢</mo><mrow id="S3.E9Xa.2.1.1.m1.2.2.2.2.1" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.cmml"><mo id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.2" stretchy="false" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.cmml">(</mo><mrow id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.cmml"><mi id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.2" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.2.cmml">s</mi><mo id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.1" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.1.cmml">−</mo><mi id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.3" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.3.cmml">m</mi><mo id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.1a" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.1.cmml">−</mo><mfrac id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.4" xref="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.4.cmml"><mrow id="S3.E9Xa.2.1.1.m1.2.2.2.2.1.1.4.2" 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xref="S3.E9Xa.2.1.1.m1.5.5.1.1.3.3">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9Xa.2.1.1.m1.5c">\displaystyle\leq C\Big{(}\sum_{k=\ell}^{n+1}\big{(}\sum_{j=-1}^{1}2^{(k+j)(s-% m-\frac{\gamma+1}{p})}\|\phi_{k+j}\ast g\|_{L^{p}(\mathbb{R}^{d-1};X)}\big{)}^% {q}\Big{)}^{\frac{1}{q}}</annotation><annotation encoding="application/x-llamapun" id="S3.E9Xa.2.1.1.m1.5d">≤ italic_C ( ∑ start_POSTSUBSCRIPT italic_k = roman_ℓ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n + 1 end_POSTSUPERSCRIPT ( ∑ start_POSTSUBSCRIPT italic_j = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT ( italic_k + italic_j ) ( italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_ϕ start_POSTSUBSCRIPT italic_k + italic_j end_POSTSUBSCRIPT ∗ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_q end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S3.E9Xb"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\Big{(}\sum_{k=\ell-1}^{n+2}\big{(}2^{k(s-m-\frac{\gamma+1}% {p})}\|\phi_{k}\ast g\|_{L^{p}(\mathbb{R}^{d-1};X)}\big{)}^{q}\Big{)}^{\frac{1% }{q}}," class="ltx_Math" display="inline" id="S3.E9Xb.2.1.1.m1.4"><semantics id="S3.E9Xb.2.1.1.m1.4a"><mrow id="S3.E9Xb.2.1.1.m1.4.4.1" xref="S3.E9Xb.2.1.1.m1.4.4.1.1.cmml"><mrow id="S3.E9Xb.2.1.1.m1.4.4.1.1" xref="S3.E9Xb.2.1.1.m1.4.4.1.1.cmml"><mi id="S3.E9Xb.2.1.1.m1.4.4.1.1.3" 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id="S3.E9Xb.2.1.1.m1.4.4.1.1.1.1.3.3.cmml" xref="S3.E9Xb.2.1.1.m1.4.4.1.1.1.1.3.3">𝑞</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E9Xb.2.1.1.m1.4c">\displaystyle\leq C\Big{(}\sum_{k=\ell-1}^{n+2}\big{(}2^{k(s-m-\frac{\gamma+1}% {p})}\|\phi_{k}\ast g\|_{L^{p}(\mathbb{R}^{d-1};X)}\big{)}^{q}\Big{)}^{\frac{1% }{q}},</annotation><annotation encoding="application/x-llamapun" id="S3.E9Xb.2.1.1.m1.4d">≤ italic_C ( ∑ start_POSTSUBSCRIPT italic_k = roman_ℓ - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n + 2 end_POSTSUPERSCRIPT ( 2 start_POSTSUPERSCRIPT italic_k ( italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_q end_ARG end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S3.SS2.3.p3.8">which converges to zero as <math alttext="\ell,n\to\infty" class="ltx_Math" display="inline" id="S3.SS2.3.p3.4.m1.2"><semantics id="S3.SS2.3.p3.4.m1.2a"><mrow id="S3.SS2.3.p3.4.m1.2.3" xref="S3.SS2.3.p3.4.m1.2.3.cmml"><mrow id="S3.SS2.3.p3.4.m1.2.3.2.2" xref="S3.SS2.3.p3.4.m1.2.3.2.1.cmml"><mi id="S3.SS2.3.p3.4.m1.1.1" mathvariant="normal" xref="S3.SS2.3.p3.4.m1.1.1.cmml">ℓ</mi><mo id="S3.SS2.3.p3.4.m1.2.3.2.2.1" xref="S3.SS2.3.p3.4.m1.2.3.2.1.cmml">,</mo><mi id="S3.SS2.3.p3.4.m1.2.2" xref="S3.SS2.3.p3.4.m1.2.2.cmml">n</mi></mrow><mo id="S3.SS2.3.p3.4.m1.2.3.1" stretchy="false" xref="S3.SS2.3.p3.4.m1.2.3.1.cmml">→</mo><mi id="S3.SS2.3.p3.4.m1.2.3.3" mathvariant="normal" xref="S3.SS2.3.p3.4.m1.2.3.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.4.m1.2b"><apply id="S3.SS2.3.p3.4.m1.2.3.cmml" xref="S3.SS2.3.p3.4.m1.2.3"><ci id="S3.SS2.3.p3.4.m1.2.3.1.cmml" xref="S3.SS2.3.p3.4.m1.2.3.1">→</ci><list id="S3.SS2.3.p3.4.m1.2.3.2.1.cmml" xref="S3.SS2.3.p3.4.m1.2.3.2.2"><ci id="S3.SS2.3.p3.4.m1.1.1.cmml" xref="S3.SS2.3.p3.4.m1.1.1">ℓ</ci><ci id="S3.SS2.3.p3.4.m1.2.2.cmml" xref="S3.SS2.3.p3.4.m1.2.2">𝑛</ci></list><infinity id="S3.SS2.3.p3.4.m1.2.3.3.cmml" xref="S3.SS2.3.p3.4.m1.2.3.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.4.m1.2c">\ell,n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.4.m1.2d">roman_ℓ , italic_n → ∞</annotation></semantics></math> since <math alttext="g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS2.3.p3.5.m2.4"><semantics id="S3.SS2.3.p3.5.m2.4a"><mrow id="S3.SS2.3.p3.5.m2.4.4" xref="S3.SS2.3.p3.5.m2.4.4.cmml"><mi id="S3.SS2.3.p3.5.m2.4.4.3" xref="S3.SS2.3.p3.5.m2.4.4.3.cmml">g</mi><mo id="S3.SS2.3.p3.5.m2.4.4.2" xref="S3.SS2.3.p3.5.m2.4.4.2.cmml">∈</mo><mrow id="S3.SS2.3.p3.5.m2.4.4.1" xref="S3.SS2.3.p3.5.m2.4.4.1.cmml"><msubsup id="S3.SS2.3.p3.5.m2.4.4.1.3" xref="S3.SS2.3.p3.5.m2.4.4.1.3.cmml"><mi id="S3.SS2.3.p3.5.m2.4.4.1.3.2.2" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.2.cmml">B</mi><mrow id="S3.SS2.3.p3.5.m2.2.2.2.4" xref="S3.SS2.3.p3.5.m2.2.2.2.3.cmml"><mi id="S3.SS2.3.p3.5.m2.1.1.1.1" xref="S3.SS2.3.p3.5.m2.1.1.1.1.cmml">p</mi><mo id="S3.SS2.3.p3.5.m2.2.2.2.4.1" xref="S3.SS2.3.p3.5.m2.2.2.2.3.cmml">,</mo><mi id="S3.SS2.3.p3.5.m2.2.2.2.2" xref="S3.SS2.3.p3.5.m2.2.2.2.2.cmml">q</mi></mrow><mrow id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.cmml"><mi id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.2" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.2.cmml">s</mi><mo id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.1" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.1.cmml">−</mo><mi id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.3" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.3.cmml">m</mi><mo id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.1a" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.cmml"><mrow id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.cmml"><mi id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.2" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.2.cmml">γ</mi><mo id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.1" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.1.cmml">+</mo><mn id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.3" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.3" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.SS2.3.p3.5.m2.4.4.1.2" xref="S3.SS2.3.p3.5.m2.4.4.1.2.cmml">⁢</mo><mrow id="S3.SS2.3.p3.5.m2.4.4.1.1.1" xref="S3.SS2.3.p3.5.m2.4.4.1.1.2.cmml"><mo id="S3.SS2.3.p3.5.m2.4.4.1.1.1.2" stretchy="false" xref="S3.SS2.3.p3.5.m2.4.4.1.1.2.cmml">(</mo><msup id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.cmml"><mi id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.2" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.cmml"><mi id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.2" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.2.cmml">d</mi><mo id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.1" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.3" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS2.3.p3.5.m2.4.4.1.1.1.3" xref="S3.SS2.3.p3.5.m2.4.4.1.1.2.cmml">;</mo><mi id="S3.SS2.3.p3.5.m2.3.3" xref="S3.SS2.3.p3.5.m2.3.3.cmml">X</mi><mo id="S3.SS2.3.p3.5.m2.4.4.1.1.1.4" stretchy="false" xref="S3.SS2.3.p3.5.m2.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.5.m2.4b"><apply id="S3.SS2.3.p3.5.m2.4.4.cmml" xref="S3.SS2.3.p3.5.m2.4.4"><in id="S3.SS2.3.p3.5.m2.4.4.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.2"></in><ci id="S3.SS2.3.p3.5.m2.4.4.3.cmml" xref="S3.SS2.3.p3.5.m2.4.4.3">𝑔</ci><apply id="S3.SS2.3.p3.5.m2.4.4.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1"><times id="S3.SS2.3.p3.5.m2.4.4.1.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.2"></times><apply id="S3.SS2.3.p3.5.m2.4.4.1.3.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS2.3.p3.5.m2.4.4.1.3.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3">subscript</csymbol><apply id="S3.SS2.3.p3.5.m2.4.4.1.3.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS2.3.p3.5.m2.4.4.1.3.2.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3">superscript</csymbol><ci id="S3.SS2.3.p3.5.m2.4.4.1.3.2.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.2">𝐵</ci><apply id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3"><minus id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.1"></minus><ci id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.2">𝑠</ci><ci id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.3.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.3">𝑚</ci><apply id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4"><divide id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4"></divide><apply id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2"><plus id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.1"></plus><ci id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.2">𝛾</ci><cn id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.3.cmml" type="integer" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.2.3">1</cn></apply><ci id="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.3.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.3.2.3.4.3">𝑝</ci></apply></apply></apply><list id="S3.SS2.3.p3.5.m2.2.2.2.3.cmml" xref="S3.SS2.3.p3.5.m2.2.2.2.4"><ci id="S3.SS2.3.p3.5.m2.1.1.1.1.cmml" xref="S3.SS2.3.p3.5.m2.1.1.1.1">𝑝</ci><ci id="S3.SS2.3.p3.5.m2.2.2.2.2.cmml" xref="S3.SS2.3.p3.5.m2.2.2.2.2">𝑞</ci></list></apply><list id="S3.SS2.3.p3.5.m2.4.4.1.1.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1"><apply id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1">superscript</csymbol><ci id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.2">ℝ</ci><apply id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3"><minus id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.1.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.1"></minus><ci id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.2.cmml" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.3.p3.5.m2.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS2.3.p3.5.m2.3.3.cmml" xref="S3.SS2.3.p3.5.m2.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.5.m2.4c">g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.5.m2.4d">italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>. Thus, <math alttext="(s_{n}g)_{n\geq 0}" class="ltx_Math" display="inline" id="S3.SS2.3.p3.6.m3.1"><semantics id="S3.SS2.3.p3.6.m3.1a"><msub id="S3.SS2.3.p3.6.m3.1.1" xref="S3.SS2.3.p3.6.m3.1.1.cmml"><mrow id="S3.SS2.3.p3.6.m3.1.1.1.1" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.cmml"><mo id="S3.SS2.3.p3.6.m3.1.1.1.1.2" stretchy="false" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.3.p3.6.m3.1.1.1.1.1" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.cmml"><msub id="S3.SS2.3.p3.6.m3.1.1.1.1.1.2" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.cmml"><mi id="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.2" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.2.cmml">s</mi><mi id="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.3" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.3.cmml">n</mi></msub><mo id="S3.SS2.3.p3.6.m3.1.1.1.1.1.1" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS2.3.p3.6.m3.1.1.1.1.1.3" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.SS2.3.p3.6.m3.1.1.1.1.3" stretchy="false" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS2.3.p3.6.m3.1.1.3" xref="S3.SS2.3.p3.6.m3.1.1.3.cmml"><mi id="S3.SS2.3.p3.6.m3.1.1.3.2" xref="S3.SS2.3.p3.6.m3.1.1.3.2.cmml">n</mi><mo id="S3.SS2.3.p3.6.m3.1.1.3.1" xref="S3.SS2.3.p3.6.m3.1.1.3.1.cmml">≥</mo><mn id="S3.SS2.3.p3.6.m3.1.1.3.3" xref="S3.SS2.3.p3.6.m3.1.1.3.3.cmml">0</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.6.m3.1b"><apply id="S3.SS2.3.p3.6.m3.1.1.cmml" xref="S3.SS2.3.p3.6.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.3.p3.6.m3.1.1.2.cmml" xref="S3.SS2.3.p3.6.m3.1.1">subscript</csymbol><apply id="S3.SS2.3.p3.6.m3.1.1.1.1.1.cmml" xref="S3.SS2.3.p3.6.m3.1.1.1.1"><times id="S3.SS2.3.p3.6.m3.1.1.1.1.1.1.cmml" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.1"></times><apply id="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.cmml" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.1.cmml" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.2">subscript</csymbol><ci id="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.2.cmml" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.2">𝑠</ci><ci id="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.3.cmml" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.2.3">𝑛</ci></apply><ci id="S3.SS2.3.p3.6.m3.1.1.1.1.1.3.cmml" xref="S3.SS2.3.p3.6.m3.1.1.1.1.1.3">𝑔</ci></apply><apply id="S3.SS2.3.p3.6.m3.1.1.3.cmml" xref="S3.SS2.3.p3.6.m3.1.1.3"><geq id="S3.SS2.3.p3.6.m3.1.1.3.1.cmml" xref="S3.SS2.3.p3.6.m3.1.1.3.1"></geq><ci id="S3.SS2.3.p3.6.m3.1.1.3.2.cmml" xref="S3.SS2.3.p3.6.m3.1.1.3.2">𝑛</ci><cn id="S3.SS2.3.p3.6.m3.1.1.3.3.cmml" type="integer" xref="S3.SS2.3.p3.6.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.6.m3.1c">(s_{n}g)_{n\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.6.m3.1d">( italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a Cauchy sequence in <math alttext="B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS2.3.p3.7.m4.5"><semantics id="S3.SS2.3.p3.7.m4.5a"><mrow id="S3.SS2.3.p3.7.m4.5.5" xref="S3.SS2.3.p3.7.m4.5.5.cmml"><msubsup id="S3.SS2.3.p3.7.m4.5.5.4" xref="S3.SS2.3.p3.7.m4.5.5.4.cmml"><mi id="S3.SS2.3.p3.7.m4.5.5.4.2.2" xref="S3.SS2.3.p3.7.m4.5.5.4.2.2.cmml">B</mi><mrow id="S3.SS2.3.p3.7.m4.2.2.2.4" xref="S3.SS2.3.p3.7.m4.2.2.2.3.cmml"><mi id="S3.SS2.3.p3.7.m4.1.1.1.1" xref="S3.SS2.3.p3.7.m4.1.1.1.1.cmml">p</mi><mo id="S3.SS2.3.p3.7.m4.2.2.2.4.1" xref="S3.SS2.3.p3.7.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS2.3.p3.7.m4.2.2.2.2" xref="S3.SS2.3.p3.7.m4.2.2.2.2.cmml">q</mi></mrow><mi id="S3.SS2.3.p3.7.m4.5.5.4.2.3" xref="S3.SS2.3.p3.7.m4.5.5.4.2.3.cmml">s</mi></msubsup><mo id="S3.SS2.3.p3.7.m4.5.5.3" xref="S3.SS2.3.p3.7.m4.5.5.3.cmml">⁢</mo><mrow id="S3.SS2.3.p3.7.m4.5.5.2.2" xref="S3.SS2.3.p3.7.m4.5.5.2.3.cmml"><mo id="S3.SS2.3.p3.7.m4.5.5.2.2.3" stretchy="false" xref="S3.SS2.3.p3.7.m4.5.5.2.3.cmml">(</mo><msup id="S3.SS2.3.p3.7.m4.4.4.1.1.1" xref="S3.SS2.3.p3.7.m4.4.4.1.1.1.cmml"><mi id="S3.SS2.3.p3.7.m4.4.4.1.1.1.2" xref="S3.SS2.3.p3.7.m4.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS2.3.p3.7.m4.4.4.1.1.1.3" xref="S3.SS2.3.p3.7.m4.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS2.3.p3.7.m4.5.5.2.2.4" xref="S3.SS2.3.p3.7.m4.5.5.2.3.cmml">,</mo><msub id="S3.SS2.3.p3.7.m4.5.5.2.2.2" xref="S3.SS2.3.p3.7.m4.5.5.2.2.2.cmml"><mi id="S3.SS2.3.p3.7.m4.5.5.2.2.2.2" xref="S3.SS2.3.p3.7.m4.5.5.2.2.2.2.cmml">w</mi><mi id="S3.SS2.3.p3.7.m4.5.5.2.2.2.3" xref="S3.SS2.3.p3.7.m4.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS2.3.p3.7.m4.5.5.2.2.5" xref="S3.SS2.3.p3.7.m4.5.5.2.3.cmml">;</mo><mi id="S3.SS2.3.p3.7.m4.3.3" xref="S3.SS2.3.p3.7.m4.3.3.cmml">X</mi><mo id="S3.SS2.3.p3.7.m4.5.5.2.2.6" stretchy="false" xref="S3.SS2.3.p3.7.m4.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.7.m4.5b"><apply id="S3.SS2.3.p3.7.m4.5.5.cmml" xref="S3.SS2.3.p3.7.m4.5.5"><times id="S3.SS2.3.p3.7.m4.5.5.3.cmml" xref="S3.SS2.3.p3.7.m4.5.5.3"></times><apply id="S3.SS2.3.p3.7.m4.5.5.4.cmml" xref="S3.SS2.3.p3.7.m4.5.5.4"><csymbol cd="ambiguous" id="S3.SS2.3.p3.7.m4.5.5.4.1.cmml" xref="S3.SS2.3.p3.7.m4.5.5.4">subscript</csymbol><apply id="S3.SS2.3.p3.7.m4.5.5.4.2.cmml" xref="S3.SS2.3.p3.7.m4.5.5.4"><csymbol cd="ambiguous" id="S3.SS2.3.p3.7.m4.5.5.4.2.1.cmml" xref="S3.SS2.3.p3.7.m4.5.5.4">superscript</csymbol><ci id="S3.SS2.3.p3.7.m4.5.5.4.2.2.cmml" xref="S3.SS2.3.p3.7.m4.5.5.4.2.2">𝐵</ci><ci id="S3.SS2.3.p3.7.m4.5.5.4.2.3.cmml" xref="S3.SS2.3.p3.7.m4.5.5.4.2.3">𝑠</ci></apply><list id="S3.SS2.3.p3.7.m4.2.2.2.3.cmml" xref="S3.SS2.3.p3.7.m4.2.2.2.4"><ci id="S3.SS2.3.p3.7.m4.1.1.1.1.cmml" xref="S3.SS2.3.p3.7.m4.1.1.1.1">𝑝</ci><ci id="S3.SS2.3.p3.7.m4.2.2.2.2.cmml" xref="S3.SS2.3.p3.7.m4.2.2.2.2">𝑞</ci></list></apply><vector id="S3.SS2.3.p3.7.m4.5.5.2.3.cmml" xref="S3.SS2.3.p3.7.m4.5.5.2.2"><apply id="S3.SS2.3.p3.7.m4.4.4.1.1.1.cmml" xref="S3.SS2.3.p3.7.m4.4.4.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.3.p3.7.m4.4.4.1.1.1.1.cmml" xref="S3.SS2.3.p3.7.m4.4.4.1.1.1">superscript</csymbol><ci id="S3.SS2.3.p3.7.m4.4.4.1.1.1.2.cmml" xref="S3.SS2.3.p3.7.m4.4.4.1.1.1.2">ℝ</ci><ci id="S3.SS2.3.p3.7.m4.4.4.1.1.1.3.cmml" xref="S3.SS2.3.p3.7.m4.4.4.1.1.1.3">𝑑</ci></apply><apply id="S3.SS2.3.p3.7.m4.5.5.2.2.2.cmml" xref="S3.SS2.3.p3.7.m4.5.5.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.3.p3.7.m4.5.5.2.2.2.1.cmml" xref="S3.SS2.3.p3.7.m4.5.5.2.2.2">subscript</csymbol><ci id="S3.SS2.3.p3.7.m4.5.5.2.2.2.2.cmml" xref="S3.SS2.3.p3.7.m4.5.5.2.2.2.2">𝑤</ci><ci id="S3.SS2.3.p3.7.m4.5.5.2.2.2.3.cmml" xref="S3.SS2.3.p3.7.m4.5.5.2.2.2.3">𝛾</ci></apply><ci id="S3.SS2.3.p3.7.m4.3.3.cmml" xref="S3.SS2.3.p3.7.m4.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.7.m4.5c">B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.7.m4.5d">italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. Hence, <math alttext="\operatorname{ext}_{m}g=\sum_{j=0}^{\infty}2^{-j}g_{j}=\lim_{n\to\infty}s_{n}g" class="ltx_Math" display="inline" id="S3.SS2.3.p3.8.m5.1"><semantics id="S3.SS2.3.p3.8.m5.1a"><mrow id="S3.SS2.3.p3.8.m5.1.1" xref="S3.SS2.3.p3.8.m5.1.1.cmml"><mrow id="S3.SS2.3.p3.8.m5.1.1.2" xref="S3.SS2.3.p3.8.m5.1.1.2.cmml"><msub id="S3.SS2.3.p3.8.m5.1.1.2.1" xref="S3.SS2.3.p3.8.m5.1.1.2.1.cmml"><mi id="S3.SS2.3.p3.8.m5.1.1.2.1.2" xref="S3.SS2.3.p3.8.m5.1.1.2.1.2.cmml">ext</mi><mi id="S3.SS2.3.p3.8.m5.1.1.2.1.3" xref="S3.SS2.3.p3.8.m5.1.1.2.1.3.cmml">m</mi></msub><mo id="S3.SS2.3.p3.8.m5.1.1.2a" lspace="0.167em" xref="S3.SS2.3.p3.8.m5.1.1.2.cmml">⁡</mo><mi id="S3.SS2.3.p3.8.m5.1.1.2.2" xref="S3.SS2.3.p3.8.m5.1.1.2.2.cmml">g</mi></mrow><mo id="S3.SS2.3.p3.8.m5.1.1.3" rspace="0.111em" xref="S3.SS2.3.p3.8.m5.1.1.3.cmml">=</mo><mrow id="S3.SS2.3.p3.8.m5.1.1.4" xref="S3.SS2.3.p3.8.m5.1.1.4.cmml"><msubsup id="S3.SS2.3.p3.8.m5.1.1.4.1" xref="S3.SS2.3.p3.8.m5.1.1.4.1.cmml"><mo id="S3.SS2.3.p3.8.m5.1.1.4.1.2.2" xref="S3.SS2.3.p3.8.m5.1.1.4.1.2.2.cmml">∑</mo><mrow id="S3.SS2.3.p3.8.m5.1.1.4.1.2.3" xref="S3.SS2.3.p3.8.m5.1.1.4.1.2.3.cmml"><mi id="S3.SS2.3.p3.8.m5.1.1.4.1.2.3.2" xref="S3.SS2.3.p3.8.m5.1.1.4.1.2.3.2.cmml">j</mi><mo id="S3.SS2.3.p3.8.m5.1.1.4.1.2.3.1" xref="S3.SS2.3.p3.8.m5.1.1.4.1.2.3.1.cmml">=</mo><mn id="S3.SS2.3.p3.8.m5.1.1.4.1.2.3.3" xref="S3.SS2.3.p3.8.m5.1.1.4.1.2.3.3.cmml">0</mn></mrow><mi id="S3.SS2.3.p3.8.m5.1.1.4.1.3" mathvariant="normal" xref="S3.SS2.3.p3.8.m5.1.1.4.1.3.cmml">∞</mi></msubsup><mrow id="S3.SS2.3.p3.8.m5.1.1.4.2" xref="S3.SS2.3.p3.8.m5.1.1.4.2.cmml"><msup id="S3.SS2.3.p3.8.m5.1.1.4.2.2" xref="S3.SS2.3.p3.8.m5.1.1.4.2.2.cmml"><mn id="S3.SS2.3.p3.8.m5.1.1.4.2.2.2" xref="S3.SS2.3.p3.8.m5.1.1.4.2.2.2.cmml">2</mn><mrow id="S3.SS2.3.p3.8.m5.1.1.4.2.2.3" xref="S3.SS2.3.p3.8.m5.1.1.4.2.2.3.cmml"><mo id="S3.SS2.3.p3.8.m5.1.1.4.2.2.3a" xref="S3.SS2.3.p3.8.m5.1.1.4.2.2.3.cmml">−</mo><mi id="S3.SS2.3.p3.8.m5.1.1.4.2.2.3.2" xref="S3.SS2.3.p3.8.m5.1.1.4.2.2.3.2.cmml">j</mi></mrow></msup><mo id="S3.SS2.3.p3.8.m5.1.1.4.2.1" xref="S3.SS2.3.p3.8.m5.1.1.4.2.1.cmml">⁢</mo><msub id="S3.SS2.3.p3.8.m5.1.1.4.2.3" xref="S3.SS2.3.p3.8.m5.1.1.4.2.3.cmml"><mi id="S3.SS2.3.p3.8.m5.1.1.4.2.3.2" xref="S3.SS2.3.p3.8.m5.1.1.4.2.3.2.cmml">g</mi><mi id="S3.SS2.3.p3.8.m5.1.1.4.2.3.3" xref="S3.SS2.3.p3.8.m5.1.1.4.2.3.3.cmml">j</mi></msub></mrow></mrow><mo id="S3.SS2.3.p3.8.m5.1.1.5" rspace="0.1389em" xref="S3.SS2.3.p3.8.m5.1.1.5.cmml">=</mo><mrow id="S3.SS2.3.p3.8.m5.1.1.6" xref="S3.SS2.3.p3.8.m5.1.1.6.cmml"><msub id="S3.SS2.3.p3.8.m5.1.1.6.1" xref="S3.SS2.3.p3.8.m5.1.1.6.1.cmml"><mo id="S3.SS2.3.p3.8.m5.1.1.6.1.2" lspace="0.1389em" rspace="0.167em" xref="S3.SS2.3.p3.8.m5.1.1.6.1.2.cmml">lim</mo><mrow id="S3.SS2.3.p3.8.m5.1.1.6.1.3" xref="S3.SS2.3.p3.8.m5.1.1.6.1.3.cmml"><mi id="S3.SS2.3.p3.8.m5.1.1.6.1.3.2" xref="S3.SS2.3.p3.8.m5.1.1.6.1.3.2.cmml">n</mi><mo id="S3.SS2.3.p3.8.m5.1.1.6.1.3.1" stretchy="false" xref="S3.SS2.3.p3.8.m5.1.1.6.1.3.1.cmml">→</mo><mi id="S3.SS2.3.p3.8.m5.1.1.6.1.3.3" mathvariant="normal" xref="S3.SS2.3.p3.8.m5.1.1.6.1.3.3.cmml">∞</mi></mrow></msub><mrow id="S3.SS2.3.p3.8.m5.1.1.6.2" xref="S3.SS2.3.p3.8.m5.1.1.6.2.cmml"><msub id="S3.SS2.3.p3.8.m5.1.1.6.2.2" xref="S3.SS2.3.p3.8.m5.1.1.6.2.2.cmml"><mi id="S3.SS2.3.p3.8.m5.1.1.6.2.2.2" xref="S3.SS2.3.p3.8.m5.1.1.6.2.2.2.cmml">s</mi><mi id="S3.SS2.3.p3.8.m5.1.1.6.2.2.3" xref="S3.SS2.3.p3.8.m5.1.1.6.2.2.3.cmml">n</mi></msub><mo id="S3.SS2.3.p3.8.m5.1.1.6.2.1" xref="S3.SS2.3.p3.8.m5.1.1.6.2.1.cmml">⁢</mo><mi id="S3.SS2.3.p3.8.m5.1.1.6.2.3" xref="S3.SS2.3.p3.8.m5.1.1.6.2.3.cmml">g</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.3.p3.8.m5.1b"><apply id="S3.SS2.3.p3.8.m5.1.1.cmml" xref="S3.SS2.3.p3.8.m5.1.1"><and id="S3.SS2.3.p3.8.m5.1.1a.cmml" xref="S3.SS2.3.p3.8.m5.1.1"></and><apply id="S3.SS2.3.p3.8.m5.1.1b.cmml" xref="S3.SS2.3.p3.8.m5.1.1"><eq id="S3.SS2.3.p3.8.m5.1.1.3.cmml" xref="S3.SS2.3.p3.8.m5.1.1.3"></eq><apply id="S3.SS2.3.p3.8.m5.1.1.2.cmml" xref="S3.SS2.3.p3.8.m5.1.1.2"><apply id="S3.SS2.3.p3.8.m5.1.1.2.1.cmml" xref="S3.SS2.3.p3.8.m5.1.1.2.1"><csymbol cd="ambiguous" id="S3.SS2.3.p3.8.m5.1.1.2.1.1.cmml" xref="S3.SS2.3.p3.8.m5.1.1.2.1">subscript</csymbol><ci id="S3.SS2.3.p3.8.m5.1.1.2.1.2.cmml" xref="S3.SS2.3.p3.8.m5.1.1.2.1.2">ext</ci><ci id="S3.SS2.3.p3.8.m5.1.1.2.1.3.cmml" xref="S3.SS2.3.p3.8.m5.1.1.2.1.3">𝑚</ci></apply><ci id="S3.SS2.3.p3.8.m5.1.1.2.2.cmml" xref="S3.SS2.3.p3.8.m5.1.1.2.2">𝑔</ci></apply><apply id="S3.SS2.3.p3.8.m5.1.1.4.cmml" xref="S3.SS2.3.p3.8.m5.1.1.4"><apply id="S3.SS2.3.p3.8.m5.1.1.4.1.cmml" xref="S3.SS2.3.p3.8.m5.1.1.4.1"><csymbol cd="ambiguous" id="S3.SS2.3.p3.8.m5.1.1.4.1.1.cmml" xref="S3.SS2.3.p3.8.m5.1.1.4.1">superscript</csymbol><apply id="S3.SS2.3.p3.8.m5.1.1.4.1.2.cmml" xref="S3.SS2.3.p3.8.m5.1.1.4.1"><csymbol cd="ambiguous" id="S3.SS2.3.p3.8.m5.1.1.4.1.2.1.cmml" xref="S3.SS2.3.p3.8.m5.1.1.4.1">subscript</csymbol><sum 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id="S3.SS2.3.p3.8.m5.1.1.6.2.2.1.cmml" xref="S3.SS2.3.p3.8.m5.1.1.6.2.2">subscript</csymbol><ci id="S3.SS2.3.p3.8.m5.1.1.6.2.2.2.cmml" xref="S3.SS2.3.p3.8.m5.1.1.6.2.2.2">𝑠</ci><ci id="S3.SS2.3.p3.8.m5.1.1.6.2.2.3.cmml" xref="S3.SS2.3.p3.8.m5.1.1.6.2.2.3">𝑛</ci></apply><ci id="S3.SS2.3.p3.8.m5.1.1.6.2.3.cmml" xref="S3.SS2.3.p3.8.m5.1.1.6.2.3">𝑔</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.3.p3.8.m5.1c">\operatorname{ext}_{m}g=\sum_{j=0}^{\infty}2^{-j}g_{j}=\lim_{n\to\infty}s_{n}g</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.3.p3.8.m5.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g = ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g</annotation></semantics></math> exists in this space and this proves <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.I2.i1" title="item i ‣ Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>.</p> </div> <div class="ltx_para" id="S3.SS2.4.p4"> <p class="ltx_p" id="S3.SS2.4.p4.2">If <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.SS2.4.p4.1.m1.2"><semantics id="S3.SS2.4.p4.1.m1.2a"><mrow id="S3.SS2.4.p4.1.m1.2.3" xref="S3.SS2.4.p4.1.m1.2.3.cmml"><mi id="S3.SS2.4.p4.1.m1.2.3.2" xref="S3.SS2.4.p4.1.m1.2.3.2.cmml">q</mi><mo id="S3.SS2.4.p4.1.m1.2.3.1" xref="S3.SS2.4.p4.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.SS2.4.p4.1.m1.2.3.3.2" xref="S3.SS2.4.p4.1.m1.2.3.3.1.cmml"><mo id="S3.SS2.4.p4.1.m1.2.3.3.2.1" stretchy="false" xref="S3.SS2.4.p4.1.m1.2.3.3.1.cmml">[</mo><mn id="S3.SS2.4.p4.1.m1.1.1" xref="S3.SS2.4.p4.1.m1.1.1.cmml">1</mn><mo id="S3.SS2.4.p4.1.m1.2.3.3.2.2" xref="S3.SS2.4.p4.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.SS2.4.p4.1.m1.2.2" mathvariant="normal" xref="S3.SS2.4.p4.1.m1.2.2.cmml">∞</mi><mo id="S3.SS2.4.p4.1.m1.2.3.3.2.3" stretchy="false" xref="S3.SS2.4.p4.1.m1.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.1.m1.2b"><apply id="S3.SS2.4.p4.1.m1.2.3.cmml" xref="S3.SS2.4.p4.1.m1.2.3"><in id="S3.SS2.4.p4.1.m1.2.3.1.cmml" xref="S3.SS2.4.p4.1.m1.2.3.1"></in><ci id="S3.SS2.4.p4.1.m1.2.3.2.cmml" xref="S3.SS2.4.p4.1.m1.2.3.2">𝑞</ci><interval closure="closed" id="S3.SS2.4.p4.1.m1.2.3.3.1.cmml" xref="S3.SS2.4.p4.1.m1.2.3.3.2"><cn id="S3.SS2.4.p4.1.m1.1.1.cmml" type="integer" xref="S3.SS2.4.p4.1.m1.1.1">1</cn><infinity id="S3.SS2.4.p4.1.m1.2.2.cmml" xref="S3.SS2.4.p4.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.1.m1.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.1.m1.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math> and <math alttext="t<s" class="ltx_Math" display="inline" id="S3.SS2.4.p4.2.m2.1"><semantics id="S3.SS2.4.p4.2.m2.1a"><mrow id="S3.SS2.4.p4.2.m2.1.1" xref="S3.SS2.4.p4.2.m2.1.1.cmml"><mi id="S3.SS2.4.p4.2.m2.1.1.2" xref="S3.SS2.4.p4.2.m2.1.1.2.cmml">t</mi><mo id="S3.SS2.4.p4.2.m2.1.1.1" xref="S3.SS2.4.p4.2.m2.1.1.1.cmml"><</mo><mi id="S3.SS2.4.p4.2.m2.1.1.3" xref="S3.SS2.4.p4.2.m2.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.2.m2.1b"><apply id="S3.SS2.4.p4.2.m2.1.1.cmml" xref="S3.SS2.4.p4.2.m2.1.1"><lt id="S3.SS2.4.p4.2.m2.1.1.1.cmml" xref="S3.SS2.4.p4.2.m2.1.1.1"></lt><ci id="S3.SS2.4.p4.2.m2.1.1.2.cmml" xref="S3.SS2.4.p4.2.m2.1.1.2">𝑡</ci><ci id="S3.SS2.4.p4.2.m2.1.1.3.cmml" xref="S3.SS2.4.p4.2.m2.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.2.m2.1c">t<s</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.2.m2.1d">italic_t < italic_s</annotation></semantics></math>, then by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Theorem 14.4.19]</cite> it follows that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)\hookrightarrow B^{t-% m-\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d-1};X)." class="ltx_Math" display="block" id="S3.Ex28.m1.7"><semantics id="S3.Ex28.m1.7a"><mrow id="S3.Ex28.m1.7.7.1" xref="S3.Ex28.m1.7.7.1.1.cmml"><mrow id="S3.Ex28.m1.7.7.1.1" xref="S3.Ex28.m1.7.7.1.1.cmml"><mi id="S3.Ex28.m1.7.7.1.1.4" xref="S3.Ex28.m1.7.7.1.1.4.cmml">g</mi><mo id="S3.Ex28.m1.7.7.1.1.5" xref="S3.Ex28.m1.7.7.1.1.5.cmml">∈</mo><mrow id="S3.Ex28.m1.7.7.1.1.1" xref="S3.Ex28.m1.7.7.1.1.1.cmml"><msubsup id="S3.Ex28.m1.7.7.1.1.1.3" xref="S3.Ex28.m1.7.7.1.1.1.3.cmml"><mi 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type="integer" xref="S3.Ex28.m1.7.7.1.1.2.1.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex28.m1.6.6.cmml" xref="S3.Ex28.m1.6.6">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex28.m1.7c">g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)\hookrightarrow B^{t-% m-\frac{\gamma+1}{p}}_{p,1}(\mathbb{R}^{d-1};X).</annotation><annotation encoding="application/x-llamapun" id="S3.Ex28.m1.7d">italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ↪ italic_B start_POSTSUPERSCRIPT italic_t - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.4.p4.4">Therefore, <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.I2.i1" title="item i ‣ Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> implies that <math alttext="\operatorname{ext}_{m}g" class="ltx_Math" display="inline" id="S3.SS2.4.p4.3.m1.1"><semantics id="S3.SS2.4.p4.3.m1.1a"><mrow id="S3.SS2.4.p4.3.m1.1.1" xref="S3.SS2.4.p4.3.m1.1.1.cmml"><msub id="S3.SS2.4.p4.3.m1.1.1.1" xref="S3.SS2.4.p4.3.m1.1.1.1.cmml"><mi id="S3.SS2.4.p4.3.m1.1.1.1.2" xref="S3.SS2.4.p4.3.m1.1.1.1.2.cmml">ext</mi><mi id="S3.SS2.4.p4.3.m1.1.1.1.3" xref="S3.SS2.4.p4.3.m1.1.1.1.3.cmml">m</mi></msub><mo id="S3.SS2.4.p4.3.m1.1.1a" lspace="0.167em" xref="S3.SS2.4.p4.3.m1.1.1.cmml">⁡</mo><mi id="S3.SS2.4.p4.3.m1.1.1.2" xref="S3.SS2.4.p4.3.m1.1.1.2.cmml">g</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.3.m1.1b"><apply id="S3.SS2.4.p4.3.m1.1.1.cmml" xref="S3.SS2.4.p4.3.m1.1.1"><apply id="S3.SS2.4.p4.3.m1.1.1.1.cmml" xref="S3.SS2.4.p4.3.m1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.4.p4.3.m1.1.1.1.1.cmml" xref="S3.SS2.4.p4.3.m1.1.1.1">subscript</csymbol><ci id="S3.SS2.4.p4.3.m1.1.1.1.2.cmml" xref="S3.SS2.4.p4.3.m1.1.1.1.2">ext</ci><ci id="S3.SS2.4.p4.3.m1.1.1.1.3.cmml" xref="S3.SS2.4.p4.3.m1.1.1.1.3">𝑚</ci></apply><ci id="S3.SS2.4.p4.3.m1.1.1.2.cmml" xref="S3.SS2.4.p4.3.m1.1.1.2">𝑔</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.3.m1.1c">\operatorname{ext}_{m}g</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.3.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g</annotation></semantics></math> exists in <math alttext="B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS2.4.p4.4.m2.5"><semantics id="S3.SS2.4.p4.4.m2.5a"><mrow id="S3.SS2.4.p4.4.m2.5.5" xref="S3.SS2.4.p4.4.m2.5.5.cmml"><msubsup id="S3.SS2.4.p4.4.m2.5.5.4" xref="S3.SS2.4.p4.4.m2.5.5.4.cmml"><mi id="S3.SS2.4.p4.4.m2.5.5.4.2.2" xref="S3.SS2.4.p4.4.m2.5.5.4.2.2.cmml">B</mi><mrow id="S3.SS2.4.p4.4.m2.2.2.2.4" xref="S3.SS2.4.p4.4.m2.2.2.2.3.cmml"><mi id="S3.SS2.4.p4.4.m2.1.1.1.1" xref="S3.SS2.4.p4.4.m2.1.1.1.1.cmml">p</mi><mo id="S3.SS2.4.p4.4.m2.2.2.2.4.1" xref="S3.SS2.4.p4.4.m2.2.2.2.3.cmml">,</mo><mn id="S3.SS2.4.p4.4.m2.2.2.2.2" xref="S3.SS2.4.p4.4.m2.2.2.2.2.cmml">1</mn></mrow><mi id="S3.SS2.4.p4.4.m2.5.5.4.2.3" xref="S3.SS2.4.p4.4.m2.5.5.4.2.3.cmml">t</mi></msubsup><mo id="S3.SS2.4.p4.4.m2.5.5.3" xref="S3.SS2.4.p4.4.m2.5.5.3.cmml">⁢</mo><mrow id="S3.SS2.4.p4.4.m2.5.5.2.2" xref="S3.SS2.4.p4.4.m2.5.5.2.3.cmml"><mo id="S3.SS2.4.p4.4.m2.5.5.2.2.3" stretchy="false" xref="S3.SS2.4.p4.4.m2.5.5.2.3.cmml">(</mo><msup id="S3.SS2.4.p4.4.m2.4.4.1.1.1" xref="S3.SS2.4.p4.4.m2.4.4.1.1.1.cmml"><mi id="S3.SS2.4.p4.4.m2.4.4.1.1.1.2" xref="S3.SS2.4.p4.4.m2.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS2.4.p4.4.m2.4.4.1.1.1.3" xref="S3.SS2.4.p4.4.m2.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS2.4.p4.4.m2.5.5.2.2.4" xref="S3.SS2.4.p4.4.m2.5.5.2.3.cmml">,</mo><msub id="S3.SS2.4.p4.4.m2.5.5.2.2.2" xref="S3.SS2.4.p4.4.m2.5.5.2.2.2.cmml"><mi id="S3.SS2.4.p4.4.m2.5.5.2.2.2.2" xref="S3.SS2.4.p4.4.m2.5.5.2.2.2.2.cmml">w</mi><mi id="S3.SS2.4.p4.4.m2.5.5.2.2.2.3" xref="S3.SS2.4.p4.4.m2.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS2.4.p4.4.m2.5.5.2.2.5" xref="S3.SS2.4.p4.4.m2.5.5.2.3.cmml">;</mo><mi id="S3.SS2.4.p4.4.m2.3.3" xref="S3.SS2.4.p4.4.m2.3.3.cmml">X</mi><mo id="S3.SS2.4.p4.4.m2.5.5.2.2.6" stretchy="false" xref="S3.SS2.4.p4.4.m2.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.4.p4.4.m2.5b"><apply id="S3.SS2.4.p4.4.m2.5.5.cmml" xref="S3.SS2.4.p4.4.m2.5.5"><times id="S3.SS2.4.p4.4.m2.5.5.3.cmml" xref="S3.SS2.4.p4.4.m2.5.5.3"></times><apply id="S3.SS2.4.p4.4.m2.5.5.4.cmml" xref="S3.SS2.4.p4.4.m2.5.5.4"><csymbol cd="ambiguous" id="S3.SS2.4.p4.4.m2.5.5.4.1.cmml" xref="S3.SS2.4.p4.4.m2.5.5.4">subscript</csymbol><apply id="S3.SS2.4.p4.4.m2.5.5.4.2.cmml" 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xref="S3.SS2.4.p4.4.m2.4.4.1.1.1.3">𝑑</ci></apply><apply id="S3.SS2.4.p4.4.m2.5.5.2.2.2.cmml" xref="S3.SS2.4.p4.4.m2.5.5.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.4.p4.4.m2.5.5.2.2.2.1.cmml" xref="S3.SS2.4.p4.4.m2.5.5.2.2.2">subscript</csymbol><ci id="S3.SS2.4.p4.4.m2.5.5.2.2.2.2.cmml" xref="S3.SS2.4.p4.4.m2.5.5.2.2.2.2">𝑤</ci><ci id="S3.SS2.4.p4.4.m2.5.5.2.2.2.3.cmml" xref="S3.SS2.4.p4.4.m2.5.5.2.2.2.3">𝛾</ci></apply><ci id="S3.SS2.4.p4.4.m2.3.3.cmml" xref="S3.SS2.4.p4.4.m2.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.4.p4.4.m2.5c">B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.4.p4.4.m2.5d">italic_B start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> and this proves <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.I2.i2" title="item ii ‣ Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>.</p> </div> <div class="ltx_para" id="S3.SS2.5.p5"> <p class="ltx_p" id="S3.SS2.5.p5.4"><span class="ltx_text ltx_font_italic" id="S3.SS2.5.p5.4.1">Step 3: continuity.</span> Let <math alttext="g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS2.5.p5.1.m1.4"><semantics id="S3.SS2.5.p5.1.m1.4a"><mrow id="S3.SS2.5.p5.1.m1.4.4" xref="S3.SS2.5.p5.1.m1.4.4.cmml"><mi id="S3.SS2.5.p5.1.m1.4.4.3" xref="S3.SS2.5.p5.1.m1.4.4.3.cmml">g</mi><mo id="S3.SS2.5.p5.1.m1.4.4.2" xref="S3.SS2.5.p5.1.m1.4.4.2.cmml">∈</mo><mrow id="S3.SS2.5.p5.1.m1.4.4.1" xref="S3.SS2.5.p5.1.m1.4.4.1.cmml"><msubsup id="S3.SS2.5.p5.1.m1.4.4.1.3" xref="S3.SS2.5.p5.1.m1.4.4.1.3.cmml"><mi id="S3.SS2.5.p5.1.m1.4.4.1.3.2.2" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.2.cmml">B</mi><mrow id="S3.SS2.5.p5.1.m1.2.2.2.4" xref="S3.SS2.5.p5.1.m1.2.2.2.3.cmml"><mi id="S3.SS2.5.p5.1.m1.1.1.1.1" xref="S3.SS2.5.p5.1.m1.1.1.1.1.cmml">p</mi><mo id="S3.SS2.5.p5.1.m1.2.2.2.4.1" xref="S3.SS2.5.p5.1.m1.2.2.2.3.cmml">,</mo><mi id="S3.SS2.5.p5.1.m1.2.2.2.2" xref="S3.SS2.5.p5.1.m1.2.2.2.2.cmml">q</mi></mrow><mrow id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.cmml"><mi id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.2" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.2.cmml">s</mi><mo id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.1" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.1.cmml">−</mo><mi id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.3" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.3.cmml">m</mi><mo id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.1a" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.4" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.4.cmml"><mrow id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.4.2" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.4.2.cmml"><mi id="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.4.2.2" xref="S3.SS2.5.p5.1.m1.4.4.1.3.2.3.4.2.2.cmml">γ</mi><mo 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xref="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.3" xref="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS2.5.p5.1.m1.4.4.1.1.1.3" xref="S3.SS2.5.p5.1.m1.4.4.1.1.2.cmml">;</mo><mi id="S3.SS2.5.p5.1.m1.3.3" xref="S3.SS2.5.p5.1.m1.3.3.cmml">X</mi><mo id="S3.SS2.5.p5.1.m1.4.4.1.1.1.4" stretchy="false" xref="S3.SS2.5.p5.1.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.1.m1.4b"><apply id="S3.SS2.5.p5.1.m1.4.4.cmml" xref="S3.SS2.5.p5.1.m1.4.4"><in id="S3.SS2.5.p5.1.m1.4.4.2.cmml" xref="S3.SS2.5.p5.1.m1.4.4.2"></in><ci id="S3.SS2.5.p5.1.m1.4.4.3.cmml" xref="S3.SS2.5.p5.1.m1.4.4.3">𝑔</ci><apply id="S3.SS2.5.p5.1.m1.4.4.1.cmml" xref="S3.SS2.5.p5.1.m1.4.4.1"><times id="S3.SS2.5.p5.1.m1.4.4.1.2.cmml" xref="S3.SS2.5.p5.1.m1.4.4.1.2"></times><apply id="S3.SS2.5.p5.1.m1.4.4.1.3.cmml" xref="S3.SS2.5.p5.1.m1.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.1.m1.4.4.1.3.1.cmml" 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xref="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.2">ℝ</ci><apply id="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.cmml" xref="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3"><minus id="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.1.cmml" xref="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.1"></minus><ci id="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.2.cmml" xref="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.5.p5.1.m1.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS2.5.p5.1.m1.3.3.cmml" xref="S3.SS2.5.p5.1.m1.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.1.m1.4c">g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.1.m1.4d">italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> with <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.SS2.5.p5.2.m2.2"><semantics id="S3.SS2.5.p5.2.m2.2a"><mrow id="S3.SS2.5.p5.2.m2.2.3" xref="S3.SS2.5.p5.2.m2.2.3.cmml"><mi id="S3.SS2.5.p5.2.m2.2.3.2" xref="S3.SS2.5.p5.2.m2.2.3.2.cmml">q</mi><mo id="S3.SS2.5.p5.2.m2.2.3.1" xref="S3.SS2.5.p5.2.m2.2.3.1.cmml">∈</mo><mrow id="S3.SS2.5.p5.2.m2.2.3.3.2" xref="S3.SS2.5.p5.2.m2.2.3.3.1.cmml"><mo id="S3.SS2.5.p5.2.m2.2.3.3.2.1" stretchy="false" xref="S3.SS2.5.p5.2.m2.2.3.3.1.cmml">[</mo><mn id="S3.SS2.5.p5.2.m2.1.1" xref="S3.SS2.5.p5.2.m2.1.1.cmml">1</mn><mo id="S3.SS2.5.p5.2.m2.2.3.3.2.2" xref="S3.SS2.5.p5.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.SS2.5.p5.2.m2.2.2" mathvariant="normal" xref="S3.SS2.5.p5.2.m2.2.2.cmml">∞</mi><mo id="S3.SS2.5.p5.2.m2.2.3.3.2.3" stretchy="false" xref="S3.SS2.5.p5.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.2.m2.2b"><apply id="S3.SS2.5.p5.2.m2.2.3.cmml" xref="S3.SS2.5.p5.2.m2.2.3"><in id="S3.SS2.5.p5.2.m2.2.3.1.cmml" xref="S3.SS2.5.p5.2.m2.2.3.1"></in><ci id="S3.SS2.5.p5.2.m2.2.3.2.cmml" xref="S3.SS2.5.p5.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S3.SS2.5.p5.2.m2.2.3.3.1.cmml" xref="S3.SS2.5.p5.2.m2.2.3.3.2"><cn id="S3.SS2.5.p5.2.m2.1.1.cmml" type="integer" xref="S3.SS2.5.p5.2.m2.1.1">1</cn><infinity id="S3.SS2.5.p5.2.m2.2.2.cmml" xref="S3.SS2.5.p5.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>. Take <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.SS2.5.p5.3.m3.1"><semantics id="S3.SS2.5.p5.3.m3.1a"><mrow id="S3.SS2.5.p5.3.m3.1.1" xref="S3.SS2.5.p5.3.m3.1.1.cmml"><mi id="S3.SS2.5.p5.3.m3.1.1.2" xref="S3.SS2.5.p5.3.m3.1.1.2.cmml">k</mi><mo id="S3.SS2.5.p5.3.m3.1.1.1" xref="S3.SS2.5.p5.3.m3.1.1.1.cmml">∈</mo><msub id="S3.SS2.5.p5.3.m3.1.1.3" xref="S3.SS2.5.p5.3.m3.1.1.3.cmml"><mi id="S3.SS2.5.p5.3.m3.1.1.3.2" xref="S3.SS2.5.p5.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S3.SS2.5.p5.3.m3.1.1.3.3" xref="S3.SS2.5.p5.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.3.m3.1b"><apply id="S3.SS2.5.p5.3.m3.1.1.cmml" xref="S3.SS2.5.p5.3.m3.1.1"><in id="S3.SS2.5.p5.3.m3.1.1.1.cmml" xref="S3.SS2.5.p5.3.m3.1.1.1"></in><ci id="S3.SS2.5.p5.3.m3.1.1.2.cmml" xref="S3.SS2.5.p5.3.m3.1.1.2">𝑘</ci><apply id="S3.SS2.5.p5.3.m3.1.1.3.cmml" xref="S3.SS2.5.p5.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.3.m3.1.1.3.1.cmml" xref="S3.SS2.5.p5.3.m3.1.1.3">subscript</csymbol><ci id="S3.SS2.5.p5.3.m3.1.1.3.2.cmml" xref="S3.SS2.5.p5.3.m3.1.1.3.2">ℕ</ci><cn id="S3.SS2.5.p5.3.m3.1.1.3.3.cmml" type="integer" xref="S3.SS2.5.p5.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.3.m3.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.3.m3.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="0<\ell<n" class="ltx_Math" display="inline" id="S3.SS2.5.p5.4.m4.1"><semantics id="S3.SS2.5.p5.4.m4.1a"><mrow id="S3.SS2.5.p5.4.m4.1.1" xref="S3.SS2.5.p5.4.m4.1.1.cmml"><mn id="S3.SS2.5.p5.4.m4.1.1.2" xref="S3.SS2.5.p5.4.m4.1.1.2.cmml">0</mn><mo id="S3.SS2.5.p5.4.m4.1.1.3" xref="S3.SS2.5.p5.4.m4.1.1.3.cmml"><</mo><mi id="S3.SS2.5.p5.4.m4.1.1.4" mathvariant="normal" xref="S3.SS2.5.p5.4.m4.1.1.4.cmml">ℓ</mi><mo id="S3.SS2.5.p5.4.m4.1.1.5" xref="S3.SS2.5.p5.4.m4.1.1.5.cmml"><</mo><mi id="S3.SS2.5.p5.4.m4.1.1.6" xref="S3.SS2.5.p5.4.m4.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.4.m4.1b"><apply id="S3.SS2.5.p5.4.m4.1.1.cmml" xref="S3.SS2.5.p5.4.m4.1.1"><and id="S3.SS2.5.p5.4.m4.1.1a.cmml" xref="S3.SS2.5.p5.4.m4.1.1"></and><apply id="S3.SS2.5.p5.4.m4.1.1b.cmml" xref="S3.SS2.5.p5.4.m4.1.1"><lt id="S3.SS2.5.p5.4.m4.1.1.3.cmml" xref="S3.SS2.5.p5.4.m4.1.1.3"></lt><cn id="S3.SS2.5.p5.4.m4.1.1.2.cmml" type="integer" xref="S3.SS2.5.p5.4.m4.1.1.2">0</cn><ci id="S3.SS2.5.p5.4.m4.1.1.4.cmml" xref="S3.SS2.5.p5.4.m4.1.1.4">ℓ</ci></apply><apply id="S3.SS2.5.p5.4.m4.1.1c.cmml" xref="S3.SS2.5.p5.4.m4.1.1"><lt id="S3.SS2.5.p5.4.m4.1.1.5.cmml" xref="S3.SS2.5.p5.4.m4.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S3.SS2.5.p5.4.m4.1.1.4.cmml" id="S3.SS2.5.p5.4.m4.1.1d.cmml" xref="S3.SS2.5.p5.4.m4.1.1"></share><ci id="S3.SS2.5.p5.4.m4.1.1.6.cmml" xref="S3.SS2.5.p5.4.m4.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.4.m4.1c">0<\ell<n</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.4.m4.1d">0 < roman_ℓ < italic_n</annotation></semantics></math>, then the estimate</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="2^{kt}\|\varphi_{k}\ast(s_{n}g-s_{\ell}g)\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X% )}\leq C\|s_{n}g-s_{\ell}g\|_{B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)}," class="ltx_Math" display="block" id="S3.Ex29.m1.9"><semantics id="S3.Ex29.m1.9a"><mrow id="S3.Ex29.m1.9.9.1" xref="S3.Ex29.m1.9.9.1.1.cmml"><mrow id="S3.Ex29.m1.9.9.1.1" xref="S3.Ex29.m1.9.9.1.1.cmml"><mrow id="S3.Ex29.m1.9.9.1.1.1" xref="S3.Ex29.m1.9.9.1.1.1.cmml"><msup id="S3.Ex29.m1.9.9.1.1.1.3" xref="S3.Ex29.m1.9.9.1.1.1.3.cmml"><mn id="S3.Ex29.m1.9.9.1.1.1.3.2" xref="S3.Ex29.m1.9.9.1.1.1.3.2.cmml">2</mn><mrow id="S3.Ex29.m1.9.9.1.1.1.3.3" xref="S3.Ex29.m1.9.9.1.1.1.3.3.cmml"><mi id="S3.Ex29.m1.9.9.1.1.1.3.3.2" xref="S3.Ex29.m1.9.9.1.1.1.3.3.2.cmml">k</mi><mo id="S3.Ex29.m1.9.9.1.1.1.3.3.1" xref="S3.Ex29.m1.9.9.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.Ex29.m1.9.9.1.1.1.3.3.3" xref="S3.Ex29.m1.9.9.1.1.1.3.3.3.cmml">t</mi></mrow></msup><mo id="S3.Ex29.m1.9.9.1.1.1.2" xref="S3.Ex29.m1.9.9.1.1.1.2.cmml">⁢</mo><msub id="S3.Ex29.m1.9.9.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.1.1.cmml"><mrow id="S3.Ex29.m1.9.9.1.1.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.1.1.1.2.cmml"><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex29.m1.9.9.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.cmml"><msub id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.3" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.3.cmml"><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.3.2" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.3.2.cmml">φ</mi><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.3.3" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.3.3.cmml">k</mi></msub><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.2" lspace="0.222em" rspace="0.222em" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.2.cmml">∗</mo><mrow id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.cmml"><mrow id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.cmml"><msub id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.2" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.2.cmml"><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.2.2" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.2.2.cmml">s</mi><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.2.3" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.2.3.cmml">n</mi></msub><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.1" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.3" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.2.3.cmml">g</mi></mrow><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.1.cmml">−</mo><mrow id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.cmml"><msub id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.2" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.2.cmml"><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.2.2" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.2.2.cmml">s</mi><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.2.3" mathvariant="normal" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.2.3.cmml">ℓ</mi></msub><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.1" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.1.cmml">⁢</mo><mi id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.3" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.3.3.cmml">g</mi></mrow></mrow><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex29.m1.9.9.1.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S3.Ex29.m1.9.9.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex29.m1.9.9.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex29.m1.3.3.3" xref="S3.Ex29.m1.3.3.3.cmml"><msup id="S3.Ex29.m1.3.3.3.5" xref="S3.Ex29.m1.3.3.3.5.cmml"><mi id="S3.Ex29.m1.3.3.3.5.2" xref="S3.Ex29.m1.3.3.3.5.2.cmml">L</mi><mi id="S3.Ex29.m1.3.3.3.5.3" xref="S3.Ex29.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S3.Ex29.m1.3.3.3.4" xref="S3.Ex29.m1.3.3.3.4.cmml">⁢</mo><mrow id="S3.Ex29.m1.3.3.3.3.2" xref="S3.Ex29.m1.3.3.3.3.3.cmml"><mo id="S3.Ex29.m1.3.3.3.3.2.3" stretchy="false" xref="S3.Ex29.m1.3.3.3.3.3.cmml">(</mo><msup id="S3.Ex29.m1.2.2.2.2.1.1" xref="S3.Ex29.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex29.m1.2.2.2.2.1.1.2" xref="S3.Ex29.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mi id="S3.Ex29.m1.2.2.2.2.1.1.3" xref="S3.Ex29.m1.2.2.2.2.1.1.3.cmml">d</mi></msup><mo id="S3.Ex29.m1.3.3.3.3.2.4" xref="S3.Ex29.m1.3.3.3.3.3.cmml">,</mo><msub id="S3.Ex29.m1.3.3.3.3.2.2" xref="S3.Ex29.m1.3.3.3.3.2.2.cmml"><mi id="S3.Ex29.m1.3.3.3.3.2.2.2" xref="S3.Ex29.m1.3.3.3.3.2.2.2.cmml">w</mi><mi id="S3.Ex29.m1.3.3.3.3.2.2.3" xref="S3.Ex29.m1.3.3.3.3.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex29.m1.3.3.3.3.2.5" xref="S3.Ex29.m1.3.3.3.3.3.cmml">;</mo><mi id="S3.Ex29.m1.1.1.1.1" xref="S3.Ex29.m1.1.1.1.1.cmml">X</mi><mo id="S3.Ex29.m1.3.3.3.3.2.6" stretchy="false" xref="S3.Ex29.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow></msub></mrow><mo id="S3.Ex29.m1.9.9.1.1.3" xref="S3.Ex29.m1.9.9.1.1.3.cmml">≤</mo><mrow id="S3.Ex29.m1.9.9.1.1.2" xref="S3.Ex29.m1.9.9.1.1.2.cmml"><mi id="S3.Ex29.m1.9.9.1.1.2.3" xref="S3.Ex29.m1.9.9.1.1.2.3.cmml">C</mi><mo id="S3.Ex29.m1.9.9.1.1.2.2" xref="S3.Ex29.m1.9.9.1.1.2.2.cmml">⁢</mo><msub id="S3.Ex29.m1.9.9.1.1.2.1" xref="S3.Ex29.m1.9.9.1.1.2.1.cmml"><mrow id="S3.Ex29.m1.9.9.1.1.2.1.1.1" xref="S3.Ex29.m1.9.9.1.1.2.1.1.2.cmml"><mo id="S3.Ex29.m1.9.9.1.1.2.1.1.1.2" stretchy="false" xref="S3.Ex29.m1.9.9.1.1.2.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.cmml"><mrow id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.cmml"><msub id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.2" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.2.cmml"><mi id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.2.2" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.2.2.cmml">s</mi><mi id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.2.3" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.2.3.cmml">n</mi></msub><mo id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.1" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.3" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.2.3.cmml">g</mi></mrow><mo id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.1" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.1.cmml">−</mo><mrow id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.3" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.3.cmml"><msub id="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.3.2" xref="S3.Ex29.m1.9.9.1.1.2.1.1.1.1.3.2.cmml"><mi 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id="S3.Ex29.m1.8.8.5.5.2.2.1.cmml" xref="S3.Ex29.m1.8.8.5.5.2.2">subscript</csymbol><ci id="S3.Ex29.m1.8.8.5.5.2.2.2.cmml" xref="S3.Ex29.m1.8.8.5.5.2.2.2">𝑤</ci><ci id="S3.Ex29.m1.8.8.5.5.2.2.3.cmml" xref="S3.Ex29.m1.8.8.5.5.2.2.3">𝛾</ci></apply><ci id="S3.Ex29.m1.6.6.3.3.cmml" xref="S3.Ex29.m1.6.6.3.3">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex29.m1.9c">2^{kt}\|\varphi_{k}\ast(s_{n}g-s_{\ell}g)\|_{L^{p}(\mathbb{R}^{d},w_{\gamma};X% )}\leq C\|s_{n}g-s_{\ell}g\|_{B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex29.m1.9d">2 start_POSTSUPERSCRIPT italic_k italic_t end_POSTSUPERSCRIPT ∥ italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ ( italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g - italic_s start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_g ) ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g - italic_s start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.5.p5.16">together with (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E9" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.9</span></a>) implies that <math alttext="(\varphi_{k}\ast s_{n}g)_{n\geq 0}" class="ltx_Math" display="inline" id="S3.SS2.5.p5.5.m1.1"><semantics id="S3.SS2.5.p5.5.m1.1a"><msub id="S3.SS2.5.p5.5.m1.1.1" xref="S3.SS2.5.p5.5.m1.1.1.cmml"><mrow id="S3.SS2.5.p5.5.m1.1.1.1.1" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.cmml"><mo id="S3.SS2.5.p5.5.m1.1.1.1.1.2" stretchy="false" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.cmml">(</mo><mrow id="S3.SS2.5.p5.5.m1.1.1.1.1.1" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.cmml"><mrow id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.cmml"><msub id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.2" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.2.cmml"><mi id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.2.2" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.2.2.cmml">φ</mi><mi id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.2.3" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.2.3.cmml">k</mi></msub><mo id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.1.cmml">∗</mo><msub id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.3" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.3.cmml"><mi id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.3.2" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.3.2.cmml">s</mi><mi id="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.3.3" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.2.3.3.cmml">n</mi></msub></mrow><mo id="S3.SS2.5.p5.5.m1.1.1.1.1.1.1" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.1.cmml">⁢</mo><mi id="S3.SS2.5.p5.5.m1.1.1.1.1.1.3" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.3.cmml">g</mi></mrow><mo id="S3.SS2.5.p5.5.m1.1.1.1.1.3" stretchy="false" xref="S3.SS2.5.p5.5.m1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S3.SS2.5.p5.5.m1.1.1.3" xref="S3.SS2.5.p5.5.m1.1.1.3.cmml"><mi id="S3.SS2.5.p5.5.m1.1.1.3.2" xref="S3.SS2.5.p5.5.m1.1.1.3.2.cmml">n</mi><mo id="S3.SS2.5.p5.5.m1.1.1.3.1" xref="S3.SS2.5.p5.5.m1.1.1.3.1.cmml">≥</mo><mn 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id="S3.SS2.5.p5.5.m1.1c">(\varphi_{k}\ast s_{n}g)_{n\geq 0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.5.m1.1d">( italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g ) start_POSTSUBSCRIPT italic_n ≥ 0 end_POSTSUBSCRIPT</annotation></semantics></math> is a Cauchy sequence in <math alttext="L^{p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS2.5.p5.6.m2.3"><semantics id="S3.SS2.5.p5.6.m2.3a"><mrow id="S3.SS2.5.p5.6.m2.3.3" xref="S3.SS2.5.p5.6.m2.3.3.cmml"><msup id="S3.SS2.5.p5.6.m2.3.3.4" xref="S3.SS2.5.p5.6.m2.3.3.4.cmml"><mi id="S3.SS2.5.p5.6.m2.3.3.4.2" xref="S3.SS2.5.p5.6.m2.3.3.4.2.cmml">L</mi><mi id="S3.SS2.5.p5.6.m2.3.3.4.3" xref="S3.SS2.5.p5.6.m2.3.3.4.3.cmml">p</mi></msup><mo id="S3.SS2.5.p5.6.m2.3.3.3" xref="S3.SS2.5.p5.6.m2.3.3.3.cmml">⁢</mo><mrow id="S3.SS2.5.p5.6.m2.3.3.2.2" xref="S3.SS2.5.p5.6.m2.3.3.2.3.cmml"><mo id="S3.SS2.5.p5.6.m2.3.3.2.2.3" 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id="S3.SS2.5.p5.6.m2.3.3.cmml" xref="S3.SS2.5.p5.6.m2.3.3"><times id="S3.SS2.5.p5.6.m2.3.3.3.cmml" xref="S3.SS2.5.p5.6.m2.3.3.3"></times><apply id="S3.SS2.5.p5.6.m2.3.3.4.cmml" xref="S3.SS2.5.p5.6.m2.3.3.4"><csymbol cd="ambiguous" id="S3.SS2.5.p5.6.m2.3.3.4.1.cmml" xref="S3.SS2.5.p5.6.m2.3.3.4">superscript</csymbol><ci id="S3.SS2.5.p5.6.m2.3.3.4.2.cmml" xref="S3.SS2.5.p5.6.m2.3.3.4.2">𝐿</ci><ci id="S3.SS2.5.p5.6.m2.3.3.4.3.cmml" xref="S3.SS2.5.p5.6.m2.3.3.4.3">𝑝</ci></apply><vector id="S3.SS2.5.p5.6.m2.3.3.2.3.cmml" xref="S3.SS2.5.p5.6.m2.3.3.2.2"><apply id="S3.SS2.5.p5.6.m2.2.2.1.1.1.cmml" xref="S3.SS2.5.p5.6.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.6.m2.2.2.1.1.1.1.cmml" xref="S3.SS2.5.p5.6.m2.2.2.1.1.1">superscript</csymbol><ci id="S3.SS2.5.p5.6.m2.2.2.1.1.1.2.cmml" xref="S3.SS2.5.p5.6.m2.2.2.1.1.1.2">ℝ</ci><ci id="S3.SS2.5.p5.6.m2.2.2.1.1.1.3.cmml" xref="S3.SS2.5.p5.6.m2.2.2.1.1.1.3">𝑑</ci></apply><apply id="S3.SS2.5.p5.6.m2.3.3.2.2.2.cmml" xref="S3.SS2.5.p5.6.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.6.m2.3.3.2.2.2.1.cmml" xref="S3.SS2.5.p5.6.m2.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.5.p5.6.m2.3.3.2.2.2.2.cmml" xref="S3.SS2.5.p5.6.m2.3.3.2.2.2.2">𝑤</ci><ci id="S3.SS2.5.p5.6.m2.3.3.2.2.2.3.cmml" xref="S3.SS2.5.p5.6.m2.3.3.2.2.2.3">𝛾</ci></apply><ci id="S3.SS2.5.p5.6.m2.1.1.cmml" xref="S3.SS2.5.p5.6.m2.1.1">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.6.m2.3c">L^{p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.6.m2.3d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for all <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.SS2.5.p5.7.m3.1"><semantics id="S3.SS2.5.p5.7.m3.1a"><mrow id="S3.SS2.5.p5.7.m3.1.1" xref="S3.SS2.5.p5.7.m3.1.1.cmml"><mi id="S3.SS2.5.p5.7.m3.1.1.2" xref="S3.SS2.5.p5.7.m3.1.1.2.cmml">k</mi><mo id="S3.SS2.5.p5.7.m3.1.1.1" xref="S3.SS2.5.p5.7.m3.1.1.1.cmml">∈</mo><msub id="S3.SS2.5.p5.7.m3.1.1.3" xref="S3.SS2.5.p5.7.m3.1.1.3.cmml"><mi id="S3.SS2.5.p5.7.m3.1.1.3.2" xref="S3.SS2.5.p5.7.m3.1.1.3.2.cmml">ℕ</mi><mn id="S3.SS2.5.p5.7.m3.1.1.3.3" xref="S3.SS2.5.p5.7.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.7.m3.1b"><apply id="S3.SS2.5.p5.7.m3.1.1.cmml" xref="S3.SS2.5.p5.7.m3.1.1"><in id="S3.SS2.5.p5.7.m3.1.1.1.cmml" xref="S3.SS2.5.p5.7.m3.1.1.1"></in><ci id="S3.SS2.5.p5.7.m3.1.1.2.cmml" xref="S3.SS2.5.p5.7.m3.1.1.2">𝑘</ci><apply id="S3.SS2.5.p5.7.m3.1.1.3.cmml" xref="S3.SS2.5.p5.7.m3.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.7.m3.1.1.3.1.cmml" xref="S3.SS2.5.p5.7.m3.1.1.3">subscript</csymbol><ci id="S3.SS2.5.p5.7.m3.1.1.3.2.cmml" xref="S3.SS2.5.p5.7.m3.1.1.3.2">ℕ</ci><cn id="S3.SS2.5.p5.7.m3.1.1.3.3.cmml" type="integer" xref="S3.SS2.5.p5.7.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.7.m3.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.7.m3.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, its limit <math alttext="w_{k}=\lim_{n\to\infty}\varphi_{k}\ast s_{n}g" class="ltx_Math" display="inline" id="S3.SS2.5.p5.8.m4.1"><semantics id="S3.SS2.5.p5.8.m4.1a"><mrow id="S3.SS2.5.p5.8.m4.1.1" xref="S3.SS2.5.p5.8.m4.1.1.cmml"><msub id="S3.SS2.5.p5.8.m4.1.1.2" xref="S3.SS2.5.p5.8.m4.1.1.2.cmml"><mi id="S3.SS2.5.p5.8.m4.1.1.2.2" xref="S3.SS2.5.p5.8.m4.1.1.2.2.cmml">w</mi><mi id="S3.SS2.5.p5.8.m4.1.1.2.3" xref="S3.SS2.5.p5.8.m4.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS2.5.p5.8.m4.1.1.1" rspace="0.1389em" xref="S3.SS2.5.p5.8.m4.1.1.1.cmml">=</mo><mrow id="S3.SS2.5.p5.8.m4.1.1.3" xref="S3.SS2.5.p5.8.m4.1.1.3.cmml"><msub id="S3.SS2.5.p5.8.m4.1.1.3.1" xref="S3.SS2.5.p5.8.m4.1.1.3.1.cmml"><mo id="S3.SS2.5.p5.8.m4.1.1.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S3.SS2.5.p5.8.m4.1.1.3.1.2.cmml">lim</mo><mrow id="S3.SS2.5.p5.8.m4.1.1.3.1.3" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3.cmml"><mi id="S3.SS2.5.p5.8.m4.1.1.3.1.3.2" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3.2.cmml">n</mi><mo id="S3.SS2.5.p5.8.m4.1.1.3.1.3.1" stretchy="false" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3.1.cmml">→</mo><mi id="S3.SS2.5.p5.8.m4.1.1.3.1.3.3" mathvariant="normal" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3.3.cmml">∞</mi></mrow></msub><mrow id="S3.SS2.5.p5.8.m4.1.1.3.2" xref="S3.SS2.5.p5.8.m4.1.1.3.2.cmml"><mrow id="S3.SS2.5.p5.8.m4.1.1.3.2.2" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.cmml"><msub id="S3.SS2.5.p5.8.m4.1.1.3.2.2.2" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.cmml"><mi id="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.2" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.2.cmml">φ</mi><mi id="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.3" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.3.cmml">k</mi></msub><mo id="S3.SS2.5.p5.8.m4.1.1.3.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.1.cmml">∗</mo><msub id="S3.SS2.5.p5.8.m4.1.1.3.2.2.3" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.cmml"><mi id="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.2" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.2.cmml">s</mi><mi id="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.3" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.3.cmml">n</mi></msub></mrow><mo id="S3.SS2.5.p5.8.m4.1.1.3.2.1" xref="S3.SS2.5.p5.8.m4.1.1.3.2.1.cmml">⁢</mo><mi id="S3.SS2.5.p5.8.m4.1.1.3.2.3" xref="S3.SS2.5.p5.8.m4.1.1.3.2.3.cmml">g</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.8.m4.1b"><apply id="S3.SS2.5.p5.8.m4.1.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1"><eq id="S3.SS2.5.p5.8.m4.1.1.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.1"></eq><apply id="S3.SS2.5.p5.8.m4.1.1.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.8.m4.1.1.2.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.2">subscript</csymbol><ci id="S3.SS2.5.p5.8.m4.1.1.2.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.2.2">𝑤</ci><ci id="S3.SS2.5.p5.8.m4.1.1.2.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.2.3">𝑘</ci></apply><apply id="S3.SS2.5.p5.8.m4.1.1.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3"><apply id="S3.SS2.5.p5.8.m4.1.1.3.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.8.m4.1.1.3.1.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.1">subscript</csymbol><limit id="S3.SS2.5.p5.8.m4.1.1.3.1.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.1.2"></limit><apply id="S3.SS2.5.p5.8.m4.1.1.3.1.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3"><ci id="S3.SS2.5.p5.8.m4.1.1.3.1.3.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3.1">→</ci><ci id="S3.SS2.5.p5.8.m4.1.1.3.1.3.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3.2">𝑛</ci><infinity id="S3.SS2.5.p5.8.m4.1.1.3.1.3.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.1.3.3"></infinity></apply></apply><apply id="S3.SS2.5.p5.8.m4.1.1.3.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2"><times id="S3.SS2.5.p5.8.m4.1.1.3.2.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.1"></times><apply id="S3.SS2.5.p5.8.m4.1.1.3.2.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2"><ci id="S3.SS2.5.p5.8.m4.1.1.3.2.2.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.1">∗</ci><apply id="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.2">subscript</csymbol><ci id="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.2">𝜑</ci><ci id="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.2.3">𝑘</ci></apply><apply id="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.1.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.3">subscript</csymbol><ci id="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.2.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.2">𝑠</ci><ci id="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.2.3.3">𝑛</ci></apply></apply><ci id="S3.SS2.5.p5.8.m4.1.1.3.2.3.cmml" xref="S3.SS2.5.p5.8.m4.1.1.3.2.3">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.8.m4.1c">w_{k}=\lim_{n\to\infty}\varphi_{k}\ast s_{n}g</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.8.m4.1d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = roman_lim start_POSTSUBSCRIPT italic_n → ∞ end_POSTSUBSCRIPT italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g</annotation></semantics></math> exists in <math alttext="L^{p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS2.5.p5.9.m5.3"><semantics id="S3.SS2.5.p5.9.m5.3a"><mrow id="S3.SS2.5.p5.9.m5.3.3" xref="S3.SS2.5.p5.9.m5.3.3.cmml"><msup id="S3.SS2.5.p5.9.m5.3.3.4" xref="S3.SS2.5.p5.9.m5.3.3.4.cmml"><mi id="S3.SS2.5.p5.9.m5.3.3.4.2" xref="S3.SS2.5.p5.9.m5.3.3.4.2.cmml">L</mi><mi id="S3.SS2.5.p5.9.m5.3.3.4.3" xref="S3.SS2.5.p5.9.m5.3.3.4.3.cmml">p</mi></msup><mo id="S3.SS2.5.p5.9.m5.3.3.3" xref="S3.SS2.5.p5.9.m5.3.3.3.cmml">⁢</mo><mrow id="S3.SS2.5.p5.9.m5.3.3.2.2" xref="S3.SS2.5.p5.9.m5.3.3.2.3.cmml"><mo id="S3.SS2.5.p5.9.m5.3.3.2.2.3" stretchy="false" xref="S3.SS2.5.p5.9.m5.3.3.2.3.cmml">(</mo><msup id="S3.SS2.5.p5.9.m5.2.2.1.1.1" xref="S3.SS2.5.p5.9.m5.2.2.1.1.1.cmml"><mi id="S3.SS2.5.p5.9.m5.2.2.1.1.1.2" xref="S3.SS2.5.p5.9.m5.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS2.5.p5.9.m5.2.2.1.1.1.3" xref="S3.SS2.5.p5.9.m5.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS2.5.p5.9.m5.3.3.2.2.4" xref="S3.SS2.5.p5.9.m5.3.3.2.3.cmml">,</mo><msub id="S3.SS2.5.p5.9.m5.3.3.2.2.2" xref="S3.SS2.5.p5.9.m5.3.3.2.2.2.cmml"><mi id="S3.SS2.5.p5.9.m5.3.3.2.2.2.2" xref="S3.SS2.5.p5.9.m5.3.3.2.2.2.2.cmml">w</mi><mi id="S3.SS2.5.p5.9.m5.3.3.2.2.2.3" xref="S3.SS2.5.p5.9.m5.3.3.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS2.5.p5.9.m5.3.3.2.2.5" xref="S3.SS2.5.p5.9.m5.3.3.2.3.cmml">;</mo><mi id="S3.SS2.5.p5.9.m5.1.1" xref="S3.SS2.5.p5.9.m5.1.1.cmml">X</mi><mo id="S3.SS2.5.p5.9.m5.3.3.2.2.6" stretchy="false" xref="S3.SS2.5.p5.9.m5.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.9.m5.3b"><apply id="S3.SS2.5.p5.9.m5.3.3.cmml" xref="S3.SS2.5.p5.9.m5.3.3"><times id="S3.SS2.5.p5.9.m5.3.3.3.cmml" xref="S3.SS2.5.p5.9.m5.3.3.3"></times><apply id="S3.SS2.5.p5.9.m5.3.3.4.cmml" xref="S3.SS2.5.p5.9.m5.3.3.4"><csymbol cd="ambiguous" id="S3.SS2.5.p5.9.m5.3.3.4.1.cmml" xref="S3.SS2.5.p5.9.m5.3.3.4">superscript</csymbol><ci id="S3.SS2.5.p5.9.m5.3.3.4.2.cmml" xref="S3.SS2.5.p5.9.m5.3.3.4.2">𝐿</ci><ci id="S3.SS2.5.p5.9.m5.3.3.4.3.cmml" xref="S3.SS2.5.p5.9.m5.3.3.4.3">𝑝</ci></apply><vector id="S3.SS2.5.p5.9.m5.3.3.2.3.cmml" xref="S3.SS2.5.p5.9.m5.3.3.2.2"><apply id="S3.SS2.5.p5.9.m5.2.2.1.1.1.cmml" xref="S3.SS2.5.p5.9.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.9.m5.2.2.1.1.1.1.cmml" xref="S3.SS2.5.p5.9.m5.2.2.1.1.1">superscript</csymbol><ci id="S3.SS2.5.p5.9.m5.2.2.1.1.1.2.cmml" xref="S3.SS2.5.p5.9.m5.2.2.1.1.1.2">ℝ</ci><ci id="S3.SS2.5.p5.9.m5.2.2.1.1.1.3.cmml" xref="S3.SS2.5.p5.9.m5.2.2.1.1.1.3">𝑑</ci></apply><apply id="S3.SS2.5.p5.9.m5.3.3.2.2.2.cmml" xref="S3.SS2.5.p5.9.m5.3.3.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.9.m5.3.3.2.2.2.1.cmml" xref="S3.SS2.5.p5.9.m5.3.3.2.2.2">subscript</csymbol><ci id="S3.SS2.5.p5.9.m5.3.3.2.2.2.2.cmml" xref="S3.SS2.5.p5.9.m5.3.3.2.2.2.2">𝑤</ci><ci id="S3.SS2.5.p5.9.m5.3.3.2.2.2.3.cmml" xref="S3.SS2.5.p5.9.m5.3.3.2.2.2.3">𝛾</ci></apply><ci id="S3.SS2.5.p5.9.m5.1.1.cmml" xref="S3.SS2.5.p5.9.m5.1.1">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.9.m5.3c">L^{p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.9.m5.3d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> and thus in <math alttext="\SS^{\prime}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S3.SS2.5.p5.10.m6.2"><semantics id="S3.SS2.5.p5.10.m6.2a"><mrow id="S3.SS2.5.p5.10.m6.2.2" xref="S3.SS2.5.p5.10.m6.2.2.cmml"><msup id="S3.SS2.5.p5.10.m6.2.2.3" xref="S3.SS2.5.p5.10.m6.2.2.3.cmml"><mi id="S3.SS2.5.p5.10.m6.2.2.3.2" xref="S3.SS2.5.p5.10.m6.2.2.3.2.cmml">SS</mi><mo id="S3.SS2.5.p5.10.m6.2.2.3.3" xref="S3.SS2.5.p5.10.m6.2.2.3.3.cmml">′</mo></msup><mo id="S3.SS2.5.p5.10.m6.2.2.2" xref="S3.SS2.5.p5.10.m6.2.2.2.cmml">⁢</mo><mrow id="S3.SS2.5.p5.10.m6.2.2.1.1" xref="S3.SS2.5.p5.10.m6.2.2.1.2.cmml"><mo id="S3.SS2.5.p5.10.m6.2.2.1.1.2" stretchy="false" xref="S3.SS2.5.p5.10.m6.2.2.1.2.cmml">(</mo><msup id="S3.SS2.5.p5.10.m6.2.2.1.1.1" xref="S3.SS2.5.p5.10.m6.2.2.1.1.1.cmml"><mi id="S3.SS2.5.p5.10.m6.2.2.1.1.1.2" xref="S3.SS2.5.p5.10.m6.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS2.5.p5.10.m6.2.2.1.1.1.3" xref="S3.SS2.5.p5.10.m6.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS2.5.p5.10.m6.2.2.1.1.3" xref="S3.SS2.5.p5.10.m6.2.2.1.2.cmml">;</mo><mi id="S3.SS2.5.p5.10.m6.1.1" xref="S3.SS2.5.p5.10.m6.1.1.cmml">X</mi><mo id="S3.SS2.5.p5.10.m6.2.2.1.1.4" stretchy="false" xref="S3.SS2.5.p5.10.m6.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.10.m6.2b"><apply id="S3.SS2.5.p5.10.m6.2.2.cmml" xref="S3.SS2.5.p5.10.m6.2.2"><times id="S3.SS2.5.p5.10.m6.2.2.2.cmml" xref="S3.SS2.5.p5.10.m6.2.2.2"></times><apply id="S3.SS2.5.p5.10.m6.2.2.3.cmml" xref="S3.SS2.5.p5.10.m6.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.10.m6.2.2.3.1.cmml" xref="S3.SS2.5.p5.10.m6.2.2.3">superscript</csymbol><ci id="S3.SS2.5.p5.10.m6.2.2.3.2.cmml" xref="S3.SS2.5.p5.10.m6.2.2.3.2">SS</ci><ci id="S3.SS2.5.p5.10.m6.2.2.3.3.cmml" xref="S3.SS2.5.p5.10.m6.2.2.3.3">′</ci></apply><list id="S3.SS2.5.p5.10.m6.2.2.1.2.cmml" xref="S3.SS2.5.p5.10.m6.2.2.1.1"><apply id="S3.SS2.5.p5.10.m6.2.2.1.1.1.cmml" xref="S3.SS2.5.p5.10.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.10.m6.2.2.1.1.1.1.cmml" xref="S3.SS2.5.p5.10.m6.2.2.1.1.1">superscript</csymbol><ci id="S3.SS2.5.p5.10.m6.2.2.1.1.1.2.cmml" xref="S3.SS2.5.p5.10.m6.2.2.1.1.1.2">ℝ</ci><ci id="S3.SS2.5.p5.10.m6.2.2.1.1.1.3.cmml" xref="S3.SS2.5.p5.10.m6.2.2.1.1.1.3">𝑑</ci></apply><ci id="S3.SS2.5.p5.10.m6.1.1.cmml" xref="S3.SS2.5.p5.10.m6.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.10.m6.2c">\SS^{\prime}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.10.m6.2d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> as well. Since <math alttext="B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)\hookrightarrow\SS^{\prime}(\mathbb{R}% ^{d};X)" class="ltx_Math" display="inline" id="S3.SS2.5.p5.11.m7.7"><semantics id="S3.SS2.5.p5.11.m7.7a"><mrow id="S3.SS2.5.p5.11.m7.7.7" xref="S3.SS2.5.p5.11.m7.7.7.cmml"><mrow id="S3.SS2.5.p5.11.m7.6.6.2" xref="S3.SS2.5.p5.11.m7.6.6.2.cmml"><msubsup id="S3.SS2.5.p5.11.m7.6.6.2.4" xref="S3.SS2.5.p5.11.m7.6.6.2.4.cmml"><mi id="S3.SS2.5.p5.11.m7.6.6.2.4.2.2" xref="S3.SS2.5.p5.11.m7.6.6.2.4.2.2.cmml">B</mi><mrow id="S3.SS2.5.p5.11.m7.2.2.2.4" xref="S3.SS2.5.p5.11.m7.2.2.2.3.cmml"><mi id="S3.SS2.5.p5.11.m7.1.1.1.1" xref="S3.SS2.5.p5.11.m7.1.1.1.1.cmml">p</mi><mo id="S3.SS2.5.p5.11.m7.2.2.2.4.1" xref="S3.SS2.5.p5.11.m7.2.2.2.3.cmml">,</mo><mn id="S3.SS2.5.p5.11.m7.2.2.2.2" xref="S3.SS2.5.p5.11.m7.2.2.2.2.cmml">1</mn></mrow><mi id="S3.SS2.5.p5.11.m7.6.6.2.4.2.3" xref="S3.SS2.5.p5.11.m7.6.6.2.4.2.3.cmml">t</mi></msubsup><mo id="S3.SS2.5.p5.11.m7.6.6.2.3" xref="S3.SS2.5.p5.11.m7.6.6.2.3.cmml">⁢</mo><mrow id="S3.SS2.5.p5.11.m7.6.6.2.2.2" xref="S3.SS2.5.p5.11.m7.6.6.2.2.3.cmml"><mo id="S3.SS2.5.p5.11.m7.6.6.2.2.2.3" stretchy="false" xref="S3.SS2.5.p5.11.m7.6.6.2.2.3.cmml">(</mo><msup id="S3.SS2.5.p5.11.m7.5.5.1.1.1.1" xref="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.cmml"><mi id="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.2" xref="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.3" xref="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS2.5.p5.11.m7.6.6.2.2.2.4" xref="S3.SS2.5.p5.11.m7.6.6.2.2.3.cmml">,</mo><msub id="S3.SS2.5.p5.11.m7.6.6.2.2.2.2" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.cmml"><mi id="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.2" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.2.cmml">w</mi><mi id="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.3" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.SS2.5.p5.11.m7.6.6.2.2.2.5" xref="S3.SS2.5.p5.11.m7.6.6.2.2.3.cmml">;</mo><mi id="S3.SS2.5.p5.11.m7.3.3" xref="S3.SS2.5.p5.11.m7.3.3.cmml">X</mi><mo id="S3.SS2.5.p5.11.m7.6.6.2.2.2.6" stretchy="false" xref="S3.SS2.5.p5.11.m7.6.6.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.SS2.5.p5.11.m7.7.7.4" stretchy="false" xref="S3.SS2.5.p5.11.m7.7.7.4.cmml">↪</mo><mrow id="S3.SS2.5.p5.11.m7.7.7.3" xref="S3.SS2.5.p5.11.m7.7.7.3.cmml"><msup id="S3.SS2.5.p5.11.m7.7.7.3.3" xref="S3.SS2.5.p5.11.m7.7.7.3.3.cmml"><mi id="S3.SS2.5.p5.11.m7.7.7.3.3.2" xref="S3.SS2.5.p5.11.m7.7.7.3.3.2.cmml">SS</mi><mo id="S3.SS2.5.p5.11.m7.7.7.3.3.3" xref="S3.SS2.5.p5.11.m7.7.7.3.3.3.cmml">′</mo></msup><mo id="S3.SS2.5.p5.11.m7.7.7.3.2" xref="S3.SS2.5.p5.11.m7.7.7.3.2.cmml">⁢</mo><mrow id="S3.SS2.5.p5.11.m7.7.7.3.1.1" xref="S3.SS2.5.p5.11.m7.7.7.3.1.2.cmml"><mo id="S3.SS2.5.p5.11.m7.7.7.3.1.1.2" stretchy="false" xref="S3.SS2.5.p5.11.m7.7.7.3.1.2.cmml">(</mo><msup id="S3.SS2.5.p5.11.m7.7.7.3.1.1.1" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.cmml"><mi id="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.2" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.2.cmml">ℝ</mi><mi id="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.3" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.3.cmml">d</mi></msup><mo id="S3.SS2.5.p5.11.m7.7.7.3.1.1.3" xref="S3.SS2.5.p5.11.m7.7.7.3.1.2.cmml">;</mo><mi id="S3.SS2.5.p5.11.m7.4.4" xref="S3.SS2.5.p5.11.m7.4.4.cmml">X</mi><mo id="S3.SS2.5.p5.11.m7.7.7.3.1.1.4" stretchy="false" xref="S3.SS2.5.p5.11.m7.7.7.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.11.m7.7b"><apply id="S3.SS2.5.p5.11.m7.7.7.cmml" xref="S3.SS2.5.p5.11.m7.7.7"><ci id="S3.SS2.5.p5.11.m7.7.7.4.cmml" xref="S3.SS2.5.p5.11.m7.7.7.4">↪</ci><apply id="S3.SS2.5.p5.11.m7.6.6.2.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2"><times id="S3.SS2.5.p5.11.m7.6.6.2.3.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.3"></times><apply id="S3.SS2.5.p5.11.m7.6.6.2.4.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.4"><csymbol cd="ambiguous" id="S3.SS2.5.p5.11.m7.6.6.2.4.1.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.4">subscript</csymbol><apply id="S3.SS2.5.p5.11.m7.6.6.2.4.2.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.4"><csymbol cd="ambiguous" id="S3.SS2.5.p5.11.m7.6.6.2.4.2.1.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.4">superscript</csymbol><ci id="S3.SS2.5.p5.11.m7.6.6.2.4.2.2.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.4.2.2">𝐵</ci><ci id="S3.SS2.5.p5.11.m7.6.6.2.4.2.3.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.4.2.3">𝑡</ci></apply><list id="S3.SS2.5.p5.11.m7.2.2.2.3.cmml" xref="S3.SS2.5.p5.11.m7.2.2.2.4"><ci id="S3.SS2.5.p5.11.m7.1.1.1.1.cmml" xref="S3.SS2.5.p5.11.m7.1.1.1.1">𝑝</ci><cn id="S3.SS2.5.p5.11.m7.2.2.2.2.cmml" type="integer" xref="S3.SS2.5.p5.11.m7.2.2.2.2">1</cn></list></apply><vector id="S3.SS2.5.p5.11.m7.6.6.2.2.3.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2"><apply id="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.cmml" xref="S3.SS2.5.p5.11.m7.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.1.cmml" xref="S3.SS2.5.p5.11.m7.5.5.1.1.1.1">superscript</csymbol><ci id="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.2.cmml" xref="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.2">ℝ</ci><ci id="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.3.cmml" xref="S3.SS2.5.p5.11.m7.5.5.1.1.1.1.3">𝑑</ci></apply><apply id="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.1.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2.2">subscript</csymbol><ci id="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.2.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.2">𝑤</ci><ci id="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.3.cmml" xref="S3.SS2.5.p5.11.m7.6.6.2.2.2.2.3">𝛾</ci></apply><ci id="S3.SS2.5.p5.11.m7.3.3.cmml" xref="S3.SS2.5.p5.11.m7.3.3">𝑋</ci></vector></apply><apply id="S3.SS2.5.p5.11.m7.7.7.3.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3"><times id="S3.SS2.5.p5.11.m7.7.7.3.2.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.2"></times><apply id="S3.SS2.5.p5.11.m7.7.7.3.3.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.11.m7.7.7.3.3.1.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.3">superscript</csymbol><ci id="S3.SS2.5.p5.11.m7.7.7.3.3.2.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.3.2">SS</ci><ci id="S3.SS2.5.p5.11.m7.7.7.3.3.3.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.3.3">′</ci></apply><list id="S3.SS2.5.p5.11.m7.7.7.3.1.2.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1"><apply id="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.1.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1.1">superscript</csymbol><ci id="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.2.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.2">ℝ</ci><ci id="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.3.cmml" xref="S3.SS2.5.p5.11.m7.7.7.3.1.1.1.3">𝑑</ci></apply><ci id="S3.SS2.5.p5.11.m7.4.4.cmml" xref="S3.SS2.5.p5.11.m7.4.4">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.11.m7.7c">B^{t}_{p,1}(\mathbb{R}^{d},w_{\gamma};X)\hookrightarrow\SS^{\prime}(\mathbb{R}% ^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.11.m7.7d">italic_B start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ↪ roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>, Step 2<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.I2.i2" title="item ii ‣ Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> implies that <math alttext="s_{n}g\to\operatorname{ext}_{m}g" class="ltx_Math" display="inline" id="S3.SS2.5.p5.12.m8.1"><semantics id="S3.SS2.5.p5.12.m8.1a"><mrow id="S3.SS2.5.p5.12.m8.1.1" xref="S3.SS2.5.p5.12.m8.1.1.cmml"><mrow id="S3.SS2.5.p5.12.m8.1.1.2" xref="S3.SS2.5.p5.12.m8.1.1.2.cmml"><msub id="S3.SS2.5.p5.12.m8.1.1.2.2" xref="S3.SS2.5.p5.12.m8.1.1.2.2.cmml"><mi id="S3.SS2.5.p5.12.m8.1.1.2.2.2" xref="S3.SS2.5.p5.12.m8.1.1.2.2.2.cmml">s</mi><mi id="S3.SS2.5.p5.12.m8.1.1.2.2.3" xref="S3.SS2.5.p5.12.m8.1.1.2.2.3.cmml">n</mi></msub><mo id="S3.SS2.5.p5.12.m8.1.1.2.1" xref="S3.SS2.5.p5.12.m8.1.1.2.1.cmml">⁢</mo><mi id="S3.SS2.5.p5.12.m8.1.1.2.3" xref="S3.SS2.5.p5.12.m8.1.1.2.3.cmml">g</mi></mrow><mo id="S3.SS2.5.p5.12.m8.1.1.1" stretchy="false" xref="S3.SS2.5.p5.12.m8.1.1.1.cmml">→</mo><mrow id="S3.SS2.5.p5.12.m8.1.1.3" xref="S3.SS2.5.p5.12.m8.1.1.3.cmml"><msub id="S3.SS2.5.p5.12.m8.1.1.3.1" xref="S3.SS2.5.p5.12.m8.1.1.3.1.cmml"><mi id="S3.SS2.5.p5.12.m8.1.1.3.1.2" xref="S3.SS2.5.p5.12.m8.1.1.3.1.2.cmml">ext</mi><mi id="S3.SS2.5.p5.12.m8.1.1.3.1.3" xref="S3.SS2.5.p5.12.m8.1.1.3.1.3.cmml">m</mi></msub><mo id="S3.SS2.5.p5.12.m8.1.1.3a" lspace="0.167em" xref="S3.SS2.5.p5.12.m8.1.1.3.cmml">⁡</mo><mi id="S3.SS2.5.p5.12.m8.1.1.3.2" xref="S3.SS2.5.p5.12.m8.1.1.3.2.cmml">g</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.12.m8.1b"><apply id="S3.SS2.5.p5.12.m8.1.1.cmml" xref="S3.SS2.5.p5.12.m8.1.1"><ci id="S3.SS2.5.p5.12.m8.1.1.1.cmml" xref="S3.SS2.5.p5.12.m8.1.1.1">→</ci><apply id="S3.SS2.5.p5.12.m8.1.1.2.cmml" xref="S3.SS2.5.p5.12.m8.1.1.2"><times id="S3.SS2.5.p5.12.m8.1.1.2.1.cmml" xref="S3.SS2.5.p5.12.m8.1.1.2.1"></times><apply id="S3.SS2.5.p5.12.m8.1.1.2.2.cmml" xref="S3.SS2.5.p5.12.m8.1.1.2.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.12.m8.1.1.2.2.1.cmml" xref="S3.SS2.5.p5.12.m8.1.1.2.2">subscript</csymbol><ci id="S3.SS2.5.p5.12.m8.1.1.2.2.2.cmml" xref="S3.SS2.5.p5.12.m8.1.1.2.2.2">𝑠</ci><ci id="S3.SS2.5.p5.12.m8.1.1.2.2.3.cmml" xref="S3.SS2.5.p5.12.m8.1.1.2.2.3">𝑛</ci></apply><ci id="S3.SS2.5.p5.12.m8.1.1.2.3.cmml" xref="S3.SS2.5.p5.12.m8.1.1.2.3">𝑔</ci></apply><apply id="S3.SS2.5.p5.12.m8.1.1.3.cmml" xref="S3.SS2.5.p5.12.m8.1.1.3"><apply id="S3.SS2.5.p5.12.m8.1.1.3.1.cmml" xref="S3.SS2.5.p5.12.m8.1.1.3.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.12.m8.1.1.3.1.1.cmml" xref="S3.SS2.5.p5.12.m8.1.1.3.1">subscript</csymbol><ci id="S3.SS2.5.p5.12.m8.1.1.3.1.2.cmml" xref="S3.SS2.5.p5.12.m8.1.1.3.1.2">ext</ci><ci id="S3.SS2.5.p5.12.m8.1.1.3.1.3.cmml" xref="S3.SS2.5.p5.12.m8.1.1.3.1.3">𝑚</ci></apply><ci id="S3.SS2.5.p5.12.m8.1.1.3.2.cmml" xref="S3.SS2.5.p5.12.m8.1.1.3.2">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.12.m8.1c">s_{n}g\to\operatorname{ext}_{m}g</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.12.m8.1d">italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g → roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g</annotation></semantics></math>, and thus also <math alttext="\varphi_{k}\ast s_{n}g\to\varphi_{k}\ast\operatorname{ext}_{m}g" class="ltx_Math" display="inline" id="S3.SS2.5.p5.13.m9.1"><semantics id="S3.SS2.5.p5.13.m9.1a"><mrow id="S3.SS2.5.p5.13.m9.1.1" xref="S3.SS2.5.p5.13.m9.1.1.cmml"><mrow id="S3.SS2.5.p5.13.m9.1.1.2" xref="S3.SS2.5.p5.13.m9.1.1.2.cmml"><mrow id="S3.SS2.5.p5.13.m9.1.1.2.2" xref="S3.SS2.5.p5.13.m9.1.1.2.2.cmml"><msub id="S3.SS2.5.p5.13.m9.1.1.2.2.2" xref="S3.SS2.5.p5.13.m9.1.1.2.2.2.cmml"><mi id="S3.SS2.5.p5.13.m9.1.1.2.2.2.2" xref="S3.SS2.5.p5.13.m9.1.1.2.2.2.2.cmml">φ</mi><mi id="S3.SS2.5.p5.13.m9.1.1.2.2.2.3" xref="S3.SS2.5.p5.13.m9.1.1.2.2.2.3.cmml">k</mi></msub><mo id="S3.SS2.5.p5.13.m9.1.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.5.p5.13.m9.1.1.2.2.1.cmml">∗</mo><msub id="S3.SS2.5.p5.13.m9.1.1.2.2.3" xref="S3.SS2.5.p5.13.m9.1.1.2.2.3.cmml"><mi id="S3.SS2.5.p5.13.m9.1.1.2.2.3.2" xref="S3.SS2.5.p5.13.m9.1.1.2.2.3.2.cmml">s</mi><mi id="S3.SS2.5.p5.13.m9.1.1.2.2.3.3" xref="S3.SS2.5.p5.13.m9.1.1.2.2.3.3.cmml">n</mi></msub></mrow><mo id="S3.SS2.5.p5.13.m9.1.1.2.1" xref="S3.SS2.5.p5.13.m9.1.1.2.1.cmml">⁢</mo><mi id="S3.SS2.5.p5.13.m9.1.1.2.3" xref="S3.SS2.5.p5.13.m9.1.1.2.3.cmml">g</mi></mrow><mo id="S3.SS2.5.p5.13.m9.1.1.1" stretchy="false" xref="S3.SS2.5.p5.13.m9.1.1.1.cmml">→</mo><mrow id="S3.SS2.5.p5.13.m9.1.1.3" xref="S3.SS2.5.p5.13.m9.1.1.3.cmml"><msub id="S3.SS2.5.p5.13.m9.1.1.3.2" xref="S3.SS2.5.p5.13.m9.1.1.3.2.cmml"><mi id="S3.SS2.5.p5.13.m9.1.1.3.2.2" xref="S3.SS2.5.p5.13.m9.1.1.3.2.2.cmml">φ</mi><mi id="S3.SS2.5.p5.13.m9.1.1.3.2.3" xref="S3.SS2.5.p5.13.m9.1.1.3.2.3.cmml">k</mi></msub><mo id="S3.SS2.5.p5.13.m9.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.5.p5.13.m9.1.1.3.1.cmml">∗</mo><mrow id="S3.SS2.5.p5.13.m9.1.1.3.3" xref="S3.SS2.5.p5.13.m9.1.1.3.3.cmml"><msub id="S3.SS2.5.p5.13.m9.1.1.3.3.1" xref="S3.SS2.5.p5.13.m9.1.1.3.3.1.cmml"><mi id="S3.SS2.5.p5.13.m9.1.1.3.3.1.2" xref="S3.SS2.5.p5.13.m9.1.1.3.3.1.2.cmml">ext</mi><mi id="S3.SS2.5.p5.13.m9.1.1.3.3.1.3" xref="S3.SS2.5.p5.13.m9.1.1.3.3.1.3.cmml">m</mi></msub><mo id="S3.SS2.5.p5.13.m9.1.1.3.3a" lspace="0.167em" xref="S3.SS2.5.p5.13.m9.1.1.3.3.cmml">⁡</mo><mi id="S3.SS2.5.p5.13.m9.1.1.3.3.2" xref="S3.SS2.5.p5.13.m9.1.1.3.3.2.cmml">g</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.13.m9.1b"><apply id="S3.SS2.5.p5.13.m9.1.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1"><ci id="S3.SS2.5.p5.13.m9.1.1.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.1">→</ci><apply id="S3.SS2.5.p5.13.m9.1.1.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2"><times id="S3.SS2.5.p5.13.m9.1.1.2.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.1"></times><apply id="S3.SS2.5.p5.13.m9.1.1.2.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2"><ci id="S3.SS2.5.p5.13.m9.1.1.2.2.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.1">∗</ci><apply id="S3.SS2.5.p5.13.m9.1.1.2.2.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.13.m9.1.1.2.2.2.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.2">subscript</csymbol><ci id="S3.SS2.5.p5.13.m9.1.1.2.2.2.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.2.2">𝜑</ci><ci id="S3.SS2.5.p5.13.m9.1.1.2.2.2.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.2.3">𝑘</ci></apply><apply id="S3.SS2.5.p5.13.m9.1.1.2.2.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.13.m9.1.1.2.2.3.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.3">subscript</csymbol><ci id="S3.SS2.5.p5.13.m9.1.1.2.2.3.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.3.2">𝑠</ci><ci id="S3.SS2.5.p5.13.m9.1.1.2.2.3.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.2.3.3">𝑛</ci></apply></apply><ci id="S3.SS2.5.p5.13.m9.1.1.2.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.2.3">𝑔</ci></apply><apply id="S3.SS2.5.p5.13.m9.1.1.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3"><ci id="S3.SS2.5.p5.13.m9.1.1.3.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.1">∗</ci><apply id="S3.SS2.5.p5.13.m9.1.1.3.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.13.m9.1.1.3.2.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.2">subscript</csymbol><ci id="S3.SS2.5.p5.13.m9.1.1.3.2.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.2.2">𝜑</ci><ci id="S3.SS2.5.p5.13.m9.1.1.3.2.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.2.3">𝑘</ci></apply><apply id="S3.SS2.5.p5.13.m9.1.1.3.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.3"><apply id="S3.SS2.5.p5.13.m9.1.1.3.3.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.3.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.13.m9.1.1.3.3.1.1.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.3.1">subscript</csymbol><ci id="S3.SS2.5.p5.13.m9.1.1.3.3.1.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.3.1.2">ext</ci><ci id="S3.SS2.5.p5.13.m9.1.1.3.3.1.3.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.3.1.3">𝑚</ci></apply><ci id="S3.SS2.5.p5.13.m9.1.1.3.3.2.cmml" xref="S3.SS2.5.p5.13.m9.1.1.3.3.2">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.13.m9.1c">\varphi_{k}\ast s_{n}g\to\varphi_{k}\ast\operatorname{ext}_{m}g</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.13.m9.1d">italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g → italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g</annotation></semantics></math>, both in <math alttext="\SS^{\prime}(\mathbb{R};X)" class="ltx_Math" display="inline" id="S3.SS2.5.p5.14.m10.2"><semantics id="S3.SS2.5.p5.14.m10.2a"><mrow id="S3.SS2.5.p5.14.m10.2.3" xref="S3.SS2.5.p5.14.m10.2.3.cmml"><msup id="S3.SS2.5.p5.14.m10.2.3.2" xref="S3.SS2.5.p5.14.m10.2.3.2.cmml"><mi id="S3.SS2.5.p5.14.m10.2.3.2.2" xref="S3.SS2.5.p5.14.m10.2.3.2.2.cmml">SS</mi><mo id="S3.SS2.5.p5.14.m10.2.3.2.3" xref="S3.SS2.5.p5.14.m10.2.3.2.3.cmml">′</mo></msup><mo id="S3.SS2.5.p5.14.m10.2.3.1" xref="S3.SS2.5.p5.14.m10.2.3.1.cmml">⁢</mo><mrow id="S3.SS2.5.p5.14.m10.2.3.3.2" xref="S3.SS2.5.p5.14.m10.2.3.3.1.cmml"><mo id="S3.SS2.5.p5.14.m10.2.3.3.2.1" stretchy="false" xref="S3.SS2.5.p5.14.m10.2.3.3.1.cmml">(</mo><mi id="S3.SS2.5.p5.14.m10.1.1" xref="S3.SS2.5.p5.14.m10.1.1.cmml">ℝ</mi><mo id="S3.SS2.5.p5.14.m10.2.3.3.2.2" xref="S3.SS2.5.p5.14.m10.2.3.3.1.cmml">;</mo><mi id="S3.SS2.5.p5.14.m10.2.2" xref="S3.SS2.5.p5.14.m10.2.2.cmml">X</mi><mo id="S3.SS2.5.p5.14.m10.2.3.3.2.3" stretchy="false" xref="S3.SS2.5.p5.14.m10.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.14.m10.2b"><apply id="S3.SS2.5.p5.14.m10.2.3.cmml" xref="S3.SS2.5.p5.14.m10.2.3"><times id="S3.SS2.5.p5.14.m10.2.3.1.cmml" xref="S3.SS2.5.p5.14.m10.2.3.1"></times><apply id="S3.SS2.5.p5.14.m10.2.3.2.cmml" xref="S3.SS2.5.p5.14.m10.2.3.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.14.m10.2.3.2.1.cmml" xref="S3.SS2.5.p5.14.m10.2.3.2">superscript</csymbol><ci id="S3.SS2.5.p5.14.m10.2.3.2.2.cmml" xref="S3.SS2.5.p5.14.m10.2.3.2.2">SS</ci><ci id="S3.SS2.5.p5.14.m10.2.3.2.3.cmml" xref="S3.SS2.5.p5.14.m10.2.3.2.3">′</ci></apply><list id="S3.SS2.5.p5.14.m10.2.3.3.1.cmml" xref="S3.SS2.5.p5.14.m10.2.3.3.2"><ci id="S3.SS2.5.p5.14.m10.1.1.cmml" xref="S3.SS2.5.p5.14.m10.1.1">ℝ</ci><ci id="S3.SS2.5.p5.14.m10.2.2.cmml" xref="S3.SS2.5.p5.14.m10.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.14.m10.2c">\SS^{\prime}(\mathbb{R};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.14.m10.2d">roman_SS start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R ; italic_X )</annotation></semantics></math> as <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S3.SS2.5.p5.15.m11.1"><semantics id="S3.SS2.5.p5.15.m11.1a"><mrow id="S3.SS2.5.p5.15.m11.1.1" xref="S3.SS2.5.p5.15.m11.1.1.cmml"><mi id="S3.SS2.5.p5.15.m11.1.1.2" xref="S3.SS2.5.p5.15.m11.1.1.2.cmml">n</mi><mo id="S3.SS2.5.p5.15.m11.1.1.1" stretchy="false" xref="S3.SS2.5.p5.15.m11.1.1.1.cmml">→</mo><mi id="S3.SS2.5.p5.15.m11.1.1.3" mathvariant="normal" xref="S3.SS2.5.p5.15.m11.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.15.m11.1b"><apply id="S3.SS2.5.p5.15.m11.1.1.cmml" xref="S3.SS2.5.p5.15.m11.1.1"><ci id="S3.SS2.5.p5.15.m11.1.1.1.cmml" xref="S3.SS2.5.p5.15.m11.1.1.1">→</ci><ci id="S3.SS2.5.p5.15.m11.1.1.2.cmml" xref="S3.SS2.5.p5.15.m11.1.1.2">𝑛</ci><infinity id="S3.SS2.5.p5.15.m11.1.1.3.cmml" xref="S3.SS2.5.p5.15.m11.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.15.m11.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.15.m11.1d">italic_n → ∞</annotation></semantics></math>. We conclude that <math alttext="w_{k}=\varphi_{k}\ast\operatorname{ext}_{m}g" class="ltx_Math" display="inline" id="S3.SS2.5.p5.16.m12.1"><semantics id="S3.SS2.5.p5.16.m12.1a"><mrow id="S3.SS2.5.p5.16.m12.1.1" xref="S3.SS2.5.p5.16.m12.1.1.cmml"><msub id="S3.SS2.5.p5.16.m12.1.1.2" xref="S3.SS2.5.p5.16.m12.1.1.2.cmml"><mi id="S3.SS2.5.p5.16.m12.1.1.2.2" xref="S3.SS2.5.p5.16.m12.1.1.2.2.cmml">w</mi><mi id="S3.SS2.5.p5.16.m12.1.1.2.3" xref="S3.SS2.5.p5.16.m12.1.1.2.3.cmml">k</mi></msub><mo id="S3.SS2.5.p5.16.m12.1.1.1" xref="S3.SS2.5.p5.16.m12.1.1.1.cmml">=</mo><mrow id="S3.SS2.5.p5.16.m12.1.1.3" xref="S3.SS2.5.p5.16.m12.1.1.3.cmml"><msub id="S3.SS2.5.p5.16.m12.1.1.3.2" xref="S3.SS2.5.p5.16.m12.1.1.3.2.cmml"><mi id="S3.SS2.5.p5.16.m12.1.1.3.2.2" xref="S3.SS2.5.p5.16.m12.1.1.3.2.2.cmml">φ</mi><mi id="S3.SS2.5.p5.16.m12.1.1.3.2.3" xref="S3.SS2.5.p5.16.m12.1.1.3.2.3.cmml">k</mi></msub><mo id="S3.SS2.5.p5.16.m12.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.SS2.5.p5.16.m12.1.1.3.1.cmml">∗</mo><mrow id="S3.SS2.5.p5.16.m12.1.1.3.3" xref="S3.SS2.5.p5.16.m12.1.1.3.3.cmml"><msub id="S3.SS2.5.p5.16.m12.1.1.3.3.1" xref="S3.SS2.5.p5.16.m12.1.1.3.3.1.cmml"><mi id="S3.SS2.5.p5.16.m12.1.1.3.3.1.2" xref="S3.SS2.5.p5.16.m12.1.1.3.3.1.2.cmml">ext</mi><mi id="S3.SS2.5.p5.16.m12.1.1.3.3.1.3" xref="S3.SS2.5.p5.16.m12.1.1.3.3.1.3.cmml">m</mi></msub><mo id="S3.SS2.5.p5.16.m12.1.1.3.3a" lspace="0.167em" xref="S3.SS2.5.p5.16.m12.1.1.3.3.cmml">⁡</mo><mi id="S3.SS2.5.p5.16.m12.1.1.3.3.2" xref="S3.SS2.5.p5.16.m12.1.1.3.3.2.cmml">g</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.16.m12.1b"><apply id="S3.SS2.5.p5.16.m12.1.1.cmml" xref="S3.SS2.5.p5.16.m12.1.1"><eq id="S3.SS2.5.p5.16.m12.1.1.1.cmml" xref="S3.SS2.5.p5.16.m12.1.1.1"></eq><apply id="S3.SS2.5.p5.16.m12.1.1.2.cmml" xref="S3.SS2.5.p5.16.m12.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.16.m12.1.1.2.1.cmml" xref="S3.SS2.5.p5.16.m12.1.1.2">subscript</csymbol><ci id="S3.SS2.5.p5.16.m12.1.1.2.2.cmml" xref="S3.SS2.5.p5.16.m12.1.1.2.2">𝑤</ci><ci id="S3.SS2.5.p5.16.m12.1.1.2.3.cmml" xref="S3.SS2.5.p5.16.m12.1.1.2.3">𝑘</ci></apply><apply id="S3.SS2.5.p5.16.m12.1.1.3.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3"><ci id="S3.SS2.5.p5.16.m12.1.1.3.1.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.1">∗</ci><apply id="S3.SS2.5.p5.16.m12.1.1.3.2.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.2"><csymbol cd="ambiguous" id="S3.SS2.5.p5.16.m12.1.1.3.2.1.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.2">subscript</csymbol><ci id="S3.SS2.5.p5.16.m12.1.1.3.2.2.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.2.2">𝜑</ci><ci id="S3.SS2.5.p5.16.m12.1.1.3.2.3.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.2.3">𝑘</ci></apply><apply id="S3.SS2.5.p5.16.m12.1.1.3.3.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.3"><apply id="S3.SS2.5.p5.16.m12.1.1.3.3.1.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.3.1"><csymbol cd="ambiguous" id="S3.SS2.5.p5.16.m12.1.1.3.3.1.1.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.3.1">subscript</csymbol><ci id="S3.SS2.5.p5.16.m12.1.1.3.3.1.2.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.3.1.2">ext</ci><ci id="S3.SS2.5.p5.16.m12.1.1.3.3.1.3.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.3.1.3">𝑚</ci></apply><ci id="S3.SS2.5.p5.16.m12.1.1.3.3.2.cmml" xref="S3.SS2.5.p5.16.m12.1.1.3.3.2">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.16.m12.1c">w_{k}=\varphi_{k}\ast\operatorname{ext}_{m}g</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.16.m12.1d">italic_w start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S3.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\varphi_{k}\ast s_{n}g\to\varphi_{k}\ast\operatorname{ext}_{m}g\quad\text{ in % }L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\,\text{ as }\,n\to\infty." class="ltx_Math" display="block" id="S3.E10.m1.2"><semantics id="S3.E10.m1.2a"><mrow id="S3.E10.m1.2.2.1"><mrow id="S3.E10.m1.2.2.1.1.2" xref="S3.E10.m1.2.2.1.1.3.cmml"><mrow id="S3.E10.m1.2.2.1.1.1.1" xref="S3.E10.m1.2.2.1.1.1.1.cmml"><mrow id="S3.E10.m1.2.2.1.1.1.1.2" xref="S3.E10.m1.2.2.1.1.1.1.2.cmml"><mrow id="S3.E10.m1.2.2.1.1.1.1.2.2" xref="S3.E10.m1.2.2.1.1.1.1.2.2.cmml"><msub id="S3.E10.m1.2.2.1.1.1.1.2.2.2" xref="S3.E10.m1.2.2.1.1.1.1.2.2.2.cmml"><mi id="S3.E10.m1.2.2.1.1.1.1.2.2.2.2" xref="S3.E10.m1.2.2.1.1.1.1.2.2.2.2.cmml">φ</mi><mi id="S3.E10.m1.2.2.1.1.1.1.2.2.2.3" xref="S3.E10.m1.2.2.1.1.1.1.2.2.2.3.cmml">k</mi></msub><mo id="S3.E10.m1.2.2.1.1.1.1.2.2.1" lspace="0.222em" rspace="0.222em" xref="S3.E10.m1.2.2.1.1.1.1.2.2.1.cmml">∗</mo><msub id="S3.E10.m1.2.2.1.1.1.1.2.2.3" xref="S3.E10.m1.2.2.1.1.1.1.2.2.3.cmml"><mi id="S3.E10.m1.2.2.1.1.1.1.2.2.3.2" xref="S3.E10.m1.2.2.1.1.1.1.2.2.3.2.cmml">s</mi><mi id="S3.E10.m1.2.2.1.1.1.1.2.2.3.3" xref="S3.E10.m1.2.2.1.1.1.1.2.2.3.3.cmml">n</mi></msub></mrow><mo id="S3.E10.m1.2.2.1.1.1.1.2.1" xref="S3.E10.m1.2.2.1.1.1.1.2.1.cmml">⁢</mo><mi id="S3.E10.m1.2.2.1.1.1.1.2.3" xref="S3.E10.m1.2.2.1.1.1.1.2.3.cmml">g</mi></mrow><mo id="S3.E10.m1.2.2.1.1.1.1.1" stretchy="false" xref="S3.E10.m1.2.2.1.1.1.1.1.cmml">→</mo><mrow id="S3.E10.m1.2.2.1.1.1.1.3" xref="S3.E10.m1.2.2.1.1.1.1.3.cmml"><msub id="S3.E10.m1.2.2.1.1.1.1.3.2" xref="S3.E10.m1.2.2.1.1.1.1.3.2.cmml"><mi id="S3.E10.m1.2.2.1.1.1.1.3.2.2" xref="S3.E10.m1.2.2.1.1.1.1.3.2.2.cmml">φ</mi><mi id="S3.E10.m1.2.2.1.1.1.1.3.2.3" xref="S3.E10.m1.2.2.1.1.1.1.3.2.3.cmml">k</mi></msub><mo id="S3.E10.m1.2.2.1.1.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S3.E10.m1.2.2.1.1.1.1.3.1.cmml">∗</mo><mrow id="S3.E10.m1.2.2.1.1.1.1.3.3" xref="S3.E10.m1.2.2.1.1.1.1.3.3.cmml"><msub id="S3.E10.m1.2.2.1.1.1.1.3.3.1" 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xref="S3.E10.m1.2.2.1.1.2.2.2.7">𝑛</ci></apply><infinity id="S3.E10.m1.2.2.1.1.2.2.4.cmml" xref="S3.E10.m1.2.2.1.1.2.2.4"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E10.m1.2c">\varphi_{k}\ast s_{n}g\to\varphi_{k}\ast\operatorname{ext}_{m}g\quad\text{ in % }L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\,\text{ as }\,n\to\infty.</annotation><annotation encoding="application/x-llamapun" id="S3.E10.m1.2d">italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_s start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g → italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g in italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) as italic_n → ∞ .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.5.p5.18">The convergence in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E10" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.10</span></a>) shows that we can let <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S3.SS2.5.p5.17.m1.1"><semantics id="S3.SS2.5.p5.17.m1.1a"><mrow id="S3.SS2.5.p5.17.m1.1.1" xref="S3.SS2.5.p5.17.m1.1.1.cmml"><mi id="S3.SS2.5.p5.17.m1.1.1.2" xref="S3.SS2.5.p5.17.m1.1.1.2.cmml">n</mi><mo id="S3.SS2.5.p5.17.m1.1.1.1" stretchy="false" xref="S3.SS2.5.p5.17.m1.1.1.1.cmml">→</mo><mi id="S3.SS2.5.p5.17.m1.1.1.3" mathvariant="normal" xref="S3.SS2.5.p5.17.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.17.m1.1b"><apply id="S3.SS2.5.p5.17.m1.1.1.cmml" xref="S3.SS2.5.p5.17.m1.1.1"><ci id="S3.SS2.5.p5.17.m1.1.1.1.cmml" xref="S3.SS2.5.p5.17.m1.1.1.1">→</ci><ci id="S3.SS2.5.p5.17.m1.1.1.2.cmml" xref="S3.SS2.5.p5.17.m1.1.1.2">𝑛</ci><infinity id="S3.SS2.5.p5.17.m1.1.1.3.cmml" xref="S3.SS2.5.p5.17.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.17.m1.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.17.m1.1d">italic_n → ∞</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E7" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.7</span></a>) to obtain for any <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.SS2.5.p5.18.m2.1"><semantics id="S3.SS2.5.p5.18.m2.1a"><mrow id="S3.SS2.5.p5.18.m2.1.1" xref="S3.SS2.5.p5.18.m2.1.1.cmml"><mi id="S3.SS2.5.p5.18.m2.1.1.2" xref="S3.SS2.5.p5.18.m2.1.1.2.cmml">k</mi><mo id="S3.SS2.5.p5.18.m2.1.1.1" xref="S3.SS2.5.p5.18.m2.1.1.1.cmml">∈</mo><msub id="S3.SS2.5.p5.18.m2.1.1.3" xref="S3.SS2.5.p5.18.m2.1.1.3.cmml"><mi id="S3.SS2.5.p5.18.m2.1.1.3.2" xref="S3.SS2.5.p5.18.m2.1.1.3.2.cmml">ℕ</mi><mn id="S3.SS2.5.p5.18.m2.1.1.3.3" xref="S3.SS2.5.p5.18.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.5.p5.18.m2.1b"><apply id="S3.SS2.5.p5.18.m2.1.1.cmml" xref="S3.SS2.5.p5.18.m2.1.1"><in id="S3.SS2.5.p5.18.m2.1.1.1.cmml" xref="S3.SS2.5.p5.18.m2.1.1.1"></in><ci id="S3.SS2.5.p5.18.m2.1.1.2.cmml" xref="S3.SS2.5.p5.18.m2.1.1.2">𝑘</ci><apply id="S3.SS2.5.p5.18.m2.1.1.3.cmml" xref="S3.SS2.5.p5.18.m2.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.5.p5.18.m2.1.1.3.1.cmml" xref="S3.SS2.5.p5.18.m2.1.1.3">subscript</csymbol><ci id="S3.SS2.5.p5.18.m2.1.1.3.2.cmml" xref="S3.SS2.5.p5.18.m2.1.1.3.2">ℕ</ci><cn id="S3.SS2.5.p5.18.m2.1.1.3.3.cmml" type="integer" xref="S3.SS2.5.p5.18.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.18.m2.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.18.m2.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex30"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="2^{ks}\|\varphi_{k}\ast\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^{d},w_{% \gamma};X)}\leq C\sum_{j=-1}^{1}2^{(k+j)(s-m-\frac{\gamma+1}{p})}\|\phi_{k+j}% \ast g\|_{L^{p}(\mathbb{R}^{d-1};X)}," class="ltx_Math" display="block" id="S3.Ex30.m1.8"><semantics id="S3.Ex30.m1.8a"><mrow id="S3.Ex30.m1.8.8.1" xref="S3.Ex30.m1.8.8.1.1.cmml"><mrow id="S3.Ex30.m1.8.8.1.1" xref="S3.Ex30.m1.8.8.1.1.cmml"><mrow id="S3.Ex30.m1.8.8.1.1.1" xref="S3.Ex30.m1.8.8.1.1.1.cmml"><msup id="S3.Ex30.m1.8.8.1.1.1.3" xref="S3.Ex30.m1.8.8.1.1.1.3.cmml"><mn id="S3.Ex30.m1.8.8.1.1.1.3.2" xref="S3.Ex30.m1.8.8.1.1.1.3.2.cmml">2</mn><mrow id="S3.Ex30.m1.8.8.1.1.1.3.3" xref="S3.Ex30.m1.8.8.1.1.1.3.3.cmml"><mi id="S3.Ex30.m1.8.8.1.1.1.3.3.2" xref="S3.Ex30.m1.8.8.1.1.1.3.3.2.cmml">k</mi><mo id="S3.Ex30.m1.8.8.1.1.1.3.3.1" xref="S3.Ex30.m1.8.8.1.1.1.3.3.1.cmml">⁢</mo><mi id="S3.Ex30.m1.8.8.1.1.1.3.3.3" xref="S3.Ex30.m1.8.8.1.1.1.3.3.3.cmml">s</mi></mrow></msup><mo id="S3.Ex30.m1.8.8.1.1.1.2" xref="S3.Ex30.m1.8.8.1.1.1.2.cmml">⁢</mo><msub id="S3.Ex30.m1.8.8.1.1.1.1" xref="S3.Ex30.m1.8.8.1.1.1.1.cmml"><mrow id="S3.Ex30.m1.8.8.1.1.1.1.1.1" xref="S3.Ex30.m1.8.8.1.1.1.1.1.2.cmml"><mo id="S3.Ex30.m1.8.8.1.1.1.1.1.1.2" stretchy="false" xref="S3.Ex30.m1.8.8.1.1.1.1.1.2.1.cmml">‖</mo><mrow id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.cmml"><msub id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.2" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.2.cmml"><mi id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.2.2" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.2.2.cmml">φ</mi><mi id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.2.3" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.2.3.cmml">k</mi></msub><mo id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.1" lspace="0.222em" rspace="0.222em" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.1.cmml">∗</mo><mrow id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.cmml"><msub id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.1" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.1.cmml"><mi id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.1.2" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.1.2.cmml">ext</mi><mi id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.1.3" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.1.3.cmml">m</mi></msub><mo id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3a" lspace="0.167em" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.cmml">⁡</mo><mi id="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.2" xref="S3.Ex30.m1.8.8.1.1.1.1.1.1.1.3.2.cmml">g</mi></mrow></mrow><mo id="S3.Ex30.m1.8.8.1.1.1.1.1.1.3" stretchy="false" xref="S3.Ex30.m1.8.8.1.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex30.m1.3.3.3" xref="S3.Ex30.m1.3.3.3.cmml"><msup id="S3.Ex30.m1.3.3.3.5" xref="S3.Ex30.m1.3.3.3.5.cmml"><mi id="S3.Ex30.m1.3.3.3.5.2" xref="S3.Ex30.m1.3.3.3.5.2.cmml">L</mi><mi id="S3.Ex30.m1.3.3.3.5.3" xref="S3.Ex30.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S3.Ex30.m1.3.3.3.4" xref="S3.Ex30.m1.3.3.3.4.cmml">⁢</mo><mrow id="S3.Ex30.m1.3.3.3.3.2" xref="S3.Ex30.m1.3.3.3.3.3.cmml"><mo id="S3.Ex30.m1.3.3.3.3.2.3" stretchy="false" xref="S3.Ex30.m1.3.3.3.3.3.cmml">(</mo><msup id="S3.Ex30.m1.2.2.2.2.1.1" xref="S3.Ex30.m1.2.2.2.2.1.1.cmml"><mi id="S3.Ex30.m1.2.2.2.2.1.1.2" xref="S3.Ex30.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mi id="S3.Ex30.m1.2.2.2.2.1.1.3" xref="S3.Ex30.m1.2.2.2.2.1.1.3.cmml">d</mi></msup><mo id="S3.Ex30.m1.3.3.3.3.2.4" xref="S3.Ex30.m1.3.3.3.3.3.cmml">,</mo><msub id="S3.Ex30.m1.3.3.3.3.2.2" xref="S3.Ex30.m1.3.3.3.3.2.2.cmml"><mi id="S3.Ex30.m1.3.3.3.3.2.2.2" xref="S3.Ex30.m1.3.3.3.3.2.2.2.cmml">w</mi><mi id="S3.Ex30.m1.3.3.3.3.2.2.3" xref="S3.Ex30.m1.3.3.3.3.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex30.m1.3.3.3.3.2.5" xref="S3.Ex30.m1.3.3.3.3.3.cmml">;</mo><mi id="S3.Ex30.m1.1.1.1.1" xref="S3.Ex30.m1.1.1.1.1.cmml">X</mi><mo id="S3.Ex30.m1.3.3.3.3.2.6" stretchy="false" xref="S3.Ex30.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow></msub></mrow><mo id="S3.Ex30.m1.8.8.1.1.3" xref="S3.Ex30.m1.8.8.1.1.3.cmml">≤</mo><mrow id="S3.Ex30.m1.8.8.1.1.2" xref="S3.Ex30.m1.8.8.1.1.2.cmml"><mi id="S3.Ex30.m1.8.8.1.1.2.3" xref="S3.Ex30.m1.8.8.1.1.2.3.cmml">C</mi><mo id="S3.Ex30.m1.8.8.1.1.2.2" xref="S3.Ex30.m1.8.8.1.1.2.2.cmml">⁢</mo><mrow id="S3.Ex30.m1.8.8.1.1.2.1" xref="S3.Ex30.m1.8.8.1.1.2.1.cmml"><munderover id="S3.Ex30.m1.8.8.1.1.2.1.2" xref="S3.Ex30.m1.8.8.1.1.2.1.2.cmml"><mo id="S3.Ex30.m1.8.8.1.1.2.1.2.2.2" movablelimits="false" xref="S3.Ex30.m1.8.8.1.1.2.1.2.2.2.cmml">∑</mo><mrow id="S3.Ex30.m1.8.8.1.1.2.1.2.2.3" xref="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.cmml"><mi id="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.2" xref="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.2.cmml">j</mi><mo id="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.1" xref="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.1.cmml">=</mo><mrow id="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.3" xref="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.3.cmml"><mo id="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.3a" xref="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.3.cmml">−</mo><mn id="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.3.2" xref="S3.Ex30.m1.8.8.1.1.2.1.2.2.3.3.2.cmml">1</mn></mrow></mrow><mn id="S3.Ex30.m1.8.8.1.1.2.1.2.3" xref="S3.Ex30.m1.8.8.1.1.2.1.2.3.cmml">1</mn></munderover><mrow id="S3.Ex30.m1.8.8.1.1.2.1.1" xref="S3.Ex30.m1.8.8.1.1.2.1.1.cmml"><msup id="S3.Ex30.m1.8.8.1.1.2.1.1.3" xref="S3.Ex30.m1.8.8.1.1.2.1.1.3.cmml"><mn id="S3.Ex30.m1.8.8.1.1.2.1.1.3.2" xref="S3.Ex30.m1.8.8.1.1.2.1.1.3.2.cmml">2</mn><mrow id="S3.Ex30.m1.5.5.2" xref="S3.Ex30.m1.5.5.2.cmml"><mrow id="S3.Ex30.m1.4.4.1.1.1" xref="S3.Ex30.m1.4.4.1.1.1.1.cmml"><mo id="S3.Ex30.m1.4.4.1.1.1.2" stretchy="false" xref="S3.Ex30.m1.4.4.1.1.1.1.cmml">(</mo><mrow id="S3.Ex30.m1.4.4.1.1.1.1" xref="S3.Ex30.m1.4.4.1.1.1.1.cmml"><mi id="S3.Ex30.m1.4.4.1.1.1.1.2" xref="S3.Ex30.m1.4.4.1.1.1.1.2.cmml">k</mi><mo id="S3.Ex30.m1.4.4.1.1.1.1.1" xref="S3.Ex30.m1.4.4.1.1.1.1.1.cmml">+</mo><mi id="S3.Ex30.m1.4.4.1.1.1.1.3" xref="S3.Ex30.m1.4.4.1.1.1.1.3.cmml">j</mi></mrow><mo id="S3.Ex30.m1.4.4.1.1.1.3" stretchy="false" xref="S3.Ex30.m1.4.4.1.1.1.1.cmml">)</mo></mrow><mo id="S3.Ex30.m1.5.5.2.3" xref="S3.Ex30.m1.5.5.2.3.cmml">⁢</mo><mrow id="S3.Ex30.m1.5.5.2.2.1" xref="S3.Ex30.m1.5.5.2.2.1.1.cmml"><mo id="S3.Ex30.m1.5.5.2.2.1.2" stretchy="false" 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C\sum_{j=-1}^{1}2^{(k+j)(s-m-\frac{\gamma+1}{p})}\|\phi_{k+j}% \ast g\|_{L^{p}(\mathbb{R}^{d-1};X)},</annotation><annotation encoding="application/x-llamapun" id="S3.Ex30.m1.8d">2 start_POSTSUPERSCRIPT italic_k italic_s end_POSTSUPERSCRIPT ∥ italic_φ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∑ start_POSTSUBSCRIPT italic_j = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT ( italic_k + italic_j ) ( italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_ϕ start_POSTSUBSCRIPT italic_k + italic_j end_POSTSUBSCRIPT ∗ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.5.p5.20">and therefore</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx12"> <tbody id="S3.Ex31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\operatorname{ext}_{m}g\|_{B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma% };X)}" class="ltx_Math" display="inline" id="S3.Ex31.m1.6"><semantics id="S3.Ex31.m1.6a"><msub id="S3.Ex31.m1.6.6" xref="S3.Ex31.m1.6.6.cmml"><mrow id="S3.Ex31.m1.6.6.1.1" xref="S3.Ex31.m1.6.6.1.2.cmml"><mo id="S3.Ex31.m1.6.6.1.1.2" stretchy="false" xref="S3.Ex31.m1.6.6.1.2.1.cmml">‖</mo><mrow id="S3.Ex31.m1.6.6.1.1.1" xref="S3.Ex31.m1.6.6.1.1.1.cmml"><msub id="S3.Ex31.m1.6.6.1.1.1.1" xref="S3.Ex31.m1.6.6.1.1.1.1.cmml"><mi id="S3.Ex31.m1.6.6.1.1.1.1.2" xref="S3.Ex31.m1.6.6.1.1.1.1.2.cmml">ext</mi><mi id="S3.Ex31.m1.6.6.1.1.1.1.3" xref="S3.Ex31.m1.6.6.1.1.1.1.3.cmml">m</mi></msub><mo id="S3.Ex31.m1.6.6.1.1.1a" lspace="0.167em" xref="S3.Ex31.m1.6.6.1.1.1.cmml">⁡</mo><mi id="S3.Ex31.m1.6.6.1.1.1.2" xref="S3.Ex31.m1.6.6.1.1.1.2.cmml">g</mi></mrow><mo id="S3.Ex31.m1.6.6.1.1.3" stretchy="false" xref="S3.Ex31.m1.6.6.1.2.1.cmml">‖</mo></mrow><mrow id="S3.Ex31.m1.5.5.5" xref="S3.Ex31.m1.5.5.5.cmml"><msubsup id="S3.Ex31.m1.5.5.5.7" xref="S3.Ex31.m1.5.5.5.7.cmml"><mi id="S3.Ex31.m1.5.5.5.7.2.2" xref="S3.Ex31.m1.5.5.5.7.2.2.cmml">B</mi><mrow id="S3.Ex31.m1.2.2.2.2.2.4" xref="S3.Ex31.m1.2.2.2.2.2.3.cmml"><mi id="S3.Ex31.m1.1.1.1.1.1.1" xref="S3.Ex31.m1.1.1.1.1.1.1.cmml">p</mi><mo id="S3.Ex31.m1.2.2.2.2.2.4.1" xref="S3.Ex31.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S3.Ex31.m1.2.2.2.2.2.2" xref="S3.Ex31.m1.2.2.2.2.2.2.cmml">q</mi></mrow><mi id="S3.Ex31.m1.5.5.5.7.2.3" xref="S3.Ex31.m1.5.5.5.7.2.3.cmml">s</mi></msubsup><mo id="S3.Ex31.m1.5.5.5.6" xref="S3.Ex31.m1.5.5.5.6.cmml">⁢</mo><mrow id="S3.Ex31.m1.5.5.5.5.2" xref="S3.Ex31.m1.5.5.5.5.3.cmml"><mo id="S3.Ex31.m1.5.5.5.5.2.3" stretchy="false" xref="S3.Ex31.m1.5.5.5.5.3.cmml">(</mo><msup id="S3.Ex31.m1.4.4.4.4.1.1" xref="S3.Ex31.m1.4.4.4.4.1.1.cmml"><mi id="S3.Ex31.m1.4.4.4.4.1.1.2" xref="S3.Ex31.m1.4.4.4.4.1.1.2.cmml">ℝ</mi><mi id="S3.Ex31.m1.4.4.4.4.1.1.3" xref="S3.Ex31.m1.4.4.4.4.1.1.3.cmml">d</mi></msup><mo id="S3.Ex31.m1.5.5.5.5.2.4" xref="S3.Ex31.m1.5.5.5.5.3.cmml">,</mo><msub id="S3.Ex31.m1.5.5.5.5.2.2" xref="S3.Ex31.m1.5.5.5.5.2.2.cmml"><mi id="S3.Ex31.m1.5.5.5.5.2.2.2" xref="S3.Ex31.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S3.Ex31.m1.5.5.5.5.2.2.3" xref="S3.Ex31.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex31.m1.5.5.5.5.2.5" xref="S3.Ex31.m1.5.5.5.5.3.cmml">;</mo><mi id="S3.Ex31.m1.3.3.3.3" xref="S3.Ex31.m1.3.3.3.3.cmml">X</mi><mo id="S3.Ex31.m1.5.5.5.5.2.6" stretchy="false" xref="S3.Ex31.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.Ex31.m1.6b"><apply id="S3.Ex31.m1.6.6.cmml" xref="S3.Ex31.m1.6.6"><csymbol cd="ambiguous" id="S3.Ex31.m1.6.6.2.cmml" xref="S3.Ex31.m1.6.6">subscript</csymbol><apply id="S3.Ex31.m1.6.6.1.2.cmml" xref="S3.Ex31.m1.6.6.1.1"><csymbol cd="latexml" id="S3.Ex31.m1.6.6.1.2.1.cmml" xref="S3.Ex31.m1.6.6.1.1.2">norm</csymbol><apply id="S3.Ex31.m1.6.6.1.1.1.cmml" xref="S3.Ex31.m1.6.6.1.1.1"><apply id="S3.Ex31.m1.6.6.1.1.1.1.cmml" xref="S3.Ex31.m1.6.6.1.1.1.1"><csymbol cd="ambiguous" id="S3.Ex31.m1.6.6.1.1.1.1.1.cmml" xref="S3.Ex31.m1.6.6.1.1.1.1">subscript</csymbol><ci id="S3.Ex31.m1.6.6.1.1.1.1.2.cmml" xref="S3.Ex31.m1.6.6.1.1.1.1.2">ext</ci><ci id="S3.Ex31.m1.6.6.1.1.1.1.3.cmml" xref="S3.Ex31.m1.6.6.1.1.1.1.3">𝑚</ci></apply><ci id="S3.Ex31.m1.6.6.1.1.1.2.cmml" 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xref="S3.Ex31.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S3.Ex31.m1.4.4.4.4.1.1.1.cmml" xref="S3.Ex31.m1.4.4.4.4.1.1">superscript</csymbol><ci id="S3.Ex31.m1.4.4.4.4.1.1.2.cmml" xref="S3.Ex31.m1.4.4.4.4.1.1.2">ℝ</ci><ci id="S3.Ex31.m1.4.4.4.4.1.1.3.cmml" xref="S3.Ex31.m1.4.4.4.4.1.1.3">𝑑</ci></apply><apply id="S3.Ex31.m1.5.5.5.5.2.2.cmml" xref="S3.Ex31.m1.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S3.Ex31.m1.5.5.5.5.2.2.1.cmml" xref="S3.Ex31.m1.5.5.5.5.2.2">subscript</csymbol><ci id="S3.Ex31.m1.5.5.5.5.2.2.2.cmml" xref="S3.Ex31.m1.5.5.5.5.2.2.2">𝑤</ci><ci id="S3.Ex31.m1.5.5.5.5.2.2.3.cmml" xref="S3.Ex31.m1.5.5.5.5.2.2.3">𝛾</ci></apply><ci id="S3.Ex31.m1.3.3.3.3.cmml" xref="S3.Ex31.m1.3.3.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex31.m1.6c">\displaystyle\|\operatorname{ext}_{m}g\|_{B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma% };X)}</annotation><annotation encoding="application/x-llamapun" id="S3.Ex31.m1.6d">∥ roman_ext 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end_POSTSUPERSCRIPT ∥ italic_ϕ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ∗ italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_k ≥ 0 end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT roman_ℓ start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S3.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|g\|_{B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)}." class="ltx_Math" display="inline" id="S3.Ex32.m1.6"><semantics id="S3.Ex32.m1.6a"><mrow id="S3.Ex32.m1.6.6.1" xref="S3.Ex32.m1.6.6.1.1.cmml"><mrow 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xref="S3.Ex32.m1.2.2.2.2.2.2">𝑞</ci></list></apply><list id="S3.Ex32.m1.4.4.4.4.2.cmml" xref="S3.Ex32.m1.4.4.4.4.1"><apply id="S3.Ex32.m1.4.4.4.4.1.1.cmml" xref="S3.Ex32.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S3.Ex32.m1.4.4.4.4.1.1.1.cmml" xref="S3.Ex32.m1.4.4.4.4.1.1">superscript</csymbol><ci id="S3.Ex32.m1.4.4.4.4.1.1.2.cmml" xref="S3.Ex32.m1.4.4.4.4.1.1.2">ℝ</ci><apply id="S3.Ex32.m1.4.4.4.4.1.1.3.cmml" xref="S3.Ex32.m1.4.4.4.4.1.1.3"><minus id="S3.Ex32.m1.4.4.4.4.1.1.3.1.cmml" xref="S3.Ex32.m1.4.4.4.4.1.1.3.1"></minus><ci id="S3.Ex32.m1.4.4.4.4.1.1.3.2.cmml" xref="S3.Ex32.m1.4.4.4.4.1.1.3.2">𝑑</ci><cn id="S3.Ex32.m1.4.4.4.4.1.1.3.3.cmml" type="integer" xref="S3.Ex32.m1.4.4.4.4.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex32.m1.3.3.3.3.cmml" xref="S3.Ex32.m1.3.3.3.3">𝑋</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex32.m1.6c">\displaystyle\leq C\|g\|_{B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)}.</annotation><annotation encoding="application/x-llamapun" id="S3.Ex32.m1.6d">≤ italic_C ∥ italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.5.p5.19">To conclude, <math alttext="\operatorname{ext}_{m}g\in B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S3.SS2.5.p5.19.m1.5"><semantics id="S3.SS2.5.p5.19.m1.5a"><mrow id="S3.SS2.5.p5.19.m1.5.5" xref="S3.SS2.5.p5.19.m1.5.5.cmml"><mrow id="S3.SS2.5.p5.19.m1.5.5.4" xref="S3.SS2.5.p5.19.m1.5.5.4.cmml"><msub 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xref="S3.SS2.5.p5.19.m1.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S3.SS2.5.p5.19.m1.3.3.cmml" xref="S3.SS2.5.p5.19.m1.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.5.p5.19.m1.5c">\operatorname{ext}_{m}g\in B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.5.p5.19.m1.5d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> and the lemma is proved. ∎</p> </div> </div> <div class="ltx_para" id="S3.SS2.p4"> <p class="ltx_p" id="S3.SS2.p4.1">Using Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.7</span></a> we can now determine the higher-order trace spaces of weighted Besov spaces and show that the corresponding extension operator is indeed given as in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem6" title="Definition 3.6. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.6</span></a>. We follow the arguments from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorem VIII.1.2.1]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.1.1.1">Theorem 3.8</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem8.p1"> <p class="ltx_p" id="S3.Thmtheorem8.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.6.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.1.1.m1.2"><semantics id="S3.Thmtheorem8.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem8.p1.1.1.m1.2.3" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem8.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem8.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem8.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem8.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem8.p1.1.1.m1.1.1" xref="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem8.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem8.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem8.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.1.1.m1.2b"><apply id="S3.Thmtheorem8.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.2.3"><in id="S3.Thmtheorem8.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.1"></in><ci id="S3.Thmtheorem8.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.2.3.3.2"><cn id="S3.Thmtheorem8.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem8.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem8.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem8.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.2.2.m2.2"><semantics id="S3.Thmtheorem8.p1.2.2.m2.2a"><mrow id="S3.Thmtheorem8.p1.2.2.m2.2.3" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.cmml"><mi id="S3.Thmtheorem8.p1.2.2.m2.2.3.2" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.2.cmml">q</mi><mo id="S3.Thmtheorem8.p1.2.2.m2.2.3.1" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem8.p1.2.2.m2.2.3.3.2" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.3.1.cmml"><mo id="S3.Thmtheorem8.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.3.1.cmml">[</mo><mn id="S3.Thmtheorem8.p1.2.2.m2.1.1" xref="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml">1</mn><mo id="S3.Thmtheorem8.p1.2.2.m2.2.3.3.2.2" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.2.2.m2.2.2" mathvariant="normal" xref="S3.Thmtheorem8.p1.2.2.m2.2.2.cmml">∞</mi><mo id="S3.Thmtheorem8.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.2.2.m2.2b"><apply id="S3.Thmtheorem8.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.2.3"><in id="S3.Thmtheorem8.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.1"></in><ci id="S3.Thmtheorem8.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S3.Thmtheorem8.p1.2.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.2.3.3.2"><cn id="S3.Thmtheorem8.p1.2.2.m2.1.1.cmml" type="integer" xref="S3.Thmtheorem8.p1.2.2.m2.1.1">1</cn><infinity id="S3.Thmtheorem8.p1.2.2.m2.2.2.cmml" xref="S3.Thmtheorem8.p1.2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.3.3.m3.1"><semantics id="S3.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem8.p1.3.3.m3.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">m</mi><mo id="S3.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S3.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.3.3.m3.1b"><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1"><in id="S3.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.1"></in><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.2">𝑚</ci><apply id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.2">ℕ</ci><cn id="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.3.3.m3.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.3.3.m3.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.4.4.m4.1"><semantics id="S3.Thmtheorem8.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem8.p1.4.4.m4.1.1" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem8.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem8.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem8.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3.cmml"><mo id="S3.Thmtheorem8.p1.4.4.m4.1.1.3a" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem8.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.4.4.m4.1b"><apply id="S3.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1"><gt id="S3.Thmtheorem8.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.1"></gt><ci id="S3.Thmtheorem8.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.2">𝛾</ci><apply id="S3.Thmtheorem8.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3"><minus id="S3.Thmtheorem8.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3"></minus><cn id="S3.Thmtheorem8.p1.4.4.m4.1.1.3.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.4.4.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.4.4.m4.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.4.4.m4.1d">italic_γ > - 1</annotation></semantics></math>, <math alttext="s>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.5.5.m5.1"><semantics id="S3.Thmtheorem8.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem8.p1.5.5.m5.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem8.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.2.cmml">s</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem8.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.2.cmml">m</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.1" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.1.cmml">+</mo><mfrac id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.cmml"><mrow id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.cmml"><mi id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.2" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.1" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.3" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.5.5.m5.1b"><apply id="S3.Thmtheorem8.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1"><gt id="S3.Thmtheorem8.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.1"></gt><ci id="S3.Thmtheorem8.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.2">𝑠</ci><apply id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3"><plus id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.1"></plus><ci id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.2">𝑚</ci><apply id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3"><divide id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3"></divide><apply id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2"><plus id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.1"></plus><ci id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.2">𝛾</ci><cn id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.2.3">1</cn></apply><ci id="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.3.cmml" xref="S3.Thmtheorem8.p1.5.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.5.5.m5.1c">s>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.5.5.m5.1d">italic_s > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.6.6.m6.1"><semantics id="S3.Thmtheorem8.p1.6.6.m6.1a"><mi id="S3.Thmtheorem8.p1.6.6.m6.1.1" xref="S3.Thmtheorem8.p1.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.6.6.m6.1b"><ci id="S3.Thmtheorem8.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem8.p1.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.6.6.m6.1d">italic_X</annotation></semantics></math> be a Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{m}:B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B^{s-m-\frac% {\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S3.Ex33.m1.9"><semantics id="S3.Ex33.m1.9a"><mrow id="S3.Ex33.m1.9.9" xref="S3.Ex33.m1.9.9.cmml"><msub id="S3.Ex33.m1.9.9.5" xref="S3.Ex33.m1.9.9.5.cmml"><mi id="S3.Ex33.m1.9.9.5.2" xref="S3.Ex33.m1.9.9.5.2.cmml">Tr</mi><mi id="S3.Ex33.m1.9.9.5.3" xref="S3.Ex33.m1.9.9.5.3.cmml">m</mi></msub><mo id="S3.Ex33.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S3.Ex33.m1.9.9.4.cmml">:</mo><mrow id="S3.Ex33.m1.9.9.3" xref="S3.Ex33.m1.9.9.3.cmml"><mrow id="S3.Ex33.m1.8.8.2.2" xref="S3.Ex33.m1.8.8.2.2.cmml"><msubsup id="S3.Ex33.m1.8.8.2.2.4" xref="S3.Ex33.m1.8.8.2.2.4.cmml"><mi id="S3.Ex33.m1.8.8.2.2.4.2.2" xref="S3.Ex33.m1.8.8.2.2.4.2.2.cmml">B</mi><mrow id="S3.Ex33.m1.2.2.2.4" xref="S3.Ex33.m1.2.2.2.3.cmml"><mi id="S3.Ex33.m1.1.1.1.1" xref="S3.Ex33.m1.1.1.1.1.cmml">p</mi><mo id="S3.Ex33.m1.2.2.2.4.1" xref="S3.Ex33.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex33.m1.2.2.2.2" xref="S3.Ex33.m1.2.2.2.2.cmml">q</mi></mrow><mi id="S3.Ex33.m1.8.8.2.2.4.2.3" xref="S3.Ex33.m1.8.8.2.2.4.2.3.cmml">s</mi></msubsup><mo id="S3.Ex33.m1.8.8.2.2.3" xref="S3.Ex33.m1.8.8.2.2.3.cmml">⁢</mo><mrow id="S3.Ex33.m1.8.8.2.2.2.2" xref="S3.Ex33.m1.8.8.2.2.2.3.cmml"><mo id="S3.Ex33.m1.8.8.2.2.2.2.3" stretchy="false" xref="S3.Ex33.m1.8.8.2.2.2.3.cmml">(</mo><msup id="S3.Ex33.m1.7.7.1.1.1.1.1" xref="S3.Ex33.m1.7.7.1.1.1.1.1.cmml"><mi id="S3.Ex33.m1.7.7.1.1.1.1.1.2" xref="S3.Ex33.m1.7.7.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.Ex33.m1.7.7.1.1.1.1.1.3" xref="S3.Ex33.m1.7.7.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.Ex33.m1.8.8.2.2.2.2.4" xref="S3.Ex33.m1.8.8.2.2.2.3.cmml">,</mo><msub 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id="S3.Ex33.m1.9c">\operatorname{Tr}_{m}:B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B^{s-m-\frac% {\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex33.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem8.p1.12"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem8.p1.12.6">is a continuous and surjective operator. Moreover, there exists a continuous right inverse <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.7.1.m1.1"><semantics id="S3.Thmtheorem8.p1.7.1.m1.1a"><msub id="S3.Thmtheorem8.p1.7.1.m1.1.1" xref="S3.Thmtheorem8.p1.7.1.m1.1.1.cmml"><mi id="S3.Thmtheorem8.p1.7.1.m1.1.1.2" xref="S3.Thmtheorem8.p1.7.1.m1.1.1.2.cmml">ext</mi><mi id="S3.Thmtheorem8.p1.7.1.m1.1.1.3" xref="S3.Thmtheorem8.p1.7.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.7.1.m1.1b"><apply id="S3.Thmtheorem8.p1.7.1.m1.1.1.cmml" xref="S3.Thmtheorem8.p1.7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.7.1.m1.1.1.1.cmml" xref="S3.Thmtheorem8.p1.7.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem8.p1.7.1.m1.1.1.2.cmml" xref="S3.Thmtheorem8.p1.7.1.m1.1.1.2">ext</ci><ci id="S3.Thmtheorem8.p1.7.1.m1.1.1.3.cmml" xref="S3.Thmtheorem8.p1.7.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.7.1.m1.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.7.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.8.2.m2.1"><semantics id="S3.Thmtheorem8.p1.8.2.m2.1a"><msub id="S3.Thmtheorem8.p1.8.2.m2.1.1" xref="S3.Thmtheorem8.p1.8.2.m2.1.1.cmml"><mi id="S3.Thmtheorem8.p1.8.2.m2.1.1.2" xref="S3.Thmtheorem8.p1.8.2.m2.1.1.2.cmml">Tr</mi><mi id="S3.Thmtheorem8.p1.8.2.m2.1.1.3" xref="S3.Thmtheorem8.p1.8.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.8.2.m2.1b"><apply id="S3.Thmtheorem8.p1.8.2.m2.1.1.cmml" xref="S3.Thmtheorem8.p1.8.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.8.2.m2.1.1.1.cmml" xref="S3.Thmtheorem8.p1.8.2.m2.1.1">subscript</csymbol><ci id="S3.Thmtheorem8.p1.8.2.m2.1.1.2.cmml" xref="S3.Thmtheorem8.p1.8.2.m2.1.1.2">Tr</ci><ci id="S3.Thmtheorem8.p1.8.2.m2.1.1.3.cmml" xref="S3.Thmtheorem8.p1.8.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.8.2.m2.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.8.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> which is independent of <math alttext="s,p,q,\gamma" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.9.3.m3.4"><semantics id="S3.Thmtheorem8.p1.9.3.m3.4a"><mrow id="S3.Thmtheorem8.p1.9.3.m3.4.5.2" xref="S3.Thmtheorem8.p1.9.3.m3.4.5.1.cmml"><mi id="S3.Thmtheorem8.p1.9.3.m3.1.1" xref="S3.Thmtheorem8.p1.9.3.m3.1.1.cmml">s</mi><mo id="S3.Thmtheorem8.p1.9.3.m3.4.5.2.1" xref="S3.Thmtheorem8.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.9.3.m3.2.2" xref="S3.Thmtheorem8.p1.9.3.m3.2.2.cmml">p</mi><mo id="S3.Thmtheorem8.p1.9.3.m3.4.5.2.2" xref="S3.Thmtheorem8.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.9.3.m3.3.3" xref="S3.Thmtheorem8.p1.9.3.m3.3.3.cmml">q</mi><mo id="S3.Thmtheorem8.p1.9.3.m3.4.5.2.3" xref="S3.Thmtheorem8.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem8.p1.9.3.m3.4.4" xref="S3.Thmtheorem8.p1.9.3.m3.4.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.9.3.m3.4b"><list id="S3.Thmtheorem8.p1.9.3.m3.4.5.1.cmml" xref="S3.Thmtheorem8.p1.9.3.m3.4.5.2"><ci id="S3.Thmtheorem8.p1.9.3.m3.1.1.cmml" xref="S3.Thmtheorem8.p1.9.3.m3.1.1">𝑠</ci><ci id="S3.Thmtheorem8.p1.9.3.m3.2.2.cmml" xref="S3.Thmtheorem8.p1.9.3.m3.2.2">𝑝</ci><ci id="S3.Thmtheorem8.p1.9.3.m3.3.3.cmml" xref="S3.Thmtheorem8.p1.9.3.m3.3.3">𝑞</ci><ci id="S3.Thmtheorem8.p1.9.3.m3.4.4.cmml" xref="S3.Thmtheorem8.p1.9.3.m3.4.4">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.9.3.m3.4c">s,p,q,\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.9.3.m3.4d">italic_s , italic_p , italic_q , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.10.4.m4.1"><semantics id="S3.Thmtheorem8.p1.10.4.m4.1a"><mi id="S3.Thmtheorem8.p1.10.4.m4.1.1" xref="S3.Thmtheorem8.p1.10.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.10.4.m4.1b"><ci id="S3.Thmtheorem8.p1.10.4.m4.1.1.cmml" xref="S3.Thmtheorem8.p1.10.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.10.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.10.4.m4.1d">italic_X</annotation></semantics></math>. For any <math alttext="0\leq j<m" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.11.5.m5.1"><semantics id="S3.Thmtheorem8.p1.11.5.m5.1a"><mrow id="S3.Thmtheorem8.p1.11.5.m5.1.1" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.cmml"><mn id="S3.Thmtheorem8.p1.11.5.m5.1.1.2" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.2.cmml">0</mn><mo id="S3.Thmtheorem8.p1.11.5.m5.1.1.3" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.3.cmml">≤</mo><mi id="S3.Thmtheorem8.p1.11.5.m5.1.1.4" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.4.cmml">j</mi><mo id="S3.Thmtheorem8.p1.11.5.m5.1.1.5" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.5.cmml"><</mo><mi id="S3.Thmtheorem8.p1.11.5.m5.1.1.6" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.11.5.m5.1b"><apply id="S3.Thmtheorem8.p1.11.5.m5.1.1.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1"><and id="S3.Thmtheorem8.p1.11.5.m5.1.1a.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1"></and><apply id="S3.Thmtheorem8.p1.11.5.m5.1.1b.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1"><leq id="S3.Thmtheorem8.p1.11.5.m5.1.1.3.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.3"></leq><cn id="S3.Thmtheorem8.p1.11.5.m5.1.1.2.cmml" type="integer" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.2">0</cn><ci id="S3.Thmtheorem8.p1.11.5.m5.1.1.4.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.4">𝑗</ci></apply><apply id="S3.Thmtheorem8.p1.11.5.m5.1.1c.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1"><lt id="S3.Thmtheorem8.p1.11.5.m5.1.1.5.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem8.p1.11.5.m5.1.1.4.cmml" id="S3.Thmtheorem8.p1.11.5.m5.1.1d.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1"></share><ci id="S3.Thmtheorem8.p1.11.5.m5.1.1.6.cmml" xref="S3.Thmtheorem8.p1.11.5.m5.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.11.5.m5.1c">0\leq j<m</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.11.5.m5.1d">0 ≤ italic_j < italic_m</annotation></semantics></math> we have <math alttext="\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0" class="ltx_Math" display="inline" id="S3.Thmtheorem8.p1.12.6.m6.1"><semantics id="S3.Thmtheorem8.p1.12.6.m6.1a"><mrow id="S3.Thmtheorem8.p1.12.6.m6.1.1" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.cmml"><mrow id="S3.Thmtheorem8.p1.12.6.m6.1.1.2" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.cmml"><msub id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.cmml"><mi id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.2" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.2.cmml">Tr</mi><mi id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.3" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.3.cmml">j</mi></msub><mo id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.1" lspace="0.167em" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.cmml"><mo id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.1" rspace="0.167em" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.1.cmml">∘</mo><msub id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.cmml"><mi id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.2" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.2.cmml">ext</mi><mi id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.3" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.3.cmml">m</mi></msub></mrow></mrow><mo id="S3.Thmtheorem8.p1.12.6.m6.1.1.1" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.1.cmml">=</mo><mn id="S3.Thmtheorem8.p1.12.6.m6.1.1.3" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem8.p1.12.6.m6.1b"><apply id="S3.Thmtheorem8.p1.12.6.m6.1.1.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1"><eq id="S3.Thmtheorem8.p1.12.6.m6.1.1.1.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.1"></eq><apply id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2"><times id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.1.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.1"></times><apply id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.1.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2">subscript</csymbol><ci id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.2.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.2">Tr</ci><ci id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.3.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.2.3">𝑗</ci></apply><apply id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3"><compose id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.1.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.1"></compose><apply id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.1.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.2.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.2">ext</ci><ci id="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.3.cmml" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.2.3.2.3">𝑚</ci></apply></apply></apply><cn id="S3.Thmtheorem8.p1.12.6.m6.1.1.3.cmml" type="integer" xref="S3.Thmtheorem8.p1.12.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem8.p1.12.6.m6.1c">\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem8.p1.12.6.m6.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS2.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS2.6.p1"> <p class="ltx_p" id="S3.SS2.6.p1.1"><span class="ltx_text ltx_font_italic" id="S3.SS2.6.p1.1.1">Step 1: trace operator.</span> Note that since <math alttext="s-m>\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S3.SS2.6.p1.1.m1.1"><semantics id="S3.SS2.6.p1.1.m1.1a"><mrow id="S3.SS2.6.p1.1.m1.1.1" xref="S3.SS2.6.p1.1.m1.1.1.cmml"><mrow id="S3.SS2.6.p1.1.m1.1.1.2" xref="S3.SS2.6.p1.1.m1.1.1.2.cmml"><mi id="S3.SS2.6.p1.1.m1.1.1.2.2" xref="S3.SS2.6.p1.1.m1.1.1.2.2.cmml">s</mi><mo id="S3.SS2.6.p1.1.m1.1.1.2.1" xref="S3.SS2.6.p1.1.m1.1.1.2.1.cmml">−</mo><mi id="S3.SS2.6.p1.1.m1.1.1.2.3" xref="S3.SS2.6.p1.1.m1.1.1.2.3.cmml">m</mi></mrow><mo id="S3.SS2.6.p1.1.m1.1.1.1" xref="S3.SS2.6.p1.1.m1.1.1.1.cmml">></mo><mfrac id="S3.SS2.6.p1.1.m1.1.1.3" xref="S3.SS2.6.p1.1.m1.1.1.3.cmml"><mrow id="S3.SS2.6.p1.1.m1.1.1.3.2" xref="S3.SS2.6.p1.1.m1.1.1.3.2.cmml"><mi id="S3.SS2.6.p1.1.m1.1.1.3.2.2" xref="S3.SS2.6.p1.1.m1.1.1.3.2.2.cmml">γ</mi><mo id="S3.SS2.6.p1.1.m1.1.1.3.2.1" xref="S3.SS2.6.p1.1.m1.1.1.3.2.1.cmml">+</mo><mn id="S3.SS2.6.p1.1.m1.1.1.3.2.3" xref="S3.SS2.6.p1.1.m1.1.1.3.2.3.cmml">1</mn></mrow><mi id="S3.SS2.6.p1.1.m1.1.1.3.3" xref="S3.SS2.6.p1.1.m1.1.1.3.3.cmml">p</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.6.p1.1.m1.1b"><apply id="S3.SS2.6.p1.1.m1.1.1.cmml" xref="S3.SS2.6.p1.1.m1.1.1"><gt id="S3.SS2.6.p1.1.m1.1.1.1.cmml" xref="S3.SS2.6.p1.1.m1.1.1.1"></gt><apply id="S3.SS2.6.p1.1.m1.1.1.2.cmml" xref="S3.SS2.6.p1.1.m1.1.1.2"><minus id="S3.SS2.6.p1.1.m1.1.1.2.1.cmml" xref="S3.SS2.6.p1.1.m1.1.1.2.1"></minus><ci id="S3.SS2.6.p1.1.m1.1.1.2.2.cmml" xref="S3.SS2.6.p1.1.m1.1.1.2.2">𝑠</ci><ci id="S3.SS2.6.p1.1.m1.1.1.2.3.cmml" xref="S3.SS2.6.p1.1.m1.1.1.2.3">𝑚</ci></apply><apply id="S3.SS2.6.p1.1.m1.1.1.3.cmml" xref="S3.SS2.6.p1.1.m1.1.1.3"><divide id="S3.SS2.6.p1.1.m1.1.1.3.1.cmml" xref="S3.SS2.6.p1.1.m1.1.1.3"></divide><apply id="S3.SS2.6.p1.1.m1.1.1.3.2.cmml" xref="S3.SS2.6.p1.1.m1.1.1.3.2"><plus id="S3.SS2.6.p1.1.m1.1.1.3.2.1.cmml" xref="S3.SS2.6.p1.1.m1.1.1.3.2.1"></plus><ci id="S3.SS2.6.p1.1.m1.1.1.3.2.2.cmml" xref="S3.SS2.6.p1.1.m1.1.1.3.2.2">𝛾</ci><cn id="S3.SS2.6.p1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S3.SS2.6.p1.1.m1.1.1.3.2.3">1</cn></apply><ci id="S3.SS2.6.p1.1.m1.1.1.3.3.cmml" xref="S3.SS2.6.p1.1.m1.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.6.p1.1.m1.1c">s-m>\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.6.p1.1.m1.1d">italic_s - italic_m > divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> it follows from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 3.10]</cite> and Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem4" title="Theorem 3.4. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.4</span></a> that</p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{m}=\operatorname{Tr}\mathop{\circ}\nolimits\partial_{1}^{m}% :B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B^{s-m-\frac{\gamma+1}{p}}_{p,q}(% \mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S3.Ex34.m1.9"><semantics id="S3.Ex34.m1.9a"><mrow id="S3.Ex34.m1.9.9" xref="S3.Ex34.m1.9.9.cmml"><mrow id="S3.Ex34.m1.9.9.5" xref="S3.Ex34.m1.9.9.5.cmml"><msub id="S3.Ex34.m1.9.9.5.2" xref="S3.Ex34.m1.9.9.5.2.cmml"><mi id="S3.Ex34.m1.9.9.5.2.2" xref="S3.Ex34.m1.9.9.5.2.2.cmml">Tr</mi><mi id="S3.Ex34.m1.9.9.5.2.3" xref="S3.Ex34.m1.9.9.5.2.3.cmml">m</mi></msub><mo id="S3.Ex34.m1.9.9.5.1" xref="S3.Ex34.m1.9.9.5.1.cmml">=</mo><mrow id="S3.Ex34.m1.9.9.5.3" xref="S3.Ex34.m1.9.9.5.3.cmml"><mi id="S3.Ex34.m1.9.9.5.3.2" xref="S3.Ex34.m1.9.9.5.3.2.cmml">Tr</mi><mo id="S3.Ex34.m1.9.9.5.3.1" lspace="0.167em" xref="S3.Ex34.m1.9.9.5.3.1.cmml">⁢</mo><mrow id="S3.Ex34.m1.9.9.5.3.3" xref="S3.Ex34.m1.9.9.5.3.3.cmml"><mo id="S3.Ex34.m1.9.9.5.3.3.1" rspace="0.0835em" xref="S3.Ex34.m1.9.9.5.3.3.1.cmml">∘</mo><msubsup id="S3.Ex34.m1.9.9.5.3.3.2" xref="S3.Ex34.m1.9.9.5.3.3.2.cmml"><mo id="S3.Ex34.m1.9.9.5.3.3.2.2.2" lspace="0.0835em" xref="S3.Ex34.m1.9.9.5.3.3.2.2.2.cmml">∂</mo><mn id="S3.Ex34.m1.9.9.5.3.3.2.2.3" xref="S3.Ex34.m1.9.9.5.3.3.2.2.3.cmml">1</mn><mi id="S3.Ex34.m1.9.9.5.3.3.2.3" xref="S3.Ex34.m1.9.9.5.3.3.2.3.cmml">m</mi></msubsup></mrow></mrow></mrow><mo id="S3.Ex34.m1.9.9.4" rspace="0.278em" xref="S3.Ex34.m1.9.9.4.cmml">:</mo><mrow id="S3.Ex34.m1.9.9.3" xref="S3.Ex34.m1.9.9.3.cmml"><mrow id="S3.Ex34.m1.8.8.2.2" xref="S3.Ex34.m1.8.8.2.2.cmml"><msubsup 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xref="S3.Ex34.m1.8.8.2.2.2.3.cmml">,</mo><msub id="S3.Ex34.m1.8.8.2.2.2.2.2" xref="S3.Ex34.m1.8.8.2.2.2.2.2.cmml"><mi id="S3.Ex34.m1.8.8.2.2.2.2.2.2" xref="S3.Ex34.m1.8.8.2.2.2.2.2.2.cmml">w</mi><mi id="S3.Ex34.m1.8.8.2.2.2.2.2.3" xref="S3.Ex34.m1.8.8.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex34.m1.8.8.2.2.2.2.5" xref="S3.Ex34.m1.8.8.2.2.2.3.cmml">;</mo><mi id="S3.Ex34.m1.5.5" xref="S3.Ex34.m1.5.5.cmml">X</mi><mo id="S3.Ex34.m1.8.8.2.2.2.2.6" stretchy="false" xref="S3.Ex34.m1.8.8.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex34.m1.9.9.3.4" stretchy="false" xref="S3.Ex34.m1.9.9.3.4.cmml">→</mo><mrow id="S3.Ex34.m1.9.9.3.3" xref="S3.Ex34.m1.9.9.3.3.cmml"><msubsup id="S3.Ex34.m1.9.9.3.3.3" xref="S3.Ex34.m1.9.9.3.3.3.cmml"><mi id="S3.Ex34.m1.9.9.3.3.3.2.2" xref="S3.Ex34.m1.9.9.3.3.3.2.2.cmml">B</mi><mrow id="S3.Ex34.m1.4.4.2.4" xref="S3.Ex34.m1.4.4.2.3.cmml"><mi id="S3.Ex34.m1.3.3.1.1" xref="S3.Ex34.m1.3.3.1.1.cmml">p</mi><mo id="S3.Ex34.m1.4.4.2.4.1" xref="S3.Ex34.m1.4.4.2.3.cmml">,</mo><mi id="S3.Ex34.m1.4.4.2.2" xref="S3.Ex34.m1.4.4.2.2.cmml">q</mi></mrow><mrow id="S3.Ex34.m1.9.9.3.3.3.2.3" xref="S3.Ex34.m1.9.9.3.3.3.2.3.cmml"><mi id="S3.Ex34.m1.9.9.3.3.3.2.3.2" xref="S3.Ex34.m1.9.9.3.3.3.2.3.2.cmml">s</mi><mo id="S3.Ex34.m1.9.9.3.3.3.2.3.1" xref="S3.Ex34.m1.9.9.3.3.3.2.3.1.cmml">−</mo><mi id="S3.Ex34.m1.9.9.3.3.3.2.3.3" xref="S3.Ex34.m1.9.9.3.3.3.2.3.3.cmml">m</mi><mo id="S3.Ex34.m1.9.9.3.3.3.2.3.1a" xref="S3.Ex34.m1.9.9.3.3.3.2.3.1.cmml">−</mo><mfrac id="S3.Ex34.m1.9.9.3.3.3.2.3.4" xref="S3.Ex34.m1.9.9.3.3.3.2.3.4.cmml"><mrow id="S3.Ex34.m1.9.9.3.3.3.2.3.4.2" xref="S3.Ex34.m1.9.9.3.3.3.2.3.4.2.cmml"><mi id="S3.Ex34.m1.9.9.3.3.3.2.3.4.2.2" xref="S3.Ex34.m1.9.9.3.3.3.2.3.4.2.2.cmml">γ</mi><mo id="S3.Ex34.m1.9.9.3.3.3.2.3.4.2.1" xref="S3.Ex34.m1.9.9.3.3.3.2.3.4.2.1.cmml">+</mo><mn id="S3.Ex34.m1.9.9.3.3.3.2.3.4.2.3" xref="S3.Ex34.m1.9.9.3.3.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S3.Ex34.m1.9.9.3.3.3.2.3.4.3" 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xref="S3.Ex34.m1.9.9.3.3.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Ex34.m1.9b"><apply id="S3.Ex34.m1.9.9.cmml" xref="S3.Ex34.m1.9.9"><ci id="S3.Ex34.m1.9.9.4.cmml" xref="S3.Ex34.m1.9.9.4">:</ci><apply id="S3.Ex34.m1.9.9.5.cmml" xref="S3.Ex34.m1.9.9.5"><eq id="S3.Ex34.m1.9.9.5.1.cmml" xref="S3.Ex34.m1.9.9.5.1"></eq><apply id="S3.Ex34.m1.9.9.5.2.cmml" xref="S3.Ex34.m1.9.9.5.2"><csymbol cd="ambiguous" id="S3.Ex34.m1.9.9.5.2.1.cmml" xref="S3.Ex34.m1.9.9.5.2">subscript</csymbol><ci id="S3.Ex34.m1.9.9.5.2.2.cmml" xref="S3.Ex34.m1.9.9.5.2.2">Tr</ci><ci id="S3.Ex34.m1.9.9.5.2.3.cmml" xref="S3.Ex34.m1.9.9.5.2.3">𝑚</ci></apply><apply id="S3.Ex34.m1.9.9.5.3.cmml" xref="S3.Ex34.m1.9.9.5.3"><times id="S3.Ex34.m1.9.9.5.3.1.cmml" xref="S3.Ex34.m1.9.9.5.3.1"></times><ci id="S3.Ex34.m1.9.9.5.3.2.cmml" xref="S3.Ex34.m1.9.9.5.3.2">Tr</ci><apply id="S3.Ex34.m1.9.9.5.3.3.cmml" xref="S3.Ex34.m1.9.9.5.3.3"><compose id="S3.Ex34.m1.9.9.5.3.3.1.cmml" 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type="integer" xref="S3.Ex34.m1.9.9.3.3.1.1.1.3.3">1</cn></apply></apply><ci id="S3.Ex34.m1.6.6.cmml" xref="S3.Ex34.m1.6.6">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex34.m1.9c">\operatorname{Tr}_{m}=\operatorname{Tr}\mathop{\circ}\nolimits\partial_{1}^{m}% :B^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B^{s-m-\frac{\gamma+1}{p}}_{p,q}(% \mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex34.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = roman_Tr ∘ ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT : italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.6.p1.2">is continuous.</p> </div> <div class="ltx_para" id="S3.SS2.7.p2"> <p class="ltx_p" id="S3.SS2.7.p2.3"><span class="ltx_text ltx_font_italic" id="S3.SS2.7.p2.3.1">Step 2: extension operator. </span> Let <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S3.SS2.7.p2.1.m1.1"><semantics id="S3.SS2.7.p2.1.m1.1a"><msub id="S3.SS2.7.p2.1.m1.1.1" xref="S3.SS2.7.p2.1.m1.1.1.cmml"><mi id="S3.SS2.7.p2.1.m1.1.1.2" xref="S3.SS2.7.p2.1.m1.1.1.2.cmml">ext</mi><mi id="S3.SS2.7.p2.1.m1.1.1.3" xref="S3.SS2.7.p2.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.1.m1.1b"><apply id="S3.SS2.7.p2.1.m1.1.1.cmml" xref="S3.SS2.7.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.7.p2.1.m1.1.1.1.cmml" xref="S3.SS2.7.p2.1.m1.1.1">subscript</csymbol><ci id="S3.SS2.7.p2.1.m1.1.1.2.cmml" xref="S3.SS2.7.p2.1.m1.1.1.2">ext</ci><ci id="S3.SS2.7.p2.1.m1.1.1.3.cmml" xref="S3.SS2.7.p2.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.1.m1.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> be the extension operator from Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem6" title="Definition 3.6. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.6</span></a>. The continuity follows from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.7</span></a> and it remains to prove that <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S3.SS2.7.p2.2.m2.1"><semantics id="S3.SS2.7.p2.2.m2.1a"><msub id="S3.SS2.7.p2.2.m2.1.1" xref="S3.SS2.7.p2.2.m2.1.1.cmml"><mi id="S3.SS2.7.p2.2.m2.1.1.2" xref="S3.SS2.7.p2.2.m2.1.1.2.cmml">ext</mi><mi id="S3.SS2.7.p2.2.m2.1.1.3" xref="S3.SS2.7.p2.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.2.m2.1b"><apply id="S3.SS2.7.p2.2.m2.1.1.cmml" xref="S3.SS2.7.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S3.SS2.7.p2.2.m2.1.1.1.cmml" xref="S3.SS2.7.p2.2.m2.1.1">subscript</csymbol><ci id="S3.SS2.7.p2.2.m2.1.1.2.cmml" xref="S3.SS2.7.p2.2.m2.1.1.2">ext</ci><ci id="S3.SS2.7.p2.2.m2.1.1.3.cmml" xref="S3.SS2.7.p2.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.2.m2.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.2.m2.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> is the right inverse of <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S3.SS2.7.p2.3.m3.1"><semantics id="S3.SS2.7.p2.3.m3.1a"><msub id="S3.SS2.7.p2.3.m3.1.1" xref="S3.SS2.7.p2.3.m3.1.1.cmml"><mi id="S3.SS2.7.p2.3.m3.1.1.2" xref="S3.SS2.7.p2.3.m3.1.1.2.cmml">Tr</mi><mi id="S3.SS2.7.p2.3.m3.1.1.3" xref="S3.SS2.7.p2.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.3.m3.1b"><apply id="S3.SS2.7.p2.3.m3.1.1.cmml" xref="S3.SS2.7.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S3.SS2.7.p2.3.m3.1.1.1.cmml" xref="S3.SS2.7.p2.3.m3.1.1">subscript</csymbol><ci id="S3.SS2.7.p2.3.m3.1.1.2.cmml" xref="S3.SS2.7.p2.3.m3.1.1.2">Tr</ci><ci id="S3.SS2.7.p2.3.m3.1.1.3.cmml" xref="S3.SS2.7.p2.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.3.m3.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.3.m3.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S3.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{j}(\operatorname{ext}_{m}g)=\delta_{jm}g,\qquad 0\leq j\leq m% ,\,\,g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)," class="ltx_Math" display="block" id="S3.E11.m1.4"><semantics id="S3.E11.m1.4a"><mrow id="S3.E11.m1.4.4.1"><mrow id="S3.E11.m1.4.4.1.1.2" xref="S3.E11.m1.4.4.1.1.3.cmml"><mrow id="S3.E11.m1.4.4.1.1.1.1.2" xref="S3.E11.m1.4.4.1.1.1.1.3.cmml"><mrow id="S3.E11.m1.4.4.1.1.1.1.1.1" 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encoding="application/x-llamapun" id="S3.E11.m1.4d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ) = italic_δ start_POSTSUBSCRIPT italic_j italic_m end_POSTSUBSCRIPT italic_g , 0 ≤ italic_j ≤ italic_m , italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.7.p2.5">where <math alttext="\delta_{jm}" class="ltx_Math" display="inline" id="S3.SS2.7.p2.4.m1.1"><semantics id="S3.SS2.7.p2.4.m1.1a"><msub id="S3.SS2.7.p2.4.m1.1.1" xref="S3.SS2.7.p2.4.m1.1.1.cmml"><mi id="S3.SS2.7.p2.4.m1.1.1.2" xref="S3.SS2.7.p2.4.m1.1.1.2.cmml">δ</mi><mrow id="S3.SS2.7.p2.4.m1.1.1.3" xref="S3.SS2.7.p2.4.m1.1.1.3.cmml"><mi id="S3.SS2.7.p2.4.m1.1.1.3.2" xref="S3.SS2.7.p2.4.m1.1.1.3.2.cmml">j</mi><mo id="S3.SS2.7.p2.4.m1.1.1.3.1" xref="S3.SS2.7.p2.4.m1.1.1.3.1.cmml">⁢</mo><mi id="S3.SS2.7.p2.4.m1.1.1.3.3" xref="S3.SS2.7.p2.4.m1.1.1.3.3.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.4.m1.1b"><apply id="S3.SS2.7.p2.4.m1.1.1.cmml" xref="S3.SS2.7.p2.4.m1.1.1"><csymbol cd="ambiguous" id="S3.SS2.7.p2.4.m1.1.1.1.cmml" xref="S3.SS2.7.p2.4.m1.1.1">subscript</csymbol><ci id="S3.SS2.7.p2.4.m1.1.1.2.cmml" xref="S3.SS2.7.p2.4.m1.1.1.2">𝛿</ci><apply id="S3.SS2.7.p2.4.m1.1.1.3.cmml" xref="S3.SS2.7.p2.4.m1.1.1.3"><times id="S3.SS2.7.p2.4.m1.1.1.3.1.cmml" xref="S3.SS2.7.p2.4.m1.1.1.3.1"></times><ci id="S3.SS2.7.p2.4.m1.1.1.3.2.cmml" xref="S3.SS2.7.p2.4.m1.1.1.3.2">𝑗</ci><ci id="S3.SS2.7.p2.4.m1.1.1.3.3.cmml" xref="S3.SS2.7.p2.4.m1.1.1.3.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.4.m1.1c">\delta_{jm}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.4.m1.1d">italic_δ start_POSTSUBSCRIPT italic_j italic_m end_POSTSUBSCRIPT</annotation></semantics></math> is the Kronecker delta. Note that from Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem6" title="Definition 3.6. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.6</span></a> and properties of the Fourier transform it follows that for <math alttext="n\geq 0" class="ltx_Math" display="inline" id="S3.SS2.7.p2.5.m2.1"><semantics id="S3.SS2.7.p2.5.m2.1a"><mrow id="S3.SS2.7.p2.5.m2.1.1" xref="S3.SS2.7.p2.5.m2.1.1.cmml"><mi id="S3.SS2.7.p2.5.m2.1.1.2" xref="S3.SS2.7.p2.5.m2.1.1.2.cmml">n</mi><mo id="S3.SS2.7.p2.5.m2.1.1.1" xref="S3.SS2.7.p2.5.m2.1.1.1.cmml">≥</mo><mn id="S3.SS2.7.p2.5.m2.1.1.3" xref="S3.SS2.7.p2.5.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.5.m2.1b"><apply id="S3.SS2.7.p2.5.m2.1.1.cmml" xref="S3.SS2.7.p2.5.m2.1.1"><geq id="S3.SS2.7.p2.5.m2.1.1.1.cmml" xref="S3.SS2.7.p2.5.m2.1.1.1"></geq><ci id="S3.SS2.7.p2.5.m2.1.1.2.cmml" xref="S3.SS2.7.p2.5.m2.1.1.2">𝑛</ci><cn id="S3.SS2.7.p2.5.m2.1.1.3.cmml" type="integer" xref="S3.SS2.7.p2.5.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.5.m2.1c">n\geq 0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.5.m2.1d">italic_n ≥ 0</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="S3.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(\mathcal{F}^{-1}\eta_{n}^{m})(x_{1})=\frac{2^{n}x_{1}^{m}}{m!}(\mathcal{F}^{-% 1}\zeta)(2^{n}x_{1}),\qquad x_{1}\in\mathbb{R}," class="ltx_Math" display="block" id="S3.E12.m1.1"><semantics id="S3.E12.m1.1a"><mrow id="S3.E12.m1.1.1.1"><mrow id="S3.E12.m1.1.1.1.1.2" xref="S3.E12.m1.1.1.1.1.3.cmml"><mrow id="S3.E12.m1.1.1.1.1.1.1" xref="S3.E12.m1.1.1.1.1.1.1.cmml"><mrow id="S3.E12.m1.1.1.1.1.1.1.2" xref="S3.E12.m1.1.1.1.1.1.1.2.cmml"><mrow id="S3.E12.m1.1.1.1.1.1.1.1.1.1" 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id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.1"></times><apply id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.cmml" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2"><csymbol cd="ambiguous" id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2">superscript</csymbol><ci id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.2.cmml" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.2">ℱ</ci><apply id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.3.cmml" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.3"><minus id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.3.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.3"></minus><cn id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.3.2.cmml" type="integer" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.2.3.2">1</cn></apply></apply><ci id="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.3.cmml" xref="S3.E12.m1.1.1.1.1.1.1.3.1.1.1.3">𝜁</ci></apply><apply id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1"><times id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.1"></times><apply id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2"><csymbol cd="ambiguous" id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2">superscript</csymbol><cn id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2.2.cmml" type="integer" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2.2">2</cn><ci id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2.3.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.2.3">𝑛</ci></apply><apply id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3"><csymbol cd="ambiguous" id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3.1.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3">subscript</csymbol><ci id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3.2.cmml" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3.2">𝑥</ci><cn id="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3.3.cmml" type="integer" xref="S3.E12.m1.1.1.1.1.1.1.4.2.1.1.3.3">1</cn></apply></apply></apply></apply><apply id="S3.E12.m1.1.1.1.1.2.2.cmml" xref="S3.E12.m1.1.1.1.1.2.2"><in id="S3.E12.m1.1.1.1.1.2.2.1.cmml" xref="S3.E12.m1.1.1.1.1.2.2.1"></in><apply id="S3.E12.m1.1.1.1.1.2.2.2.cmml" xref="S3.E12.m1.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S3.E12.m1.1.1.1.1.2.2.2.1.cmml" xref="S3.E12.m1.1.1.1.1.2.2.2">subscript</csymbol><ci id="S3.E12.m1.1.1.1.1.2.2.2.2.cmml" xref="S3.E12.m1.1.1.1.1.2.2.2.2">𝑥</ci><cn id="S3.E12.m1.1.1.1.1.2.2.2.3.cmml" type="integer" xref="S3.E12.m1.1.1.1.1.2.2.2.3">1</cn></apply><ci id="S3.E12.m1.1.1.1.1.2.2.3.cmml" xref="S3.E12.m1.1.1.1.1.2.2.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E12.m1.1c">(\mathcal{F}^{-1}\eta_{n}^{m})(x_{1})=\frac{2^{n}x_{1}^{m}}{m!}(\mathcal{F}^{-% 1}\zeta)(2^{n}x_{1}),\qquad x_{1}\in\mathbb{R},</annotation><annotation encoding="application/x-llamapun" id="S3.E12.m1.1d">( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = divide start_ARG 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT end_ARG start_ARG italic_m ! end_ARG ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ζ ) ( 2 start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) , italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_R ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.7.p2.12">where <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S3.SS2.7.p2.6.m1.1"><semantics id="S3.SS2.7.p2.6.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S3.SS2.7.p2.6.m1.1.1" xref="S3.SS2.7.p2.6.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.6.m1.1b"><ci id="S3.SS2.7.p2.6.m1.1.1.cmml" xref="S3.SS2.7.p2.6.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.6.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.6.m1.1d">caligraphic_F</annotation></semantics></math> is the one-dimensional Fourier transform and <math alttext="\zeta:=\eta" class="ltx_Math" display="inline" id="S3.SS2.7.p2.7.m2.1"><semantics id="S3.SS2.7.p2.7.m2.1a"><mrow id="S3.SS2.7.p2.7.m2.1.1" xref="S3.SS2.7.p2.7.m2.1.1.cmml"><mi id="S3.SS2.7.p2.7.m2.1.1.2" xref="S3.SS2.7.p2.7.m2.1.1.2.cmml">ζ</mi><mo id="S3.SS2.7.p2.7.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.7.p2.7.m2.1.1.1.cmml">:=</mo><mi id="S3.SS2.7.p2.7.m2.1.1.3" xref="S3.SS2.7.p2.7.m2.1.1.3.cmml">η</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.7.m2.1b"><apply id="S3.SS2.7.p2.7.m2.1.1.cmml" xref="S3.SS2.7.p2.7.m2.1.1"><csymbol cd="latexml" id="S3.SS2.7.p2.7.m2.1.1.1.cmml" xref="S3.SS2.7.p2.7.m2.1.1.1">assign</csymbol><ci id="S3.SS2.7.p2.7.m2.1.1.2.cmml" xref="S3.SS2.7.p2.7.m2.1.1.2">𝜁</ci><ci id="S3.SS2.7.p2.7.m2.1.1.3.cmml" xref="S3.SS2.7.p2.7.m2.1.1.3">𝜂</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.7.m2.1c">\zeta:=\eta</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.7.m2.1d">italic_ζ := italic_η</annotation></semantics></math> if <math alttext="n\geq 1" class="ltx_Math" display="inline" id="S3.SS2.7.p2.8.m3.1"><semantics id="S3.SS2.7.p2.8.m3.1a"><mrow id="S3.SS2.7.p2.8.m3.1.1" xref="S3.SS2.7.p2.8.m3.1.1.cmml"><mi id="S3.SS2.7.p2.8.m3.1.1.2" xref="S3.SS2.7.p2.8.m3.1.1.2.cmml">n</mi><mo id="S3.SS2.7.p2.8.m3.1.1.1" xref="S3.SS2.7.p2.8.m3.1.1.1.cmml">≥</mo><mn id="S3.SS2.7.p2.8.m3.1.1.3" xref="S3.SS2.7.p2.8.m3.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.8.m3.1b"><apply id="S3.SS2.7.p2.8.m3.1.1.cmml" xref="S3.SS2.7.p2.8.m3.1.1"><geq id="S3.SS2.7.p2.8.m3.1.1.1.cmml" xref="S3.SS2.7.p2.8.m3.1.1.1"></geq><ci id="S3.SS2.7.p2.8.m3.1.1.2.cmml" xref="S3.SS2.7.p2.8.m3.1.1.2">𝑛</ci><cn id="S3.SS2.7.p2.8.m3.1.1.3.cmml" type="integer" xref="S3.SS2.7.p2.8.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.8.m3.1c">n\geq 1</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.8.m3.1d">italic_n ≥ 1</annotation></semantics></math> and <math alttext="\zeta:=\eta_{0}" class="ltx_Math" display="inline" id="S3.SS2.7.p2.9.m4.1"><semantics id="S3.SS2.7.p2.9.m4.1a"><mrow id="S3.SS2.7.p2.9.m4.1.1" xref="S3.SS2.7.p2.9.m4.1.1.cmml"><mi id="S3.SS2.7.p2.9.m4.1.1.2" xref="S3.SS2.7.p2.9.m4.1.1.2.cmml">ζ</mi><mo id="S3.SS2.7.p2.9.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S3.SS2.7.p2.9.m4.1.1.1.cmml">:=</mo><msub id="S3.SS2.7.p2.9.m4.1.1.3" xref="S3.SS2.7.p2.9.m4.1.1.3.cmml"><mi id="S3.SS2.7.p2.9.m4.1.1.3.2" xref="S3.SS2.7.p2.9.m4.1.1.3.2.cmml">η</mi><mn id="S3.SS2.7.p2.9.m4.1.1.3.3" xref="S3.SS2.7.p2.9.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.9.m4.1b"><apply id="S3.SS2.7.p2.9.m4.1.1.cmml" xref="S3.SS2.7.p2.9.m4.1.1"><csymbol cd="latexml" id="S3.SS2.7.p2.9.m4.1.1.1.cmml" xref="S3.SS2.7.p2.9.m4.1.1.1">assign</csymbol><ci id="S3.SS2.7.p2.9.m4.1.1.2.cmml" xref="S3.SS2.7.p2.9.m4.1.1.2">𝜁</ci><apply id="S3.SS2.7.p2.9.m4.1.1.3.cmml" xref="S3.SS2.7.p2.9.m4.1.1.3"><csymbol cd="ambiguous" id="S3.SS2.7.p2.9.m4.1.1.3.1.cmml" xref="S3.SS2.7.p2.9.m4.1.1.3">subscript</csymbol><ci id="S3.SS2.7.p2.9.m4.1.1.3.2.cmml" xref="S3.SS2.7.p2.9.m4.1.1.3.2">𝜂</ci><cn id="S3.SS2.7.p2.9.m4.1.1.3.3.cmml" type="integer" xref="S3.SS2.7.p2.9.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.9.m4.1c">\zeta:=\eta_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.9.m4.1d">italic_ζ := italic_η start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> if <math alttext="n=0" class="ltx_Math" display="inline" id="S3.SS2.7.p2.10.m5.1"><semantics id="S3.SS2.7.p2.10.m5.1a"><mrow id="S3.SS2.7.p2.10.m5.1.1" xref="S3.SS2.7.p2.10.m5.1.1.cmml"><mi id="S3.SS2.7.p2.10.m5.1.1.2" xref="S3.SS2.7.p2.10.m5.1.1.2.cmml">n</mi><mo id="S3.SS2.7.p2.10.m5.1.1.1" xref="S3.SS2.7.p2.10.m5.1.1.1.cmml">=</mo><mn id="S3.SS2.7.p2.10.m5.1.1.3" xref="S3.SS2.7.p2.10.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.10.m5.1b"><apply id="S3.SS2.7.p2.10.m5.1.1.cmml" xref="S3.SS2.7.p2.10.m5.1.1"><eq id="S3.SS2.7.p2.10.m5.1.1.1.cmml" xref="S3.SS2.7.p2.10.m5.1.1.1"></eq><ci id="S3.SS2.7.p2.10.m5.1.1.2.cmml" xref="S3.SS2.7.p2.10.m5.1.1.2">𝑛</ci><cn id="S3.SS2.7.p2.10.m5.1.1.3.cmml" type="integer" xref="S3.SS2.7.p2.10.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.10.m5.1c">n=0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.10.m5.1d">italic_n = 0</annotation></semantics></math>. Therefore, if <math alttext="g\in\SS(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS2.7.p2.11.m6.2"><semantics id="S3.SS2.7.p2.11.m6.2a"><mrow id="S3.SS2.7.p2.11.m6.2.2" xref="S3.SS2.7.p2.11.m6.2.2.cmml"><mi id="S3.SS2.7.p2.11.m6.2.2.3" xref="S3.SS2.7.p2.11.m6.2.2.3.cmml">g</mi><mo id="S3.SS2.7.p2.11.m6.2.2.2" xref="S3.SS2.7.p2.11.m6.2.2.2.cmml">∈</mo><mrow id="S3.SS2.7.p2.11.m6.2.2.1" xref="S3.SS2.7.p2.11.m6.2.2.1.cmml"><mi id="S3.SS2.7.p2.11.m6.2.2.1.3" xref="S3.SS2.7.p2.11.m6.2.2.1.3.cmml">SS</mi><mo id="S3.SS2.7.p2.11.m6.2.2.1.2" xref="S3.SS2.7.p2.11.m6.2.2.1.2.cmml">⁢</mo><mrow id="S3.SS2.7.p2.11.m6.2.2.1.1.1" xref="S3.SS2.7.p2.11.m6.2.2.1.1.2.cmml"><mo id="S3.SS2.7.p2.11.m6.2.2.1.1.1.2" stretchy="false" xref="S3.SS2.7.p2.11.m6.2.2.1.1.2.cmml">(</mo><msup id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.cmml"><mi id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.2" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.cmml"><mi id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.2" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.2.cmml">d</mi><mo id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.1" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.3" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS2.7.p2.11.m6.2.2.1.1.1.3" xref="S3.SS2.7.p2.11.m6.2.2.1.1.2.cmml">;</mo><mi id="S3.SS2.7.p2.11.m6.1.1" xref="S3.SS2.7.p2.11.m6.1.1.cmml">X</mi><mo id="S3.SS2.7.p2.11.m6.2.2.1.1.1.4" stretchy="false" xref="S3.SS2.7.p2.11.m6.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.11.m6.2b"><apply id="S3.SS2.7.p2.11.m6.2.2.cmml" xref="S3.SS2.7.p2.11.m6.2.2"><in id="S3.SS2.7.p2.11.m6.2.2.2.cmml" xref="S3.SS2.7.p2.11.m6.2.2.2"></in><ci id="S3.SS2.7.p2.11.m6.2.2.3.cmml" xref="S3.SS2.7.p2.11.m6.2.2.3">𝑔</ci><apply id="S3.SS2.7.p2.11.m6.2.2.1.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1"><times id="S3.SS2.7.p2.11.m6.2.2.1.2.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.2"></times><ci id="S3.SS2.7.p2.11.m6.2.2.1.3.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.3">SS</ci><list id="S3.SS2.7.p2.11.m6.2.2.1.1.2.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1"><apply id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.1.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1">superscript</csymbol><ci id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.2.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.2">ℝ</ci><apply id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3"><minus id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.1.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.1"></minus><ci id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.2.cmml" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.7.p2.11.m6.2.2.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS2.7.p2.11.m6.1.1.cmml" xref="S3.SS2.7.p2.11.m6.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.11.m6.2c">g\in\SS(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.11.m6.2d">italic_g ∈ roman_SS ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>, then using <math alttext="\operatorname{Tr}_{j}=\operatorname{Tr}\mathop{\circ}\nolimits\partial_{1}^{j}" class="ltx_Math" display="inline" id="S3.SS2.7.p2.12.m7.1"><semantics id="S3.SS2.7.p2.12.m7.1a"><mrow id="S3.SS2.7.p2.12.m7.1.1" xref="S3.SS2.7.p2.12.m7.1.1.cmml"><msub id="S3.SS2.7.p2.12.m7.1.1.2" xref="S3.SS2.7.p2.12.m7.1.1.2.cmml"><mi id="S3.SS2.7.p2.12.m7.1.1.2.2" xref="S3.SS2.7.p2.12.m7.1.1.2.2.cmml">Tr</mi><mi id="S3.SS2.7.p2.12.m7.1.1.2.3" xref="S3.SS2.7.p2.12.m7.1.1.2.3.cmml">j</mi></msub><mo id="S3.SS2.7.p2.12.m7.1.1.1" xref="S3.SS2.7.p2.12.m7.1.1.1.cmml">=</mo><mrow id="S3.SS2.7.p2.12.m7.1.1.3" xref="S3.SS2.7.p2.12.m7.1.1.3.cmml"><mi id="S3.SS2.7.p2.12.m7.1.1.3.2" xref="S3.SS2.7.p2.12.m7.1.1.3.2.cmml">Tr</mi><mo id="S3.SS2.7.p2.12.m7.1.1.3.1" lspace="0.167em" xref="S3.SS2.7.p2.12.m7.1.1.3.1.cmml">⁢</mo><mrow id="S3.SS2.7.p2.12.m7.1.1.3.3" xref="S3.SS2.7.p2.12.m7.1.1.3.3.cmml"><mo id="S3.SS2.7.p2.12.m7.1.1.3.3.1" rspace="0.0835em" xref="S3.SS2.7.p2.12.m7.1.1.3.3.1.cmml">∘</mo><msubsup id="S3.SS2.7.p2.12.m7.1.1.3.3.2" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2.cmml"><mo id="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.2" lspace="0.0835em" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.2.cmml">∂</mo><mn id="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.3" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.3.cmml">1</mn><mi id="S3.SS2.7.p2.12.m7.1.1.3.3.2.3" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2.3.cmml">j</mi></msubsup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.12.m7.1b"><apply id="S3.SS2.7.p2.12.m7.1.1.cmml" xref="S3.SS2.7.p2.12.m7.1.1"><eq id="S3.SS2.7.p2.12.m7.1.1.1.cmml" xref="S3.SS2.7.p2.12.m7.1.1.1"></eq><apply id="S3.SS2.7.p2.12.m7.1.1.2.cmml" xref="S3.SS2.7.p2.12.m7.1.1.2"><csymbol cd="ambiguous" id="S3.SS2.7.p2.12.m7.1.1.2.1.cmml" xref="S3.SS2.7.p2.12.m7.1.1.2">subscript</csymbol><ci id="S3.SS2.7.p2.12.m7.1.1.2.2.cmml" xref="S3.SS2.7.p2.12.m7.1.1.2.2">Tr</ci><ci id="S3.SS2.7.p2.12.m7.1.1.2.3.cmml" xref="S3.SS2.7.p2.12.m7.1.1.2.3">𝑗</ci></apply><apply id="S3.SS2.7.p2.12.m7.1.1.3.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3"><times id="S3.SS2.7.p2.12.m7.1.1.3.1.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.1"></times><ci id="S3.SS2.7.p2.12.m7.1.1.3.2.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.2">Tr</ci><apply id="S3.SS2.7.p2.12.m7.1.1.3.3.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3"><compose id="S3.SS2.7.p2.12.m7.1.1.3.3.1.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3.1"></compose><apply id="S3.SS2.7.p2.12.m7.1.1.3.3.2.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.7.p2.12.m7.1.1.3.3.2.1.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2">superscript</csymbol><apply id="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2"><csymbol cd="ambiguous" id="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.1.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2">subscript</csymbol><partialdiff id="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.2.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.2"></partialdiff><cn id="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.3.cmml" type="integer" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2.2.3">1</cn></apply><ci id="S3.SS2.7.p2.12.m7.1.1.3.3.2.3.cmml" xref="S3.SS2.7.p2.12.m7.1.1.3.3.2.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.12.m7.1c">\operatorname{Tr}_{j}=\operatorname{Tr}\mathop{\circ}\nolimits\partial_{1}^{j}</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.12.m7.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = roman_Tr ∘ ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT</annotation></semantics></math>, the product rule and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E3" title="In Definition 3.6. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.3</span></a>), gives</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx13"> <tbody id="S3.E13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{Tr}_{j}\big{(}2^{-n}(\mathcal{F}^{-1}\eta_{n}^{m})(% \phi_{n}\ast g)\big{)}" class="ltx_Math" display="inline" id="S3.E13.m1.2"><semantics id="S3.E13.m1.2a"><mrow id="S3.E13.m1.2.2.2" xref="S3.E13.m1.2.2.3.cmml"><msub id="S3.E13.m1.1.1.1.1" xref="S3.E13.m1.1.1.1.1.cmml"><mi id="S3.E13.m1.1.1.1.1.2" xref="S3.E13.m1.1.1.1.1.2.cmml">Tr</mi><mi id="S3.E13.m1.1.1.1.1.3" xref="S3.E13.m1.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.E13.m1.2.2.2a" xref="S3.E13.m1.2.2.3.cmml">⁡</mo><mrow id="S3.E13.m1.2.2.2.2" xref="S3.E13.m1.2.2.3.cmml"><mo id="S3.E13.m1.2.2.2.2.2" maxsize="120%" minsize="120%" xref="S3.E13.m1.2.2.3.cmml">(</mo><mrow id="S3.E13.m1.2.2.2.2.1" xref="S3.E13.m1.2.2.2.2.1.cmml"><msup id="S3.E13.m1.2.2.2.2.1.4" xref="S3.E13.m1.2.2.2.2.1.4.cmml"><mn id="S3.E13.m1.2.2.2.2.1.4.2" xref="S3.E13.m1.2.2.2.2.1.4.2.cmml">2</mn><mrow id="S3.E13.m1.2.2.2.2.1.4.3" xref="S3.E13.m1.2.2.2.2.1.4.3.cmml"><mo id="S3.E13.m1.2.2.2.2.1.4.3a" xref="S3.E13.m1.2.2.2.2.1.4.3.cmml">−</mo><mi id="S3.E13.m1.2.2.2.2.1.4.3.2" xref="S3.E13.m1.2.2.2.2.1.4.3.2.cmml">n</mi></mrow></msup><mo id="S3.E13.m1.2.2.2.2.1.3" xref="S3.E13.m1.2.2.2.2.1.3.cmml">⁢</mo><mrow id="S3.E13.m1.2.2.2.2.1.1.1" xref="S3.E13.m1.2.2.2.2.1.1.1.1.cmml"><mo id="S3.E13.m1.2.2.2.2.1.1.1.2" stretchy="false" xref="S3.E13.m1.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S3.E13.m1.2.2.2.2.1.1.1.1" xref="S3.E13.m1.2.2.2.2.1.1.1.1.cmml"><msup id="S3.E13.m1.2.2.2.2.1.1.1.1.2" xref="S3.E13.m1.2.2.2.2.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" 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xref="S3.E13.m1.2.2.2.2.1.1.1.1.3">superscript</csymbol><apply id="S3.E13.m1.2.2.2.2.1.1.1.1.3.2.cmml" xref="S3.E13.m1.2.2.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.2.2.1.1.1.1.3.2.1.cmml" xref="S3.E13.m1.2.2.2.2.1.1.1.1.3">subscript</csymbol><ci id="S3.E13.m1.2.2.2.2.1.1.1.1.3.2.2.cmml" xref="S3.E13.m1.2.2.2.2.1.1.1.1.3.2.2">𝜂</ci><ci id="S3.E13.m1.2.2.2.2.1.1.1.1.3.2.3.cmml" xref="S3.E13.m1.2.2.2.2.1.1.1.1.3.2.3">𝑛</ci></apply><ci id="S3.E13.m1.2.2.2.2.1.1.1.1.3.3.cmml" xref="S3.E13.m1.2.2.2.2.1.1.1.1.3.3">𝑚</ci></apply></apply><apply id="S3.E13.m1.2.2.2.2.1.2.1.1.cmml" xref="S3.E13.m1.2.2.2.2.1.2.1"><ci id="S3.E13.m1.2.2.2.2.1.2.1.1.1.cmml" xref="S3.E13.m1.2.2.2.2.1.2.1.1.1">∗</ci><apply id="S3.E13.m1.2.2.2.2.1.2.1.1.2.cmml" xref="S3.E13.m1.2.2.2.2.1.2.1.1.2"><csymbol cd="ambiguous" id="S3.E13.m1.2.2.2.2.1.2.1.1.2.1.cmml" xref="S3.E13.m1.2.2.2.2.1.2.1.1.2">subscript</csymbol><ci id="S3.E13.m1.2.2.2.2.1.2.1.1.2.2.cmml" xref="S3.E13.m1.2.2.2.2.1.2.1.1.2.2">italic-ϕ</ci><ci id="S3.E13.m1.2.2.2.2.1.2.1.1.2.3.cmml" xref="S3.E13.m1.2.2.2.2.1.2.1.1.2.3">𝑛</ci></apply><ci id="S3.E13.m1.2.2.2.2.1.2.1.1.3.cmml" xref="S3.E13.m1.2.2.2.2.1.2.1.1.3">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m1.2c">\displaystyle\operatorname{Tr}_{j}\big{(}2^{-n}(\mathcal{F}^{-1}\eta_{n}^{m})(% \phi_{n}\ast g)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m1.2d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_g ) )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S3.E13.m2.1.1.1.1.1.1.2.2.3.cmml" xref="S3.E13.m2.1.1.1.1.1.1.2.2.3"></leq><cn id="S3.E13.m2.1.1.1.1.1.1.2.2.2.cmml" type="integer" xref="S3.E13.m2.1.1.1.1.1.1.2.2.2">0</cn><ci id="S3.E13.m2.1.1.1.1.1.1.2.2.4.cmml" xref="S3.E13.m2.1.1.1.1.1.1.2.2.4">𝑗</ci></apply><apply id="S3.E13.m2.1.1.1.1.1.1.2.2c.cmml" xref="S3.E13.m2.1.1.1.1.1.1.2.2"><leq id="S3.E13.m2.1.1.1.1.1.1.2.2.5.cmml" xref="S3.E13.m2.1.1.1.1.1.1.2.2.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S3.E13.m2.1.1.1.1.1.1.2.2.4.cmml" id="S3.E13.m2.1.1.1.1.1.1.2.2d.cmml" xref="S3.E13.m2.1.1.1.1.1.1.2.2"></share><ci id="S3.E13.m2.1.1.1.1.1.1.2.2.6.cmml" xref="S3.E13.m2.1.1.1.1.1.1.2.2.6">𝑚</ci></apply></apply></apply><apply id="S3.E13.m2.1.1.1.1.2.2.cmml" xref="S3.E13.m2.1.1.1.1.2.2"><geq id="S3.E13.m2.1.1.1.1.2.2.1.cmml" xref="S3.E13.m2.1.1.1.1.2.2.1"></geq><ci id="S3.E13.m2.1.1.1.1.2.2.2.cmml" xref="S3.E13.m2.1.1.1.1.2.2.2">𝑛</ci><cn id="S3.E13.m2.1.1.1.1.2.2.3.cmml" type="integer" xref="S3.E13.m2.1.1.1.1.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.E13.m2.1c">\displaystyle=\delta_{jm}(\phi_{n}\ast g),\qquad 0\leq j\leq m,\,n\geq 0.</annotation><annotation encoding="application/x-llamapun" id="S3.E13.m2.1d">= italic_δ start_POSTSUBSCRIPT italic_j italic_m end_POSTSUBSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_g ) , 0 ≤ italic_j ≤ italic_m , italic_n ≥ 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3.13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.7.p2.13">Therefore, by continuity of the trace operator (Step 1) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E13" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.13</span></a>), we find for <math alttext="0\leq j\leq m" class="ltx_Math" display="inline" id="S3.SS2.7.p2.13.m1.1"><semantics id="S3.SS2.7.p2.13.m1.1a"><mrow id="S3.SS2.7.p2.13.m1.1.1" xref="S3.SS2.7.p2.13.m1.1.1.cmml"><mn id="S3.SS2.7.p2.13.m1.1.1.2" xref="S3.SS2.7.p2.13.m1.1.1.2.cmml">0</mn><mo id="S3.SS2.7.p2.13.m1.1.1.3" xref="S3.SS2.7.p2.13.m1.1.1.3.cmml">≤</mo><mi id="S3.SS2.7.p2.13.m1.1.1.4" xref="S3.SS2.7.p2.13.m1.1.1.4.cmml">j</mi><mo id="S3.SS2.7.p2.13.m1.1.1.5" xref="S3.SS2.7.p2.13.m1.1.1.5.cmml">≤</mo><mi id="S3.SS2.7.p2.13.m1.1.1.6" xref="S3.SS2.7.p2.13.m1.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.13.m1.1b"><apply id="S3.SS2.7.p2.13.m1.1.1.cmml" xref="S3.SS2.7.p2.13.m1.1.1"><and id="S3.SS2.7.p2.13.m1.1.1a.cmml" xref="S3.SS2.7.p2.13.m1.1.1"></and><apply id="S3.SS2.7.p2.13.m1.1.1b.cmml" xref="S3.SS2.7.p2.13.m1.1.1"><leq id="S3.SS2.7.p2.13.m1.1.1.3.cmml" xref="S3.SS2.7.p2.13.m1.1.1.3"></leq><cn id="S3.SS2.7.p2.13.m1.1.1.2.cmml" type="integer" xref="S3.SS2.7.p2.13.m1.1.1.2">0</cn><ci id="S3.SS2.7.p2.13.m1.1.1.4.cmml" xref="S3.SS2.7.p2.13.m1.1.1.4">𝑗</ci></apply><apply id="S3.SS2.7.p2.13.m1.1.1c.cmml" xref="S3.SS2.7.p2.13.m1.1.1"><leq id="S3.SS2.7.p2.13.m1.1.1.5.cmml" xref="S3.SS2.7.p2.13.m1.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S3.SS2.7.p2.13.m1.1.1.4.cmml" id="S3.SS2.7.p2.13.m1.1.1d.cmml" xref="S3.SS2.7.p2.13.m1.1.1"></share><ci id="S3.SS2.7.p2.13.m1.1.1.6.cmml" xref="S3.SS2.7.p2.13.m1.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.13.m1.1c">0\leq j\leq m</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.13.m1.1d">0 ≤ italic_j ≤ italic_m</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex35"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{j}(\operatorname{ext}_{m}g)=\sum_{n=0}^{\infty}% \operatorname{Tr}_{j}\big{(}2^{-n}(\mathcal{F}^{-1}\eta_{n}^{m})(\phi_{n}\ast g% )\big{)}=\delta_{jm}\sum_{n=0}^{\infty}(\phi_{n}\ast g)=\delta_{jm}g," class="ltx_Math" display="block" id="S3.Ex35.m1.1"><semantics id="S3.Ex35.m1.1a"><mrow id="S3.Ex35.m1.1.1.1" xref="S3.Ex35.m1.1.1.1.1.cmml"><mrow id="S3.Ex35.m1.1.1.1.1" xref="S3.Ex35.m1.1.1.1.1.cmml"><mrow id="S3.Ex35.m1.1.1.1.1.2.2" xref="S3.Ex35.m1.1.1.1.1.2.3.cmml"><msub id="S3.Ex35.m1.1.1.1.1.1.1.1" xref="S3.Ex35.m1.1.1.1.1.1.1.1.cmml"><mi id="S3.Ex35.m1.1.1.1.1.1.1.1.2" xref="S3.Ex35.m1.1.1.1.1.1.1.1.2.cmml">Tr</mi><mi id="S3.Ex35.m1.1.1.1.1.1.1.1.3" xref="S3.Ex35.m1.1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S3.Ex35.m1.1.1.1.1.2.2a" xref="S3.Ex35.m1.1.1.1.1.2.3.cmml">⁡</mo><mrow 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xref="S3.Ex35.m1.1.1.1.1.10.3">𝑔</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Ex35.m1.1c">\operatorname{Tr}_{j}(\operatorname{ext}_{m}g)=\sum_{n=0}^{\infty}% \operatorname{Tr}_{j}\big{(}2^{-n}(\mathcal{F}^{-1}\eta_{n}^{m})(\phi_{n}\ast g% )\big{)}=\delta_{jm}\sum_{n=0}^{\infty}(\phi_{n}\ast g)=\delta_{jm}g,</annotation><annotation encoding="application/x-llamapun" id="S3.Ex35.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ) = ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 2 start_POSTSUPERSCRIPT - italic_n end_POSTSUPERSCRIPT ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ) ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_g ) ) = italic_δ start_POSTSUBSCRIPT italic_j italic_m end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_g ) = italic_δ start_POSTSUBSCRIPT italic_j italic_m end_POSTSUBSCRIPT italic_g ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.SS2.7.p2.18">using properties of the Littlewood-Paley sequence in the last identity. By density (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Lemma 3.8]</cite>) this extends to all <math alttext="g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS2.7.p2.14.m1.4"><semantics id="S3.SS2.7.p2.14.m1.4a"><mrow id="S3.SS2.7.p2.14.m1.4.4" xref="S3.SS2.7.p2.14.m1.4.4.cmml"><mi id="S3.SS2.7.p2.14.m1.4.4.3" xref="S3.SS2.7.p2.14.m1.4.4.3.cmml">g</mi><mo id="S3.SS2.7.p2.14.m1.4.4.2" xref="S3.SS2.7.p2.14.m1.4.4.2.cmml">∈</mo><mrow id="S3.SS2.7.p2.14.m1.4.4.1" xref="S3.SS2.7.p2.14.m1.4.4.1.cmml"><msubsup id="S3.SS2.7.p2.14.m1.4.4.1.3" xref="S3.SS2.7.p2.14.m1.4.4.1.3.cmml"><mi id="S3.SS2.7.p2.14.m1.4.4.1.3.2.2" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.2.cmml">B</mi><mrow id="S3.SS2.7.p2.14.m1.2.2.2.4" xref="S3.SS2.7.p2.14.m1.2.2.2.3.cmml"><mi id="S3.SS2.7.p2.14.m1.1.1.1.1" xref="S3.SS2.7.p2.14.m1.1.1.1.1.cmml">p</mi><mo id="S3.SS2.7.p2.14.m1.2.2.2.4.1" 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xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.3" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S3.SS2.7.p2.14.m1.4.4.1.2" xref="S3.SS2.7.p2.14.m1.4.4.1.2.cmml">⁢</mo><mrow id="S3.SS2.7.p2.14.m1.4.4.1.1.1" xref="S3.SS2.7.p2.14.m1.4.4.1.1.2.cmml"><mo id="S3.SS2.7.p2.14.m1.4.4.1.1.1.2" stretchy="false" xref="S3.SS2.7.p2.14.m1.4.4.1.1.2.cmml">(</mo><msup id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.cmml"><mi id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.2" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.2.cmml">ℝ</mi><mrow id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.cmml"><mi id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.2" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.2.cmml">d</mi><mo id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.1" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.1.cmml">−</mo><mn id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.3" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S3.SS2.7.p2.14.m1.4.4.1.1.1.3" xref="S3.SS2.7.p2.14.m1.4.4.1.1.2.cmml">;</mo><mi id="S3.SS2.7.p2.14.m1.3.3" xref="S3.SS2.7.p2.14.m1.3.3.cmml">X</mi><mo id="S3.SS2.7.p2.14.m1.4.4.1.1.1.4" stretchy="false" xref="S3.SS2.7.p2.14.m1.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.14.m1.4b"><apply id="S3.SS2.7.p2.14.m1.4.4.cmml" xref="S3.SS2.7.p2.14.m1.4.4"><in id="S3.SS2.7.p2.14.m1.4.4.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.2"></in><ci id="S3.SS2.7.p2.14.m1.4.4.3.cmml" xref="S3.SS2.7.p2.14.m1.4.4.3">𝑔</ci><apply id="S3.SS2.7.p2.14.m1.4.4.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1"><times id="S3.SS2.7.p2.14.m1.4.4.1.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.2"></times><apply id="S3.SS2.7.p2.14.m1.4.4.1.3.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS2.7.p2.14.m1.4.4.1.3.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3">subscript</csymbol><apply id="S3.SS2.7.p2.14.m1.4.4.1.3.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3"><csymbol cd="ambiguous" id="S3.SS2.7.p2.14.m1.4.4.1.3.2.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3">superscript</csymbol><ci id="S3.SS2.7.p2.14.m1.4.4.1.3.2.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.2">𝐵</ci><apply id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3"><minus id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.1"></minus><ci id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.2">𝑠</ci><ci id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.3.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.3">𝑚</ci><apply id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4"><divide id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4"></divide><apply id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2"><plus id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.1"></plus><ci id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.2">𝛾</ci><cn id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.3.cmml" type="integer" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.2.3">1</cn></apply><ci id="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.3.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.3.2.3.4.3">𝑝</ci></apply></apply></apply><list id="S3.SS2.7.p2.14.m1.2.2.2.3.cmml" xref="S3.SS2.7.p2.14.m1.2.2.2.4"><ci id="S3.SS2.7.p2.14.m1.1.1.1.1.cmml" xref="S3.SS2.7.p2.14.m1.1.1.1.1">𝑝</ci><ci id="S3.SS2.7.p2.14.m1.2.2.2.2.cmml" xref="S3.SS2.7.p2.14.m1.2.2.2.2">𝑞</ci></list></apply><list id="S3.SS2.7.p2.14.m1.4.4.1.1.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1"><apply id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1">superscript</csymbol><ci id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.2">ℝ</ci><apply id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3"><minus id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.1.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.1"></minus><ci id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.2.cmml" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.7.p2.14.m1.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS2.7.p2.14.m1.3.3.cmml" xref="S3.SS2.7.p2.14.m1.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.14.m1.4c">g\in B^{s-m-\frac{\gamma+1}{p}}_{p,q}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.14.m1.4d">italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> with <math alttext="q<\infty" class="ltx_Math" display="inline" id="S3.SS2.7.p2.15.m2.1"><semantics id="S3.SS2.7.p2.15.m2.1a"><mrow id="S3.SS2.7.p2.15.m2.1.1" xref="S3.SS2.7.p2.15.m2.1.1.cmml"><mi id="S3.SS2.7.p2.15.m2.1.1.2" xref="S3.SS2.7.p2.15.m2.1.1.2.cmml">q</mi><mo id="S3.SS2.7.p2.15.m2.1.1.1" xref="S3.SS2.7.p2.15.m2.1.1.1.cmml"><</mo><mi id="S3.SS2.7.p2.15.m2.1.1.3" mathvariant="normal" xref="S3.SS2.7.p2.15.m2.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.15.m2.1b"><apply id="S3.SS2.7.p2.15.m2.1.1.cmml" xref="S3.SS2.7.p2.15.m2.1.1"><lt id="S3.SS2.7.p2.15.m2.1.1.1.cmml" xref="S3.SS2.7.p2.15.m2.1.1.1"></lt><ci id="S3.SS2.7.p2.15.m2.1.1.2.cmml" xref="S3.SS2.7.p2.15.m2.1.1.2">𝑞</ci><infinity id="S3.SS2.7.p2.15.m2.1.1.3.cmml" xref="S3.SS2.7.p2.15.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.15.m2.1c">q<\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.15.m2.1d">italic_q < ∞</annotation></semantics></math>. If <math alttext="q=\infty" class="ltx_Math" display="inline" id="S3.SS2.7.p2.16.m3.1"><semantics id="S3.SS2.7.p2.16.m3.1a"><mrow id="S3.SS2.7.p2.16.m3.1.1" xref="S3.SS2.7.p2.16.m3.1.1.cmml"><mi id="S3.SS2.7.p2.16.m3.1.1.2" xref="S3.SS2.7.p2.16.m3.1.1.2.cmml">q</mi><mo id="S3.SS2.7.p2.16.m3.1.1.1" xref="S3.SS2.7.p2.16.m3.1.1.1.cmml">=</mo><mi id="S3.SS2.7.p2.16.m3.1.1.3" mathvariant="normal" xref="S3.SS2.7.p2.16.m3.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.16.m3.1b"><apply id="S3.SS2.7.p2.16.m3.1.1.cmml" xref="S3.SS2.7.p2.16.m3.1.1"><eq id="S3.SS2.7.p2.16.m3.1.1.1.cmml" xref="S3.SS2.7.p2.16.m3.1.1.1"></eq><ci id="S3.SS2.7.p2.16.m3.1.1.2.cmml" xref="S3.SS2.7.p2.16.m3.1.1.2">𝑞</ci><infinity id="S3.SS2.7.p2.16.m3.1.1.3.cmml" xref="S3.SS2.7.p2.16.m3.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.16.m3.1c">q=\infty</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.16.m3.1d">italic_q = ∞</annotation></semantics></math>, then we have <math alttext="B^{s-m-\frac{\gamma+1}{p}}_{p,\infty}(\mathbb{R}^{d-1};X)\hookrightarrow B^{s-% m-\frac{\gamma+1}{p}-\varepsilon}_{p,1}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S3.SS2.7.p2.17.m4.8"><semantics id="S3.SS2.7.p2.17.m4.8a"><mrow id="S3.SS2.7.p2.17.m4.8.8" xref="S3.SS2.7.p2.17.m4.8.8.cmml"><mrow id="S3.SS2.7.p2.17.m4.7.7.1" xref="S3.SS2.7.p2.17.m4.7.7.1.cmml"><msubsup id="S3.SS2.7.p2.17.m4.7.7.1.3" xref="S3.SS2.7.p2.17.m4.7.7.1.3.cmml"><mi id="S3.SS2.7.p2.17.m4.7.7.1.3.2.2" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.2.cmml">B</mi><mrow id="S3.SS2.7.p2.17.m4.2.2.2.4" xref="S3.SS2.7.p2.17.m4.2.2.2.3.cmml"><mi id="S3.SS2.7.p2.17.m4.1.1.1.1" xref="S3.SS2.7.p2.17.m4.1.1.1.1.cmml">p</mi><mo id="S3.SS2.7.p2.17.m4.2.2.2.4.1" xref="S3.SS2.7.p2.17.m4.2.2.2.3.cmml">,</mo><mi id="S3.SS2.7.p2.17.m4.2.2.2.2" mathvariant="normal" xref="S3.SS2.7.p2.17.m4.2.2.2.2.cmml">∞</mi></mrow><mrow id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.cmml"><mi id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.2" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.2.cmml">s</mi><mo id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.1" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.1.cmml">−</mo><mi id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.3" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.3.cmml">m</mi><mo id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.1a" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.1.cmml">−</mo><mfrac id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.cmml"><mrow id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2.cmml"><mi id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2.2" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2.2.cmml">γ</mi><mo id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2.1" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2.1.cmml">+</mo><mn id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2.3" xref="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S3.SS2.7.p2.17.m4.7.7.1.3.2.3.4.3" 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xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.2">𝐵</ci><apply id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3"><minus id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.1.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.1"></minus><ci id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.2.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.2">𝑠</ci><ci id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.3.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.3">𝑚</ci><apply id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4"><divide id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.1.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4"></divide><apply id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2"><plus id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2.1.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2.1"></plus><ci id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2.2.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2.2">𝛾</ci><cn id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2.3.cmml" type="integer" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.2.3">1</cn></apply><ci id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.3.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.4.3">𝑝</ci></apply><ci id="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.5.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.3.2.3.5">𝜀</ci></apply></apply><list id="S3.SS2.7.p2.17.m4.4.4.2.3.cmml" xref="S3.SS2.7.p2.17.m4.4.4.2.4"><ci id="S3.SS2.7.p2.17.m4.3.3.1.1.cmml" xref="S3.SS2.7.p2.17.m4.3.3.1.1">𝑝</ci><cn id="S3.SS2.7.p2.17.m4.4.4.2.2.cmml" type="integer" xref="S3.SS2.7.p2.17.m4.4.4.2.2">1</cn></list></apply><list id="S3.SS2.7.p2.17.m4.8.8.2.1.2.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1"><apply id="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1.1"><csymbol cd="ambiguous" id="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.1.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1.1">superscript</csymbol><ci id="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.2.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.2">ℝ</ci><apply id="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3"><minus id="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3.1.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3.1"></minus><ci id="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3.2.cmml" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3.2">𝑑</ci><cn id="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3.3.cmml" type="integer" xref="S3.SS2.7.p2.17.m4.8.8.2.1.1.1.3.3">1</cn></apply></apply><ci id="S3.SS2.7.p2.17.m4.6.6.cmml" xref="S3.SS2.7.p2.17.m4.6.6">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.17.m4.8c">B^{s-m-\frac{\gamma+1}{p}}_{p,\infty}(\mathbb{R}^{d-1};X)\hookrightarrow B^{s-% m-\frac{\gamma+1}{p}-\varepsilon}_{p,1}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.17.m4.8d">italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ↪ italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG - italic_ε end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> for all <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S3.SS2.7.p2.18.m5.1"><semantics id="S3.SS2.7.p2.18.m5.1a"><mrow id="S3.SS2.7.p2.18.m5.1.1" xref="S3.SS2.7.p2.18.m5.1.1.cmml"><mi id="S3.SS2.7.p2.18.m5.1.1.2" xref="S3.SS2.7.p2.18.m5.1.1.2.cmml">ε</mi><mo id="S3.SS2.7.p2.18.m5.1.1.1" xref="S3.SS2.7.p2.18.m5.1.1.1.cmml">></mo><mn id="S3.SS2.7.p2.18.m5.1.1.3" xref="S3.SS2.7.p2.18.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.SS2.7.p2.18.m5.1b"><apply id="S3.SS2.7.p2.18.m5.1.1.cmml" xref="S3.SS2.7.p2.18.m5.1.1"><gt id="S3.SS2.7.p2.18.m5.1.1.1.cmml" xref="S3.SS2.7.p2.18.m5.1.1.1"></gt><ci id="S3.SS2.7.p2.18.m5.1.1.2.cmml" xref="S3.SS2.7.p2.18.m5.1.1.2">𝜀</ci><cn id="S3.SS2.7.p2.18.m5.1.1.3.cmml" type="integer" xref="S3.SS2.7.p2.18.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.SS2.7.p2.18.m5.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S3.SS2.7.p2.18.m5.1d">italic_ε > 0</annotation></semantics></math> (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Theorem 14.4.19]</cite>). This proves (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E11" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.11</span></a>). ∎</p> </div> </div> <div class="ltx_para" id="S3.SS2.p5"> <p class="ltx_p" id="S3.SS2.p5.1">We continue with the characterisation of higher-order trace spaces for Triebel-Lizorkin spaces.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S3.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.1.1.1">Theorem 3.9</span></span><span class="ltx_text ltx_font_bold" id="S3.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S3.Thmtheorem9.p1"> <p class="ltx_p" id="S3.Thmtheorem9.p1.6"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem9.p1.6.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.1.1.m1.2"><semantics id="S3.Thmtheorem9.p1.1.1.m1.2a"><mrow id="S3.Thmtheorem9.p1.1.1.m1.2.3" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.cmml"><mi id="S3.Thmtheorem9.p1.1.1.m1.2.3.2" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S3.Thmtheorem9.p1.1.1.m1.2.3.1" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem9.p1.1.1.m1.2.3.3.2" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml"><mo id="S3.Thmtheorem9.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S3.Thmtheorem9.p1.1.1.m1.1.1" xref="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml">1</mn><mo id="S3.Thmtheorem9.p1.1.1.m1.2.3.3.2.2" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.1.1.m1.2.2" mathvariant="normal" xref="S3.Thmtheorem9.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S3.Thmtheorem9.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.1.1.m1.2b"><apply id="S3.Thmtheorem9.p1.1.1.m1.2.3.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.2.3"><in id="S3.Thmtheorem9.p1.1.1.m1.2.3.1.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.1"></in><ci id="S3.Thmtheorem9.p1.1.1.m1.2.3.2.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S3.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.2.3.3.2"><cn id="S3.Thmtheorem9.p1.1.1.m1.1.1.cmml" type="integer" xref="S3.Thmtheorem9.p1.1.1.m1.1.1">1</cn><infinity id="S3.Thmtheorem9.p1.1.1.m1.2.2.cmml" xref="S3.Thmtheorem9.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="q\in[1,\infty]" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.2.2.m2.2"><semantics id="S3.Thmtheorem9.p1.2.2.m2.2a"><mrow id="S3.Thmtheorem9.p1.2.2.m2.2.3" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.cmml"><mi id="S3.Thmtheorem9.p1.2.2.m2.2.3.2" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.2.cmml">q</mi><mo id="S3.Thmtheorem9.p1.2.2.m2.2.3.1" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.1.cmml">∈</mo><mrow id="S3.Thmtheorem9.p1.2.2.m2.2.3.3.2" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.3.1.cmml"><mo id="S3.Thmtheorem9.p1.2.2.m2.2.3.3.2.1" stretchy="false" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.3.1.cmml">[</mo><mn id="S3.Thmtheorem9.p1.2.2.m2.1.1" xref="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml">1</mn><mo id="S3.Thmtheorem9.p1.2.2.m2.2.3.3.2.2" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.3.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.2.2.m2.2.2" mathvariant="normal" xref="S3.Thmtheorem9.p1.2.2.m2.2.2.cmml">∞</mi><mo id="S3.Thmtheorem9.p1.2.2.m2.2.3.3.2.3" stretchy="false" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.3.1.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.2.2.m2.2b"><apply id="S3.Thmtheorem9.p1.2.2.m2.2.3.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.2.3"><in id="S3.Thmtheorem9.p1.2.2.m2.2.3.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.1"></in><ci id="S3.Thmtheorem9.p1.2.2.m2.2.3.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.2">𝑞</ci><interval closure="closed" id="S3.Thmtheorem9.p1.2.2.m2.2.3.3.1.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.2.3.3.2"><cn id="S3.Thmtheorem9.p1.2.2.m2.1.1.cmml" type="integer" xref="S3.Thmtheorem9.p1.2.2.m2.1.1">1</cn><infinity id="S3.Thmtheorem9.p1.2.2.m2.2.2.cmml" xref="S3.Thmtheorem9.p1.2.2.m2.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.2.2.m2.2c">q\in[1,\infty]</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.2.2.m2.2d">italic_q ∈ [ 1 , ∞ ]</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.3.3.m3.1"><semantics id="S3.Thmtheorem9.p1.3.3.m3.1a"><mrow id="S3.Thmtheorem9.p1.3.3.m3.1.1" 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xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.2">ℕ</ci><cn id="S3.Thmtheorem9.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.3.3.m3.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.3.3.m3.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma>-1" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.4.4.m4.1"><semantics id="S3.Thmtheorem9.p1.4.4.m4.1a"><mrow id="S3.Thmtheorem9.p1.4.4.m4.1.1" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml"><mi id="S3.Thmtheorem9.p1.4.4.m4.1.1.2" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.2.cmml">γ</mi><mo id="S3.Thmtheorem9.p1.4.4.m4.1.1.1" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem9.p1.4.4.m4.1.1.3" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.3.cmml"><mo id="S3.Thmtheorem9.p1.4.4.m4.1.1.3a" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.3.cmml">−</mo><mn id="S3.Thmtheorem9.p1.4.4.m4.1.1.3.2" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.4.4.m4.1b"><apply id="S3.Thmtheorem9.p1.4.4.m4.1.1.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1"><gt id="S3.Thmtheorem9.p1.4.4.m4.1.1.1.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.1"></gt><ci id="S3.Thmtheorem9.p1.4.4.m4.1.1.2.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.2">𝛾</ci><apply id="S3.Thmtheorem9.p1.4.4.m4.1.1.3.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.3"><minus id="S3.Thmtheorem9.p1.4.4.m4.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.3"></minus><cn id="S3.Thmtheorem9.p1.4.4.m4.1.1.3.2.cmml" type="integer" xref="S3.Thmtheorem9.p1.4.4.m4.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.4.4.m4.1c">\gamma>-1</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.4.4.m4.1d">italic_γ > - 1</annotation></semantics></math>, <math alttext="s>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.5.5.m5.1"><semantics id="S3.Thmtheorem9.p1.5.5.m5.1a"><mrow id="S3.Thmtheorem9.p1.5.5.m5.1.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml"><mi id="S3.Thmtheorem9.p1.5.5.m5.1.1.2" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml">s</mi><mo id="S3.Thmtheorem9.p1.5.5.m5.1.1.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.1.cmml">></mo><mrow id="S3.Thmtheorem9.p1.5.5.m5.1.1.3" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.cmml"><mi id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.2" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.2.cmml">m</mi><mo id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.1.cmml">+</mo><mfrac id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.cmml"><mrow id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.cmml"><mi id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.2" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.1" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.3" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.3" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.5.5.m5.1b"><apply id="S3.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1"><gt id="S3.Thmtheorem9.p1.5.5.m5.1.1.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.1"></gt><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.2.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.2">𝑠</ci><apply id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3"><plus id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.1"></plus><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.2.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.2">𝑚</ci><apply id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3"><divide id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3"></divide><apply id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2"><plus id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.1.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.1"></plus><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.2.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.2">𝛾</ci><cn id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.2.3">1</cn></apply><ci id="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.3.cmml" xref="S3.Thmtheorem9.p1.5.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.5.5.m5.1c">s>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.5.5.m5.1d">italic_s > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.6.6.m6.1"><semantics id="S3.Thmtheorem9.p1.6.6.m6.1a"><mi id="S3.Thmtheorem9.p1.6.6.m6.1.1" xref="S3.Thmtheorem9.p1.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.6.6.m6.1b"><ci id="S3.Thmtheorem9.p1.6.6.m6.1.1.cmml" xref="S3.Thmtheorem9.p1.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.6.6.m6.1d">italic_X</annotation></semantics></math> be a Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S3.Ex36"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{m}:F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B^{s-m-\frac% {\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S3.Ex36.m1.9"><semantics id="S3.Ex36.m1.9a"><mrow id="S3.Ex36.m1.9.9" xref="S3.Ex36.m1.9.9.cmml"><msub id="S3.Ex36.m1.9.9.5" xref="S3.Ex36.m1.9.9.5.cmml"><mi id="S3.Ex36.m1.9.9.5.2" xref="S3.Ex36.m1.9.9.5.2.cmml">Tr</mi><mi id="S3.Ex36.m1.9.9.5.3" xref="S3.Ex36.m1.9.9.5.3.cmml">m</mi></msub><mo id="S3.Ex36.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S3.Ex36.m1.9.9.4.cmml">:</mo><mrow id="S3.Ex36.m1.9.9.3" xref="S3.Ex36.m1.9.9.3.cmml"><mrow id="S3.Ex36.m1.8.8.2.2" xref="S3.Ex36.m1.8.8.2.2.cmml"><msubsup id="S3.Ex36.m1.8.8.2.2.4" xref="S3.Ex36.m1.8.8.2.2.4.cmml"><mi id="S3.Ex36.m1.8.8.2.2.4.2.2" xref="S3.Ex36.m1.8.8.2.2.4.2.2.cmml">F</mi><mrow id="S3.Ex36.m1.2.2.2.4" xref="S3.Ex36.m1.2.2.2.3.cmml"><mi id="S3.Ex36.m1.1.1.1.1" xref="S3.Ex36.m1.1.1.1.1.cmml">p</mi><mo id="S3.Ex36.m1.2.2.2.4.1" xref="S3.Ex36.m1.2.2.2.3.cmml">,</mo><mi id="S3.Ex36.m1.2.2.2.2" xref="S3.Ex36.m1.2.2.2.2.cmml">q</mi></mrow><mi id="S3.Ex36.m1.8.8.2.2.4.2.3" xref="S3.Ex36.m1.8.8.2.2.4.2.3.cmml">s</mi></msubsup><mo id="S3.Ex36.m1.8.8.2.2.3" xref="S3.Ex36.m1.8.8.2.2.3.cmml">⁢</mo><mrow id="S3.Ex36.m1.8.8.2.2.2.2" xref="S3.Ex36.m1.8.8.2.2.2.3.cmml"><mo id="S3.Ex36.m1.8.8.2.2.2.2.3" stretchy="false" xref="S3.Ex36.m1.8.8.2.2.2.3.cmml">(</mo><msup id="S3.Ex36.m1.7.7.1.1.1.1.1" xref="S3.Ex36.m1.7.7.1.1.1.1.1.cmml"><mi id="S3.Ex36.m1.7.7.1.1.1.1.1.2" xref="S3.Ex36.m1.7.7.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S3.Ex36.m1.7.7.1.1.1.1.1.3" xref="S3.Ex36.m1.7.7.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S3.Ex36.m1.8.8.2.2.2.2.4" xref="S3.Ex36.m1.8.8.2.2.2.3.cmml">,</mo><msub id="S3.Ex36.m1.8.8.2.2.2.2.2" xref="S3.Ex36.m1.8.8.2.2.2.2.2.cmml"><mi id="S3.Ex36.m1.8.8.2.2.2.2.2.2" xref="S3.Ex36.m1.8.8.2.2.2.2.2.2.cmml">w</mi><mi id="S3.Ex36.m1.8.8.2.2.2.2.2.3" xref="S3.Ex36.m1.8.8.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S3.Ex36.m1.8.8.2.2.2.2.5" xref="S3.Ex36.m1.8.8.2.2.2.3.cmml">;</mo><mi id="S3.Ex36.m1.5.5" xref="S3.Ex36.m1.5.5.cmml">X</mi><mo id="S3.Ex36.m1.8.8.2.2.2.2.6" stretchy="false" xref="S3.Ex36.m1.8.8.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S3.Ex36.m1.9.9.3.4" stretchy="false" xref="S3.Ex36.m1.9.9.3.4.cmml">→</mo><mrow id="S3.Ex36.m1.9.9.3.3" xref="S3.Ex36.m1.9.9.3.3.cmml"><msubsup id="S3.Ex36.m1.9.9.3.3.3" xref="S3.Ex36.m1.9.9.3.3.3.cmml"><mi id="S3.Ex36.m1.9.9.3.3.3.2.2" xref="S3.Ex36.m1.9.9.3.3.3.2.2.cmml">B</mi><mrow id="S3.Ex36.m1.4.4.2.4" xref="S3.Ex36.m1.4.4.2.3.cmml"><mi id="S3.Ex36.m1.3.3.1.1" xref="S3.Ex36.m1.3.3.1.1.cmml">p</mi><mo id="S3.Ex36.m1.4.4.2.4.1" xref="S3.Ex36.m1.4.4.2.3.cmml">,</mo><mi id="S3.Ex36.m1.4.4.2.2" xref="S3.Ex36.m1.4.4.2.2.cmml">p</mi></mrow><mrow id="S3.Ex36.m1.9.9.3.3.3.2.3" xref="S3.Ex36.m1.9.9.3.3.3.2.3.cmml"><mi id="S3.Ex36.m1.9.9.3.3.3.2.3.2" xref="S3.Ex36.m1.9.9.3.3.3.2.3.2.cmml">s</mi><mo id="S3.Ex36.m1.9.9.3.3.3.2.3.1" xref="S3.Ex36.m1.9.9.3.3.3.2.3.1.cmml">−</mo><mi id="S3.Ex36.m1.9.9.3.3.3.2.3.3" xref="S3.Ex36.m1.9.9.3.3.3.2.3.3.cmml">m</mi><mo id="S3.Ex36.m1.9.9.3.3.3.2.3.1a" xref="S3.Ex36.m1.9.9.3.3.3.2.3.1.cmml">−</mo><mfrac id="S3.Ex36.m1.9.9.3.3.3.2.3.4" xref="S3.Ex36.m1.9.9.3.3.3.2.3.4.cmml"><mrow id="S3.Ex36.m1.9.9.3.3.3.2.3.4.2" xref="S3.Ex36.m1.9.9.3.3.3.2.3.4.2.cmml"><mi id="S3.Ex36.m1.9.9.3.3.3.2.3.4.2.2" xref="S3.Ex36.m1.9.9.3.3.3.2.3.4.2.2.cmml">γ</mi><mo id="S3.Ex36.m1.9.9.3.3.3.2.3.4.2.1" xref="S3.Ex36.m1.9.9.3.3.3.2.3.4.2.1.cmml">+</mo><mn id="S3.Ex36.m1.9.9.3.3.3.2.3.4.2.3" xref="S3.Ex36.m1.9.9.3.3.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S3.Ex36.m1.9.9.3.3.3.2.3.4.3" xref="S3.Ex36.m1.9.9.3.3.3.2.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo 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id="S3.Ex36.m1.9c">\operatorname{Tr}_{m}:F^{s}_{p,q}(\mathbb{R}^{d},w_{\gamma};X)\to B^{s-m-\frac% {\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S3.Ex36.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_q end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S3.Thmtheorem9.p1.12"><span class="ltx_text ltx_font_italic" id="S3.Thmtheorem9.p1.12.6">is a continuous and surjective operator. Moreover, there exists a continuous right inverse <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.7.1.m1.1"><semantics id="S3.Thmtheorem9.p1.7.1.m1.1a"><msub id="S3.Thmtheorem9.p1.7.1.m1.1.1" xref="S3.Thmtheorem9.p1.7.1.m1.1.1.cmml"><mi id="S3.Thmtheorem9.p1.7.1.m1.1.1.2" xref="S3.Thmtheorem9.p1.7.1.m1.1.1.2.cmml">ext</mi><mi id="S3.Thmtheorem9.p1.7.1.m1.1.1.3" xref="S3.Thmtheorem9.p1.7.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.7.1.m1.1b"><apply id="S3.Thmtheorem9.p1.7.1.m1.1.1.cmml" xref="S3.Thmtheorem9.p1.7.1.m1.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.7.1.m1.1.1.1.cmml" xref="S3.Thmtheorem9.p1.7.1.m1.1.1">subscript</csymbol><ci id="S3.Thmtheorem9.p1.7.1.m1.1.1.2.cmml" xref="S3.Thmtheorem9.p1.7.1.m1.1.1.2">ext</ci><ci id="S3.Thmtheorem9.p1.7.1.m1.1.1.3.cmml" xref="S3.Thmtheorem9.p1.7.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.7.1.m1.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.7.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.8.2.m2.1"><semantics id="S3.Thmtheorem9.p1.8.2.m2.1a"><msub id="S3.Thmtheorem9.p1.8.2.m2.1.1" xref="S3.Thmtheorem9.p1.8.2.m2.1.1.cmml"><mi id="S3.Thmtheorem9.p1.8.2.m2.1.1.2" xref="S3.Thmtheorem9.p1.8.2.m2.1.1.2.cmml">Tr</mi><mi id="S3.Thmtheorem9.p1.8.2.m2.1.1.3" xref="S3.Thmtheorem9.p1.8.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.8.2.m2.1b"><apply id="S3.Thmtheorem9.p1.8.2.m2.1.1.cmml" xref="S3.Thmtheorem9.p1.8.2.m2.1.1"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.8.2.m2.1.1.1.cmml" xref="S3.Thmtheorem9.p1.8.2.m2.1.1">subscript</csymbol><ci id="S3.Thmtheorem9.p1.8.2.m2.1.1.2.cmml" xref="S3.Thmtheorem9.p1.8.2.m2.1.1.2">Tr</ci><ci id="S3.Thmtheorem9.p1.8.2.m2.1.1.3.cmml" xref="S3.Thmtheorem9.p1.8.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.8.2.m2.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.8.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> which is independent of <math alttext="s,p,q,\gamma" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.9.3.m3.4"><semantics id="S3.Thmtheorem9.p1.9.3.m3.4a"><mrow id="S3.Thmtheorem9.p1.9.3.m3.4.5.2" xref="S3.Thmtheorem9.p1.9.3.m3.4.5.1.cmml"><mi id="S3.Thmtheorem9.p1.9.3.m3.1.1" xref="S3.Thmtheorem9.p1.9.3.m3.1.1.cmml">s</mi><mo id="S3.Thmtheorem9.p1.9.3.m3.4.5.2.1" xref="S3.Thmtheorem9.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.9.3.m3.2.2" xref="S3.Thmtheorem9.p1.9.3.m3.2.2.cmml">p</mi><mo id="S3.Thmtheorem9.p1.9.3.m3.4.5.2.2" xref="S3.Thmtheorem9.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.9.3.m3.3.3" xref="S3.Thmtheorem9.p1.9.3.m3.3.3.cmml">q</mi><mo id="S3.Thmtheorem9.p1.9.3.m3.4.5.2.3" xref="S3.Thmtheorem9.p1.9.3.m3.4.5.1.cmml">,</mo><mi id="S3.Thmtheorem9.p1.9.3.m3.4.4" xref="S3.Thmtheorem9.p1.9.3.m3.4.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.9.3.m3.4b"><list id="S3.Thmtheorem9.p1.9.3.m3.4.5.1.cmml" xref="S3.Thmtheorem9.p1.9.3.m3.4.5.2"><ci id="S3.Thmtheorem9.p1.9.3.m3.1.1.cmml" xref="S3.Thmtheorem9.p1.9.3.m3.1.1">𝑠</ci><ci id="S3.Thmtheorem9.p1.9.3.m3.2.2.cmml" xref="S3.Thmtheorem9.p1.9.3.m3.2.2">𝑝</ci><ci id="S3.Thmtheorem9.p1.9.3.m3.3.3.cmml" xref="S3.Thmtheorem9.p1.9.3.m3.3.3">𝑞</ci><ci id="S3.Thmtheorem9.p1.9.3.m3.4.4.cmml" xref="S3.Thmtheorem9.p1.9.3.m3.4.4">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.9.3.m3.4c">s,p,q,\gamma</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.9.3.m3.4d">italic_s , italic_p , italic_q , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.10.4.m4.1"><semantics id="S3.Thmtheorem9.p1.10.4.m4.1a"><mi id="S3.Thmtheorem9.p1.10.4.m4.1.1" xref="S3.Thmtheorem9.p1.10.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.10.4.m4.1b"><ci id="S3.Thmtheorem9.p1.10.4.m4.1.1.cmml" xref="S3.Thmtheorem9.p1.10.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.10.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.10.4.m4.1d">italic_X</annotation></semantics></math>. For any <math alttext="0\leq j<m" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.11.5.m5.1"><semantics id="S3.Thmtheorem9.p1.11.5.m5.1a"><mrow id="S3.Thmtheorem9.p1.11.5.m5.1.1" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.cmml"><mn id="S3.Thmtheorem9.p1.11.5.m5.1.1.2" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.2.cmml">0</mn><mo id="S3.Thmtheorem9.p1.11.5.m5.1.1.3" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.3.cmml">≤</mo><mi id="S3.Thmtheorem9.p1.11.5.m5.1.1.4" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.4.cmml">j</mi><mo id="S3.Thmtheorem9.p1.11.5.m5.1.1.5" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.5.cmml"><</mo><mi id="S3.Thmtheorem9.p1.11.5.m5.1.1.6" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.11.5.m5.1b"><apply id="S3.Thmtheorem9.p1.11.5.m5.1.1.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1"><and id="S3.Thmtheorem9.p1.11.5.m5.1.1a.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1"></and><apply id="S3.Thmtheorem9.p1.11.5.m5.1.1b.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1"><leq id="S3.Thmtheorem9.p1.11.5.m5.1.1.3.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.3"></leq><cn id="S3.Thmtheorem9.p1.11.5.m5.1.1.2.cmml" type="integer" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.2">0</cn><ci id="S3.Thmtheorem9.p1.11.5.m5.1.1.4.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.4">𝑗</ci></apply><apply id="S3.Thmtheorem9.p1.11.5.m5.1.1c.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1"><lt id="S3.Thmtheorem9.p1.11.5.m5.1.1.5.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem9.p1.11.5.m5.1.1.4.cmml" id="S3.Thmtheorem9.p1.11.5.m5.1.1d.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1"></share><ci id="S3.Thmtheorem9.p1.11.5.m5.1.1.6.cmml" xref="S3.Thmtheorem9.p1.11.5.m5.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.11.5.m5.1c">0\leq j<m</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.11.5.m5.1d">0 ≤ italic_j < italic_m</annotation></semantics></math> we have <math alttext="\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0" class="ltx_Math" display="inline" id="S3.Thmtheorem9.p1.12.6.m6.1"><semantics id="S3.Thmtheorem9.p1.12.6.m6.1a"><mrow id="S3.Thmtheorem9.p1.12.6.m6.1.1" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.cmml"><mrow id="S3.Thmtheorem9.p1.12.6.m6.1.1.2" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.cmml"><msub id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.cmml"><mi id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.2" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.2.cmml">Tr</mi><mi id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.3" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.3.cmml">j</mi></msub><mo id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.1" lspace="0.167em" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.1.cmml">⁢</mo><mrow id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.cmml"><mo id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.1" rspace="0.167em" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.1.cmml">∘</mo><msub id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.cmml"><mi id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.2" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.2.cmml">ext</mi><mi id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.3" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.3.cmml">m</mi></msub></mrow></mrow><mo id="S3.Thmtheorem9.p1.12.6.m6.1.1.1" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.1.cmml">=</mo><mn id="S3.Thmtheorem9.p1.12.6.m6.1.1.3" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S3.Thmtheorem9.p1.12.6.m6.1b"><apply id="S3.Thmtheorem9.p1.12.6.m6.1.1.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1"><eq id="S3.Thmtheorem9.p1.12.6.m6.1.1.1.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.1"></eq><apply id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2"><times id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.1.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.1"></times><apply id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.1.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2">subscript</csymbol><ci id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.2.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.2">Tr</ci><ci id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.3.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.2.3">𝑗</ci></apply><apply id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3"><compose id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.1.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.1"></compose><apply id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2"><csymbol cd="ambiguous" id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.1.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2">subscript</csymbol><ci id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.2.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.2">ext</ci><ci id="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.3.cmml" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.2.3.2.3">𝑚</ci></apply></apply></apply><cn id="S3.Thmtheorem9.p1.12.6.m6.1.1.3.cmml" type="integer" xref="S3.Thmtheorem9.p1.12.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.Thmtheorem9.p1.12.6.m6.1c">\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0</annotation><annotation encoding="application/x-llamapun" id="S3.Thmtheorem9.p1.12.6.m6.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S3.SS2.8"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S3.SS2.8.p1"> <p class="ltx_p" id="S3.SS2.8.p1.1">The proof is similar to the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem5" title="Theorem 3.5. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.5</span></a> using Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem8" title="Theorem 3.8. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.8</span></a> instead of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem4" title="Theorem 3.4. ‣ 3.1. The zeroth-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.4</span></a>. ∎</p> </div> </div> </section> </section> <section class="ltx_section" id="S4"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">4. </span>Trace spaces of Bessel potential and Sobolev spaces</h2> <div class="ltx_para" id="S4.p1"> <p class="ltx_p" id="S4.p1.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.p1.1.m1.2"><semantics id="S4.p1.1.m1.2a"><mrow id="S4.p1.1.m1.2.3" xref="S4.p1.1.m1.2.3.cmml"><mi id="S4.p1.1.m1.2.3.2" xref="S4.p1.1.m1.2.3.2.cmml">p</mi><mo id="S4.p1.1.m1.2.3.1" xref="S4.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.p1.1.m1.2.3.3.2" xref="S4.p1.1.m1.2.3.3.1.cmml"><mo id="S4.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.p1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.p1.1.m1.1.1" xref="S4.p1.1.m1.1.1.cmml">1</mn><mo id="S4.p1.1.m1.2.3.3.2.2" xref="S4.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.p1.1.m1.2.2" mathvariant="normal" xref="S4.p1.1.m1.2.2.cmml">∞</mi><mo id="S4.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.1.m1.2b"><apply id="S4.p1.1.m1.2.3.cmml" xref="S4.p1.1.m1.2.3"><in id="S4.p1.1.m1.2.3.1.cmml" xref="S4.p1.1.m1.2.3.1"></in><ci id="S4.p1.1.m1.2.3.2.cmml" xref="S4.p1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.p1.1.m1.2.3.3.1.cmml" xref="S4.p1.1.m1.2.3.3.2"><cn id="S4.p1.1.m1.1.1.cmml" type="integer" xref="S4.p1.1.m1.1.1">1</cn><infinity id="S4.p1.1.m1.2.2.cmml" xref="S4.p1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.p1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="X" class="ltx_Math" display="inline" id="S4.p1.2.m2.1"><semantics id="S4.p1.2.m2.1a"><mi id="S4.p1.2.m2.1.1" xref="S4.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.p1.2.m2.1b"><ci id="S4.p1.2.m2.1.1.cmml" xref="S4.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.p1.2.m2.1d">italic_X</annotation></semantics></math> a Banach space and let <math alttext="\mathcal{O}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.p1.3.m3.1"><semantics id="S4.p1.3.m3.1a"><mrow id="S4.p1.3.m3.1.1" xref="S4.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p1.3.m3.1.1.2" xref="S4.p1.3.m3.1.1.2.cmml">𝒪</mi><mo id="S4.p1.3.m3.1.1.1" xref="S4.p1.3.m3.1.1.1.cmml">⊆</mo><msup id="S4.p1.3.m3.1.1.3" xref="S4.p1.3.m3.1.1.3.cmml"><mi id="S4.p1.3.m3.1.1.3.2" xref="S4.p1.3.m3.1.1.3.2.cmml">ℝ</mi><mi id="S4.p1.3.m3.1.1.3.3" xref="S4.p1.3.m3.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.3.m3.1b"><apply id="S4.p1.3.m3.1.1.cmml" xref="S4.p1.3.m3.1.1"><subset id="S4.p1.3.m3.1.1.1.cmml" xref="S4.p1.3.m3.1.1.1"></subset><ci id="S4.p1.3.m3.1.1.2.cmml" xref="S4.p1.3.m3.1.1.2">𝒪</ci><apply id="S4.p1.3.m3.1.1.3.cmml" xref="S4.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S4.p1.3.m3.1.1.3.1.cmml" xref="S4.p1.3.m3.1.1.3">superscript</csymbol><ci id="S4.p1.3.m3.1.1.3.2.cmml" xref="S4.p1.3.m3.1.1.3.2">ℝ</ci><ci id="S4.p1.3.m3.1.1.3.3.cmml" xref="S4.p1.3.m3.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.3.m3.1c">\mathcal{O}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.3.m3.1d">caligraphic_O ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> be open. We define the power weight <math alttext="w_{\gamma}^{\partial\mathcal{O}}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}" class="ltx_math_unparsed" display="inline" id="S4.p1.4.m4.3"><semantics id="S4.p1.4.m4.3a"><mrow id="S4.p1.4.m4.3b"><msubsup id="S4.p1.4.m4.3.4"><mi id="S4.p1.4.m4.3.4.2.2">w</mi><mi id="S4.p1.4.m4.3.4.2.3">γ</mi><mrow id="S4.p1.4.m4.3.4.3"><mo id="S4.p1.4.m4.3.4.3.1" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S4.p1.4.m4.3.4.3.2">𝒪</mi></mrow></msubsup><mrow id="S4.p1.4.m4.3.5"><mo id="S4.p1.4.m4.3.5.1" stretchy="false">(</mo><mi id="S4.p1.4.m4.1.1">x</mi><mo id="S4.p1.4.m4.3.5.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S4.p1.4.m4.3.6" rspace="0.278em">:=</mo><mi id="S4.p1.4.m4.2.2">dist</mi><msup id="S4.p1.4.m4.3.7"><mrow id="S4.p1.4.m4.3.7.2"><mo id="S4.p1.4.m4.3.7.2.1" stretchy="false">(</mo><mi id="S4.p1.4.m4.3.3">x</mi><mo id="S4.p1.4.m4.3.7.2.2">,</mo><mo id="S4.p1.4.m4.3.7.2.3" lspace="0em" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S4.p1.4.m4.3.7.2.4">𝒪</mi><mo id="S4.p1.4.m4.3.7.2.5" stretchy="false">)</mo></mrow><mi id="S4.p1.4.m4.3.7.3">γ</mi></msup></mrow><annotation encoding="application/x-tex" id="S4.p1.4.m4.3c">w_{\gamma}^{\partial\mathcal{O}}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.4.m4.3d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT ( italic_x ) := roman_dist ( italic_x , ∂ caligraphic_O ) start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="x\in\mathcal{O}" class="ltx_Math" display="inline" id="S4.p1.5.m5.1"><semantics id="S4.p1.5.m5.1a"><mrow id="S4.p1.5.m5.1.1" xref="S4.p1.5.m5.1.1.cmml"><mi id="S4.p1.5.m5.1.1.2" xref="S4.p1.5.m5.1.1.2.cmml">x</mi><mo id="S4.p1.5.m5.1.1.1" xref="S4.p1.5.m5.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S4.p1.5.m5.1.1.3" xref="S4.p1.5.m5.1.1.3.cmml">𝒪</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.5.m5.1b"><apply id="S4.p1.5.m5.1.1.cmml" xref="S4.p1.5.m5.1.1"><in id="S4.p1.5.m5.1.1.1.cmml" xref="S4.p1.5.m5.1.1.1"></in><ci id="S4.p1.5.m5.1.1.2.cmml" xref="S4.p1.5.m5.1.1.2">𝑥</ci><ci id="S4.p1.5.m5.1.1.3.cmml" xref="S4.p1.5.m5.1.1.3">𝒪</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.5.m5.1c">x\in\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.5.m5.1d">italic_x ∈ caligraphic_O</annotation></semantics></math> and <math alttext="\gamma\in\mathbb{R}" class="ltx_Math" display="inline" id="S4.p1.6.m6.1"><semantics id="S4.p1.6.m6.1a"><mrow id="S4.p1.6.m6.1.1" xref="S4.p1.6.m6.1.1.cmml"><mi id="S4.p1.6.m6.1.1.2" xref="S4.p1.6.m6.1.1.2.cmml">γ</mi><mo id="S4.p1.6.m6.1.1.1" xref="S4.p1.6.m6.1.1.1.cmml">∈</mo><mi id="S4.p1.6.m6.1.1.3" xref="S4.p1.6.m6.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.6.m6.1b"><apply id="S4.p1.6.m6.1.1.cmml" xref="S4.p1.6.m6.1.1"><in id="S4.p1.6.m6.1.1.1.cmml" xref="S4.p1.6.m6.1.1.1"></in><ci id="S4.p1.6.m6.1.1.2.cmml" xref="S4.p1.6.m6.1.1.2">𝛾</ci><ci id="S4.p1.6.m6.1.1.3.cmml" xref="S4.p1.6.m6.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.6.m6.1c">\gamma\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.6.m6.1d">italic_γ ∈ blackboard_R</annotation></semantics></math>. In the rest of this paper, we mainly consider the following two weighted spaces:</p> <ol class="ltx_enumerate" id="S4.I1"> <li class="ltx_item" id="S4.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S4.I1.i1.p1"> <p class="ltx_p" id="S4.I1.i1.p1.3">Bessel potential spaces <math alttext="H^{s,p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.1.m1.5"><semantics id="S4.I1.i1.p1.1.m1.5a"><mrow id="S4.I1.i1.p1.1.m1.5.5" xref="S4.I1.i1.p1.1.m1.5.5.cmml"><msup id="S4.I1.i1.p1.1.m1.5.5.3" xref="S4.I1.i1.p1.1.m1.5.5.3.cmml"><mi id="S4.I1.i1.p1.1.m1.5.5.3.2" xref="S4.I1.i1.p1.1.m1.5.5.3.2.cmml">H</mi><mrow id="S4.I1.i1.p1.1.m1.2.2.2.4" xref="S4.I1.i1.p1.1.m1.2.2.2.3.cmml"><mi id="S4.I1.i1.p1.1.m1.1.1.1.1" xref="S4.I1.i1.p1.1.m1.1.1.1.1.cmml">s</mi><mo id="S4.I1.i1.p1.1.m1.2.2.2.4.1" xref="S4.I1.i1.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.I1.i1.p1.1.m1.2.2.2.2" xref="S4.I1.i1.p1.1.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.I1.i1.p1.1.m1.5.5.2" xref="S4.I1.i1.p1.1.m1.5.5.2.cmml">⁢</mo><mrow id="S4.I1.i1.p1.1.m1.5.5.1.1" xref="S4.I1.i1.p1.1.m1.5.5.1.2.cmml"><mo id="S4.I1.i1.p1.1.m1.5.5.1.1.2" stretchy="false" xref="S4.I1.i1.p1.1.m1.5.5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.I1.i1.p1.1.m1.3.3" xref="S4.I1.i1.p1.1.m1.3.3.cmml">𝒪</mi><mo id="S4.I1.i1.p1.1.m1.5.5.1.1.3" xref="S4.I1.i1.p1.1.m1.5.5.1.2.cmml">,</mo><msubsup id="S4.I1.i1.p1.1.m1.5.5.1.1.1" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.cmml"><mi id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.2" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.2.cmml">w</mi><mi id="S4.I1.i1.p1.1.m1.5.5.1.1.1.3" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.3.cmml">γ</mi><mrow id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.cmml"><mo id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.1" rspace="0em" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.1.cmml">∂</mo><mi class="ltx_font_mathcaligraphic" id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.2" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.2.cmml">𝒪</mi></mrow></msubsup><mo id="S4.I1.i1.p1.1.m1.5.5.1.1.4" xref="S4.I1.i1.p1.1.m1.5.5.1.2.cmml">;</mo><mi id="S4.I1.i1.p1.1.m1.4.4" xref="S4.I1.i1.p1.1.m1.4.4.cmml">X</mi><mo id="S4.I1.i1.p1.1.m1.5.5.1.1.5" stretchy="false" xref="S4.I1.i1.p1.1.m1.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.1.m1.5b"><apply id="S4.I1.i1.p1.1.m1.5.5.cmml" xref="S4.I1.i1.p1.1.m1.5.5"><times id="S4.I1.i1.p1.1.m1.5.5.2.cmml" xref="S4.I1.i1.p1.1.m1.5.5.2"></times><apply id="S4.I1.i1.p1.1.m1.5.5.3.cmml" xref="S4.I1.i1.p1.1.m1.5.5.3"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.5.5.3.1.cmml" xref="S4.I1.i1.p1.1.m1.5.5.3">superscript</csymbol><ci id="S4.I1.i1.p1.1.m1.5.5.3.2.cmml" xref="S4.I1.i1.p1.1.m1.5.5.3.2">𝐻</ci><list id="S4.I1.i1.p1.1.m1.2.2.2.3.cmml" xref="S4.I1.i1.p1.1.m1.2.2.2.4"><ci id="S4.I1.i1.p1.1.m1.1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.1.1.1.1">𝑠</ci><ci id="S4.I1.i1.p1.1.m1.2.2.2.2.cmml" xref="S4.I1.i1.p1.1.m1.2.2.2.2">𝑝</ci></list></apply><vector id="S4.I1.i1.p1.1.m1.5.5.1.2.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1"><ci id="S4.I1.i1.p1.1.m1.3.3.cmml" xref="S4.I1.i1.p1.1.m1.3.3">𝒪</ci><apply id="S4.I1.i1.p1.1.m1.5.5.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.5.5.1.1.1.1.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1">subscript</csymbol><apply id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.1.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1">superscript</csymbol><ci id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.2.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.2">𝑤</ci><apply id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3"><partialdiff id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.1.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.1"></partialdiff><ci id="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.2.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.2.3.2">𝒪</ci></apply></apply><ci id="S4.I1.i1.p1.1.m1.5.5.1.1.1.3.cmml" xref="S4.I1.i1.p1.1.m1.5.5.1.1.1.3">𝛾</ci></apply><ci id="S4.I1.i1.p1.1.m1.4.4.cmml" xref="S4.I1.i1.p1.1.m1.4.4">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.1.m1.5c">H^{s,p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.1.m1.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> with <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S4.I1.i1.p1.2.m2.1"><semantics id="S4.I1.i1.p1.2.m2.1a"><mrow id="S4.I1.i1.p1.2.m2.1.1" xref="S4.I1.i1.p1.2.m2.1.1.cmml"><mi id="S4.I1.i1.p1.2.m2.1.1.2" xref="S4.I1.i1.p1.2.m2.1.1.2.cmml">s</mi><mo id="S4.I1.i1.p1.2.m2.1.1.1" xref="S4.I1.i1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S4.I1.i1.p1.2.m2.1.1.3" xref="S4.I1.i1.p1.2.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.2.m2.1b"><apply id="S4.I1.i1.p1.2.m2.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1"><in id="S4.I1.i1.p1.2.m2.1.1.1.cmml" xref="S4.I1.i1.p1.2.m2.1.1.1"></in><ci id="S4.I1.i1.p1.2.m2.1.1.2.cmml" xref="S4.I1.i1.p1.2.m2.1.1.2">𝑠</ci><ci id="S4.I1.i1.p1.2.m2.1.1.3.cmml" xref="S4.I1.i1.p1.2.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.2.m2.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.2.m2.1d">italic_s ∈ blackboard_R</annotation></semantics></math> and <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.I1.i1.p1.3.m3.2"><semantics id="S4.I1.i1.p1.3.m3.2a"><mrow id="S4.I1.i1.p1.3.m3.2.2" xref="S4.I1.i1.p1.3.m3.2.2.cmml"><mi id="S4.I1.i1.p1.3.m3.2.2.4" xref="S4.I1.i1.p1.3.m3.2.2.4.cmml">γ</mi><mo id="S4.I1.i1.p1.3.m3.2.2.3" xref="S4.I1.i1.p1.3.m3.2.2.3.cmml">∈</mo><mrow id="S4.I1.i1.p1.3.m3.2.2.2.2" xref="S4.I1.i1.p1.3.m3.2.2.2.3.cmml"><mo id="S4.I1.i1.p1.3.m3.2.2.2.2.3" stretchy="false" xref="S4.I1.i1.p1.3.m3.2.2.2.3.cmml">(</mo><mrow id="S4.I1.i1.p1.3.m3.1.1.1.1.1" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.cmml"><mo id="S4.I1.i1.p1.3.m3.1.1.1.1.1a" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.cmml">−</mo><mn id="S4.I1.i1.p1.3.m3.1.1.1.1.1.2" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.I1.i1.p1.3.m3.2.2.2.2.4" xref="S4.I1.i1.p1.3.m3.2.2.2.3.cmml">,</mo><mrow id="S4.I1.i1.p1.3.m3.2.2.2.2.2" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2.cmml"><mi id="S4.I1.i1.p1.3.m3.2.2.2.2.2.2" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2.2.cmml">p</mi><mo id="S4.I1.i1.p1.3.m3.2.2.2.2.2.1" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2.1.cmml">−</mo><mn id="S4.I1.i1.p1.3.m3.2.2.2.2.2.3" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.I1.i1.p1.3.m3.2.2.2.2.5" stretchy="false" xref="S4.I1.i1.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i1.p1.3.m3.2b"><apply id="S4.I1.i1.p1.3.m3.2.2.cmml" xref="S4.I1.i1.p1.3.m3.2.2"><in id="S4.I1.i1.p1.3.m3.2.2.3.cmml" xref="S4.I1.i1.p1.3.m3.2.2.3"></in><ci id="S4.I1.i1.p1.3.m3.2.2.4.cmml" xref="S4.I1.i1.p1.3.m3.2.2.4">𝛾</ci><interval closure="open" id="S4.I1.i1.p1.3.m3.2.2.2.3.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.2"><apply id="S4.I1.i1.p1.3.m3.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1"><minus id="S4.I1.i1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1"></minus><cn id="S4.I1.i1.p1.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.I1.i1.p1.3.m3.1.1.1.1.1.2">1</cn></apply><apply id="S4.I1.i1.p1.3.m3.2.2.2.2.2.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2"><minus id="S4.I1.i1.p1.3.m3.2.2.2.2.2.1.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2.1"></minus><ci id="S4.I1.i1.p1.3.m3.2.2.2.2.2.2.cmml" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2.2">𝑝</ci><cn id="S4.I1.i1.p1.3.m3.2.2.2.2.2.3.cmml" type="integer" xref="S4.I1.i1.p1.3.m3.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i1.p1.3.m3.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i1.p1.3.m3.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S4.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S4.I1.i2.p1"> <p class="ltx_p" id="S4.I1.i2.p1.3">Sobolev spaces <math alttext="W^{k,p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X)" class="ltx_Math" display="inline" id="S4.I1.i2.p1.1.m1.5"><semantics id="S4.I1.i2.p1.1.m1.5a"><mrow id="S4.I1.i2.p1.1.m1.5.5" xref="S4.I1.i2.p1.1.m1.5.5.cmml"><msup id="S4.I1.i2.p1.1.m1.5.5.3" xref="S4.I1.i2.p1.1.m1.5.5.3.cmml"><mi id="S4.I1.i2.p1.1.m1.5.5.3.2" xref="S4.I1.i2.p1.1.m1.5.5.3.2.cmml">W</mi><mrow id="S4.I1.i2.p1.1.m1.2.2.2.4" xref="S4.I1.i2.p1.1.m1.2.2.2.3.cmml"><mi id="S4.I1.i2.p1.1.m1.1.1.1.1" xref="S4.I1.i2.p1.1.m1.1.1.1.1.cmml">k</mi><mo id="S4.I1.i2.p1.1.m1.2.2.2.4.1" xref="S4.I1.i2.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.I1.i2.p1.1.m1.2.2.2.2" xref="S4.I1.i2.p1.1.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.I1.i2.p1.1.m1.5.5.2" xref="S4.I1.i2.p1.1.m1.5.5.2.cmml">⁢</mo><mrow id="S4.I1.i2.p1.1.m1.5.5.1.1" xref="S4.I1.i2.p1.1.m1.5.5.1.2.cmml"><mo id="S4.I1.i2.p1.1.m1.5.5.1.1.2" stretchy="false" xref="S4.I1.i2.p1.1.m1.5.5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.I1.i2.p1.1.m1.3.3" xref="S4.I1.i2.p1.1.m1.3.3.cmml">𝒪</mi><mo id="S4.I1.i2.p1.1.m1.5.5.1.1.3" xref="S4.I1.i2.p1.1.m1.5.5.1.2.cmml">,</mo><msubsup id="S4.I1.i2.p1.1.m1.5.5.1.1.1" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.cmml"><mi id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.2" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.2.cmml">w</mi><mi id="S4.I1.i2.p1.1.m1.5.5.1.1.1.3" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.3.cmml">γ</mi><mrow id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.cmml"><mo id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.1" rspace="0em" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.1.cmml">∂</mo><mi class="ltx_font_mathcaligraphic" id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.2" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.2.cmml">𝒪</mi></mrow></msubsup><mo id="S4.I1.i2.p1.1.m1.5.5.1.1.4" xref="S4.I1.i2.p1.1.m1.5.5.1.2.cmml">;</mo><mi id="S4.I1.i2.p1.1.m1.4.4" xref="S4.I1.i2.p1.1.m1.4.4.cmml">X</mi><mo id="S4.I1.i2.p1.1.m1.5.5.1.1.5" stretchy="false" xref="S4.I1.i2.p1.1.m1.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I1.i2.p1.1.m1.5b"><apply id="S4.I1.i2.p1.1.m1.5.5.cmml" xref="S4.I1.i2.p1.1.m1.5.5"><times id="S4.I1.i2.p1.1.m1.5.5.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.2"></times><apply id="S4.I1.i2.p1.1.m1.5.5.3.cmml" xref="S4.I1.i2.p1.1.m1.5.5.3"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.5.5.3.1.cmml" xref="S4.I1.i2.p1.1.m1.5.5.3">superscript</csymbol><ci id="S4.I1.i2.p1.1.m1.5.5.3.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.3.2">𝑊</ci><list id="S4.I1.i2.p1.1.m1.2.2.2.3.cmml" xref="S4.I1.i2.p1.1.m1.2.2.2.4"><ci id="S4.I1.i2.p1.1.m1.1.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.1.1.1.1">𝑘</ci><ci id="S4.I1.i2.p1.1.m1.2.2.2.2.cmml" xref="S4.I1.i2.p1.1.m1.2.2.2.2">𝑝</ci></list></apply><vector id="S4.I1.i2.p1.1.m1.5.5.1.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1"><ci id="S4.I1.i2.p1.1.m1.3.3.cmml" xref="S4.I1.i2.p1.1.m1.3.3">𝒪</ci><apply id="S4.I1.i2.p1.1.m1.5.5.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.5.5.1.1.1.1.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1">subscript</csymbol><apply id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.1.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1">superscript</csymbol><ci id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.2">𝑤</ci><apply id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3"><partialdiff id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.1.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.1"></partialdiff><ci id="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.2.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.2.3.2">𝒪</ci></apply></apply><ci id="S4.I1.i2.p1.1.m1.5.5.1.1.1.3.cmml" xref="S4.I1.i2.p1.1.m1.5.5.1.1.1.3">𝛾</ci></apply><ci id="S4.I1.i2.p1.1.m1.4.4.cmml" xref="S4.I1.i2.p1.1.m1.4.4">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I1.i2.p1.1.m1.5c">W^{k,p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.1.m1.5d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> with <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" 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id="S4.I1.i2.p1.3.m3.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.I1.i2.p1.3.m3.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S4.p1.8">To finally prove our main results in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6" title="6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6</span></a> about complex interpolation for these weighted spaces on bounded domains, it suffices to consider the half-space <math alttext="\mathcal{O}=\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.p1.7.m1.1"><semantics id="S4.p1.7.m1.1a"><mrow id="S4.p1.7.m1.1.1" xref="S4.p1.7.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.p1.7.m1.1.1.2" xref="S4.p1.7.m1.1.1.2.cmml">𝒪</mi><mo id="S4.p1.7.m1.1.1.1" xref="S4.p1.7.m1.1.1.1.cmml">=</mo><msubsup id="S4.p1.7.m1.1.1.3" xref="S4.p1.7.m1.1.1.3.cmml"><mi id="S4.p1.7.m1.1.1.3.2.2" xref="S4.p1.7.m1.1.1.3.2.2.cmml">ℝ</mi><mo id="S4.p1.7.m1.1.1.3.3" xref="S4.p1.7.m1.1.1.3.3.cmml">+</mo><mi id="S4.p1.7.m1.1.1.3.2.3" xref="S4.p1.7.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.7.m1.1b"><apply id="S4.p1.7.m1.1.1.cmml" xref="S4.p1.7.m1.1.1"><eq id="S4.p1.7.m1.1.1.1.cmml" xref="S4.p1.7.m1.1.1.1"></eq><ci id="S4.p1.7.m1.1.1.2.cmml" xref="S4.p1.7.m1.1.1.2">𝒪</ci><apply id="S4.p1.7.m1.1.1.3.cmml" xref="S4.p1.7.m1.1.1.3"><csymbol cd="ambiguous" id="S4.p1.7.m1.1.1.3.1.cmml" xref="S4.p1.7.m1.1.1.3">subscript</csymbol><apply id="S4.p1.7.m1.1.1.3.2.cmml" xref="S4.p1.7.m1.1.1.3"><csymbol cd="ambiguous" id="S4.p1.7.m1.1.1.3.2.1.cmml" xref="S4.p1.7.m1.1.1.3">superscript</csymbol><ci id="S4.p1.7.m1.1.1.3.2.2.cmml" xref="S4.p1.7.m1.1.1.3.2.2">ℝ</ci><ci id="S4.p1.7.m1.1.1.3.2.3.cmml" xref="S4.p1.7.m1.1.1.3.2.3">𝑑</ci></apply><plus id="S4.p1.7.m1.1.1.3.3.cmml" xref="S4.p1.7.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.7.m1.1c">\mathcal{O}=\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.7.m1.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> by localisation arguments. In this case we write <math alttext="w_{\gamma}=w_{\gamma}^{\partial\mathbb{R}^{d}_{+}}=|x_{1}|^{\gamma}" class="ltx_Math" display="inline" id="S4.p1.8.m2.1"><semantics id="S4.p1.8.m2.1a"><mrow id="S4.p1.8.m2.1.1" xref="S4.p1.8.m2.1.1.cmml"><msub id="S4.p1.8.m2.1.1.3" xref="S4.p1.8.m2.1.1.3.cmml"><mi id="S4.p1.8.m2.1.1.3.2" xref="S4.p1.8.m2.1.1.3.2.cmml">w</mi><mi id="S4.p1.8.m2.1.1.3.3" xref="S4.p1.8.m2.1.1.3.3.cmml">γ</mi></msub><mo id="S4.p1.8.m2.1.1.4" xref="S4.p1.8.m2.1.1.4.cmml">=</mo><msubsup id="S4.p1.8.m2.1.1.5" xref="S4.p1.8.m2.1.1.5.cmml"><mi id="S4.p1.8.m2.1.1.5.2.2" xref="S4.p1.8.m2.1.1.5.2.2.cmml">w</mi><mi id="S4.p1.8.m2.1.1.5.2.3" xref="S4.p1.8.m2.1.1.5.2.3.cmml">γ</mi><mrow id="S4.p1.8.m2.1.1.5.3" xref="S4.p1.8.m2.1.1.5.3.cmml"><mo id="S4.p1.8.m2.1.1.5.3.1" rspace="0em" xref="S4.p1.8.m2.1.1.5.3.1.cmml">∂</mo><msubsup id="S4.p1.8.m2.1.1.5.3.2" xref="S4.p1.8.m2.1.1.5.3.2.cmml"><mi id="S4.p1.8.m2.1.1.5.3.2.2.2" xref="S4.p1.8.m2.1.1.5.3.2.2.2.cmml">ℝ</mi><mo id="S4.p1.8.m2.1.1.5.3.2.3" xref="S4.p1.8.m2.1.1.5.3.2.3.cmml">+</mo><mi id="S4.p1.8.m2.1.1.5.3.2.2.3" xref="S4.p1.8.m2.1.1.5.3.2.2.3.cmml">d</mi></msubsup></mrow></msubsup><mo id="S4.p1.8.m2.1.1.6" xref="S4.p1.8.m2.1.1.6.cmml">=</mo><msup id="S4.p1.8.m2.1.1.1" xref="S4.p1.8.m2.1.1.1.cmml"><mrow id="S4.p1.8.m2.1.1.1.1.1" xref="S4.p1.8.m2.1.1.1.1.2.cmml"><mo id="S4.p1.8.m2.1.1.1.1.1.2" stretchy="false" xref="S4.p1.8.m2.1.1.1.1.2.1.cmml">|</mo><msub id="S4.p1.8.m2.1.1.1.1.1.1" xref="S4.p1.8.m2.1.1.1.1.1.1.cmml"><mi id="S4.p1.8.m2.1.1.1.1.1.1.2" xref="S4.p1.8.m2.1.1.1.1.1.1.2.cmml">x</mi><mn id="S4.p1.8.m2.1.1.1.1.1.1.3" xref="S4.p1.8.m2.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.p1.8.m2.1.1.1.1.1.3" stretchy="false" xref="S4.p1.8.m2.1.1.1.1.2.1.cmml">|</mo></mrow><mi id="S4.p1.8.m2.1.1.1.3" xref="S4.p1.8.m2.1.1.1.3.cmml">γ</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.p1.8.m2.1b"><apply id="S4.p1.8.m2.1.1.cmml" xref="S4.p1.8.m2.1.1"><and id="S4.p1.8.m2.1.1a.cmml" xref="S4.p1.8.m2.1.1"></and><apply id="S4.p1.8.m2.1.1b.cmml" xref="S4.p1.8.m2.1.1"><eq id="S4.p1.8.m2.1.1.4.cmml" xref="S4.p1.8.m2.1.1.4"></eq><apply id="S4.p1.8.m2.1.1.3.cmml" xref="S4.p1.8.m2.1.1.3"><csymbol cd="ambiguous" id="S4.p1.8.m2.1.1.3.1.cmml" xref="S4.p1.8.m2.1.1.3">subscript</csymbol><ci id="S4.p1.8.m2.1.1.3.2.cmml" xref="S4.p1.8.m2.1.1.3.2">𝑤</ci><ci id="S4.p1.8.m2.1.1.3.3.cmml" xref="S4.p1.8.m2.1.1.3.3">𝛾</ci></apply><apply id="S4.p1.8.m2.1.1.5.cmml" xref="S4.p1.8.m2.1.1.5"><csymbol cd="ambiguous" id="S4.p1.8.m2.1.1.5.1.cmml" xref="S4.p1.8.m2.1.1.5">superscript</csymbol><apply id="S4.p1.8.m2.1.1.5.2.cmml" xref="S4.p1.8.m2.1.1.5"><csymbol cd="ambiguous" id="S4.p1.8.m2.1.1.5.2.1.cmml" xref="S4.p1.8.m2.1.1.5">subscript</csymbol><ci id="S4.p1.8.m2.1.1.5.2.2.cmml" xref="S4.p1.8.m2.1.1.5.2.2">𝑤</ci><ci id="S4.p1.8.m2.1.1.5.2.3.cmml" xref="S4.p1.8.m2.1.1.5.2.3">𝛾</ci></apply><apply id="S4.p1.8.m2.1.1.5.3.cmml" xref="S4.p1.8.m2.1.1.5.3"><partialdiff id="S4.p1.8.m2.1.1.5.3.1.cmml" xref="S4.p1.8.m2.1.1.5.3.1"></partialdiff><apply id="S4.p1.8.m2.1.1.5.3.2.cmml" xref="S4.p1.8.m2.1.1.5.3.2"><csymbol cd="ambiguous" id="S4.p1.8.m2.1.1.5.3.2.1.cmml" xref="S4.p1.8.m2.1.1.5.3.2">subscript</csymbol><apply id="S4.p1.8.m2.1.1.5.3.2.2.cmml" xref="S4.p1.8.m2.1.1.5.3.2"><csymbol cd="ambiguous" id="S4.p1.8.m2.1.1.5.3.2.2.1.cmml" xref="S4.p1.8.m2.1.1.5.3.2">superscript</csymbol><ci id="S4.p1.8.m2.1.1.5.3.2.2.2.cmml" xref="S4.p1.8.m2.1.1.5.3.2.2.2">ℝ</ci><ci id="S4.p1.8.m2.1.1.5.3.2.2.3.cmml" xref="S4.p1.8.m2.1.1.5.3.2.2.3">𝑑</ci></apply><plus id="S4.p1.8.m2.1.1.5.3.2.3.cmml" xref="S4.p1.8.m2.1.1.5.3.2.3"></plus></apply></apply></apply></apply><apply id="S4.p1.8.m2.1.1c.cmml" xref="S4.p1.8.m2.1.1"><eq id="S4.p1.8.m2.1.1.6.cmml" xref="S4.p1.8.m2.1.1.6"></eq><share href="https://arxiv.org/html/2503.14636v1#S4.p1.8.m2.1.1.5.cmml" id="S4.p1.8.m2.1.1d.cmml" xref="S4.p1.8.m2.1.1"></share><apply id="S4.p1.8.m2.1.1.1.cmml" xref="S4.p1.8.m2.1.1.1"><csymbol cd="ambiguous" id="S4.p1.8.m2.1.1.1.2.cmml" xref="S4.p1.8.m2.1.1.1">superscript</csymbol><apply id="S4.p1.8.m2.1.1.1.1.2.cmml" xref="S4.p1.8.m2.1.1.1.1.1"><abs id="S4.p1.8.m2.1.1.1.1.2.1.cmml" xref="S4.p1.8.m2.1.1.1.1.1.2"></abs><apply id="S4.p1.8.m2.1.1.1.1.1.1.cmml" xref="S4.p1.8.m2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.p1.8.m2.1.1.1.1.1.1.1.cmml" xref="S4.p1.8.m2.1.1.1.1.1.1">subscript</csymbol><ci id="S4.p1.8.m2.1.1.1.1.1.1.2.cmml" xref="S4.p1.8.m2.1.1.1.1.1.1.2">𝑥</ci><cn id="S4.p1.8.m2.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.p1.8.m2.1.1.1.1.1.1.3">1</cn></apply></apply><ci id="S4.p1.8.m2.1.1.1.3.cmml" xref="S4.p1.8.m2.1.1.1.3">𝛾</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p1.8.m2.1c">w_{\gamma}=w_{\gamma}^{\partial\mathbb{R}^{d}_{+}}=|x_{1}|^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S4.p1.8.m2.1d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT = italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = | italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> as before. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S4.p2"> <p class="ltx_p" id="S4.p2.2">We start with the trace spaces for the above-mentioned weighted spaces on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.p2.1.m1.1"><semantics id="S4.p2.1.m1.1a"><msup id="S4.p2.1.m1.1.1" xref="S4.p2.1.m1.1.1.cmml"><mi id="S4.p2.1.m1.1.1.2" xref="S4.p2.1.m1.1.1.2.cmml">ℝ</mi><mi id="S4.p2.1.m1.1.1.3" xref="S4.p2.1.m1.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S4.p2.1.m1.1b"><apply id="S4.p2.1.m1.1.1.cmml" xref="S4.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.p2.1.m1.1.1.1.cmml" xref="S4.p2.1.m1.1.1">superscript</csymbol><ci id="S4.p2.1.m1.1.1.2.cmml" xref="S4.p2.1.m1.1.1.2">ℝ</ci><ci id="S4.p2.1.m1.1.1.3.cmml" xref="S4.p2.1.m1.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.1.m1.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.1.m1.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.p2.2.m2.1"><semantics id="S4.p2.2.m2.1a"><msubsup id="S4.p2.2.m2.1.1" xref="S4.p2.2.m2.1.1.cmml"><mi id="S4.p2.2.m2.1.1.2.2" xref="S4.p2.2.m2.1.1.2.2.cmml">ℝ</mi><mo id="S4.p2.2.m2.1.1.3" xref="S4.p2.2.m2.1.1.3.cmml">+</mo><mi id="S4.p2.2.m2.1.1.2.3" xref="S4.p2.2.m2.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.p2.2.m2.1b"><apply id="S4.p2.2.m2.1.1.cmml" xref="S4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.2.m2.1.1.1.cmml" xref="S4.p2.2.m2.1.1">subscript</csymbol><apply id="S4.p2.2.m2.1.1.2.cmml" xref="S4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.p2.2.m2.1.1.2.1.cmml" xref="S4.p2.2.m2.1.1">superscript</csymbol><ci id="S4.p2.2.m2.1.1.2.2.cmml" xref="S4.p2.2.m2.1.1.2.2">ℝ</ci><ci id="S4.p2.2.m2.1.1.2.3.cmml" xref="S4.p2.2.m2.1.1.2.3">𝑑</ci></apply><plus id="S4.p2.2.m2.1.1.3.cmml" xref="S4.p2.2.m2.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.p2.2.m2.1c">\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.p2.2.m2.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS1" title="4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a>. Afterwards, we apply the trace theorems to prove some density results in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS2" title="4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.2</span></a>.</p> </div> <section class="ltx_subsection" id="S4.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.1. </span>Trace spaces of weighted spaces on the half-space</h3> <div class="ltx_para" id="S4.SS1.p1"> <p class="ltx_p" id="S4.SS1.p1.1">Using the results from Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.SS2" title="3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.2</span></a> about weighted Besov and Triebel-Lizorkin spaces, we can now characterise higher-order trace spaces of weighted Bessel potential and Sobolev spaces. We start with the case <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.SS1.p1.1.m1.2"><semantics id="S4.SS1.p1.1.m1.2a"><mrow id="S4.SS1.p1.1.m1.2.2" xref="S4.SS1.p1.1.m1.2.2.cmml"><mi id="S4.SS1.p1.1.m1.2.2.4" xref="S4.SS1.p1.1.m1.2.2.4.cmml">γ</mi><mo id="S4.SS1.p1.1.m1.2.2.3" xref="S4.SS1.p1.1.m1.2.2.3.cmml">∈</mo><mrow id="S4.SS1.p1.1.m1.2.2.2.2" xref="S4.SS1.p1.1.m1.2.2.2.3.cmml"><mo id="S4.SS1.p1.1.m1.2.2.2.2.3" stretchy="false" xref="S4.SS1.p1.1.m1.2.2.2.3.cmml">(</mo><mrow id="S4.SS1.p1.1.m1.1.1.1.1.1" xref="S4.SS1.p1.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS1.p1.1.m1.1.1.1.1.1a" xref="S4.SS1.p1.1.m1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS1.p1.1.m1.1.1.1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS1.p1.1.m1.2.2.2.2.4" xref="S4.SS1.p1.1.m1.2.2.2.3.cmml">,</mo><mrow id="S4.SS1.p1.1.m1.2.2.2.2.2" xref="S4.SS1.p1.1.m1.2.2.2.2.2.cmml"><mi id="S4.SS1.p1.1.m1.2.2.2.2.2.2" xref="S4.SS1.p1.1.m1.2.2.2.2.2.2.cmml">p</mi><mo id="S4.SS1.p1.1.m1.2.2.2.2.2.1" xref="S4.SS1.p1.1.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S4.SS1.p1.1.m1.2.2.2.2.2.3" xref="S4.SS1.p1.1.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.SS1.p1.1.m1.2.2.2.2.5" stretchy="false" xref="S4.SS1.p1.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.2b"><apply id="S4.SS1.p1.1.m1.2.2.cmml" xref="S4.SS1.p1.1.m1.2.2"><in id="S4.SS1.p1.1.m1.2.2.3.cmml" xref="S4.SS1.p1.1.m1.2.2.3"></in><ci id="S4.SS1.p1.1.m1.2.2.4.cmml" xref="S4.SS1.p1.1.m1.2.2.4">𝛾</ci><interval closure="open" id="S4.SS1.p1.1.m1.2.2.2.3.cmml" xref="S4.SS1.p1.1.m1.2.2.2.2"><apply id="S4.SS1.p1.1.m1.1.1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1.1.1.1"><minus id="S4.SS1.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1.1.1.1"></minus><cn id="S4.SS1.p1.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.p1.1.m1.1.1.1.1.1.2">1</cn></apply><apply id="S4.SS1.p1.1.m1.2.2.2.2.2.cmml" xref="S4.SS1.p1.1.m1.2.2.2.2.2"><minus id="S4.SS1.p1.1.m1.2.2.2.2.2.1.cmml" xref="S4.SS1.p1.1.m1.2.2.2.2.2.1"></minus><ci id="S4.SS1.p1.1.m1.2.2.2.2.2.2.cmml" xref="S4.SS1.p1.1.m1.2.2.2.2.2.2">𝑝</ci><cn id="S4.SS1.p1.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS1.p1.1.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p1.1.m1.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.1.1.1">Theorem 4.1</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem1.p1"> <p class="ltx_p" id="S4.Thmtheorem1.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.5.5">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.1.1.m1.2"><semantics id="S4.Thmtheorem1.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem1.p1.1.1.m1.2.3" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem1.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem1.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem1.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem1.p1.1.1.m1.1.1" xref="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem1.p1.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem1.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem1.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem1.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.1.1.m1.2b"><apply id="S4.Thmtheorem1.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.2.3"><in id="S4.Thmtheorem1.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.1"></in><ci id="S4.Thmtheorem1.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.2.3.3.2"><cn id="S4.Thmtheorem1.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem1.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem1.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.2.2.m2.1"><semantics id="S4.Thmtheorem1.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem1.p1.2.2.m2.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem1.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2.cmml">m</mi><mo id="S4.Thmtheorem1.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem1.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.2" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.2.2.m2.1b"><apply id="S4.Thmtheorem1.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1"><in id="S4.Thmtheorem1.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.2">𝑚</ci><apply id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3.2">ℕ</ci><cn id="S4.Thmtheorem1.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.2.2.m2.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.2.2.m2.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.3.3.m3.2"><semantics id="S4.Thmtheorem1.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem1.p1.3.3.m3.2.2" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.cmml"><mi id="S4.Thmtheorem1.p1.3.3.m3.2.2.4" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.4.cmml">γ</mi><mo id="S4.Thmtheorem1.p1.3.3.m3.2.2.3" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.3.cmml">∈</mo><mrow id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml"><mo id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">(</mo><mrow id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1a" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.4" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">,</mo><mrow id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.cmml">p</mi><mo id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.1" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.1.cmml">−</mo><mn id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.3.3.m3.2b"><apply id="S4.Thmtheorem1.p1.3.3.m3.2.2.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2"><in id="S4.Thmtheorem1.p1.3.3.m3.2.2.3.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.3"></in><ci id="S4.Thmtheorem1.p1.3.3.m3.2.2.4.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.4">𝛾</ci><interval closure="open" id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2"><apply id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"><minus id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"></minus><cn id="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2">1</cn></apply><apply id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2"><minus id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.1"></minus><ci id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2">𝑝</ci><cn id="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.3.3.m3.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.3.3.m3.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>, <math alttext="\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.4.4.m4.2"><semantics id="S4.Thmtheorem1.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem1.p1.4.4.m4.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem1.p1.4.4.m4.2.2.4" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.4.cmml">𝒪</mi><mo id="S4.Thmtheorem1.p1.4.4.m4.2.2.3" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.3.cmml">∈</mo><mrow id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml"><mo id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml">{</mo><msup id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.4" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml">,</mo><msubsup id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.3" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.3.cmml">+</mo><mi id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.4.4.m4.2b"><apply id="S4.Thmtheorem1.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2"><in id="S4.Thmtheorem1.p1.4.4.m4.2.2.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.3"></in><ci id="S4.Thmtheorem1.p1.4.4.m4.2.2.4.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.4">𝒪</ci><set id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2"><apply id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.2">ℝ</ci><ci id="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.1.1.1.1.1.3">𝑑</ci></apply><apply id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2">subscript</csymbol><apply id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2">superscript</csymbol><ci id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.2">ℝ</ci><ci id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem1.p1.4.4.m4.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.4.4.m4.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.4.4.m4.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.5.5.m5.1"><semantics id="S4.Thmtheorem1.p1.5.5.m5.1a"><mi id="S4.Thmtheorem1.p1.5.5.m5.1.1" xref="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.5.5.m5.1b"><ci id="S4.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space.</span></p> <ol class="ltx_enumerate" id="S4.I2"> <li class="ltx_item" id="S4.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S4.I2.i1.p1"> <p class="ltx_p" id="S4.I2.i1.p1.2"><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.2.1">If </span><math alttext="s>0" class="ltx_Math" display="inline" id="S4.I2.i1.p1.1.m1.1"><semantics id="S4.I2.i1.p1.1.m1.1a"><mrow id="S4.I2.i1.p1.1.m1.1.1" xref="S4.I2.i1.p1.1.m1.1.1.cmml"><mi id="S4.I2.i1.p1.1.m1.1.1.2" xref="S4.I2.i1.p1.1.m1.1.1.2.cmml">s</mi><mo id="S4.I2.i1.p1.1.m1.1.1.1" xref="S4.I2.i1.p1.1.m1.1.1.1.cmml">></mo><mn id="S4.I2.i1.p1.1.m1.1.1.3" xref="S4.I2.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.1.m1.1b"><apply id="S4.I2.i1.p1.1.m1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.1.1"><gt id="S4.I2.i1.p1.1.m1.1.1.1.cmml" xref="S4.I2.i1.p1.1.m1.1.1.1"></gt><ci id="S4.I2.i1.p1.1.m1.1.1.2.cmml" xref="S4.I2.i1.p1.1.m1.1.1.2">𝑠</ci><cn id="S4.I2.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I2.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.1.m1.1c">s>0</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.1.m1.1d">italic_s > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.2.2"> satisfies </span><math alttext="s>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.I2.i1.p1.2.m2.1"><semantics id="S4.I2.i1.p1.2.m2.1a"><mrow id="S4.I2.i1.p1.2.m2.1.1" xref="S4.I2.i1.p1.2.m2.1.1.cmml"><mi id="S4.I2.i1.p1.2.m2.1.1.2" xref="S4.I2.i1.p1.2.m2.1.1.2.cmml">s</mi><mo id="S4.I2.i1.p1.2.m2.1.1.1" xref="S4.I2.i1.p1.2.m2.1.1.1.cmml">></mo><mrow id="S4.I2.i1.p1.2.m2.1.1.3" xref="S4.I2.i1.p1.2.m2.1.1.3.cmml"><mi id="S4.I2.i1.p1.2.m2.1.1.3.2" xref="S4.I2.i1.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="S4.I2.i1.p1.2.m2.1.1.3.1" xref="S4.I2.i1.p1.2.m2.1.1.3.1.cmml">+</mo><mfrac id="S4.I2.i1.p1.2.m2.1.1.3.3" xref="S4.I2.i1.p1.2.m2.1.1.3.3.cmml"><mrow id="S4.I2.i1.p1.2.m2.1.1.3.3.2" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2.cmml"><mi id="S4.I2.i1.p1.2.m2.1.1.3.3.2.2" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.I2.i1.p1.2.m2.1.1.3.3.2.1" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2.1.cmml">+</mo><mn id="S4.I2.i1.p1.2.m2.1.1.3.3.2.3" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.I2.i1.p1.2.m2.1.1.3.3.3" xref="S4.I2.i1.p1.2.m2.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i1.p1.2.m2.1b"><apply id="S4.I2.i1.p1.2.m2.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1"><gt id="S4.I2.i1.p1.2.m2.1.1.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1.1"></gt><ci id="S4.I2.i1.p1.2.m2.1.1.2.cmml" xref="S4.I2.i1.p1.2.m2.1.1.2">𝑠</ci><apply id="S4.I2.i1.p1.2.m2.1.1.3.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3"><plus id="S4.I2.i1.p1.2.m2.1.1.3.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.1"></plus><ci id="S4.I2.i1.p1.2.m2.1.1.3.2.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.2">𝑚</ci><apply id="S4.I2.i1.p1.2.m2.1.1.3.3.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.3"><divide id="S4.I2.i1.p1.2.m2.1.1.3.3.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.3"></divide><apply id="S4.I2.i1.p1.2.m2.1.1.3.3.2.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2"><plus id="S4.I2.i1.p1.2.m2.1.1.3.3.2.1.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2.1"></plus><ci id="S4.I2.i1.p1.2.m2.1.1.3.3.2.2.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2.2">𝛾</ci><cn id="S4.I2.i1.p1.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S4.I2.i1.p1.2.m2.1.1.3.3.2.3">1</cn></apply><ci id="S4.I2.i1.p1.2.m2.1.1.3.3.3.cmml" xref="S4.I2.i1.p1.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i1.p1.2.m2.1c">s>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i1.p1.2.m2.1d">italic_s > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.2.3">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{m}:H^{s,p}(\mathcal{O},w_{\gamma};X)\to B_{p,p}^{s-m-\frac{% \gamma+1}{p}}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S4.Ex1.m1.9"><semantics id="S4.Ex1.m1.9a"><mrow id="S4.Ex1.m1.9.9" xref="S4.Ex1.m1.9.9.cmml"><msub id="S4.Ex1.m1.9.9.4" xref="S4.Ex1.m1.9.9.4.cmml"><mi id="S4.Ex1.m1.9.9.4.2" xref="S4.Ex1.m1.9.9.4.2.cmml">Tr</mi><mi id="S4.Ex1.m1.9.9.4.3" xref="S4.Ex1.m1.9.9.4.3.cmml">m</mi></msub><mo id="S4.Ex1.m1.9.9.3" lspace="0.278em" rspace="0.278em" xref="S4.Ex1.m1.9.9.3.cmml">:</mo><mrow id="S4.Ex1.m1.9.9.2" xref="S4.Ex1.m1.9.9.2.cmml"><mrow id="S4.Ex1.m1.8.8.1.1" xref="S4.Ex1.m1.8.8.1.1.cmml"><msup id="S4.Ex1.m1.8.8.1.1.3" xref="S4.Ex1.m1.8.8.1.1.3.cmml"><mi id="S4.Ex1.m1.8.8.1.1.3.2" xref="S4.Ex1.m1.8.8.1.1.3.2.cmml">H</mi><mrow id="S4.Ex1.m1.2.2.2.4" xref="S4.Ex1.m1.2.2.2.3.cmml"><mi id="S4.Ex1.m1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.cmml">s</mi><mo id="S4.Ex1.m1.2.2.2.4.1" xref="S4.Ex1.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex1.m1.2.2.2.2" xref="S4.Ex1.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex1.m1.8.8.1.1.2" xref="S4.Ex1.m1.8.8.1.1.2.cmml">⁢</mo><mrow id="S4.Ex1.m1.8.8.1.1.1.1" xref="S4.Ex1.m1.8.8.1.1.1.2.cmml"><mo id="S4.Ex1.m1.8.8.1.1.1.1.2" stretchy="false" xref="S4.Ex1.m1.8.8.1.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex1.m1.5.5" xref="S4.Ex1.m1.5.5.cmml">𝒪</mi><mo id="S4.Ex1.m1.8.8.1.1.1.1.3" xref="S4.Ex1.m1.8.8.1.1.1.2.cmml">,</mo><msub id="S4.Ex1.m1.8.8.1.1.1.1.1" xref="S4.Ex1.m1.8.8.1.1.1.1.1.cmml"><mi id="S4.Ex1.m1.8.8.1.1.1.1.1.2" xref="S4.Ex1.m1.8.8.1.1.1.1.1.2.cmml">w</mi><mi id="S4.Ex1.m1.8.8.1.1.1.1.1.3" xref="S4.Ex1.m1.8.8.1.1.1.1.1.3.cmml">γ</mi></msub><mo id="S4.Ex1.m1.8.8.1.1.1.1.4" xref="S4.Ex1.m1.8.8.1.1.1.2.cmml">;</mo><mi id="S4.Ex1.m1.6.6" xref="S4.Ex1.m1.6.6.cmml">X</mi><mo id="S4.Ex1.m1.8.8.1.1.1.1.5" stretchy="false" xref="S4.Ex1.m1.8.8.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="S4.Ex1.m1.9.9.2.3" stretchy="false" xref="S4.Ex1.m1.9.9.2.3.cmml">→</mo><mrow id="S4.Ex1.m1.9.9.2.2" xref="S4.Ex1.m1.9.9.2.2.cmml"><msubsup id="S4.Ex1.m1.9.9.2.2.3" xref="S4.Ex1.m1.9.9.2.2.3.cmml"><mi id="S4.Ex1.m1.9.9.2.2.3.2.2" xref="S4.Ex1.m1.9.9.2.2.3.2.2.cmml">B</mi><mrow id="S4.Ex1.m1.4.4.2.4" xref="S4.Ex1.m1.4.4.2.3.cmml"><mi id="S4.Ex1.m1.3.3.1.1" xref="S4.Ex1.m1.3.3.1.1.cmml">p</mi><mo id="S4.Ex1.m1.4.4.2.4.1" xref="S4.Ex1.m1.4.4.2.3.cmml">,</mo><mi 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xref="S4.Ex1.m1.9.9.2.2.2.cmml">⁢</mo><mrow id="S4.Ex1.m1.9.9.2.2.1.1" xref="S4.Ex1.m1.9.9.2.2.1.2.cmml"><mo id="S4.Ex1.m1.9.9.2.2.1.1.2" stretchy="false" xref="S4.Ex1.m1.9.9.2.2.1.2.cmml">(</mo><msup id="S4.Ex1.m1.9.9.2.2.1.1.1" xref="S4.Ex1.m1.9.9.2.2.1.1.1.cmml"><mi id="S4.Ex1.m1.9.9.2.2.1.1.1.2" xref="S4.Ex1.m1.9.9.2.2.1.1.1.2.cmml">ℝ</mi><mrow id="S4.Ex1.m1.9.9.2.2.1.1.1.3" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3.cmml"><mi id="S4.Ex1.m1.9.9.2.2.1.1.1.3.2" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3.2.cmml">d</mi><mo id="S4.Ex1.m1.9.9.2.2.1.1.1.3.1" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3.1.cmml">−</mo><mn id="S4.Ex1.m1.9.9.2.2.1.1.1.3.3" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Ex1.m1.9.9.2.2.1.1.3" xref="S4.Ex1.m1.9.9.2.2.1.2.cmml">;</mo><mi id="S4.Ex1.m1.7.7" xref="S4.Ex1.m1.7.7.cmml">X</mi><mo id="S4.Ex1.m1.9.9.2.2.1.1.4" stretchy="false" xref="S4.Ex1.m1.9.9.2.2.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex1.m1.9b"><apply id="S4.Ex1.m1.9.9.cmml" xref="S4.Ex1.m1.9.9"><ci id="S4.Ex1.m1.9.9.3.cmml" xref="S4.Ex1.m1.9.9.3">:</ci><apply id="S4.Ex1.m1.9.9.4.cmml" xref="S4.Ex1.m1.9.9.4"><csymbol cd="ambiguous" id="S4.Ex1.m1.9.9.4.1.cmml" xref="S4.Ex1.m1.9.9.4">subscript</csymbol><ci id="S4.Ex1.m1.9.9.4.2.cmml" xref="S4.Ex1.m1.9.9.4.2">Tr</ci><ci id="S4.Ex1.m1.9.9.4.3.cmml" xref="S4.Ex1.m1.9.9.4.3">𝑚</ci></apply><apply id="S4.Ex1.m1.9.9.2.cmml" xref="S4.Ex1.m1.9.9.2"><ci id="S4.Ex1.m1.9.9.2.3.cmml" xref="S4.Ex1.m1.9.9.2.3">→</ci><apply id="S4.Ex1.m1.8.8.1.1.cmml" xref="S4.Ex1.m1.8.8.1.1"><times id="S4.Ex1.m1.8.8.1.1.2.cmml" xref="S4.Ex1.m1.8.8.1.1.2"></times><apply id="S4.Ex1.m1.8.8.1.1.3.cmml" xref="S4.Ex1.m1.8.8.1.1.3"><csymbol cd="ambiguous" id="S4.Ex1.m1.8.8.1.1.3.1.cmml" xref="S4.Ex1.m1.8.8.1.1.3">superscript</csymbol><ci id="S4.Ex1.m1.8.8.1.1.3.2.cmml" xref="S4.Ex1.m1.8.8.1.1.3.2">𝐻</ci><list id="S4.Ex1.m1.2.2.2.3.cmml" xref="S4.Ex1.m1.2.2.2.4"><ci id="S4.Ex1.m1.1.1.1.1.cmml" 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xref="S4.Ex1.m1.9.9.2.2.3.3.4.2.1"></plus><ci id="S4.Ex1.m1.9.9.2.2.3.3.4.2.2.cmml" xref="S4.Ex1.m1.9.9.2.2.3.3.4.2.2">𝛾</ci><cn id="S4.Ex1.m1.9.9.2.2.3.3.4.2.3.cmml" type="integer" xref="S4.Ex1.m1.9.9.2.2.3.3.4.2.3">1</cn></apply><ci id="S4.Ex1.m1.9.9.2.2.3.3.4.3.cmml" xref="S4.Ex1.m1.9.9.2.2.3.3.4.3">𝑝</ci></apply></apply></apply><list id="S4.Ex1.m1.9.9.2.2.1.2.cmml" xref="S4.Ex1.m1.9.9.2.2.1.1"><apply id="S4.Ex1.m1.9.9.2.2.1.1.1.cmml" xref="S4.Ex1.m1.9.9.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Ex1.m1.9.9.2.2.1.1.1.1.cmml" xref="S4.Ex1.m1.9.9.2.2.1.1.1">superscript</csymbol><ci id="S4.Ex1.m1.9.9.2.2.1.1.1.2.cmml" xref="S4.Ex1.m1.9.9.2.2.1.1.1.2">ℝ</ci><apply id="S4.Ex1.m1.9.9.2.2.1.1.1.3.cmml" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3"><minus id="S4.Ex1.m1.9.9.2.2.1.1.1.3.1.cmml" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3.1"></minus><ci id="S4.Ex1.m1.9.9.2.2.1.1.1.3.2.cmml" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3.2">𝑑</ci><cn id="S4.Ex1.m1.9.9.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.Ex1.m1.9.9.2.2.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex1.m1.7.7.cmml" xref="S4.Ex1.m1.7.7">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex1.m1.9c">\operatorname{Tr}_{m}:H^{s,p}(\mathcal{O},w_{\gamma};X)\to B_{p,p}^{s-m-\frac{% \gamma+1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex1.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.I2.i1.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I2.i1.p1.3.1">is a continuous and surjective operator.</span></p> </div> </li> <li class="ltx_item" id="S4.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S4.I2.i2.p1"> <p class="ltx_p" id="S4.I2.i2.p1.2"><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.2.1">If </span><math alttext="k\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S4.I2.i2.p1.1.m1.1"><semantics id="S4.I2.i2.p1.1.m1.1a"><mrow id="S4.I2.i2.p1.1.m1.1.1" xref="S4.I2.i2.p1.1.m1.1.1.cmml"><mi id="S4.I2.i2.p1.1.m1.1.1.2" xref="S4.I2.i2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S4.I2.i2.p1.1.m1.1.1.1" xref="S4.I2.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.I2.i2.p1.1.m1.1.1.3" xref="S4.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S4.I2.i2.p1.1.m1.1.1.3.2" xref="S4.I2.i2.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S4.I2.i2.p1.1.m1.1.1.3.3" xref="S4.I2.i2.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.1.m1.1b"><apply id="S4.I2.i2.p1.1.m1.1.1.cmml" xref="S4.I2.i2.p1.1.m1.1.1"><in id="S4.I2.i2.p1.1.m1.1.1.1.cmml" xref="S4.I2.i2.p1.1.m1.1.1.1"></in><ci id="S4.I2.i2.p1.1.m1.1.1.2.cmml" xref="S4.I2.i2.p1.1.m1.1.1.2">𝑘</ci><apply id="S4.I2.i2.p1.1.m1.1.1.3.cmml" xref="S4.I2.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="S4.I2.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="S4.I2.i2.p1.1.m1.1.1.3.2">ℕ</ci><cn id="S4.I2.i2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.I2.i2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.1.m1.1c">k\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.1.m1.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.2.2"> satisfies </span><math alttext="k>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.I2.i2.p1.2.m2.1"><semantics id="S4.I2.i2.p1.2.m2.1a"><mrow id="S4.I2.i2.p1.2.m2.1.1" xref="S4.I2.i2.p1.2.m2.1.1.cmml"><mi id="S4.I2.i2.p1.2.m2.1.1.2" xref="S4.I2.i2.p1.2.m2.1.1.2.cmml">k</mi><mo id="S4.I2.i2.p1.2.m2.1.1.1" xref="S4.I2.i2.p1.2.m2.1.1.1.cmml">></mo><mrow id="S4.I2.i2.p1.2.m2.1.1.3" xref="S4.I2.i2.p1.2.m2.1.1.3.cmml"><mi id="S4.I2.i2.p1.2.m2.1.1.3.2" xref="S4.I2.i2.p1.2.m2.1.1.3.2.cmml">m</mi><mo id="S4.I2.i2.p1.2.m2.1.1.3.1" xref="S4.I2.i2.p1.2.m2.1.1.3.1.cmml">+</mo><mfrac id="S4.I2.i2.p1.2.m2.1.1.3.3" xref="S4.I2.i2.p1.2.m2.1.1.3.3.cmml"><mrow id="S4.I2.i2.p1.2.m2.1.1.3.3.2" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2.cmml"><mi id="S4.I2.i2.p1.2.m2.1.1.3.3.2.2" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.I2.i2.p1.2.m2.1.1.3.3.2.1" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2.1.cmml">+</mo><mn id="S4.I2.i2.p1.2.m2.1.1.3.3.2.3" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.I2.i2.p1.2.m2.1.1.3.3.3" xref="S4.I2.i2.p1.2.m2.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I2.i2.p1.2.m2.1b"><apply id="S4.I2.i2.p1.2.m2.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1"><gt id="S4.I2.i2.p1.2.m2.1.1.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.1"></gt><ci id="S4.I2.i2.p1.2.m2.1.1.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.2">𝑘</ci><apply id="S4.I2.i2.p1.2.m2.1.1.3.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3"><plus id="S4.I2.i2.p1.2.m2.1.1.3.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.1"></plus><ci id="S4.I2.i2.p1.2.m2.1.1.3.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.2">𝑚</ci><apply id="S4.I2.i2.p1.2.m2.1.1.3.3.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.3"><divide id="S4.I2.i2.p1.2.m2.1.1.3.3.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.3"></divide><apply id="S4.I2.i2.p1.2.m2.1.1.3.3.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2"><plus id="S4.I2.i2.p1.2.m2.1.1.3.3.2.1.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2.1"></plus><ci id="S4.I2.i2.p1.2.m2.1.1.3.3.2.2.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2.2">𝛾</ci><cn id="S4.I2.i2.p1.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S4.I2.i2.p1.2.m2.1.1.3.3.2.3">1</cn></apply><ci id="S4.I2.i2.p1.2.m2.1.1.3.3.3.cmml" xref="S4.I2.i2.p1.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I2.i2.p1.2.m2.1c">k>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.I2.i2.p1.2.m2.1d">italic_k > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.2.3">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{m}:W^{k,p}(\mathcal{O},w_{\gamma};X)\to B_{p,p}^{k-m-\frac{% 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divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.I2.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I2.i2.p1.3.1">is a continuous and surjective operator.</span></p> </div> </li> </ol> <p class="ltx_p" id="S4.Thmtheorem1.p1.11"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem1.p1.11.6">In both cases, there exists a continuous right inverse <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.6.1.m1.1"><semantics id="S4.Thmtheorem1.p1.6.1.m1.1a"><msub id="S4.Thmtheorem1.p1.6.1.m1.1.1" xref="S4.Thmtheorem1.p1.6.1.m1.1.1.cmml"><mi id="S4.Thmtheorem1.p1.6.1.m1.1.1.2" xref="S4.Thmtheorem1.p1.6.1.m1.1.1.2.cmml">ext</mi><mi id="S4.Thmtheorem1.p1.6.1.m1.1.1.3" xref="S4.Thmtheorem1.p1.6.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.6.1.m1.1b"><apply id="S4.Thmtheorem1.p1.6.1.m1.1.1.cmml" xref="S4.Thmtheorem1.p1.6.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.6.1.m1.1.1.1.cmml" xref="S4.Thmtheorem1.p1.6.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem1.p1.6.1.m1.1.1.2.cmml" xref="S4.Thmtheorem1.p1.6.1.m1.1.1.2">ext</ci><ci id="S4.Thmtheorem1.p1.6.1.m1.1.1.3.cmml" xref="S4.Thmtheorem1.p1.6.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.6.1.m1.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.6.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.7.2.m2.1"><semantics id="S4.Thmtheorem1.p1.7.2.m2.1a"><msub id="S4.Thmtheorem1.p1.7.2.m2.1.1" xref="S4.Thmtheorem1.p1.7.2.m2.1.1.cmml"><mi id="S4.Thmtheorem1.p1.7.2.m2.1.1.2" xref="S4.Thmtheorem1.p1.7.2.m2.1.1.2.cmml">Tr</mi><mi id="S4.Thmtheorem1.p1.7.2.m2.1.1.3" xref="S4.Thmtheorem1.p1.7.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.7.2.m2.1b"><apply id="S4.Thmtheorem1.p1.7.2.m2.1.1.cmml" xref="S4.Thmtheorem1.p1.7.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.7.2.m2.1.1.1.cmml" xref="S4.Thmtheorem1.p1.7.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem1.p1.7.2.m2.1.1.2.cmml" xref="S4.Thmtheorem1.p1.7.2.m2.1.1.2">Tr</ci><ci id="S4.Thmtheorem1.p1.7.2.m2.1.1.3.cmml" xref="S4.Thmtheorem1.p1.7.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.7.2.m2.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.7.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> which is independent of <math alttext="k,s,p,\gamma" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.8.3.m3.4"><semantics id="S4.Thmtheorem1.p1.8.3.m3.4a"><mrow id="S4.Thmtheorem1.p1.8.3.m3.4.5.2" xref="S4.Thmtheorem1.p1.8.3.m3.4.5.1.cmml"><mi id="S4.Thmtheorem1.p1.8.3.m3.1.1" xref="S4.Thmtheorem1.p1.8.3.m3.1.1.cmml">k</mi><mo id="S4.Thmtheorem1.p1.8.3.m3.4.5.2.1" xref="S4.Thmtheorem1.p1.8.3.m3.4.5.1.cmml">,</mo><mi id="S4.Thmtheorem1.p1.8.3.m3.2.2" xref="S4.Thmtheorem1.p1.8.3.m3.2.2.cmml">s</mi><mo id="S4.Thmtheorem1.p1.8.3.m3.4.5.2.2" xref="S4.Thmtheorem1.p1.8.3.m3.4.5.1.cmml">,</mo><mi id="S4.Thmtheorem1.p1.8.3.m3.3.3" xref="S4.Thmtheorem1.p1.8.3.m3.3.3.cmml">p</mi><mo id="S4.Thmtheorem1.p1.8.3.m3.4.5.2.3" xref="S4.Thmtheorem1.p1.8.3.m3.4.5.1.cmml">,</mo><mi id="S4.Thmtheorem1.p1.8.3.m3.4.4" xref="S4.Thmtheorem1.p1.8.3.m3.4.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.8.3.m3.4b"><list id="S4.Thmtheorem1.p1.8.3.m3.4.5.1.cmml" xref="S4.Thmtheorem1.p1.8.3.m3.4.5.2"><ci id="S4.Thmtheorem1.p1.8.3.m3.1.1.cmml" xref="S4.Thmtheorem1.p1.8.3.m3.1.1">𝑘</ci><ci id="S4.Thmtheorem1.p1.8.3.m3.2.2.cmml" xref="S4.Thmtheorem1.p1.8.3.m3.2.2">𝑠</ci><ci id="S4.Thmtheorem1.p1.8.3.m3.3.3.cmml" xref="S4.Thmtheorem1.p1.8.3.m3.3.3">𝑝</ci><ci id="S4.Thmtheorem1.p1.8.3.m3.4.4.cmml" xref="S4.Thmtheorem1.p1.8.3.m3.4.4">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.8.3.m3.4c">k,s,p,\gamma</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.8.3.m3.4d">italic_k , italic_s , italic_p , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.9.4.m4.1"><semantics id="S4.Thmtheorem1.p1.9.4.m4.1a"><mi id="S4.Thmtheorem1.p1.9.4.m4.1.1" xref="S4.Thmtheorem1.p1.9.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.9.4.m4.1b"><ci id="S4.Thmtheorem1.p1.9.4.m4.1.1.cmml" xref="S4.Thmtheorem1.p1.9.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.9.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.9.4.m4.1d">italic_X</annotation></semantics></math>. For any <math alttext="0\leq j<m" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.10.5.m5.1"><semantics id="S4.Thmtheorem1.p1.10.5.m5.1a"><mrow id="S4.Thmtheorem1.p1.10.5.m5.1.1" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.cmml"><mn id="S4.Thmtheorem1.p1.10.5.m5.1.1.2" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.2.cmml">0</mn><mo id="S4.Thmtheorem1.p1.10.5.m5.1.1.3" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.3.cmml">≤</mo><mi id="S4.Thmtheorem1.p1.10.5.m5.1.1.4" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.4.cmml">j</mi><mo id="S4.Thmtheorem1.p1.10.5.m5.1.1.5" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.5.cmml"><</mo><mi id="S4.Thmtheorem1.p1.10.5.m5.1.1.6" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.10.5.m5.1b"><apply id="S4.Thmtheorem1.p1.10.5.m5.1.1.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1"><and id="S4.Thmtheorem1.p1.10.5.m5.1.1a.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1"></and><apply id="S4.Thmtheorem1.p1.10.5.m5.1.1b.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1"><leq id="S4.Thmtheorem1.p1.10.5.m5.1.1.3.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.3"></leq><cn id="S4.Thmtheorem1.p1.10.5.m5.1.1.2.cmml" type="integer" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.2">0</cn><ci id="S4.Thmtheorem1.p1.10.5.m5.1.1.4.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.4">𝑗</ci></apply><apply id="S4.Thmtheorem1.p1.10.5.m5.1.1c.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1"><lt id="S4.Thmtheorem1.p1.10.5.m5.1.1.5.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1.p1.10.5.m5.1.1.4.cmml" id="S4.Thmtheorem1.p1.10.5.m5.1.1d.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1"></share><ci id="S4.Thmtheorem1.p1.10.5.m5.1.1.6.cmml" xref="S4.Thmtheorem1.p1.10.5.m5.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.10.5.m5.1c">0\leq j<m</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.10.5.m5.1d">0 ≤ italic_j < italic_m</annotation></semantics></math> we have <math alttext="\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0" class="ltx_Math" display="inline" id="S4.Thmtheorem1.p1.11.6.m6.1"><semantics id="S4.Thmtheorem1.p1.11.6.m6.1a"><mrow id="S4.Thmtheorem1.p1.11.6.m6.1.1" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.cmml"><mrow id="S4.Thmtheorem1.p1.11.6.m6.1.1.2" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.cmml"><msub id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.cmml"><mi id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.2" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.2.cmml">Tr</mi><mi id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.3" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.3.cmml">j</mi></msub><mo id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.1" lspace="0.167em" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.cmml"><mo id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.1" rspace="0.167em" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.1.cmml">∘</mo><msub id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.cmml"><mi id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.2" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.2.cmml">ext</mi><mi id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.3" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.3.cmml">m</mi></msub></mrow></mrow><mo id="S4.Thmtheorem1.p1.11.6.m6.1.1.1" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.1.cmml">=</mo><mn id="S4.Thmtheorem1.p1.11.6.m6.1.1.3" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem1.p1.11.6.m6.1b"><apply id="S4.Thmtheorem1.p1.11.6.m6.1.1.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1"><eq id="S4.Thmtheorem1.p1.11.6.m6.1.1.1.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.1"></eq><apply id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2"><times id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.1"></times><apply id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.1.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.2.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.2">Tr</ci><ci id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.3.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.2.3">𝑗</ci></apply><apply id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3"><compose id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.1.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.1"></compose><apply id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.1.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.2.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.2">ext</ci><ci id="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.3.cmml" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.2.3.2.3">𝑚</ci></apply></apply></apply><cn id="S4.Thmtheorem1.p1.11.6.m6.1.1.3.cmml" type="integer" xref="S4.Thmtheorem1.p1.11.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem1.p1.11.6.m6.1c">\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem1.p1.11.6.m6.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S4.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.1.1.1">Remark 4.2</span></span><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem2.p1"> <p class="ltx_p" id="S4.Thmtheorem2.p1.13">If the conditions of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> hold and <math alttext="f_{1},f_{2}\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.1.m1.7"><semantics id="S4.Thmtheorem2.p1.1.m1.7a"><mrow id="S4.Thmtheorem2.p1.1.m1.7.7" xref="S4.Thmtheorem2.p1.1.m1.7.7.cmml"><mrow id="S4.Thmtheorem2.p1.1.m1.5.5.2.2" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.3.cmml"><msub id="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1" xref="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.2" xref="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.2.cmml">f</mi><mn id="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.3" xref="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.3" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.3.cmml">,</mo><msub id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.cmml"><mi id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.2" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.2.cmml">f</mi><mn id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.3" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S4.Thmtheorem2.p1.1.m1.7.7.5" xref="S4.Thmtheorem2.p1.1.m1.7.7.5.cmml">∈</mo><mrow id="S4.Thmtheorem2.p1.1.m1.7.7.4" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.cmml"><msup id="S4.Thmtheorem2.p1.1.m1.7.7.4.4" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.4.cmml"><mi id="S4.Thmtheorem2.p1.1.m1.7.7.4.4.2" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.4.2.cmml">H</mi><mrow id="S4.Thmtheorem2.p1.1.m1.2.2.2.4" xref="S4.Thmtheorem2.p1.1.m1.2.2.2.3.cmml"><mi id="S4.Thmtheorem2.p1.1.m1.1.1.1.1" xref="S4.Thmtheorem2.p1.1.m1.1.1.1.1.cmml">s</mi><mo id="S4.Thmtheorem2.p1.1.m1.2.2.2.4.1" xref="S4.Thmtheorem2.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem2.p1.1.m1.2.2.2.2" xref="S4.Thmtheorem2.p1.1.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Thmtheorem2.p1.1.m1.7.7.4.3" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.3.cmml">⁢</mo><mrow id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.3.cmml"><mo id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.3.cmml">(</mo><msup id="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1" xref="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.2" xref="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.2.cmml">ℝ</mi><mi id="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.3" xref="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.3.cmml">d</mi></msup><mo id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.4" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.3.cmml">,</mo><msub id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.cmml"><mi id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.2" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.2.cmml">w</mi><mi id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.3" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.3.cmml">γ</mi></msub><mo id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.5" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.3.cmml">;</mo><mi id="S4.Thmtheorem2.p1.1.m1.3.3" xref="S4.Thmtheorem2.p1.1.m1.3.3.cmml">X</mi><mo id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.6" stretchy="false" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.1.m1.7b"><apply id="S4.Thmtheorem2.p1.1.m1.7.7.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7"><in id="S4.Thmtheorem2.p1.1.m1.7.7.5.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.5"></in><list id="S4.Thmtheorem2.p1.1.m1.5.5.2.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2"><apply id="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.2">𝑓</ci><cn id="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.1.m1.4.4.1.1.1.3">1</cn></apply><apply id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.2">𝑓</ci><cn id="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.1.m1.5.5.2.2.2.3">2</cn></apply></list><apply id="S4.Thmtheorem2.p1.1.m1.7.7.4.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4"><times id="S4.Thmtheorem2.p1.1.m1.7.7.4.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.3"></times><apply id="S4.Thmtheorem2.p1.1.m1.7.7.4.4.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.4"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.m1.7.7.4.4.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.4">superscript</csymbol><ci id="S4.Thmtheorem2.p1.1.m1.7.7.4.4.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.4.2">𝐻</ci><list id="S4.Thmtheorem2.p1.1.m1.2.2.2.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.2.2.2.4"><ci id="S4.Thmtheorem2.p1.1.m1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.1.1.1.1">𝑠</ci><ci id="S4.Thmtheorem2.p1.1.m1.2.2.2.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.2.2.2.2">𝑝</ci></list></apply><vector id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2"><apply id="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.2">ℝ</ci><ci id="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.6.6.3.1.1.1.3">𝑑</ci></apply><apply id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.1.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.2.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.2">𝑤</ci><ci id="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.7.7.4.2.2.2.3">𝛾</ci></apply><ci id="S4.Thmtheorem2.p1.1.m1.3.3.cmml" xref="S4.Thmtheorem2.p1.1.m1.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.1.m1.7c">f_{1},f_{2}\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.1.m1.7d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> satisfy <math alttext="f_{1}|_{\mathbb{R}^{d}_{+}}=f_{2}|_{\mathbb{R}^{d}_{+}}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.2.m2.4"><semantics id="S4.Thmtheorem2.p1.2.m2.4a"><mrow id="S4.Thmtheorem2.p1.2.m2.4.4" xref="S4.Thmtheorem2.p1.2.m2.4.4.cmml"><msub id="S4.Thmtheorem2.p1.2.m2.3.3.1.1" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.2.cmml"><mrow id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.2.cmml"><msub id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.2" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.2.cmml">f</mi><mn id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.3" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.2.1.cmml">|</mo></mrow><msubsup id="S4.Thmtheorem2.p1.2.m2.1.1.1" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.1.1.1.2.2" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.Thmtheorem2.p1.2.m2.1.1.1.3" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.3.cmml">+</mo><mi id="S4.Thmtheorem2.p1.2.m2.1.1.1.2.3" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.2.3.cmml">d</mi></msubsup></msub><mo id="S4.Thmtheorem2.p1.2.m2.4.4.3" xref="S4.Thmtheorem2.p1.2.m2.4.4.3.cmml">=</mo><msub id="S4.Thmtheorem2.p1.2.m2.4.4.2.1" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.2.cmml"><mrow id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.2.cmml"><msub id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.2" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.2.cmml">f</mi><mn id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.3" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.3.cmml">2</mn></msub><mo id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.2" stretchy="false" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.2.1.cmml">|</mo></mrow><msubsup id="S4.Thmtheorem2.p1.2.m2.2.2.1" xref="S4.Thmtheorem2.p1.2.m2.2.2.1.cmml"><mi id="S4.Thmtheorem2.p1.2.m2.2.2.1.2.2" xref="S4.Thmtheorem2.p1.2.m2.2.2.1.2.2.cmml">ℝ</mi><mo id="S4.Thmtheorem2.p1.2.m2.2.2.1.3" xref="S4.Thmtheorem2.p1.2.m2.2.2.1.3.cmml">+</mo><mi id="S4.Thmtheorem2.p1.2.m2.2.2.1.2.3" xref="S4.Thmtheorem2.p1.2.m2.2.2.1.2.3.cmml">d</mi></msubsup></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.2.m2.4b"><apply id="S4.Thmtheorem2.p1.2.m2.4.4.cmml" xref="S4.Thmtheorem2.p1.2.m2.4.4"><eq id="S4.Thmtheorem2.p1.2.m2.4.4.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.4.4.3"></eq><apply id="S4.Thmtheorem2.p1.2.m2.3.3.1.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1"><csymbol cd="latexml" id="S4.Thmtheorem2.p1.2.m2.3.3.1.2.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.2">evaluated-at</csymbol><apply id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.2">𝑓</ci><cn id="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.2.m2.3.3.1.1.1.1.3">1</cn></apply><apply id="S4.Thmtheorem2.p1.2.m2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1">subscript</csymbol><apply id="S4.Thmtheorem2.p1.2.m2.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.1.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.2.m2.1.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.2.2">ℝ</ci><ci id="S4.Thmtheorem2.p1.2.m2.1.1.1.2.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.2.3">𝑑</ci></apply><plus id="S4.Thmtheorem2.p1.2.m2.1.1.1.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.1.1.1.3"></plus></apply></apply><apply id="S4.Thmtheorem2.p1.2.m2.4.4.2.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1"><csymbol cd="latexml" id="S4.Thmtheorem2.p1.2.m2.4.4.2.2.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.2">evaluated-at</csymbol><apply id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.2">𝑓</ci><cn id="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.2.m2.4.4.2.1.1.1.3">2</cn></apply><apply id="S4.Thmtheorem2.p1.2.m2.2.2.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.2.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.2.2.1.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.2.2.1">subscript</csymbol><apply id="S4.Thmtheorem2.p1.2.m2.2.2.1.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.2.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.2.m2.2.2.1.2.1.cmml" xref="S4.Thmtheorem2.p1.2.m2.2.2.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.2.m2.2.2.1.2.2.cmml" xref="S4.Thmtheorem2.p1.2.m2.2.2.1.2.2">ℝ</ci><ci id="S4.Thmtheorem2.p1.2.m2.2.2.1.2.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.2.2.1.2.3">𝑑</ci></apply><plus id="S4.Thmtheorem2.p1.2.m2.2.2.1.3.cmml" xref="S4.Thmtheorem2.p1.2.m2.2.2.1.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.2.m2.4c">f_{1}|_{\mathbb{R}^{d}_{+}}=f_{2}|_{\mathbb{R}^{d}_{+}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.2.m2.4d">italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math>, then we have <math alttext="\operatorname{Tr}_{j}f_{1}=\operatorname{Tr}_{j}f_{2}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.3.m3.1"><semantics id="S4.Thmtheorem2.p1.3.m3.1a"><mrow id="S4.Thmtheorem2.p1.3.m3.1.1" xref="S4.Thmtheorem2.p1.3.m3.1.1.cmml"><mrow id="S4.Thmtheorem2.p1.3.m3.1.1.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.cmml"><msub id="S4.Thmtheorem2.p1.3.m3.1.1.2.1" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.1.cmml"><mi id="S4.Thmtheorem2.p1.3.m3.1.1.2.1.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.1.2.cmml">Tr</mi><mi id="S4.Thmtheorem2.p1.3.m3.1.1.2.1.3" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.1.3.cmml">j</mi></msub><mo id="S4.Thmtheorem2.p1.3.m3.1.1.2a" lspace="0.167em" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.cmml">⁡</mo><msub id="S4.Thmtheorem2.p1.3.m3.1.1.2.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.2.cmml"><mi id="S4.Thmtheorem2.p1.3.m3.1.1.2.2.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.2.2.cmml">f</mi><mn id="S4.Thmtheorem2.p1.3.m3.1.1.2.2.3" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.2.3.cmml">1</mn></msub></mrow><mo id="S4.Thmtheorem2.p1.3.m3.1.1.1" xref="S4.Thmtheorem2.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S4.Thmtheorem2.p1.3.m3.1.1.3" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.cmml"><msub id="S4.Thmtheorem2.p1.3.m3.1.1.3.1" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.1.cmml"><mi id="S4.Thmtheorem2.p1.3.m3.1.1.3.1.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.1.2.cmml">Tr</mi><mi id="S4.Thmtheorem2.p1.3.m3.1.1.3.1.3" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.1.3.cmml">j</mi></msub><mo id="S4.Thmtheorem2.p1.3.m3.1.1.3a" lspace="0.167em" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.cmml">⁡</mo><msub id="S4.Thmtheorem2.p1.3.m3.1.1.3.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.2.cmml"><mi id="S4.Thmtheorem2.p1.3.m3.1.1.3.2.2" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.2.2.cmml">f</mi><mn id="S4.Thmtheorem2.p1.3.m3.1.1.3.2.3" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.2.3.cmml">2</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.3.m3.1b"><apply id="S4.Thmtheorem2.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1"><eq id="S4.Thmtheorem2.p1.3.m3.1.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.1"></eq><apply id="S4.Thmtheorem2.p1.3.m3.1.1.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2"><apply id="S4.Thmtheorem2.p1.3.m3.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.3.m3.1.1.2.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.3.m3.1.1.2.1.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.1.2">Tr</ci><ci id="S4.Thmtheorem2.p1.3.m3.1.1.2.1.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.1.3">𝑗</ci></apply><apply id="S4.Thmtheorem2.p1.3.m3.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.3.m3.1.1.2.2.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.3.m3.1.1.2.2.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.2.2">𝑓</ci><cn id="S4.Thmtheorem2.p1.3.m3.1.1.2.2.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.3.m3.1.1.2.2.3">1</cn></apply></apply><apply id="S4.Thmtheorem2.p1.3.m3.1.1.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3"><apply id="S4.Thmtheorem2.p1.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.3.m3.1.1.3.1.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.3.m3.1.1.3.1.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.1.2">Tr</ci><ci id="S4.Thmtheorem2.p1.3.m3.1.1.3.1.3.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.1.3">𝑗</ci></apply><apply id="S4.Thmtheorem2.p1.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.3.m3.1.1.3.2.1.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.2">subscript</csymbol><ci id="S4.Thmtheorem2.p1.3.m3.1.1.3.2.2.cmml" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.2.2">𝑓</ci><cn id="S4.Thmtheorem2.p1.3.m3.1.1.3.2.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.3.m3.1.1.3.2.3">2</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.3.m3.1c">\operatorname{Tr}_{j}f_{1}=\operatorname{Tr}_{j}f_{2}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.3.m3.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math> for all <math alttext="j\in\{0,\dots,m\}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.4.m4.3"><semantics id="S4.Thmtheorem2.p1.4.m4.3a"><mrow id="S4.Thmtheorem2.p1.4.m4.3.4" xref="S4.Thmtheorem2.p1.4.m4.3.4.cmml"><mi id="S4.Thmtheorem2.p1.4.m4.3.4.2" xref="S4.Thmtheorem2.p1.4.m4.3.4.2.cmml">j</mi><mo id="S4.Thmtheorem2.p1.4.m4.3.4.1" xref="S4.Thmtheorem2.p1.4.m4.3.4.1.cmml">∈</mo><mrow id="S4.Thmtheorem2.p1.4.m4.3.4.3.2" xref="S4.Thmtheorem2.p1.4.m4.3.4.3.1.cmml"><mo id="S4.Thmtheorem2.p1.4.m4.3.4.3.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.3.4.3.1.cmml">{</mo><mn id="S4.Thmtheorem2.p1.4.m4.1.1" xref="S4.Thmtheorem2.p1.4.m4.1.1.cmml">0</mn><mo id="S4.Thmtheorem2.p1.4.m4.3.4.3.2.2" xref="S4.Thmtheorem2.p1.4.m4.3.4.3.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.4.m4.2.2" mathvariant="normal" xref="S4.Thmtheorem2.p1.4.m4.2.2.cmml">…</mi><mo id="S4.Thmtheorem2.p1.4.m4.3.4.3.2.3" xref="S4.Thmtheorem2.p1.4.m4.3.4.3.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.4.m4.3.3" xref="S4.Thmtheorem2.p1.4.m4.3.3.cmml">m</mi><mo id="S4.Thmtheorem2.p1.4.m4.3.4.3.2.4" stretchy="false" xref="S4.Thmtheorem2.p1.4.m4.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.4.m4.3b"><apply id="S4.Thmtheorem2.p1.4.m4.3.4.cmml" xref="S4.Thmtheorem2.p1.4.m4.3.4"><in id="S4.Thmtheorem2.p1.4.m4.3.4.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.3.4.1"></in><ci id="S4.Thmtheorem2.p1.4.m4.3.4.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.3.4.2">𝑗</ci><set id="S4.Thmtheorem2.p1.4.m4.3.4.3.1.cmml" xref="S4.Thmtheorem2.p1.4.m4.3.4.3.2"><cn id="S4.Thmtheorem2.p1.4.m4.1.1.cmml" type="integer" xref="S4.Thmtheorem2.p1.4.m4.1.1">0</cn><ci id="S4.Thmtheorem2.p1.4.m4.2.2.cmml" xref="S4.Thmtheorem2.p1.4.m4.2.2">…</ci><ci id="S4.Thmtheorem2.p1.4.m4.3.3.cmml" xref="S4.Thmtheorem2.p1.4.m4.3.3">𝑚</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.4.m4.3c">j\in\{0,\dots,m\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.4.m4.3d">italic_j ∈ { 0 , … , italic_m }</annotation></semantics></math>. This can be proved similarly as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 6.3(2)]</cite>. This implies that the trace operator <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.5.m5.1"><semantics id="S4.Thmtheorem2.p1.5.m5.1a"><msub id="S4.Thmtheorem2.p1.5.m5.1.1" xref="S4.Thmtheorem2.p1.5.m5.1.1.cmml"><mi id="S4.Thmtheorem2.p1.5.m5.1.1.2" xref="S4.Thmtheorem2.p1.5.m5.1.1.2.cmml">Tr</mi><mi id="S4.Thmtheorem2.p1.5.m5.1.1.3" xref="S4.Thmtheorem2.p1.5.m5.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.5.m5.1b"><apply id="S4.Thmtheorem2.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.5.m5.1.1.1.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.5.m5.1.1.2.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.2">Tr</ci><ci id="S4.Thmtheorem2.p1.5.m5.1.1.3.cmml" xref="S4.Thmtheorem2.p1.5.m5.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.5.m5.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.5.m5.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.6.m6.1"><semantics id="S4.Thmtheorem2.p1.6.m6.1a"><msup id="S4.Thmtheorem2.p1.6.m6.1.1" xref="S4.Thmtheorem2.p1.6.m6.1.1.cmml"><mi id="S4.Thmtheorem2.p1.6.m6.1.1.2" xref="S4.Thmtheorem2.p1.6.m6.1.1.2.cmml">ℝ</mi><mi id="S4.Thmtheorem2.p1.6.m6.1.1.3" xref="S4.Thmtheorem2.p1.6.m6.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.6.m6.1b"><apply id="S4.Thmtheorem2.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.6.m6.1.1.1.cmml" xref="S4.Thmtheorem2.p1.6.m6.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.6.m6.1.1.2.cmml" xref="S4.Thmtheorem2.p1.6.m6.1.1.2">ℝ</ci><ci id="S4.Thmtheorem2.p1.6.m6.1.1.3.cmml" xref="S4.Thmtheorem2.p1.6.m6.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.6.m6.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> gives rise to a well-defined trace operator on <math alttext="\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.7.m7.1"><semantics id="S4.Thmtheorem2.p1.7.m7.1a"><msubsup id="S4.Thmtheorem2.p1.7.m7.1.1" xref="S4.Thmtheorem2.p1.7.m7.1.1.cmml"><mi id="S4.Thmtheorem2.p1.7.m7.1.1.2.2" xref="S4.Thmtheorem2.p1.7.m7.1.1.2.2.cmml">ℝ</mi><mo id="S4.Thmtheorem2.p1.7.m7.1.1.3" xref="S4.Thmtheorem2.p1.7.m7.1.1.3.cmml">+</mo><mi id="S4.Thmtheorem2.p1.7.m7.1.1.2.3" xref="S4.Thmtheorem2.p1.7.m7.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.7.m7.1b"><apply id="S4.Thmtheorem2.p1.7.m7.1.1.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.7.m7.1.1.1.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1">subscript</csymbol><apply id="S4.Thmtheorem2.p1.7.m7.1.1.2.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.7.m7.1.1.2.1.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1">superscript</csymbol><ci id="S4.Thmtheorem2.p1.7.m7.1.1.2.2.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1.2.2">ℝ</ci><ci id="S4.Thmtheorem2.p1.7.m7.1.1.2.3.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1.2.3">𝑑</ci></apply><plus id="S4.Thmtheorem2.p1.7.m7.1.1.3.cmml" xref="S4.Thmtheorem2.p1.7.m7.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.7.m7.1c">\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>, which we denote by <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.8.m8.1"><semantics id="S4.Thmtheorem2.p1.8.m8.1a"><msub id="S4.Thmtheorem2.p1.8.m8.1.1" xref="S4.Thmtheorem2.p1.8.m8.1.1.cmml"><mi id="S4.Thmtheorem2.p1.8.m8.1.1.2" xref="S4.Thmtheorem2.p1.8.m8.1.1.2.cmml">Tr</mi><mi id="S4.Thmtheorem2.p1.8.m8.1.1.3" xref="S4.Thmtheorem2.p1.8.m8.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.8.m8.1b"><apply id="S4.Thmtheorem2.p1.8.m8.1.1.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.8.m8.1.1.1.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1">subscript</csymbol><ci id="S4.Thmtheorem2.p1.8.m8.1.1.2.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1.2">Tr</ci><ci id="S4.Thmtheorem2.p1.8.m8.1.1.3.cmml" xref="S4.Thmtheorem2.p1.8.m8.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.8.m8.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.8.m8.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> as well. Moreover, if <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.9.m9.1"><semantics id="S4.Thmtheorem2.p1.9.m9.1a"><mi id="S4.Thmtheorem2.p1.9.m9.1.1" xref="S4.Thmtheorem2.p1.9.m9.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.9.m9.1b"><ci id="S4.Thmtheorem2.p1.9.m9.1.1.cmml" xref="S4.Thmtheorem2.p1.9.m9.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.9.m9.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.9.m9.1d">italic_f</annotation></semantics></math> is continuous on <math alttext="[0,\delta)\times\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.10.m10.2"><semantics id="S4.Thmtheorem2.p1.10.m10.2a"><mrow id="S4.Thmtheorem2.p1.10.m10.2.3" xref="S4.Thmtheorem2.p1.10.m10.2.3.cmml"><mrow id="S4.Thmtheorem2.p1.10.m10.2.3.2.2" xref="S4.Thmtheorem2.p1.10.m10.2.3.2.1.cmml"><mo id="S4.Thmtheorem2.p1.10.m10.2.3.2.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.10.m10.2.3.2.1.cmml">[</mo><mn id="S4.Thmtheorem2.p1.10.m10.1.1" xref="S4.Thmtheorem2.p1.10.m10.1.1.cmml">0</mn><mo id="S4.Thmtheorem2.p1.10.m10.2.3.2.2.2" xref="S4.Thmtheorem2.p1.10.m10.2.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem2.p1.10.m10.2.2" xref="S4.Thmtheorem2.p1.10.m10.2.2.cmml">δ</mi><mo id="S4.Thmtheorem2.p1.10.m10.2.3.2.2.3" rspace="0.055em" stretchy="false" xref="S4.Thmtheorem2.p1.10.m10.2.3.2.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem2.p1.10.m10.2.3.1" rspace="0.222em" xref="S4.Thmtheorem2.p1.10.m10.2.3.1.cmml">×</mo><msup id="S4.Thmtheorem2.p1.10.m10.2.3.3" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.cmml"><mi id="S4.Thmtheorem2.p1.10.m10.2.3.3.2" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.2.cmml">ℝ</mi><mrow id="S4.Thmtheorem2.p1.10.m10.2.3.3.3" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3.cmml"><mi id="S4.Thmtheorem2.p1.10.m10.2.3.3.3.2" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3.2.cmml">d</mi><mo id="S4.Thmtheorem2.p1.10.m10.2.3.3.3.1" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3.1.cmml">−</mo><mn id="S4.Thmtheorem2.p1.10.m10.2.3.3.3.3" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.10.m10.2b"><apply id="S4.Thmtheorem2.p1.10.m10.2.3.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3"><times id="S4.Thmtheorem2.p1.10.m10.2.3.1.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.1"></times><interval closure="closed-open" id="S4.Thmtheorem2.p1.10.m10.2.3.2.1.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.2.2"><cn id="S4.Thmtheorem2.p1.10.m10.1.1.cmml" type="integer" xref="S4.Thmtheorem2.p1.10.m10.1.1">0</cn><ci id="S4.Thmtheorem2.p1.10.m10.2.2.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.2">𝛿</ci></interval><apply id="S4.Thmtheorem2.p1.10.m10.2.3.3.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.10.m10.2.3.3.1.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.3">superscript</csymbol><ci id="S4.Thmtheorem2.p1.10.m10.2.3.3.2.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.2">ℝ</ci><apply id="S4.Thmtheorem2.p1.10.m10.2.3.3.3.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3"><minus id="S4.Thmtheorem2.p1.10.m10.2.3.3.3.1.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3.1"></minus><ci id="S4.Thmtheorem2.p1.10.m10.2.3.3.3.2.cmml" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3.2">𝑑</ci><cn id="S4.Thmtheorem2.p1.10.m10.2.3.3.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.10.m10.2.3.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.10.m10.2c">[0,\delta)\times\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.10.m10.2d">[ 0 , italic_δ ) × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> for some <math alttext="\delta>0" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.11.m11.1"><semantics id="S4.Thmtheorem2.p1.11.m11.1a"><mrow id="S4.Thmtheorem2.p1.11.m11.1.1" xref="S4.Thmtheorem2.p1.11.m11.1.1.cmml"><mi id="S4.Thmtheorem2.p1.11.m11.1.1.2" xref="S4.Thmtheorem2.p1.11.m11.1.1.2.cmml">δ</mi><mo id="S4.Thmtheorem2.p1.11.m11.1.1.1" xref="S4.Thmtheorem2.p1.11.m11.1.1.1.cmml">></mo><mn id="S4.Thmtheorem2.p1.11.m11.1.1.3" xref="S4.Thmtheorem2.p1.11.m11.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.11.m11.1b"><apply id="S4.Thmtheorem2.p1.11.m11.1.1.cmml" xref="S4.Thmtheorem2.p1.11.m11.1.1"><gt id="S4.Thmtheorem2.p1.11.m11.1.1.1.cmml" xref="S4.Thmtheorem2.p1.11.m11.1.1.1"></gt><ci id="S4.Thmtheorem2.p1.11.m11.1.1.2.cmml" xref="S4.Thmtheorem2.p1.11.m11.1.1.2">𝛿</ci><cn id="S4.Thmtheorem2.p1.11.m11.1.1.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.11.m11.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.11.m11.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.11.m11.1d">italic_δ > 0</annotation></semantics></math>, then the trace operator coincides with the restriction of <math alttext="f" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.12.m12.1"><semantics id="S4.Thmtheorem2.p1.12.m12.1a"><mi id="S4.Thmtheorem2.p1.12.m12.1.1" xref="S4.Thmtheorem2.p1.12.m12.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.12.m12.1b"><ci id="S4.Thmtheorem2.p1.12.m12.1.1.cmml" xref="S4.Thmtheorem2.p1.12.m12.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.12.m12.1c">f</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.12.m12.1d">italic_f</annotation></semantics></math> to <math alttext="\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem2.p1.13.m13.4"><semantics id="S4.Thmtheorem2.p1.13.m13.4a"><mrow id="S4.Thmtheorem2.p1.13.m13.4.4.2" xref="S4.Thmtheorem2.p1.13.m13.4.4.3.cmml"><mo id="S4.Thmtheorem2.p1.13.m13.4.4.2.3" stretchy="false" xref="S4.Thmtheorem2.p1.13.m13.4.4.3.1.cmml">{</mo><mrow id="S4.Thmtheorem2.p1.13.m13.3.3.1.1.2" xref="S4.Thmtheorem2.p1.13.m13.3.3.1.1.1.cmml"><mo id="S4.Thmtheorem2.p1.13.m13.3.3.1.1.2.1" stretchy="false" xref="S4.Thmtheorem2.p1.13.m13.3.3.1.1.1.cmml">(</mo><mn id="S4.Thmtheorem2.p1.13.m13.1.1" xref="S4.Thmtheorem2.p1.13.m13.1.1.cmml">0</mn><mo id="S4.Thmtheorem2.p1.13.m13.3.3.1.1.2.2" xref="S4.Thmtheorem2.p1.13.m13.3.3.1.1.1.cmml">,</mo><mover accent="true" id="S4.Thmtheorem2.p1.13.m13.2.2" xref="S4.Thmtheorem2.p1.13.m13.2.2.cmml"><mi id="S4.Thmtheorem2.p1.13.m13.2.2.2" xref="S4.Thmtheorem2.p1.13.m13.2.2.2.cmml">x</mi><mo id="S4.Thmtheorem2.p1.13.m13.2.2.1" xref="S4.Thmtheorem2.p1.13.m13.2.2.1.cmml">~</mo></mover><mo id="S4.Thmtheorem2.p1.13.m13.3.3.1.1.2.3" rspace="0.278em" stretchy="false" xref="S4.Thmtheorem2.p1.13.m13.3.3.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem2.p1.13.m13.4.4.2.4" rspace="0.278em" xref="S4.Thmtheorem2.p1.13.m13.4.4.3.1.cmml">:</mo><mrow id="S4.Thmtheorem2.p1.13.m13.4.4.2.2" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.cmml"><mover accent="true" id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.cmml"><mi id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.2" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.2.cmml">x</mi><mo id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.1" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.1.cmml">~</mo></mover><mo id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.1" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.1.cmml">∈</mo><msup id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.cmml"><mi id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.2" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.2.cmml">ℝ</mi><mrow id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.cmml"><mi id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.2" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.2.cmml">d</mi><mo id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.1" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.1.cmml">−</mo><mn id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.3" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.3.cmml">1</mn></mrow></msup></mrow><mo id="S4.Thmtheorem2.p1.13.m13.4.4.2.5" stretchy="false" xref="S4.Thmtheorem2.p1.13.m13.4.4.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem2.p1.13.m13.4b"><apply id="S4.Thmtheorem2.p1.13.m13.4.4.3.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2"><csymbol cd="latexml" id="S4.Thmtheorem2.p1.13.m13.4.4.3.1.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.3">conditional-set</csymbol><interval closure="open" id="S4.Thmtheorem2.p1.13.m13.3.3.1.1.1.cmml" xref="S4.Thmtheorem2.p1.13.m13.3.3.1.1.2"><cn id="S4.Thmtheorem2.p1.13.m13.1.1.cmml" type="integer" xref="S4.Thmtheorem2.p1.13.m13.1.1">0</cn><apply id="S4.Thmtheorem2.p1.13.m13.2.2.cmml" xref="S4.Thmtheorem2.p1.13.m13.2.2"><ci id="S4.Thmtheorem2.p1.13.m13.2.2.1.cmml" xref="S4.Thmtheorem2.p1.13.m13.2.2.1">~</ci><ci id="S4.Thmtheorem2.p1.13.m13.2.2.2.cmml" xref="S4.Thmtheorem2.p1.13.m13.2.2.2">𝑥</ci></apply></interval><apply id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2"><in id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.1.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.1"></in><apply id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2"><ci id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.1.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.1">~</ci><ci id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.2.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.2.2">𝑥</ci></apply><apply id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.1.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3">superscript</csymbol><ci id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.2.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.2">ℝ</ci><apply id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3"><minus id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.1.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.1"></minus><ci id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.2.cmml" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.2">𝑑</ci><cn id="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem2.p1.13.m13.4.4.2.2.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem2.p1.13.m13.4c">\{(0,\widetilde{x}):\widetilde{x}\in\mathbb{R}^{d-1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem2.p1.13.m13.4d">{ ( 0 , over~ start_ARG italic_x end_ARG ) : over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math>.</p> </div> </div> <div class="ltx_proof" id="S4.SS1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a>.</h6> <div class="ltx_para" id="S4.SS1.1.p1"> <p class="ltx_p" id="S4.SS1.1.p1.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.1.p1.1.1">Step 1: <math alttext="\mathcal{O}=\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.SS1.1.p1.1.1.m1.1"><semantics id="S4.SS1.1.p1.1.1.m1.1a"><mrow id="S4.SS1.1.p1.1.1.m1.1.1" xref="S4.SS1.1.p1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.1.p1.1.1.m1.1.1.2" xref="S4.SS1.1.p1.1.1.m1.1.1.2.cmml">𝒪</mi><mo id="S4.SS1.1.p1.1.1.m1.1.1.1" xref="S4.SS1.1.p1.1.1.m1.1.1.1.cmml">=</mo><msup id="S4.SS1.1.p1.1.1.m1.1.1.3" xref="S4.SS1.1.p1.1.1.m1.1.1.3.cmml"><mi id="S4.SS1.1.p1.1.1.m1.1.1.3.2" xref="S4.SS1.1.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S4.SS1.1.p1.1.1.m1.1.1.3.3" xref="S4.SS1.1.p1.1.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.1.p1.1.1.m1.1b"><apply id="S4.SS1.1.p1.1.1.m1.1.1.cmml" xref="S4.SS1.1.p1.1.1.m1.1.1"><eq id="S4.SS1.1.p1.1.1.m1.1.1.1.cmml" xref="S4.SS1.1.p1.1.1.m1.1.1.1"></eq><ci id="S4.SS1.1.p1.1.1.m1.1.1.2.cmml" xref="S4.SS1.1.p1.1.1.m1.1.1.2">𝒪</ci><apply id="S4.SS1.1.p1.1.1.m1.1.1.3.cmml" xref="S4.SS1.1.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.1.p1.1.1.m1.1.1.3.1.cmml" xref="S4.SS1.1.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS1.1.p1.1.1.m1.1.1.3.2.cmml" xref="S4.SS1.1.p1.1.1.m1.1.1.3.2">ℝ</ci><ci id="S4.SS1.1.p1.1.1.m1.1.1.3.3.cmml" xref="S4.SS1.1.p1.1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.1.p1.1.1.m1.1c">\mathcal{O}=\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.1.p1.1.1.m1.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>.</span> This follows immediately from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem9" title="Theorem 3.9. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.9</span></a> and the embeddings (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E4.1" title="In 2.4 ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.4a</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E4.2" title="In 2.4 ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.4b</span></a>).</p> </div> <div class="ltx_para" id="S4.SS1.2.p2"> <p class="ltx_p" id="S4.SS1.2.p2.2"><span class="ltx_text ltx_font_italic" id="S4.SS1.2.p2.1.1">Step 2: <math alttext="\mathcal{O}=\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS1.2.p2.1.1.m1.1"><semantics id="S4.SS1.2.p2.1.1.m1.1a"><mrow id="S4.SS1.2.p2.1.1.m1.1.1" xref="S4.SS1.2.p2.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p2.1.1.m1.1.1.2" xref="S4.SS1.2.p2.1.1.m1.1.1.2.cmml">𝒪</mi><mo id="S4.SS1.2.p2.1.1.m1.1.1.1" xref="S4.SS1.2.p2.1.1.m1.1.1.1.cmml">=</mo><msubsup id="S4.SS1.2.p2.1.1.m1.1.1.3" xref="S4.SS1.2.p2.1.1.m1.1.1.3.cmml"><mi id="S4.SS1.2.p2.1.1.m1.1.1.3.2.2" xref="S4.SS1.2.p2.1.1.m1.1.1.3.2.2.cmml">ℝ</mi><mo id="S4.SS1.2.p2.1.1.m1.1.1.3.3" xref="S4.SS1.2.p2.1.1.m1.1.1.3.3.cmml">+</mo><mi id="S4.SS1.2.p2.1.1.m1.1.1.3.2.3" xref="S4.SS1.2.p2.1.1.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.1.1.m1.1b"><apply id="S4.SS1.2.p2.1.1.m1.1.1.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1"><eq id="S4.SS1.2.p2.1.1.m1.1.1.1.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.1"></eq><ci id="S4.SS1.2.p2.1.1.m1.1.1.2.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.2">𝒪</ci><apply id="S4.SS1.2.p2.1.1.m1.1.1.3.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.2.p2.1.1.m1.1.1.3.1.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.3">subscript</csymbol><apply id="S4.SS1.2.p2.1.1.m1.1.1.3.2.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.2.p2.1.1.m1.1.1.3.2.1.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS1.2.p2.1.1.m1.1.1.3.2.2.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.3.2.2">ℝ</ci><ci id="S4.SS1.2.p2.1.1.m1.1.1.3.2.3.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.3.2.3">𝑑</ci></apply><plus id="S4.SS1.2.p2.1.1.m1.1.1.3.3.cmml" xref="S4.SS1.2.p2.1.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.1.1.m1.1c">\mathcal{O}=\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.1.1.m1.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>.</span> Statement <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i2" title="item ii ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> with <math alttext="\mathcal{O}=\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS1.2.p2.2.m1.1"><semantics id="S4.SS1.2.p2.2.m1.1a"><mrow id="S4.SS1.2.p2.2.m1.1.1" xref="S4.SS1.2.p2.2.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.2.p2.2.m1.1.1.2" xref="S4.SS1.2.p2.2.m1.1.1.2.cmml">𝒪</mi><mo id="S4.SS1.2.p2.2.m1.1.1.1" xref="S4.SS1.2.p2.2.m1.1.1.1.cmml">=</mo><msubsup id="S4.SS1.2.p2.2.m1.1.1.3" xref="S4.SS1.2.p2.2.m1.1.1.3.cmml"><mi id="S4.SS1.2.p2.2.m1.1.1.3.2.2" xref="S4.SS1.2.p2.2.m1.1.1.3.2.2.cmml">ℝ</mi><mo id="S4.SS1.2.p2.2.m1.1.1.3.3" xref="S4.SS1.2.p2.2.m1.1.1.3.3.cmml">+</mo><mi id="S4.SS1.2.p2.2.m1.1.1.3.2.3" xref="S4.SS1.2.p2.2.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.2.p2.2.m1.1b"><apply id="S4.SS1.2.p2.2.m1.1.1.cmml" xref="S4.SS1.2.p2.2.m1.1.1"><eq id="S4.SS1.2.p2.2.m1.1.1.1.cmml" xref="S4.SS1.2.p2.2.m1.1.1.1"></eq><ci id="S4.SS1.2.p2.2.m1.1.1.2.cmml" xref="S4.SS1.2.p2.2.m1.1.1.2">𝒪</ci><apply id="S4.SS1.2.p2.2.m1.1.1.3.cmml" xref="S4.SS1.2.p2.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.2.p2.2.m1.1.1.3.1.cmml" xref="S4.SS1.2.p2.2.m1.1.1.3">subscript</csymbol><apply id="S4.SS1.2.p2.2.m1.1.1.3.2.cmml" xref="S4.SS1.2.p2.2.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.2.p2.2.m1.1.1.3.2.1.cmml" xref="S4.SS1.2.p2.2.m1.1.1.3">superscript</csymbol><ci id="S4.SS1.2.p2.2.m1.1.1.3.2.2.cmml" xref="S4.SS1.2.p2.2.m1.1.1.3.2.2">ℝ</ci><ci id="S4.SS1.2.p2.2.m1.1.1.3.2.3.cmml" xref="S4.SS1.2.p2.2.m1.1.1.3.2.3">𝑑</ci></apply><plus id="S4.SS1.2.p2.2.m1.1.1.3.3.cmml" xref="S4.SS1.2.p2.2.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.2.p2.2.m1.1c">\mathcal{O}=\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.2.p2.2.m1.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> follows from Step 1 and a higher-order reflection argument, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 5.1]</cite>.</p> </div> <div class="ltx_para" id="S4.SS1.3.p3"> <p class="ltx_p" id="S4.SS1.3.p3.5">To prove <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> with <math alttext="\mathcal{O}=\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.1.m1.1"><semantics id="S4.SS1.3.p3.1.m1.1a"><mrow id="S4.SS1.3.p3.1.m1.1.1" xref="S4.SS1.3.p3.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.1.m1.1.1.2" xref="S4.SS1.3.p3.1.m1.1.1.2.cmml">𝒪</mi><mo id="S4.SS1.3.p3.1.m1.1.1.1" xref="S4.SS1.3.p3.1.m1.1.1.1.cmml">=</mo><msubsup id="S4.SS1.3.p3.1.m1.1.1.3" xref="S4.SS1.3.p3.1.m1.1.1.3.cmml"><mi id="S4.SS1.3.p3.1.m1.1.1.3.2.2" xref="S4.SS1.3.p3.1.m1.1.1.3.2.2.cmml">ℝ</mi><mo id="S4.SS1.3.p3.1.m1.1.1.3.3" xref="S4.SS1.3.p3.1.m1.1.1.3.3.cmml">+</mo><mi id="S4.SS1.3.p3.1.m1.1.1.3.2.3" xref="S4.SS1.3.p3.1.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.1.m1.1b"><apply id="S4.SS1.3.p3.1.m1.1.1.cmml" xref="S4.SS1.3.p3.1.m1.1.1"><eq id="S4.SS1.3.p3.1.m1.1.1.1.cmml" xref="S4.SS1.3.p3.1.m1.1.1.1"></eq><ci id="S4.SS1.3.p3.1.m1.1.1.2.cmml" xref="S4.SS1.3.p3.1.m1.1.1.2">𝒪</ci><apply id="S4.SS1.3.p3.1.m1.1.1.3.cmml" xref="S4.SS1.3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.1.m1.1.1.3.1.cmml" xref="S4.SS1.3.p3.1.m1.1.1.3">subscript</csymbol><apply id="S4.SS1.3.p3.1.m1.1.1.3.2.cmml" xref="S4.SS1.3.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.3.p3.1.m1.1.1.3.2.1.cmml" xref="S4.SS1.3.p3.1.m1.1.1.3">superscript</csymbol><ci id="S4.SS1.3.p3.1.m1.1.1.3.2.2.cmml" xref="S4.SS1.3.p3.1.m1.1.1.3.2.2">ℝ</ci><ci id="S4.SS1.3.p3.1.m1.1.1.3.2.3.cmml" xref="S4.SS1.3.p3.1.m1.1.1.3.2.3">𝑑</ci></apply><plus id="S4.SS1.3.p3.1.m1.1.1.3.3.cmml" xref="S4.SS1.3.p3.1.m1.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.1.m1.1c">\mathcal{O}=\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.1.m1.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> we will not construct a reflection operator from <math alttext="H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS1.3.p3.2.m2.5"><semantics id="S4.SS1.3.p3.2.m2.5a"><mrow id="S4.SS1.3.p3.2.m2.5.5" xref="S4.SS1.3.p3.2.m2.5.5.cmml"><msup id="S4.SS1.3.p3.2.m2.5.5.4" xref="S4.SS1.3.p3.2.m2.5.5.4.cmml"><mi id="S4.SS1.3.p3.2.m2.5.5.4.2" xref="S4.SS1.3.p3.2.m2.5.5.4.2.cmml">H</mi><mrow id="S4.SS1.3.p3.2.m2.2.2.2.4" xref="S4.SS1.3.p3.2.m2.2.2.2.3.cmml"><mi id="S4.SS1.3.p3.2.m2.1.1.1.1" xref="S4.SS1.3.p3.2.m2.1.1.1.1.cmml">s</mi><mo id="S4.SS1.3.p3.2.m2.2.2.2.4.1" xref="S4.SS1.3.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.3.p3.2.m2.2.2.2.2" xref="S4.SS1.3.p3.2.m2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS1.3.p3.2.m2.5.5.3" xref="S4.SS1.3.p3.2.m2.5.5.3.cmml">⁢</mo><mrow 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id="S4.SS1.3.p3.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.4.m4.1d">italic_X</annotation></semantics></math> to be <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S4.SS1.3.p3.5.m5.1"><semantics id="S4.SS1.3.p3.5.m5.1a"><mi id="S4.SS1.3.p3.5.m5.1.1" xref="S4.SS1.3.p3.5.m5.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.3.p3.5.m5.1b"><ci id="S4.SS1.3.p3.5.m5.1.1.cmml" xref="S4.SS1.3.p3.5.m5.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.3.p3.5.m5.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.3.p3.5.m5.1d">roman_UMD</annotation></semantics></math>, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.6]</cite>. Instead, we do the reflection argument on Besov spaces. We claim that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{m}:B^{s}_{p,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^{s-m-% \frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S4.E1.m1.9"><semantics id="S4.E1.m1.9a"><mrow id="S4.E1.m1.9.9" xref="S4.E1.m1.9.9.cmml"><msub id="S4.E1.m1.9.9.5" xref="S4.E1.m1.9.9.5.cmml"><mi id="S4.E1.m1.9.9.5.2" xref="S4.E1.m1.9.9.5.2.cmml">Tr</mi><mi id="S4.E1.m1.9.9.5.3" xref="S4.E1.m1.9.9.5.3.cmml">m</mi></msub><mo id="S4.E1.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S4.E1.m1.9.9.4.cmml">:</mo><mrow id="S4.E1.m1.9.9.3" xref="S4.E1.m1.9.9.3.cmml"><mrow id="S4.E1.m1.8.8.2.2" xref="S4.E1.m1.8.8.2.2.cmml"><msubsup id="S4.E1.m1.8.8.2.2.4" xref="S4.E1.m1.8.8.2.2.4.cmml"><mi id="S4.E1.m1.8.8.2.2.4.2.2" 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\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.E1.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.3.p3.6">is a continuous and surjective operator and has a right inverse. Note that <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 5.4]</cite> holds for real interpolation as well, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>, Section 1.2.4]</cite>. Therefore, by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 6.1]</cite> we find that there exists a higher-order reflection operator <math alttext="\mathcal{E}:B^{s}_{p,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^{s}_{p,p}(% \mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS1.3.p3.6.m1.10"><semantics id="S4.SS1.3.p3.6.m1.10a"><mrow id="S4.SS1.3.p3.6.m1.10.10" xref="S4.SS1.3.p3.6.m1.10.10.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.3.p3.6.m1.10.10.6" xref="S4.SS1.3.p3.6.m1.10.10.6.cmml">ℰ</mi><mo id="S4.SS1.3.p3.6.m1.10.10.5" lspace="0.278em" rspace="0.278em" xref="S4.SS1.3.p3.6.m1.10.10.5.cmml">:</mo><mrow id="S4.SS1.3.p3.6.m1.10.10.4" xref="S4.SS1.3.p3.6.m1.10.10.4.cmml"><mrow id="S4.SS1.3.p3.6.m1.8.8.2.2" xref="S4.SS1.3.p3.6.m1.8.8.2.2.cmml"><msubsup id="S4.SS1.3.p3.6.m1.8.8.2.2.4" xref="S4.SS1.3.p3.6.m1.8.8.2.2.4.cmml"><mi 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encoding="application/x-llamapun" id="S4.SS1.3.p3.6.m1.10d">caligraphic_E : italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. The claim now follows from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem8" title="Theorem 3.8. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.8</span></a>.</p> </div> <div class="ltx_para" id="S4.SS1.4.p4"> <p class="ltx_p" id="S4.SS1.4.p4.7">Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.SS1.4.p4.1.m1.1"><semantics id="S4.SS1.4.p4.1.m1.1a"><mrow id="S4.SS1.4.p4.1.m1.1.1" xref="S4.SS1.4.p4.1.m1.1.1.cmml"><mi id="S4.SS1.4.p4.1.m1.1.1.2" xref="S4.SS1.4.p4.1.m1.1.1.2.cmml">ε</mi><mo id="S4.SS1.4.p4.1.m1.1.1.1" xref="S4.SS1.4.p4.1.m1.1.1.1.cmml">></mo><mn id="S4.SS1.4.p4.1.m1.1.1.3" xref="S4.SS1.4.p4.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.1.m1.1b"><apply id="S4.SS1.4.p4.1.m1.1.1.cmml" xref="S4.SS1.4.p4.1.m1.1.1"><gt id="S4.SS1.4.p4.1.m1.1.1.1.cmml" xref="S4.SS1.4.p4.1.m1.1.1.1"></gt><ci id="S4.SS1.4.p4.1.m1.1.1.2.cmml" xref="S4.SS1.4.p4.1.m1.1.1.2">𝜀</ci><cn id="S4.SS1.4.p4.1.m1.1.1.3.cmml" type="integer" xref="S4.SS1.4.p4.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.1.m1.1c">\varepsilon>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.1.m1.1d">italic_ε > 0</annotation></semantics></math> be small and set <math alttext="\gamma_{0}:=\gamma+\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.4.p4.2.m2.1"><semantics id="S4.SS1.4.p4.2.m2.1a"><mrow id="S4.SS1.4.p4.2.m2.1.1" xref="S4.SS1.4.p4.2.m2.1.1.cmml"><msub id="S4.SS1.4.p4.2.m2.1.1.2" xref="S4.SS1.4.p4.2.m2.1.1.2.cmml"><mi id="S4.SS1.4.p4.2.m2.1.1.2.2" xref="S4.SS1.4.p4.2.m2.1.1.2.2.cmml">γ</mi><mn id="S4.SS1.4.p4.2.m2.1.1.2.3" xref="S4.SS1.4.p4.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS1.4.p4.2.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.4.p4.2.m2.1.1.1.cmml">:=</mo><mrow id="S4.SS1.4.p4.2.m2.1.1.3" xref="S4.SS1.4.p4.2.m2.1.1.3.cmml"><mi id="S4.SS1.4.p4.2.m2.1.1.3.2" xref="S4.SS1.4.p4.2.m2.1.1.3.2.cmml">γ</mi><mo id="S4.SS1.4.p4.2.m2.1.1.3.1" xref="S4.SS1.4.p4.2.m2.1.1.3.1.cmml">+</mo><mi id="S4.SS1.4.p4.2.m2.1.1.3.3" xref="S4.SS1.4.p4.2.m2.1.1.3.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.2.m2.1b"><apply id="S4.SS1.4.p4.2.m2.1.1.cmml" xref="S4.SS1.4.p4.2.m2.1.1"><csymbol cd="latexml" id="S4.SS1.4.p4.2.m2.1.1.1.cmml" xref="S4.SS1.4.p4.2.m2.1.1.1">assign</csymbol><apply id="S4.SS1.4.p4.2.m2.1.1.2.cmml" xref="S4.SS1.4.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.4.p4.2.m2.1.1.2.1.cmml" xref="S4.SS1.4.p4.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS1.4.p4.2.m2.1.1.2.2.cmml" xref="S4.SS1.4.p4.2.m2.1.1.2.2">𝛾</ci><cn id="S4.SS1.4.p4.2.m2.1.1.2.3.cmml" type="integer" xref="S4.SS1.4.p4.2.m2.1.1.2.3">0</cn></apply><apply id="S4.SS1.4.p4.2.m2.1.1.3.cmml" xref="S4.SS1.4.p4.2.m2.1.1.3"><plus id="S4.SS1.4.p4.2.m2.1.1.3.1.cmml" xref="S4.SS1.4.p4.2.m2.1.1.3.1"></plus><ci id="S4.SS1.4.p4.2.m2.1.1.3.2.cmml" xref="S4.SS1.4.p4.2.m2.1.1.3.2">𝛾</ci><ci id="S4.SS1.4.p4.2.m2.1.1.3.3.cmml" xref="S4.SS1.4.p4.2.m2.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.2.m2.1c">\gamma_{0}:=\gamma+\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.2.m2.1d">italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := italic_γ + italic_ε</annotation></semantics></math>, <math alttext="\gamma_{1}:=\gamma-\varepsilon" class="ltx_Math" display="inline" id="S4.SS1.4.p4.3.m3.1"><semantics id="S4.SS1.4.p4.3.m3.1a"><mrow id="S4.SS1.4.p4.3.m3.1.1" xref="S4.SS1.4.p4.3.m3.1.1.cmml"><msub id="S4.SS1.4.p4.3.m3.1.1.2" xref="S4.SS1.4.p4.3.m3.1.1.2.cmml"><mi id="S4.SS1.4.p4.3.m3.1.1.2.2" xref="S4.SS1.4.p4.3.m3.1.1.2.2.cmml">γ</mi><mn id="S4.SS1.4.p4.3.m3.1.1.2.3" xref="S4.SS1.4.p4.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS1.4.p4.3.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.4.p4.3.m3.1.1.1.cmml">:=</mo><mrow id="S4.SS1.4.p4.3.m3.1.1.3" xref="S4.SS1.4.p4.3.m3.1.1.3.cmml"><mi id="S4.SS1.4.p4.3.m3.1.1.3.2" xref="S4.SS1.4.p4.3.m3.1.1.3.2.cmml">γ</mi><mo id="S4.SS1.4.p4.3.m3.1.1.3.1" xref="S4.SS1.4.p4.3.m3.1.1.3.1.cmml">−</mo><mi id="S4.SS1.4.p4.3.m3.1.1.3.3" xref="S4.SS1.4.p4.3.m3.1.1.3.3.cmml">ε</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.3.m3.1b"><apply id="S4.SS1.4.p4.3.m3.1.1.cmml" xref="S4.SS1.4.p4.3.m3.1.1"><csymbol cd="latexml" id="S4.SS1.4.p4.3.m3.1.1.1.cmml" xref="S4.SS1.4.p4.3.m3.1.1.1">assign</csymbol><apply id="S4.SS1.4.p4.3.m3.1.1.2.cmml" xref="S4.SS1.4.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.4.p4.3.m3.1.1.2.1.cmml" xref="S4.SS1.4.p4.3.m3.1.1.2">subscript</csymbol><ci id="S4.SS1.4.p4.3.m3.1.1.2.2.cmml" xref="S4.SS1.4.p4.3.m3.1.1.2.2">𝛾</ci><cn id="S4.SS1.4.p4.3.m3.1.1.2.3.cmml" type="integer" xref="S4.SS1.4.p4.3.m3.1.1.2.3">1</cn></apply><apply id="S4.SS1.4.p4.3.m3.1.1.3.cmml" xref="S4.SS1.4.p4.3.m3.1.1.3"><minus id="S4.SS1.4.p4.3.m3.1.1.3.1.cmml" xref="S4.SS1.4.p4.3.m3.1.1.3.1"></minus><ci id="S4.SS1.4.p4.3.m3.1.1.3.2.cmml" xref="S4.SS1.4.p4.3.m3.1.1.3.2">𝛾</ci><ci id="S4.SS1.4.p4.3.m3.1.1.3.3.cmml" xref="S4.SS1.4.p4.3.m3.1.1.3.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.3.m3.1c">\gamma_{1}:=\gamma-\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.3.m3.1d">italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := italic_γ - italic_ε</annotation></semantics></math>, <math alttext="s_{0}:=s+\varepsilon/p" class="ltx_Math" display="inline" id="S4.SS1.4.p4.4.m4.1"><semantics id="S4.SS1.4.p4.4.m4.1a"><mrow id="S4.SS1.4.p4.4.m4.1.1" xref="S4.SS1.4.p4.4.m4.1.1.cmml"><msub id="S4.SS1.4.p4.4.m4.1.1.2" xref="S4.SS1.4.p4.4.m4.1.1.2.cmml"><mi id="S4.SS1.4.p4.4.m4.1.1.2.2" xref="S4.SS1.4.p4.4.m4.1.1.2.2.cmml">s</mi><mn id="S4.SS1.4.p4.4.m4.1.1.2.3" xref="S4.SS1.4.p4.4.m4.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS1.4.p4.4.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.4.p4.4.m4.1.1.1.cmml">:=</mo><mrow id="S4.SS1.4.p4.4.m4.1.1.3" xref="S4.SS1.4.p4.4.m4.1.1.3.cmml"><mi id="S4.SS1.4.p4.4.m4.1.1.3.2" xref="S4.SS1.4.p4.4.m4.1.1.3.2.cmml">s</mi><mo id="S4.SS1.4.p4.4.m4.1.1.3.1" xref="S4.SS1.4.p4.4.m4.1.1.3.1.cmml">+</mo><mrow id="S4.SS1.4.p4.4.m4.1.1.3.3" xref="S4.SS1.4.p4.4.m4.1.1.3.3.cmml"><mi id="S4.SS1.4.p4.4.m4.1.1.3.3.2" xref="S4.SS1.4.p4.4.m4.1.1.3.3.2.cmml">ε</mi><mo id="S4.SS1.4.p4.4.m4.1.1.3.3.1" xref="S4.SS1.4.p4.4.m4.1.1.3.3.1.cmml">/</mo><mi id="S4.SS1.4.p4.4.m4.1.1.3.3.3" xref="S4.SS1.4.p4.4.m4.1.1.3.3.3.cmml">p</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.4.m4.1b"><apply id="S4.SS1.4.p4.4.m4.1.1.cmml" xref="S4.SS1.4.p4.4.m4.1.1"><csymbol cd="latexml" id="S4.SS1.4.p4.4.m4.1.1.1.cmml" xref="S4.SS1.4.p4.4.m4.1.1.1">assign</csymbol><apply id="S4.SS1.4.p4.4.m4.1.1.2.cmml" xref="S4.SS1.4.p4.4.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.4.p4.4.m4.1.1.2.1.cmml" xref="S4.SS1.4.p4.4.m4.1.1.2">subscript</csymbol><ci id="S4.SS1.4.p4.4.m4.1.1.2.2.cmml" xref="S4.SS1.4.p4.4.m4.1.1.2.2">𝑠</ci><cn id="S4.SS1.4.p4.4.m4.1.1.2.3.cmml" type="integer" xref="S4.SS1.4.p4.4.m4.1.1.2.3">0</cn></apply><apply id="S4.SS1.4.p4.4.m4.1.1.3.cmml" xref="S4.SS1.4.p4.4.m4.1.1.3"><plus id="S4.SS1.4.p4.4.m4.1.1.3.1.cmml" xref="S4.SS1.4.p4.4.m4.1.1.3.1"></plus><ci id="S4.SS1.4.p4.4.m4.1.1.3.2.cmml" xref="S4.SS1.4.p4.4.m4.1.1.3.2">𝑠</ci><apply id="S4.SS1.4.p4.4.m4.1.1.3.3.cmml" xref="S4.SS1.4.p4.4.m4.1.1.3.3"><divide id="S4.SS1.4.p4.4.m4.1.1.3.3.1.cmml" xref="S4.SS1.4.p4.4.m4.1.1.3.3.1"></divide><ci id="S4.SS1.4.p4.4.m4.1.1.3.3.2.cmml" xref="S4.SS1.4.p4.4.m4.1.1.3.3.2">𝜀</ci><ci id="S4.SS1.4.p4.4.m4.1.1.3.3.3.cmml" xref="S4.SS1.4.p4.4.m4.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.4.m4.1c">s_{0}:=s+\varepsilon/p</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.4.m4.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := italic_s + italic_ε / italic_p</annotation></semantics></math> and <math alttext="s_{1}:=s-\varepsilon/p" class="ltx_Math" display="inline" id="S4.SS1.4.p4.5.m5.1"><semantics id="S4.SS1.4.p4.5.m5.1a"><mrow id="S4.SS1.4.p4.5.m5.1.1" xref="S4.SS1.4.p4.5.m5.1.1.cmml"><msub id="S4.SS1.4.p4.5.m5.1.1.2" xref="S4.SS1.4.p4.5.m5.1.1.2.cmml"><mi id="S4.SS1.4.p4.5.m5.1.1.2.2" xref="S4.SS1.4.p4.5.m5.1.1.2.2.cmml">s</mi><mn id="S4.SS1.4.p4.5.m5.1.1.2.3" xref="S4.SS1.4.p4.5.m5.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS1.4.p4.5.m5.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.4.p4.5.m5.1.1.1.cmml">:=</mo><mrow id="S4.SS1.4.p4.5.m5.1.1.3" xref="S4.SS1.4.p4.5.m5.1.1.3.cmml"><mi id="S4.SS1.4.p4.5.m5.1.1.3.2" xref="S4.SS1.4.p4.5.m5.1.1.3.2.cmml">s</mi><mo id="S4.SS1.4.p4.5.m5.1.1.3.1" xref="S4.SS1.4.p4.5.m5.1.1.3.1.cmml">−</mo><mrow id="S4.SS1.4.p4.5.m5.1.1.3.3" xref="S4.SS1.4.p4.5.m5.1.1.3.3.cmml"><mi id="S4.SS1.4.p4.5.m5.1.1.3.3.2" xref="S4.SS1.4.p4.5.m5.1.1.3.3.2.cmml">ε</mi><mo id="S4.SS1.4.p4.5.m5.1.1.3.3.1" xref="S4.SS1.4.p4.5.m5.1.1.3.3.1.cmml">/</mo><mi id="S4.SS1.4.p4.5.m5.1.1.3.3.3" xref="S4.SS1.4.p4.5.m5.1.1.3.3.3.cmml">p</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.5.m5.1b"><apply id="S4.SS1.4.p4.5.m5.1.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1"><csymbol cd="latexml" id="S4.SS1.4.p4.5.m5.1.1.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1.1">assign</csymbol><apply id="S4.SS1.4.p4.5.m5.1.1.2.cmml" xref="S4.SS1.4.p4.5.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.4.p4.5.m5.1.1.2.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1.2">subscript</csymbol><ci id="S4.SS1.4.p4.5.m5.1.1.2.2.cmml" xref="S4.SS1.4.p4.5.m5.1.1.2.2">𝑠</ci><cn id="S4.SS1.4.p4.5.m5.1.1.2.3.cmml" type="integer" xref="S4.SS1.4.p4.5.m5.1.1.2.3">1</cn></apply><apply id="S4.SS1.4.p4.5.m5.1.1.3.cmml" xref="S4.SS1.4.p4.5.m5.1.1.3"><minus id="S4.SS1.4.p4.5.m5.1.1.3.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1.3.1"></minus><ci id="S4.SS1.4.p4.5.m5.1.1.3.2.cmml" xref="S4.SS1.4.p4.5.m5.1.1.3.2">𝑠</ci><apply id="S4.SS1.4.p4.5.m5.1.1.3.3.cmml" xref="S4.SS1.4.p4.5.m5.1.1.3.3"><divide id="S4.SS1.4.p4.5.m5.1.1.3.3.1.cmml" xref="S4.SS1.4.p4.5.m5.1.1.3.3.1"></divide><ci id="S4.SS1.4.p4.5.m5.1.1.3.3.2.cmml" xref="S4.SS1.4.p4.5.m5.1.1.3.3.2">𝜀</ci><ci id="S4.SS1.4.p4.5.m5.1.1.3.3.3.cmml" xref="S4.SS1.4.p4.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.5.m5.1c">s_{1}:=s-\varepsilon/p</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.5.m5.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := italic_s - italic_ε / italic_p</annotation></semantics></math>. From Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.Thmtheorem3" title="Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E4.1" title="In 2.4 ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.4a</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E5" title="In 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.5</span></a>) (which also hold on <math alttext="\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS1.4.p4.6.m6.1"><semantics id="S4.SS1.4.p4.6.m6.1a"><msubsup id="S4.SS1.4.p4.6.m6.1.1" xref="S4.SS1.4.p4.6.m6.1.1.cmml"><mi id="S4.SS1.4.p4.6.m6.1.1.2.2" xref="S4.SS1.4.p4.6.m6.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS1.4.p4.6.m6.1.1.3" xref="S4.SS1.4.p4.6.m6.1.1.3.cmml">+</mo><mi id="S4.SS1.4.p4.6.m6.1.1.2.3" xref="S4.SS1.4.p4.6.m6.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.6.m6.1b"><apply id="S4.SS1.4.p4.6.m6.1.1.cmml" xref="S4.SS1.4.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.6.m6.1.1.1.cmml" xref="S4.SS1.4.p4.6.m6.1.1">subscript</csymbol><apply id="S4.SS1.4.p4.6.m6.1.1.2.cmml" xref="S4.SS1.4.p4.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.6.m6.1.1.2.1.cmml" xref="S4.SS1.4.p4.6.m6.1.1">superscript</csymbol><ci id="S4.SS1.4.p4.6.m6.1.1.2.2.cmml" xref="S4.SS1.4.p4.6.m6.1.1.2.2">ℝ</ci><ci id="S4.SS1.4.p4.6.m6.1.1.2.3.cmml" xref="S4.SS1.4.p4.6.m6.1.1.2.3">𝑑</ci></apply><plus id="S4.SS1.4.p4.6.m6.1.1.3.cmml" xref="S4.SS1.4.p4.6.m6.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.6.m6.1c">\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.6.m6.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> since the spaces on <math alttext="\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS1.4.p4.7.m7.1"><semantics id="S4.SS1.4.p4.7.m7.1a"><msubsup id="S4.SS1.4.p4.7.m7.1.1" xref="S4.SS1.4.p4.7.m7.1.1.cmml"><mi id="S4.SS1.4.p4.7.m7.1.1.2.2" xref="S4.SS1.4.p4.7.m7.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS1.4.p4.7.m7.1.1.3" xref="S4.SS1.4.p4.7.m7.1.1.3.cmml">+</mo><mi id="S4.SS1.4.p4.7.m7.1.1.2.3" xref="S4.SS1.4.p4.7.m7.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.7.m7.1b"><apply id="S4.SS1.4.p4.7.m7.1.1.cmml" xref="S4.SS1.4.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.7.m7.1.1.1.cmml" xref="S4.SS1.4.p4.7.m7.1.1">subscript</csymbol><apply id="S4.SS1.4.p4.7.m7.1.1.2.cmml" xref="S4.SS1.4.p4.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS1.4.p4.7.m7.1.1.2.1.cmml" xref="S4.SS1.4.p4.7.m7.1.1">superscript</csymbol><ci id="S4.SS1.4.p4.7.m7.1.1.2.2.cmml" xref="S4.SS1.4.p4.7.m7.1.1.2.2">ℝ</ci><ci id="S4.SS1.4.p4.7.m7.1.1.2.3.cmml" xref="S4.SS1.4.p4.7.m7.1.1.2.3">𝑑</ci></apply><plus id="S4.SS1.4.p4.7.m7.1.1.3.cmml" xref="S4.SS1.4.p4.7.m7.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.4.p4.7.m7.1c">\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.4.p4.7.m7.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> are defined as factor spaces), we have the chain of embeddings</p> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E2"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E2X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle B^{s_{0}}_{p,p}(\mathbb{R}^{d}_{+},w_{\gamma_{0}};X)" class="ltx_Math" display="inline" id="S4.E2X.2.1.1.m1.5"><semantics id="S4.E2X.2.1.1.m1.5a"><mrow id="S4.E2X.2.1.1.m1.5.5" xref="S4.E2X.2.1.1.m1.5.5.cmml"><msubsup id="S4.E2X.2.1.1.m1.5.5.4" xref="S4.E2X.2.1.1.m1.5.5.4.cmml"><mi id="S4.E2X.2.1.1.m1.5.5.4.2.2" xref="S4.E2X.2.1.1.m1.5.5.4.2.2.cmml">B</mi><mrow id="S4.E2X.2.1.1.m1.2.2.2.4" xref="S4.E2X.2.1.1.m1.2.2.2.3.cmml"><mi id="S4.E2X.2.1.1.m1.1.1.1.1" xref="S4.E2X.2.1.1.m1.1.1.1.1.cmml">p</mi><mo id="S4.E2X.2.1.1.m1.2.2.2.4.1" xref="S4.E2X.2.1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.E2X.2.1.1.m1.2.2.2.2" xref="S4.E2X.2.1.1.m1.2.2.2.2.cmml">p</mi></mrow><msub id="S4.E2X.2.1.1.m1.5.5.4.2.3" xref="S4.E2X.2.1.1.m1.5.5.4.2.3.cmml"><mi id="S4.E2X.2.1.1.m1.5.5.4.2.3.2" xref="S4.E2X.2.1.1.m1.5.5.4.2.3.2.cmml">s</mi><mn id="S4.E2X.2.1.1.m1.5.5.4.2.3.3" xref="S4.E2X.2.1.1.m1.5.5.4.2.3.3.cmml">0</mn></msub></msubsup><mo id="S4.E2X.2.1.1.m1.5.5.3" xref="S4.E2X.2.1.1.m1.5.5.3.cmml">⁢</mo><mrow id="S4.E2X.2.1.1.m1.5.5.2.2" xref="S4.E2X.2.1.1.m1.5.5.2.3.cmml"><mo id="S4.E2X.2.1.1.m1.5.5.2.2.3" stretchy="false" xref="S4.E2X.2.1.1.m1.5.5.2.3.cmml">(</mo><msubsup id="S4.E2X.2.1.1.m1.4.4.1.1.1" xref="S4.E2X.2.1.1.m1.4.4.1.1.1.cmml"><mi id="S4.E2X.2.1.1.m1.4.4.1.1.1.2.2" xref="S4.E2X.2.1.1.m1.4.4.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.E2X.2.1.1.m1.4.4.1.1.1.3" xref="S4.E2X.2.1.1.m1.4.4.1.1.1.3.cmml">+</mo><mi id="S4.E2X.2.1.1.m1.4.4.1.1.1.2.3" xref="S4.E2X.2.1.1.m1.4.4.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.E2X.2.1.1.m1.5.5.2.2.4" xref="S4.E2X.2.1.1.m1.5.5.2.3.cmml">,</mo><msub id="S4.E2X.2.1.1.m1.5.5.2.2.2" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.cmml"><mi id="S4.E2X.2.1.1.m1.5.5.2.2.2.2" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.2.cmml">w</mi><msub id="S4.E2X.2.1.1.m1.5.5.2.2.2.3" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.3.cmml"><mi id="S4.E2X.2.1.1.m1.5.5.2.2.2.3.2" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.3.2.cmml">γ</mi><mn id="S4.E2X.2.1.1.m1.5.5.2.2.2.3.3" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.3.3.cmml">0</mn></msub></msub><mo id="S4.E2X.2.1.1.m1.5.5.2.2.5" xref="S4.E2X.2.1.1.m1.5.5.2.3.cmml">;</mo><mi id="S4.E2X.2.1.1.m1.3.3" xref="S4.E2X.2.1.1.m1.3.3.cmml">X</mi><mo id="S4.E2X.2.1.1.m1.5.5.2.2.6" stretchy="false" xref="S4.E2X.2.1.1.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E2X.2.1.1.m1.5b"><apply id="S4.E2X.2.1.1.m1.5.5.cmml" xref="S4.E2X.2.1.1.m1.5.5"><times id="S4.E2X.2.1.1.m1.5.5.3.cmml" xref="S4.E2X.2.1.1.m1.5.5.3"></times><apply id="S4.E2X.2.1.1.m1.5.5.4.cmml" xref="S4.E2X.2.1.1.m1.5.5.4"><csymbol cd="ambiguous" id="S4.E2X.2.1.1.m1.5.5.4.1.cmml" xref="S4.E2X.2.1.1.m1.5.5.4">subscript</csymbol><apply id="S4.E2X.2.1.1.m1.5.5.4.2.cmml" xref="S4.E2X.2.1.1.m1.5.5.4"><csymbol cd="ambiguous" id="S4.E2X.2.1.1.m1.5.5.4.2.1.cmml" xref="S4.E2X.2.1.1.m1.5.5.4">superscript</csymbol><ci id="S4.E2X.2.1.1.m1.5.5.4.2.2.cmml" xref="S4.E2X.2.1.1.m1.5.5.4.2.2">𝐵</ci><apply id="S4.E2X.2.1.1.m1.5.5.4.2.3.cmml" xref="S4.E2X.2.1.1.m1.5.5.4.2.3"><csymbol cd="ambiguous" id="S4.E2X.2.1.1.m1.5.5.4.2.3.1.cmml" xref="S4.E2X.2.1.1.m1.5.5.4.2.3">subscript</csymbol><ci id="S4.E2X.2.1.1.m1.5.5.4.2.3.2.cmml" xref="S4.E2X.2.1.1.m1.5.5.4.2.3.2">𝑠</ci><cn id="S4.E2X.2.1.1.m1.5.5.4.2.3.3.cmml" type="integer" xref="S4.E2X.2.1.1.m1.5.5.4.2.3.3">0</cn></apply></apply><list id="S4.E2X.2.1.1.m1.2.2.2.3.cmml" xref="S4.E2X.2.1.1.m1.2.2.2.4"><ci id="S4.E2X.2.1.1.m1.1.1.1.1.cmml" xref="S4.E2X.2.1.1.m1.1.1.1.1">𝑝</ci><ci id="S4.E2X.2.1.1.m1.2.2.2.2.cmml" xref="S4.E2X.2.1.1.m1.2.2.2.2">𝑝</ci></list></apply><vector 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xref="S4.E2X.2.1.1.m1.5.5.2.2.2.2">𝑤</ci><apply id="S4.E2X.2.1.1.m1.5.5.2.2.2.3.cmml" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.3"><csymbol cd="ambiguous" id="S4.E2X.2.1.1.m1.5.5.2.2.2.3.1.cmml" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.3">subscript</csymbol><ci id="S4.E2X.2.1.1.m1.5.5.2.2.2.3.2.cmml" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.3.2">𝛾</ci><cn id="S4.E2X.2.1.1.m1.5.5.2.2.2.3.3.cmml" type="integer" xref="S4.E2X.2.1.1.m1.5.5.2.2.2.3.3">0</cn></apply></apply><ci id="S4.E2X.2.1.1.m1.3.3.cmml" xref="S4.E2X.2.1.1.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E2X.2.1.1.m1.5c">\displaystyle B^{s_{0}}_{p,p}(\mathbb{R}^{d}_{+},w_{\gamma_{0}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.E2X.2.1.1.m1.5d">italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=F^{s_{0}}_{p,p}(\mathbb{R}^{d}_{+},w_{\gamma_{0}};X)% \hookrightarrow F^{s}_{p,1}(\mathbb{R}^{d}_{+},w_{\gamma};X)\hookrightarrow H^% {s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.E2X.3.2.2.m1.15"><semantics id="S4.E2X.3.2.2.m1.15a"><mrow id="S4.E2X.3.2.2.m1.15.15" xref="S4.E2X.3.2.2.m1.15.15.cmml"><mi id="S4.E2X.3.2.2.m1.15.15.8" xref="S4.E2X.3.2.2.m1.15.15.8.cmml"></mi><mo id="S4.E2X.3.2.2.m1.15.15.9" xref="S4.E2X.3.2.2.m1.15.15.9.cmml">=</mo><mrow id="S4.E2X.3.2.2.m1.11.11.2" xref="S4.E2X.3.2.2.m1.11.11.2.cmml"><msubsup id="S4.E2X.3.2.2.m1.11.11.2.4" xref="S4.E2X.3.2.2.m1.11.11.2.4.cmml"><mi id="S4.E2X.3.2.2.m1.11.11.2.4.2.2" xref="S4.E2X.3.2.2.m1.11.11.2.4.2.2.cmml">F</mi><mrow 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italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) ↪ italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , 1 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ↪ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(4.2)</span></td> </tr> <tr 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_{p,p}(\mathbb{R}^{d}_{+},w_{\gamma_{1}};X).</annotation><annotation encoding="application/x-llamapun" id="S4.E2Xa.2.1.1.m1.10d">↪ italic_F start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , ∞ end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ↪ italic_F start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) = italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.SS1.4.p4.8">From (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E1" title="In Proof of Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E2" title="In Proof of Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.2</span></a>) it follows that for <math alttext="f\in H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS1.4.p4.8.m1.5"><semantics id="S4.SS1.4.p4.8.m1.5a"><mrow id="S4.SS1.4.p4.8.m1.5.5" xref="S4.SS1.4.p4.8.m1.5.5.cmml"><mi id="S4.SS1.4.p4.8.m1.5.5.4" xref="S4.SS1.4.p4.8.m1.5.5.4.cmml">f</mi><mo id="S4.SS1.4.p4.8.m1.5.5.3" xref="S4.SS1.4.p4.8.m1.5.5.3.cmml">∈</mo><mrow id="S4.SS1.4.p4.8.m1.5.5.2" xref="S4.SS1.4.p4.8.m1.5.5.2.cmml"><msup id="S4.SS1.4.p4.8.m1.5.5.2.4" xref="S4.SS1.4.p4.8.m1.5.5.2.4.cmml"><mi id="S4.SS1.4.p4.8.m1.5.5.2.4.2" xref="S4.SS1.4.p4.8.m1.5.5.2.4.2.cmml">H</mi><mrow id="S4.SS1.4.p4.8.m1.2.2.2.4" xref="S4.SS1.4.p4.8.m1.2.2.2.3.cmml"><mi id="S4.SS1.4.p4.8.m1.1.1.1.1" xref="S4.SS1.4.p4.8.m1.1.1.1.1.cmml">s</mi><mo id="S4.SS1.4.p4.8.m1.2.2.2.4.1" xref="S4.SS1.4.p4.8.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS1.4.p4.8.m1.2.2.2.2" xref="S4.SS1.4.p4.8.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS1.4.p4.8.m1.5.5.2.3" xref="S4.SS1.4.p4.8.m1.5.5.2.3.cmml">⁢</mo><mrow id="S4.SS1.4.p4.8.m1.5.5.2.2.2" xref="S4.SS1.4.p4.8.m1.5.5.2.2.3.cmml"><mo id="S4.SS1.4.p4.8.m1.5.5.2.2.2.3" stretchy="false" xref="S4.SS1.4.p4.8.m1.5.5.2.2.3.cmml">(</mo><msubsup id="S4.SS1.4.p4.8.m1.4.4.1.1.1.1" xref="S4.SS1.4.p4.8.m1.4.4.1.1.1.1.cmml"><mi id="S4.SS1.4.p4.8.m1.4.4.1.1.1.1.2.2" xref="S4.SS1.4.p4.8.m1.4.4.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS1.4.p4.8.m1.4.4.1.1.1.1.3" xref="S4.SS1.4.p4.8.m1.4.4.1.1.1.1.3.cmml">+</mo><mi id="S4.SS1.4.p4.8.m1.4.4.1.1.1.1.2.3" xref="S4.SS1.4.p4.8.m1.4.4.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.SS1.4.p4.8.m1.5.5.2.2.2.4" xref="S4.SS1.4.p4.8.m1.5.5.2.2.3.cmml">,</mo><msub id="S4.SS1.4.p4.8.m1.5.5.2.2.2.2" xref="S4.SS1.4.p4.8.m1.5.5.2.2.2.2.cmml"><mi id="S4.SS1.4.p4.8.m1.5.5.2.2.2.2.2" xref="S4.SS1.4.p4.8.m1.5.5.2.2.2.2.2.cmml">w</mi><mi id="S4.SS1.4.p4.8.m1.5.5.2.2.2.2.3" xref="S4.SS1.4.p4.8.m1.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS1.4.p4.8.m1.5.5.2.2.2.5" xref="S4.SS1.4.p4.8.m1.5.5.2.2.3.cmml">;</mo><mi id="S4.SS1.4.p4.8.m1.3.3" xref="S4.SS1.4.p4.8.m1.3.3.cmml">X</mi><mo id="S4.SS1.4.p4.8.m1.5.5.2.2.2.6" stretchy="false" xref="S4.SS1.4.p4.8.m1.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.4.p4.8.m1.5b"><apply id="S4.SS1.4.p4.8.m1.5.5.cmml" xref="S4.SS1.4.p4.8.m1.5.5"><in id="S4.SS1.4.p4.8.m1.5.5.3.cmml" xref="S4.SS1.4.p4.8.m1.5.5.3"></in><ci id="S4.SS1.4.p4.8.m1.5.5.4.cmml" xref="S4.SS1.4.p4.8.m1.5.5.4">𝑓</ci><apply id="S4.SS1.4.p4.8.m1.5.5.2.cmml" xref="S4.SS1.4.p4.8.m1.5.5.2"><times id="S4.SS1.4.p4.8.m1.5.5.2.3.cmml" xref="S4.SS1.4.p4.8.m1.5.5.2.3"></times><apply id="S4.SS1.4.p4.8.m1.5.5.2.4.cmml" xref="S4.SS1.4.p4.8.m1.5.5.2.4"><csymbol cd="ambiguous" id="S4.SS1.4.p4.8.m1.5.5.2.4.1.cmml" 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start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx14"> <tbody id="S4.Ex3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\operatorname{Tr}_{m}f\|_{B^{s-m-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};X)}" class="ltx_Math" display="inline" id="S4.Ex3.m1.5"><semantics id="S4.Ex3.m1.5a"><msub id="S4.Ex3.m1.5.5" xref="S4.Ex3.m1.5.5.cmml"><mrow id="S4.Ex3.m1.5.5.1.1" xref="S4.Ex3.m1.5.5.1.2.cmml"><mo id="S4.Ex3.m1.5.5.1.1.2" stretchy="false" xref="S4.Ex3.m1.5.5.1.2.1.cmml">‖</mo><mrow id="S4.Ex3.m1.5.5.1.1.1" xref="S4.Ex3.m1.5.5.1.1.1.cmml"><msub id="S4.Ex3.m1.5.5.1.1.1.1" xref="S4.Ex3.m1.5.5.1.1.1.1.cmml"><mi id="S4.Ex3.m1.5.5.1.1.1.1.2" xref="S4.Ex3.m1.5.5.1.1.1.1.2.cmml">Tr</mi><mi id="S4.Ex3.m1.5.5.1.1.1.1.3" xref="S4.Ex3.m1.5.5.1.1.1.1.3.cmml">m</mi></msub><mo id="S4.Ex3.m1.5.5.1.1.1a" lspace="0.167em" xref="S4.Ex3.m1.5.5.1.1.1.cmml">⁡</mo><mi id="S4.Ex3.m1.5.5.1.1.1.2" xref="S4.Ex3.m1.5.5.1.1.1.2.cmml">f</mi></mrow><mo id="S4.Ex3.m1.5.5.1.1.3" stretchy="false" xref="S4.Ex3.m1.5.5.1.2.1.cmml">‖</mo></mrow><mrow id="S4.Ex3.m1.4.4.4" xref="S4.Ex3.m1.4.4.4.cmml"><msubsup id="S4.Ex3.m1.4.4.4.6" xref="S4.Ex3.m1.4.4.4.6.cmml"><mi id="S4.Ex3.m1.4.4.4.6.2.2" xref="S4.Ex3.m1.4.4.4.6.2.2.cmml">B</mi><mrow id="S4.Ex3.m1.2.2.2.2.2.4" xref="S4.Ex3.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex3.m1.1.1.1.1.1.1" xref="S4.Ex3.m1.1.1.1.1.1.1.cmml">p</mi><mo id="S4.Ex3.m1.2.2.2.2.2.4.1" xref="S4.Ex3.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.Ex3.m1.2.2.2.2.2.2" xref="S4.Ex3.m1.2.2.2.2.2.2.cmml">p</mi></mrow><mrow id="S4.Ex3.m1.4.4.4.6.2.3" xref="S4.Ex3.m1.4.4.4.6.2.3.cmml"><mi id="S4.Ex3.m1.4.4.4.6.2.3.2" xref="S4.Ex3.m1.4.4.4.6.2.3.2.cmml">s</mi><mo id="S4.Ex3.m1.4.4.4.6.2.3.1" xref="S4.Ex3.m1.4.4.4.6.2.3.1.cmml">−</mo><mi id="S4.Ex3.m1.4.4.4.6.2.3.3" xref="S4.Ex3.m1.4.4.4.6.2.3.3.cmml">m</mi><mo id="S4.Ex3.m1.4.4.4.6.2.3.1a" xref="S4.Ex3.m1.4.4.4.6.2.3.1.cmml">−</mo><mfrac id="S4.Ex3.m1.4.4.4.6.2.3.4" xref="S4.Ex3.m1.4.4.4.6.2.3.4.cmml"><mrow id="S4.Ex3.m1.4.4.4.6.2.3.4.2" xref="S4.Ex3.m1.4.4.4.6.2.3.4.2.cmml"><mi id="S4.Ex3.m1.4.4.4.6.2.3.4.2.2" xref="S4.Ex3.m1.4.4.4.6.2.3.4.2.2.cmml">γ</mi><mo id="S4.Ex3.m1.4.4.4.6.2.3.4.2.1" xref="S4.Ex3.m1.4.4.4.6.2.3.4.2.1.cmml">+</mo><mn id="S4.Ex3.m1.4.4.4.6.2.3.4.2.3" xref="S4.Ex3.m1.4.4.4.6.2.3.4.2.3.cmml">1</mn></mrow><mi id="S4.Ex3.m1.4.4.4.6.2.3.4.3" xref="S4.Ex3.m1.4.4.4.6.2.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S4.Ex3.m1.4.4.4.5" xref="S4.Ex3.m1.4.4.4.5.cmml">⁢</mo><mrow id="S4.Ex3.m1.4.4.4.4.1" xref="S4.Ex3.m1.4.4.4.4.2.cmml"><mo id="S4.Ex3.m1.4.4.4.4.1.2" stretchy="false" xref="S4.Ex3.m1.4.4.4.4.2.cmml">(</mo><msup id="S4.Ex3.m1.4.4.4.4.1.1" xref="S4.Ex3.m1.4.4.4.4.1.1.cmml"><mi id="S4.Ex3.m1.4.4.4.4.1.1.2" xref="S4.Ex3.m1.4.4.4.4.1.1.2.cmml">ℝ</mi><mrow id="S4.Ex3.m1.4.4.4.4.1.1.3" xref="S4.Ex3.m1.4.4.4.4.1.1.3.cmml"><mi id="S4.Ex3.m1.4.4.4.4.1.1.3.2" xref="S4.Ex3.m1.4.4.4.4.1.1.3.2.cmml">d</mi><mo id="S4.Ex3.m1.4.4.4.4.1.1.3.1" xref="S4.Ex3.m1.4.4.4.4.1.1.3.1.cmml">−</mo><mn id="S4.Ex3.m1.4.4.4.4.1.1.3.3" xref="S4.Ex3.m1.4.4.4.4.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Ex3.m1.4.4.4.4.1.3" xref="S4.Ex3.m1.4.4.4.4.2.cmml">;</mo><mi id="S4.Ex3.m1.3.3.3.3" xref="S4.Ex3.m1.3.3.3.3.cmml">X</mi><mo id="S4.Ex3.m1.4.4.4.4.1.4" stretchy="false" xref="S4.Ex3.m1.4.4.4.4.2.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Ex3.m1.5b"><apply id="S4.Ex3.m1.5.5.cmml" xref="S4.Ex3.m1.5.5"><csymbol cd="ambiguous" id="S4.Ex3.m1.5.5.2.cmml" xref="S4.Ex3.m1.5.5">subscript</csymbol><apply id="S4.Ex3.m1.5.5.1.2.cmml" xref="S4.Ex3.m1.5.5.1.1"><csymbol cd="latexml" 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id="S4.Ex3.m2.5d">= ∥ roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_m - divide start_ARG italic_γ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|f\|_{B^{s_{1}}_{p,p}(\mathbb{R}^{d}_{+},w_{\gamma_{1}};X)% }\leq C\|f\|_{H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}" class="ltx_Math" display="inline" 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end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.4.p4.9">and for <math alttext="g\in B^{s-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S4.SS1.4.p4.9.m1.4"><semantics id="S4.SS1.4.p4.9.m1.4a"><mrow id="S4.SS1.4.p4.9.m1.4.4" xref="S4.SS1.4.p4.9.m1.4.4.cmml"><mi id="S4.SS1.4.p4.9.m1.4.4.3" xref="S4.SS1.4.p4.9.m1.4.4.3.cmml">g</mi><mo id="S4.SS1.4.p4.9.m1.4.4.2" xref="S4.SS1.4.p4.9.m1.4.4.2.cmml">∈</mo><mrow id="S4.SS1.4.p4.9.m1.4.4.1" xref="S4.SS1.4.p4.9.m1.4.4.1.cmml"><msubsup 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end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx15"> <tbody id="S4.Ex5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\operatorname{ext}_{m}g\|_{H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma% };X)}" class="ltx_Math" display="inline" id="S4.Ex5.m1.6"><semantics id="S4.Ex5.m1.6a"><msub id="S4.Ex5.m1.6.6" xref="S4.Ex5.m1.6.6.cmml"><mrow id="S4.Ex5.m1.6.6.1.1" xref="S4.Ex5.m1.6.6.1.2.cmml"><mo id="S4.Ex5.m1.6.6.1.1.2" stretchy="false" xref="S4.Ex5.m1.6.6.1.2.1.cmml">‖</mo><mrow id="S4.Ex5.m1.6.6.1.1.1" xref="S4.Ex5.m1.6.6.1.1.1.cmml"><msub id="S4.Ex5.m1.6.6.1.1.1.1" xref="S4.Ex5.m1.6.6.1.1.1.1.cmml"><mi id="S4.Ex5.m1.6.6.1.1.1.1.2" xref="S4.Ex5.m1.6.6.1.1.1.1.2.cmml">ext</mi><mi id="S4.Ex5.m1.6.6.1.1.1.1.3" xref="S4.Ex5.m1.6.6.1.1.1.1.3.cmml">m</mi></msub><mo id="S4.Ex5.m1.6.6.1.1.1a" lspace="0.167em" xref="S4.Ex5.m1.6.6.1.1.1.cmml">⁡</mo><mi id="S4.Ex5.m1.6.6.1.1.1.2" xref="S4.Ex5.m1.6.6.1.1.1.2.cmml">g</mi></mrow><mo id="S4.Ex5.m1.6.6.1.1.3" stretchy="false" xref="S4.Ex5.m1.6.6.1.2.1.cmml">‖</mo></mrow><mrow id="S4.Ex5.m1.5.5.5" xref="S4.Ex5.m1.5.5.5.cmml"><msup id="S4.Ex5.m1.5.5.5.7" xref="S4.Ex5.m1.5.5.5.7.cmml"><mi id="S4.Ex5.m1.5.5.5.7.2" xref="S4.Ex5.m1.5.5.5.7.2.cmml">H</mi><mrow id="S4.Ex5.m1.2.2.2.2.2.4" xref="S4.Ex5.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex5.m1.1.1.1.1.1.1" xref="S4.Ex5.m1.1.1.1.1.1.1.cmml">s</mi><mo id="S4.Ex5.m1.2.2.2.2.2.4.1" xref="S4.Ex5.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.Ex5.m1.2.2.2.2.2.2" xref="S4.Ex5.m1.2.2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex5.m1.5.5.5.6" xref="S4.Ex5.m1.5.5.5.6.cmml">⁢</mo><mrow id="S4.Ex5.m1.5.5.5.5.2" xref="S4.Ex5.m1.5.5.5.5.3.cmml"><mo id="S4.Ex5.m1.5.5.5.5.2.3" stretchy="false" xref="S4.Ex5.m1.5.5.5.5.3.cmml">(</mo><msubsup id="S4.Ex5.m1.4.4.4.4.1.1" xref="S4.Ex5.m1.4.4.4.4.1.1.cmml"><mi id="S4.Ex5.m1.4.4.4.4.1.1.2.2" xref="S4.Ex5.m1.4.4.4.4.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex5.m1.4.4.4.4.1.1.3" xref="S4.Ex5.m1.4.4.4.4.1.1.3.cmml">+</mo><mi id="S4.Ex5.m1.4.4.4.4.1.1.2.3" xref="S4.Ex5.m1.4.4.4.4.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.Ex5.m1.5.5.5.5.2.4" xref="S4.Ex5.m1.5.5.5.5.3.cmml">,</mo><msub id="S4.Ex5.m1.5.5.5.5.2.2" xref="S4.Ex5.m1.5.5.5.5.2.2.cmml"><mi id="S4.Ex5.m1.5.5.5.5.2.2.2" xref="S4.Ex5.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S4.Ex5.m1.5.5.5.5.2.2.3" xref="S4.Ex5.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex5.m1.5.5.5.5.2.5" xref="S4.Ex5.m1.5.5.5.5.3.cmml">;</mo><mi id="S4.Ex5.m1.3.3.3.3" xref="S4.Ex5.m1.3.3.3.3.cmml">X</mi><mo id="S4.Ex5.m1.5.5.5.5.2.6" stretchy="false" xref="S4.Ex5.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Ex5.m1.6b"><apply id="S4.Ex5.m1.6.6.cmml" xref="S4.Ex5.m1.6.6"><csymbol cd="ambiguous" id="S4.Ex5.m1.6.6.2.cmml" xref="S4.Ex5.m1.6.6">subscript</csymbol><apply id="S4.Ex5.m1.6.6.1.2.cmml" xref="S4.Ex5.m1.6.6.1.1"><csymbol cd="latexml" id="S4.Ex5.m1.6.6.1.2.1.cmml" xref="S4.Ex5.m1.6.6.1.1.2">norm</csymbol><apply id="S4.Ex5.m1.6.6.1.1.1.cmml" xref="S4.Ex5.m1.6.6.1.1.1"><apply id="S4.Ex5.m1.6.6.1.1.1.1.cmml" xref="S4.Ex5.m1.6.6.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex5.m1.6.6.1.1.1.1.1.cmml" xref="S4.Ex5.m1.6.6.1.1.1.1">subscript</csymbol><ci id="S4.Ex5.m1.6.6.1.1.1.1.2.cmml" xref="S4.Ex5.m1.6.6.1.1.1.1.2">ext</ci><ci id="S4.Ex5.m1.6.6.1.1.1.1.3.cmml" xref="S4.Ex5.m1.6.6.1.1.1.1.3">𝑚</ci></apply><ci id="S4.Ex5.m1.6.6.1.1.1.2.cmml" xref="S4.Ex5.m1.6.6.1.1.1.2">𝑔</ci></apply></apply><apply id="S4.Ex5.m1.5.5.5.cmml" xref="S4.Ex5.m1.5.5.5"><times id="S4.Ex5.m1.5.5.5.6.cmml" xref="S4.Ex5.m1.5.5.5.6"></times><apply id="S4.Ex5.m1.5.5.5.7.cmml" xref="S4.Ex5.m1.5.5.5.7"><csymbol cd="ambiguous" id="S4.Ex5.m1.5.5.5.7.1.cmml" xref="S4.Ex5.m1.5.5.5.7">superscript</csymbol><ci id="S4.Ex5.m1.5.5.5.7.2.cmml" xref="S4.Ex5.m1.5.5.5.7.2">𝐻</ci><list id="S4.Ex5.m1.2.2.2.2.2.3.cmml" xref="S4.Ex5.m1.2.2.2.2.2.4"><ci id="S4.Ex5.m1.1.1.1.1.1.1.cmml" xref="S4.Ex5.m1.1.1.1.1.1.1">𝑠</ci><ci id="S4.Ex5.m1.2.2.2.2.2.2.cmml" xref="S4.Ex5.m1.2.2.2.2.2.2">𝑝</ci></list></apply><vector id="S4.Ex5.m1.5.5.5.5.3.cmml" xref="S4.Ex5.m1.5.5.5.5.2"><apply id="S4.Ex5.m1.4.4.4.4.1.1.cmml" xref="S4.Ex5.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S4.Ex5.m1.4.4.4.4.1.1.1.cmml" xref="S4.Ex5.m1.4.4.4.4.1.1">subscript</csymbol><apply id="S4.Ex5.m1.4.4.4.4.1.1.2.cmml" xref="S4.Ex5.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S4.Ex5.m1.4.4.4.4.1.1.2.1.cmml" xref="S4.Ex5.m1.4.4.4.4.1.1">superscript</csymbol><ci id="S4.Ex5.m1.4.4.4.4.1.1.2.2.cmml" xref="S4.Ex5.m1.4.4.4.4.1.1.2.2">ℝ</ci><ci id="S4.Ex5.m1.4.4.4.4.1.1.2.3.cmml" xref="S4.Ex5.m1.4.4.4.4.1.1.2.3">𝑑</ci></apply><plus id="S4.Ex5.m1.4.4.4.4.1.1.3.cmml" xref="S4.Ex5.m1.4.4.4.4.1.1.3"></plus></apply><apply id="S4.Ex5.m1.5.5.5.5.2.2.cmml" xref="S4.Ex5.m1.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S4.Ex5.m1.5.5.5.5.2.2.1.cmml" xref="S4.Ex5.m1.5.5.5.5.2.2">subscript</csymbol><ci id="S4.Ex5.m1.5.5.5.5.2.2.2.cmml" xref="S4.Ex5.m1.5.5.5.5.2.2.2">𝑤</ci><ci id="S4.Ex5.m1.5.5.5.5.2.2.3.cmml" xref="S4.Ex5.m1.5.5.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex5.m1.3.3.3.3.cmml" xref="S4.Ex5.m1.3.3.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex5.m1.6c">\displaystyle\|\operatorname{ext}_{m}g\|_{H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma% };X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex5.m1.6d">∥ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|\operatorname{ext}_{m}g\|_{B^{s_{0}}_{p,p}(\mathbb{R}^{d}% _{+},w_{\gamma_{0}};X)}" class="ltx_Math" display="inline" id="S4.Ex5.m2.6"><semantics id="S4.Ex5.m2.6a"><mrow id="S4.Ex5.m2.6.6" xref="S4.Ex5.m2.6.6.cmml"><mi id="S4.Ex5.m2.6.6.3" xref="S4.Ex5.m2.6.6.3.cmml"></mi><mo id="S4.Ex5.m2.6.6.2" xref="S4.Ex5.m2.6.6.2.cmml">≤</mo><mrow id="S4.Ex5.m2.6.6.1" xref="S4.Ex5.m2.6.6.1.cmml"><mi id="S4.Ex5.m2.6.6.1.3" xref="S4.Ex5.m2.6.6.1.3.cmml">C</mi><mo id="S4.Ex5.m2.6.6.1.2" xref="S4.Ex5.m2.6.6.1.2.cmml">⁢</mo><msub id="S4.Ex5.m2.6.6.1.1" xref="S4.Ex5.m2.6.6.1.1.cmml"><mrow id="S4.Ex5.m2.6.6.1.1.1.1" xref="S4.Ex5.m2.6.6.1.1.1.2.cmml"><mo id="S4.Ex5.m2.6.6.1.1.1.1.2" stretchy="false" 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start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|g\|_{B^{s_{0}-m-\frac{\gamma_{0}+1}{p}}_{p,p}(\mathbb{R}^% {d-1};X)}=\|g\|_{B^{s-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)}." class="ltx_Math" display="inline" id="S4.Ex6.m1.11"><semantics id="S4.Ex6.m1.11a"><mrow id="S4.Ex6.m1.11.11.1" xref="S4.Ex6.m1.11.11.1.1.cmml"><mrow id="S4.Ex6.m1.11.11.1.1" xref="S4.Ex6.m1.11.11.1.1.cmml"><mi id="S4.Ex6.m1.11.11.1.1.2" xref="S4.Ex6.m1.11.11.1.1.2.cmml"></mi><mo 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xref="S4.Ex6.m1.8.8.4.4.1.1">superscript</csymbol><ci id="S4.Ex6.m1.8.8.4.4.1.1.2.cmml" xref="S4.Ex6.m1.8.8.4.4.1.1.2">ℝ</ci><apply id="S4.Ex6.m1.8.8.4.4.1.1.3.cmml" xref="S4.Ex6.m1.8.8.4.4.1.1.3"><minus id="S4.Ex6.m1.8.8.4.4.1.1.3.1.cmml" xref="S4.Ex6.m1.8.8.4.4.1.1.3.1"></minus><ci id="S4.Ex6.m1.8.8.4.4.1.1.3.2.cmml" xref="S4.Ex6.m1.8.8.4.4.1.1.3.2">𝑑</ci><cn id="S4.Ex6.m1.8.8.4.4.1.1.3.3.cmml" type="integer" xref="S4.Ex6.m1.8.8.4.4.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex6.m1.7.7.3.3.cmml" xref="S4.Ex6.m1.7.7.3.3">𝑋</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex6.m1.11c">\displaystyle\leq C\|g\|_{B^{s_{0}-m-\frac{\gamma_{0}+1}{p}}_{p,p}(\mathbb{R}^% {d-1};X)}=\|g\|_{B^{s-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex6.m1.11d">≤ italic_C ∥ italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_m - divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT = ∥ italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.4.p4.10">This finishes the proof. ∎</p> </div> </div> <div class="ltx_theorem ltx_theorem_remark" id="S4.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.1.1.1">Remark 4.3</span></span><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem3.p1"> <p class="ltx_p" id="S4.Thmtheorem3.p1.13">For <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.1.m1.1"><semantics id="S4.Thmtheorem3.p1.1.m1.1a"><mrow id="S4.Thmtheorem3.p1.1.m1.1.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.1.m1.1.1.2" xref="S4.Thmtheorem3.p1.1.m1.1.1.2.cmml">m</mi><mo id="S4.Thmtheorem3.p1.1.m1.1.1.1" xref="S4.Thmtheorem3.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem3.p1.1.m1.1.1.3" xref="S4.Thmtheorem3.p1.1.m1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.1.m1.1.1.3.2" xref="S4.Thmtheorem3.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S4.Thmtheorem3.p1.1.m1.1.1.3.3" xref="S4.Thmtheorem3.p1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.1.m1.1b"><apply id="S4.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1"><in id="S4.Thmtheorem3.p1.1.m1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.1"></in><ci id="S4.Thmtheorem3.p1.1.m1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.2">𝑚</ci><apply id="S4.Thmtheorem3.p1.1.m1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.1.m1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.1.m1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.1.m1.1.1.3.2">ℕ</ci><cn id="S4.Thmtheorem3.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.1.m1.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.1.m1.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.2.m2.2"><semantics id="S4.Thmtheorem3.p1.2.m2.2a"><mrow id="S4.Thmtheorem3.p1.2.m2.2.2" xref="S4.Thmtheorem3.p1.2.m2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.2.m2.2.2.4" xref="S4.Thmtheorem3.p1.2.m2.2.2.4.cmml">𝒪</mi><mo id="S4.Thmtheorem3.p1.2.m2.2.2.3" xref="S4.Thmtheorem3.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S4.Thmtheorem3.p1.2.m2.2.2.2.2" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.3.cmml"><mo id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.3.cmml">{</mo><msup id="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1" xref="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.2" xref="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.3" xref="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.4" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.3.cmml">,</mo><msubsup id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.2" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.3" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.3.cmml">+</mo><mi id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.3" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.2.m2.2b"><apply id="S4.Thmtheorem3.p1.2.m2.2.2.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2"><in id="S4.Thmtheorem3.p1.2.m2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.3"></in><ci id="S4.Thmtheorem3.p1.2.m2.2.2.4.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.4">𝒪</ci><set id="S4.Thmtheorem3.p1.2.m2.2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2"><apply id="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.2">ℝ</ci><ci id="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.2.m2.1.1.1.1.1.3">𝑑</ci></apply><apply id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2">subscript</csymbol><apply id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2">superscript</csymbol><ci id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.2">ℝ</ci><ci id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.2.m2.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.2.m2.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.2.m2.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math> it is clear that the trace operator <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.3.m3.1"><semantics id="S4.Thmtheorem3.p1.3.m3.1a"><msub id="S4.Thmtheorem3.p1.3.m3.1.1" xref="S4.Thmtheorem3.p1.3.m3.1.1.cmml"><mi id="S4.Thmtheorem3.p1.3.m3.1.1.2" xref="S4.Thmtheorem3.p1.3.m3.1.1.2.cmml">Tr</mi><mi id="S4.Thmtheorem3.p1.3.m3.1.1.3" xref="S4.Thmtheorem3.p1.3.m3.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.3.m3.1b"><apply id="S4.Thmtheorem3.p1.3.m3.1.1.cmml" xref="S4.Thmtheorem3.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.3.m3.1.1.1.cmml" xref="S4.Thmtheorem3.p1.3.m3.1.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.3.m3.1.1.2.cmml" xref="S4.Thmtheorem3.p1.3.m3.1.1.2">Tr</ci><ci id="S4.Thmtheorem3.p1.3.m3.1.1.3.cmml" xref="S4.Thmtheorem3.p1.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.3.m3.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.3.m3.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> maps <math alttext="C_{\mathrm{c}}^{\infty}(\overline{\mathcal{O}};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.4.m4.2"><semantics id="S4.Thmtheorem3.p1.4.m4.2a"><mrow id="S4.Thmtheorem3.p1.4.m4.2.3" xref="S4.Thmtheorem3.p1.4.m4.2.3.cmml"><msubsup id="S4.Thmtheorem3.p1.4.m4.2.3.2" xref="S4.Thmtheorem3.p1.4.m4.2.3.2.cmml"><mi id="S4.Thmtheorem3.p1.4.m4.2.3.2.2.2" xref="S4.Thmtheorem3.p1.4.m4.2.3.2.2.2.cmml">C</mi><mi id="S4.Thmtheorem3.p1.4.m4.2.3.2.2.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.4.m4.2.3.2.2.3.cmml">c</mi><mi id="S4.Thmtheorem3.p1.4.m4.2.3.2.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.4.m4.2.3.2.3.cmml">∞</mi></msubsup><mo id="S4.Thmtheorem3.p1.4.m4.2.3.1" xref="S4.Thmtheorem3.p1.4.m4.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.4.m4.2.3.3.2" xref="S4.Thmtheorem3.p1.4.m4.2.3.3.1.cmml"><mo id="S4.Thmtheorem3.p1.4.m4.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.4.m4.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem3.p1.4.m4.1.1" xref="S4.Thmtheorem3.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.4.m4.1.1.2" xref="S4.Thmtheorem3.p1.4.m4.1.1.2.cmml">𝒪</mi><mo id="S4.Thmtheorem3.p1.4.m4.1.1.1" xref="S4.Thmtheorem3.p1.4.m4.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem3.p1.4.m4.2.3.3.2.2" xref="S4.Thmtheorem3.p1.4.m4.2.3.3.1.cmml">;</mo><mi id="S4.Thmtheorem3.p1.4.m4.2.2" xref="S4.Thmtheorem3.p1.4.m4.2.2.cmml">X</mi><mo id="S4.Thmtheorem3.p1.4.m4.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.4.m4.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.4.m4.2b"><apply id="S4.Thmtheorem3.p1.4.m4.2.3.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3"><times id="S4.Thmtheorem3.p1.4.m4.2.3.1.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.1"></times><apply id="S4.Thmtheorem3.p1.4.m4.2.3.2.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.4.m4.2.3.2.1.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.2">superscript</csymbol><apply id="S4.Thmtheorem3.p1.4.m4.2.3.2.2.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.4.m4.2.3.2.2.1.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem3.p1.4.m4.2.3.2.2.2.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.2.2.2">𝐶</ci><ci id="S4.Thmtheorem3.p1.4.m4.2.3.2.2.3.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.2.2.3">c</ci></apply><infinity id="S4.Thmtheorem3.p1.4.m4.2.3.2.3.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.2.3"></infinity></apply><list id="S4.Thmtheorem3.p1.4.m4.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.3.3.2"><apply id="S4.Thmtheorem3.p1.4.m4.1.1.cmml" xref="S4.Thmtheorem3.p1.4.m4.1.1"><ci id="S4.Thmtheorem3.p1.4.m4.1.1.1.cmml" xref="S4.Thmtheorem3.p1.4.m4.1.1.1">¯</ci><ci id="S4.Thmtheorem3.p1.4.m4.1.1.2.cmml" xref="S4.Thmtheorem3.p1.4.m4.1.1.2">𝒪</ci></apply><ci id="S4.Thmtheorem3.p1.4.m4.2.2.cmml" xref="S4.Thmtheorem3.p1.4.m4.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.4.m4.2c">C_{\mathrm{c}}^{\infty}(\overline{\mathcal{O}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.4.m4.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG caligraphic_O end_ARG ; italic_X )</annotation></semantics></math> into <math alttext="C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.5.m5.2"><semantics id="S4.Thmtheorem3.p1.5.m5.2a"><mrow id="S4.Thmtheorem3.p1.5.m5.2.2" xref="S4.Thmtheorem3.p1.5.m5.2.2.cmml"><msubsup id="S4.Thmtheorem3.p1.5.m5.2.2.3" xref="S4.Thmtheorem3.p1.5.m5.2.2.3.cmml"><mi id="S4.Thmtheorem3.p1.5.m5.2.2.3.2.2" xref="S4.Thmtheorem3.p1.5.m5.2.2.3.2.2.cmml">C</mi><mi id="S4.Thmtheorem3.p1.5.m5.2.2.3.2.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.5.m5.2.2.3.2.3.cmml">c</mi><mi id="S4.Thmtheorem3.p1.5.m5.2.2.3.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.5.m5.2.2.3.3.cmml">∞</mi></msubsup><mo id="S4.Thmtheorem3.p1.5.m5.2.2.2" xref="S4.Thmtheorem3.p1.5.m5.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.5.m5.2.2.1.1" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.2.cmml"><mo id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.2.cmml">(</mo><msup id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.2" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.2.cmml">ℝ</mi><mrow id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.2" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.2.cmml">d</mi><mo id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.1" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.1.cmml">−</mo><mn id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.3" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.3" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.2.cmml">;</mo><mi id="S4.Thmtheorem3.p1.5.m5.1.1" xref="S4.Thmtheorem3.p1.5.m5.1.1.cmml">X</mi><mo id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.4" stretchy="false" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.5.m5.2b"><apply id="S4.Thmtheorem3.p1.5.m5.2.2.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2"><times id="S4.Thmtheorem3.p1.5.m5.2.2.2.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.2"></times><apply id="S4.Thmtheorem3.p1.5.m5.2.2.3.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.m5.2.2.3.1.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.3">superscript</csymbol><apply id="S4.Thmtheorem3.p1.5.m5.2.2.3.2.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.m5.2.2.3.2.1.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.5.m5.2.2.3.2.2.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.3.2.2">𝐶</ci><ci id="S4.Thmtheorem3.p1.5.m5.2.2.3.2.3.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.3.2.3">c</ci></apply><infinity id="S4.Thmtheorem3.p1.5.m5.2.2.3.3.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.3.3"></infinity></apply><list id="S4.Thmtheorem3.p1.5.m5.2.2.1.2.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1"><apply id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.2">ℝ</ci><apply id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3"><minus id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.1"></minus><ci id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.2">𝑑</ci><cn id="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.5.m5.2.2.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Thmtheorem3.p1.5.m5.1.1.cmml" xref="S4.Thmtheorem3.p1.5.m5.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.5.m5.2c">C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.5.m5.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>. Moreover, the extension operator <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.6.m6.1"><semantics id="S4.Thmtheorem3.p1.6.m6.1a"><msub id="S4.Thmtheorem3.p1.6.m6.1.1" xref="S4.Thmtheorem3.p1.6.m6.1.1.cmml"><mi id="S4.Thmtheorem3.p1.6.m6.1.1.2" xref="S4.Thmtheorem3.p1.6.m6.1.1.2.cmml">ext</mi><mi id="S4.Thmtheorem3.p1.6.m6.1.1.3" xref="S4.Thmtheorem3.p1.6.m6.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.6.m6.1b"><apply id="S4.Thmtheorem3.p1.6.m6.1.1.cmml" xref="S4.Thmtheorem3.p1.6.m6.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.6.m6.1.1.1.cmml" xref="S4.Thmtheorem3.p1.6.m6.1.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.6.m6.1.1.2.cmml" xref="S4.Thmtheorem3.p1.6.m6.1.1.2">ext</ci><ci id="S4.Thmtheorem3.p1.6.m6.1.1.3.cmml" xref="S4.Thmtheorem3.p1.6.m6.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.6.m6.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.6.m6.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> maps <math alttext="C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.7.m7.2"><semantics id="S4.Thmtheorem3.p1.7.m7.2a"><mrow id="S4.Thmtheorem3.p1.7.m7.2.2" xref="S4.Thmtheorem3.p1.7.m7.2.2.cmml"><msubsup id="S4.Thmtheorem3.p1.7.m7.2.2.3" xref="S4.Thmtheorem3.p1.7.m7.2.2.3.cmml"><mi id="S4.Thmtheorem3.p1.7.m7.2.2.3.2.2" xref="S4.Thmtheorem3.p1.7.m7.2.2.3.2.2.cmml">C</mi><mi id="S4.Thmtheorem3.p1.7.m7.2.2.3.2.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.7.m7.2.2.3.2.3.cmml">c</mi><mi id="S4.Thmtheorem3.p1.7.m7.2.2.3.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.7.m7.2.2.3.3.cmml">∞</mi></msubsup><mo id="S4.Thmtheorem3.p1.7.m7.2.2.2" xref="S4.Thmtheorem3.p1.7.m7.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.7.m7.2.2.1.1" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.2.cmml"><mo id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.2.cmml">(</mo><msup id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.2" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.2.cmml">ℝ</mi><mrow id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.2" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.2.cmml">d</mi><mo id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.1" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.1.cmml">−</mo><mn id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.3" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.3" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.2.cmml">;</mo><mi id="S4.Thmtheorem3.p1.7.m7.1.1" xref="S4.Thmtheorem3.p1.7.m7.1.1.cmml">X</mi><mo id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.4" stretchy="false" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.7.m7.2b"><apply id="S4.Thmtheorem3.p1.7.m7.2.2.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2"><times id="S4.Thmtheorem3.p1.7.m7.2.2.2.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.2"></times><apply id="S4.Thmtheorem3.p1.7.m7.2.2.3.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.7.m7.2.2.3.1.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.3">superscript</csymbol><apply id="S4.Thmtheorem3.p1.7.m7.2.2.3.2.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.7.m7.2.2.3.2.1.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.7.m7.2.2.3.2.2.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.3.2.2">𝐶</ci><ci id="S4.Thmtheorem3.p1.7.m7.2.2.3.2.3.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.3.2.3">c</ci></apply><infinity id="S4.Thmtheorem3.p1.7.m7.2.2.3.3.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.3.3"></infinity></apply><list id="S4.Thmtheorem3.p1.7.m7.2.2.1.2.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1"><apply id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.2">ℝ</ci><apply id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3"><minus id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.1"></minus><ci id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.2">𝑑</ci><cn id="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.7.m7.2.2.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Thmtheorem3.p1.7.m7.1.1.cmml" xref="S4.Thmtheorem3.p1.7.m7.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.7.m7.2c">C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.7.m7.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> into <math alttext="C^{\infty}(\overline{\mathcal{O}};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.8.m8.2"><semantics id="S4.Thmtheorem3.p1.8.m8.2a"><mrow id="S4.Thmtheorem3.p1.8.m8.2.3" xref="S4.Thmtheorem3.p1.8.m8.2.3.cmml"><msup id="S4.Thmtheorem3.p1.8.m8.2.3.2" xref="S4.Thmtheorem3.p1.8.m8.2.3.2.cmml"><mi id="S4.Thmtheorem3.p1.8.m8.2.3.2.2" xref="S4.Thmtheorem3.p1.8.m8.2.3.2.2.cmml">C</mi><mi id="S4.Thmtheorem3.p1.8.m8.2.3.2.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.8.m8.2.3.2.3.cmml">∞</mi></msup><mo id="S4.Thmtheorem3.p1.8.m8.2.3.1" xref="S4.Thmtheorem3.p1.8.m8.2.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.8.m8.2.3.3.2" xref="S4.Thmtheorem3.p1.8.m8.2.3.3.1.cmml"><mo id="S4.Thmtheorem3.p1.8.m8.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.8.m8.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem3.p1.8.m8.1.1" xref="S4.Thmtheorem3.p1.8.m8.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.8.m8.1.1.2" xref="S4.Thmtheorem3.p1.8.m8.1.1.2.cmml">𝒪</mi><mo id="S4.Thmtheorem3.p1.8.m8.1.1.1" xref="S4.Thmtheorem3.p1.8.m8.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem3.p1.8.m8.2.3.3.2.2" xref="S4.Thmtheorem3.p1.8.m8.2.3.3.1.cmml">;</mo><mi id="S4.Thmtheorem3.p1.8.m8.2.2" xref="S4.Thmtheorem3.p1.8.m8.2.2.cmml">X</mi><mo id="S4.Thmtheorem3.p1.8.m8.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.8.m8.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.8.m8.2b"><apply id="S4.Thmtheorem3.p1.8.m8.2.3.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.3"><times id="S4.Thmtheorem3.p1.8.m8.2.3.1.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.3.1"></times><apply id="S4.Thmtheorem3.p1.8.m8.2.3.2.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.8.m8.2.3.2.1.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.3.2">superscript</csymbol><ci id="S4.Thmtheorem3.p1.8.m8.2.3.2.2.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.3.2.2">𝐶</ci><infinity id="S4.Thmtheorem3.p1.8.m8.2.3.2.3.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.3.2.3"></infinity></apply><list id="S4.Thmtheorem3.p1.8.m8.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.3.3.2"><apply id="S4.Thmtheorem3.p1.8.m8.1.1.cmml" xref="S4.Thmtheorem3.p1.8.m8.1.1"><ci id="S4.Thmtheorem3.p1.8.m8.1.1.1.cmml" xref="S4.Thmtheorem3.p1.8.m8.1.1.1">¯</ci><ci id="S4.Thmtheorem3.p1.8.m8.1.1.2.cmml" xref="S4.Thmtheorem3.p1.8.m8.1.1.2">𝒪</ci></apply><ci id="S4.Thmtheorem3.p1.8.m8.2.2.cmml" xref="S4.Thmtheorem3.p1.8.m8.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.8.m8.2c">C^{\infty}(\overline{\mathcal{O}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.8.m8.2d">italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG caligraphic_O end_ARG ; italic_X )</annotation></semantics></math>. Indeed, let <math alttext="g\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.9.m9.2"><semantics id="S4.Thmtheorem3.p1.9.m9.2a"><mrow id="S4.Thmtheorem3.p1.9.m9.2.2" xref="S4.Thmtheorem3.p1.9.m9.2.2.cmml"><mi id="S4.Thmtheorem3.p1.9.m9.2.2.3" xref="S4.Thmtheorem3.p1.9.m9.2.2.3.cmml">g</mi><mo id="S4.Thmtheorem3.p1.9.m9.2.2.2" xref="S4.Thmtheorem3.p1.9.m9.2.2.2.cmml">∈</mo><mrow id="S4.Thmtheorem3.p1.9.m9.2.2.1" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.cmml"><msubsup id="S4.Thmtheorem3.p1.9.m9.2.2.1.3" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3.cmml"><mi id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.2" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.2.cmml">C</mi><mi id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.3.cmml">c</mi><mi id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3.3.cmml">∞</mi></msubsup><mo id="S4.Thmtheorem3.p1.9.m9.2.2.1.2" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.2.cmml"><mo id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.2.cmml">(</mo><msup id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.2" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.2.cmml">ℝ</mi><mrow id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.2" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.2.cmml">d</mi><mo id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.1" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.1.cmml">−</mo><mn id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.3" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.3" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.2.cmml">;</mo><mi id="S4.Thmtheorem3.p1.9.m9.1.1" xref="S4.Thmtheorem3.p1.9.m9.1.1.cmml">X</mi><mo id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.4" stretchy="false" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.9.m9.2b"><apply id="S4.Thmtheorem3.p1.9.m9.2.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2"><in id="S4.Thmtheorem3.p1.9.m9.2.2.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.2"></in><ci id="S4.Thmtheorem3.p1.9.m9.2.2.3.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.3">𝑔</ci><apply id="S4.Thmtheorem3.p1.9.m9.2.2.1.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1"><times id="S4.Thmtheorem3.p1.9.m9.2.2.1.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.2"></times><apply id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.1.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3">superscript</csymbol><apply id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.1.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.2">𝐶</ci><ci id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.3.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3.2.3">c</ci></apply><infinity id="S4.Thmtheorem3.p1.9.m9.2.2.1.3.3.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.3.3"></infinity></apply><list id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1"><apply id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1">superscript</csymbol><ci id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.2">ℝ</ci><apply id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3"><minus id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.1"></minus><ci id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.2">𝑑</ci><cn id="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.9.m9.2.2.1.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Thmtheorem3.p1.9.m9.1.1.cmml" xref="S4.Thmtheorem3.p1.9.m9.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.9.m9.2c">g\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.9.m9.2d">italic_g ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>, then <math alttext="g\in B_{p,p}^{s-m-\frac{1}{p}}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.10.m10.4"><semantics id="S4.Thmtheorem3.p1.10.m10.4a"><mrow id="S4.Thmtheorem3.p1.10.m10.4.4" xref="S4.Thmtheorem3.p1.10.m10.4.4.cmml"><mi id="S4.Thmtheorem3.p1.10.m10.4.4.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.3.cmml">g</mi><mo id="S4.Thmtheorem3.p1.10.m10.4.4.2" xref="S4.Thmtheorem3.p1.10.m10.4.4.2.cmml">∈</mo><mrow id="S4.Thmtheorem3.p1.10.m10.4.4.1" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.cmml"><msubsup id="S4.Thmtheorem3.p1.10.m10.4.4.1.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.cmml"><mi id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.2.2" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.2.2.cmml">B</mi><mrow id="S4.Thmtheorem3.p1.10.m10.2.2.2.4" xref="S4.Thmtheorem3.p1.10.m10.2.2.2.3.cmml"><mi id="S4.Thmtheorem3.p1.10.m10.1.1.1.1" xref="S4.Thmtheorem3.p1.10.m10.1.1.1.1.cmml">p</mi><mo id="S4.Thmtheorem3.p1.10.m10.2.2.2.4.1" xref="S4.Thmtheorem3.p1.10.m10.2.2.2.3.cmml">,</mo><mi id="S4.Thmtheorem3.p1.10.m10.2.2.2.2" xref="S4.Thmtheorem3.p1.10.m10.2.2.2.2.cmml">p</mi></mrow><mrow id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.cmml"><mi id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.2" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.2.cmml">s</mi><mo id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.1" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.1.cmml">−</mo><mi id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.3.cmml">m</mi><mo id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.1a" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.1.cmml">−</mo><mfrac id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4.cmml"><mn id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4.2" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4.2.cmml">1</mn><mi id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S4.Thmtheorem3.p1.10.m10.4.4.1.2" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.2.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.2.cmml"><mo id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.2" stretchy="false" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.2.cmml">(</mo><msup id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.cmml"><mi id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.2" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.2.cmml">ℝ</mi><mrow id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.2" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.2.cmml">d</mi><mo id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.1" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.1.cmml">−</mo><mn id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.3" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.2.cmml">;</mo><mi id="S4.Thmtheorem3.p1.10.m10.3.3" xref="S4.Thmtheorem3.p1.10.m10.3.3.cmml">X</mi><mo id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.4" stretchy="false" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.10.m10.4b"><apply id="S4.Thmtheorem3.p1.10.m10.4.4.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4"><in id="S4.Thmtheorem3.p1.10.m10.4.4.2.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.2"></in><ci id="S4.Thmtheorem3.p1.10.m10.4.4.3.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.3">𝑔</ci><apply id="S4.Thmtheorem3.p1.10.m10.4.4.1.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1"><times id="S4.Thmtheorem3.p1.10.m10.4.4.1.2.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.2"></times><apply id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.1.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3">superscript</csymbol><apply id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.2.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.2.1.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3">subscript</csymbol><ci id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.2.2.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.2.2">𝐵</ci><list id="S4.Thmtheorem3.p1.10.m10.2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.10.m10.2.2.2.4"><ci id="S4.Thmtheorem3.p1.10.m10.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.10.m10.1.1.1.1">𝑝</ci><ci id="S4.Thmtheorem3.p1.10.m10.2.2.2.2.cmml" xref="S4.Thmtheorem3.p1.10.m10.2.2.2.2">𝑝</ci></list></apply><apply id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3"><minus id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.1.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.1"></minus><ci id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.2.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.2">𝑠</ci><ci id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.3.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.3">𝑚</ci><apply id="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.3.3.4"><divide 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xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.1"></minus><ci id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem3.p1.10.m10.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Thmtheorem3.p1.10.m10.3.3.cmml" xref="S4.Thmtheorem3.p1.10.m10.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.10.m10.4c">g\in B_{p,p}^{s-m-\frac{1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.10.m10.4d">italic_g ∈ italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> and <math alttext="\operatorname{ext}_{m}g\in H^{s,p}(\mathcal{O};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.11.m11.4"><semantics id="S4.Thmtheorem3.p1.11.m11.4a"><mrow id="S4.Thmtheorem3.p1.11.m11.4.5" xref="S4.Thmtheorem3.p1.11.m11.4.5.cmml"><mrow id="S4.Thmtheorem3.p1.11.m11.4.5.2" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.cmml"><msub id="S4.Thmtheorem3.p1.11.m11.4.5.2.1" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.1.cmml"><mi id="S4.Thmtheorem3.p1.11.m11.4.5.2.1.2" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.1.2.cmml">ext</mi><mi id="S4.Thmtheorem3.p1.11.m11.4.5.2.1.3" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.1.3.cmml">m</mi></msub><mo id="S4.Thmtheorem3.p1.11.m11.4.5.2a" lspace="0.167em" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.cmml">⁡</mo><mi id="S4.Thmtheorem3.p1.11.m11.4.5.2.2" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.2.cmml">g</mi></mrow><mo id="S4.Thmtheorem3.p1.11.m11.4.5.1" xref="S4.Thmtheorem3.p1.11.m11.4.5.1.cmml">∈</mo><mrow id="S4.Thmtheorem3.p1.11.m11.4.5.3" 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xref="S4.Thmtheorem3.p1.11.m11.3.3.cmml">𝒪</mi><mo id="S4.Thmtheorem3.p1.11.m11.4.5.3.3.2.2" xref="S4.Thmtheorem3.p1.11.m11.4.5.3.3.1.cmml">;</mo><mi id="S4.Thmtheorem3.p1.11.m11.4.4" xref="S4.Thmtheorem3.p1.11.m11.4.4.cmml">X</mi><mo id="S4.Thmtheorem3.p1.11.m11.4.5.3.3.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.11.m11.4.5.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.11.m11.4b"><apply id="S4.Thmtheorem3.p1.11.m11.4.5.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5"><in id="S4.Thmtheorem3.p1.11.m11.4.5.1.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.1"></in><apply id="S4.Thmtheorem3.p1.11.m11.4.5.2.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.2"><apply id="S4.Thmtheorem3.p1.11.m11.4.5.2.1.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.11.m11.4.5.2.1.1.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.11.m11.4.5.2.1.2.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.1.2">ext</ci><ci id="S4.Thmtheorem3.p1.11.m11.4.5.2.1.3.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.1.3">𝑚</ci></apply><ci id="S4.Thmtheorem3.p1.11.m11.4.5.2.2.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.2.2">𝑔</ci></apply><apply id="S4.Thmtheorem3.p1.11.m11.4.5.3.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.3"><times id="S4.Thmtheorem3.p1.11.m11.4.5.3.1.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.3.1"></times><apply id="S4.Thmtheorem3.p1.11.m11.4.5.3.2.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.11.m11.4.5.3.2.1.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.3.2">superscript</csymbol><ci id="S4.Thmtheorem3.p1.11.m11.4.5.3.2.2.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.3.2.2">𝐻</ci><list id="S4.Thmtheorem3.p1.11.m11.2.2.2.3.cmml" xref="S4.Thmtheorem3.p1.11.m11.2.2.2.4"><ci id="S4.Thmtheorem3.p1.11.m11.1.1.1.1.cmml" xref="S4.Thmtheorem3.p1.11.m11.1.1.1.1">𝑠</ci><ci id="S4.Thmtheorem3.p1.11.m11.2.2.2.2.cmml" xref="S4.Thmtheorem3.p1.11.m11.2.2.2.2">𝑝</ci></list></apply><list id="S4.Thmtheorem3.p1.11.m11.4.5.3.3.1.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.5.3.3.2"><ci id="S4.Thmtheorem3.p1.11.m11.3.3.cmml" xref="S4.Thmtheorem3.p1.11.m11.3.3">𝒪</ci><ci id="S4.Thmtheorem3.p1.11.m11.4.4.cmml" xref="S4.Thmtheorem3.p1.11.m11.4.4">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.11.m11.4c">\operatorname{ext}_{m}g\in H^{s,p}(\mathcal{O};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.11.m11.4d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O ; italic_X )</annotation></semantics></math> for all <math alttext="s>m+\frac{1}{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.12.m12.1"><semantics id="S4.Thmtheorem3.p1.12.m12.1a"><mrow id="S4.Thmtheorem3.p1.12.m12.1.1" xref="S4.Thmtheorem3.p1.12.m12.1.1.cmml"><mi id="S4.Thmtheorem3.p1.12.m12.1.1.2" xref="S4.Thmtheorem3.p1.12.m12.1.1.2.cmml">s</mi><mo id="S4.Thmtheorem3.p1.12.m12.1.1.1" xref="S4.Thmtheorem3.p1.12.m12.1.1.1.cmml">></mo><mrow id="S4.Thmtheorem3.p1.12.m12.1.1.3" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.cmml"><mi id="S4.Thmtheorem3.p1.12.m12.1.1.3.2" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.2.cmml">m</mi><mo id="S4.Thmtheorem3.p1.12.m12.1.1.3.1" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.1.cmml">+</mo><mfrac id="S4.Thmtheorem3.p1.12.m12.1.1.3.3" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.3.cmml"><mn id="S4.Thmtheorem3.p1.12.m12.1.1.3.3.2" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.3.2.cmml">1</mn><mi id="S4.Thmtheorem3.p1.12.m12.1.1.3.3.3" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.12.m12.1b"><apply id="S4.Thmtheorem3.p1.12.m12.1.1.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1"><gt id="S4.Thmtheorem3.p1.12.m12.1.1.1.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.1"></gt><ci id="S4.Thmtheorem3.p1.12.m12.1.1.2.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.2">𝑠</ci><apply id="S4.Thmtheorem3.p1.12.m12.1.1.3.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.3"><plus id="S4.Thmtheorem3.p1.12.m12.1.1.3.1.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.1"></plus><ci id="S4.Thmtheorem3.p1.12.m12.1.1.3.2.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.2">𝑚</ci><apply id="S4.Thmtheorem3.p1.12.m12.1.1.3.3.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.3"><divide id="S4.Thmtheorem3.p1.12.m12.1.1.3.3.1.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.3"></divide><cn id="S4.Thmtheorem3.p1.12.m12.1.1.3.3.2.cmml" type="integer" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.3.2">1</cn><ci id="S4.Thmtheorem3.p1.12.m12.1.1.3.3.3.cmml" xref="S4.Thmtheorem3.p1.12.m12.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.12.m12.1c">s>m+\frac{1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.12.m12.1d">italic_s > italic_m + divide start_ARG 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. The Sobolev embedding implies that <math alttext="\operatorname{ext}_{m}g\in C^{\infty}(\overline{\mathcal{O}};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem3.p1.13.m13.2"><semantics id="S4.Thmtheorem3.p1.13.m13.2a"><mrow id="S4.Thmtheorem3.p1.13.m13.2.3" xref="S4.Thmtheorem3.p1.13.m13.2.3.cmml"><mrow id="S4.Thmtheorem3.p1.13.m13.2.3.2" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.cmml"><msub id="S4.Thmtheorem3.p1.13.m13.2.3.2.1" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.1.cmml"><mi id="S4.Thmtheorem3.p1.13.m13.2.3.2.1.2" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.1.2.cmml">ext</mi><mi id="S4.Thmtheorem3.p1.13.m13.2.3.2.1.3" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.1.3.cmml">m</mi></msub><mo id="S4.Thmtheorem3.p1.13.m13.2.3.2a" lspace="0.167em" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.cmml">⁡</mo><mi id="S4.Thmtheorem3.p1.13.m13.2.3.2.2" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.2.cmml">g</mi></mrow><mo id="S4.Thmtheorem3.p1.13.m13.2.3.1" xref="S4.Thmtheorem3.p1.13.m13.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem3.p1.13.m13.2.3.3" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.cmml"><msup id="S4.Thmtheorem3.p1.13.m13.2.3.3.2" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.2.cmml"><mi id="S4.Thmtheorem3.p1.13.m13.2.3.3.2.2" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.2.2.cmml">C</mi><mi id="S4.Thmtheorem3.p1.13.m13.2.3.3.2.3" mathvariant="normal" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.2.3.cmml">∞</mi></msup><mo id="S4.Thmtheorem3.p1.13.m13.2.3.3.1" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.1.cmml">⁢</mo><mrow id="S4.Thmtheorem3.p1.13.m13.2.3.3.3.2" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.3.1.cmml"><mo id="S4.Thmtheorem3.p1.13.m13.2.3.3.3.2.1" stretchy="false" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.3.1.cmml">(</mo><mover accent="true" id="S4.Thmtheorem3.p1.13.m13.1.1" xref="S4.Thmtheorem3.p1.13.m13.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Thmtheorem3.p1.13.m13.1.1.2" xref="S4.Thmtheorem3.p1.13.m13.1.1.2.cmml">𝒪</mi><mo id="S4.Thmtheorem3.p1.13.m13.1.1.1" xref="S4.Thmtheorem3.p1.13.m13.1.1.1.cmml">¯</mo></mover><mo id="S4.Thmtheorem3.p1.13.m13.2.3.3.3.2.2" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.3.1.cmml">;</mo><mi id="S4.Thmtheorem3.p1.13.m13.2.2" xref="S4.Thmtheorem3.p1.13.m13.2.2.cmml">X</mi><mo id="S4.Thmtheorem3.p1.13.m13.2.3.3.3.2.3" stretchy="false" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem3.p1.13.m13.2b"><apply id="S4.Thmtheorem3.p1.13.m13.2.3.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3"><in id="S4.Thmtheorem3.p1.13.m13.2.3.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.1"></in><apply id="S4.Thmtheorem3.p1.13.m13.2.3.2.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.2"><apply id="S4.Thmtheorem3.p1.13.m13.2.3.2.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.1"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.13.m13.2.3.2.1.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.1">subscript</csymbol><ci id="S4.Thmtheorem3.p1.13.m13.2.3.2.1.2.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.1.2">ext</ci><ci id="S4.Thmtheorem3.p1.13.m13.2.3.2.1.3.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.1.3">𝑚</ci></apply><ci id="S4.Thmtheorem3.p1.13.m13.2.3.2.2.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.2.2">𝑔</ci></apply><apply id="S4.Thmtheorem3.p1.13.m13.2.3.3.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.3"><times id="S4.Thmtheorem3.p1.13.m13.2.3.3.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.1"></times><apply id="S4.Thmtheorem3.p1.13.m13.2.3.3.2.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem3.p1.13.m13.2.3.3.2.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.2">superscript</csymbol><ci id="S4.Thmtheorem3.p1.13.m13.2.3.3.2.2.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.2.2">𝐶</ci><infinity id="S4.Thmtheorem3.p1.13.m13.2.3.3.2.3.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.2.3"></infinity></apply><list id="S4.Thmtheorem3.p1.13.m13.2.3.3.3.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.3.3.3.2"><apply id="S4.Thmtheorem3.p1.13.m13.1.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.1.1"><ci id="S4.Thmtheorem3.p1.13.m13.1.1.1.cmml" xref="S4.Thmtheorem3.p1.13.m13.1.1.1">¯</ci><ci id="S4.Thmtheorem3.p1.13.m13.1.1.2.cmml" xref="S4.Thmtheorem3.p1.13.m13.1.1.2">𝒪</ci></apply><ci id="S4.Thmtheorem3.p1.13.m13.2.2.cmml" xref="S4.Thmtheorem3.p1.13.m13.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem3.p1.13.m13.2c">\operatorname{ext}_{m}g\in C^{\infty}(\overline{\mathcal{O}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem3.p1.13.m13.2d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG caligraphic_O end_ARG ; italic_X )</annotation></semantics></math>.</p> </div> </div> <div class="ltx_para" id="S4.SS1.p2"> <p class="ltx_p" id="S4.SS1.p2.2">We extend the result for Sobolev spaces in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a> from <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.SS1.p2.1.m1.2"><semantics id="S4.SS1.p2.1.m1.2a"><mrow id="S4.SS1.p2.1.m1.2.2" xref="S4.SS1.p2.1.m1.2.2.cmml"><mi id="S4.SS1.p2.1.m1.2.2.4" xref="S4.SS1.p2.1.m1.2.2.4.cmml">γ</mi><mo id="S4.SS1.p2.1.m1.2.2.3" xref="S4.SS1.p2.1.m1.2.2.3.cmml">∈</mo><mrow id="S4.SS1.p2.1.m1.2.2.2.2" xref="S4.SS1.p2.1.m1.2.2.2.3.cmml"><mo id="S4.SS1.p2.1.m1.2.2.2.2.3" stretchy="false" xref="S4.SS1.p2.1.m1.2.2.2.3.cmml">(</mo><mrow id="S4.SS1.p2.1.m1.1.1.1.1.1" xref="S4.SS1.p2.1.m1.1.1.1.1.1.cmml"><mo id="S4.SS1.p2.1.m1.1.1.1.1.1a" xref="S4.SS1.p2.1.m1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS1.p2.1.m1.1.1.1.1.1.2" xref="S4.SS1.p2.1.m1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS1.p2.1.m1.2.2.2.2.4" xref="S4.SS1.p2.1.m1.2.2.2.3.cmml">,</mo><mrow id="S4.SS1.p2.1.m1.2.2.2.2.2" xref="S4.SS1.p2.1.m1.2.2.2.2.2.cmml"><mi id="S4.SS1.p2.1.m1.2.2.2.2.2.2" xref="S4.SS1.p2.1.m1.2.2.2.2.2.2.cmml">p</mi><mo id="S4.SS1.p2.1.m1.2.2.2.2.2.1" xref="S4.SS1.p2.1.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S4.SS1.p2.1.m1.2.2.2.2.2.3" xref="S4.SS1.p2.1.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.SS1.p2.1.m1.2.2.2.2.5" stretchy="false" xref="S4.SS1.p2.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.1.m1.2b"><apply id="S4.SS1.p2.1.m1.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2"><in id="S4.SS1.p2.1.m1.2.2.3.cmml" xref="S4.SS1.p2.1.m1.2.2.3"></in><ci id="S4.SS1.p2.1.m1.2.2.4.cmml" xref="S4.SS1.p2.1.m1.2.2.4">𝛾</ci><interval closure="open" id="S4.SS1.p2.1.m1.2.2.2.3.cmml" xref="S4.SS1.p2.1.m1.2.2.2.2"><apply id="S4.SS1.p2.1.m1.1.1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1.1.1"><minus id="S4.SS1.p2.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.p2.1.m1.1.1.1.1.1"></minus><cn id="S4.SS1.p2.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.p2.1.m1.1.1.1.1.1.2">1</cn></apply><apply id="S4.SS1.p2.1.m1.2.2.2.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2.2.2.2"><minus id="S4.SS1.p2.1.m1.2.2.2.2.2.1.cmml" xref="S4.SS1.p2.1.m1.2.2.2.2.2.1"></minus><ci id="S4.SS1.p2.1.m1.2.2.2.2.2.2.cmml" xref="S4.SS1.p2.1.m1.2.2.2.2.2.2">𝑝</ci><cn id="S4.SS1.p2.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS1.p2.1.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.1.m1.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.1.m1.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> to <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S4.SS1.p2.2.m2.4"><semantics id="S4.SS1.p2.2.m2.4a"><mrow id="S4.SS1.p2.2.m2.4.4" xref="S4.SS1.p2.2.m2.4.4.cmml"><mi id="S4.SS1.p2.2.m2.4.4.5" xref="S4.SS1.p2.2.m2.4.4.5.cmml">γ</mi><mo id="S4.SS1.p2.2.m2.4.4.4" xref="S4.SS1.p2.2.m2.4.4.4.cmml">∈</mo><mrow id="S4.SS1.p2.2.m2.4.4.3" xref="S4.SS1.p2.2.m2.4.4.3.cmml"><mrow id="S4.SS1.p2.2.m2.2.2.1.1.1" xref="S4.SS1.p2.2.m2.2.2.1.1.2.cmml"><mo id="S4.SS1.p2.2.m2.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.p2.2.m2.2.2.1.1.2.cmml">(</mo><mrow id="S4.SS1.p2.2.m2.2.2.1.1.1.1" xref="S4.SS1.p2.2.m2.2.2.1.1.1.1.cmml"><mo id="S4.SS1.p2.2.m2.2.2.1.1.1.1a" xref="S4.SS1.p2.2.m2.2.2.1.1.1.1.cmml">−</mo><mn id="S4.SS1.p2.2.m2.2.2.1.1.1.1.2" xref="S4.SS1.p2.2.m2.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS1.p2.2.m2.2.2.1.1.1.3" xref="S4.SS1.p2.2.m2.2.2.1.1.2.cmml">,</mo><mi id="S4.SS1.p2.2.m2.1.1" mathvariant="normal" xref="S4.SS1.p2.2.m2.1.1.cmml">∞</mi><mo id="S4.SS1.p2.2.m2.2.2.1.1.1.4" stretchy="false" xref="S4.SS1.p2.2.m2.2.2.1.1.2.cmml">)</mo></mrow><mo id="S4.SS1.p2.2.m2.4.4.3.4" xref="S4.SS1.p2.2.m2.4.4.3.4.cmml">∖</mo><mrow id="S4.SS1.p2.2.m2.4.4.3.3.2" xref="S4.SS1.p2.2.m2.4.4.3.3.3.cmml"><mo id="S4.SS1.p2.2.m2.4.4.3.3.2.3" stretchy="false" xref="S4.SS1.p2.2.m2.4.4.3.3.3.1.cmml">{</mo><mrow id="S4.SS1.p2.2.m2.3.3.2.2.1.1" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.cmml"><mrow id="S4.SS1.p2.2.m2.3.3.2.2.1.1.2" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.cmml"><mi id="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.2" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.1" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.3" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S4.SS1.p2.2.m2.3.3.2.2.1.1.1" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.1.cmml">−</mo><mn id="S4.SS1.p2.2.m2.3.3.2.2.1.1.3" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.p2.2.m2.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS1.p2.2.m2.4.4.3.3.3.1.cmml">:</mo><mrow id="S4.SS1.p2.2.m2.4.4.3.3.2.2" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.cmml"><mi id="S4.SS1.p2.2.m2.4.4.3.3.2.2.2" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.2.cmml">j</mi><mo id="S4.SS1.p2.2.m2.4.4.3.3.2.2.1" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S4.SS1.p2.2.m2.4.4.3.3.2.2.3" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.cmml"><mi id="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.2" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.3" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.SS1.p2.2.m2.4.4.3.3.2.5" stretchy="false" xref="S4.SS1.p2.2.m2.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p2.2.m2.4b"><apply id="S4.SS1.p2.2.m2.4.4.cmml" xref="S4.SS1.p2.2.m2.4.4"><in id="S4.SS1.p2.2.m2.4.4.4.cmml" xref="S4.SS1.p2.2.m2.4.4.4"></in><ci id="S4.SS1.p2.2.m2.4.4.5.cmml" xref="S4.SS1.p2.2.m2.4.4.5">𝛾</ci><apply id="S4.SS1.p2.2.m2.4.4.3.cmml" xref="S4.SS1.p2.2.m2.4.4.3"><setdiff id="S4.SS1.p2.2.m2.4.4.3.4.cmml" xref="S4.SS1.p2.2.m2.4.4.3.4"></setdiff><interval closure="open" id="S4.SS1.p2.2.m2.2.2.1.1.2.cmml" xref="S4.SS1.p2.2.m2.2.2.1.1.1"><apply id="S4.SS1.p2.2.m2.2.2.1.1.1.1.cmml" 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xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S4.SS1.p2.2.m2.3.3.2.2.1.1.3.cmml" type="integer" xref="S4.SS1.p2.2.m2.3.3.2.2.1.1.3">1</cn></apply><apply id="S4.SS1.p2.2.m2.4.4.3.3.2.2.cmml" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2"><in id="S4.SS1.p2.2.m2.4.4.3.3.2.2.1.cmml" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.1"></in><ci id="S4.SS1.p2.2.m2.4.4.3.3.2.2.2.cmml" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.2">𝑗</ci><apply id="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.cmml" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.1.cmml" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.3">subscript</csymbol><ci id="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.2.cmml" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S4.SS1.p2.2.m2.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p2.2.m2.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p2.2.m2.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S4.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.1.1.1">Theorem 4.4</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem4.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem4.p1"> <p class="ltx_p" id="S4.Thmtheorem4.p1.6"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.6.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.1.1.m1.2"><semantics id="S4.Thmtheorem4.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem4.p1.1.1.m1.2.3" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem4.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem4.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem4.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem4.p1.1.1.m1.1.1" xref="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem4.p1.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem4.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem4.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.1.1.m1.2b"><apply id="S4.Thmtheorem4.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.2.3"><in id="S4.Thmtheorem4.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.1"></in><ci id="S4.Thmtheorem4.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.2.3.3.2"><cn id="S4.Thmtheorem4.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem4.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem4.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem4.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.2.2.m2.1"><semantics id="S4.Thmtheorem4.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem4.p1.2.2.m2.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml">m</mi><mo id="S4.Thmtheorem4.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem4.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.2.2.m2.1b"><apply id="S4.Thmtheorem4.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1"><in id="S4.Thmtheorem4.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.2">𝑚</ci><apply id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.2">ℕ</ci><cn id="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.2.2.m2.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.2.2.m2.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="k\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.3.3.m3.1"><semantics id="S4.Thmtheorem4.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem4.p1.3.3.m3.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem4.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem4.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem4.p1.3.3.m3.1.1.1.cmml">∈</mo><msub 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xref="S4.Thmtheorem4.p1.3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.3.3.m3.1c">k\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.3.3.m3.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.4.4.m4.4"><semantics id="S4.Thmtheorem4.p1.4.4.m4.4a"><mrow id="S4.Thmtheorem4.p1.4.4.m4.4.4" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.4.4.5" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.5.cmml">γ</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.4.4.4" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.4.cmml">∈</mo><mrow id="S4.Thmtheorem4.p1.4.4.m4.4.4.3" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.cmml"><mrow id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.2.cmml"><mo id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.2.cmml">(</mo><mrow id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1a" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.2" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.3" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.2.cmml">,</mo><mi id="S4.Thmtheorem4.p1.4.4.m4.1.1" mathvariant="normal" xref="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml">∞</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.4" stretchy="false" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.2.cmml">)</mo></mrow><mo id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.4" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.4.cmml">∖</mo><mrow id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.3.cmml"><mo id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.3" stretchy="false" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.3.1.cmml">{</mo><mrow id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.cmml"><mrow id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.2" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.1" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.3" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.1" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.3" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.3.1.cmml">:</mo><mrow id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.2" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.2.cmml">j</mi><mo id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.1" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.cmml"><mi id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.2" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.3" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.5" stretchy="false" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.4.4.m4.4b"><apply id="S4.Thmtheorem4.p1.4.4.m4.4.4.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4"><in id="S4.Thmtheorem4.p1.4.4.m4.4.4.4.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.4"></in><ci id="S4.Thmtheorem4.p1.4.4.m4.4.4.5.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.5">𝛾</ci><apply id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3"><setdiff id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.4.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.4"></setdiff><interval closure="open" id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1"><apply id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1"><minus id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1"></minus><cn id="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.4.m4.2.2.1.1.1.1.2">1</cn></apply><infinity id="S4.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.1.1"></infinity></interval><apply id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2"><csymbol cd="latexml" id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.3.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.3">conditional-set</csymbol><apply id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1"><minus id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.1"></minus><apply id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2"><times id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.1"></times><ci id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.4.m4.3.3.2.2.1.1.3">1</cn></apply><apply id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2"><in id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.1"></in><ci id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.2">𝑗</ci><apply id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.1.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3">subscript</csymbol><ci id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.2.cmml" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.4.4.m4.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.4.4.m4.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.4.4.m4.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> such that <math alttext="k>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.5.5.m5.1"><semantics id="S4.Thmtheorem4.p1.5.5.m5.1a"><mrow id="S4.Thmtheorem4.p1.5.5.m5.1.1" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem4.p1.5.5.m5.1.1.2" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem4.p1.5.5.m5.1.1.1" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.1.cmml">></mo><mrow id="S4.Thmtheorem4.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.cmml"><mi id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.2" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.2.cmml">m</mi><mo id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.1" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.1.cmml">+</mo><mfrac id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.cmml"><mrow id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.cmml"><mi id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.2" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.1" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.3" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.3" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.5.5.m5.1b"><apply id="S4.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1"><gt id="S4.Thmtheorem4.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.1"></gt><ci id="S4.Thmtheorem4.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.2">𝑘</ci><apply id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3"><plus id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.1"></plus><ci id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.2">𝑚</ci><apply id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3"><divide id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3"></divide><apply id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2"><plus id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.1.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.1"></plus><ci id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.2.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.2">𝛾</ci><cn id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.2.3">1</cn></apply><ci id="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.3.cmml" xref="S4.Thmtheorem4.p1.5.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.5.5.m5.1c">k>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.5.5.m5.1d">italic_k > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.6.6.m6.1"><semantics id="S4.Thmtheorem4.p1.6.6.m6.1a"><mi id="S4.Thmtheorem4.p1.6.6.m6.1.1" xref="S4.Thmtheorem4.p1.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.6.6.m6.1b"><ci id="S4.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem4.p1.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.6.6.m6.1d">italic_X</annotation></semantics></math> be a Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{m}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B_{p,p}^{k-m% -\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S4.Ex7.m1.9"><semantics id="S4.Ex7.m1.9a"><mrow id="S4.Ex7.m1.9.9" xref="S4.Ex7.m1.9.9.cmml"><msub id="S4.Ex7.m1.9.9.5" xref="S4.Ex7.m1.9.9.5.cmml"><mi id="S4.Ex7.m1.9.9.5.2" xref="S4.Ex7.m1.9.9.5.2.cmml">Tr</mi><mi id="S4.Ex7.m1.9.9.5.3" xref="S4.Ex7.m1.9.9.5.3.cmml">m</mi></msub><mo id="S4.Ex7.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex7.m1.9.9.4.cmml">:</mo><mrow id="S4.Ex7.m1.9.9.3" xref="S4.Ex7.m1.9.9.3.cmml"><mrow id="S4.Ex7.m1.8.8.2.2" xref="S4.Ex7.m1.8.8.2.2.cmml"><msup id="S4.Ex7.m1.8.8.2.2.4" xref="S4.Ex7.m1.8.8.2.2.4.cmml"><mi id="S4.Ex7.m1.8.8.2.2.4.2" xref="S4.Ex7.m1.8.8.2.2.4.2.cmml">W</mi><mrow id="S4.Ex7.m1.2.2.2.4" xref="S4.Ex7.m1.2.2.2.3.cmml"><mi id="S4.Ex7.m1.1.1.1.1" xref="S4.Ex7.m1.1.1.1.1.cmml">k</mi><mo id="S4.Ex7.m1.2.2.2.4.1" xref="S4.Ex7.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex7.m1.2.2.2.2" xref="S4.Ex7.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex7.m1.8.8.2.2.3" xref="S4.Ex7.m1.8.8.2.2.3.cmml">⁢</mo><mrow id="S4.Ex7.m1.8.8.2.2.2.2" xref="S4.Ex7.m1.8.8.2.2.2.3.cmml"><mo id="S4.Ex7.m1.8.8.2.2.2.2.3" stretchy="false" xref="S4.Ex7.m1.8.8.2.2.2.3.cmml">(</mo><msubsup id="S4.Ex7.m1.7.7.1.1.1.1.1" xref="S4.Ex7.m1.7.7.1.1.1.1.1.cmml"><mi id="S4.Ex7.m1.7.7.1.1.1.1.1.2.2" xref="S4.Ex7.m1.7.7.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex7.m1.7.7.1.1.1.1.1.3" xref="S4.Ex7.m1.7.7.1.1.1.1.1.3.cmml">+</mo><mi id="S4.Ex7.m1.7.7.1.1.1.1.1.2.3" xref="S4.Ex7.m1.7.7.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.Ex7.m1.8.8.2.2.2.2.4" xref="S4.Ex7.m1.8.8.2.2.2.3.cmml">,</mo><msub id="S4.Ex7.m1.8.8.2.2.2.2.2" 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xref="S4.Ex7.m1.9.9.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.Ex7.m1.9.9.3.3.1.1.1.1.cmml" xref="S4.Ex7.m1.9.9.3.3.1.1.1">superscript</csymbol><ci id="S4.Ex7.m1.9.9.3.3.1.1.1.2.cmml" xref="S4.Ex7.m1.9.9.3.3.1.1.1.2">ℝ</ci><apply id="S4.Ex7.m1.9.9.3.3.1.1.1.3.cmml" xref="S4.Ex7.m1.9.9.3.3.1.1.1.3"><minus id="S4.Ex7.m1.9.9.3.3.1.1.1.3.1.cmml" xref="S4.Ex7.m1.9.9.3.3.1.1.1.3.1"></minus><ci id="S4.Ex7.m1.9.9.3.3.1.1.1.3.2.cmml" xref="S4.Ex7.m1.9.9.3.3.1.1.1.3.2">𝑑</ci><cn id="S4.Ex7.m1.9.9.3.3.1.1.1.3.3.cmml" type="integer" xref="S4.Ex7.m1.9.9.3.3.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex7.m1.6.6.cmml" xref="S4.Ex7.m1.6.6">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex7.m1.9c">\operatorname{Tr}_{m}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B_{p,p}^{k-m% -\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex7.m1.9d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.Thmtheorem4.p1.12"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem4.p1.12.6">is a continuous and surjective operator. Moreover, there exists a continuous right inverse <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.7.1.m1.1"><semantics id="S4.Thmtheorem4.p1.7.1.m1.1a"><msub id="S4.Thmtheorem4.p1.7.1.m1.1.1" xref="S4.Thmtheorem4.p1.7.1.m1.1.1.cmml"><mi id="S4.Thmtheorem4.p1.7.1.m1.1.1.2" xref="S4.Thmtheorem4.p1.7.1.m1.1.1.2.cmml">ext</mi><mi id="S4.Thmtheorem4.p1.7.1.m1.1.1.3" xref="S4.Thmtheorem4.p1.7.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.7.1.m1.1b"><apply id="S4.Thmtheorem4.p1.7.1.m1.1.1.cmml" xref="S4.Thmtheorem4.p1.7.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.7.1.m1.1.1.1.cmml" xref="S4.Thmtheorem4.p1.7.1.m1.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.7.1.m1.1.1.2.cmml" xref="S4.Thmtheorem4.p1.7.1.m1.1.1.2">ext</ci><ci id="S4.Thmtheorem4.p1.7.1.m1.1.1.3.cmml" xref="S4.Thmtheorem4.p1.7.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.7.1.m1.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.7.1.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.8.2.m2.1"><semantics id="S4.Thmtheorem4.p1.8.2.m2.1a"><msub id="S4.Thmtheorem4.p1.8.2.m2.1.1" xref="S4.Thmtheorem4.p1.8.2.m2.1.1.cmml"><mi id="S4.Thmtheorem4.p1.8.2.m2.1.1.2" xref="S4.Thmtheorem4.p1.8.2.m2.1.1.2.cmml">Tr</mi><mi id="S4.Thmtheorem4.p1.8.2.m2.1.1.3" xref="S4.Thmtheorem4.p1.8.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.8.2.m2.1b"><apply id="S4.Thmtheorem4.p1.8.2.m2.1.1.cmml" xref="S4.Thmtheorem4.p1.8.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.8.2.m2.1.1.1.cmml" xref="S4.Thmtheorem4.p1.8.2.m2.1.1">subscript</csymbol><ci id="S4.Thmtheorem4.p1.8.2.m2.1.1.2.cmml" xref="S4.Thmtheorem4.p1.8.2.m2.1.1.2">Tr</ci><ci id="S4.Thmtheorem4.p1.8.2.m2.1.1.3.cmml" xref="S4.Thmtheorem4.p1.8.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.8.2.m2.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.8.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> which is independent of <math alttext="k,p,\gamma" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.9.3.m3.3"><semantics id="S4.Thmtheorem4.p1.9.3.m3.3a"><mrow id="S4.Thmtheorem4.p1.9.3.m3.3.4.2" xref="S4.Thmtheorem4.p1.9.3.m3.3.4.1.cmml"><mi id="S4.Thmtheorem4.p1.9.3.m3.1.1" xref="S4.Thmtheorem4.p1.9.3.m3.1.1.cmml">k</mi><mo id="S4.Thmtheorem4.p1.9.3.m3.3.4.2.1" xref="S4.Thmtheorem4.p1.9.3.m3.3.4.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.9.3.m3.2.2" xref="S4.Thmtheorem4.p1.9.3.m3.2.2.cmml">p</mi><mo id="S4.Thmtheorem4.p1.9.3.m3.3.4.2.2" xref="S4.Thmtheorem4.p1.9.3.m3.3.4.1.cmml">,</mo><mi id="S4.Thmtheorem4.p1.9.3.m3.3.3" xref="S4.Thmtheorem4.p1.9.3.m3.3.3.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.9.3.m3.3b"><list id="S4.Thmtheorem4.p1.9.3.m3.3.4.1.cmml" xref="S4.Thmtheorem4.p1.9.3.m3.3.4.2"><ci id="S4.Thmtheorem4.p1.9.3.m3.1.1.cmml" xref="S4.Thmtheorem4.p1.9.3.m3.1.1">𝑘</ci><ci id="S4.Thmtheorem4.p1.9.3.m3.2.2.cmml" xref="S4.Thmtheorem4.p1.9.3.m3.2.2">𝑝</ci><ci id="S4.Thmtheorem4.p1.9.3.m3.3.3.cmml" xref="S4.Thmtheorem4.p1.9.3.m3.3.3">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.9.3.m3.3c">k,p,\gamma</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.9.3.m3.3d">italic_k , italic_p , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.10.4.m4.1"><semantics id="S4.Thmtheorem4.p1.10.4.m4.1a"><mi id="S4.Thmtheorem4.p1.10.4.m4.1.1" xref="S4.Thmtheorem4.p1.10.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.10.4.m4.1b"><ci id="S4.Thmtheorem4.p1.10.4.m4.1.1.cmml" xref="S4.Thmtheorem4.p1.10.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.10.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.10.4.m4.1d">italic_X</annotation></semantics></math>. For any <math alttext="0\leq j<m" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.11.5.m5.1"><semantics id="S4.Thmtheorem4.p1.11.5.m5.1a"><mrow id="S4.Thmtheorem4.p1.11.5.m5.1.1" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.cmml"><mn id="S4.Thmtheorem4.p1.11.5.m5.1.1.2" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.2.cmml">0</mn><mo id="S4.Thmtheorem4.p1.11.5.m5.1.1.3" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.3.cmml">≤</mo><mi id="S4.Thmtheorem4.p1.11.5.m5.1.1.4" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.4.cmml">j</mi><mo id="S4.Thmtheorem4.p1.11.5.m5.1.1.5" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.5.cmml"><</mo><mi id="S4.Thmtheorem4.p1.11.5.m5.1.1.6" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.6.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.11.5.m5.1b"><apply id="S4.Thmtheorem4.p1.11.5.m5.1.1.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1"><and id="S4.Thmtheorem4.p1.11.5.m5.1.1a.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1"></and><apply id="S4.Thmtheorem4.p1.11.5.m5.1.1b.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1"><leq id="S4.Thmtheorem4.p1.11.5.m5.1.1.3.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.3"></leq><cn id="S4.Thmtheorem4.p1.11.5.m5.1.1.2.cmml" type="integer" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.2">0</cn><ci id="S4.Thmtheorem4.p1.11.5.m5.1.1.4.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.4">𝑗</ci></apply><apply id="S4.Thmtheorem4.p1.11.5.m5.1.1c.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1"><lt id="S4.Thmtheorem4.p1.11.5.m5.1.1.5.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4.p1.11.5.m5.1.1.4.cmml" id="S4.Thmtheorem4.p1.11.5.m5.1.1d.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1"></share><ci id="S4.Thmtheorem4.p1.11.5.m5.1.1.6.cmml" xref="S4.Thmtheorem4.p1.11.5.m5.1.1.6">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.11.5.m5.1c">0\leq j<m</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.11.5.m5.1d">0 ≤ italic_j < italic_m</annotation></semantics></math> we have <math alttext="\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0" class="ltx_Math" display="inline" id="S4.Thmtheorem4.p1.12.6.m6.1"><semantics id="S4.Thmtheorem4.p1.12.6.m6.1a"><mrow id="S4.Thmtheorem4.p1.12.6.m6.1.1" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.cmml"><mrow id="S4.Thmtheorem4.p1.12.6.m6.1.1.2" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.cmml"><msub id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.cmml"><mi id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.2" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.2.cmml">Tr</mi><mi id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.3" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.3.cmml">j</mi></msub><mo id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.1" lspace="0.167em" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.1.cmml">⁢</mo><mrow id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.cmml"><mo id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.1" rspace="0.167em" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.1.cmml">∘</mo><msub id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.cmml"><mi id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.2" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.2.cmml">ext</mi><mi id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.3" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.3.cmml">m</mi></msub></mrow></mrow><mo id="S4.Thmtheorem4.p1.12.6.m6.1.1.1" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.1.cmml">=</mo><mn id="S4.Thmtheorem4.p1.12.6.m6.1.1.3" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem4.p1.12.6.m6.1b"><apply id="S4.Thmtheorem4.p1.12.6.m6.1.1.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1"><eq id="S4.Thmtheorem4.p1.12.6.m6.1.1.1.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.1"></eq><apply id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2"><times id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.1.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.1"></times><apply id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.1.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.2.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.2">Tr</ci><ci id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.3.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.2.3">𝑗</ci></apply><apply id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3"><compose id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.1.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.1"></compose><apply id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.1.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2">subscript</csymbol><ci id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.2.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.2">ext</ci><ci id="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.3.cmml" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.2.3.2.3">𝑚</ci></apply></apply></apply><cn id="S4.Thmtheorem4.p1.12.6.m6.1.1.3.cmml" type="integer" xref="S4.Thmtheorem4.p1.12.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem4.p1.12.6.m6.1c">\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{m}=0</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem4.p1.12.6.m6.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS1.7"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.5.p1"> <p class="ltx_p" id="S4.SS1.5.p1.4">Let <math alttext="j\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.SS1.5.p1.1.m1.1"><semantics id="S4.SS1.5.p1.1.m1.1a"><mrow id="S4.SS1.5.p1.1.m1.1.1" xref="S4.SS1.5.p1.1.m1.1.1.cmml"><mi id="S4.SS1.5.p1.1.m1.1.1.2" xref="S4.SS1.5.p1.1.m1.1.1.2.cmml">j</mi><mo id="S4.SS1.5.p1.1.m1.1.1.1" xref="S4.SS1.5.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S4.SS1.5.p1.1.m1.1.1.3" xref="S4.SS1.5.p1.1.m1.1.1.3.cmml"><mi id="S4.SS1.5.p1.1.m1.1.1.3.2" xref="S4.SS1.5.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS1.5.p1.1.m1.1.1.3.3" xref="S4.SS1.5.p1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p1.1.m1.1b"><apply id="S4.SS1.5.p1.1.m1.1.1.cmml" xref="S4.SS1.5.p1.1.m1.1.1"><in id="S4.SS1.5.p1.1.m1.1.1.1.cmml" xref="S4.SS1.5.p1.1.m1.1.1.1"></in><ci id="S4.SS1.5.p1.1.m1.1.1.2.cmml" xref="S4.SS1.5.p1.1.m1.1.1.2">𝑗</ci><apply id="S4.SS1.5.p1.1.m1.1.1.3.cmml" xref="S4.SS1.5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.5.p1.1.m1.1.1.3.1.cmml" xref="S4.SS1.5.p1.1.m1.1.1.3">subscript</csymbol><ci id="S4.SS1.5.p1.1.m1.1.1.3.2.cmml" xref="S4.SS1.5.p1.1.m1.1.1.3.2">ℕ</ci><cn id="S4.SS1.5.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S4.SS1.5.p1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p1.1.m1.1c">j\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p1.1.m1.1d">italic_j ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be such that <math alttext="\gamma\in(jp-1,(j+1)p-1)" class="ltx_Math" display="inline" id="S4.SS1.5.p1.2.m2.2"><semantics id="S4.SS1.5.p1.2.m2.2a"><mrow id="S4.SS1.5.p1.2.m2.2.2" xref="S4.SS1.5.p1.2.m2.2.2.cmml"><mi id="S4.SS1.5.p1.2.m2.2.2.4" xref="S4.SS1.5.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S4.SS1.5.p1.2.m2.2.2.3" xref="S4.SS1.5.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S4.SS1.5.p1.2.m2.2.2.2.2" xref="S4.SS1.5.p1.2.m2.2.2.2.3.cmml"><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S4.SS1.5.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S4.SS1.5.p1.2.m2.1.1.1.1.1" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.cmml"><mrow id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.cmml"><mi id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.2" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.2.cmml">j</mi><mo id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.1" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.3" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.3.cmml">p</mi></mrow><mo id="S4.SS1.5.p1.2.m2.1.1.1.1.1.1" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS1.5.p1.2.m2.1.1.1.1.1.3" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.4" xref="S4.SS1.5.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="S4.SS1.5.p1.2.m2.2.2.2.2.2" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.cmml"><mrow id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.cmml"><mrow id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.cmml"><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.2" stretchy="false" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.cmml"><mi id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.2" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.2.cmml">j</mi><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.1" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.1.cmml">+</mo><mn id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.3" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.3" stretchy="false" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.2" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.2.cmml">⁢</mo><mi id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.3" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.3.cmml">p</mi></mrow><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.2.2" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.2.cmml">−</mo><mn id="S4.SS1.5.p1.2.m2.2.2.2.2.2.3" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.SS1.5.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S4.SS1.5.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p1.2.m2.2b"><apply id="S4.SS1.5.p1.2.m2.2.2.cmml" xref="S4.SS1.5.p1.2.m2.2.2"><in id="S4.SS1.5.p1.2.m2.2.2.3.cmml" xref="S4.SS1.5.p1.2.m2.2.2.3"></in><ci id="S4.SS1.5.p1.2.m2.2.2.4.cmml" xref="S4.SS1.5.p1.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S4.SS1.5.p1.2.m2.2.2.2.3.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2"><apply id="S4.SS1.5.p1.2.m2.1.1.1.1.1.cmml" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1"><minus id="S4.SS1.5.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.1"></minus><apply id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.cmml" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2"><times id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.1"></times><ci id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.2.cmml" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.2">𝑗</ci><ci id="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.3.cmml" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.2.3">𝑝</ci></apply><cn id="S4.SS1.5.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.5.p1.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="S4.SS1.5.p1.2.m2.2.2.2.2.2.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2"><minus id="S4.SS1.5.p1.2.m2.2.2.2.2.2.2.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.2"></minus><apply id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1"><times id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.2.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.2"></times><apply id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1"><plus id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.1.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.1"></plus><ci id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.2.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.2">𝑗</ci><cn id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.1.1.1.3">1</cn></apply><ci id="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.3.cmml" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.1.3">𝑝</ci></apply><cn id="S4.SS1.5.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS1.5.p1.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p1.2.m2.2c">\gamma\in(jp-1,(j+1)p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p1.2.m2.2d">italic_γ ∈ ( italic_j italic_p - 1 , ( italic_j + 1 ) italic_p - 1 )</annotation></semantics></math>. The case <math alttext="j=0" class="ltx_Math" display="inline" id="S4.SS1.5.p1.3.m3.1"><semantics id="S4.SS1.5.p1.3.m3.1a"><mrow id="S4.SS1.5.p1.3.m3.1.1" xref="S4.SS1.5.p1.3.m3.1.1.cmml"><mi id="S4.SS1.5.p1.3.m3.1.1.2" xref="S4.SS1.5.p1.3.m3.1.1.2.cmml">j</mi><mo id="S4.SS1.5.p1.3.m3.1.1.1" xref="S4.SS1.5.p1.3.m3.1.1.1.cmml">=</mo><mn id="S4.SS1.5.p1.3.m3.1.1.3" xref="S4.SS1.5.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p1.3.m3.1b"><apply id="S4.SS1.5.p1.3.m3.1.1.cmml" xref="S4.SS1.5.p1.3.m3.1.1"><eq id="S4.SS1.5.p1.3.m3.1.1.1.cmml" xref="S4.SS1.5.p1.3.m3.1.1.1"></eq><ci id="S4.SS1.5.p1.3.m3.1.1.2.cmml" xref="S4.SS1.5.p1.3.m3.1.1.2">𝑗</ci><cn id="S4.SS1.5.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.SS1.5.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p1.3.m3.1c">j=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p1.3.m3.1d">italic_j = 0</annotation></semantics></math> follows from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i2" title="item ii ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>, so we can assume <math alttext="j\geq 1" class="ltx_Math" display="inline" id="S4.SS1.5.p1.4.m4.1"><semantics id="S4.SS1.5.p1.4.m4.1a"><mrow id="S4.SS1.5.p1.4.m4.1.1" xref="S4.SS1.5.p1.4.m4.1.1.cmml"><mi id="S4.SS1.5.p1.4.m4.1.1.2" xref="S4.SS1.5.p1.4.m4.1.1.2.cmml">j</mi><mo id="S4.SS1.5.p1.4.m4.1.1.1" xref="S4.SS1.5.p1.4.m4.1.1.1.cmml">≥</mo><mn id="S4.SS1.5.p1.4.m4.1.1.3" xref="S4.SS1.5.p1.4.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.5.p1.4.m4.1b"><apply id="S4.SS1.5.p1.4.m4.1.1.cmml" xref="S4.SS1.5.p1.4.m4.1.1"><geq id="S4.SS1.5.p1.4.m4.1.1.1.cmml" xref="S4.SS1.5.p1.4.m4.1.1.1"></geq><ci id="S4.SS1.5.p1.4.m4.1.1.2.cmml" xref="S4.SS1.5.p1.4.m4.1.1.2">𝑗</ci><cn id="S4.SS1.5.p1.4.m4.1.1.3.cmml" type="integer" xref="S4.SS1.5.p1.4.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.5.p1.4.m4.1c">j\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.5.p1.4.m4.1d">italic_j ≥ 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS1.6.p2"> <p class="ltx_p" id="S4.SS1.6.p2.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.6.p2.1.1">Step 1: trace operator.</span> Let <math alttext="f\in W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS1.6.p2.1.m1.5"><semantics id="S4.SS1.6.p2.1.m1.5a"><mrow id="S4.SS1.6.p2.1.m1.5.5" xref="S4.SS1.6.p2.1.m1.5.5.cmml"><mi id="S4.SS1.6.p2.1.m1.5.5.4" xref="S4.SS1.6.p2.1.m1.5.5.4.cmml">f</mi><mo id="S4.SS1.6.p2.1.m1.5.5.3" xref="S4.SS1.6.p2.1.m1.5.5.3.cmml">∈</mo><mrow id="S4.SS1.6.p2.1.m1.5.5.2" xref="S4.SS1.6.p2.1.m1.5.5.2.cmml"><msup id="S4.SS1.6.p2.1.m1.5.5.2.4" xref="S4.SS1.6.p2.1.m1.5.5.2.4.cmml"><mi id="S4.SS1.6.p2.1.m1.5.5.2.4.2" 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xref="S4.Ex8.m1.4.4.4.4.1.1.3.2">𝑑</ci><cn id="S4.Ex8.m1.4.4.4.4.1.1.3.3.cmml" type="integer" xref="S4.Ex8.m1.4.4.4.4.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex8.m1.3.3.3.3.cmml" xref="S4.Ex8.m1.3.3.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m1.5c">\displaystyle\|\operatorname{Tr}_{m}f\|_{B^{k-m-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m1.5d">∥ roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="S4.Ex8.m2.4.4.4.4.1.1.3.1"></minus><ci id="S4.Ex8.m2.4.4.4.4.1.1.3.2.cmml" xref="S4.Ex8.m2.4.4.4.4.1.1.3.2">𝑑</ci><cn id="S4.Ex8.m2.4.4.4.4.1.1.3.3.cmml" type="integer" xref="S4.Ex8.m2.4.4.4.4.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex8.m2.3.3.3.3.cmml" xref="S4.Ex8.m2.3.3.3.3">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex8.m2.5c">\displaystyle=\|\operatorname{Tr}_{m}f\|_{B^{k-j-m-\frac{\gamma-jp+1}{p}}_{p,p% }(\mathbb{R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex8.m2.5d">= ∥ roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k - italic_j - italic_m - divide start_ARG italic_γ - italic_j italic_p + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|f\|_{W^{k-j,p}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X)}\leq C% \|f\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}," class="ltx_Math" display="inline" id="S4.Ex9.m1.13"><semantics id="S4.Ex9.m1.13a"><mrow id="S4.Ex9.m1.13.13.1" xref="S4.Ex9.m1.13.13.1.1.cmml"><mrow id="S4.Ex9.m1.13.13.1.1" xref="S4.Ex9.m1.13.13.1.1.cmml"><mi id="S4.Ex9.m1.13.13.1.1.2" xref="S4.Ex9.m1.13.13.1.1.2.cmml"></mi><mo id="S4.Ex9.m1.13.13.1.1.3" xref="S4.Ex9.m1.13.13.1.1.3.cmml">≤</mo><mrow id="S4.Ex9.m1.13.13.1.1.4" xref="S4.Ex9.m1.13.13.1.1.4.cmml"><mi id="S4.Ex9.m1.13.13.1.1.4.2" xref="S4.Ex9.m1.13.13.1.1.4.2.cmml">C</mi><mo 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xref="S4.Ex9.m1.9.9.4.4.1.1.2.3">𝑑</ci></apply><plus id="S4.Ex9.m1.9.9.4.4.1.1.3.cmml" xref="S4.Ex9.m1.9.9.4.4.1.1.3"></plus></apply><apply id="S4.Ex9.m1.10.10.5.5.2.2.cmml" xref="S4.Ex9.m1.10.10.5.5.2.2"><csymbol cd="ambiguous" id="S4.Ex9.m1.10.10.5.5.2.2.1.cmml" xref="S4.Ex9.m1.10.10.5.5.2.2">subscript</csymbol><ci id="S4.Ex9.m1.10.10.5.5.2.2.2.cmml" xref="S4.Ex9.m1.10.10.5.5.2.2.2">𝑤</ci><ci id="S4.Ex9.m1.10.10.5.5.2.2.3.cmml" xref="S4.Ex9.m1.10.10.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex9.m1.8.8.3.3.cmml" xref="S4.Ex9.m1.8.8.3.3">𝑋</ci></vector></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex9.m1.13c">\displaystyle\leq C\|f\|_{W^{k-j,p}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X)}\leq C% \|f\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex9.m1.13d">≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k - italic_j , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_f ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.6.p2.3">since <math alttext="\gamma-jp\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.SS1.6.p2.2.m1.2"><semantics id="S4.SS1.6.p2.2.m1.2a"><mrow id="S4.SS1.6.p2.2.m1.2.2" xref="S4.SS1.6.p2.2.m1.2.2.cmml"><mrow id="S4.SS1.6.p2.2.m1.2.2.4" xref="S4.SS1.6.p2.2.m1.2.2.4.cmml"><mi id="S4.SS1.6.p2.2.m1.2.2.4.2" xref="S4.SS1.6.p2.2.m1.2.2.4.2.cmml">γ</mi><mo id="S4.SS1.6.p2.2.m1.2.2.4.1" xref="S4.SS1.6.p2.2.m1.2.2.4.1.cmml">−</mo><mrow id="S4.SS1.6.p2.2.m1.2.2.4.3" xref="S4.SS1.6.p2.2.m1.2.2.4.3.cmml"><mi id="S4.SS1.6.p2.2.m1.2.2.4.3.2" xref="S4.SS1.6.p2.2.m1.2.2.4.3.2.cmml">j</mi><mo id="S4.SS1.6.p2.2.m1.2.2.4.3.1" xref="S4.SS1.6.p2.2.m1.2.2.4.3.1.cmml">⁢</mo><mi id="S4.SS1.6.p2.2.m1.2.2.4.3.3" xref="S4.SS1.6.p2.2.m1.2.2.4.3.3.cmml">p</mi></mrow></mrow><mo id="S4.SS1.6.p2.2.m1.2.2.3" xref="S4.SS1.6.p2.2.m1.2.2.3.cmml">∈</mo><mrow id="S4.SS1.6.p2.2.m1.2.2.2.2" xref="S4.SS1.6.p2.2.m1.2.2.2.3.cmml"><mo id="S4.SS1.6.p2.2.m1.2.2.2.2.3" stretchy="false" xref="S4.SS1.6.p2.2.m1.2.2.2.3.cmml">(</mo><mrow id="S4.SS1.6.p2.2.m1.1.1.1.1.1" xref="S4.SS1.6.p2.2.m1.1.1.1.1.1.cmml"><mo id="S4.SS1.6.p2.2.m1.1.1.1.1.1a" xref="S4.SS1.6.p2.2.m1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS1.6.p2.2.m1.1.1.1.1.1.2" xref="S4.SS1.6.p2.2.m1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS1.6.p2.2.m1.2.2.2.2.4" xref="S4.SS1.6.p2.2.m1.2.2.2.3.cmml">,</mo><mrow id="S4.SS1.6.p2.2.m1.2.2.2.2.2" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2.cmml"><mi id="S4.SS1.6.p2.2.m1.2.2.2.2.2.2" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2.2.cmml">p</mi><mo id="S4.SS1.6.p2.2.m1.2.2.2.2.2.1" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S4.SS1.6.p2.2.m1.2.2.2.2.2.3" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.SS1.6.p2.2.m1.2.2.2.2.5" stretchy="false" xref="S4.SS1.6.p2.2.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p2.2.m1.2b"><apply id="S4.SS1.6.p2.2.m1.2.2.cmml" xref="S4.SS1.6.p2.2.m1.2.2"><in id="S4.SS1.6.p2.2.m1.2.2.3.cmml" xref="S4.SS1.6.p2.2.m1.2.2.3"></in><apply id="S4.SS1.6.p2.2.m1.2.2.4.cmml" xref="S4.SS1.6.p2.2.m1.2.2.4"><minus id="S4.SS1.6.p2.2.m1.2.2.4.1.cmml" xref="S4.SS1.6.p2.2.m1.2.2.4.1"></minus><ci id="S4.SS1.6.p2.2.m1.2.2.4.2.cmml" xref="S4.SS1.6.p2.2.m1.2.2.4.2">𝛾</ci><apply id="S4.SS1.6.p2.2.m1.2.2.4.3.cmml" xref="S4.SS1.6.p2.2.m1.2.2.4.3"><times id="S4.SS1.6.p2.2.m1.2.2.4.3.1.cmml" xref="S4.SS1.6.p2.2.m1.2.2.4.3.1"></times><ci id="S4.SS1.6.p2.2.m1.2.2.4.3.2.cmml" xref="S4.SS1.6.p2.2.m1.2.2.4.3.2">𝑗</ci><ci id="S4.SS1.6.p2.2.m1.2.2.4.3.3.cmml" xref="S4.SS1.6.p2.2.m1.2.2.4.3.3">𝑝</ci></apply></apply><interval closure="open" id="S4.SS1.6.p2.2.m1.2.2.2.3.cmml" xref="S4.SS1.6.p2.2.m1.2.2.2.2"><apply id="S4.SS1.6.p2.2.m1.1.1.1.1.1.cmml" xref="S4.SS1.6.p2.2.m1.1.1.1.1.1"><minus id="S4.SS1.6.p2.2.m1.1.1.1.1.1.1.cmml" xref="S4.SS1.6.p2.2.m1.1.1.1.1.1"></minus><cn id="S4.SS1.6.p2.2.m1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS1.6.p2.2.m1.1.1.1.1.1.2">1</cn></apply><apply id="S4.SS1.6.p2.2.m1.2.2.2.2.2.cmml" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2"><minus id="S4.SS1.6.p2.2.m1.2.2.2.2.2.1.cmml" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2.1"></minus><ci id="S4.SS1.6.p2.2.m1.2.2.2.2.2.2.cmml" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2.2">𝑝</ci><cn id="S4.SS1.6.p2.2.m1.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS1.6.p2.2.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p2.2.m1.2c">\gamma-jp\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p2.2.m1.2d">italic_γ - italic_j italic_p ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> and <math alttext="k-j>m\geq 0" class="ltx_Math" display="inline" id="S4.SS1.6.p2.3.m2.1"><semantics id="S4.SS1.6.p2.3.m2.1a"><mrow id="S4.SS1.6.p2.3.m2.1.1" xref="S4.SS1.6.p2.3.m2.1.1.cmml"><mrow id="S4.SS1.6.p2.3.m2.1.1.2" xref="S4.SS1.6.p2.3.m2.1.1.2.cmml"><mi id="S4.SS1.6.p2.3.m2.1.1.2.2" xref="S4.SS1.6.p2.3.m2.1.1.2.2.cmml">k</mi><mo id="S4.SS1.6.p2.3.m2.1.1.2.1" xref="S4.SS1.6.p2.3.m2.1.1.2.1.cmml">−</mo><mi id="S4.SS1.6.p2.3.m2.1.1.2.3" xref="S4.SS1.6.p2.3.m2.1.1.2.3.cmml">j</mi></mrow><mo id="S4.SS1.6.p2.3.m2.1.1.3" xref="S4.SS1.6.p2.3.m2.1.1.3.cmml">></mo><mi id="S4.SS1.6.p2.3.m2.1.1.4" xref="S4.SS1.6.p2.3.m2.1.1.4.cmml">m</mi><mo id="S4.SS1.6.p2.3.m2.1.1.5" xref="S4.SS1.6.p2.3.m2.1.1.5.cmml">≥</mo><mn id="S4.SS1.6.p2.3.m2.1.1.6" xref="S4.SS1.6.p2.3.m2.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.6.p2.3.m2.1b"><apply id="S4.SS1.6.p2.3.m2.1.1.cmml" xref="S4.SS1.6.p2.3.m2.1.1"><and id="S4.SS1.6.p2.3.m2.1.1a.cmml" xref="S4.SS1.6.p2.3.m2.1.1"></and><apply id="S4.SS1.6.p2.3.m2.1.1b.cmml" xref="S4.SS1.6.p2.3.m2.1.1"><gt id="S4.SS1.6.p2.3.m2.1.1.3.cmml" xref="S4.SS1.6.p2.3.m2.1.1.3"></gt><apply id="S4.SS1.6.p2.3.m2.1.1.2.cmml" xref="S4.SS1.6.p2.3.m2.1.1.2"><minus id="S4.SS1.6.p2.3.m2.1.1.2.1.cmml" xref="S4.SS1.6.p2.3.m2.1.1.2.1"></minus><ci id="S4.SS1.6.p2.3.m2.1.1.2.2.cmml" xref="S4.SS1.6.p2.3.m2.1.1.2.2">𝑘</ci><ci id="S4.SS1.6.p2.3.m2.1.1.2.3.cmml" xref="S4.SS1.6.p2.3.m2.1.1.2.3">𝑗</ci></apply><ci id="S4.SS1.6.p2.3.m2.1.1.4.cmml" xref="S4.SS1.6.p2.3.m2.1.1.4">𝑚</ci></apply><apply id="S4.SS1.6.p2.3.m2.1.1c.cmml" xref="S4.SS1.6.p2.3.m2.1.1"><geq id="S4.SS1.6.p2.3.m2.1.1.5.cmml" xref="S4.SS1.6.p2.3.m2.1.1.5"></geq><share href="https://arxiv.org/html/2503.14636v1#S4.SS1.6.p2.3.m2.1.1.4.cmml" id="S4.SS1.6.p2.3.m2.1.1d.cmml" xref="S4.SS1.6.p2.3.m2.1.1"></share><cn id="S4.SS1.6.p2.3.m2.1.1.6.cmml" type="integer" xref="S4.SS1.6.p2.3.m2.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.6.p2.3.m2.1c">k-j>m\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.6.p2.3.m2.1d">italic_k - italic_j > italic_m ≥ 0</annotation></semantics></math> so that Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i2" title="item ii ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> applies. The last estimate follows from Hardy’s inequality, see for example <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Corollary 3.3]</cite>.</p> </div> <div class="ltx_para" id="S4.SS1.7.p3"> <p class="ltx_p" id="S4.SS1.7.p3.5"><span class="ltx_text ltx_font_italic" id="S4.SS1.7.p3.5.1">Step 2: extension operator.</span> Let <math alttext="\alpha=(\alpha_{1},\widetilde{\alpha})\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{d% -1}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.1.m1.2"><semantics id="S4.SS1.7.p3.1.m1.2a"><mrow id="S4.SS1.7.p3.1.m1.2.2" xref="S4.SS1.7.p3.1.m1.2.2.cmml"><mi id="S4.SS1.7.p3.1.m1.2.2.3" xref="S4.SS1.7.p3.1.m1.2.2.3.cmml">α</mi><mo id="S4.SS1.7.p3.1.m1.2.2.4" xref="S4.SS1.7.p3.1.m1.2.2.4.cmml">=</mo><mrow id="S4.SS1.7.p3.1.m1.2.2.1.1" xref="S4.SS1.7.p3.1.m1.2.2.1.2.cmml"><mo id="S4.SS1.7.p3.1.m1.2.2.1.1.2" stretchy="false" xref="S4.SS1.7.p3.1.m1.2.2.1.2.cmml">(</mo><msub id="S4.SS1.7.p3.1.m1.2.2.1.1.1" xref="S4.SS1.7.p3.1.m1.2.2.1.1.1.cmml"><mi id="S4.SS1.7.p3.1.m1.2.2.1.1.1.2" xref="S4.SS1.7.p3.1.m1.2.2.1.1.1.2.cmml">α</mi><mn id="S4.SS1.7.p3.1.m1.2.2.1.1.1.3" xref="S4.SS1.7.p3.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS1.7.p3.1.m1.2.2.1.1.3" xref="S4.SS1.7.p3.1.m1.2.2.1.2.cmml">,</mo><mover accent="true" id="S4.SS1.7.p3.1.m1.1.1" xref="S4.SS1.7.p3.1.m1.1.1.cmml"><mi id="S4.SS1.7.p3.1.m1.1.1.2" xref="S4.SS1.7.p3.1.m1.1.1.2.cmml">α</mi><mo id="S4.SS1.7.p3.1.m1.1.1.1" xref="S4.SS1.7.p3.1.m1.1.1.1.cmml">~</mo></mover><mo id="S4.SS1.7.p3.1.m1.2.2.1.1.4" stretchy="false" xref="S4.SS1.7.p3.1.m1.2.2.1.2.cmml">)</mo></mrow><mo id="S4.SS1.7.p3.1.m1.2.2.5" xref="S4.SS1.7.p3.1.m1.2.2.5.cmml">∈</mo><mrow id="S4.SS1.7.p3.1.m1.2.2.6" xref="S4.SS1.7.p3.1.m1.2.2.6.cmml"><msub id="S4.SS1.7.p3.1.m1.2.2.6.2" xref="S4.SS1.7.p3.1.m1.2.2.6.2.cmml"><mi id="S4.SS1.7.p3.1.m1.2.2.6.2.2" xref="S4.SS1.7.p3.1.m1.2.2.6.2.2.cmml">ℕ</mi><mn id="S4.SS1.7.p3.1.m1.2.2.6.2.3" xref="S4.SS1.7.p3.1.m1.2.2.6.2.3.cmml">0</mn></msub><mo id="S4.SS1.7.p3.1.m1.2.2.6.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.7.p3.1.m1.2.2.6.1.cmml">×</mo><msubsup id="S4.SS1.7.p3.1.m1.2.2.6.3" xref="S4.SS1.7.p3.1.m1.2.2.6.3.cmml"><mi id="S4.SS1.7.p3.1.m1.2.2.6.3.2.2" xref="S4.SS1.7.p3.1.m1.2.2.6.3.2.2.cmml">ℕ</mi><mn id="S4.SS1.7.p3.1.m1.2.2.6.3.2.3" xref="S4.SS1.7.p3.1.m1.2.2.6.3.2.3.cmml">0</mn><mrow id="S4.SS1.7.p3.1.m1.2.2.6.3.3" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3.cmml"><mi id="S4.SS1.7.p3.1.m1.2.2.6.3.3.2" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3.2.cmml">d</mi><mo id="S4.SS1.7.p3.1.m1.2.2.6.3.3.1" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3.1.cmml">−</mo><mn id="S4.SS1.7.p3.1.m1.2.2.6.3.3.3" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.1.m1.2b"><apply id="S4.SS1.7.p3.1.m1.2.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2"><and id="S4.SS1.7.p3.1.m1.2.2a.cmml" xref="S4.SS1.7.p3.1.m1.2.2"></and><apply id="S4.SS1.7.p3.1.m1.2.2b.cmml" xref="S4.SS1.7.p3.1.m1.2.2"><eq id="S4.SS1.7.p3.1.m1.2.2.4.cmml" xref="S4.SS1.7.p3.1.m1.2.2.4"></eq><ci id="S4.SS1.7.p3.1.m1.2.2.3.cmml" xref="S4.SS1.7.p3.1.m1.2.2.3">𝛼</ci><interval closure="open" id="S4.SS1.7.p3.1.m1.2.2.1.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2.1.1"><apply id="S4.SS1.7.p3.1.m1.2.2.1.1.1.cmml" xref="S4.SS1.7.p3.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.1.m1.2.2.1.1.1.1.cmml" xref="S4.SS1.7.p3.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S4.SS1.7.p3.1.m1.2.2.1.1.1.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2.1.1.1.2">𝛼</ci><cn id="S4.SS1.7.p3.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS1.7.p3.1.m1.2.2.1.1.1.3">1</cn></apply><apply id="S4.SS1.7.p3.1.m1.1.1.cmml" xref="S4.SS1.7.p3.1.m1.1.1"><ci id="S4.SS1.7.p3.1.m1.1.1.1.cmml" xref="S4.SS1.7.p3.1.m1.1.1.1">~</ci><ci id="S4.SS1.7.p3.1.m1.1.1.2.cmml" xref="S4.SS1.7.p3.1.m1.1.1.2">𝛼</ci></apply></interval></apply><apply id="S4.SS1.7.p3.1.m1.2.2c.cmml" xref="S4.SS1.7.p3.1.m1.2.2"><in id="S4.SS1.7.p3.1.m1.2.2.5.cmml" xref="S4.SS1.7.p3.1.m1.2.2.5"></in><share href="https://arxiv.org/html/2503.14636v1#S4.SS1.7.p3.1.m1.2.2.1.cmml" id="S4.SS1.7.p3.1.m1.2.2d.cmml" xref="S4.SS1.7.p3.1.m1.2.2"></share><apply id="S4.SS1.7.p3.1.m1.2.2.6.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6"><times id="S4.SS1.7.p3.1.m1.2.2.6.1.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.1"></times><apply id="S4.SS1.7.p3.1.m1.2.2.6.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.2"><csymbol cd="ambiguous" id="S4.SS1.7.p3.1.m1.2.2.6.2.1.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.2">subscript</csymbol><ci id="S4.SS1.7.p3.1.m1.2.2.6.2.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.2.2">ℕ</ci><cn id="S4.SS1.7.p3.1.m1.2.2.6.2.3.cmml" type="integer" xref="S4.SS1.7.p3.1.m1.2.2.6.2.3">0</cn></apply><apply id="S4.SS1.7.p3.1.m1.2.2.6.3.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3"><csymbol cd="ambiguous" id="S4.SS1.7.p3.1.m1.2.2.6.3.1.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3">superscript</csymbol><apply id="S4.SS1.7.p3.1.m1.2.2.6.3.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3"><csymbol cd="ambiguous" id="S4.SS1.7.p3.1.m1.2.2.6.3.2.1.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3">subscript</csymbol><ci id="S4.SS1.7.p3.1.m1.2.2.6.3.2.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3.2.2">ℕ</ci><cn id="S4.SS1.7.p3.1.m1.2.2.6.3.2.3.cmml" type="integer" xref="S4.SS1.7.p3.1.m1.2.2.6.3.2.3">0</cn></apply><apply id="S4.SS1.7.p3.1.m1.2.2.6.3.3.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3"><minus id="S4.SS1.7.p3.1.m1.2.2.6.3.3.1.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3.1"></minus><ci id="S4.SS1.7.p3.1.m1.2.2.6.3.3.2.cmml" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3.2">𝑑</ci><cn id="S4.SS1.7.p3.1.m1.2.2.6.3.3.3.cmml" type="integer" xref="S4.SS1.7.p3.1.m1.2.2.6.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.1.m1.2c">\alpha=(\alpha_{1},\widetilde{\alpha})\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{d% -1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.1.m1.2d">italic_α = ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_α end_ARG ) ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="|\alpha|\leq k" class="ltx_Math" display="inline" id="S4.SS1.7.p3.2.m2.1"><semantics id="S4.SS1.7.p3.2.m2.1a"><mrow id="S4.SS1.7.p3.2.m2.1.2" xref="S4.SS1.7.p3.2.m2.1.2.cmml"><mrow id="S4.SS1.7.p3.2.m2.1.2.2.2" xref="S4.SS1.7.p3.2.m2.1.2.2.1.cmml"><mo id="S4.SS1.7.p3.2.m2.1.2.2.2.1" stretchy="false" xref="S4.SS1.7.p3.2.m2.1.2.2.1.1.cmml">|</mo><mi id="S4.SS1.7.p3.2.m2.1.1" xref="S4.SS1.7.p3.2.m2.1.1.cmml">α</mi><mo id="S4.SS1.7.p3.2.m2.1.2.2.2.2" stretchy="false" xref="S4.SS1.7.p3.2.m2.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS1.7.p3.2.m2.1.2.1" xref="S4.SS1.7.p3.2.m2.1.2.1.cmml">≤</mo><mi id="S4.SS1.7.p3.2.m2.1.2.3" xref="S4.SS1.7.p3.2.m2.1.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.2.m2.1b"><apply id="S4.SS1.7.p3.2.m2.1.2.cmml" xref="S4.SS1.7.p3.2.m2.1.2"><leq id="S4.SS1.7.p3.2.m2.1.2.1.cmml" xref="S4.SS1.7.p3.2.m2.1.2.1"></leq><apply id="S4.SS1.7.p3.2.m2.1.2.2.1.cmml" xref="S4.SS1.7.p3.2.m2.1.2.2.2"><abs id="S4.SS1.7.p3.2.m2.1.2.2.1.1.cmml" xref="S4.SS1.7.p3.2.m2.1.2.2.2.1"></abs><ci id="S4.SS1.7.p3.2.m2.1.1.cmml" xref="S4.SS1.7.p3.2.m2.1.1">𝛼</ci></apply><ci id="S4.SS1.7.p3.2.m2.1.2.3.cmml" xref="S4.SS1.7.p3.2.m2.1.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.2.m2.1c">|\alpha|\leq k</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.2.m2.1d">| italic_α | ≤ italic_k</annotation></semantics></math> and let <math alttext="g\in B^{k-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S4.SS1.7.p3.3.m3.4"><semantics id="S4.SS1.7.p3.3.m3.4a"><mrow id="S4.SS1.7.p3.3.m3.4.4" xref="S4.SS1.7.p3.3.m3.4.4.cmml"><mi id="S4.SS1.7.p3.3.m3.4.4.3" xref="S4.SS1.7.p3.3.m3.4.4.3.cmml">g</mi><mo id="S4.SS1.7.p3.3.m3.4.4.2" xref="S4.SS1.7.p3.3.m3.4.4.2.cmml">∈</mo><mrow id="S4.SS1.7.p3.3.m3.4.4.1" xref="S4.SS1.7.p3.3.m3.4.4.1.cmml"><msubsup id="S4.SS1.7.p3.3.m3.4.4.1.3" xref="S4.SS1.7.p3.3.m3.4.4.1.3.cmml"><mi id="S4.SS1.7.p3.3.m3.4.4.1.3.2.2" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.2.cmml">B</mi><mrow id="S4.SS1.7.p3.3.m3.2.2.2.4" xref="S4.SS1.7.p3.3.m3.2.2.2.3.cmml"><mi id="S4.SS1.7.p3.3.m3.1.1.1.1" xref="S4.SS1.7.p3.3.m3.1.1.1.1.cmml">p</mi><mo id="S4.SS1.7.p3.3.m3.2.2.2.4.1" xref="S4.SS1.7.p3.3.m3.2.2.2.3.cmml">,</mo><mi id="S4.SS1.7.p3.3.m3.2.2.2.2" xref="S4.SS1.7.p3.3.m3.2.2.2.2.cmml">p</mi></mrow><mrow id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.cmml"><mi id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.2" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.2.cmml">k</mi><mo id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.1" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.1.cmml">−</mo><mi id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.3" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.3.cmml">m</mi><mo id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.1a" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.1.cmml">−</mo><mfrac id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.cmml"><mrow id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2.cmml"><mi id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2.2" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2.2.cmml">γ</mi><mo id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2.1" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2.1.cmml">+</mo><mn id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2.3" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.3" xref="S4.SS1.7.p3.3.m3.4.4.1.3.2.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S4.SS1.7.p3.3.m3.4.4.1.2" xref="S4.SS1.7.p3.3.m3.4.4.1.2.cmml">⁢</mo><mrow id="S4.SS1.7.p3.3.m3.4.4.1.1.1" xref="S4.SS1.7.p3.3.m3.4.4.1.1.2.cmml"><mo id="S4.SS1.7.p3.3.m3.4.4.1.1.1.2" stretchy="false" xref="S4.SS1.7.p3.3.m3.4.4.1.1.2.cmml">(</mo><msup 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xref="S4.SS1.7.p3.3.m3.2.2.2.2">𝑝</ci></list></apply><list id="S4.SS1.7.p3.3.m3.4.4.1.1.2.cmml" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1"><apply id="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.cmml" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.1.cmml" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1.1">superscript</csymbol><ci id="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.2.cmml" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.2">ℝ</ci><apply id="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3.cmml" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3"><minus id="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3.1.cmml" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3.1"></minus><ci id="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3.2.cmml" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3.2">𝑑</ci><cn id="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3.3.cmml" type="integer" xref="S4.SS1.7.p3.3.m3.4.4.1.1.1.1.3.3">1</cn></apply></apply><ci id="S4.SS1.7.p3.3.m3.3.3.cmml" xref="S4.SS1.7.p3.3.m3.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.3.m3.4c">g\in B^{k-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.3.m3.4d">italic_g ∈ italic_B start_POSTSUPERSCRIPT italic_k - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>. Then estimating the norm on <math alttext="\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.4.m4.1"><semantics id="S4.SS1.7.p3.4.m4.1a"><msubsup id="S4.SS1.7.p3.4.m4.1.1" xref="S4.SS1.7.p3.4.m4.1.1.cmml"><mi id="S4.SS1.7.p3.4.m4.1.1.2.2" xref="S4.SS1.7.p3.4.m4.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS1.7.p3.4.m4.1.1.3" xref="S4.SS1.7.p3.4.m4.1.1.3.cmml">+</mo><mi id="S4.SS1.7.p3.4.m4.1.1.2.3" xref="S4.SS1.7.p3.4.m4.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.4.m4.1b"><apply id="S4.SS1.7.p3.4.m4.1.1.cmml" xref="S4.SS1.7.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.4.m4.1.1.1.cmml" xref="S4.SS1.7.p3.4.m4.1.1">subscript</csymbol><apply id="S4.SS1.7.p3.4.m4.1.1.2.cmml" xref="S4.SS1.7.p3.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.4.m4.1.1.2.1.cmml" xref="S4.SS1.7.p3.4.m4.1.1">superscript</csymbol><ci id="S4.SS1.7.p3.4.m4.1.1.2.2.cmml" xref="S4.SS1.7.p3.4.m4.1.1.2.2">ℝ</ci><ci id="S4.SS1.7.p3.4.m4.1.1.2.3.cmml" xref="S4.SS1.7.p3.4.m4.1.1.2.3">𝑑</ci></apply><plus id="S4.SS1.7.p3.4.m4.1.1.3.cmml" xref="S4.SS1.7.p3.4.m4.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.4.m4.1c">\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.4.m4.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> by the norm on <math alttext="\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.5.m5.1"><semantics id="S4.SS1.7.p3.5.m5.1a"><msup id="S4.SS1.7.p3.5.m5.1.1" xref="S4.SS1.7.p3.5.m5.1.1.cmml"><mi id="S4.SS1.7.p3.5.m5.1.1.2" xref="S4.SS1.7.p3.5.m5.1.1.2.cmml">ℝ</mi><mi id="S4.SS1.7.p3.5.m5.1.1.3" xref="S4.SS1.7.p3.5.m5.1.1.3.cmml">d</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.5.m5.1b"><apply id="S4.SS1.7.p3.5.m5.1.1.cmml" xref="S4.SS1.7.p3.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.5.m5.1.1.1.cmml" xref="S4.SS1.7.p3.5.m5.1.1">superscript</csymbol><ci id="S4.SS1.7.p3.5.m5.1.1.2.cmml" xref="S4.SS1.7.p3.5.m5.1.1.2">ℝ</ci><ci id="S4.SS1.7.p3.5.m5.1.1.3.cmml" xref="S4.SS1.7.p3.5.m5.1.1.3">𝑑</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.5.m5.1c">\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.5.m5.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math> and using Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.Thmtheorem3" title="Theorem 2.3. ‣ 2.4. Embeddings ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a> gives</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^{d}_{+},w_{% \gamma};X)}\leq\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^% {d},w_{\gamma};X)}\leq C\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{B^{s_{0}% }_{p,p}(\mathbb{R}^{d},w_{\gamma_{0}};X)}," class="ltx_Math" display="block" id="S4.Ex10.m1.12"><semantics id="S4.Ex10.m1.12a"><mrow id="S4.Ex10.m1.12.12.1" xref="S4.Ex10.m1.12.12.1.1.cmml"><mrow id="S4.Ex10.m1.12.12.1.1" xref="S4.Ex10.m1.12.12.1.1.cmml"><msub id="S4.Ex10.m1.12.12.1.1.1" xref="S4.Ex10.m1.12.12.1.1.1.cmml"><mrow id="S4.Ex10.m1.12.12.1.1.1.1.1" xref="S4.Ex10.m1.12.12.1.1.1.1.2.cmml"><mo id="S4.Ex10.m1.12.12.1.1.1.1.1.2" stretchy="false" xref="S4.Ex10.m1.12.12.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex10.m1.12.12.1.1.1.1.1.1" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.cmml"><msup id="S4.Ex10.m1.12.12.1.1.1.1.1.1.1" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex10.m1.12.12.1.1.1.1.1.1.1.2" lspace="0em" rspace="0.167em" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.1.2.cmml">∂</mo><mi id="S4.Ex10.m1.12.12.1.1.1.1.1.1.1.3" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.1.3.cmml">α</mi></msup><mrow id="S4.Ex10.m1.12.12.1.1.1.1.1.1.2" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.2.cmml"><msub id="S4.Ex10.m1.12.12.1.1.1.1.1.1.2.1" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.2.1.cmml"><mi id="S4.Ex10.m1.12.12.1.1.1.1.1.1.2.1.2" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.2.1.2.cmml">ext</mi><mi id="S4.Ex10.m1.12.12.1.1.1.1.1.1.2.1.3" xref="S4.Ex10.m1.12.12.1.1.1.1.1.1.2.1.3.cmml">m</mi></msub><mo id="S4.Ex10.m1.12.12.1.1.1.1.1.1.2a" lspace="0.167em" 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xref="S4.Ex10.m1.9.9.3.3">𝑋</ci></vector></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex10.m1.12c">\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^{d}_{+},w_{% \gamma};X)}\leq\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^% {d},w_{\gamma};X)}\leq C\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{B^{s_{0}% }_{p,p}(\mathbb{R}^{d},w_{\gamma_{0}};X)},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex10.m1.12d">∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ ∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.7.p3.8">where <math alttext="\gamma_{0}>\gamma" class="ltx_Math" 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id="S4.SS1.7.p3.6.m1.1.1.2.3.cmml" type="integer" xref="S4.SS1.7.p3.6.m1.1.1.2.3">0</cn></apply><ci id="S4.SS1.7.p3.6.m1.1.1.3.cmml" xref="S4.SS1.7.p3.6.m1.1.1.3">𝛾</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.6.m1.1c">\gamma_{0}>\gamma</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.6.m1.1d">italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > italic_γ</annotation></semantics></math> and <math alttext="s_{0}>0" class="ltx_Math" display="inline" id="S4.SS1.7.p3.7.m2.1"><semantics id="S4.SS1.7.p3.7.m2.1a"><mrow id="S4.SS1.7.p3.7.m2.1.1" xref="S4.SS1.7.p3.7.m2.1.1.cmml"><msub id="S4.SS1.7.p3.7.m2.1.1.2" xref="S4.SS1.7.p3.7.m2.1.1.2.cmml"><mi id="S4.SS1.7.p3.7.m2.1.1.2.2" xref="S4.SS1.7.p3.7.m2.1.1.2.2.cmml">s</mi><mn id="S4.SS1.7.p3.7.m2.1.1.2.3" xref="S4.SS1.7.p3.7.m2.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS1.7.p3.7.m2.1.1.1" xref="S4.SS1.7.p3.7.m2.1.1.1.cmml">></mo><mn id="S4.SS1.7.p3.7.m2.1.1.3" xref="S4.SS1.7.p3.7.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.7.m2.1b"><apply id="S4.SS1.7.p3.7.m2.1.1.cmml" xref="S4.SS1.7.p3.7.m2.1.1"><gt id="S4.SS1.7.p3.7.m2.1.1.1.cmml" xref="S4.SS1.7.p3.7.m2.1.1.1"></gt><apply id="S4.SS1.7.p3.7.m2.1.1.2.cmml" xref="S4.SS1.7.p3.7.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.7.p3.7.m2.1.1.2.1.cmml" xref="S4.SS1.7.p3.7.m2.1.1.2">subscript</csymbol><ci id="S4.SS1.7.p3.7.m2.1.1.2.2.cmml" xref="S4.SS1.7.p3.7.m2.1.1.2.2">𝑠</ci><cn id="S4.SS1.7.p3.7.m2.1.1.2.3.cmml" type="integer" xref="S4.SS1.7.p3.7.m2.1.1.2.3">0</cn></apply><cn id="S4.SS1.7.p3.7.m2.1.1.3.cmml" type="integer" xref="S4.SS1.7.p3.7.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.7.m2.1c">s_{0}>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.7.m2.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math> are such that <math alttext="s_{0}-\frac{\gamma_{0}}{p}=-\frac{\gamma}{p}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.8.m3.1"><semantics id="S4.SS1.7.p3.8.m3.1a"><mrow id="S4.SS1.7.p3.8.m3.1.1" xref="S4.SS1.7.p3.8.m3.1.1.cmml"><mrow id="S4.SS1.7.p3.8.m3.1.1.2" xref="S4.SS1.7.p3.8.m3.1.1.2.cmml"><msub id="S4.SS1.7.p3.8.m3.1.1.2.2" xref="S4.SS1.7.p3.8.m3.1.1.2.2.cmml"><mi id="S4.SS1.7.p3.8.m3.1.1.2.2.2" xref="S4.SS1.7.p3.8.m3.1.1.2.2.2.cmml">s</mi><mn id="S4.SS1.7.p3.8.m3.1.1.2.2.3" xref="S4.SS1.7.p3.8.m3.1.1.2.2.3.cmml">0</mn></msub><mo id="S4.SS1.7.p3.8.m3.1.1.2.1" xref="S4.SS1.7.p3.8.m3.1.1.2.1.cmml">−</mo><mfrac id="S4.SS1.7.p3.8.m3.1.1.2.3" xref="S4.SS1.7.p3.8.m3.1.1.2.3.cmml"><msub id="S4.SS1.7.p3.8.m3.1.1.2.3.2" xref="S4.SS1.7.p3.8.m3.1.1.2.3.2.cmml"><mi id="S4.SS1.7.p3.8.m3.1.1.2.3.2.2" xref="S4.SS1.7.p3.8.m3.1.1.2.3.2.2.cmml">γ</mi><mn id="S4.SS1.7.p3.8.m3.1.1.2.3.2.3" xref="S4.SS1.7.p3.8.m3.1.1.2.3.2.3.cmml">0</mn></msub><mi id="S4.SS1.7.p3.8.m3.1.1.2.3.3" xref="S4.SS1.7.p3.8.m3.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S4.SS1.7.p3.8.m3.1.1.1" xref="S4.SS1.7.p3.8.m3.1.1.1.cmml">=</mo><mrow id="S4.SS1.7.p3.8.m3.1.1.3" xref="S4.SS1.7.p3.8.m3.1.1.3.cmml"><mo id="S4.SS1.7.p3.8.m3.1.1.3a" xref="S4.SS1.7.p3.8.m3.1.1.3.cmml">−</mo><mfrac id="S4.SS1.7.p3.8.m3.1.1.3.2" xref="S4.SS1.7.p3.8.m3.1.1.3.2.cmml"><mi id="S4.SS1.7.p3.8.m3.1.1.3.2.2" xref="S4.SS1.7.p3.8.m3.1.1.3.2.2.cmml">γ</mi><mi id="S4.SS1.7.p3.8.m3.1.1.3.2.3" xref="S4.SS1.7.p3.8.m3.1.1.3.2.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.8.m3.1b"><apply id="S4.SS1.7.p3.8.m3.1.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1"><eq id="S4.SS1.7.p3.8.m3.1.1.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1.1"></eq><apply id="S4.SS1.7.p3.8.m3.1.1.2.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2"><minus id="S4.SS1.7.p3.8.m3.1.1.2.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.1"></minus><apply id="S4.SS1.7.p3.8.m3.1.1.2.2.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.7.p3.8.m3.1.1.2.2.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.2">subscript</csymbol><ci id="S4.SS1.7.p3.8.m3.1.1.2.2.2.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.2.2">𝑠</ci><cn id="S4.SS1.7.p3.8.m3.1.1.2.2.3.cmml" type="integer" xref="S4.SS1.7.p3.8.m3.1.1.2.2.3">0</cn></apply><apply id="S4.SS1.7.p3.8.m3.1.1.2.3.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.3"><divide id="S4.SS1.7.p3.8.m3.1.1.2.3.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.3"></divide><apply id="S4.SS1.7.p3.8.m3.1.1.2.3.2.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.3.2"><csymbol cd="ambiguous" id="S4.SS1.7.p3.8.m3.1.1.2.3.2.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.3.2">subscript</csymbol><ci id="S4.SS1.7.p3.8.m3.1.1.2.3.2.2.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.3.2.2">𝛾</ci><cn id="S4.SS1.7.p3.8.m3.1.1.2.3.2.3.cmml" type="integer" xref="S4.SS1.7.p3.8.m3.1.1.2.3.2.3">0</cn></apply><ci id="S4.SS1.7.p3.8.m3.1.1.2.3.3.cmml" xref="S4.SS1.7.p3.8.m3.1.1.2.3.3">𝑝</ci></apply></apply><apply id="S4.SS1.7.p3.8.m3.1.1.3.cmml" xref="S4.SS1.7.p3.8.m3.1.1.3"><minus id="S4.SS1.7.p3.8.m3.1.1.3.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1.3"></minus><apply id="S4.SS1.7.p3.8.m3.1.1.3.2.cmml" xref="S4.SS1.7.p3.8.m3.1.1.3.2"><divide id="S4.SS1.7.p3.8.m3.1.1.3.2.1.cmml" xref="S4.SS1.7.p3.8.m3.1.1.3.2"></divide><ci id="S4.SS1.7.p3.8.m3.1.1.3.2.2.cmml" xref="S4.SS1.7.p3.8.m3.1.1.3.2.2">𝛾</ci><ci id="S4.SS1.7.p3.8.m3.1.1.3.2.3.cmml" xref="S4.SS1.7.p3.8.m3.1.1.3.2.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.8.m3.1c">s_{0}-\frac{\gamma_{0}}{p}=-\frac{\gamma}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.8.m3.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_p end_ARG = - divide start_ARG italic_γ end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Therefore, similar to the proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.Thmtheorem7" title="Lemma 3.7. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.7</span></a>, appealing to (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E3" title="In 2.2. Weighted function spaces ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.3</span></a>), yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx17"> <tbody id="S4.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^% {d}_{+},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.Ex11.m1.4"><semantics id="S4.Ex11.m1.4a"><msub id="S4.Ex11.m1.4.4" xref="S4.Ex11.m1.4.4.cmml"><mrow id="S4.Ex11.m1.4.4.1.1" xref="S4.Ex11.m1.4.4.1.2.cmml"><mo id="S4.Ex11.m1.4.4.1.1.2" stretchy="false" xref="S4.Ex11.m1.4.4.1.2.1.cmml">‖</mo><mrow id="S4.Ex11.m1.4.4.1.1.1" xref="S4.Ex11.m1.4.4.1.1.1.cmml"><msup id="S4.Ex11.m1.4.4.1.1.1.1" xref="S4.Ex11.m1.4.4.1.1.1.1.cmml"><mo 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xref="S4.Ex11.m1.4.4.1.1.1.2"><apply id="S4.Ex11.m1.4.4.1.1.1.2.1.cmml" xref="S4.Ex11.m1.4.4.1.1.1.2.1"><csymbol cd="ambiguous" id="S4.Ex11.m1.4.4.1.1.1.2.1.1.cmml" xref="S4.Ex11.m1.4.4.1.1.1.2.1">subscript</csymbol><ci id="S4.Ex11.m1.4.4.1.1.1.2.1.2.cmml" xref="S4.Ex11.m1.4.4.1.1.1.2.1.2">ext</ci><ci id="S4.Ex11.m1.4.4.1.1.1.2.1.3.cmml" xref="S4.Ex11.m1.4.4.1.1.1.2.1.3">𝑚</ci></apply><ci id="S4.Ex11.m1.4.4.1.1.1.2.2.cmml" xref="S4.Ex11.m1.4.4.1.1.1.2.2">𝑔</ci></apply></apply></apply><apply id="S4.Ex11.m1.3.3.3.cmml" xref="S4.Ex11.m1.3.3.3"><times id="S4.Ex11.m1.3.3.3.4.cmml" xref="S4.Ex11.m1.3.3.3.4"></times><apply id="S4.Ex11.m1.3.3.3.5.cmml" xref="S4.Ex11.m1.3.3.3.5"><csymbol cd="ambiguous" id="S4.Ex11.m1.3.3.3.5.1.cmml" xref="S4.Ex11.m1.3.3.3.5">superscript</csymbol><ci id="S4.Ex11.m1.3.3.3.5.2.cmml" xref="S4.Ex11.m1.3.3.3.5.2">𝐿</ci><ci id="S4.Ex11.m1.3.3.3.5.3.cmml" xref="S4.Ex11.m1.3.3.3.5.3">𝑝</ci></apply><vector id="S4.Ex11.m1.3.3.3.3.3.cmml" xref="S4.Ex11.m1.3.3.3.3.2"><apply 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xref="S4.Ex11.m1.1.1.1.1">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex11.m1.4c">\displaystyle\;\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^% {d}_{+},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex11.m1.4d">∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex12"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex12.m1.1"><semantics id="S4.Ex12.m1.1a"><mo id="S4.Ex12.m1.1.1" xref="S4.Ex12.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex12.m1.1b"><leq id="S4.Ex12.m1.1.1.cmml" xref="S4.Ex12.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex12.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex12.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;C\Big{(}\sum_{\ell=0}^{\infty}\sum_{i=-1}^{1}2^{\ell p(s_{0}-1)% }\|\partial_{1}^{\alpha_{1}}\mathcal{F}^{-1}\eta^{m}_{\ell+i}\|^{p}_{L^{p}(% \mathbb{R},w_{\gamma_{0}})}\|\partial^{\widetilde{\alpha}}(\phi_{\ell+i}\ast g% )\|^{p}_{L^{p}(\mathbb{R}^{d-1};X)}\Big{)}^{\frac{1}{p}}," class="ltx_Math" display="inline" id="S4.Ex12.m2.6"><semantics id="S4.Ex12.m2.6a"><mrow id="S4.Ex12.m2.6.6.1" xref="S4.Ex12.m2.6.6.1.1.cmml"><mrow id="S4.Ex12.m2.6.6.1.1" xref="S4.Ex12.m2.6.6.1.1.cmml"><mi id="S4.Ex12.m2.6.6.1.1.3" xref="S4.Ex12.m2.6.6.1.1.3.cmml">C</mi><mo id="S4.Ex12.m2.6.6.1.1.2" xref="S4.Ex12.m2.6.6.1.1.2.cmml">⁢</mo><msup id="S4.Ex12.m2.6.6.1.1.1" xref="S4.Ex12.m2.6.6.1.1.1.cmml"><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.cmml"><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.2" maxsize="160%" minsize="160%" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.cmml"><mstyle displaystyle="true" id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.cmml"><munderover id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3a" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.cmml"><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.2" movablelimits="false" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.2.cmml">∑</mo><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3.cmml"><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3.2" mathvariant="normal" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3.2.cmml">ℓ</mi><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3.1.cmml">=</mo><mn id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.2.3.3.cmml">0</mn></mrow><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.3" mathvariant="normal" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.3.3.cmml">∞</mi></munderover></mstyle><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.cmml"><mstyle displaystyle="true" id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3.cmml"><munderover id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3a" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3.cmml"><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3.2.2" movablelimits="false" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3.2.2.cmml">∑</mo><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3.2.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3.2.3.cmml"><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.3.2.3.2" 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xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.2.3.2.cmml">1</mn></mrow></msup><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><msubsup id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.2.2" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.2.2.cmml">η</mi><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.cmml"><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.2" mathvariant="normal" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.2.cmml">ℓ</mi><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.1.cmml">+</mo><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.1.1.1.2.3.3.3.cmml">i</mi></mrow><mi 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xref="S4.Ex12.m2.3.3.2.2.1.1.2.cmml">w</mi><msub id="S4.Ex12.m2.3.3.2.2.1.1.3" xref="S4.Ex12.m2.3.3.2.2.1.1.3.cmml"><mi id="S4.Ex12.m2.3.3.2.2.1.1.3.2" xref="S4.Ex12.m2.3.3.2.2.1.1.3.2.cmml">γ</mi><mn id="S4.Ex12.m2.3.3.2.2.1.1.3.3" xref="S4.Ex12.m2.3.3.2.2.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Ex12.m2.3.3.2.2.1.4" stretchy="false" xref="S4.Ex12.m2.3.3.2.2.2.cmml">)</mo></mrow></mrow><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.1.1.1.1.3.cmml">p</mi></msubsup><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.3a" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.3.cmml">⁢</mo><msubsup id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.cmml"><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.2.cmml"><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.2" stretchy="false" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.cmml"><msup id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.cmml"><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.2" lspace="0em" rspace="0em" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.2.cmml">∂</mo><mover accent="true" id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.3" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.3.cmml"><mi id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.3.2" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.3.2.cmml">α</mi><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.3.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.2.3.1.cmml">~</mo></mover></msup><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.1.cmml"><mo id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.2" stretchy="false" xref="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex12.m2.6.6.1.1.1.1.1.1.2.2.2.1.1.1.1.1.1.1" 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roman_ℓ = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_i = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT roman_ℓ italic_p ( italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - 1 ) end_POSTSUPERSCRIPT ∥ ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_η start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ + italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ∥ ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT roman_ℓ + italic_i end_POSTSUBSCRIPT ∗ italic_g ) ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.7.p3.13">where <math alttext="\mathcal{F}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.9.m1.1"><semantics id="S4.SS1.7.p3.9.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS1.7.p3.9.m1.1.1" xref="S4.SS1.7.p3.9.m1.1.1.cmml">ℱ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.9.m1.1b"><ci id="S4.SS1.7.p3.9.m1.1.1.cmml" xref="S4.SS1.7.p3.9.m1.1.1">ℱ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.9.m1.1c">\mathcal{F}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.9.m1.1d">caligraphic_F</annotation></semantics></math> is the one-dimensional Fourier transform and <math alttext="(\phi_{\ell})_{\ell\geq 0}\in\Phi(\mathbb{R}^{d-1})" class="ltx_Math" display="inline" id="S4.SS1.7.p3.10.m2.2"><semantics id="S4.SS1.7.p3.10.m2.2a"><mrow id="S4.SS1.7.p3.10.m2.2.2" xref="S4.SS1.7.p3.10.m2.2.2.cmml"><msub id="S4.SS1.7.p3.10.m2.1.1.1" xref="S4.SS1.7.p3.10.m2.1.1.1.cmml"><mrow id="S4.SS1.7.p3.10.m2.1.1.1.1.1" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.cmml"><mo id="S4.SS1.7.p3.10.m2.1.1.1.1.1.2" stretchy="false" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS1.7.p3.10.m2.1.1.1.1.1.1" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.cmml"><mi id="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.2" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.2.cmml">ϕ</mi><mi id="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.3" mathvariant="normal" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.3.cmml">ℓ</mi></msub><mo 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id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.cmml"><mi id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.2" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.2.cmml">ℝ</mi><mrow id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.cmml"><mi id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.2" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.2.cmml">d</mi><mo id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.1" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.1.cmml">−</mo><mn id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.3" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S4.SS1.7.p3.10.m2.2.2.2.1.1.3" stretchy="false" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.10.m2.2b"><apply id="S4.SS1.7.p3.10.m2.2.2.cmml" xref="S4.SS1.7.p3.10.m2.2.2"><in id="S4.SS1.7.p3.10.m2.2.2.3.cmml" xref="S4.SS1.7.p3.10.m2.2.2.3"></in><apply id="S4.SS1.7.p3.10.m2.1.1.1.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.10.m2.1.1.1.2.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1">subscript</csymbol><apply id="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.1.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.2.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.2">italic-ϕ</ci><ci id="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.3.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1.1.1.1.3">ℓ</ci></apply><apply id="S4.SS1.7.p3.10.m2.1.1.1.3.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1.3"><geq id="S4.SS1.7.p3.10.m2.1.1.1.3.1.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1.3.1"></geq><ci id="S4.SS1.7.p3.10.m2.1.1.1.3.2.cmml" xref="S4.SS1.7.p3.10.m2.1.1.1.3.2">ℓ</ci><cn id="S4.SS1.7.p3.10.m2.1.1.1.3.3.cmml" type="integer" xref="S4.SS1.7.p3.10.m2.1.1.1.3.3">0</cn></apply></apply><apply id="S4.SS1.7.p3.10.m2.2.2.2.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2"><times id="S4.SS1.7.p3.10.m2.2.2.2.2.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.2"></times><ci id="S4.SS1.7.p3.10.m2.2.2.2.3.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.3">Φ</ci><apply id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.1.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1">superscript</csymbol><ci id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.2.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.2">ℝ</ci><apply id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3"><minus id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.1.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.1"></minus><ci id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.2.cmml" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.2">𝑑</ci><cn id="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.SS1.7.p3.10.m2.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.10.m2.2c">(\phi_{\ell})_{\ell\geq 0}\in\Phi(\mathbb{R}^{d-1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.10.m2.2d">( italic_ϕ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT roman_ℓ ≥ 0 end_POSTSUBSCRIPT ∈ roman_Φ ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT )</annotation></semantics></math>. We estimate the two <math alttext="L^{p}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.11.m3.1"><semantics id="S4.SS1.7.p3.11.m3.1a"><msup id="S4.SS1.7.p3.11.m3.1.1" xref="S4.SS1.7.p3.11.m3.1.1.cmml"><mi id="S4.SS1.7.p3.11.m3.1.1.2" xref="S4.SS1.7.p3.11.m3.1.1.2.cmml">L</mi><mi id="S4.SS1.7.p3.11.m3.1.1.3" xref="S4.SS1.7.p3.11.m3.1.1.3.cmml">p</mi></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.11.m3.1b"><apply id="S4.SS1.7.p3.11.m3.1.1.cmml" xref="S4.SS1.7.p3.11.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.11.m3.1.1.1.cmml" xref="S4.SS1.7.p3.11.m3.1.1">superscript</csymbol><ci id="S4.SS1.7.p3.11.m3.1.1.2.cmml" xref="S4.SS1.7.p3.11.m3.1.1.2">𝐿</ci><ci id="S4.SS1.7.p3.11.m3.1.1.3.cmml" xref="S4.SS1.7.p3.11.m3.1.1.3">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.11.m3.1c">L^{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.11.m3.1d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>-norms in the summation above separately. First, by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S3.E12" title="In Proof. ‣ 3.2. The higher-order trace operator ‣ 3. Trace spaces of Besov and Triebel-Lizorkin spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">3.12</span></a>), the product rule and Hardy’s inequality, we obtain for <math alttext="\ell\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.12.m4.1"><semantics id="S4.SS1.7.p3.12.m4.1a"><mrow id="S4.SS1.7.p3.12.m4.1.1" xref="S4.SS1.7.p3.12.m4.1.1.cmml"><mi id="S4.SS1.7.p3.12.m4.1.1.2" mathvariant="normal" xref="S4.SS1.7.p3.12.m4.1.1.2.cmml">ℓ</mi><mo id="S4.SS1.7.p3.12.m4.1.1.1" xref="S4.SS1.7.p3.12.m4.1.1.1.cmml">∈</mo><msub id="S4.SS1.7.p3.12.m4.1.1.3" xref="S4.SS1.7.p3.12.m4.1.1.3.cmml"><mi id="S4.SS1.7.p3.12.m4.1.1.3.2" xref="S4.SS1.7.p3.12.m4.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS1.7.p3.12.m4.1.1.3.3" xref="S4.SS1.7.p3.12.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.12.m4.1b"><apply id="S4.SS1.7.p3.12.m4.1.1.cmml" xref="S4.SS1.7.p3.12.m4.1.1"><in id="S4.SS1.7.p3.12.m4.1.1.1.cmml" xref="S4.SS1.7.p3.12.m4.1.1.1"></in><ci id="S4.SS1.7.p3.12.m4.1.1.2.cmml" xref="S4.SS1.7.p3.12.m4.1.1.2">ℓ</ci><apply id="S4.SS1.7.p3.12.m4.1.1.3.cmml" xref="S4.SS1.7.p3.12.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.7.p3.12.m4.1.1.3.1.cmml" xref="S4.SS1.7.p3.12.m4.1.1.3">subscript</csymbol><ci id="S4.SS1.7.p3.12.m4.1.1.3.2.cmml" xref="S4.SS1.7.p3.12.m4.1.1.3.2">ℕ</ci><cn id="S4.SS1.7.p3.12.m4.1.1.3.3.cmml" type="integer" xref="S4.SS1.7.p3.12.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.12.m4.1c">\ell\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.12.m4.1d">roman_ℓ ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="i\in\{-1,0,1\}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.13.m5.3"><semantics id="S4.SS1.7.p3.13.m5.3a"><mrow id="S4.SS1.7.p3.13.m5.3.3" xref="S4.SS1.7.p3.13.m5.3.3.cmml"><mi id="S4.SS1.7.p3.13.m5.3.3.3" xref="S4.SS1.7.p3.13.m5.3.3.3.cmml">i</mi><mo id="S4.SS1.7.p3.13.m5.3.3.2" xref="S4.SS1.7.p3.13.m5.3.3.2.cmml">∈</mo><mrow id="S4.SS1.7.p3.13.m5.3.3.1.1" xref="S4.SS1.7.p3.13.m5.3.3.1.2.cmml"><mo id="S4.SS1.7.p3.13.m5.3.3.1.1.2" stretchy="false" xref="S4.SS1.7.p3.13.m5.3.3.1.2.cmml">{</mo><mrow id="S4.SS1.7.p3.13.m5.3.3.1.1.1" xref="S4.SS1.7.p3.13.m5.3.3.1.1.1.cmml"><mo id="S4.SS1.7.p3.13.m5.3.3.1.1.1a" xref="S4.SS1.7.p3.13.m5.3.3.1.1.1.cmml">−</mo><mn id="S4.SS1.7.p3.13.m5.3.3.1.1.1.2" xref="S4.SS1.7.p3.13.m5.3.3.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS1.7.p3.13.m5.3.3.1.1.3" xref="S4.SS1.7.p3.13.m5.3.3.1.2.cmml">,</mo><mn id="S4.SS1.7.p3.13.m5.1.1" xref="S4.SS1.7.p3.13.m5.1.1.cmml">0</mn><mo id="S4.SS1.7.p3.13.m5.3.3.1.1.4" xref="S4.SS1.7.p3.13.m5.3.3.1.2.cmml">,</mo><mn id="S4.SS1.7.p3.13.m5.2.2" xref="S4.SS1.7.p3.13.m5.2.2.cmml">1</mn><mo id="S4.SS1.7.p3.13.m5.3.3.1.1.5" stretchy="false" xref="S4.SS1.7.p3.13.m5.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.13.m5.3b"><apply id="S4.SS1.7.p3.13.m5.3.3.cmml" xref="S4.SS1.7.p3.13.m5.3.3"><in id="S4.SS1.7.p3.13.m5.3.3.2.cmml" xref="S4.SS1.7.p3.13.m5.3.3.2"></in><ci id="S4.SS1.7.p3.13.m5.3.3.3.cmml" xref="S4.SS1.7.p3.13.m5.3.3.3">𝑖</ci><set id="S4.SS1.7.p3.13.m5.3.3.1.2.cmml" xref="S4.SS1.7.p3.13.m5.3.3.1.1"><apply id="S4.SS1.7.p3.13.m5.3.3.1.1.1.cmml" xref="S4.SS1.7.p3.13.m5.3.3.1.1.1"><minus id="S4.SS1.7.p3.13.m5.3.3.1.1.1.1.cmml" xref="S4.SS1.7.p3.13.m5.3.3.1.1.1"></minus><cn id="S4.SS1.7.p3.13.m5.3.3.1.1.1.2.cmml" type="integer" xref="S4.SS1.7.p3.13.m5.3.3.1.1.1.2">1</cn></apply><cn id="S4.SS1.7.p3.13.m5.1.1.cmml" type="integer" xref="S4.SS1.7.p3.13.m5.1.1">0</cn><cn id="S4.SS1.7.p3.13.m5.2.2.cmml" type="integer" xref="S4.SS1.7.p3.13.m5.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.13.m5.3c">i\in\{-1,0,1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.13.m5.3d">italic_i ∈ { - 1 , 0 , 1 }</annotation></semantics></math> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx18"> <tbody id="S4.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\partial_{1}^{\alpha_{1}}\mathcal{F}^{-1}" class="ltx_math_unparsed" display="inline" id="S4.Ex13.m1.1"><semantics id="S4.Ex13.m1.1a"><mrow id="S4.Ex13.m1.1b"><mo id="S4.Ex13.m1.1.1" rspace="0.0835em">∥</mo><msubsup id="S4.Ex13.m1.1.2"><mo id="S4.Ex13.m1.1.2.2.2" lspace="0.0835em" rspace="0em">∂</mo><mn id="S4.Ex13.m1.1.2.2.3">1</mn><msub id="S4.Ex13.m1.1.2.3"><mi id="S4.Ex13.m1.1.2.3.2">α</mi><mn id="S4.Ex13.m1.1.2.3.3">1</mn></msub></msubsup><msup id="S4.Ex13.m1.1.3"><mi class="ltx_font_mathcaligraphic" id="S4.Ex13.m1.1.3.2">ℱ</mi><mrow id="S4.Ex13.m1.1.3.3"><mo id="S4.Ex13.m1.1.3.3a">−</mo><mn id="S4.Ex13.m1.1.3.3.2">1</mn></mrow></msup></mrow><annotation encoding="application/x-tex" id="S4.Ex13.m1.1c">\displaystyle\|\partial_{1}^{\alpha_{1}}\mathcal{F}^{-1}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m1.1d">∥ ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\eta^{m}_{\ell+i}\|^{p}_{L^{p}(\mathbb{R},w_{\gamma_{0}})}" class="ltx_Math" display="inline" id="S4.Ex13.m2.4"><semantics id="S4.Ex13.m2.4a"><msubsup id="S4.Ex13.m2.4.4.1" xref="S4.Ex13.m2.4.4.2.cmml"><mrow id="S4.Ex13.m2.4.4.1.1.1" xref="S4.Ex13.m2.4.4.2.cmml"><msubsup id="S4.Ex13.m2.4.4.1.1.1.1" xref="S4.Ex13.m2.4.4.1.1.1.1.cmml"><mi id="S4.Ex13.m2.4.4.1.1.1.1.2.2" xref="S4.Ex13.m2.4.4.1.1.1.1.2.2.cmml">η</mi><mrow id="S4.Ex13.m2.4.4.1.1.1.1.3" xref="S4.Ex13.m2.4.4.1.1.1.1.3.cmml"><mi id="S4.Ex13.m2.4.4.1.1.1.1.3.2" mathvariant="normal" xref="S4.Ex13.m2.4.4.1.1.1.1.3.2.cmml">ℓ</mi><mo id="S4.Ex13.m2.4.4.1.1.1.1.3.1" xref="S4.Ex13.m2.4.4.1.1.1.1.3.1.cmml">+</mo><mi id="S4.Ex13.m2.4.4.1.1.1.1.3.3" xref="S4.Ex13.m2.4.4.1.1.1.1.3.3.cmml">i</mi></mrow><mi id="S4.Ex13.m2.4.4.1.1.1.1.2.3" xref="S4.Ex13.m2.4.4.1.1.1.1.2.3.cmml">m</mi></msubsup><mo fence="true" id="S4.Ex13.m2.4.4.1.1.1.2" lspace="0em" xref="S4.Ex13.m2.4.4.2.1.cmml">∥</mo></mrow><mrow id="S4.Ex13.m2.2.2.2" xref="S4.Ex13.m2.2.2.2.cmml"><msup id="S4.Ex13.m2.2.2.2.4" xref="S4.Ex13.m2.2.2.2.4.cmml"><mi id="S4.Ex13.m2.2.2.2.4.2" xref="S4.Ex13.m2.2.2.2.4.2.cmml">L</mi><mi id="S4.Ex13.m2.2.2.2.4.3" xref="S4.Ex13.m2.2.2.2.4.3.cmml">p</mi></msup><mo id="S4.Ex13.m2.2.2.2.3" xref="S4.Ex13.m2.2.2.2.3.cmml">⁢</mo><mrow id="S4.Ex13.m2.2.2.2.2.1" xref="S4.Ex13.m2.2.2.2.2.2.cmml"><mo id="S4.Ex13.m2.2.2.2.2.1.2" stretchy="false" xref="S4.Ex13.m2.2.2.2.2.2.cmml">(</mo><mi id="S4.Ex13.m2.1.1.1.1" xref="S4.Ex13.m2.1.1.1.1.cmml">ℝ</mi><mo id="S4.Ex13.m2.2.2.2.2.1.3" xref="S4.Ex13.m2.2.2.2.2.2.cmml">,</mo><msub id="S4.Ex13.m2.2.2.2.2.1.1" xref="S4.Ex13.m2.2.2.2.2.1.1.cmml"><mi id="S4.Ex13.m2.2.2.2.2.1.1.2" xref="S4.Ex13.m2.2.2.2.2.1.1.2.cmml">w</mi><msub id="S4.Ex13.m2.2.2.2.2.1.1.3" xref="S4.Ex13.m2.2.2.2.2.1.1.3.cmml"><mi id="S4.Ex13.m2.2.2.2.2.1.1.3.2" xref="S4.Ex13.m2.2.2.2.2.1.1.3.2.cmml">γ</mi><mn id="S4.Ex13.m2.2.2.2.2.1.1.3.3" xref="S4.Ex13.m2.2.2.2.2.1.1.3.3.cmml">0</mn></msub></msub><mo id="S4.Ex13.m2.2.2.2.2.1.4" stretchy="false" xref="S4.Ex13.m2.2.2.2.2.2.cmml">)</mo></mrow></mrow><mi id="S4.Ex13.m2.3.3.1" xref="S4.Ex13.m2.3.3.1.cmml">p</mi></msubsup><annotation-xml encoding="MathML-Content" id="S4.Ex13.m2.4b"><apply id="S4.Ex13.m2.4.4.2.cmml" xref="S4.Ex13.m2.4.4.1"><csymbol cd="latexml" id="S4.Ex13.m2.4.4.2.1.cmml" xref="S4.Ex13.m2.4.4.1.1.1.2">evaluated-at</csymbol><apply id="S4.Ex13.m2.4.4.1.1.1.1.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex13.m2.4.4.1.1.1.1.1.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1">subscript</csymbol><apply id="S4.Ex13.m2.4.4.1.1.1.1.2.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex13.m2.4.4.1.1.1.1.2.1.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1">superscript</csymbol><ci id="S4.Ex13.m2.4.4.1.1.1.1.2.2.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1.2.2">𝜂</ci><ci id="S4.Ex13.m2.4.4.1.1.1.1.2.3.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1.2.3">𝑚</ci></apply><apply id="S4.Ex13.m2.4.4.1.1.1.1.3.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1.3"><plus id="S4.Ex13.m2.4.4.1.1.1.1.3.1.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1.3.1"></plus><ci id="S4.Ex13.m2.4.4.1.1.1.1.3.2.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1.3.2">ℓ</ci><ci id="S4.Ex13.m2.4.4.1.1.1.1.3.3.cmml" xref="S4.Ex13.m2.4.4.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S4.Ex13.m2.2.2.2.cmml" xref="S4.Ex13.m2.2.2.2"><times id="S4.Ex13.m2.2.2.2.3.cmml" xref="S4.Ex13.m2.2.2.2.3"></times><apply id="S4.Ex13.m2.2.2.2.4.cmml" xref="S4.Ex13.m2.2.2.2.4"><csymbol cd="ambiguous" id="S4.Ex13.m2.2.2.2.4.1.cmml" xref="S4.Ex13.m2.2.2.2.4">superscript</csymbol><ci id="S4.Ex13.m2.2.2.2.4.2.cmml" xref="S4.Ex13.m2.2.2.2.4.2">𝐿</ci><ci id="S4.Ex13.m2.2.2.2.4.3.cmml" xref="S4.Ex13.m2.2.2.2.4.3">𝑝</ci></apply><interval closure="open" id="S4.Ex13.m2.2.2.2.2.2.cmml" xref="S4.Ex13.m2.2.2.2.2.1"><ci id="S4.Ex13.m2.1.1.1.1.cmml" xref="S4.Ex13.m2.1.1.1.1">ℝ</ci><apply id="S4.Ex13.m2.2.2.2.2.1.1.cmml" xref="S4.Ex13.m2.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.Ex13.m2.2.2.2.2.1.1.1.cmml" xref="S4.Ex13.m2.2.2.2.2.1.1">subscript</csymbol><ci id="S4.Ex13.m2.2.2.2.2.1.1.2.cmml" xref="S4.Ex13.m2.2.2.2.2.1.1.2">𝑤</ci><apply id="S4.Ex13.m2.2.2.2.2.1.1.3.cmml" xref="S4.Ex13.m2.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.Ex13.m2.2.2.2.2.1.1.3.1.cmml" xref="S4.Ex13.m2.2.2.2.2.1.1.3">subscript</csymbol><ci id="S4.Ex13.m2.2.2.2.2.1.1.3.2.cmml" xref="S4.Ex13.m2.2.2.2.2.1.1.3.2">𝛾</ci><cn id="S4.Ex13.m2.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S4.Ex13.m2.2.2.2.2.1.1.3.3">0</cn></apply></apply></interval></apply><ci id="S4.Ex13.m2.3.3.1.cmml" xref="S4.Ex13.m2.3.3.1">𝑝</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex13.m2.4c">\displaystyle\eta^{m}_{\ell+i}\|^{p}_{L^{p}(\mathbb{R},w_{\gamma_{0}})}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex13.m2.4d">italic_η start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_ℓ + italic_i end_POSTSUBSCRIPT ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\,2^{\ell p}\big{\|}x_{1}\mapsto\partial_{1}^{\alpha_{1}}x_% {1}^{m}(\mathcal{F}^{-1}\zeta)(2^{\ell+i}x_{1})\big{\|}^{p}_{L^{p}(\mathbb{R},% w_{\gamma_{0}})}" class="ltx_Math" display="inline" id="S4.Ex14.m1.4"><semantics id="S4.Ex14.m1.4a"><mrow id="S4.Ex14.m1.4.4" xref="S4.Ex14.m1.4.4.cmml"><mi id="S4.Ex14.m1.4.4.3" xref="S4.Ex14.m1.4.4.3.cmml"></mi><mo id="S4.Ex14.m1.4.4.4" xref="S4.Ex14.m1.4.4.4.cmml">≤</mo><mrow id="S4.Ex14.m1.4.4.5" xref="S4.Ex14.m1.4.4.5.cmml"><mrow id="S4.Ex14.m1.4.4.5.2" xref="S4.Ex14.m1.4.4.5.2.cmml"><mi id="S4.Ex14.m1.4.4.5.2.2" xref="S4.Ex14.m1.4.4.5.2.2.cmml">C</mi><mo id="S4.Ex14.m1.4.4.5.2.1" xref="S4.Ex14.m1.4.4.5.2.1.cmml">⁢</mo><msup id="S4.Ex14.m1.4.4.5.2.3" xref="S4.Ex14.m1.4.4.5.2.3.cmml"><mn id="S4.Ex14.m1.4.4.5.2.3.2" xref="S4.Ex14.m1.4.4.5.2.3.2.cmml"> 2</mn><mrow id="S4.Ex14.m1.4.4.5.2.3.3" xref="S4.Ex14.m1.4.4.5.2.3.3.cmml"><mi id="S4.Ex14.m1.4.4.5.2.3.3.2" mathvariant="normal" xref="S4.Ex14.m1.4.4.5.2.3.3.2.cmml">ℓ</mi><mo id="S4.Ex14.m1.4.4.5.2.3.3.1" xref="S4.Ex14.m1.4.4.5.2.3.3.1.cmml">⁢</mo><mi id="S4.Ex14.m1.4.4.5.2.3.3.3" xref="S4.Ex14.m1.4.4.5.2.3.3.3.cmml">p</mi></mrow></msup></mrow><mo id="S4.Ex14.m1.4.4.5.1" mathsize="120%" xref="S4.Ex14.m1.4.4.5.1.cmml">∥</mo><msub id="S4.Ex14.m1.4.4.5.3" xref="S4.Ex14.m1.4.4.5.3.cmml"><mi id="S4.Ex14.m1.4.4.5.3.2" xref="S4.Ex14.m1.4.4.5.3.2.cmml">x</mi><mn id="S4.Ex14.m1.4.4.5.3.3" xref="S4.Ex14.m1.4.4.5.3.3.cmml">1</mn></msub></mrow><mo id="S4.Ex14.m1.4.4.6" rspace="0.1389em" stretchy="false" xref="S4.Ex14.m1.4.4.6.cmml">↦</mo><msubsup id="S4.Ex14.m1.4.4.1.1" xref="S4.Ex14.m1.4.4.1.2.cmml"><mrow id="S4.Ex14.m1.4.4.1.1.1.1" xref="S4.Ex14.m1.4.4.1.2.cmml"><mrow id="S4.Ex14.m1.4.4.1.1.1.1.1" xref="S4.Ex14.m1.4.4.1.1.1.1.1.cmml"><msubsup id="S4.Ex14.m1.4.4.1.1.1.1.1.3" xref="S4.Ex14.m1.4.4.1.1.1.1.1.3.cmml"><mo id="S4.Ex14.m1.4.4.1.1.1.1.1.3.2.2" lspace="0.1389em" rspace="0em" xref="S4.Ex14.m1.4.4.1.1.1.1.1.3.2.2.cmml">∂</mo><mn id="S4.Ex14.m1.4.4.1.1.1.1.1.3.2.3" xref="S4.Ex14.m1.4.4.1.1.1.1.1.3.2.3.cmml">1</mn><msub id="S4.Ex14.m1.4.4.1.1.1.1.1.3.3" xref="S4.Ex14.m1.4.4.1.1.1.1.1.3.3.cmml"><mi id="S4.Ex14.m1.4.4.1.1.1.1.1.3.3.2" xref="S4.Ex14.m1.4.4.1.1.1.1.1.3.3.2.cmml">α</mi><mn id="S4.Ex14.m1.4.4.1.1.1.1.1.3.3.3" xref="S4.Ex14.m1.4.4.1.1.1.1.1.3.3.3.cmml">1</mn></msub></msubsup><mrow id="S4.Ex14.m1.4.4.1.1.1.1.1.2" xref="S4.Ex14.m1.4.4.1.1.1.1.1.2.cmml"><msubsup id="S4.Ex14.m1.4.4.1.1.1.1.1.2.4" xref="S4.Ex14.m1.4.4.1.1.1.1.1.2.4.cmml"><mi id="S4.Ex14.m1.4.4.1.1.1.1.1.2.4.2.2" xref="S4.Ex14.m1.4.4.1.1.1.1.1.2.4.2.2.cmml">x</mi><mn id="S4.Ex14.m1.4.4.1.1.1.1.1.2.4.2.3" xref="S4.Ex14.m1.4.4.1.1.1.1.1.2.4.2.3.cmml">1</mn><mi id="S4.Ex14.m1.4.4.1.1.1.1.1.2.4.3" 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xref="S4.Ex14.m1.2.2.2.4"><csymbol cd="ambiguous" id="S4.Ex14.m1.2.2.2.4.1.cmml" xref="S4.Ex14.m1.2.2.2.4">superscript</csymbol><ci id="S4.Ex14.m1.2.2.2.4.2.cmml" xref="S4.Ex14.m1.2.2.2.4.2">𝐿</ci><ci id="S4.Ex14.m1.2.2.2.4.3.cmml" xref="S4.Ex14.m1.2.2.2.4.3">𝑝</ci></apply><interval closure="open" id="S4.Ex14.m1.2.2.2.2.2.cmml" xref="S4.Ex14.m1.2.2.2.2.1"><ci id="S4.Ex14.m1.1.1.1.1.cmml" xref="S4.Ex14.m1.1.1.1.1">ℝ</ci><apply id="S4.Ex14.m1.2.2.2.2.1.1.cmml" xref="S4.Ex14.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.Ex14.m1.2.2.2.2.1.1.1.cmml" xref="S4.Ex14.m1.2.2.2.2.1.1">subscript</csymbol><ci id="S4.Ex14.m1.2.2.2.2.1.1.2.cmml" xref="S4.Ex14.m1.2.2.2.2.1.1.2">𝑤</ci><apply id="S4.Ex14.m1.2.2.2.2.1.1.3.cmml" xref="S4.Ex14.m1.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S4.Ex14.m1.2.2.2.2.1.1.3.1.cmml" xref="S4.Ex14.m1.2.2.2.2.1.1.3">subscript</csymbol><ci id="S4.Ex14.m1.2.2.2.2.1.1.3.2.cmml" xref="S4.Ex14.m1.2.2.2.2.1.1.3.2">𝛾</ci><cn id="S4.Ex14.m1.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S4.Ex14.m1.2.2.2.2.1.1.3.3">0</cn></apply></apply></interval></apply><ci id="S4.Ex14.m1.3.3.1.cmml" xref="S4.Ex14.m1.3.3.1">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex14.m1.4c">\displaystyle\leq C\,2^{\ell p}\big{\|}x_{1}\mapsto\partial_{1}^{\alpha_{1}}x_% {1}^{m}(\mathcal{F}^{-1}\zeta)(2^{\ell+i}x_{1})\big{\|}^{p}_{L^{p}(\mathbb{R},% w_{\gamma_{0}})}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex14.m1.4d">≤ italic_C 2 start_POSTSUPERSCRIPT roman_ℓ italic_p end_POSTSUPERSCRIPT ∥ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↦ ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ζ ) ( 2 start_POSTSUPERSCRIPT roman_ℓ + italic_i end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\,2^{\ell p}\sum_{n=0}^{\alpha_{1}}2^{(\ell+i)(\alpha_{1}-n% )p}\big{\|}x_{1}\mapsto x_{1}^{m-n}(\mathcal{F}^{-1}\zeta)^{(\alpha_{1}-n)}(2^% {\ell+i}x_{1})\big{\|}^{p}_{L^{p}(\mathbb{R},w_{\gamma_{0}})}" class="ltx_Math" display="inline" id="S4.Ex15.m1.7"><semantics id="S4.Ex15.m1.7a"><mrow id="S4.Ex15.m1.7.7" xref="S4.Ex15.m1.7.7.cmml"><mi id="S4.Ex15.m1.7.7.3" xref="S4.Ex15.m1.7.7.3.cmml"></mi><mo id="S4.Ex15.m1.7.7.4" xref="S4.Ex15.m1.7.7.4.cmml">≤</mo><mrow id="S4.Ex15.m1.7.7.5" xref="S4.Ex15.m1.7.7.5.cmml"><mrow id="S4.Ex15.m1.7.7.5.2" xref="S4.Ex15.m1.7.7.5.2.cmml"><mi id="S4.Ex15.m1.7.7.5.2.2" xref="S4.Ex15.m1.7.7.5.2.2.cmml">C</mi><mo id="S4.Ex15.m1.7.7.5.2.1" xref="S4.Ex15.m1.7.7.5.2.1.cmml">⁢</mo><msup id="S4.Ex15.m1.7.7.5.2.3" xref="S4.Ex15.m1.7.7.5.2.3.cmml"><mn id="S4.Ex15.m1.7.7.5.2.3.2" xref="S4.Ex15.m1.7.7.5.2.3.2.cmml"> 2</mn><mrow id="S4.Ex15.m1.7.7.5.2.3.3" xref="S4.Ex15.m1.7.7.5.2.3.3.cmml"><mi id="S4.Ex15.m1.7.7.5.2.3.3.2" mathvariant="normal" xref="S4.Ex15.m1.7.7.5.2.3.3.2.cmml">ℓ</mi><mo id="S4.Ex15.m1.7.7.5.2.3.3.1" xref="S4.Ex15.m1.7.7.5.2.3.3.1.cmml">⁢</mo><mi id="S4.Ex15.m1.7.7.5.2.3.3.3" xref="S4.Ex15.m1.7.7.5.2.3.3.3.cmml">p</mi></mrow></msup><mo id="S4.Ex15.m1.7.7.5.2.1a" xref="S4.Ex15.m1.7.7.5.2.1.cmml">⁢</mo><mrow id="S4.Ex15.m1.7.7.5.2.4" xref="S4.Ex15.m1.7.7.5.2.4.cmml"><mstyle displaystyle="true" id="S4.Ex15.m1.7.7.5.2.4.1" xref="S4.Ex15.m1.7.7.5.2.4.1.cmml"><munderover id="S4.Ex15.m1.7.7.5.2.4.1a" xref="S4.Ex15.m1.7.7.5.2.4.1.cmml"><mo id="S4.Ex15.m1.7.7.5.2.4.1.2.2" movablelimits="false" xref="S4.Ex15.m1.7.7.5.2.4.1.2.2.cmml">∑</mo><mrow id="S4.Ex15.m1.7.7.5.2.4.1.2.3" xref="S4.Ex15.m1.7.7.5.2.4.1.2.3.cmml"><mi id="S4.Ex15.m1.7.7.5.2.4.1.2.3.2" xref="S4.Ex15.m1.7.7.5.2.4.1.2.3.2.cmml">n</mi><mo id="S4.Ex15.m1.7.7.5.2.4.1.2.3.1" xref="S4.Ex15.m1.7.7.5.2.4.1.2.3.1.cmml">=</mo><mn id="S4.Ex15.m1.7.7.5.2.4.1.2.3.3" xref="S4.Ex15.m1.7.7.5.2.4.1.2.3.3.cmml">0</mn></mrow><msub id="S4.Ex15.m1.7.7.5.2.4.1.3" xref="S4.Ex15.m1.7.7.5.2.4.1.3.cmml"><mi id="S4.Ex15.m1.7.7.5.2.4.1.3.2" xref="S4.Ex15.m1.7.7.5.2.4.1.3.2.cmml">α</mi><mn id="S4.Ex15.m1.7.7.5.2.4.1.3.3" xref="S4.Ex15.m1.7.7.5.2.4.1.3.3.cmml">1</mn></msub></munderover></mstyle><msup id="S4.Ex15.m1.7.7.5.2.4.2" xref="S4.Ex15.m1.7.7.5.2.4.2.cmml"><mn 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cd="ambiguous" id="S4.Ex15.m1.5.5.2.2.1.1.3.1.cmml" xref="S4.Ex15.m1.5.5.2.2.1.1.3">subscript</csymbol><ci id="S4.Ex15.m1.5.5.2.2.1.1.3.2.cmml" xref="S4.Ex15.m1.5.5.2.2.1.1.3.2">𝛾</ci><cn id="S4.Ex15.m1.5.5.2.2.1.1.3.3.cmml" type="integer" xref="S4.Ex15.m1.5.5.2.2.1.1.3.3">0</cn></apply></apply></interval></apply><ci id="S4.Ex15.m1.6.6.1.cmml" xref="S4.Ex15.m1.6.6.1">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex15.m1.7c">\displaystyle\leq C\,2^{\ell p}\sum_{n=0}^{\alpha_{1}}2^{(\ell+i)(\alpha_{1}-n% )p}\big{\|}x_{1}\mapsto x_{1}^{m-n}(\mathcal{F}^{-1}\zeta)^{(\alpha_{1}-n)}(2^% {\ell+i}x_{1})\big{\|}^{p}_{L^{p}(\mathbb{R},w_{\gamma_{0}})}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex15.m1.7d">≤ italic_C 2 start_POSTSUPERSCRIPT roman_ℓ italic_p end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_n = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT ( roman_ℓ + italic_i ) ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_n ) italic_p end_POSTSUPERSCRIPT ∥ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↦ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - italic_n end_POSTSUPERSCRIPT ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ζ ) start_POSTSUPERSCRIPT ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_n ) end_POSTSUPERSCRIPT ( 2 start_POSTSUPERSCRIPT roman_ℓ + italic_i end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex16"><tr 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\mathbb{R},w_{\gamma_{0}})}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex16.m1.6d">≤ italic_C 2 start_POSTSUPERSCRIPT roman_ℓ italic_p ( 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ∥ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ↦ italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ( caligraphic_F start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT italic_ζ ) start_POSTSUPERSCRIPT ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) end_POSTSUPERSCRIPT ( 2 start_POSTSUPERSCRIPT roman_ℓ + italic_i end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R , italic_w start_POSTSUBSCRIPT italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\,2^{\ell p(1+\alpha_{1}-m-\frac{\gamma_{0}+1}{p})}." class="ltx_Math" display="inline" id="S4.Ex17.m1.2"><semantics id="S4.Ex17.m1.2a"><mrow id="S4.Ex17.m1.2.2.1" xref="S4.Ex17.m1.2.2.1.1.cmml"><mrow id="S4.Ex17.m1.2.2.1.1" xref="S4.Ex17.m1.2.2.1.1.cmml"><mi id="S4.Ex17.m1.2.2.1.1.2" xref="S4.Ex17.m1.2.2.1.1.2.cmml"></mi><mo id="S4.Ex17.m1.2.2.1.1.1" xref="S4.Ex17.m1.2.2.1.1.1.cmml">≤</mo><mrow id="S4.Ex17.m1.2.2.1.1.3" xref="S4.Ex17.m1.2.2.1.1.3.cmml"><mi id="S4.Ex17.m1.2.2.1.1.3.2" xref="S4.Ex17.m1.2.2.1.1.3.2.cmml">C</mi><mo id="S4.Ex17.m1.2.2.1.1.3.1" xref="S4.Ex17.m1.2.2.1.1.3.1.cmml">⁢</mo><msup id="S4.Ex17.m1.2.2.1.1.3.3" xref="S4.Ex17.m1.2.2.1.1.3.3.cmml"><mn 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id="S4.Ex17.m1.1.1.1.1.1.1.4.2.3.cmml" type="integer" xref="S4.Ex17.m1.1.1.1.1.1.1.4.2.3">1</cn></apply><ci id="S4.Ex17.m1.1.1.1.1.1.1.4.3.cmml" xref="S4.Ex17.m1.1.1.1.1.1.1.4.3">𝑝</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex17.m1.2c">\displaystyle\leq C\,2^{\ell p(1+\alpha_{1}-m-\frac{\gamma_{0}+1}{p})}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex17.m1.2d">≤ italic_C 2 start_POSTSUPERSCRIPT roman_ℓ italic_p ( 1 + italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_m - divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.7.p3.15">For the norm on <math alttext="\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S4.SS1.7.p3.14.m1.1"><semantics id="S4.SS1.7.p3.14.m1.1a"><msup id="S4.SS1.7.p3.14.m1.1.1" xref="S4.SS1.7.p3.14.m1.1.1.cmml"><mi id="S4.SS1.7.p3.14.m1.1.1.2" xref="S4.SS1.7.p3.14.m1.1.1.2.cmml">ℝ</mi><mrow id="S4.SS1.7.p3.14.m1.1.1.3" xref="S4.SS1.7.p3.14.m1.1.1.3.cmml"><mi id="S4.SS1.7.p3.14.m1.1.1.3.2" xref="S4.SS1.7.p3.14.m1.1.1.3.2.cmml">d</mi><mo id="S4.SS1.7.p3.14.m1.1.1.3.1" xref="S4.SS1.7.p3.14.m1.1.1.3.1.cmml">−</mo><mn id="S4.SS1.7.p3.14.m1.1.1.3.3" xref="S4.SS1.7.p3.14.m1.1.1.3.3.cmml">1</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.14.m1.1b"><apply id="S4.SS1.7.p3.14.m1.1.1.cmml" xref="S4.SS1.7.p3.14.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.7.p3.14.m1.1.1.1.cmml" xref="S4.SS1.7.p3.14.m1.1.1">superscript</csymbol><ci id="S4.SS1.7.p3.14.m1.1.1.2.cmml" xref="S4.SS1.7.p3.14.m1.1.1.2">ℝ</ci><apply id="S4.SS1.7.p3.14.m1.1.1.3.cmml" xref="S4.SS1.7.p3.14.m1.1.1.3"><minus id="S4.SS1.7.p3.14.m1.1.1.3.1.cmml" xref="S4.SS1.7.p3.14.m1.1.1.3.1"></minus><ci id="S4.SS1.7.p3.14.m1.1.1.3.2.cmml" xref="S4.SS1.7.p3.14.m1.1.1.3.2">𝑑</ci><cn id="S4.SS1.7.p3.14.m1.1.1.3.3.cmml" type="integer" xref="S4.SS1.7.p3.14.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.14.m1.1c">\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.14.m1.1d">blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> we estimate with the notation <math alttext="T_{n}g:=\phi_{n}\ast g" class="ltx_Math" display="inline" id="S4.SS1.7.p3.15.m2.1"><semantics id="S4.SS1.7.p3.15.m2.1a"><mrow id="S4.SS1.7.p3.15.m2.1.1" xref="S4.SS1.7.p3.15.m2.1.1.cmml"><mrow id="S4.SS1.7.p3.15.m2.1.1.2" xref="S4.SS1.7.p3.15.m2.1.1.2.cmml"><msub id="S4.SS1.7.p3.15.m2.1.1.2.2" xref="S4.SS1.7.p3.15.m2.1.1.2.2.cmml"><mi id="S4.SS1.7.p3.15.m2.1.1.2.2.2" xref="S4.SS1.7.p3.15.m2.1.1.2.2.2.cmml">T</mi><mi id="S4.SS1.7.p3.15.m2.1.1.2.2.3" xref="S4.SS1.7.p3.15.m2.1.1.2.2.3.cmml">n</mi></msub><mo id="S4.SS1.7.p3.15.m2.1.1.2.1" xref="S4.SS1.7.p3.15.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.SS1.7.p3.15.m2.1.1.2.3" xref="S4.SS1.7.p3.15.m2.1.1.2.3.cmml">g</mi></mrow><mo id="S4.SS1.7.p3.15.m2.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.7.p3.15.m2.1.1.1.cmml">:=</mo><mrow id="S4.SS1.7.p3.15.m2.1.1.3" xref="S4.SS1.7.p3.15.m2.1.1.3.cmml"><msub id="S4.SS1.7.p3.15.m2.1.1.3.2" xref="S4.SS1.7.p3.15.m2.1.1.3.2.cmml"><mi id="S4.SS1.7.p3.15.m2.1.1.3.2.2" xref="S4.SS1.7.p3.15.m2.1.1.3.2.2.cmml">ϕ</mi><mi id="S4.SS1.7.p3.15.m2.1.1.3.2.3" xref="S4.SS1.7.p3.15.m2.1.1.3.2.3.cmml">n</mi></msub><mo id="S4.SS1.7.p3.15.m2.1.1.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS1.7.p3.15.m2.1.1.3.1.cmml">∗</mo><mi id="S4.SS1.7.p3.15.m2.1.1.3.3" xref="S4.SS1.7.p3.15.m2.1.1.3.3.cmml">g</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.15.m2.1b"><apply id="S4.SS1.7.p3.15.m2.1.1.cmml" xref="S4.SS1.7.p3.15.m2.1.1"><csymbol 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xref="S4.SS1.7.p3.15.m2.1.1.3.2">subscript</csymbol><ci id="S4.SS1.7.p3.15.m2.1.1.3.2.2.cmml" xref="S4.SS1.7.p3.15.m2.1.1.3.2.2">italic-ϕ</ci><ci id="S4.SS1.7.p3.15.m2.1.1.3.2.3.cmml" xref="S4.SS1.7.p3.15.m2.1.1.3.2.3">𝑛</ci></apply><ci id="S4.SS1.7.p3.15.m2.1.1.3.3.cmml" xref="S4.SS1.7.p3.15.m2.1.1.3.3">𝑔</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.15.m2.1c">T_{n}g:=\phi_{n}\ast g</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.15.m2.1d">italic_T start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT italic_g := italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∗ italic_g</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx19"> <tbody id="S4.Ex18"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math 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id="S4.Ex18.m2.3c">\displaystyle\leq\sum_{n=-1}^{1}\|\partial^{\widetilde{\alpha}}(\phi_{\ell+i}% \ast T_{\ell+i+n}g)\|^{p}_{L^{p}(\mathbb{R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex18.m2.3d">≤ ∑ start_POSTSUBSCRIPT italic_n = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∥ ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT roman_ℓ + italic_i end_POSTSUBSCRIPT ∗ italic_T start_POSTSUBSCRIPT roman_ℓ + italic_i + italic_n end_POSTSUBSCRIPT italic_g ) ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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id="S4.Ex19.m1.3.3.2.2.1.1.1.cmml" xref="S4.Ex19.m1.3.3.2.2.1.1">superscript</csymbol><ci id="S4.Ex19.m1.3.3.2.2.1.1.2.cmml" xref="S4.Ex19.m1.3.3.2.2.1.1.2">ℝ</ci><apply id="S4.Ex19.m1.3.3.2.2.1.1.3.cmml" xref="S4.Ex19.m1.3.3.2.2.1.1.3"><minus id="S4.Ex19.m1.3.3.2.2.1.1.3.1.cmml" xref="S4.Ex19.m1.3.3.2.2.1.1.3.1"></minus><ci id="S4.Ex19.m1.3.3.2.2.1.1.3.2.cmml" xref="S4.Ex19.m1.3.3.2.2.1.1.3.2">𝑑</ci><cn id="S4.Ex19.m1.3.3.2.2.1.1.3.3.cmml" type="integer" xref="S4.Ex19.m1.3.3.2.2.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex19.m1.2.2.1.1.cmml" xref="S4.Ex19.m1.2.2.1.1">𝑋</ci></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex19.m1.4c">\displaystyle\leq C\,2^{\ell p|\widetilde{\alpha}|}\sum_{n=-1}^{1}\|T_{\ell+i+% n}g\|^{p}_{L^{p}(\mathbb{R}^{d-1};X)}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex19.m1.4d">≤ italic_C 2 start_POSTSUPERSCRIPT roman_ℓ italic_p | over~ start_ARG italic_α end_ARG | end_POSTSUPERSCRIPT ∑ start_POSTSUBSCRIPT italic_n = - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT roman_ℓ + italic_i + italic_n end_POSTSUBSCRIPT italic_g ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.7.p3.18">Combining the estimates above yields</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx20"> <tbody id="S4.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^{d% },w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.Ex20.m1.4"><semantics id="S4.Ex20.m1.4a"><msub id="S4.Ex20.m1.4.4" xref="S4.Ex20.m1.4.4.cmml"><mrow id="S4.Ex20.m1.4.4.1.1" xref="S4.Ex20.m1.4.4.1.2.cmml"><mo id="S4.Ex20.m1.4.4.1.1.2" stretchy="false" xref="S4.Ex20.m1.4.4.1.2.1.cmml">‖</mo><mrow id="S4.Ex20.m1.4.4.1.1.1" xref="S4.Ex20.m1.4.4.1.1.1.cmml"><msup id="S4.Ex20.m1.4.4.1.1.1.1" xref="S4.Ex20.m1.4.4.1.1.1.1.cmml"><mo id="S4.Ex20.m1.4.4.1.1.1.1.2" lspace="0em" rspace="0.167em" xref="S4.Ex20.m1.4.4.1.1.1.1.2.cmml">∂</mo><mi id="S4.Ex20.m1.4.4.1.1.1.1.3" xref="S4.Ex20.m1.4.4.1.1.1.1.3.cmml">α</mi></msup><mrow id="S4.Ex20.m1.4.4.1.1.1.2" xref="S4.Ex20.m1.4.4.1.1.1.2.cmml"><msub id="S4.Ex20.m1.4.4.1.1.1.2.1" xref="S4.Ex20.m1.4.4.1.1.1.2.1.cmml"><mi id="S4.Ex20.m1.4.4.1.1.1.2.1.2" xref="S4.Ex20.m1.4.4.1.1.1.2.1.2.cmml">ext</mi><mi id="S4.Ex20.m1.4.4.1.1.1.2.1.3" xref="S4.Ex20.m1.4.4.1.1.1.2.1.3.cmml">m</mi></msub><mo id="S4.Ex20.m1.4.4.1.1.1.2a" lspace="0.167em" xref="S4.Ex20.m1.4.4.1.1.1.2.cmml">⁡</mo><mi id="S4.Ex20.m1.4.4.1.1.1.2.2" xref="S4.Ex20.m1.4.4.1.1.1.2.2.cmml">g</mi></mrow></mrow><mo id="S4.Ex20.m1.4.4.1.1.3" stretchy="false" xref="S4.Ex20.m1.4.4.1.2.1.cmml">‖</mo></mrow><mrow id="S4.Ex20.m1.3.3.3" xref="S4.Ex20.m1.3.3.3.cmml"><msup id="S4.Ex20.m1.3.3.3.5" xref="S4.Ex20.m1.3.3.3.5.cmml"><mi id="S4.Ex20.m1.3.3.3.5.2" xref="S4.Ex20.m1.3.3.3.5.2.cmml">L</mi><mi id="S4.Ex20.m1.3.3.3.5.3" xref="S4.Ex20.m1.3.3.3.5.3.cmml">p</mi></msup><mo id="S4.Ex20.m1.3.3.3.4" xref="S4.Ex20.m1.3.3.3.4.cmml">⁢</mo><mrow id="S4.Ex20.m1.3.3.3.3.2" xref="S4.Ex20.m1.3.3.3.3.3.cmml"><mo id="S4.Ex20.m1.3.3.3.3.2.3" stretchy="false" xref="S4.Ex20.m1.3.3.3.3.3.cmml">(</mo><msup id="S4.Ex20.m1.2.2.2.2.1.1" xref="S4.Ex20.m1.2.2.2.2.1.1.cmml"><mi id="S4.Ex20.m1.2.2.2.2.1.1.2" xref="S4.Ex20.m1.2.2.2.2.1.1.2.cmml">ℝ</mi><mi id="S4.Ex20.m1.2.2.2.2.1.1.3" xref="S4.Ex20.m1.2.2.2.2.1.1.3.cmml">d</mi></msup><mo id="S4.Ex20.m1.3.3.3.3.2.4" xref="S4.Ex20.m1.3.3.3.3.3.cmml">,</mo><msub id="S4.Ex20.m1.3.3.3.3.2.2" xref="S4.Ex20.m1.3.3.3.3.2.2.cmml"><mi id="S4.Ex20.m1.3.3.3.3.2.2.2" xref="S4.Ex20.m1.3.3.3.3.2.2.2.cmml">w</mi><mi id="S4.Ex20.m1.3.3.3.3.2.2.3" xref="S4.Ex20.m1.3.3.3.3.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex20.m1.3.3.3.3.2.5" xref="S4.Ex20.m1.3.3.3.3.3.cmml">;</mo><mi id="S4.Ex20.m1.1.1.1.1" xref="S4.Ex20.m1.1.1.1.1.cmml">X</mi><mo id="S4.Ex20.m1.3.3.3.3.2.6" stretchy="false" xref="S4.Ex20.m1.3.3.3.3.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Ex20.m1.4b"><apply id="S4.Ex20.m1.4.4.cmml" xref="S4.Ex20.m1.4.4"><csymbol cd="ambiguous" id="S4.Ex20.m1.4.4.2.cmml" xref="S4.Ex20.m1.4.4">subscript</csymbol><apply id="S4.Ex20.m1.4.4.1.2.cmml" xref="S4.Ex20.m1.4.4.1.1"><csymbol cd="latexml" id="S4.Ex20.m1.4.4.1.2.1.cmml" xref="S4.Ex20.m1.4.4.1.1.2">norm</csymbol><apply id="S4.Ex20.m1.4.4.1.1.1.cmml" xref="S4.Ex20.m1.4.4.1.1.1"><apply id="S4.Ex20.m1.4.4.1.1.1.1.cmml" 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id="S4.Ex20.m1.3.3.3.5.cmml" xref="S4.Ex20.m1.3.3.3.5"><csymbol cd="ambiguous" id="S4.Ex20.m1.3.3.3.5.1.cmml" xref="S4.Ex20.m1.3.3.3.5">superscript</csymbol><ci id="S4.Ex20.m1.3.3.3.5.2.cmml" xref="S4.Ex20.m1.3.3.3.5.2">𝐿</ci><ci id="S4.Ex20.m1.3.3.3.5.3.cmml" xref="S4.Ex20.m1.3.3.3.5.3">𝑝</ci></apply><vector id="S4.Ex20.m1.3.3.3.3.3.cmml" xref="S4.Ex20.m1.3.3.3.3.2"><apply id="S4.Ex20.m1.2.2.2.2.1.1.cmml" xref="S4.Ex20.m1.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.Ex20.m1.2.2.2.2.1.1.1.cmml" xref="S4.Ex20.m1.2.2.2.2.1.1">superscript</csymbol><ci id="S4.Ex20.m1.2.2.2.2.1.1.2.cmml" xref="S4.Ex20.m1.2.2.2.2.1.1.2">ℝ</ci><ci id="S4.Ex20.m1.2.2.2.2.1.1.3.cmml" xref="S4.Ex20.m1.2.2.2.2.1.1.3">𝑑</ci></apply><apply id="S4.Ex20.m1.3.3.3.3.2.2.cmml" xref="S4.Ex20.m1.3.3.3.3.2.2"><csymbol cd="ambiguous" id="S4.Ex20.m1.3.3.3.3.2.2.1.cmml" xref="S4.Ex20.m1.3.3.3.3.2.2">subscript</csymbol><ci id="S4.Ex20.m1.3.3.3.3.2.2.2.cmml" xref="S4.Ex20.m1.3.3.3.3.2.2.2">𝑤</ci><ci id="S4.Ex20.m1.3.3.3.3.2.2.3.cmml" xref="S4.Ex20.m1.3.3.3.3.2.2.3">𝛾</ci></apply><ci id="S4.Ex20.m1.1.1.1.1.cmml" xref="S4.Ex20.m1.1.1.1.1">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex20.m1.4c">\displaystyle\|\partial^{\alpha}\operatorname{ext}_{m}g\|_{L^{p}(\mathbb{R}^{d% },w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex20.m1.4d">∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\Big{(}\sum_{\ell=0}^{\infty}2^{\ell p(s_{0}+|\alpha|-m-% 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id="S4.Ex20.m2.4.4.2.2.1.1.2.cmml" xref="S4.Ex20.m2.4.4.2.2.1.1.2">ℝ</ci><apply id="S4.Ex20.m2.4.4.2.2.1.1.3.cmml" xref="S4.Ex20.m2.4.4.2.2.1.1.3"><minus id="S4.Ex20.m2.4.4.2.2.1.1.3.1.cmml" xref="S4.Ex20.m2.4.4.2.2.1.1.3.1"></minus><ci id="S4.Ex20.m2.4.4.2.2.1.1.3.2.cmml" xref="S4.Ex20.m2.4.4.2.2.1.1.3.2">𝑑</ci><cn id="S4.Ex20.m2.4.4.2.2.1.1.3.3.cmml" type="integer" xref="S4.Ex20.m2.4.4.2.2.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex20.m2.3.3.1.1.cmml" xref="S4.Ex20.m2.3.3.1.1">𝑋</ci></list></apply></apply></apply></apply><apply id="S4.Ex20.m2.5.5.1.1.3.cmml" xref="S4.Ex20.m2.5.5.1.1.3"><divide id="S4.Ex20.m2.5.5.1.1.3.1.cmml" xref="S4.Ex20.m2.5.5.1.1.3"></divide><cn id="S4.Ex20.m2.5.5.1.1.3.2.cmml" type="integer" xref="S4.Ex20.m2.5.5.1.1.3.2">1</cn><ci id="S4.Ex20.m2.5.5.1.1.3.3.cmml" xref="S4.Ex20.m2.5.5.1.1.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex20.m2.5c">\displaystyle\leq C\Big{(}\sum_{\ell=0}^{\infty}2^{\ell p(s_{0}+|\alpha|-m-% \frac{\gamma_{0}+1}{p})}\|T_{\ell}g\|^{p}_{L^{p}(\mathbb{R}^{d-1};X)}\Big{)}^{% \frac{1}{p}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex20.m2.5d">≤ italic_C ( ∑ start_POSTSUBSCRIPT roman_ℓ = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT roman_ℓ italic_p ( italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + | italic_α | - italic_m - divide start_ARG italic_γ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_T start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_g ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex21"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\Big{(}\sum_{\ell=0}^{\infty}2^{\ell p(k-m-\frac{\gamma+1}{% p})}\|\phi_{\ell}\ast g\|^{p}_{L^{p}(\mathbb{R}^{d-1};X)}\Big{)}^{\frac{1}{p}}" class="ltx_Math" display="inline" id="S4.Ex21.m1.4"><semantics id="S4.Ex21.m1.4a"><mrow id="S4.Ex21.m1.4.4" xref="S4.Ex21.m1.4.4.cmml"><mi id="S4.Ex21.m1.4.4.3" xref="S4.Ex21.m1.4.4.3.cmml"></mi><mo id="S4.Ex21.m1.4.4.2" xref="S4.Ex21.m1.4.4.2.cmml">≤</mo><mrow id="S4.Ex21.m1.4.4.1" xref="S4.Ex21.m1.4.4.1.cmml"><mi id="S4.Ex21.m1.4.4.1.3" xref="S4.Ex21.m1.4.4.1.3.cmml">C</mi><mo id="S4.Ex21.m1.4.4.1.2" xref="S4.Ex21.m1.4.4.1.2.cmml">⁢</mo><msup id="S4.Ex21.m1.4.4.1.1" xref="S4.Ex21.m1.4.4.1.1.cmml"><mrow id="S4.Ex21.m1.4.4.1.1.1.1" 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id="S4.Ex21.m1.4.4.1.1.3.cmml" xref="S4.Ex21.m1.4.4.1.1.3"><divide id="S4.Ex21.m1.4.4.1.1.3.1.cmml" xref="S4.Ex21.m1.4.4.1.1.3"></divide><cn id="S4.Ex21.m1.4.4.1.1.3.2.cmml" type="integer" xref="S4.Ex21.m1.4.4.1.1.3.2">1</cn><ci id="S4.Ex21.m1.4.4.1.1.3.3.cmml" xref="S4.Ex21.m1.4.4.1.1.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex21.m1.4c">\displaystyle\leq C\Big{(}\sum_{\ell=0}^{\infty}2^{\ell p(k-m-\frac{\gamma+1}{% p})}\|\phi_{\ell}\ast g\|^{p}_{L^{p}(\mathbb{R}^{d-1};X)}\Big{)}^{\frac{1}{p}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex21.m1.4d">≤ italic_C ( ∑ start_POSTSUBSCRIPT roman_ℓ = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT 2 start_POSTSUPERSCRIPT roman_ℓ italic_p ( italic_k - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ) end_POSTSUPERSCRIPT ∥ italic_ϕ start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∗ italic_g ∥ start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex22"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=C\|g\|_{B^{k-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)}," class="ltx_Math" display="inline" id="S4.Ex22.m1.6"><semantics id="S4.Ex22.m1.6a"><mrow id="S4.Ex22.m1.6.6.1" xref="S4.Ex22.m1.6.6.1.1.cmml"><mrow id="S4.Ex22.m1.6.6.1.1" xref="S4.Ex22.m1.6.6.1.1.cmml"><mi id="S4.Ex22.m1.6.6.1.1.2" xref="S4.Ex22.m1.6.6.1.1.2.cmml"></mi><mo 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italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.7.p3.17">where the constants are independent of <math alttext="g" class="ltx_Math" display="inline" id="S4.SS1.7.p3.16.m1.1"><semantics id="S4.SS1.7.p3.16.m1.1a"><mi id="S4.SS1.7.p3.16.m1.1.1" xref="S4.SS1.7.p3.16.m1.1.1.cmml">g</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.16.m1.1b"><ci id="S4.SS1.7.p3.16.m1.1.1.cmml" xref="S4.SS1.7.p3.16.m1.1.1">𝑔</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.16.m1.1c">g</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.16.m1.1d">italic_g</annotation></semantics></math> and <math alttext="\ell" class="ltx_Math" display="inline" id="S4.SS1.7.p3.17.m2.1"><semantics id="S4.SS1.7.p3.17.m2.1a"><mi id="S4.SS1.7.p3.17.m2.1.1" mathvariant="normal" xref="S4.SS1.7.p3.17.m2.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S4.SS1.7.p3.17.m2.1b"><ci id="S4.SS1.7.p3.17.m2.1.1.cmml" xref="S4.SS1.7.p3.17.m2.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.7.p3.17.m2.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.7.p3.17.m2.1d">roman_ℓ</annotation></semantics></math> and we have used that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="s_{0}+|\alpha|-m-\frac{\gamma_{0}+1}{p}=|\alpha|-m-\frac{\gamma+1}{p}\leq k-m-% \frac{\gamma+1}{p}." class="ltx_Math" display="block" id="S4.Ex23.m1.3"><semantics id="S4.Ex23.m1.3a"><mrow id="S4.Ex23.m1.3.3.1" 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start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG ≤ italic_k - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.7.p3.19">This proves the continuity for the extension operator. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS1.p3"> <p class="ltx_p" id="S4.SS1.p3.3">To close this section we state a related result for the vector of traces given by</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\overline{\operatorname{Tr}}_{m}:=(\operatorname{Tr}_{0},\dots,\operatorname{% Tr}_{m}),\qquad m\in\mathbb{N}_{0}." class="ltx_Math" display="block" id="S4.Ex24.m1.2"><semantics id="S4.Ex24.m1.2a"><mrow id="S4.Ex24.m1.2.2.1"><mrow 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xref="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.2.cmml">Tr</mi><mn id="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.3" xref="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.Ex24.m1.2.2.1.1.1.1.2.2.4" xref="S4.Ex24.m1.2.2.1.1.1.1.2.3.cmml">,</mo><mi id="S4.Ex24.m1.1.1" mathvariant="normal" xref="S4.Ex24.m1.1.1.cmml">…</mi><mo id="S4.Ex24.m1.2.2.1.1.1.1.2.2.5" xref="S4.Ex24.m1.2.2.1.1.1.1.2.3.cmml">,</mo><msub id="S4.Ex24.m1.2.2.1.1.1.1.2.2.2" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.cmml"><mi id="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.2" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.2.cmml">Tr</mi><mi id="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.3" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.3.cmml">m</mi></msub><mo id="S4.Ex24.m1.2.2.1.1.1.1.2.2.6" stretchy="false" xref="S4.Ex24.m1.2.2.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex24.m1.2.2.1.1.2.3" rspace="2.167em" xref="S4.Ex24.m1.2.2.1.1.3a.cmml">,</mo><mrow id="S4.Ex24.m1.2.2.1.1.2.2" xref="S4.Ex24.m1.2.2.1.1.2.2.cmml"><mi id="S4.Ex24.m1.2.2.1.1.2.2.2" xref="S4.Ex24.m1.2.2.1.1.2.2.2.cmml">m</mi><mo id="S4.Ex24.m1.2.2.1.1.2.2.1" xref="S4.Ex24.m1.2.2.1.1.2.2.1.cmml">∈</mo><msub id="S4.Ex24.m1.2.2.1.1.2.2.3" xref="S4.Ex24.m1.2.2.1.1.2.2.3.cmml"><mi id="S4.Ex24.m1.2.2.1.1.2.2.3.2" xref="S4.Ex24.m1.2.2.1.1.2.2.3.2.cmml">ℕ</mi><mn id="S4.Ex24.m1.2.2.1.1.2.2.3.3" xref="S4.Ex24.m1.2.2.1.1.2.2.3.3.cmml">0</mn></msub></mrow></mrow><mo id="S4.Ex24.m1.2.2.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex24.m1.2b"><apply id="S4.Ex24.m1.2.2.1.1.3.cmml" xref="S4.Ex24.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="S4.Ex24.m1.2.2.1.1.3a.cmml" xref="S4.Ex24.m1.2.2.1.1.2.3">formulae-sequence</csymbol><apply id="S4.Ex24.m1.2.2.1.1.1.1.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1"><csymbol cd="latexml" id="S4.Ex24.m1.2.2.1.1.1.1.3.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.3">assign</csymbol><apply id="S4.Ex24.m1.2.2.1.1.1.1.4.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.4"><csymbol cd="ambiguous" id="S4.Ex24.m1.2.2.1.1.1.1.4.1.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.4">subscript</csymbol><apply id="S4.Ex24.m1.2.2.1.1.1.1.4.2.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.4.2"><ci id="S4.Ex24.m1.2.2.1.1.1.1.4.2.1.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.4.2.1">¯</ci><ci id="S4.Ex24.m1.2.2.1.1.1.1.4.2.2.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.4.2.2">Tr</ci></apply><ci id="S4.Ex24.m1.2.2.1.1.1.1.4.3.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.4.3">𝑚</ci></apply><vector id="S4.Ex24.m1.2.2.1.1.1.1.2.3.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2"><apply id="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.1.1.1">subscript</csymbol><ci id="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.2">Tr</ci><cn id="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.Ex24.m1.2.2.1.1.1.1.1.1.1.3">0</cn></apply><ci id="S4.Ex24.m1.1.1.cmml" xref="S4.Ex24.m1.1.1">…</ci><apply id="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2.2"><csymbol cd="ambiguous" id="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.1.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2.2">subscript</csymbol><ci id="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.2.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.2">Tr</ci><ci id="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.3.cmml" xref="S4.Ex24.m1.2.2.1.1.1.1.2.2.2.3">𝑚</ci></apply></vector></apply><apply id="S4.Ex24.m1.2.2.1.1.2.2.cmml" xref="S4.Ex24.m1.2.2.1.1.2.2"><in id="S4.Ex24.m1.2.2.1.1.2.2.1.cmml" xref="S4.Ex24.m1.2.2.1.1.2.2.1"></in><ci id="S4.Ex24.m1.2.2.1.1.2.2.2.cmml" xref="S4.Ex24.m1.2.2.1.1.2.2.2">𝑚</ci><apply id="S4.Ex24.m1.2.2.1.1.2.2.3.cmml" xref="S4.Ex24.m1.2.2.1.1.2.2.3"><csymbol cd="ambiguous" id="S4.Ex24.m1.2.2.1.1.2.2.3.1.cmml" xref="S4.Ex24.m1.2.2.1.1.2.2.3">subscript</csymbol><ci id="S4.Ex24.m1.2.2.1.1.2.2.3.2.cmml" xref="S4.Ex24.m1.2.2.1.1.2.2.3.2">ℕ</ci><cn id="S4.Ex24.m1.2.2.1.1.2.2.3.3.cmml" type="integer" xref="S4.Ex24.m1.2.2.1.1.2.2.3.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex24.m1.2c">\overline{\operatorname{Tr}}_{m}:=(\operatorname{Tr}_{0},\dots,\operatorname{% Tr}_{m}),\qquad m\in\mathbb{N}_{0}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex24.m1.2d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT := ( roman_Tr start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) , italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.p3.2">We follow the proof of <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorem VIII.1.3.2]</cite> to determine the trace space for <math alttext="\overline{\operatorname{Tr}}_{m}" class="ltx_Math" display="inline" id="S4.SS1.p3.1.m1.1"><semantics id="S4.SS1.p3.1.m1.1a"><msub id="S4.SS1.p3.1.m1.1.1" xref="S4.SS1.p3.1.m1.1.1.cmml"><mover accent="true" id="S4.SS1.p3.1.m1.1.1.2" xref="S4.SS1.p3.1.m1.1.1.2.cmml"><mi id="S4.SS1.p3.1.m1.1.1.2.2" xref="S4.SS1.p3.1.m1.1.1.2.2.cmml">Tr</mi><mo id="S4.SS1.p3.1.m1.1.1.2.1" xref="S4.SS1.p3.1.m1.1.1.2.1.cmml">¯</mo></mover><mi id="S4.SS1.p3.1.m1.1.1.3" xref="S4.SS1.p3.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.1.m1.1b"><apply id="S4.SS1.p3.1.m1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.1.m1.1.1.1.cmml" xref="S4.SS1.p3.1.m1.1.1">subscript</csymbol><apply id="S4.SS1.p3.1.m1.1.1.2.cmml" xref="S4.SS1.p3.1.m1.1.1.2"><ci id="S4.SS1.p3.1.m1.1.1.2.1.cmml" xref="S4.SS1.p3.1.m1.1.1.2.1">¯</ci><ci id="S4.SS1.p3.1.m1.1.1.2.2.cmml" xref="S4.SS1.p3.1.m1.1.1.2.2">Tr</ci></apply><ci id="S4.SS1.p3.1.m1.1.1.3.cmml" xref="S4.SS1.p3.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.1.m1.1c">\overline{\operatorname{Tr}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.1.m1.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> from the trace space for <math alttext="\operatorname{Tr}_{m}" class="ltx_Math" display="inline" id="S4.SS1.p3.2.m2.1"><semantics id="S4.SS1.p3.2.m2.1a"><msub id="S4.SS1.p3.2.m2.1.1" xref="S4.SS1.p3.2.m2.1.1.cmml"><mi id="S4.SS1.p3.2.m2.1.1.2" xref="S4.SS1.p3.2.m2.1.1.2.cmml">Tr</mi><mi id="S4.SS1.p3.2.m2.1.1.3" xref="S4.SS1.p3.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.p3.2.m2.1b"><apply id="S4.SS1.p3.2.m2.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.p3.2.m2.1.1.1.cmml" xref="S4.SS1.p3.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.p3.2.m2.1.1.2.cmml" xref="S4.SS1.p3.2.m2.1.1.2">Tr</ci><ci id="S4.SS1.p3.2.m2.1.1.3.cmml" xref="S4.SS1.p3.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p3.2.m2.1c">\operatorname{Tr}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.p3.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. A similar result is contained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>, Section 2.9.2]</cite> for weighted scalar-valued Sobolev spaces.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.1.1.1">Proposition 4.5</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem5.p1"> <p class="ltx_p" id="S4.Thmtheorem5.p1.3"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.3.3">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.1.1.m1.2"><semantics id="S4.Thmtheorem5.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem5.p1.1.1.m1.2.3" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem5.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem5.p1.1.1.m1.1.1" xref="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.1.1.m1.2b"><apply id="S4.Thmtheorem5.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3"><in id="S4.Thmtheorem5.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.1"></in><ci id="S4.Thmtheorem5.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.3.3.2"><cn id="S4.Thmtheorem5.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem5.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem5.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.2.2.m2.1"><semantics id="S4.Thmtheorem5.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem5.p1.2.2.m2.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml">m</mi><mo id="S4.Thmtheorem5.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S4.Thmtheorem5.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml"><mi id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.2.2.m2.1b"><apply id="S4.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1"><in id="S4.Thmtheorem5.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.1"></in><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.2">𝑚</ci><apply id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.2">ℕ</ci><cn id="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem5.p1.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.2.2.m2.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.2.2.m2.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.3.3.m3.1"><semantics id="S4.Thmtheorem5.p1.3.3.m3.1a"><mi id="S4.Thmtheorem5.p1.3.3.m3.1.1" xref="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.3.3.m3.1b"><ci id="S4.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.3.3.m3.1d">italic_X</annotation></semantics></math> be a Banach space.</span></p> <ol class="ltx_enumerate" id="S4.I3"> <li class="ltx_item" id="S4.I3.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S4.I3.i1.p1"> <p class="ltx_p" id="S4.I3.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S4.I3.i1.p1.4.1">If </span><math alttext="s>0" class="ltx_Math" display="inline" id="S4.I3.i1.p1.1.m1.1"><semantics id="S4.I3.i1.p1.1.m1.1a"><mrow id="S4.I3.i1.p1.1.m1.1.1" xref="S4.I3.i1.p1.1.m1.1.1.cmml"><mi id="S4.I3.i1.p1.1.m1.1.1.2" xref="S4.I3.i1.p1.1.m1.1.1.2.cmml">s</mi><mo id="S4.I3.i1.p1.1.m1.1.1.1" xref="S4.I3.i1.p1.1.m1.1.1.1.cmml">></mo><mn id="S4.I3.i1.p1.1.m1.1.1.3" xref="S4.I3.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.1.m1.1b"><apply id="S4.I3.i1.p1.1.m1.1.1.cmml" xref="S4.I3.i1.p1.1.m1.1.1"><gt id="S4.I3.i1.p1.1.m1.1.1.1.cmml" xref="S4.I3.i1.p1.1.m1.1.1.1"></gt><ci id="S4.I3.i1.p1.1.m1.1.1.2.cmml" xref="S4.I3.i1.p1.1.m1.1.1.2">𝑠</ci><cn id="S4.I3.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I3.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.1.m1.1c">s>0</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.1.m1.1d">italic_s > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I3.i1.p1.4.2">, </span><math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.I3.i1.p1.2.m2.2"><semantics id="S4.I3.i1.p1.2.m2.2a"><mrow id="S4.I3.i1.p1.2.m2.2.2" xref="S4.I3.i1.p1.2.m2.2.2.cmml"><mi id="S4.I3.i1.p1.2.m2.2.2.4" xref="S4.I3.i1.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S4.I3.i1.p1.2.m2.2.2.3" xref="S4.I3.i1.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S4.I3.i1.p1.2.m2.2.2.2.2" xref="S4.I3.i1.p1.2.m2.2.2.2.3.cmml"><mo id="S4.I3.i1.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S4.I3.i1.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S4.I3.i1.p1.2.m2.1.1.1.1.1" xref="S4.I3.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S4.I3.i1.p1.2.m2.1.1.1.1.1a" xref="S4.I3.i1.p1.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S4.I3.i1.p1.2.m2.1.1.1.1.1.2" xref="S4.I3.i1.p1.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.I3.i1.p1.2.m2.2.2.2.2.4" xref="S4.I3.i1.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="S4.I3.i1.p1.2.m2.2.2.2.2.2" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2.cmml"><mi id="S4.I3.i1.p1.2.m2.2.2.2.2.2.2" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2.2.cmml">p</mi><mo id="S4.I3.i1.p1.2.m2.2.2.2.2.2.1" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2.1.cmml">−</mo><mn id="S4.I3.i1.p1.2.m2.2.2.2.2.2.3" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.I3.i1.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S4.I3.i1.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.2.m2.2b"><apply id="S4.I3.i1.p1.2.m2.2.2.cmml" xref="S4.I3.i1.p1.2.m2.2.2"><in id="S4.I3.i1.p1.2.m2.2.2.3.cmml" xref="S4.I3.i1.p1.2.m2.2.2.3"></in><ci id="S4.I3.i1.p1.2.m2.2.2.4.cmml" xref="S4.I3.i1.p1.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S4.I3.i1.p1.2.m2.2.2.2.3.cmml" xref="S4.I3.i1.p1.2.m2.2.2.2.2"><apply id="S4.I3.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S4.I3.i1.p1.2.m2.1.1.1.1.1"><minus id="S4.I3.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S4.I3.i1.p1.2.m2.1.1.1.1.1"></minus><cn id="S4.I3.i1.p1.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S4.I3.i1.p1.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S4.I3.i1.p1.2.m2.2.2.2.2.2.cmml" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2"><minus id="S4.I3.i1.p1.2.m2.2.2.2.2.2.1.cmml" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2.1"></minus><ci id="S4.I3.i1.p1.2.m2.2.2.2.2.2.2.cmml" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2.2">𝑝</ci><cn id="S4.I3.i1.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S4.I3.i1.p1.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.2.m2.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.2.m2.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I3.i1.p1.4.3">, </span><math alttext="s>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.I3.i1.p1.3.m3.1"><semantics id="S4.I3.i1.p1.3.m3.1a"><mrow id="S4.I3.i1.p1.3.m3.1.1" xref="S4.I3.i1.p1.3.m3.1.1.cmml"><mi id="S4.I3.i1.p1.3.m3.1.1.2" xref="S4.I3.i1.p1.3.m3.1.1.2.cmml">s</mi><mo id="S4.I3.i1.p1.3.m3.1.1.1" xref="S4.I3.i1.p1.3.m3.1.1.1.cmml">></mo><mrow id="S4.I3.i1.p1.3.m3.1.1.3" xref="S4.I3.i1.p1.3.m3.1.1.3.cmml"><mi id="S4.I3.i1.p1.3.m3.1.1.3.2" xref="S4.I3.i1.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="S4.I3.i1.p1.3.m3.1.1.3.1" xref="S4.I3.i1.p1.3.m3.1.1.3.1.cmml">+</mo><mfrac id="S4.I3.i1.p1.3.m3.1.1.3.3" xref="S4.I3.i1.p1.3.m3.1.1.3.3.cmml"><mrow id="S4.I3.i1.p1.3.m3.1.1.3.3.2" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2.cmml"><mi id="S4.I3.i1.p1.3.m3.1.1.3.3.2.2" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.I3.i1.p1.3.m3.1.1.3.3.2.1" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2.1.cmml">+</mo><mn id="S4.I3.i1.p1.3.m3.1.1.3.3.2.3" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.I3.i1.p1.3.m3.1.1.3.3.3" xref="S4.I3.i1.p1.3.m3.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.3.m3.1b"><apply id="S4.I3.i1.p1.3.m3.1.1.cmml" xref="S4.I3.i1.p1.3.m3.1.1"><gt id="S4.I3.i1.p1.3.m3.1.1.1.cmml" xref="S4.I3.i1.p1.3.m3.1.1.1"></gt><ci id="S4.I3.i1.p1.3.m3.1.1.2.cmml" xref="S4.I3.i1.p1.3.m3.1.1.2">𝑠</ci><apply id="S4.I3.i1.p1.3.m3.1.1.3.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3"><plus id="S4.I3.i1.p1.3.m3.1.1.3.1.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.1"></plus><ci id="S4.I3.i1.p1.3.m3.1.1.3.2.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.2">𝑚</ci><apply id="S4.I3.i1.p1.3.m3.1.1.3.3.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.3"><divide id="S4.I3.i1.p1.3.m3.1.1.3.3.1.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.3"></divide><apply id="S4.I3.i1.p1.3.m3.1.1.3.3.2.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2"><plus id="S4.I3.i1.p1.3.m3.1.1.3.3.2.1.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2.1"></plus><ci id="S4.I3.i1.p1.3.m3.1.1.3.3.2.2.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2.2">𝛾</ci><cn id="S4.I3.i1.p1.3.m3.1.1.3.3.2.3.cmml" type="integer" xref="S4.I3.i1.p1.3.m3.1.1.3.3.2.3">1</cn></apply><ci id="S4.I3.i1.p1.3.m3.1.1.3.3.3.cmml" xref="S4.I3.i1.p1.3.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.3.m3.1c">s>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.3.m3.1d">italic_s > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I3.i1.p1.4.4"> and </span><math alttext="\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}" class="ltx_Math" display="inline" id="S4.I3.i1.p1.4.m4.2"><semantics id="S4.I3.i1.p1.4.m4.2a"><mrow id="S4.I3.i1.p1.4.m4.2.2" xref="S4.I3.i1.p1.4.m4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.I3.i1.p1.4.m4.2.2.4" xref="S4.I3.i1.p1.4.m4.2.2.4.cmml">𝒪</mi><mo id="S4.I3.i1.p1.4.m4.2.2.3" xref="S4.I3.i1.p1.4.m4.2.2.3.cmml">∈</mo><mrow id="S4.I3.i1.p1.4.m4.2.2.2.2" xref="S4.I3.i1.p1.4.m4.2.2.2.3.cmml"><mo id="S4.I3.i1.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S4.I3.i1.p1.4.m4.2.2.2.3.cmml">{</mo><msup id="S4.I3.i1.p1.4.m4.1.1.1.1.1" xref="S4.I3.i1.p1.4.m4.1.1.1.1.1.cmml"><mi id="S4.I3.i1.p1.4.m4.1.1.1.1.1.2" xref="S4.I3.i1.p1.4.m4.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.I3.i1.p1.4.m4.1.1.1.1.1.3" xref="S4.I3.i1.p1.4.m4.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.I3.i1.p1.4.m4.2.2.2.2.4" xref="S4.I3.i1.p1.4.m4.2.2.2.3.cmml">,</mo><msubsup id="S4.I3.i1.p1.4.m4.2.2.2.2.2" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2.cmml"><mi id="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.2" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S4.I3.i1.p1.4.m4.2.2.2.2.2.3" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2.3.cmml">+</mo><mi id="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.3" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S4.I3.i1.p1.4.m4.2.2.2.2.5" stretchy="false" xref="S4.I3.i1.p1.4.m4.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i1.p1.4.m4.2b"><apply id="S4.I3.i1.p1.4.m4.2.2.cmml" xref="S4.I3.i1.p1.4.m4.2.2"><in id="S4.I3.i1.p1.4.m4.2.2.3.cmml" xref="S4.I3.i1.p1.4.m4.2.2.3"></in><ci id="S4.I3.i1.p1.4.m4.2.2.4.cmml" xref="S4.I3.i1.p1.4.m4.2.2.4">𝒪</ci><set id="S4.I3.i1.p1.4.m4.2.2.2.3.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2"><apply id="S4.I3.i1.p1.4.m4.1.1.1.1.1.cmml" xref="S4.I3.i1.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.I3.i1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.I3.i1.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S4.I3.i1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S4.I3.i1.p1.4.m4.1.1.1.1.1.2">ℝ</ci><ci id="S4.I3.i1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S4.I3.i1.p1.4.m4.1.1.1.1.1.3">𝑑</ci></apply><apply id="S4.I3.i1.p1.4.m4.2.2.2.2.2.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.I3.i1.p1.4.m4.2.2.2.2.2.1.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2">subscript</csymbol><apply id="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2">superscript</csymbol><ci id="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.2">ℝ</ci><ci id="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S4.I3.i1.p1.4.m4.2.2.2.2.2.3.cmml" xref="S4.I3.i1.p1.4.m4.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i1.p1.4.m4.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i1.p1.4.m4.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I3.i1.p1.4.5">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex25"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\overline{\operatorname{Tr}}_{m}:H^{s,p}(\mathcal{O},w_{\gamma};X)\to\prod_{j=% 0}^{m}B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S4.Ex25.m1.9"><semantics id="S4.Ex25.m1.9a"><mrow id="S4.Ex25.m1.9.9" xref="S4.Ex25.m1.9.9.cmml"><msub id="S4.Ex25.m1.9.9.4" xref="S4.Ex25.m1.9.9.4.cmml"><mover accent="true" id="S4.Ex25.m1.9.9.4.2" xref="S4.Ex25.m1.9.9.4.2.cmml"><mi id="S4.Ex25.m1.9.9.4.2.2" xref="S4.Ex25.m1.9.9.4.2.2.cmml">Tr</mi><mo id="S4.Ex25.m1.9.9.4.2.1" xref="S4.Ex25.m1.9.9.4.2.1.cmml">¯</mo></mover><mi id="S4.Ex25.m1.9.9.4.3" xref="S4.Ex25.m1.9.9.4.3.cmml">m</mi></msub><mo id="S4.Ex25.m1.9.9.3" lspace="0.278em" rspace="0.278em" xref="S4.Ex25.m1.9.9.3.cmml">:</mo><mrow id="S4.Ex25.m1.9.9.2" xref="S4.Ex25.m1.9.9.2.cmml"><mrow id="S4.Ex25.m1.8.8.1.1" xref="S4.Ex25.m1.8.8.1.1.cmml"><msup id="S4.Ex25.m1.8.8.1.1.3" xref="S4.Ex25.m1.8.8.1.1.3.cmml"><mi id="S4.Ex25.m1.8.8.1.1.3.2" xref="S4.Ex25.m1.8.8.1.1.3.2.cmml">H</mi><mrow id="S4.Ex25.m1.2.2.2.4" xref="S4.Ex25.m1.2.2.2.3.cmml"><mi id="S4.Ex25.m1.1.1.1.1" xref="S4.Ex25.m1.1.1.1.1.cmml">s</mi><mo id="S4.Ex25.m1.2.2.2.4.1" xref="S4.Ex25.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex25.m1.2.2.2.2" xref="S4.Ex25.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex25.m1.8.8.1.1.2" xref="S4.Ex25.m1.8.8.1.1.2.cmml">⁢</mo><mrow id="S4.Ex25.m1.8.8.1.1.1.1" xref="S4.Ex25.m1.8.8.1.1.1.2.cmml"><mo id="S4.Ex25.m1.8.8.1.1.1.1.2" stretchy="false" xref="S4.Ex25.m1.8.8.1.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex25.m1.5.5" xref="S4.Ex25.m1.5.5.cmml">𝒪</mi><mo id="S4.Ex25.m1.8.8.1.1.1.1.3" xref="S4.Ex25.m1.8.8.1.1.1.2.cmml">,</mo><msub 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id="S4.Ex25.m1.9.9.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S4.Ex25.m1.9.9.2.2.1.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex25.m1.7.7.cmml" xref="S4.Ex25.m1.7.7">𝑋</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex25.m1.9c">\overline{\operatorname{Tr}}_{m}:H^{s,p}(\mathcal{O},w_{\gamma};X)\to\prod_{j=% 0}^{m}B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex25.m1.9d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.I3.i1.p1.5"><span class="ltx_text ltx_font_italic" id="S4.I3.i1.p1.5.1">is a continuous and surjective operator.</span></p> </div> </li> <li class="ltx_item" id="S4.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S4.I3.i2.p1"> <p class="ltx_p" id="S4.I3.i2.p1.3"><span class="ltx_text ltx_font_italic" id="S4.I3.i2.p1.3.1">If </span><math alttext="k\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S4.I3.i2.p1.1.m1.1"><semantics id="S4.I3.i2.p1.1.m1.1a"><mrow id="S4.I3.i2.p1.1.m1.1.1" xref="S4.I3.i2.p1.1.m1.1.1.cmml"><mi id="S4.I3.i2.p1.1.m1.1.1.2" xref="S4.I3.i2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S4.I3.i2.p1.1.m1.1.1.1" 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xref="S4.I3.i2.p1.2.m2.3.3.2.2.1.1.3">1</cn></apply><apply id="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.cmml" xref="S4.I3.i2.p1.2.m2.4.4.3.3.2.2"><in id="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.1.cmml" xref="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.1"></in><ci id="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.2.cmml" xref="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.2">𝑗</ci><apply id="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3.cmml" xref="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3.1.cmml" xref="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3">subscript</csymbol><ci id="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3.2.cmml" xref="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S4.I3.i2.p1.2.m2.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.2.m2.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.2.m2.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I3.i2.p1.3.3"> and </span><math alttext="k>m+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.I3.i2.p1.3.m3.1"><semantics id="S4.I3.i2.p1.3.m3.1a"><mrow id="S4.I3.i2.p1.3.m3.1.1" xref="S4.I3.i2.p1.3.m3.1.1.cmml"><mi id="S4.I3.i2.p1.3.m3.1.1.2" xref="S4.I3.i2.p1.3.m3.1.1.2.cmml">k</mi><mo id="S4.I3.i2.p1.3.m3.1.1.1" xref="S4.I3.i2.p1.3.m3.1.1.1.cmml">></mo><mrow id="S4.I3.i2.p1.3.m3.1.1.3" xref="S4.I3.i2.p1.3.m3.1.1.3.cmml"><mi id="S4.I3.i2.p1.3.m3.1.1.3.2" xref="S4.I3.i2.p1.3.m3.1.1.3.2.cmml">m</mi><mo id="S4.I3.i2.p1.3.m3.1.1.3.1" xref="S4.I3.i2.p1.3.m3.1.1.3.1.cmml">+</mo><mfrac id="S4.I3.i2.p1.3.m3.1.1.3.3" xref="S4.I3.i2.p1.3.m3.1.1.3.3.cmml"><mrow id="S4.I3.i2.p1.3.m3.1.1.3.3.2" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2.cmml"><mi id="S4.I3.i2.p1.3.m3.1.1.3.3.2.2" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.I3.i2.p1.3.m3.1.1.3.3.2.1" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2.1.cmml">+</mo><mn id="S4.I3.i2.p1.3.m3.1.1.3.3.2.3" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.I3.i2.p1.3.m3.1.1.3.3.3" xref="S4.I3.i2.p1.3.m3.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I3.i2.p1.3.m3.1b"><apply id="S4.I3.i2.p1.3.m3.1.1.cmml" xref="S4.I3.i2.p1.3.m3.1.1"><gt id="S4.I3.i2.p1.3.m3.1.1.1.cmml" xref="S4.I3.i2.p1.3.m3.1.1.1"></gt><ci id="S4.I3.i2.p1.3.m3.1.1.2.cmml" xref="S4.I3.i2.p1.3.m3.1.1.2">𝑘</ci><apply id="S4.I3.i2.p1.3.m3.1.1.3.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3"><plus id="S4.I3.i2.p1.3.m3.1.1.3.1.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.1"></plus><ci id="S4.I3.i2.p1.3.m3.1.1.3.2.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.2">𝑚</ci><apply id="S4.I3.i2.p1.3.m3.1.1.3.3.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.3"><divide id="S4.I3.i2.p1.3.m3.1.1.3.3.1.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.3"></divide><apply id="S4.I3.i2.p1.3.m3.1.1.3.3.2.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2"><plus id="S4.I3.i2.p1.3.m3.1.1.3.3.2.1.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2.1"></plus><ci id="S4.I3.i2.p1.3.m3.1.1.3.3.2.2.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2.2">𝛾</ci><cn id="S4.I3.i2.p1.3.m3.1.1.3.3.2.3.cmml" type="integer" xref="S4.I3.i2.p1.3.m3.1.1.3.3.2.3">1</cn></apply><ci id="S4.I3.i2.p1.3.m3.1.1.3.3.3.cmml" xref="S4.I3.i2.p1.3.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I3.i2.p1.3.m3.1c">k>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.I3.i2.p1.3.m3.1d">italic_k > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I3.i2.p1.3.4">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex26"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\overline{\operatorname{Tr}}_{m}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to% \prod_{j=0}^{m}B_{p,p}^{k-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S4.Ex26.m1.9"><semantics id="S4.Ex26.m1.9a"><mrow id="S4.Ex26.m1.9.9" xref="S4.Ex26.m1.9.9.cmml"><msub id="S4.Ex26.m1.9.9.5" xref="S4.Ex26.m1.9.9.5.cmml"><mover accent="true" id="S4.Ex26.m1.9.9.5.2" xref="S4.Ex26.m1.9.9.5.2.cmml"><mi id="S4.Ex26.m1.9.9.5.2.2" xref="S4.Ex26.m1.9.9.5.2.2.cmml">Tr</mi><mo id="S4.Ex26.m1.9.9.5.2.1" xref="S4.Ex26.m1.9.9.5.2.1.cmml">¯</mo></mover><mi id="S4.Ex26.m1.9.9.5.3" xref="S4.Ex26.m1.9.9.5.3.cmml">m</mi></msub><mo id="S4.Ex26.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S4.Ex26.m1.9.9.4.cmml">:</mo><mrow id="S4.Ex26.m1.9.9.3" xref="S4.Ex26.m1.9.9.3.cmml"><mrow id="S4.Ex26.m1.8.8.2.2" xref="S4.Ex26.m1.8.8.2.2.cmml"><msup id="S4.Ex26.m1.8.8.2.2.4" xref="S4.Ex26.m1.8.8.2.2.4.cmml"><mi id="S4.Ex26.m1.8.8.2.2.4.2" xref="S4.Ex26.m1.8.8.2.2.4.2.cmml">W</mi><mrow id="S4.Ex26.m1.2.2.2.4" xref="S4.Ex26.m1.2.2.2.3.cmml"><mi id="S4.Ex26.m1.1.1.1.1" xref="S4.Ex26.m1.1.1.1.1.cmml">k</mi><mo id="S4.Ex26.m1.2.2.2.4.1" xref="S4.Ex26.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex26.m1.2.2.2.2" xref="S4.Ex26.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex26.m1.8.8.2.2.3" xref="S4.Ex26.m1.8.8.2.2.3.cmml">⁢</mo><mrow id="S4.Ex26.m1.8.8.2.2.2.2" xref="S4.Ex26.m1.8.8.2.2.2.3.cmml"><mo id="S4.Ex26.m1.8.8.2.2.2.2.3" stretchy="false" xref="S4.Ex26.m1.8.8.2.2.2.3.cmml">(</mo><msubsup id="S4.Ex26.m1.7.7.1.1.1.1.1" xref="S4.Ex26.m1.7.7.1.1.1.1.1.cmml"><mi id="S4.Ex26.m1.7.7.1.1.1.1.1.2.2" xref="S4.Ex26.m1.7.7.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex26.m1.7.7.1.1.1.1.1.3" xref="S4.Ex26.m1.7.7.1.1.1.1.1.3.cmml">+</mo><mi id="S4.Ex26.m1.7.7.1.1.1.1.1.2.3" xref="S4.Ex26.m1.7.7.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.Ex26.m1.8.8.2.2.2.2.4" xref="S4.Ex26.m1.8.8.2.2.2.3.cmml">,</mo><msub id="S4.Ex26.m1.8.8.2.2.2.2.2" xref="S4.Ex26.m1.8.8.2.2.2.2.2.cmml"><mi id="S4.Ex26.m1.8.8.2.2.2.2.2.2" xref="S4.Ex26.m1.8.8.2.2.2.2.2.2.cmml">w</mi><mi id="S4.Ex26.m1.8.8.2.2.2.2.2.3" xref="S4.Ex26.m1.8.8.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex26.m1.8.8.2.2.2.2.5" xref="S4.Ex26.m1.8.8.2.2.2.3.cmml">;</mo><mi id="S4.Ex26.m1.5.5" xref="S4.Ex26.m1.5.5.cmml">X</mi><mo id="S4.Ex26.m1.8.8.2.2.2.2.6" stretchy="false" xref="S4.Ex26.m1.8.8.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.Ex26.m1.9.9.3.4" rspace="0.111em" stretchy="false" xref="S4.Ex26.m1.9.9.3.4.cmml">→</mo><mrow id="S4.Ex26.m1.9.9.3.3" xref="S4.Ex26.m1.9.9.3.3.cmml"><munderover id="S4.Ex26.m1.9.9.3.3.2" xref="S4.Ex26.m1.9.9.3.3.2.cmml"><mo id="S4.Ex26.m1.9.9.3.3.2.2.2" movablelimits="false" xref="S4.Ex26.m1.9.9.3.3.2.2.2.cmml">∏</mo><mrow id="S4.Ex26.m1.9.9.3.3.2.2.3" xref="S4.Ex26.m1.9.9.3.3.2.2.3.cmml"><mi id="S4.Ex26.m1.9.9.3.3.2.2.3.2" 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id="S4.Ex26.m1.9c">\overline{\operatorname{Tr}}_{m}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to% \prod_{j=0}^{m}B_{p,p}^{k-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex26.m1.9d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT : italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.I3.i2.p1.4"><span class="ltx_text ltx_font_italic" id="S4.I3.i2.p1.4.1">is a continuous and surjective operator.</span></p> </div> </li> </ol> <p class="ltx_p" id="S4.Thmtheorem5.p1.7"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem5.p1.7.4">In both cases, there exists a continuous right inverse <math alttext="\overline{\operatorname{ext}}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.4.1.m1.1"><semantics id="S4.Thmtheorem5.p1.4.1.m1.1a"><msub id="S4.Thmtheorem5.p1.4.1.m1.1.1" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p1.4.1.m1.1.1.2" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.2.cmml"><mi id="S4.Thmtheorem5.p1.4.1.m1.1.1.2.2" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.2.2.cmml">ext</mi><mo id="S4.Thmtheorem5.p1.4.1.m1.1.1.2.1" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.2.1.cmml">¯</mo></mover><mi id="S4.Thmtheorem5.p1.4.1.m1.1.1.3" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.4.1.m1.1b"><apply id="S4.Thmtheorem5.p1.4.1.m1.1.1.cmml" xref="S4.Thmtheorem5.p1.4.1.m1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.4.1.m1.1.1.1.cmml" xref="S4.Thmtheorem5.p1.4.1.m1.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p1.4.1.m1.1.1.2.cmml" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.2"><ci id="S4.Thmtheorem5.p1.4.1.m1.1.1.2.1.cmml" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p1.4.1.m1.1.1.2.2.cmml" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.2.2">ext</ci></apply><ci id="S4.Thmtheorem5.p1.4.1.m1.1.1.3.cmml" xref="S4.Thmtheorem5.p1.4.1.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.4.1.m1.1c">\overline{\operatorname{ext}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.4.1.m1.1d">over¯ start_ARG roman_ext end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\overline{\operatorname{Tr}}_{m}" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.5.2.m2.1"><semantics id="S4.Thmtheorem5.p1.5.2.m2.1a"><msub id="S4.Thmtheorem5.p1.5.2.m2.1.1" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.cmml"><mover accent="true" id="S4.Thmtheorem5.p1.5.2.m2.1.1.2" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.2.cmml"><mi id="S4.Thmtheorem5.p1.5.2.m2.1.1.2.2" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.2.2.cmml">Tr</mi><mo id="S4.Thmtheorem5.p1.5.2.m2.1.1.2.1" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.2.1.cmml">¯</mo></mover><mi id="S4.Thmtheorem5.p1.5.2.m2.1.1.3" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.5.2.m2.1b"><apply id="S4.Thmtheorem5.p1.5.2.m2.1.1.cmml" xref="S4.Thmtheorem5.p1.5.2.m2.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem5.p1.5.2.m2.1.1.1.cmml" xref="S4.Thmtheorem5.p1.5.2.m2.1.1">subscript</csymbol><apply id="S4.Thmtheorem5.p1.5.2.m2.1.1.2.cmml" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.2"><ci id="S4.Thmtheorem5.p1.5.2.m2.1.1.2.1.cmml" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.2.1">¯</ci><ci id="S4.Thmtheorem5.p1.5.2.m2.1.1.2.2.cmml" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.2.2">Tr</ci></apply><ci id="S4.Thmtheorem5.p1.5.2.m2.1.1.3.cmml" xref="S4.Thmtheorem5.p1.5.2.m2.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.5.2.m2.1c">\overline{\operatorname{Tr}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.5.2.m2.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> which is independent of <math alttext="k,s,p,\gamma" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.6.3.m3.4"><semantics id="S4.Thmtheorem5.p1.6.3.m3.4a"><mrow id="S4.Thmtheorem5.p1.6.3.m3.4.5.2" xref="S4.Thmtheorem5.p1.6.3.m3.4.5.1.cmml"><mi id="S4.Thmtheorem5.p1.6.3.m3.1.1" xref="S4.Thmtheorem5.p1.6.3.m3.1.1.cmml">k</mi><mo id="S4.Thmtheorem5.p1.6.3.m3.4.5.2.1" xref="S4.Thmtheorem5.p1.6.3.m3.4.5.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.6.3.m3.2.2" xref="S4.Thmtheorem5.p1.6.3.m3.2.2.cmml">s</mi><mo id="S4.Thmtheorem5.p1.6.3.m3.4.5.2.2" xref="S4.Thmtheorem5.p1.6.3.m3.4.5.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.6.3.m3.3.3" xref="S4.Thmtheorem5.p1.6.3.m3.3.3.cmml">p</mi><mo id="S4.Thmtheorem5.p1.6.3.m3.4.5.2.3" xref="S4.Thmtheorem5.p1.6.3.m3.4.5.1.cmml">,</mo><mi id="S4.Thmtheorem5.p1.6.3.m3.4.4" xref="S4.Thmtheorem5.p1.6.3.m3.4.4.cmml">γ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.6.3.m3.4b"><list id="S4.Thmtheorem5.p1.6.3.m3.4.5.1.cmml" xref="S4.Thmtheorem5.p1.6.3.m3.4.5.2"><ci id="S4.Thmtheorem5.p1.6.3.m3.1.1.cmml" xref="S4.Thmtheorem5.p1.6.3.m3.1.1">𝑘</ci><ci id="S4.Thmtheorem5.p1.6.3.m3.2.2.cmml" xref="S4.Thmtheorem5.p1.6.3.m3.2.2">𝑠</ci><ci id="S4.Thmtheorem5.p1.6.3.m3.3.3.cmml" xref="S4.Thmtheorem5.p1.6.3.m3.3.3">𝑝</ci><ci id="S4.Thmtheorem5.p1.6.3.m3.4.4.cmml" xref="S4.Thmtheorem5.p1.6.3.m3.4.4">𝛾</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.6.3.m3.4c">k,s,p,\gamma</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.6.3.m3.4d">italic_k , italic_s , italic_p , italic_γ</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem5.p1.7.4.m4.1"><semantics id="S4.Thmtheorem5.p1.7.4.m4.1a"><mi id="S4.Thmtheorem5.p1.7.4.m4.1.1" xref="S4.Thmtheorem5.p1.7.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem5.p1.7.4.m4.1b"><ci id="S4.Thmtheorem5.p1.7.4.m4.1.1.cmml" xref="S4.Thmtheorem5.p1.7.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem5.p1.7.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem5.p1.7.4.m4.1d">italic_X</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S4.SS1.10"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS1.8.p1"> <p class="ltx_p" id="S4.SS1.8.p1.1">The proof is similar to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorem VIII.1.3.2]</cite> and for the convenience of the reader, we sketch the proof.</p> </div> <div class="ltx_para" id="S4.SS1.9.p2"> <p class="ltx_p" id="S4.SS1.9.p2.5"><span class="ltx_text ltx_font_italic" id="S4.SS1.9.p2.5.1">Step 1: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I3.i1" title="item i ‣ Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>.</span> For <math alttext="j\in\{0,\dots,m\}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.1.m1.3"><semantics id="S4.SS1.9.p2.1.m1.3a"><mrow id="S4.SS1.9.p2.1.m1.3.4" xref="S4.SS1.9.p2.1.m1.3.4.cmml"><mi id="S4.SS1.9.p2.1.m1.3.4.2" xref="S4.SS1.9.p2.1.m1.3.4.2.cmml">j</mi><mo id="S4.SS1.9.p2.1.m1.3.4.1" xref="S4.SS1.9.p2.1.m1.3.4.1.cmml">∈</mo><mrow id="S4.SS1.9.p2.1.m1.3.4.3.2" xref="S4.SS1.9.p2.1.m1.3.4.3.1.cmml"><mo id="S4.SS1.9.p2.1.m1.3.4.3.2.1" stretchy="false" xref="S4.SS1.9.p2.1.m1.3.4.3.1.cmml">{</mo><mn id="S4.SS1.9.p2.1.m1.1.1" xref="S4.SS1.9.p2.1.m1.1.1.cmml">0</mn><mo id="S4.SS1.9.p2.1.m1.3.4.3.2.2" xref="S4.SS1.9.p2.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.1.m1.2.2" mathvariant="normal" xref="S4.SS1.9.p2.1.m1.2.2.cmml">…</mi><mo id="S4.SS1.9.p2.1.m1.3.4.3.2.3" xref="S4.SS1.9.p2.1.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.1.m1.3.3" xref="S4.SS1.9.p2.1.m1.3.3.cmml">m</mi><mo id="S4.SS1.9.p2.1.m1.3.4.3.2.4" stretchy="false" xref="S4.SS1.9.p2.1.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.1.m1.3b"><apply id="S4.SS1.9.p2.1.m1.3.4.cmml" xref="S4.SS1.9.p2.1.m1.3.4"><in id="S4.SS1.9.p2.1.m1.3.4.1.cmml" xref="S4.SS1.9.p2.1.m1.3.4.1"></in><ci id="S4.SS1.9.p2.1.m1.3.4.2.cmml" xref="S4.SS1.9.p2.1.m1.3.4.2">𝑗</ci><set id="S4.SS1.9.p2.1.m1.3.4.3.1.cmml" xref="S4.SS1.9.p2.1.m1.3.4.3.2"><cn id="S4.SS1.9.p2.1.m1.1.1.cmml" type="integer" xref="S4.SS1.9.p2.1.m1.1.1">0</cn><ci id="S4.SS1.9.p2.1.m1.2.2.cmml" xref="S4.SS1.9.p2.1.m1.2.2">…</ci><ci id="S4.SS1.9.p2.1.m1.3.3.cmml" xref="S4.SS1.9.p2.1.m1.3.3">𝑚</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.1.m1.3c">j\in\{0,\dots,m\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.1.m1.3d">italic_j ∈ { 0 , … , italic_m }</annotation></semantics></math> let <math alttext="\operatorname{Tr}_{j}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.2.m2.1"><semantics id="S4.SS1.9.p2.2.m2.1a"><msub id="S4.SS1.9.p2.2.m2.1.1" xref="S4.SS1.9.p2.2.m2.1.1.cmml"><mi id="S4.SS1.9.p2.2.m2.1.1.2" xref="S4.SS1.9.p2.2.m2.1.1.2.cmml">Tr</mi><mi id="S4.SS1.9.p2.2.m2.1.1.3" xref="S4.SS1.9.p2.2.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.2.m2.1b"><apply id="S4.SS1.9.p2.2.m2.1.1.cmml" xref="S4.SS1.9.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.2.m2.1.1.1.cmml" xref="S4.SS1.9.p2.2.m2.1.1">subscript</csymbol><ci id="S4.SS1.9.p2.2.m2.1.1.2.cmml" xref="S4.SS1.9.p2.2.m2.1.1.2">Tr</ci><ci id="S4.SS1.9.p2.2.m2.1.1.3.cmml" xref="S4.SS1.9.p2.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.2.m2.1c">\operatorname{Tr}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.2.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\operatorname{ext}_{j}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.3.m3.1"><semantics id="S4.SS1.9.p2.3.m3.1a"><msub id="S4.SS1.9.p2.3.m3.1.1" xref="S4.SS1.9.p2.3.m3.1.1.cmml"><mi id="S4.SS1.9.p2.3.m3.1.1.2" xref="S4.SS1.9.p2.3.m3.1.1.2.cmml">ext</mi><mi id="S4.SS1.9.p2.3.m3.1.1.3" xref="S4.SS1.9.p2.3.m3.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.3.m3.1b"><apply id="S4.SS1.9.p2.3.m3.1.1.cmml" xref="S4.SS1.9.p2.3.m3.1.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.3.m3.1.1.1.cmml" xref="S4.SS1.9.p2.3.m3.1.1">subscript</csymbol><ci id="S4.SS1.9.p2.3.m3.1.1.2.cmml" xref="S4.SS1.9.p2.3.m3.1.1.2">ext</ci><ci id="S4.SS1.9.p2.3.m3.1.1.3.cmml" xref="S4.SS1.9.p2.3.m3.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.3.m3.1c">\operatorname{ext}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.3.m3.1d">roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> be as in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>. The continuity of <math alttext="\overline{\operatorname{Tr}}_{m}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.4.m4.1"><semantics id="S4.SS1.9.p2.4.m4.1a"><msub id="S4.SS1.9.p2.4.m4.1.1" xref="S4.SS1.9.p2.4.m4.1.1.cmml"><mover accent="true" id="S4.SS1.9.p2.4.m4.1.1.2" xref="S4.SS1.9.p2.4.m4.1.1.2.cmml"><mi id="S4.SS1.9.p2.4.m4.1.1.2.2" xref="S4.SS1.9.p2.4.m4.1.1.2.2.cmml">Tr</mi><mo id="S4.SS1.9.p2.4.m4.1.1.2.1" xref="S4.SS1.9.p2.4.m4.1.1.2.1.cmml">¯</mo></mover><mi id="S4.SS1.9.p2.4.m4.1.1.3" xref="S4.SS1.9.p2.4.m4.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.4.m4.1b"><apply id="S4.SS1.9.p2.4.m4.1.1.cmml" xref="S4.SS1.9.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.4.m4.1.1.1.cmml" xref="S4.SS1.9.p2.4.m4.1.1">subscript</csymbol><apply id="S4.SS1.9.p2.4.m4.1.1.2.cmml" xref="S4.SS1.9.p2.4.m4.1.1.2"><ci id="S4.SS1.9.p2.4.m4.1.1.2.1.cmml" xref="S4.SS1.9.p2.4.m4.1.1.2.1">¯</ci><ci id="S4.SS1.9.p2.4.m4.1.1.2.2.cmml" xref="S4.SS1.9.p2.4.m4.1.1.2.2">Tr</ci></apply><ci id="S4.SS1.9.p2.4.m4.1.1.3.cmml" xref="S4.SS1.9.p2.4.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.4.m4.1c">\overline{\operatorname{Tr}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.4.m4.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> follows from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>. We continue with constructing an extension operator <math alttext="\overline{\operatorname{ext}}_{m}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.5.m5.1"><semantics id="S4.SS1.9.p2.5.m5.1a"><msub id="S4.SS1.9.p2.5.m5.1.1" xref="S4.SS1.9.p2.5.m5.1.1.cmml"><mover accent="true" id="S4.SS1.9.p2.5.m5.1.1.2" xref="S4.SS1.9.p2.5.m5.1.1.2.cmml"><mi id="S4.SS1.9.p2.5.m5.1.1.2.2" xref="S4.SS1.9.p2.5.m5.1.1.2.2.cmml">ext</mi><mo id="S4.SS1.9.p2.5.m5.1.1.2.1" xref="S4.SS1.9.p2.5.m5.1.1.2.1.cmml">¯</mo></mover><mi id="S4.SS1.9.p2.5.m5.1.1.3" xref="S4.SS1.9.p2.5.m5.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.5.m5.1b"><apply id="S4.SS1.9.p2.5.m5.1.1.cmml" xref="S4.SS1.9.p2.5.m5.1.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.5.m5.1.1.1.cmml" xref="S4.SS1.9.p2.5.m5.1.1">subscript</csymbol><apply id="S4.SS1.9.p2.5.m5.1.1.2.cmml" xref="S4.SS1.9.p2.5.m5.1.1.2"><ci id="S4.SS1.9.p2.5.m5.1.1.2.1.cmml" xref="S4.SS1.9.p2.5.m5.1.1.2.1">¯</ci><ci id="S4.SS1.9.p2.5.m5.1.1.2.2.cmml" xref="S4.SS1.9.p2.5.m5.1.1.2.2">ext</ci></apply><ci id="S4.SS1.9.p2.5.m5.1.1.3.cmml" xref="S4.SS1.9.p2.5.m5.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.5.m5.1c">\overline{\operatorname{ext}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.5.m5.1d">over¯ start_ARG roman_ext end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math>. Let</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex27"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="(g_{0},\dots,g_{m})\in\prod_{j=0}^{m}B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{% R}^{d-1};X)" class="ltx_Math" display="block" id="S4.Ex27.m1.7"><semantics id="S4.Ex27.m1.7a"><mrow id="S4.Ex27.m1.7.7" xref="S4.Ex27.m1.7.7.cmml"><mrow id="S4.Ex27.m1.6.6.2.2" xref="S4.Ex27.m1.6.6.2.3.cmml"><mo id="S4.Ex27.m1.6.6.2.2.3" stretchy="false" xref="S4.Ex27.m1.6.6.2.3.cmml">(</mo><msub id="S4.Ex27.m1.5.5.1.1.1" xref="S4.Ex27.m1.5.5.1.1.1.cmml"><mi id="S4.Ex27.m1.5.5.1.1.1.2" xref="S4.Ex27.m1.5.5.1.1.1.2.cmml">g</mi><mn id="S4.Ex27.m1.5.5.1.1.1.3" xref="S4.Ex27.m1.5.5.1.1.1.3.cmml">0</mn></msub><mo id="S4.Ex27.m1.6.6.2.2.4" xref="S4.Ex27.m1.6.6.2.3.cmml">,</mo><mi id="S4.Ex27.m1.3.3" mathvariant="normal" xref="S4.Ex27.m1.3.3.cmml">…</mi><mo id="S4.Ex27.m1.6.6.2.2.5" xref="S4.Ex27.m1.6.6.2.3.cmml">,</mo><msub id="S4.Ex27.m1.6.6.2.2.2" xref="S4.Ex27.m1.6.6.2.2.2.cmml"><mi id="S4.Ex27.m1.6.6.2.2.2.2" xref="S4.Ex27.m1.6.6.2.2.2.2.cmml">g</mi><mi id="S4.Ex27.m1.6.6.2.2.2.3" xref="S4.Ex27.m1.6.6.2.2.2.3.cmml">m</mi></msub><mo id="S4.Ex27.m1.6.6.2.2.6" stretchy="false" xref="S4.Ex27.m1.6.6.2.3.cmml">)</mo></mrow><mo id="S4.Ex27.m1.7.7.4" rspace="0.111em" xref="S4.Ex27.m1.7.7.4.cmml">∈</mo><mrow id="S4.Ex27.m1.7.7.3" xref="S4.Ex27.m1.7.7.3.cmml"><munderover id="S4.Ex27.m1.7.7.3.2" xref="S4.Ex27.m1.7.7.3.2.cmml"><mo id="S4.Ex27.m1.7.7.3.2.2.2" movablelimits="false" xref="S4.Ex27.m1.7.7.3.2.2.2.cmml">∏</mo><mrow id="S4.Ex27.m1.7.7.3.2.2.3" xref="S4.Ex27.m1.7.7.3.2.2.3.cmml"><mi id="S4.Ex27.m1.7.7.3.2.2.3.2" xref="S4.Ex27.m1.7.7.3.2.2.3.2.cmml">j</mi><mo id="S4.Ex27.m1.7.7.3.2.2.3.1" xref="S4.Ex27.m1.7.7.3.2.2.3.1.cmml">=</mo><mn id="S4.Ex27.m1.7.7.3.2.2.3.3" xref="S4.Ex27.m1.7.7.3.2.2.3.3.cmml">0</mn></mrow><mi id="S4.Ex27.m1.7.7.3.2.3" xref="S4.Ex27.m1.7.7.3.2.3.cmml">m</mi></munderover><mrow id="S4.Ex27.m1.7.7.3.1" xref="S4.Ex27.m1.7.7.3.1.cmml"><msubsup id="S4.Ex27.m1.7.7.3.1.3" xref="S4.Ex27.m1.7.7.3.1.3.cmml"><mi id="S4.Ex27.m1.7.7.3.1.3.2.2" xref="S4.Ex27.m1.7.7.3.1.3.2.2.cmml">B</mi><mrow id="S4.Ex27.m1.2.2.2.4" xref="S4.Ex27.m1.2.2.2.3.cmml"><mi id="S4.Ex27.m1.1.1.1.1" xref="S4.Ex27.m1.1.1.1.1.cmml">p</mi><mo id="S4.Ex27.m1.2.2.2.4.1" xref="S4.Ex27.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex27.m1.2.2.2.2" xref="S4.Ex27.m1.2.2.2.2.cmml">p</mi></mrow><mrow id="S4.Ex27.m1.7.7.3.1.3.3" xref="S4.Ex27.m1.7.7.3.1.3.3.cmml"><mi id="S4.Ex27.m1.7.7.3.1.3.3.2" xref="S4.Ex27.m1.7.7.3.1.3.3.2.cmml">s</mi><mo id="S4.Ex27.m1.7.7.3.1.3.3.1" xref="S4.Ex27.m1.7.7.3.1.3.3.1.cmml">−</mo><mi id="S4.Ex27.m1.7.7.3.1.3.3.3" xref="S4.Ex27.m1.7.7.3.1.3.3.3.cmml">j</mi><mo id="S4.Ex27.m1.7.7.3.1.3.3.1a" xref="S4.Ex27.m1.7.7.3.1.3.3.1.cmml">−</mo><mfrac id="S4.Ex27.m1.7.7.3.1.3.3.4" 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id="S4.Ex27.m1.1.1.1.1.cmml" xref="S4.Ex27.m1.1.1.1.1">𝑝</ci><ci id="S4.Ex27.m1.2.2.2.2.cmml" xref="S4.Ex27.m1.2.2.2.2">𝑝</ci></list></apply><apply id="S4.Ex27.m1.7.7.3.1.3.3.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3"><minus id="S4.Ex27.m1.7.7.3.1.3.3.1.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.1"></minus><ci id="S4.Ex27.m1.7.7.3.1.3.3.2.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.2">𝑠</ci><ci id="S4.Ex27.m1.7.7.3.1.3.3.3.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.3">𝑗</ci><apply id="S4.Ex27.m1.7.7.3.1.3.3.4.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.4"><divide id="S4.Ex27.m1.7.7.3.1.3.3.4.1.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.4"></divide><apply id="S4.Ex27.m1.7.7.3.1.3.3.4.2.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.4.2"><plus id="S4.Ex27.m1.7.7.3.1.3.3.4.2.1.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.4.2.1"></plus><ci id="S4.Ex27.m1.7.7.3.1.3.3.4.2.2.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.4.2.2">𝛾</ci><cn id="S4.Ex27.m1.7.7.3.1.3.3.4.2.3.cmml" type="integer" xref="S4.Ex27.m1.7.7.3.1.3.3.4.2.3">1</cn></apply><ci id="S4.Ex27.m1.7.7.3.1.3.3.4.3.cmml" xref="S4.Ex27.m1.7.7.3.1.3.3.4.3">𝑝</ci></apply></apply></apply><list id="S4.Ex27.m1.7.7.3.1.1.2.cmml" xref="S4.Ex27.m1.7.7.3.1.1.1"><apply id="S4.Ex27.m1.7.7.3.1.1.1.1.cmml" xref="S4.Ex27.m1.7.7.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.Ex27.m1.7.7.3.1.1.1.1.1.cmml" xref="S4.Ex27.m1.7.7.3.1.1.1.1">superscript</csymbol><ci id="S4.Ex27.m1.7.7.3.1.1.1.1.2.cmml" xref="S4.Ex27.m1.7.7.3.1.1.1.1.2">ℝ</ci><apply id="S4.Ex27.m1.7.7.3.1.1.1.1.3.cmml" xref="S4.Ex27.m1.7.7.3.1.1.1.1.3"><minus id="S4.Ex27.m1.7.7.3.1.1.1.1.3.1.cmml" xref="S4.Ex27.m1.7.7.3.1.1.1.1.3.1"></minus><ci id="S4.Ex27.m1.7.7.3.1.1.1.1.3.2.cmml" xref="S4.Ex27.m1.7.7.3.1.1.1.1.3.2">𝑑</ci><cn id="S4.Ex27.m1.7.7.3.1.1.1.1.3.3.cmml" type="integer" xref="S4.Ex27.m1.7.7.3.1.1.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex27.m1.4.4.cmml" xref="S4.Ex27.m1.4.4">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex27.m1.7c">(g_{0},\dots,g_{m})\in\prod_{j=0}^{m}B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{% R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex27.m1.7d">( italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_g start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ∈ ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.9.p2.7">and define <math alttext="f_{0}:=\operatorname{ext}_{0}g_{0}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.6.m1.1"><semantics id="S4.SS1.9.p2.6.m1.1a"><mrow id="S4.SS1.9.p2.6.m1.1.1" xref="S4.SS1.9.p2.6.m1.1.1.cmml"><msub id="S4.SS1.9.p2.6.m1.1.1.2" xref="S4.SS1.9.p2.6.m1.1.1.2.cmml"><mi id="S4.SS1.9.p2.6.m1.1.1.2.2" xref="S4.SS1.9.p2.6.m1.1.1.2.2.cmml">f</mi><mn id="S4.SS1.9.p2.6.m1.1.1.2.3" xref="S4.SS1.9.p2.6.m1.1.1.2.3.cmml">0</mn></msub><mo id="S4.SS1.9.p2.6.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.9.p2.6.m1.1.1.1.cmml">:=</mo><mrow id="S4.SS1.9.p2.6.m1.1.1.3" xref="S4.SS1.9.p2.6.m1.1.1.3.cmml"><msub id="S4.SS1.9.p2.6.m1.1.1.3.1" xref="S4.SS1.9.p2.6.m1.1.1.3.1.cmml"><mi id="S4.SS1.9.p2.6.m1.1.1.3.1.2" xref="S4.SS1.9.p2.6.m1.1.1.3.1.2.cmml">ext</mi><mn id="S4.SS1.9.p2.6.m1.1.1.3.1.3" xref="S4.SS1.9.p2.6.m1.1.1.3.1.3.cmml">0</mn></msub><mo id="S4.SS1.9.p2.6.m1.1.1.3a" lspace="0.167em" xref="S4.SS1.9.p2.6.m1.1.1.3.cmml">⁡</mo><msub id="S4.SS1.9.p2.6.m1.1.1.3.2" xref="S4.SS1.9.p2.6.m1.1.1.3.2.cmml"><mi id="S4.SS1.9.p2.6.m1.1.1.3.2.2" xref="S4.SS1.9.p2.6.m1.1.1.3.2.2.cmml">g</mi><mn id="S4.SS1.9.p2.6.m1.1.1.3.2.3" xref="S4.SS1.9.p2.6.m1.1.1.3.2.3.cmml">0</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.6.m1.1b"><apply id="S4.SS1.9.p2.6.m1.1.1.cmml" xref="S4.SS1.9.p2.6.m1.1.1"><csymbol cd="latexml" id="S4.SS1.9.p2.6.m1.1.1.1.cmml" xref="S4.SS1.9.p2.6.m1.1.1.1">assign</csymbol><apply id="S4.SS1.9.p2.6.m1.1.1.2.cmml" xref="S4.SS1.9.p2.6.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.9.p2.6.m1.1.1.2.1.cmml" xref="S4.SS1.9.p2.6.m1.1.1.2">subscript</csymbol><ci id="S4.SS1.9.p2.6.m1.1.1.2.2.cmml" xref="S4.SS1.9.p2.6.m1.1.1.2.2">𝑓</ci><cn id="S4.SS1.9.p2.6.m1.1.1.2.3.cmml" type="integer" xref="S4.SS1.9.p2.6.m1.1.1.2.3">0</cn></apply><apply id="S4.SS1.9.p2.6.m1.1.1.3.cmml" xref="S4.SS1.9.p2.6.m1.1.1.3"><apply id="S4.SS1.9.p2.6.m1.1.1.3.1.cmml" xref="S4.SS1.9.p2.6.m1.1.1.3.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.6.m1.1.1.3.1.1.cmml" xref="S4.SS1.9.p2.6.m1.1.1.3.1">subscript</csymbol><ci id="S4.SS1.9.p2.6.m1.1.1.3.1.2.cmml" xref="S4.SS1.9.p2.6.m1.1.1.3.1.2">ext</ci><cn id="S4.SS1.9.p2.6.m1.1.1.3.1.3.cmml" type="integer" xref="S4.SS1.9.p2.6.m1.1.1.3.1.3">0</cn></apply><apply id="S4.SS1.9.p2.6.m1.1.1.3.2.cmml" xref="S4.SS1.9.p2.6.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.SS1.9.p2.6.m1.1.1.3.2.1.cmml" xref="S4.SS1.9.p2.6.m1.1.1.3.2">subscript</csymbol><ci id="S4.SS1.9.p2.6.m1.1.1.3.2.2.cmml" xref="S4.SS1.9.p2.6.m1.1.1.3.2.2">𝑔</ci><cn id="S4.SS1.9.p2.6.m1.1.1.3.2.3.cmml" type="integer" xref="S4.SS1.9.p2.6.m1.1.1.3.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.6.m1.1c">f_{0}:=\operatorname{ext}_{0}g_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.6.m1.1d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := roman_ext start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. For <math alttext="j\in\{1,\dots,m\}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.7.m2.3"><semantics id="S4.SS1.9.p2.7.m2.3a"><mrow id="S4.SS1.9.p2.7.m2.3.4" xref="S4.SS1.9.p2.7.m2.3.4.cmml"><mi id="S4.SS1.9.p2.7.m2.3.4.2" xref="S4.SS1.9.p2.7.m2.3.4.2.cmml">j</mi><mo id="S4.SS1.9.p2.7.m2.3.4.1" xref="S4.SS1.9.p2.7.m2.3.4.1.cmml">∈</mo><mrow id="S4.SS1.9.p2.7.m2.3.4.3.2" xref="S4.SS1.9.p2.7.m2.3.4.3.1.cmml"><mo id="S4.SS1.9.p2.7.m2.3.4.3.2.1" stretchy="false" xref="S4.SS1.9.p2.7.m2.3.4.3.1.cmml">{</mo><mn id="S4.SS1.9.p2.7.m2.1.1" xref="S4.SS1.9.p2.7.m2.1.1.cmml">1</mn><mo id="S4.SS1.9.p2.7.m2.3.4.3.2.2" xref="S4.SS1.9.p2.7.m2.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.7.m2.2.2" mathvariant="normal" xref="S4.SS1.9.p2.7.m2.2.2.cmml">…</mi><mo id="S4.SS1.9.p2.7.m2.3.4.3.2.3" xref="S4.SS1.9.p2.7.m2.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.7.m2.3.3" xref="S4.SS1.9.p2.7.m2.3.3.cmml">m</mi><mo id="S4.SS1.9.p2.7.m2.3.4.3.2.4" stretchy="false" xref="S4.SS1.9.p2.7.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.7.m2.3b"><apply id="S4.SS1.9.p2.7.m2.3.4.cmml" xref="S4.SS1.9.p2.7.m2.3.4"><in id="S4.SS1.9.p2.7.m2.3.4.1.cmml" xref="S4.SS1.9.p2.7.m2.3.4.1"></in><ci id="S4.SS1.9.p2.7.m2.3.4.2.cmml" xref="S4.SS1.9.p2.7.m2.3.4.2">𝑗</ci><set id="S4.SS1.9.p2.7.m2.3.4.3.1.cmml" xref="S4.SS1.9.p2.7.m2.3.4.3.2"><cn id="S4.SS1.9.p2.7.m2.1.1.cmml" type="integer" xref="S4.SS1.9.p2.7.m2.1.1">1</cn><ci id="S4.SS1.9.p2.7.m2.2.2.cmml" xref="S4.SS1.9.p2.7.m2.2.2">…</ci><ci id="S4.SS1.9.p2.7.m2.3.3.cmml" xref="S4.SS1.9.p2.7.m2.3.3">𝑚</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.7.m2.3c">j\in\{1,\dots,m\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.7.m2.3d">italic_j ∈ { 1 , … , italic_m }</annotation></semantics></math> we recursively define</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex28"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f_{j}:=f_{j-1}+\operatorname{ext}_{j}(g_{j}-\operatorname{Tr}_{j}f_{j-1})=h_{j% }f_{j-1}+\operatorname{ext}_{j}g_{j}," class="ltx_Math" display="block" id="S4.Ex28.m1.1"><semantics id="S4.Ex28.m1.1a"><mrow id="S4.Ex28.m1.1.1.1" xref="S4.Ex28.m1.1.1.1.1.cmml"><mrow id="S4.Ex28.m1.1.1.1.1" xref="S4.Ex28.m1.1.1.1.1.cmml"><msub id="S4.Ex28.m1.1.1.1.1.4" xref="S4.Ex28.m1.1.1.1.1.4.cmml"><mi id="S4.Ex28.m1.1.1.1.1.4.2" xref="S4.Ex28.m1.1.1.1.1.4.2.cmml">f</mi><mi id="S4.Ex28.m1.1.1.1.1.4.3" xref="S4.Ex28.m1.1.1.1.1.4.3.cmml">j</mi></msub><mo id="S4.Ex28.m1.1.1.1.1.5" lspace="0.278em" rspace="0.278em" xref="S4.Ex28.m1.1.1.1.1.5.cmml">:=</mo><mrow id="S4.Ex28.m1.1.1.1.1.2" xref="S4.Ex28.m1.1.1.1.1.2.cmml"><msub id="S4.Ex28.m1.1.1.1.1.2.4" xref="S4.Ex28.m1.1.1.1.1.2.4.cmml"><mi id="S4.Ex28.m1.1.1.1.1.2.4.2" xref="S4.Ex28.m1.1.1.1.1.2.4.2.cmml">f</mi><mrow id="S4.Ex28.m1.1.1.1.1.2.4.3" xref="S4.Ex28.m1.1.1.1.1.2.4.3.cmml"><mi id="S4.Ex28.m1.1.1.1.1.2.4.3.2" xref="S4.Ex28.m1.1.1.1.1.2.4.3.2.cmml">j</mi><mo id="S4.Ex28.m1.1.1.1.1.2.4.3.1" xref="S4.Ex28.m1.1.1.1.1.2.4.3.1.cmml">−</mo><mn id="S4.Ex28.m1.1.1.1.1.2.4.3.3" xref="S4.Ex28.m1.1.1.1.1.2.4.3.3.cmml">1</mn></mrow></msub><mo id="S4.Ex28.m1.1.1.1.1.2.3" xref="S4.Ex28.m1.1.1.1.1.2.3.cmml">+</mo><mrow id="S4.Ex28.m1.1.1.1.1.2.2.2" xref="S4.Ex28.m1.1.1.1.1.2.2.3.cmml"><msub id="S4.Ex28.m1.1.1.1.1.1.1.1.1" xref="S4.Ex28.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S4.Ex28.m1.1.1.1.1.1.1.1.1.2" xref="S4.Ex28.m1.1.1.1.1.1.1.1.1.2.cmml">ext</mi><mi id="S4.Ex28.m1.1.1.1.1.1.1.1.1.3" xref="S4.Ex28.m1.1.1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S4.Ex28.m1.1.1.1.1.2.2.2a" xref="S4.Ex28.m1.1.1.1.1.2.2.3.cmml">⁡</mo><mrow id="S4.Ex28.m1.1.1.1.1.2.2.2.2" xref="S4.Ex28.m1.1.1.1.1.2.2.3.cmml"><mo id="S4.Ex28.m1.1.1.1.1.2.2.2.2.2" stretchy="false" 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id="S4.Ex28.m1.1c">f_{j}:=f_{j-1}+\operatorname{ext}_{j}(g_{j}-\operatorname{Tr}_{j}f_{j-1})=h_{j% }f_{j-1}+\operatorname{ext}_{j}g_{j},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex28.m1.1d">italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT := italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT + roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT ) = italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT + roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.9.p2.9">where <math alttext="h_{j}:=1-\operatorname{ext}_{j}\mathop{\circ}\nolimits\operatorname{Tr}_{j}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.8.m1.1"><semantics id="S4.SS1.9.p2.8.m1.1a"><mrow id="S4.SS1.9.p2.8.m1.1.1" xref="S4.SS1.9.p2.8.m1.1.1.cmml"><msub id="S4.SS1.9.p2.8.m1.1.1.2" xref="S4.SS1.9.p2.8.m1.1.1.2.cmml"><mi id="S4.SS1.9.p2.8.m1.1.1.2.2" xref="S4.SS1.9.p2.8.m1.1.1.2.2.cmml">h</mi><mi id="S4.SS1.9.p2.8.m1.1.1.2.3" xref="S4.SS1.9.p2.8.m1.1.1.2.3.cmml">j</mi></msub><mo id="S4.SS1.9.p2.8.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S4.SS1.9.p2.8.m1.1.1.1.cmml">:=</mo><mrow id="S4.SS1.9.p2.8.m1.1.1.3" xref="S4.SS1.9.p2.8.m1.1.1.3.cmml"><mn id="S4.SS1.9.p2.8.m1.1.1.3.2" xref="S4.SS1.9.p2.8.m1.1.1.3.2.cmml">1</mn><mo id="S4.SS1.9.p2.8.m1.1.1.3.1" xref="S4.SS1.9.p2.8.m1.1.1.3.1.cmml">−</mo><mrow id="S4.SS1.9.p2.8.m1.1.1.3.3" xref="S4.SS1.9.p2.8.m1.1.1.3.3.cmml"><msub id="S4.SS1.9.p2.8.m1.1.1.3.3.2" xref="S4.SS1.9.p2.8.m1.1.1.3.3.2.cmml"><mi id="S4.SS1.9.p2.8.m1.1.1.3.3.2.2" xref="S4.SS1.9.p2.8.m1.1.1.3.3.2.2.cmml">ext</mi><mi id="S4.SS1.9.p2.8.m1.1.1.3.3.2.3" xref="S4.SS1.9.p2.8.m1.1.1.3.3.2.3.cmml">j</mi></msub><mo id="S4.SS1.9.p2.8.m1.1.1.3.3.1" lspace="0.167em" xref="S4.SS1.9.p2.8.m1.1.1.3.3.1.cmml">⁢</mo><mrow id="S4.SS1.9.p2.8.m1.1.1.3.3.3" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.cmml"><mo id="S4.SS1.9.p2.8.m1.1.1.3.3.3.1" rspace="0.167em" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.1.cmml">∘</mo><msub id="S4.SS1.9.p2.8.m1.1.1.3.3.3.2" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.cmml"><mi id="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.2" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.2.cmml">Tr</mi><mi id="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.3" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.3.cmml">j</mi></msub></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.8.m1.1b"><apply id="S4.SS1.9.p2.8.m1.1.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1"><csymbol cd="latexml" id="S4.SS1.9.p2.8.m1.1.1.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1.1">assign</csymbol><apply id="S4.SS1.9.p2.8.m1.1.1.2.cmml" xref="S4.SS1.9.p2.8.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS1.9.p2.8.m1.1.1.2.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1.2">subscript</csymbol><ci id="S4.SS1.9.p2.8.m1.1.1.2.2.cmml" xref="S4.SS1.9.p2.8.m1.1.1.2.2">ℎ</ci><ci id="S4.SS1.9.p2.8.m1.1.1.2.3.cmml" xref="S4.SS1.9.p2.8.m1.1.1.2.3">𝑗</ci></apply><apply id="S4.SS1.9.p2.8.m1.1.1.3.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3"><minus id="S4.SS1.9.p2.8.m1.1.1.3.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.1"></minus><cn id="S4.SS1.9.p2.8.m1.1.1.3.2.cmml" type="integer" xref="S4.SS1.9.p2.8.m1.1.1.3.2">1</cn><apply id="S4.SS1.9.p2.8.m1.1.1.3.3.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3"><times id="S4.SS1.9.p2.8.m1.1.1.3.3.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.1"></times><apply id="S4.SS1.9.p2.8.m1.1.1.3.3.2.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.9.p2.8.m1.1.1.3.3.2.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.2">subscript</csymbol><ci id="S4.SS1.9.p2.8.m1.1.1.3.3.2.2.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.2.2">ext</ci><ci id="S4.SS1.9.p2.8.m1.1.1.3.3.2.3.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.2.3">𝑗</ci></apply><apply id="S4.SS1.9.p2.8.m1.1.1.3.3.3.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3"><compose id="S4.SS1.9.p2.8.m1.1.1.3.3.3.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.1"></compose><apply id="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.2"><csymbol cd="ambiguous" id="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.1.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.2">subscript</csymbol><ci id="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.2.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.2">Tr</ci><ci id="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.3.cmml" xref="S4.SS1.9.p2.8.m1.1.1.3.3.3.2.3">𝑗</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.8.m1.1c">h_{j}:=1-\operatorname{ext}_{j}\mathop{\circ}\nolimits\operatorname{Tr}_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.8.m1.1d">italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT := 1 - roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> is a bounded operator on <math alttext="H^{s,p}(\mathcal{O},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS1.9.p2.9.m2.5"><semantics id="S4.SS1.9.p2.9.m2.5a"><mrow id="S4.SS1.9.p2.9.m2.5.5" xref="S4.SS1.9.p2.9.m2.5.5.cmml"><msup id="S4.SS1.9.p2.9.m2.5.5.3" xref="S4.SS1.9.p2.9.m2.5.5.3.cmml"><mi id="S4.SS1.9.p2.9.m2.5.5.3.2" xref="S4.SS1.9.p2.9.m2.5.5.3.2.cmml">H</mi><mrow id="S4.SS1.9.p2.9.m2.2.2.2.4" xref="S4.SS1.9.p2.9.m2.2.2.2.3.cmml"><mi id="S4.SS1.9.p2.9.m2.1.1.1.1" xref="S4.SS1.9.p2.9.m2.1.1.1.1.cmml">s</mi><mo id="S4.SS1.9.p2.9.m2.2.2.2.4.1" xref="S4.SS1.9.p2.9.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS1.9.p2.9.m2.2.2.2.2" xref="S4.SS1.9.p2.9.m2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS1.9.p2.9.m2.5.5.2" xref="S4.SS1.9.p2.9.m2.5.5.2.cmml">⁢</mo><mrow id="S4.SS1.9.p2.9.m2.5.5.1.1" xref="S4.SS1.9.p2.9.m2.5.5.1.2.cmml"><mo id="S4.SS1.9.p2.9.m2.5.5.1.1.2" stretchy="false" xref="S4.SS1.9.p2.9.m2.5.5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.SS1.9.p2.9.m2.3.3" xref="S4.SS1.9.p2.9.m2.3.3.cmml">𝒪</mi><mo id="S4.SS1.9.p2.9.m2.5.5.1.1.3" xref="S4.SS1.9.p2.9.m2.5.5.1.2.cmml">,</mo><msub id="S4.SS1.9.p2.9.m2.5.5.1.1.1" xref="S4.SS1.9.p2.9.m2.5.5.1.1.1.cmml"><mi id="S4.SS1.9.p2.9.m2.5.5.1.1.1.2" xref="S4.SS1.9.p2.9.m2.5.5.1.1.1.2.cmml">w</mi><mi id="S4.SS1.9.p2.9.m2.5.5.1.1.1.3" xref="S4.SS1.9.p2.9.m2.5.5.1.1.1.3.cmml">γ</mi></msub><mo id="S4.SS1.9.p2.9.m2.5.5.1.1.4" xref="S4.SS1.9.p2.9.m2.5.5.1.2.cmml">;</mo><mi id="S4.SS1.9.p2.9.m2.4.4" xref="S4.SS1.9.p2.9.m2.4.4.cmml">X</mi><mo id="S4.SS1.9.p2.9.m2.5.5.1.1.5" stretchy="false" xref="S4.SS1.9.p2.9.m2.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.9.m2.5b"><apply id="S4.SS1.9.p2.9.m2.5.5.cmml" xref="S4.SS1.9.p2.9.m2.5.5"><times id="S4.SS1.9.p2.9.m2.5.5.2.cmml" xref="S4.SS1.9.p2.9.m2.5.5.2"></times><apply id="S4.SS1.9.p2.9.m2.5.5.3.cmml" xref="S4.SS1.9.p2.9.m2.5.5.3"><csymbol cd="ambiguous" id="S4.SS1.9.p2.9.m2.5.5.3.1.cmml" xref="S4.SS1.9.p2.9.m2.5.5.3">superscript</csymbol><ci id="S4.SS1.9.p2.9.m2.5.5.3.2.cmml" xref="S4.SS1.9.p2.9.m2.5.5.3.2">𝐻</ci><list id="S4.SS1.9.p2.9.m2.2.2.2.3.cmml" xref="S4.SS1.9.p2.9.m2.2.2.2.4"><ci id="S4.SS1.9.p2.9.m2.1.1.1.1.cmml" xref="S4.SS1.9.p2.9.m2.1.1.1.1">𝑠</ci><ci id="S4.SS1.9.p2.9.m2.2.2.2.2.cmml" xref="S4.SS1.9.p2.9.m2.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS1.9.p2.9.m2.5.5.1.2.cmml" xref="S4.SS1.9.p2.9.m2.5.5.1.1"><ci id="S4.SS1.9.p2.9.m2.3.3.cmml" xref="S4.SS1.9.p2.9.m2.3.3">𝒪</ci><apply id="S4.SS1.9.p2.9.m2.5.5.1.1.1.cmml" xref="S4.SS1.9.p2.9.m2.5.5.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.9.m2.5.5.1.1.1.1.cmml" xref="S4.SS1.9.p2.9.m2.5.5.1.1.1">subscript</csymbol><ci id="S4.SS1.9.p2.9.m2.5.5.1.1.1.2.cmml" xref="S4.SS1.9.p2.9.m2.5.5.1.1.1.2">𝑤</ci><ci id="S4.SS1.9.p2.9.m2.5.5.1.1.1.3.cmml" xref="S4.SS1.9.p2.9.m2.5.5.1.1.1.3">𝛾</ci></apply><ci id="S4.SS1.9.p2.9.m2.4.4.cmml" xref="S4.SS1.9.p2.9.m2.4.4">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.9.m2.5c">H^{s,p}(\mathcal{O},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.9.m2.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>. It is straightforward to verify that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f_{m}=\operatorname{ext}_{m}g_{m}+\sum_{j=0}^{m-1}\Big{(}\prod_{n=j+1}^{m}h_{n% }\Big{)}\operatorname{ext}_{j}g_{j}." class="ltx_Math" display="block" id="S4.E3.m1.1"><semantics id="S4.E3.m1.1a"><mrow id="S4.E3.m1.1.1.1" xref="S4.E3.m1.1.1.1.1.cmml"><mrow id="S4.E3.m1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.cmml"><msub id="S4.E3.m1.1.1.1.1.3" xref="S4.E3.m1.1.1.1.1.3.cmml"><mi id="S4.E3.m1.1.1.1.1.3.2" xref="S4.E3.m1.1.1.1.1.3.2.cmml">f</mi><mi id="S4.E3.m1.1.1.1.1.3.3" xref="S4.E3.m1.1.1.1.1.3.3.cmml">m</mi></msub><mo id="S4.E3.m1.1.1.1.1.2" xref="S4.E3.m1.1.1.1.1.2.cmml">=</mo><mrow id="S4.E3.m1.1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.1.cmml"><mrow id="S4.E3.m1.1.1.1.1.1.3" xref="S4.E3.m1.1.1.1.1.1.3.cmml"><msub id="S4.E3.m1.1.1.1.1.1.3.1" xref="S4.E3.m1.1.1.1.1.1.3.1.cmml"><mi id="S4.E3.m1.1.1.1.1.1.3.1.2" xref="S4.E3.m1.1.1.1.1.1.3.1.2.cmml">ext</mi><mi id="S4.E3.m1.1.1.1.1.1.3.1.3" xref="S4.E3.m1.1.1.1.1.1.3.1.3.cmml">m</mi></msub><mo id="S4.E3.m1.1.1.1.1.1.3a" lspace="0.167em" xref="S4.E3.m1.1.1.1.1.1.3.cmml">⁡</mo><msub id="S4.E3.m1.1.1.1.1.1.3.2" xref="S4.E3.m1.1.1.1.1.1.3.2.cmml"><mi id="S4.E3.m1.1.1.1.1.1.3.2.2" xref="S4.E3.m1.1.1.1.1.1.3.2.2.cmml">g</mi><mi id="S4.E3.m1.1.1.1.1.1.3.2.3" xref="S4.E3.m1.1.1.1.1.1.3.2.3.cmml">m</mi></msub></mrow><mo id="S4.E3.m1.1.1.1.1.1.2" rspace="0.055em" xref="S4.E3.m1.1.1.1.1.1.2.cmml">+</mo><mrow id="S4.E3.m1.1.1.1.1.1.1" xref="S4.E3.m1.1.1.1.1.1.1.cmml"><munderover id="S4.E3.m1.1.1.1.1.1.1.2" xref="S4.E3.m1.1.1.1.1.1.1.2.cmml"><mo id="S4.E3.m1.1.1.1.1.1.1.2.2.2" movablelimits="false" rspace="0em" xref="S4.E3.m1.1.1.1.1.1.1.2.2.2.cmml">∑</mo><mrow id="S4.E3.m1.1.1.1.1.1.1.2.2.3" xref="S4.E3.m1.1.1.1.1.1.1.2.2.3.cmml"><mi 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xref="S4.E3.m1.1.1.1.1.1.1.1.3.2.3">𝑗</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E3.m1.1c">f_{m}=\operatorname{ext}_{m}g_{m}+\sum_{j=0}^{m-1}\Big{(}\prod_{n=j+1}^{m}h_{n% }\Big{)}\operatorname{ext}_{j}g_{j}.</annotation><annotation encoding="application/x-llamapun" id="S4.E3.m1.1d">italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ( ∏ start_POSTSUBSCRIPT italic_n = italic_j + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_h start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS1.9.p2.10">Moreover, note that from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> we have for <math alttext="j\in\{0,\dots,m\}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.10.m1.3"><semantics id="S4.SS1.9.p2.10.m1.3a"><mrow id="S4.SS1.9.p2.10.m1.3.4" xref="S4.SS1.9.p2.10.m1.3.4.cmml"><mi id="S4.SS1.9.p2.10.m1.3.4.2" xref="S4.SS1.9.p2.10.m1.3.4.2.cmml">j</mi><mo id="S4.SS1.9.p2.10.m1.3.4.1" xref="S4.SS1.9.p2.10.m1.3.4.1.cmml">∈</mo><mrow id="S4.SS1.9.p2.10.m1.3.4.3.2" xref="S4.SS1.9.p2.10.m1.3.4.3.1.cmml"><mo id="S4.SS1.9.p2.10.m1.3.4.3.2.1" stretchy="false" xref="S4.SS1.9.p2.10.m1.3.4.3.1.cmml">{</mo><mn id="S4.SS1.9.p2.10.m1.1.1" xref="S4.SS1.9.p2.10.m1.1.1.cmml">0</mn><mo id="S4.SS1.9.p2.10.m1.3.4.3.2.2" xref="S4.SS1.9.p2.10.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.10.m1.2.2" mathvariant="normal" xref="S4.SS1.9.p2.10.m1.2.2.cmml">…</mi><mo id="S4.SS1.9.p2.10.m1.3.4.3.2.3" xref="S4.SS1.9.p2.10.m1.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.10.m1.3.3" xref="S4.SS1.9.p2.10.m1.3.3.cmml">m</mi><mo id="S4.SS1.9.p2.10.m1.3.4.3.2.4" stretchy="false" xref="S4.SS1.9.p2.10.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.10.m1.3b"><apply id="S4.SS1.9.p2.10.m1.3.4.cmml" xref="S4.SS1.9.p2.10.m1.3.4"><in id="S4.SS1.9.p2.10.m1.3.4.1.cmml" xref="S4.SS1.9.p2.10.m1.3.4.1"></in><ci id="S4.SS1.9.p2.10.m1.3.4.2.cmml" xref="S4.SS1.9.p2.10.m1.3.4.2">𝑗</ci><set id="S4.SS1.9.p2.10.m1.3.4.3.1.cmml" xref="S4.SS1.9.p2.10.m1.3.4.3.2"><cn id="S4.SS1.9.p2.10.m1.1.1.cmml" type="integer" xref="S4.SS1.9.p2.10.m1.1.1">0</cn><ci id="S4.SS1.9.p2.10.m1.2.2.cmml" xref="S4.SS1.9.p2.10.m1.2.2">…</ci><ci id="S4.SS1.9.p2.10.m1.3.3.cmml" xref="S4.SS1.9.p2.10.m1.3.3">𝑚</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.10.m1.3c">j\in\{0,\dots,m\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.10.m1.3d">italic_j ∈ { 0 , … , italic_m }</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E4"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E4X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{Tr}_{j}\mathop{\circ}\nolimits h_{j}" class="ltx_Math" display="inline" id="S4.E4X.2.1.1.m1.1"><semantics id="S4.E4X.2.1.1.m1.1a"><mrow id="S4.E4X.2.1.1.m1.1.1" xref="S4.E4X.2.1.1.m1.1.1.cmml"><msub id="S4.E4X.2.1.1.m1.1.1.2" xref="S4.E4X.2.1.1.m1.1.1.2.cmml"><mi id="S4.E4X.2.1.1.m1.1.1.2.2" xref="S4.E4X.2.1.1.m1.1.1.2.2.cmml">Tr</mi><mi id="S4.E4X.2.1.1.m1.1.1.2.3" xref="S4.E4X.2.1.1.m1.1.1.2.3.cmml">j</mi></msub><mo id="S4.E4X.2.1.1.m1.1.1.1" lspace="0.167em" xref="S4.E4X.2.1.1.m1.1.1.1.cmml">⁢</mo><mrow id="S4.E4X.2.1.1.m1.1.1.3" xref="S4.E4X.2.1.1.m1.1.1.3.cmml"><mo id="S4.E4X.2.1.1.m1.1.1.3.1" rspace="0.167em" xref="S4.E4X.2.1.1.m1.1.1.3.1.cmml">∘</mo><msub id="S4.E4X.2.1.1.m1.1.1.3.2" xref="S4.E4X.2.1.1.m1.1.1.3.2.cmml"><mi id="S4.E4X.2.1.1.m1.1.1.3.2.2" xref="S4.E4X.2.1.1.m1.1.1.3.2.2.cmml">h</mi><mi id="S4.E4X.2.1.1.m1.1.1.3.2.3" xref="S4.E4X.2.1.1.m1.1.1.3.2.3.cmml">j</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E4X.2.1.1.m1.1b"><apply id="S4.E4X.2.1.1.m1.1.1.cmml" xref="S4.E4X.2.1.1.m1.1.1"><times id="S4.E4X.2.1.1.m1.1.1.1.cmml" xref="S4.E4X.2.1.1.m1.1.1.1"></times><apply id="S4.E4X.2.1.1.m1.1.1.2.cmml" xref="S4.E4X.2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.E4X.2.1.1.m1.1.1.2.1.cmml" xref="S4.E4X.2.1.1.m1.1.1.2">subscript</csymbol><ci id="S4.E4X.2.1.1.m1.1.1.2.2.cmml" xref="S4.E4X.2.1.1.m1.1.1.2.2">Tr</ci><ci id="S4.E4X.2.1.1.m1.1.1.2.3.cmml" xref="S4.E4X.2.1.1.m1.1.1.2.3">𝑗</ci></apply><apply id="S4.E4X.2.1.1.m1.1.1.3.cmml" xref="S4.E4X.2.1.1.m1.1.1.3"><compose id="S4.E4X.2.1.1.m1.1.1.3.1.cmml" xref="S4.E4X.2.1.1.m1.1.1.3.1"></compose><apply id="S4.E4X.2.1.1.m1.1.1.3.2.cmml" xref="S4.E4X.2.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.E4X.2.1.1.m1.1.1.3.2.1.cmml" xref="S4.E4X.2.1.1.m1.1.1.3.2">subscript</csymbol><ci id="S4.E4X.2.1.1.m1.1.1.3.2.2.cmml" xref="S4.E4X.2.1.1.m1.1.1.3.2.2">ℎ</ci><ci id="S4.E4X.2.1.1.m1.1.1.3.2.3.cmml" xref="S4.E4X.2.1.1.m1.1.1.3.2.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4X.2.1.1.m1.1c">\displaystyle\operatorname{Tr}_{j}\mathop{\circ}\nolimits h_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.E4X.2.1.1.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{Tr}_{j}\mathop{\circ}\nolimits\,(1-\operatorname{% ext}_{j}\mathop{\circ}\nolimits\operatorname{Tr}_{j})=\operatorname{Tr}_{j}-% \operatorname{Tr}_{j}=0\quad\text{ and }" 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xref="S4.E4X.3.2.2.m1.2.2.1.1.6.2.3">𝑗</ci></apply><apply id="S4.E4X.3.2.2.m1.2.2.1.1.6.3.cmml" xref="S4.E4X.3.2.2.m1.2.2.1.1.6.3"><csymbol cd="ambiguous" id="S4.E4X.3.2.2.m1.2.2.1.1.6.3.1.cmml" xref="S4.E4X.3.2.2.m1.2.2.1.1.6.3">subscript</csymbol><ci id="S4.E4X.3.2.2.m1.2.2.1.1.6.3.2.cmml" xref="S4.E4X.3.2.2.m1.2.2.1.1.6.3.2">Tr</ci><ci id="S4.E4X.3.2.2.m1.2.2.1.1.6.3.3.cmml" xref="S4.E4X.3.2.2.m1.2.2.1.1.6.3.3">𝑗</ci></apply></apply></apply><apply id="S4.E4X.3.2.2.m1.2.2.1.1e.cmml" xref="S4.E4X.3.2.2.m1.2.2.1.1"><eq id="S4.E4X.3.2.2.m1.2.2.1.1.7.cmml" xref="S4.E4X.3.2.2.m1.2.2.1.1.7"></eq><share href="https://arxiv.org/html/2503.14636v1#S4.E4X.3.2.2.m1.2.2.1.1.6.cmml" id="S4.E4X.3.2.2.m1.2.2.1.1f.cmml" xref="S4.E4X.3.2.2.m1.2.2.1.1"></share><cn id="S4.E4X.3.2.2.m1.2.2.1.1.8.cmml" type="integer" xref="S4.E4X.3.2.2.m1.2.2.1.1.8">0</cn></apply></apply><ci id="S4.E4X.3.2.2.m1.1.1a.cmml" xref="S4.E4X.3.2.2.m1.1.1"><mtext id="S4.E4X.3.2.2.m1.1.1.cmml" xref="S4.E4X.3.2.2.m1.1.1"> and</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4X.3.2.2.m1.2c">\displaystyle=\operatorname{Tr}_{j}\mathop{\circ}\nolimits\,(1-\operatorname{% ext}_{j}\mathop{\circ}\nolimits\operatorname{Tr}_{j})=\operatorname{Tr}_{j}-% \operatorname{Tr}_{j}=0\quad\text{ and }</annotation><annotation encoding="application/x-llamapun" id="S4.E4X.3.2.2.m1.2d">= roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ ( 1 - roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0 and</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(4.4)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row 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xref="S4.E4Xa.2.1.1.m1.1.1.2.3.2">𝑗</ci><cn id="S4.E4Xa.2.1.1.m1.1.1.2.3.3.cmml" type="integer" xref="S4.E4Xa.2.1.1.m1.1.1.2.3.3">1</cn></apply></apply><apply id="S4.E4Xa.2.1.1.m1.1.1.3.cmml" xref="S4.E4Xa.2.1.1.m1.1.1.3"><compose id="S4.E4Xa.2.1.1.m1.1.1.3.1.cmml" xref="S4.E4Xa.2.1.1.m1.1.1.3.1"></compose><apply id="S4.E4Xa.2.1.1.m1.1.1.3.2.cmml" xref="S4.E4Xa.2.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S4.E4Xa.2.1.1.m1.1.1.3.2.1.cmml" xref="S4.E4Xa.2.1.1.m1.1.1.3.2">subscript</csymbol><ci id="S4.E4Xa.2.1.1.m1.1.1.3.2.2.cmml" xref="S4.E4Xa.2.1.1.m1.1.1.3.2.2">ℎ</ci><ci id="S4.E4Xa.2.1.1.m1.1.1.3.2.3.cmml" xref="S4.E4Xa.2.1.1.m1.1.1.3.2.3">𝑗</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4Xa.2.1.1.m1.1c">\displaystyle\operatorname{Tr}_{j_{1}}\mathop{\circ}\nolimits h_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.E4Xa.2.1.1.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT 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id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.3.3.1.cmml" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.3.3">subscript</csymbol><ci id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.3.3.2.cmml" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.3.3.2">𝑗</ci><cn id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.3.3.3.cmml" type="integer" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.3.3.3">1</cn></apply></apply><set id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.2.cmml" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1"><cn id="S4.E4Xa.3.2.2.m1.1.1.cmml" type="integer" xref="S4.E4Xa.3.2.2.m1.1.1">0</cn><ci id="S4.E4Xa.3.2.2.m1.2.2.cmml" xref="S4.E4Xa.3.2.2.m1.2.2">…</ci><apply id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1.cmml" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1"><minus id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1.1.cmml" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1.1"></minus><ci id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1.2.cmml" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1.2">𝑗</ci><cn id="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1.3.cmml" type="integer" xref="S4.E4Xa.3.2.2.m1.3.3.1.1.2.2.1.1.1.3">1</cn></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4Xa.3.2.2.m1.3c">\displaystyle=\operatorname{Tr}_{j_{1}}\mathop{\circ}\nolimits\,(1-% \operatorname{ext}_{j}\mathop{\circ}\nolimits\operatorname{Tr}_{j})=% \operatorname{Tr}_{j_{1}}\qquad\text{ for }j_{1}\in\{0,\dots,j-1\}.</annotation><annotation encoding="application/x-llamapun" id="S4.E4Xa.3.2.2.m1.3d">= roman_Tr start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∘ ( 1 - roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) = roman_Tr start_POSTSUBSCRIPT italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT for italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ { 0 , … , italic_j - 1 } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.SS1.9.p2.13">From (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E3" title="In Proof. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.3</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E4" title="In Proof. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a>) it follows that <math alttext="\operatorname{Tr}_{j}f_{m}=g_{j}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.11.m1.1"><semantics id="S4.SS1.9.p2.11.m1.1a"><mrow id="S4.SS1.9.p2.11.m1.1.1" xref="S4.SS1.9.p2.11.m1.1.1.cmml"><mrow id="S4.SS1.9.p2.11.m1.1.1.2" xref="S4.SS1.9.p2.11.m1.1.1.2.cmml"><msub id="S4.SS1.9.p2.11.m1.1.1.2.1" xref="S4.SS1.9.p2.11.m1.1.1.2.1.cmml"><mi id="S4.SS1.9.p2.11.m1.1.1.2.1.2" xref="S4.SS1.9.p2.11.m1.1.1.2.1.2.cmml">Tr</mi><mi id="S4.SS1.9.p2.11.m1.1.1.2.1.3" xref="S4.SS1.9.p2.11.m1.1.1.2.1.3.cmml">j</mi></msub><mo id="S4.SS1.9.p2.11.m1.1.1.2a" lspace="0.167em" xref="S4.SS1.9.p2.11.m1.1.1.2.cmml">⁡</mo><msub id="S4.SS1.9.p2.11.m1.1.1.2.2" xref="S4.SS1.9.p2.11.m1.1.1.2.2.cmml"><mi id="S4.SS1.9.p2.11.m1.1.1.2.2.2" xref="S4.SS1.9.p2.11.m1.1.1.2.2.2.cmml">f</mi><mi id="S4.SS1.9.p2.11.m1.1.1.2.2.3" xref="S4.SS1.9.p2.11.m1.1.1.2.2.3.cmml">m</mi></msub></mrow><mo id="S4.SS1.9.p2.11.m1.1.1.1" xref="S4.SS1.9.p2.11.m1.1.1.1.cmml">=</mo><msub id="S4.SS1.9.p2.11.m1.1.1.3" xref="S4.SS1.9.p2.11.m1.1.1.3.cmml"><mi id="S4.SS1.9.p2.11.m1.1.1.3.2" xref="S4.SS1.9.p2.11.m1.1.1.3.2.cmml">g</mi><mi id="S4.SS1.9.p2.11.m1.1.1.3.3" xref="S4.SS1.9.p2.11.m1.1.1.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.11.m1.1b"><apply id="S4.SS1.9.p2.11.m1.1.1.cmml" xref="S4.SS1.9.p2.11.m1.1.1"><eq id="S4.SS1.9.p2.11.m1.1.1.1.cmml" xref="S4.SS1.9.p2.11.m1.1.1.1"></eq><apply id="S4.SS1.9.p2.11.m1.1.1.2.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2"><apply id="S4.SS1.9.p2.11.m1.1.1.2.1.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.11.m1.1.1.2.1.1.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.1">subscript</csymbol><ci id="S4.SS1.9.p2.11.m1.1.1.2.1.2.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.1.2">Tr</ci><ci id="S4.SS1.9.p2.11.m1.1.1.2.1.3.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.1.3">𝑗</ci></apply><apply id="S4.SS1.9.p2.11.m1.1.1.2.2.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS1.9.p2.11.m1.1.1.2.2.1.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.2">subscript</csymbol><ci id="S4.SS1.9.p2.11.m1.1.1.2.2.2.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.2.2">𝑓</ci><ci id="S4.SS1.9.p2.11.m1.1.1.2.2.3.cmml" xref="S4.SS1.9.p2.11.m1.1.1.2.2.3">𝑚</ci></apply></apply><apply id="S4.SS1.9.p2.11.m1.1.1.3.cmml" xref="S4.SS1.9.p2.11.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS1.9.p2.11.m1.1.1.3.1.cmml" xref="S4.SS1.9.p2.11.m1.1.1.3">subscript</csymbol><ci id="S4.SS1.9.p2.11.m1.1.1.3.2.cmml" xref="S4.SS1.9.p2.11.m1.1.1.3.2">𝑔</ci><ci id="S4.SS1.9.p2.11.m1.1.1.3.3.cmml" xref="S4.SS1.9.p2.11.m1.1.1.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.11.m1.1c">\operatorname{Tr}_{j}f_{m}=g_{j}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.11.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT = italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="j\in\{0,\dots,m\}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.12.m2.3"><semantics id="S4.SS1.9.p2.12.m2.3a"><mrow id="S4.SS1.9.p2.12.m2.3.4" xref="S4.SS1.9.p2.12.m2.3.4.cmml"><mi id="S4.SS1.9.p2.12.m2.3.4.2" xref="S4.SS1.9.p2.12.m2.3.4.2.cmml">j</mi><mo id="S4.SS1.9.p2.12.m2.3.4.1" xref="S4.SS1.9.p2.12.m2.3.4.1.cmml">∈</mo><mrow id="S4.SS1.9.p2.12.m2.3.4.3.2" xref="S4.SS1.9.p2.12.m2.3.4.3.1.cmml"><mo id="S4.SS1.9.p2.12.m2.3.4.3.2.1" stretchy="false" xref="S4.SS1.9.p2.12.m2.3.4.3.1.cmml">{</mo><mn id="S4.SS1.9.p2.12.m2.1.1" xref="S4.SS1.9.p2.12.m2.1.1.cmml">0</mn><mo id="S4.SS1.9.p2.12.m2.3.4.3.2.2" xref="S4.SS1.9.p2.12.m2.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.12.m2.2.2" mathvariant="normal" xref="S4.SS1.9.p2.12.m2.2.2.cmml">…</mi><mo id="S4.SS1.9.p2.12.m2.3.4.3.2.3" xref="S4.SS1.9.p2.12.m2.3.4.3.1.cmml">,</mo><mi id="S4.SS1.9.p2.12.m2.3.3" xref="S4.SS1.9.p2.12.m2.3.3.cmml">m</mi><mo id="S4.SS1.9.p2.12.m2.3.4.3.2.4" stretchy="false" xref="S4.SS1.9.p2.12.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.12.m2.3b"><apply id="S4.SS1.9.p2.12.m2.3.4.cmml" xref="S4.SS1.9.p2.12.m2.3.4"><in id="S4.SS1.9.p2.12.m2.3.4.1.cmml" xref="S4.SS1.9.p2.12.m2.3.4.1"></in><ci id="S4.SS1.9.p2.12.m2.3.4.2.cmml" xref="S4.SS1.9.p2.12.m2.3.4.2">𝑗</ci><set id="S4.SS1.9.p2.12.m2.3.4.3.1.cmml" xref="S4.SS1.9.p2.12.m2.3.4.3.2"><cn id="S4.SS1.9.p2.12.m2.1.1.cmml" type="integer" xref="S4.SS1.9.p2.12.m2.1.1">0</cn><ci id="S4.SS1.9.p2.12.m2.2.2.cmml" xref="S4.SS1.9.p2.12.m2.2.2">…</ci><ci id="S4.SS1.9.p2.12.m2.3.3.cmml" xref="S4.SS1.9.p2.12.m2.3.3">𝑚</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.12.m2.3c">j\in\{0,\dots,m\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.12.m2.3d">italic_j ∈ { 0 , … , italic_m }</annotation></semantics></math>. Hence, we can define <math alttext="\overline{\operatorname{ext}}_{m}(g_{0},\dots,g_{m}):=f_{m}" class="ltx_Math" display="inline" id="S4.SS1.9.p2.13.m3.3"><semantics id="S4.SS1.9.p2.13.m3.3a"><mrow id="S4.SS1.9.p2.13.m3.3.3" xref="S4.SS1.9.p2.13.m3.3.3.cmml"><mrow id="S4.SS1.9.p2.13.m3.3.3.2" xref="S4.SS1.9.p2.13.m3.3.3.2.cmml"><msub id="S4.SS1.9.p2.13.m3.3.3.2.4" xref="S4.SS1.9.p2.13.m3.3.3.2.4.cmml"><mover accent="true" id="S4.SS1.9.p2.13.m3.3.3.2.4.2" xref="S4.SS1.9.p2.13.m3.3.3.2.4.2.cmml"><mi id="S4.SS1.9.p2.13.m3.3.3.2.4.2.2" xref="S4.SS1.9.p2.13.m3.3.3.2.4.2.2.cmml">ext</mi><mo id="S4.SS1.9.p2.13.m3.3.3.2.4.2.1" xref="S4.SS1.9.p2.13.m3.3.3.2.4.2.1.cmml">¯</mo></mover><mi id="S4.SS1.9.p2.13.m3.3.3.2.4.3" xref="S4.SS1.9.p2.13.m3.3.3.2.4.3.cmml">m</mi></msub><mo id="S4.SS1.9.p2.13.m3.3.3.2.3" xref="S4.SS1.9.p2.13.m3.3.3.2.3.cmml">⁢</mo><mrow id="S4.SS1.9.p2.13.m3.3.3.2.2.2" xref="S4.SS1.9.p2.13.m3.3.3.2.2.3.cmml"><mo id="S4.SS1.9.p2.13.m3.3.3.2.2.2.3" stretchy="false" xref="S4.SS1.9.p2.13.m3.3.3.2.2.3.cmml">(</mo><msub id="S4.SS1.9.p2.13.m3.2.2.1.1.1.1" xref="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.cmml"><mi id="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.2" xref="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.2.cmml">g</mi><mn id="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.3" xref="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.3.cmml">0</mn></msub><mo id="S4.SS1.9.p2.13.m3.3.3.2.2.2.4" xref="S4.SS1.9.p2.13.m3.3.3.2.2.3.cmml">,</mo><mi id="S4.SS1.9.p2.13.m3.1.1" mathvariant="normal" xref="S4.SS1.9.p2.13.m3.1.1.cmml">…</mi><mo id="S4.SS1.9.p2.13.m3.3.3.2.2.2.5" xref="S4.SS1.9.p2.13.m3.3.3.2.2.3.cmml">,</mo><msub id="S4.SS1.9.p2.13.m3.3.3.2.2.2.2" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.cmml"><mi id="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.2" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.2.cmml">g</mi><mi id="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.3" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.3.cmml">m</mi></msub><mo id="S4.SS1.9.p2.13.m3.3.3.2.2.2.6" rspace="0.278em" stretchy="false" xref="S4.SS1.9.p2.13.m3.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS1.9.p2.13.m3.3.3.3" rspace="0.278em" xref="S4.SS1.9.p2.13.m3.3.3.3.cmml">:=</mo><msub id="S4.SS1.9.p2.13.m3.3.3.4" xref="S4.SS1.9.p2.13.m3.3.3.4.cmml"><mi id="S4.SS1.9.p2.13.m3.3.3.4.2" xref="S4.SS1.9.p2.13.m3.3.3.4.2.cmml">f</mi><mi id="S4.SS1.9.p2.13.m3.3.3.4.3" xref="S4.SS1.9.p2.13.m3.3.3.4.3.cmml">m</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.9.p2.13.m3.3b"><apply id="S4.SS1.9.p2.13.m3.3.3.cmml" xref="S4.SS1.9.p2.13.m3.3.3"><csymbol cd="latexml" id="S4.SS1.9.p2.13.m3.3.3.3.cmml" xref="S4.SS1.9.p2.13.m3.3.3.3">assign</csymbol><apply id="S4.SS1.9.p2.13.m3.3.3.2.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2"><times id="S4.SS1.9.p2.13.m3.3.3.2.3.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.3"></times><apply id="S4.SS1.9.p2.13.m3.3.3.2.4.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.4"><csymbol cd="ambiguous" id="S4.SS1.9.p2.13.m3.3.3.2.4.1.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.4">subscript</csymbol><apply id="S4.SS1.9.p2.13.m3.3.3.2.4.2.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.4.2"><ci id="S4.SS1.9.p2.13.m3.3.3.2.4.2.1.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.4.2.1">¯</ci><ci id="S4.SS1.9.p2.13.m3.3.3.2.4.2.2.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.4.2.2">ext</ci></apply><ci id="S4.SS1.9.p2.13.m3.3.3.2.4.3.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.4.3">𝑚</ci></apply><vector id="S4.SS1.9.p2.13.m3.3.3.2.2.3.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2"><apply id="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.cmml" xref="S4.SS1.9.p2.13.m3.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.1.cmml" xref="S4.SS1.9.p2.13.m3.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.2.cmml" xref="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.2">𝑔</ci><cn id="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.3.cmml" type="integer" xref="S4.SS1.9.p2.13.m3.2.2.1.1.1.1.3">0</cn></apply><ci id="S4.SS1.9.p2.13.m3.1.1.cmml" xref="S4.SS1.9.p2.13.m3.1.1">…</ci><apply id="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.1.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2.2">subscript</csymbol><ci id="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.2.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.2">𝑔</ci><ci id="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.3.cmml" xref="S4.SS1.9.p2.13.m3.3.3.2.2.2.2.3">𝑚</ci></apply></vector></apply><apply id="S4.SS1.9.p2.13.m3.3.3.4.cmml" xref="S4.SS1.9.p2.13.m3.3.3.4"><csymbol cd="ambiguous" id="S4.SS1.9.p2.13.m3.3.3.4.1.cmml" xref="S4.SS1.9.p2.13.m3.3.3.4">subscript</csymbol><ci id="S4.SS1.9.p2.13.m3.3.3.4.2.cmml" xref="S4.SS1.9.p2.13.m3.3.3.4.2">𝑓</ci><ci id="S4.SS1.9.p2.13.m3.3.3.4.3.cmml" xref="S4.SS1.9.p2.13.m3.3.3.4.3">𝑚</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.9.p2.13.m3.3c">\overline{\operatorname{ext}}_{m}(g_{0},\dots,g_{m}):=f_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS1.9.p2.13.m3.3d">over¯ start_ARG roman_ext end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_g start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) := italic_f start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> and it is straightforward to verify that this operator satisfies the desired properties.</p> </div> <div class="ltx_para" id="S4.SS1.10.p3"> <p class="ltx_p" id="S4.SS1.10.p3.1"><span class="ltx_text ltx_font_italic" id="S4.SS1.10.p3.1.1">Step 2: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I3.i2" title="item ii ‣ Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>.</span> This case is similar to Step 1 using Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4" title="Theorem 4.4. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a> instead of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S4.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">4.2. </span>Density results</h3> <div class="ltx_para" id="S4.SS2.p1"> <p class="ltx_p" id="S4.SS2.p1.1">As an application of the trace theorems in the previous section, we prove density of certain classes of test functions in weighted spaces with zero boundary conditions.</p> </div> <section class="ltx_subsubsection" id="S4.SS2.SSS1"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.2.1. </span>Bessel potential spaces</h4> <div class="ltx_para" id="S4.SS2.SSS1.p1"> <p class="ltx_p" id="S4.SS2.SSS1.p1.5">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.1.m1.2"><semantics id="S4.SS2.SSS1.p1.1.m1.2a"><mrow id="S4.SS2.SSS1.p1.1.m1.2.3" xref="S4.SS2.SSS1.p1.1.m1.2.3.cmml"><mi id="S4.SS2.SSS1.p1.1.m1.2.3.2" xref="S4.SS2.SSS1.p1.1.m1.2.3.2.cmml">p</mi><mo id="S4.SS2.SSS1.p1.1.m1.2.3.1" xref="S4.SS2.SSS1.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.SS2.SSS1.p1.1.m1.2.3.3.2" xref="S4.SS2.SSS1.p1.1.m1.2.3.3.1.cmml"><mo id="S4.SS2.SSS1.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS1.p1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.SS2.SSS1.p1.1.m1.1.1" xref="S4.SS2.SSS1.p1.1.m1.1.1.cmml">1</mn><mo id="S4.SS2.SSS1.p1.1.m1.2.3.3.2.2" xref="S4.SS2.SSS1.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS1.p1.1.m1.2.2" mathvariant="normal" xref="S4.SS2.SSS1.p1.1.m1.2.2.cmml">∞</mi><mo id="S4.SS2.SSS1.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS1.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.1.m1.2b"><apply id="S4.SS2.SSS1.p1.1.m1.2.3.cmml" xref="S4.SS2.SSS1.p1.1.m1.2.3"><in id="S4.SS2.SSS1.p1.1.m1.2.3.1.cmml" xref="S4.SS2.SSS1.p1.1.m1.2.3.1"></in><ci id="S4.SS2.SSS1.p1.1.m1.2.3.2.cmml" xref="S4.SS2.SSS1.p1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.SS2.SSS1.p1.1.m1.2.3.3.1.cmml" xref="S4.SS2.SSS1.p1.1.m1.2.3.3.2"><cn id="S4.SS2.SSS1.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.p1.1.m1.1.1">1</cn><infinity id="S4.SS2.SSS1.p1.1.m1.2.2.cmml" xref="S4.SS2.SSS1.p1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.2.m2.1"><semantics id="S4.SS2.SSS1.p1.2.m2.1a"><mrow id="S4.SS2.SSS1.p1.2.m2.1.1" xref="S4.SS2.SSS1.p1.2.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p1.2.m2.1.1.2" xref="S4.SS2.SSS1.p1.2.m2.1.1.2.cmml">s</mi><mo id="S4.SS2.SSS1.p1.2.m2.1.1.1" xref="S4.SS2.SSS1.p1.2.m2.1.1.1.cmml">∈</mo><mi id="S4.SS2.SSS1.p1.2.m2.1.1.3" xref="S4.SS2.SSS1.p1.2.m2.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.2.m2.1b"><apply id="S4.SS2.SSS1.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1"><in id="S4.SS2.SSS1.p1.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1.1"></in><ci id="S4.SS2.SSS1.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1.2">𝑠</ci><ci id="S4.SS2.SSS1.p1.2.m2.1.1.3.cmml" xref="S4.SS2.SSS1.p1.2.m2.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.2.m2.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.2.m2.1d">italic_s ∈ blackboard_R</annotation></semantics></math>, <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.3.m3.2"><semantics id="S4.SS2.SSS1.p1.3.m3.2a"><mrow id="S4.SS2.SSS1.p1.3.m3.2.2" xref="S4.SS2.SSS1.p1.3.m3.2.2.cmml"><mi id="S4.SS2.SSS1.p1.3.m3.2.2.4" xref="S4.SS2.SSS1.p1.3.m3.2.2.4.cmml">γ</mi><mo id="S4.SS2.SSS1.p1.3.m3.2.2.3" xref="S4.SS2.SSS1.p1.3.m3.2.2.3.cmml">∈</mo><mrow id="S4.SS2.SSS1.p1.3.m3.2.2.2.2" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.3.cmml"><mo id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.3.cmml">(</mo><mrow id="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1" xref="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1a" xref="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.2" xref="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.4" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.3.cmml">,</mo><mrow id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.2" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.2.cmml">p</mi><mo id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.1" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.1.cmml">−</mo><mn id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.3" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.5" stretchy="false" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.3.m3.2b"><apply id="S4.SS2.SSS1.p1.3.m3.2.2.cmml" xref="S4.SS2.SSS1.p1.3.m3.2.2"><in id="S4.SS2.SSS1.p1.3.m3.2.2.3.cmml" xref="S4.SS2.SSS1.p1.3.m3.2.2.3"></in><ci id="S4.SS2.SSS1.p1.3.m3.2.2.4.cmml" xref="S4.SS2.SSS1.p1.3.m3.2.2.4">𝛾</ci><interval closure="open" id="S4.SS2.SSS1.p1.3.m3.2.2.2.3.cmml" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2"><apply id="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1"><minus id="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1"></minus><cn id="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.SSS1.p1.3.m3.1.1.1.1.1.2">1</cn></apply><apply id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2"><minus id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.1"></minus><ci id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.2">𝑝</ci><cn id="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.3.m3.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.3.m3.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.3.m3.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>, <math alttext="\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.4.m4.2"><semantics id="S4.SS2.SSS1.p1.4.m4.2a"><mrow id="S4.SS2.SSS1.p1.4.m4.2.2" xref="S4.SS2.SSS1.p1.4.m4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS1.p1.4.m4.2.2.4" xref="S4.SS2.SSS1.p1.4.m4.2.2.4.cmml">𝒪</mi><mo id="S4.SS2.SSS1.p1.4.m4.2.2.3" xref="S4.SS2.SSS1.p1.4.m4.2.2.3.cmml">∈</mo><mrow id="S4.SS2.SSS1.p1.4.m4.2.2.2.2" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.3.cmml"><mo id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.3.cmml">{</mo><msup id="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.2" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.3" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.4" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.3.cmml">,</mo><msubsup id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.2" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.3" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.3.cmml">+</mo><mi id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.3" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.5" stretchy="false" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.4.m4.2b"><apply id="S4.SS2.SSS1.p1.4.m4.2.2.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2"><in id="S4.SS2.SSS1.p1.4.m4.2.2.3.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.3"></in><ci id="S4.SS2.SSS1.p1.4.m4.2.2.4.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.4">𝒪</ci><set id="S4.SS2.SSS1.p1.4.m4.2.2.2.3.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2"><apply id="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p1.4.m4.1.1.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2">superscript</csymbol><ci id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.2">ℝ</ci><ci id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.p1.4.m4.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.4.m4.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.4.m4.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.5.m5.1"><semantics id="S4.SS2.SSS1.p1.5.m5.1a"><mi id="S4.SS2.SSS1.p1.5.m5.1.1" xref="S4.SS2.SSS1.p1.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.5.m5.1b"><ci id="S4.SS2.SSS1.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS1.p1.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Then we define</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex29"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{s,p}_{0}(\mathcal{O},w_{\gamma};X):=\Big{\{}f\in H^{s,p}(\mathcal{O},w_{% \gamma};X):\operatorname{Tr}(\partial^{\alpha}f)=0\text{ if }s-|\alpha|>\tfrac% {\gamma+1}{p}\Big{\}}." class="ltx_Math" display="block" id="S4.Ex29.m1.11"><semantics id="S4.Ex29.m1.11a"><mrow id="S4.Ex29.m1.11.11.1" xref="S4.Ex29.m1.11.11.1.1.cmml"><mrow id="S4.Ex29.m1.11.11.1.1" xref="S4.Ex29.m1.11.11.1.1.cmml"><mrow id="S4.Ex29.m1.11.11.1.1.1" xref="S4.Ex29.m1.11.11.1.1.1.cmml"><msubsup id="S4.Ex29.m1.11.11.1.1.1.3" xref="S4.Ex29.m1.11.11.1.1.1.3.cmml"><mi id="S4.Ex29.m1.11.11.1.1.1.3.2.2" xref="S4.Ex29.m1.11.11.1.1.1.3.2.2.cmml">H</mi><mn id="S4.Ex29.m1.11.11.1.1.1.3.3" xref="S4.Ex29.m1.11.11.1.1.1.3.3.cmml">0</mn><mrow id="S4.Ex29.m1.2.2.2.4" 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href="https://arxiv.org/html/2503.14636v1#S4.Ex29.m1.11.11.1.1.3.2.2.4.cmml" id="S4.Ex29.m1.11.11.1.1.3.2.2d.cmml" xref="S4.Ex29.m1.11.11.1.1.3.2.2"></share><apply id="S4.Ex29.m1.11.11.1.1.3.2.2.6.cmml" xref="S4.Ex29.m1.11.11.1.1.3.2.2.6"><divide id="S4.Ex29.m1.11.11.1.1.3.2.2.6.1.cmml" xref="S4.Ex29.m1.11.11.1.1.3.2.2.6"></divide><apply id="S4.Ex29.m1.11.11.1.1.3.2.2.6.2.cmml" xref="S4.Ex29.m1.11.11.1.1.3.2.2.6.2"><plus id="S4.Ex29.m1.11.11.1.1.3.2.2.6.2.1.cmml" xref="S4.Ex29.m1.11.11.1.1.3.2.2.6.2.1"></plus><ci id="S4.Ex29.m1.11.11.1.1.3.2.2.6.2.2.cmml" xref="S4.Ex29.m1.11.11.1.1.3.2.2.6.2.2">𝛾</ci><cn id="S4.Ex29.m1.11.11.1.1.3.2.2.6.2.3.cmml" type="integer" xref="S4.Ex29.m1.11.11.1.1.3.2.2.6.2.3">1</cn></apply><ci id="S4.Ex29.m1.11.11.1.1.3.2.2.6.3.cmml" xref="S4.Ex29.m1.11.11.1.1.3.2.2.6.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex29.m1.11c">H^{s,p}_{0}(\mathcal{O},w_{\gamma};X):=\Big{\{}f\in H^{s,p}(\mathcal{O},w_{% \gamma};X):\operatorname{Tr}(\partial^{\alpha}f)=0\text{ if }s-|\alpha|>\tfrac% {\gamma+1}{p}\Big{\}}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex29.m1.11d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) := { italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) : roman_Tr ( ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_f ) = 0 if italic_s - | italic_α | > divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.p1.8">All the traces in the above definition are well defined by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a>. 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id="S4.SS2.SSS1.p1.6.m1.9.9.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.6.m1.9.9.1.3.3">0</cn></apply><vector id="S4.SS2.SSS1.p1.6.m1.9.9.1.1.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1"><ci id="S4.SS2.SSS1.p1.6.m1.5.5.cmml" xref="S4.SS2.SSS1.p1.6.m1.5.5">𝒪</ci><apply id="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1.2">𝑤</ci><ci id="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.p1.6.m1.9.9.1.1.1.1.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.p1.6.m1.6.6.cmml" xref="S4.SS2.SSS1.p1.6.m1.6.6">𝑋</ci></vector></apply><apply id="S4.SS2.SSS1.p1.6.m1.10.10.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2"><times id="S4.SS2.SSS1.p1.6.m1.10.10.2.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.2"></times><apply id="S4.SS2.SSS1.p1.6.m1.10.10.2.3.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.6.m1.10.10.2.3.1.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.3">superscript</csymbol><ci id="S4.SS2.SSS1.p1.6.m1.10.10.2.3.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.3.2">𝐻</ci><list id="S4.SS2.SSS1.p1.6.m1.4.4.2.3.cmml" xref="S4.SS2.SSS1.p1.6.m1.4.4.2.4"><ci id="S4.SS2.SSS1.p1.6.m1.3.3.1.1.cmml" xref="S4.SS2.SSS1.p1.6.m1.3.3.1.1">𝑠</ci><ci id="S4.SS2.SSS1.p1.6.m1.4.4.2.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.4.4.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS1.p1.6.m1.10.10.2.1.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1"><ci id="S4.SS2.SSS1.p1.6.m1.7.7.cmml" xref="S4.SS2.SSS1.p1.6.m1.7.7">𝒪</ci><apply id="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1.2.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1.2">𝑤</ci><ci id="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.p1.6.m1.10.10.2.1.1.1.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.p1.6.m1.8.8.cmml" xref="S4.SS2.SSS1.p1.6.m1.8.8">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.6.m1.10c">H^{s,p}_{0}(\mathcal{O},w_{\gamma};X)=H^{s,p}(\mathcal{O},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.6.m1.10d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> if <math alttext="s\leq\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.7.m2.1"><semantics id="S4.SS2.SSS1.p1.7.m2.1a"><mrow id="S4.SS2.SSS1.p1.7.m2.1.1" xref="S4.SS2.SSS1.p1.7.m2.1.1.cmml"><mi id="S4.SS2.SSS1.p1.7.m2.1.1.2" xref="S4.SS2.SSS1.p1.7.m2.1.1.2.cmml">s</mi><mo id="S4.SS2.SSS1.p1.7.m2.1.1.1" xref="S4.SS2.SSS1.p1.7.m2.1.1.1.cmml">≤</mo><mfrac id="S4.SS2.SSS1.p1.7.m2.1.1.3" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.cmml"><mrow id="S4.SS2.SSS1.p1.7.m2.1.1.3.2" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2.cmml"><mi id="S4.SS2.SSS1.p1.7.m2.1.1.3.2.2" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2.2.cmml">γ</mi><mo id="S4.SS2.SSS1.p1.7.m2.1.1.3.2.1" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2.1.cmml">+</mo><mn id="S4.SS2.SSS1.p1.7.m2.1.1.3.2.3" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2.3.cmml">1</mn></mrow><mi id="S4.SS2.SSS1.p1.7.m2.1.1.3.3" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.3.cmml">p</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.p1.7.m2.1b"><apply id="S4.SS2.SSS1.p1.7.m2.1.1.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1"><leq id="S4.SS2.SSS1.p1.7.m2.1.1.1.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.1"></leq><ci id="S4.SS2.SSS1.p1.7.m2.1.1.2.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.2">𝑠</ci><apply id="S4.SS2.SSS1.p1.7.m2.1.1.3.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.3"><divide id="S4.SS2.SSS1.p1.7.m2.1.1.3.1.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.3"></divide><apply id="S4.SS2.SSS1.p1.7.m2.1.1.3.2.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2"><plus id="S4.SS2.SSS1.p1.7.m2.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2.1"></plus><ci id="S4.SS2.SSS1.p1.7.m2.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2.2">𝛾</ci><cn id="S4.SS2.SSS1.p1.7.m2.1.1.3.2.3.cmml" type="integer" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.2.3">1</cn></apply><ci id="S4.SS2.SSS1.p1.7.m2.1.1.3.3.cmml" xref="S4.SS2.SSS1.p1.7.m2.1.1.3.3">𝑝</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.7.m2.1c">s\leq\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.7.m2.1d">italic_s ≤ divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Moreover, we set <math alttext="\mathbb{R}_{\bullet}^{d}:=\mathbb{R}^{d}\setminus\{(0,\widetilde{x}):% \widetilde{x}\in\mathbb{R}^{d-1}\}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p1.8.m3.4"><semantics id="S4.SS2.SSS1.p1.8.m3.4a"><mrow id="S4.SS2.SSS1.p1.8.m3.4.4" xref="S4.SS2.SSS1.p1.8.m3.4.4.cmml"><msubsup id="S4.SS2.SSS1.p1.8.m3.4.4.4" xref="S4.SS2.SSS1.p1.8.m3.4.4.4.cmml"><mi id="S4.SS2.SSS1.p1.8.m3.4.4.4.2.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.4.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.p1.8.m3.4.4.4.2.3" xref="S4.SS2.SSS1.p1.8.m3.4.4.4.2.3.cmml">∙</mo><mi id="S4.SS2.SSS1.p1.8.m3.4.4.4.3" xref="S4.SS2.SSS1.p1.8.m3.4.4.4.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.p1.8.m3.4.4.3" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS1.p1.8.m3.4.4.3.cmml">:=</mo><mrow id="S4.SS2.SSS1.p1.8.m3.4.4.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.cmml"><msup id="S4.SS2.SSS1.p1.8.m3.4.4.2.4" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.4.cmml"><mi id="S4.SS2.SSS1.p1.8.m3.4.4.2.4.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.4.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.p1.8.m3.4.4.2.4.3" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.4.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.p1.8.m3.4.4.2.3" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.3.cmml">∖</mo><mrow id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.3.cmml"><mo id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.3.1.cmml">{</mo><mrow id="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.2" xref="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.2.1" stretchy="false" xref="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.1.cmml">(</mo><mn id="S4.SS2.SSS1.p1.8.m3.1.1" xref="S4.SS2.SSS1.p1.8.m3.1.1.cmml">0</mn><mo id="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.2.2" xref="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.1.cmml">,</mo><mover accent="true" id="S4.SS2.SSS1.p1.8.m3.2.2" xref="S4.SS2.SSS1.p1.8.m3.2.2.cmml"><mi id="S4.SS2.SSS1.p1.8.m3.2.2.2" xref="S4.SS2.SSS1.p1.8.m3.2.2.2.cmml">x</mi><mo id="S4.SS2.SSS1.p1.8.m3.2.2.1" xref="S4.SS2.SSS1.p1.8.m3.2.2.1.cmml">~</mo></mover><mo id="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.2.3" rspace="0.278em" stretchy="false" xref="S4.SS2.SSS1.p1.8.m3.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.4" rspace="0.278em" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.3.1.cmml">:</mo><mrow id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.2.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.2.2.cmml">x</mi><mo id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.2.1" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.2.1.cmml">~</mo></mover><mo id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.1" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.1.cmml">∈</mo><msup id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.3" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.3.2" xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.3.2.cmml">ℝ</mi><mrow 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xref="S4.SS2.SSS1.p1.8.m3.4.4.2.2.2.2.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p1.8.m3.4c">\mathbb{R}_{\bullet}^{d}:=\mathbb{R}^{d}\setminus\{(0,\widetilde{x}):% \widetilde{x}\in\mathbb{R}^{d-1}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p1.8.m3.4d">blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT := blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ∖ { ( 0 , over~ start_ARG italic_x end_ARG ) : over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT }</annotation></semantics></math>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S4.SS2.SSS1.p2"> <p class="ltx_p" id="S4.SS2.SSS1.p2.2">To prove a density result for <math alttext="H^{s,p}_{0}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p2.1.m1.2"><semantics 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xref="S4.SS2.SSS1.p2.1.m1.2.3">superscript</csymbol><ci id="S4.SS2.SSS1.p2.1.m1.2.3.2.2.cmml" xref="S4.SS2.SSS1.p2.1.m1.2.3.2.2">𝐻</ci><list id="S4.SS2.SSS1.p2.1.m1.2.2.2.3.cmml" xref="S4.SS2.SSS1.p2.1.m1.2.2.2.4"><ci id="S4.SS2.SSS1.p2.1.m1.1.1.1.1.cmml" xref="S4.SS2.SSS1.p2.1.m1.1.1.1.1">𝑠</ci><ci id="S4.SS2.SSS1.p2.1.m1.2.2.2.2.cmml" xref="S4.SS2.SSS1.p2.1.m1.2.2.2.2">𝑝</ci></list></apply><cn id="S4.SS2.SSS1.p2.1.m1.2.3.3.cmml" type="integer" xref="S4.SS2.SSS1.p2.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.p2.1.m1.2c">H^{s,p}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.p2.1.m1.2d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> we record the following proposition about the boundedness of the pointwise multiplication operator <math alttext="\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}" 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end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.1.1.1">Proposition 4.6</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem6.p1"> <p class="ltx_p" id="S4.Thmtheorem6.p1.7"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem6.p1.7.7">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.1.1.m1.2"><semantics id="S4.Thmtheorem6.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem6.p1.1.1.m1.2.3" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem6.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem6.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem6.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem6.p1.1.1.m1.1.1" xref="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem6.p1.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem6.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem6.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem6.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.1.1.m1.2b"><apply id="S4.Thmtheorem6.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.2.3"><in id="S4.Thmtheorem6.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.1"></in><ci id="S4.Thmtheorem6.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.2.3.3.2"><cn id="S4.Thmtheorem6.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem6.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem6.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem6.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="s>-1+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.2.2.m2.1"><semantics id="S4.Thmtheorem6.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.2.cmml">s</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.1.cmml">></mo><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.cmml"><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.cmml"><mo id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2a" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.cmml">−</mo><mn id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.1.cmml">+</mo><mfrac id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.cmml"><mrow id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.cmml"><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.2" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.1" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.1.cmml">+</mo><mn id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.3" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.2.2.m2.1b"><apply id="S4.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1"><gt id="S4.Thmtheorem6.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.1"></gt><ci id="S4.Thmtheorem6.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.2">𝑠</ci><apply id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3"><plus id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.1"></plus><apply id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2"><minus id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2"></minus><cn id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.2.cmml" type="integer" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.2.2">1</cn></apply><apply id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3"><divide id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3"></divide><apply id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2"><plus id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.1.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.1"></plus><ci id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.2.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.2">𝛾</ci><cn id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.2.3">1</cn></apply><ci id="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.3.cmml" xref="S4.Thmtheorem6.p1.2.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.2.2.m2.1c">s>-1+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.2.2.m2.1d">italic_s > - 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>, <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.3.3.m3.2"><semantics id="S4.Thmtheorem6.p1.3.3.m3.2a"><mrow id="S4.Thmtheorem6.p1.3.3.m3.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.2.2.4" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.4.cmml">γ</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.2.2.3" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.3.cmml">∈</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.3.cmml"><mo id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.3.cmml">(</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1a" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.2" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.4" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.3.cmml">,</mo><mrow id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.2" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.2.cmml">p</mi><mo id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.1" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.1.cmml">−</mo><mn id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.3" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.3.3.m3.2b"><apply id="S4.Thmtheorem6.p1.3.3.m3.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2"><in id="S4.Thmtheorem6.p1.3.3.m3.2.2.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.3"></in><ci id="S4.Thmtheorem6.p1.3.3.m3.2.2.4.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.4">𝛾</ci><interval closure="open" id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.3.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2"><apply id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1"><minus id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1"></minus><cn id="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem6.p1.3.3.m3.1.1.1.1.1.2">1</cn></apply><apply id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2"><minus id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.1"></minus><ci id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.2">𝑝</ci><cn id="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem6.p1.3.3.m3.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.3.3.m3.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.3.3.m3.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.4.4.m4.1"><semantics id="S4.Thmtheorem6.p1.4.4.m4.1a"><mi id="S4.Thmtheorem6.p1.4.4.m4.1.1" xref="S4.Thmtheorem6.p1.4.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.4.4.m4.1b"><ci id="S4.Thmtheorem6.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem6.p1.4.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.4.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.4.4.m4.1d">italic_X</annotation></semantics></math> be a Banach space. Then <math alttext="\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.5.5.m5.1"><semantics id="S4.Thmtheorem6.p1.5.5.m5.1a"><msub id="S4.Thmtheorem6.p1.5.5.m5.1.1" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.cmml"><mi id="S4.Thmtheorem6.p1.5.5.m5.1.1.2" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.2.cmml">𝟏</mi><msubsup id="S4.Thmtheorem6.p1.5.5.m5.1.1.3" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3.cmml"><mi id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.2" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.2.cmml">ℝ</mi><mo id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.3" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3.3.cmml">+</mo><mi id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.3" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.3.cmml">d</mi></msubsup></msub><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem6.p1.5.5.m5.1b"><apply id="S4.Thmtheorem6.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.5.5.m5.1.1.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1">subscript</csymbol><csymbol cd="latexml" id="S4.Thmtheorem6.p1.5.5.m5.1.1.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.2">1</csymbol><apply id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3">subscript</csymbol><apply id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.1.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3">superscript</csymbol><ci id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.2.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.2">ℝ</ci><ci id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.3.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3.2.3">𝑑</ci></apply><plus id="S4.Thmtheorem6.p1.5.5.m5.1.1.3.3.cmml" xref="S4.Thmtheorem6.p1.5.5.m5.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.5.5.m5.1c">\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.5.5.m5.1d">bold_1 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is a pointwise multiplier on the closure of <math alttext="C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.6.6.m6.2"><semantics id="S4.Thmtheorem6.p1.6.6.m6.2a"><mrow id="S4.Thmtheorem6.p1.6.6.m6.2.2" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.cmml"><msubsup id="S4.Thmtheorem6.p1.6.6.m6.2.2.3" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.3.cmml"><mi id="S4.Thmtheorem6.p1.6.6.m6.2.2.3.2.2" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.3.2.2.cmml">C</mi><mi id="S4.Thmtheorem6.p1.6.6.m6.2.2.3.2.3" mathvariant="normal" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.3.2.3.cmml">c</mi><mi id="S4.Thmtheorem6.p1.6.6.m6.2.2.3.3" mathvariant="normal" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.3.3.cmml">∞</mi></msubsup><mo id="S4.Thmtheorem6.p1.6.6.m6.2.2.2" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.2.cmml">⁢</mo><mrow id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.2.cmml"><mo id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.2.cmml">(</mo><msubsup id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.2" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.3" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.3.cmml">∙</mo><mi id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.3" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.3" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.2.cmml">;</mo><mi id="S4.Thmtheorem6.p1.6.6.m6.1.1" xref="S4.Thmtheorem6.p1.6.6.m6.1.1.cmml">X</mi><mo 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xref="S4.Thmtheorem6.p1.6.6.m6.2.2.3.2.3">c</ci></apply><infinity id="S4.Thmtheorem6.p1.6.6.m6.2.2.3.3.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.3.3"></infinity></apply><list id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.2.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1"><apply id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1">superscript</csymbol><apply id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.1.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.2.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.2">ℝ</ci><ci id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.3.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.2.3">∙</ci></apply><ci id="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.2.2.1.1.1.3">𝑑</ci></apply><ci id="S4.Thmtheorem6.p1.6.6.m6.1.1.cmml" xref="S4.Thmtheorem6.p1.6.6.m6.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem6.p1.6.6.m6.2c">C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem6.p1.6.6.m6.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> with respect to the norm of <math alttext="H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.Thmtheorem6.p1.7.7.m7.5"><semantics id="S4.Thmtheorem6.p1.7.7.m7.5a"><mrow id="S4.Thmtheorem6.p1.7.7.m7.5.5" xref="S4.Thmtheorem6.p1.7.7.m7.5.5.cmml"><msup id="S4.Thmtheorem6.p1.7.7.m7.5.5.4" 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id="S4.Thmtheorem6.p1.7.7.m7.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. Moreover, it holds that</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex30"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\partial^{\alpha}(\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}f)=% \operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}(\partial^{\alpha}f)\qquad\text{% for }|\alpha|\leq m\,\text{ and }\,f\in\overline{C_{\mathrm{c}}^{\infty}(% \mathbb{R}_{\bullet}^{d};X)}^{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}." class="ltx_Math" display="block" id="S4.Ex30.m1.9"><semantics id="S4.Ex30.m1.9a"><mrow id="S4.Ex30.m1.9.9.1"><mrow id="S4.Ex30.m1.9.9.1.1.2" xref="S4.Ex30.m1.9.9.1.1.3.cmml"><mrow id="S4.Ex30.m1.9.9.1.1.1.1" xref="S4.Ex30.m1.9.9.1.1.1.1.cmml"><mrow id="S4.Ex30.m1.9.9.1.1.1.1.1" xref="S4.Ex30.m1.9.9.1.1.1.1.1.cmml"><msup id="S4.Ex30.m1.9.9.1.1.1.1.1.2" xref="S4.Ex30.m1.9.9.1.1.1.1.1.2.cmml"><mo id="S4.Ex30.m1.9.9.1.1.1.1.1.2.2" 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}\,f\in\overline{C_{\mathrm{c}}^{\infty}(% \mathbb{R}_{\bullet}^{d};X)}^{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex30.m1.9d">∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT ( bold_1 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f ) = bold_1 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_f ) for | italic_α | ≤ italic_m and italic_f ∈ over¯ start_ARG italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X ) end_ARG start_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS2.SSS1.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.SSS1.1.p1"> <p class="ltx_p" id="S4.SS2.SSS1.1.p1.13">By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Theorem 4.1]</cite> it holds that <math alttext="\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.1.m1.1"><semantics id="S4.SS2.SSS1.1.p1.1.m1.1a"><msub id="S4.SS2.SSS1.1.p1.1.m1.1.1" xref="S4.SS2.SSS1.1.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.1.p1.1.m1.1.1.2" xref="S4.SS2.SSS1.1.p1.1.m1.1.1.2.cmml">𝟏</mi><msubsup 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id="S4.SS2.SSS1.1.p1.2.m2.5"><semantics id="S4.SS2.SSS1.1.p1.2.m2.5a"><mrow id="S4.SS2.SSS1.1.p1.2.m2.5.5" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.cmml"><msup id="S4.SS2.SSS1.1.p1.2.m2.5.5.4" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.4.cmml"><mi id="S4.SS2.SSS1.1.p1.2.m2.5.5.4.2" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.1.p1.2.m2.2.2.2.4" xref="S4.SS2.SSS1.1.p1.2.m2.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.1.p1.2.m2.1.1.1.1" xref="S4.SS2.SSS1.1.p1.2.m2.1.1.1.1.cmml">σ</mi><mo id="S4.SS2.SSS1.1.p1.2.m2.2.2.2.4.1" xref="S4.SS2.SSS1.1.p1.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.1.p1.2.m2.2.2.2.2" xref="S4.SS2.SSS1.1.p1.2.m2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS2.SSS1.1.p1.2.m2.5.5.3" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.3.cmml"><mo id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.3" stretchy="false" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.3.cmml">(</mo><msup id="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1" xref="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.cmml"><mi id="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.2" xref="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.3" xref="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.4" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.cmml"><mi id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.2" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.3" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.5" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.3.cmml">;</mo><mi id="S4.SS2.SSS1.1.p1.2.m2.3.3" xref="S4.SS2.SSS1.1.p1.2.m2.3.3.cmml">X</mi><mo id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.6" stretchy="false" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.2.m2.5b"><apply id="S4.SS2.SSS1.1.p1.2.m2.5.5.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5"><times id="S4.SS2.SSS1.1.p1.2.m2.5.5.3.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.3"></times><apply id="S4.SS2.SSS1.1.p1.2.m2.5.5.4.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.1.p1.2.m2.5.5.4.1.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.4">superscript</csymbol><ci id="S4.SS2.SSS1.1.p1.2.m2.5.5.4.2.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.4.2">𝐻</ci><list id="S4.SS2.SSS1.1.p1.2.m2.2.2.2.3.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.2.2.2.4"><ci id="S4.SS2.SSS1.1.p1.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.1.1.1.1">𝜎</ci><ci id="S4.SS2.SSS1.1.p1.2.m2.2.2.2.2.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.3.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2"><apply id="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.2.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.3.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.4.4.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.1.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.3.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.1.p1.2.m2.3.3.cmml" xref="S4.SS2.SSS1.1.p1.2.m2.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.2.m2.5c">H^{\sigma,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.2.m2.5d">italic_H start_POSTSUPERSCRIPT italic_σ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for <math alttext="-1+\frac{\gamma+1}{p}<\sigma<\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.3.m3.1"><semantics id="S4.SS2.SSS1.1.p1.3.m3.1a"><mrow id="S4.SS2.SSS1.1.p1.3.m3.1.1" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.cmml"><mrow id="S4.SS2.SSS1.1.p1.3.m3.1.1.2" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.cmml"><mrow id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.cmml"><mo id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2a" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.cmml">−</mo><mn id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.2" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.2.cmml">1</mn></mrow><mo id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.1" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.1.cmml">+</mo><mfrac id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.cmml"><mrow id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.cmml"><mi id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.2" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.2.cmml">γ</mi><mo id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.1" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.1.cmml">+</mo><mn id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.3" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.3" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S4.SS2.SSS1.1.p1.3.m3.1.1.3" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS1.1.p1.3.m3.1.1.4" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.4.cmml">σ</mi><mo id="S4.SS2.SSS1.1.p1.3.m3.1.1.5" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.5.cmml"><</mo><mfrac id="S4.SS2.SSS1.1.p1.3.m3.1.1.6" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.cmml"><mrow id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.cmml"><mi id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.2" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.2.cmml">γ</mi><mo id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.1" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.1.cmml">+</mo><mn id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.3" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.3.cmml">1</mn></mrow><mi id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.3" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.3.cmml">p</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.3.m3.1b"><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1"><and id="S4.SS2.SSS1.1.p1.3.m3.1.1a.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1"></and><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1b.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1"><lt id="S4.SS2.SSS1.1.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.3"></lt><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2"><plus id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.1.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.1"></plus><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2"><minus id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2"></minus><cn id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.2.cmml" type="integer" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.2.2">1</cn></apply><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3"><divide id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.1.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3"></divide><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2"><plus id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.1.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.1"></plus><ci id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.2.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.2">𝛾</ci><cn id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.3.cmml" type="integer" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.2.3">1</cn></apply><ci id="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.3.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.2.3.3">𝑝</ci></apply></apply><ci id="S4.SS2.SSS1.1.p1.3.m3.1.1.4.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.4">𝜎</ci></apply><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1c.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1"><lt id="S4.SS2.SSS1.1.p1.3.m3.1.1.5.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S4.SS2.SSS1.1.p1.3.m3.1.1.4.cmml" id="S4.SS2.SSS1.1.p1.3.m3.1.1d.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1"></share><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6"><divide id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.1.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6"></divide><apply id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2"><plus id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.1.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.1"></plus><ci id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.2.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.2">𝛾</ci><cn id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.3.cmml" type="integer" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.2.3">1</cn></apply><ci id="S4.SS2.SSS1.1.p1.3.m3.1.1.6.3.cmml" xref="S4.SS2.SSS1.1.p1.3.m3.1.1.6.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.3.m3.1c">-1+\frac{\gamma+1}{p}<\sigma<\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.3.m3.1d">- 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_σ < divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. We note that <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Theorem 4.1]</cite> actually holds for any Banach space <math alttext="X" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.4.m4.1"><semantics id="S4.SS2.SSS1.1.p1.4.m4.1a"><mi id="S4.SS2.SSS1.1.p1.4.m4.1.1" xref="S4.SS2.SSS1.1.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.4.m4.1b"><ci id="S4.SS2.SSS1.1.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS1.1.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.4.m4.1d">italic_X</annotation></semantics></math> and the <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.5.m5.1"><semantics id="S4.SS2.SSS1.1.p1.5.m5.1a"><mi id="S4.SS2.SSS1.1.p1.5.m5.1.1" xref="S4.SS2.SSS1.1.p1.5.m5.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.5.m5.1b"><ci id="S4.SS2.SSS1.1.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS1.1.p1.5.m5.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.5.m5.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.5.m5.1d">roman_UMD</annotation></semantics></math> condition in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 4.2]</cite> is redundant. The norm equivalence in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 4.2]</cite> follows from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Proposition 15.2.12]</cite> using that <math alttext="-\Delta" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.6.m6.1"><semantics id="S4.SS2.SSS1.1.p1.6.m6.1a"><mrow id="S4.SS2.SSS1.1.p1.6.m6.1.1" xref="S4.SS2.SSS1.1.p1.6.m6.1.1.cmml"><mo id="S4.SS2.SSS1.1.p1.6.m6.1.1a" xref="S4.SS2.SSS1.1.p1.6.m6.1.1.cmml">−</mo><mi id="S4.SS2.SSS1.1.p1.6.m6.1.1.2" mathvariant="normal" xref="S4.SS2.SSS1.1.p1.6.m6.1.1.2.cmml">Δ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.6.m6.1b"><apply id="S4.SS2.SSS1.1.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS1.1.p1.6.m6.1.1"><minus id="S4.SS2.SSS1.1.p1.6.m6.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.6.m6.1.1"></minus><ci id="S4.SS2.SSS1.1.p1.6.m6.1.1.2.cmml" xref="S4.SS2.SSS1.1.p1.6.m6.1.1.2">Δ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.6.m6.1c">-\Delta</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.6.m6.1d">- roman_Δ</annotation></semantics></math> is sectorial on <math alttext="H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.7.m7.5"><semantics id="S4.SS2.SSS1.1.p1.7.m7.5a"><mrow id="S4.SS2.SSS1.1.p1.7.m7.5.5" xref="S4.SS2.SSS1.1.p1.7.m7.5.5.cmml"><msup id="S4.SS2.SSS1.1.p1.7.m7.5.5.4" xref="S4.SS2.SSS1.1.p1.7.m7.5.5.4.cmml"><mi id="S4.SS2.SSS1.1.p1.7.m7.5.5.4.2" xref="S4.SS2.SSS1.1.p1.7.m7.5.5.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.1.p1.7.m7.2.2.2.4" xref="S4.SS2.SSS1.1.p1.7.m7.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.1.p1.7.m7.1.1.1.1" xref="S4.SS2.SSS1.1.p1.7.m7.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.1.p1.7.m7.2.2.2.4.1" xref="S4.SS2.SSS1.1.p1.7.m7.2.2.2.3.cmml">,</mo><mi 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xref="S4.SS2.SSS1.1.p1.7.m7.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.1.p1.7.m7.3.3.cmml" xref="S4.SS2.SSS1.1.p1.7.m7.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.7.m7.5c">H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.7.m7.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> with domain <math alttext="H^{s+2,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.8.m8.5"><semantics id="S4.SS2.SSS1.1.p1.8.m8.5a"><mrow id="S4.SS2.SSS1.1.p1.8.m8.5.5" xref="S4.SS2.SSS1.1.p1.8.m8.5.5.cmml"><msup id="S4.SS2.SSS1.1.p1.8.m8.5.5.4" xref="S4.SS2.SSS1.1.p1.8.m8.5.5.4.cmml"><mi id="S4.SS2.SSS1.1.p1.8.m8.5.5.4.2" 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xref="S4.SS2.SSS1.1.p1.8.m8.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.1.p1.8.m8.3.3.cmml" xref="S4.SS2.SSS1.1.p1.8.m8.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.8.m8.5c">H^{s+2,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.8.m8.5d">italic_H start_POSTSUPERSCRIPT italic_s + 2 , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for any Banach space <math alttext="X" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.9.m9.1"><semantics id="S4.SS2.SSS1.1.p1.9.m9.1a"><mi id="S4.SS2.SSS1.1.p1.9.m9.1.1" xref="S4.SS2.SSS1.1.p1.9.m9.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.9.m9.1b"><ci id="S4.SS2.SSS1.1.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS1.1.p1.9.m9.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.9.m9.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.9.m9.1d">italic_X</annotation></semantics></math> and <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.10.m10.1"><semantics id="S4.SS2.SSS1.1.p1.10.m10.1a"><mrow id="S4.SS2.SSS1.1.p1.10.m10.1.1" xref="S4.SS2.SSS1.1.p1.10.m10.1.1.cmml"><mi id="S4.SS2.SSS1.1.p1.10.m10.1.1.2" xref="S4.SS2.SSS1.1.p1.10.m10.1.1.2.cmml">s</mi><mo id="S4.SS2.SSS1.1.p1.10.m10.1.1.1" xref="S4.SS2.SSS1.1.p1.10.m10.1.1.1.cmml">∈</mo><mi id="S4.SS2.SSS1.1.p1.10.m10.1.1.3" xref="S4.SS2.SSS1.1.p1.10.m10.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.10.m10.1b"><apply id="S4.SS2.SSS1.1.p1.10.m10.1.1.cmml" xref="S4.SS2.SSS1.1.p1.10.m10.1.1"><in id="S4.SS2.SSS1.1.p1.10.m10.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.10.m10.1.1.1"></in><ci id="S4.SS2.SSS1.1.p1.10.m10.1.1.2.cmml" xref="S4.SS2.SSS1.1.p1.10.m10.1.1.2">𝑠</ci><ci id="S4.SS2.SSS1.1.p1.10.m10.1.1.3.cmml" xref="S4.SS2.SSS1.1.p1.10.m10.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.10.m10.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.10.m10.1d">italic_s ∈ blackboard_R</annotation></semantics></math> (by the <math alttext="L^{p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.11.m11.3"><semantics id="S4.SS2.SSS1.1.p1.11.m11.3a"><mrow id="S4.SS2.SSS1.1.p1.11.m11.3.3" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.cmml"><msup id="S4.SS2.SSS1.1.p1.11.m11.3.3.4" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.4.cmml"><mi id="S4.SS2.SSS1.1.p1.11.m11.3.3.4.2" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.4.2.cmml">L</mi><mi id="S4.SS2.SSS1.1.p1.11.m11.3.3.4.3" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.4.3.cmml">p</mi></msup><mo id="S4.SS2.SSS1.1.p1.11.m11.3.3.3" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.3.cmml"><mo id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.3" stretchy="false" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.3.cmml">(</mo><msup id="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1" xref="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.2" xref="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.3" xref="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.4" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.cmml"><mi id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.2" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.3" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.5" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.3.cmml">;</mo><mi id="S4.SS2.SSS1.1.p1.11.m11.1.1" xref="S4.SS2.SSS1.1.p1.11.m11.1.1.cmml">X</mi><mo id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.6" stretchy="false" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.11.m11.3b"><apply id="S4.SS2.SSS1.1.p1.11.m11.3.3.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3"><times id="S4.SS2.SSS1.1.p1.11.m11.3.3.3.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.3"></times><apply id="S4.SS2.SSS1.1.p1.11.m11.3.3.4.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.1.p1.11.m11.3.3.4.1.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.4">superscript</csymbol><ci id="S4.SS2.SSS1.1.p1.11.m11.3.3.4.2.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.4.2">𝐿</ci><ci id="S4.SS2.SSS1.1.p1.11.m11.3.3.4.3.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.4.3">𝑝</ci></apply><vector id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.3.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2"><apply id="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.2.2.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.1.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.2.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.3.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.3.3.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.1.p1.11.m11.1.1.cmml" xref="S4.SS2.SSS1.1.p1.11.m11.1.1">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.11.m11.3c">L^{p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.11.m11.3d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>-case and lifting). Indeed, <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Proposition 15.2.12]</cite> yields for any <math alttext="s\in\mathbb{R}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.12.m12.1"><semantics id="S4.SS2.SSS1.1.p1.12.m12.1a"><mrow id="S4.SS2.SSS1.1.p1.12.m12.1.1" xref="S4.SS2.SSS1.1.p1.12.m12.1.1.cmml"><mi id="S4.SS2.SSS1.1.p1.12.m12.1.1.2" xref="S4.SS2.SSS1.1.p1.12.m12.1.1.2.cmml">s</mi><mo id="S4.SS2.SSS1.1.p1.12.m12.1.1.1" xref="S4.SS2.SSS1.1.p1.12.m12.1.1.1.cmml">∈</mo><mi id="S4.SS2.SSS1.1.p1.12.m12.1.1.3" xref="S4.SS2.SSS1.1.p1.12.m12.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.12.m12.1b"><apply id="S4.SS2.SSS1.1.p1.12.m12.1.1.cmml" xref="S4.SS2.SSS1.1.p1.12.m12.1.1"><in id="S4.SS2.SSS1.1.p1.12.m12.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.12.m12.1.1.1"></in><ci id="S4.SS2.SSS1.1.p1.12.m12.1.1.2.cmml" xref="S4.SS2.SSS1.1.p1.12.m12.1.1.2">𝑠</ci><ci id="S4.SS2.SSS1.1.p1.12.m12.1.1.3.cmml" xref="S4.SS2.SSS1.1.p1.12.m12.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.12.m12.1c">s\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.12.m12.1d">italic_s ∈ blackboard_R</annotation></semantics></math> and <math alttext="\sigma\geq 0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.13.m13.1"><semantics id="S4.SS2.SSS1.1.p1.13.m13.1a"><mrow id="S4.SS2.SSS1.1.p1.13.m13.1.1" xref="S4.SS2.SSS1.1.p1.13.m13.1.1.cmml"><mi id="S4.SS2.SSS1.1.p1.13.m13.1.1.2" xref="S4.SS2.SSS1.1.p1.13.m13.1.1.2.cmml">σ</mi><mo id="S4.SS2.SSS1.1.p1.13.m13.1.1.1" xref="S4.SS2.SSS1.1.p1.13.m13.1.1.1.cmml">≥</mo><mn id="S4.SS2.SSS1.1.p1.13.m13.1.1.3" xref="S4.SS2.SSS1.1.p1.13.m13.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.13.m13.1b"><apply id="S4.SS2.SSS1.1.p1.13.m13.1.1.cmml" xref="S4.SS2.SSS1.1.p1.13.m13.1.1"><geq id="S4.SS2.SSS1.1.p1.13.m13.1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.13.m13.1.1.1"></geq><ci id="S4.SS2.SSS1.1.p1.13.m13.1.1.2.cmml" xref="S4.SS2.SSS1.1.p1.13.m13.1.1.2">𝜎</ci><cn id="S4.SS2.SSS1.1.p1.13.m13.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.1.p1.13.m13.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.13.m13.1c">\sigma\geq 0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.13.m13.1d">italic_σ ≥ 0</annotation></semantics></math> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx21"> <tbody id="S4.Ex31"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|f\|_{H^{s+\sigma,p}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.Ex31.m1.6"><semantics id="S4.Ex31.m1.6a"><msub id="S4.Ex31.m1.6.7" 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id="S4.Ex31.m1.1.1.1.1.1.1" xref="S4.Ex31.m1.1.1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S4.Ex31.m1.5.5.5.6" xref="S4.Ex31.m1.5.5.5.6.cmml">⁢</mo><mrow id="S4.Ex31.m1.5.5.5.5.2" xref="S4.Ex31.m1.5.5.5.5.3.cmml"><mo id="S4.Ex31.m1.5.5.5.5.2.3" stretchy="false" xref="S4.Ex31.m1.5.5.5.5.3.cmml">(</mo><msup id="S4.Ex31.m1.4.4.4.4.1.1" xref="S4.Ex31.m1.4.4.4.4.1.1.cmml"><mi id="S4.Ex31.m1.4.4.4.4.1.1.2" xref="S4.Ex31.m1.4.4.4.4.1.1.2.cmml">ℝ</mi><mi id="S4.Ex31.m1.4.4.4.4.1.1.3" xref="S4.Ex31.m1.4.4.4.4.1.1.3.cmml">d</mi></msup><mo id="S4.Ex31.m1.5.5.5.5.2.4" xref="S4.Ex31.m1.5.5.5.5.3.cmml">,</mo><msub id="S4.Ex31.m1.5.5.5.5.2.2" xref="S4.Ex31.m1.5.5.5.5.2.2.cmml"><mi id="S4.Ex31.m1.5.5.5.5.2.2.2" xref="S4.Ex31.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S4.Ex31.m1.5.5.5.5.2.2.3" xref="S4.Ex31.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex31.m1.5.5.5.5.2.5" xref="S4.Ex31.m1.5.5.5.5.3.cmml">;</mo><mi id="S4.Ex31.m1.3.3.3.3" xref="S4.Ex31.m1.3.3.3.3.cmml">X</mi><mo id="S4.Ex31.m1.5.5.5.5.2.6" stretchy="false" xref="S4.Ex31.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Ex31.m1.6b"><apply id="S4.Ex31.m1.6.7.cmml" xref="S4.Ex31.m1.6.7"><csymbol cd="ambiguous" id="S4.Ex31.m1.6.7.1.cmml" xref="S4.Ex31.m1.6.7">subscript</csymbol><apply id="S4.Ex31.m1.6.7.2.1.cmml" xref="S4.Ex31.m1.6.7.2.2"><csymbol cd="latexml" id="S4.Ex31.m1.6.7.2.1.1.cmml" xref="S4.Ex31.m1.6.7.2.2.1">norm</csymbol><ci id="S4.Ex31.m1.6.6.cmml" xref="S4.Ex31.m1.6.6">𝑓</ci></apply><apply id="S4.Ex31.m1.5.5.5.cmml" xref="S4.Ex31.m1.5.5.5"><times id="S4.Ex31.m1.5.5.5.6.cmml" xref="S4.Ex31.m1.5.5.5.6"></times><apply id="S4.Ex31.m1.5.5.5.7.cmml" xref="S4.Ex31.m1.5.5.5.7"><csymbol cd="ambiguous" id="S4.Ex31.m1.5.5.5.7.1.cmml" xref="S4.Ex31.m1.5.5.5.7">superscript</csymbol><ci id="S4.Ex31.m1.5.5.5.7.2.cmml" xref="S4.Ex31.m1.5.5.5.7.2">𝐻</ci><list id="S4.Ex31.m1.2.2.2.2.2.3.cmml" xref="S4.Ex31.m1.2.2.2.2.2.2"><apply id="S4.Ex31.m1.2.2.2.2.2.2.1.cmml" xref="S4.Ex31.m1.2.2.2.2.2.2.1"><plus id="S4.Ex31.m1.2.2.2.2.2.2.1.1.cmml" xref="S4.Ex31.m1.2.2.2.2.2.2.1.1"></plus><ci id="S4.Ex31.m1.2.2.2.2.2.2.1.2.cmml" xref="S4.Ex31.m1.2.2.2.2.2.2.1.2">𝑠</ci><ci id="S4.Ex31.m1.2.2.2.2.2.2.1.3.cmml" xref="S4.Ex31.m1.2.2.2.2.2.2.1.3">𝜎</ci></apply><ci id="S4.Ex31.m1.1.1.1.1.1.1.cmml" xref="S4.Ex31.m1.1.1.1.1.1.1">𝑝</ci></list></apply><vector id="S4.Ex31.m1.5.5.5.5.3.cmml" xref="S4.Ex31.m1.5.5.5.5.2"><apply id="S4.Ex31.m1.4.4.4.4.1.1.cmml" xref="S4.Ex31.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S4.Ex31.m1.4.4.4.4.1.1.1.cmml" xref="S4.Ex31.m1.4.4.4.4.1.1">superscript</csymbol><ci id="S4.Ex31.m1.4.4.4.4.1.1.2.cmml" xref="S4.Ex31.m1.4.4.4.4.1.1.2">ℝ</ci><ci id="S4.Ex31.m1.4.4.4.4.1.1.3.cmml" xref="S4.Ex31.m1.4.4.4.4.1.1.3">𝑑</ci></apply><apply id="S4.Ex31.m1.5.5.5.5.2.2.cmml" xref="S4.Ex31.m1.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S4.Ex31.m1.5.5.5.5.2.2.1.cmml" xref="S4.Ex31.m1.5.5.5.5.2.2">subscript</csymbol><ci id="S4.Ex31.m1.5.5.5.5.2.2.2.cmml" xref="S4.Ex31.m1.5.5.5.5.2.2.2">𝑤</ci><ci id="S4.Ex31.m1.5.5.5.5.2.2.3.cmml" xref="S4.Ex31.m1.5.5.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex31.m1.3.3.3.3.cmml" xref="S4.Ex31.m1.3.3.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex31.m1.6c">\displaystyle\|f\|_{H^{s+\sigma,p}(\mathbb{R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex31.m1.6d">∥ italic_f ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s + italic_σ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{\|}(1-\Delta)^{\frac{\sigma}{2}}f\big{\|}_{H^{s,p}(\mathbb{% R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.Ex31.m2.6"><semantics id="S4.Ex31.m2.6a"><mrow 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xref="S4.Ex31.m2.4.4.4.4.1.1.2">ℝ</ci><ci id="S4.Ex31.m2.4.4.4.4.1.1.3.cmml" xref="S4.Ex31.m2.4.4.4.4.1.1.3">𝑑</ci></apply><apply id="S4.Ex31.m2.5.5.5.5.2.2.cmml" xref="S4.Ex31.m2.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S4.Ex31.m2.5.5.5.5.2.2.1.cmml" xref="S4.Ex31.m2.5.5.5.5.2.2">subscript</csymbol><ci id="S4.Ex31.m2.5.5.5.5.2.2.2.cmml" xref="S4.Ex31.m2.5.5.5.5.2.2.2">𝑤</ci><ci id="S4.Ex31.m2.5.5.5.5.2.2.3.cmml" xref="S4.Ex31.m2.5.5.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex31.m2.3.3.3.3.cmml" xref="S4.Ex31.m2.3.3.3.3">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex31.m2.6c">\displaystyle=\big{\|}(1-\Delta)^{\frac{\sigma}{2}}f\big{\|}_{H^{s,p}(\mathbb{% R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex31.m2.6d">= ∥ ( 1 - roman_Δ ) start_POSTSUPERSCRIPT divide start_ARG italic_σ end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex32"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\eqsim\|f\|_{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}+\big{\|}(-% \Delta)^{\frac{\sigma}{2}}f\big{\|}_{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)},% \qquad f\in H^{s+\sigma,p}(\mathbb{R}^{d},w_{\gamma};X)," class="ltx_Math" display="inline" id="S4.Ex32.m1.15"><semantics id="S4.Ex32.m1.15a"><mrow id="S4.Ex32.m1.15.15.1"><mrow id="S4.Ex32.m1.15.15.1.1.2" xref="S4.Ex32.m1.15.15.1.1.3.cmml"><mrow id="S4.Ex32.m1.15.15.1.1.1.1" 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start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT + ∥ ( - roman_Δ ) start_POSTSUPERSCRIPT divide start_ARG italic_σ end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT italic_f ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT , italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s + italic_σ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.1.p1.14">which proves <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 4.2]</cite> for any Banach space <math alttext="X" class="ltx_Math" display="inline" id="S4.SS2.SSS1.1.p1.14.m1.1"><semantics id="S4.SS2.SSS1.1.p1.14.m1.1a"><mi id="S4.SS2.SSS1.1.p1.14.m1.1.1" xref="S4.SS2.SSS1.1.p1.14.m1.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.1.p1.14.m1.1b"><ci id="S4.SS2.SSS1.1.p1.14.m1.1.1.cmml" xref="S4.SS2.SSS1.1.p1.14.m1.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.1.p1.14.m1.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.1.p1.14.m1.1d">italic_X</annotation></semantics></math>. In particular, this answers <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Problem Q.12]</cite>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.2.p2"> <p class="ltx_p" id="S4.SS2.SSS1.2.p2.2">To complete the proof of the proposition we can argue similarly as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 6.1 & Proposition 6.2]</cite> to extend the result from <math alttext="d=1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.2.p2.1.m1.1"><semantics id="S4.SS2.SSS1.2.p2.1.m1.1a"><mrow id="S4.SS2.SSS1.2.p2.1.m1.1.1" xref="S4.SS2.SSS1.2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.2.p2.1.m1.1.1.2" xref="S4.SS2.SSS1.2.p2.1.m1.1.1.2.cmml">d</mi><mo id="S4.SS2.SSS1.2.p2.1.m1.1.1.1" xref="S4.SS2.SSS1.2.p2.1.m1.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.2.p2.1.m1.1.1.3" xref="S4.SS2.SSS1.2.p2.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.2.p2.1.m1.1b"><apply id="S4.SS2.SSS1.2.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS1.2.p2.1.m1.1.1"><eq id="S4.SS2.SSS1.2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.2.p2.1.m1.1.1.1"></eq><ci id="S4.SS2.SSS1.2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.2.p2.1.m1.1.1.2">𝑑</ci><cn id="S4.SS2.SSS1.2.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.2.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.2.p2.1.m1.1c">d=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.2.p2.1.m1.1d">italic_d = 1</annotation></semantics></math> to <math alttext="d\geq 1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.2.p2.2.m2.1"><semantics id="S4.SS2.SSS1.2.p2.2.m2.1a"><mrow id="S4.SS2.SSS1.2.p2.2.m2.1.1" xref="S4.SS2.SSS1.2.p2.2.m2.1.1.cmml"><mi id="S4.SS2.SSS1.2.p2.2.m2.1.1.2" xref="S4.SS2.SSS1.2.p2.2.m2.1.1.2.cmml">d</mi><mo id="S4.SS2.SSS1.2.p2.2.m2.1.1.1" xref="S4.SS2.SSS1.2.p2.2.m2.1.1.1.cmml">≥</mo><mn id="S4.SS2.SSS1.2.p2.2.m2.1.1.3" xref="S4.SS2.SSS1.2.p2.2.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.2.p2.2.m2.1b"><apply id="S4.SS2.SSS1.2.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS1.2.p2.2.m2.1.1"><geq id="S4.SS2.SSS1.2.p2.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.2.p2.2.m2.1.1.1"></geq><ci id="S4.SS2.SSS1.2.p2.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.2.p2.2.m2.1.1.2">𝑑</ci><cn id="S4.SS2.SSS1.2.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.2.p2.2.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.2.p2.2.m2.1c">d\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.2.p2.2.m2.1d">italic_d ≥ 1</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.SSS1.p3"> <p class="ltx_p" id="S4.SS2.SSS1.p3.1">We can now prove a density result for <math alttext="H^{s,p}_{0}(\mathcal{O},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.p3.1.m1.5"><semantics id="S4.SS2.SSS1.p3.1.m1.5a"><mrow id="S4.SS2.SSS1.p3.1.m1.5.5" xref="S4.SS2.SSS1.p3.1.m1.5.5.cmml"><msubsup id="S4.SS2.SSS1.p3.1.m1.5.5.3" xref="S4.SS2.SSS1.p3.1.m1.5.5.3.cmml"><mi id="S4.SS2.SSS1.p3.1.m1.5.5.3.2.2" xref="S4.SS2.SSS1.p3.1.m1.5.5.3.2.2.cmml">H</mi><mn id="S4.SS2.SSS1.p3.1.m1.5.5.3.3" xref="S4.SS2.SSS1.p3.1.m1.5.5.3.3.cmml">0</mn><mrow id="S4.SS2.SSS1.p3.1.m1.2.2.2.4" xref="S4.SS2.SSS1.p3.1.m1.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.p3.1.m1.1.1.1.1" xref="S4.SS2.SSS1.p3.1.m1.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.p3.1.m1.2.2.2.4.1" xref="S4.SS2.SSS1.p3.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.p3.1.m1.2.2.2.2" xref="S4.SS2.SSS1.p3.1.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S4.SS2.SSS1.p3.1.m1.5.5.2" xref="S4.SS2.SSS1.p3.1.m1.5.5.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.p3.1.m1.5.5.1.1" xref="S4.SS2.SSS1.p3.1.m1.5.5.1.2.cmml"><mo 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which will be needed in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6" title="6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6</span></a> to characterise complex interpolation spaces of Bessel potential spaces with vanishing boundary conditions.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.1.1.1">Proposition 4.7</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem7.p1"> <p class="ltx_p" id="S4.Thmtheorem7.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem7.p1.5.5">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.1.1.m1.2"><semantics id="S4.Thmtheorem7.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem7.p1.1.1.m1.2.3" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem7.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S4.Thmtheorem7.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem7.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem7.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem7.p1.1.1.m1.1.1" xref="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem7.p1.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem7.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem7.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem7.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.1.1.m1.2b"><apply id="S4.Thmtheorem7.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.2.3"><in id="S4.Thmtheorem7.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.1"></in><ci id="S4.Thmtheorem7.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.Thmtheorem7.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.2.3.3.2"><cn id="S4.Thmtheorem7.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem7.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem7.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem7.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="s>-1+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.2.2.m2.1"><semantics id="S4.Thmtheorem7.p1.2.2.m2.1a"><mrow id="S4.Thmtheorem7.p1.2.2.m2.1.1" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S4.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">s</mi><mo id="S4.Thmtheorem7.p1.2.2.m2.1.1.1" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.1.cmml">></mo><mrow id="S4.Thmtheorem7.p1.2.2.m2.1.1.3" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.cmml"><mrow id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.cmml"><mo id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2a" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.cmml">−</mo><mn id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.2" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.1" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.1.cmml">+</mo><mfrac id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.cmml"><mrow id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.cmml"><mi id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.2" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.1" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.1.cmml">+</mo><mn id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.3" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.3" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.2.2.m2.1b"><apply id="S4.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1"><gt id="S4.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.1"></gt><ci id="S4.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.2">𝑠</ci><apply id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3"><plus id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.1"></plus><apply id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2"><minus id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2"></minus><cn id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.2.cmml" type="integer" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.2.2">1</cn></apply><apply id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3"><divide id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3"></divide><apply id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2"><plus id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.1.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.1"></plus><ci id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.2.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.2">𝛾</ci><cn id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.2.3">1</cn></apply><ci id="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.3.cmml" xref="S4.Thmtheorem7.p1.2.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.2.2.m2.1c">s>-1+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.2.2.m2.1d">italic_s > - 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> such that <math alttext="s\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.3.3.m3.1"><semantics id="S4.Thmtheorem7.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem7.p1.3.3.m3.1.1" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem7.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.2.cmml">s</mi><mo id="S4.Thmtheorem7.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.1.cmml">∉</mo><mrow id="S4.Thmtheorem7.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.cmml"><msub id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.cmml"><mi id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.2" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.2.cmml">ℕ</mi><mn id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.3" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.3.cmml">0</mn></msub><mo id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.1" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.1.cmml">+</mo><mfrac id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.cmml"><mrow id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.cmml"><mi id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.2" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.2.cmml">γ</mi><mo id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.1" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.1.cmml">+</mo><mn id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.3" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.3" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.3.3.m3.1b"><apply id="S4.Thmtheorem7.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1"><notin id="S4.Thmtheorem7.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.1"></notin><ci id="S4.Thmtheorem7.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.2">𝑠</ci><apply id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3"><plus id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.1"></plus><apply id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2">subscript</csymbol><ci id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.2">ℕ</ci><cn id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.2.3">0</cn></apply><apply id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3"><divide id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3"></divide><apply id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2"><plus id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.1.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.1"></plus><ci id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.2.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.2">𝛾</ci><cn id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.2.3">1</cn></apply><ci id="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.3.cmml" xref="S4.Thmtheorem7.p1.3.3.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.3.3.m3.1c">s\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.3.3.m3.1d">italic_s ∉ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>, <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.4.4.m4.2"><semantics id="S4.Thmtheorem7.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m4.2.2.4" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.4.cmml">γ</mi><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.3" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.3.cmml">∈</mo><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml"><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.3" stretchy="false" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml">(</mo><mrow id="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1a" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.2" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.4" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml">,</mo><mrow id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.cmml"><mi id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.2" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.2.cmml">p</mi><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.1" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.1.cmml">−</mo><mn id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.3" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.5" stretchy="false" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.4.4.m4.2b"><apply id="S4.Thmtheorem7.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2"><in id="S4.Thmtheorem7.p1.4.4.m4.2.2.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.3"></in><ci id="S4.Thmtheorem7.p1.4.4.m4.2.2.4.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.4">𝛾</ci><interval closure="open" id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.3.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2"><apply id="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1"><minus id="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1"></minus><cn id="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem7.p1.4.4.m4.1.1.1.1.1.2">1</cn></apply><apply id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2"><minus id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.1.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.1"></minus><ci id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.2.cmml" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.2">𝑝</ci><cn id="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem7.p1.4.4.m4.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.4.4.m4.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.4.4.m4.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem7.p1.5.5.m5.1"><semantics id="S4.Thmtheorem7.p1.5.5.m5.1a"><mi id="S4.Thmtheorem7.p1.5.5.m5.1.1" xref="S4.Thmtheorem7.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem7.p1.5.5.m5.1b"><ci id="S4.Thmtheorem7.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem7.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem7.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem7.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Then</span></p> <ol class="ltx_enumerate" id="S4.I4"> <li class="ltx_item" id="S4.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S4.I4.i1.p1"> <p class="ltx_p" id="S4.I4.i1.p1.1"><math alttext="H^{s,p}_{0}(\mathbb{R}^{d},w_{\gamma};X)=\overline{C_{\mathrm{c}}^{\infty}(% \mathbb{R}_{\bullet}^{d};X)}^{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.I4.i1.p1.1.m1.12"><semantics id="S4.I4.i1.p1.1.m1.12a"><mrow id="S4.I4.i1.p1.1.m1.12.12" xref="S4.I4.i1.p1.1.m1.12.12.cmml"><mrow id="S4.I4.i1.p1.1.m1.12.12.2" xref="S4.I4.i1.p1.1.m1.12.12.2.cmml"><msubsup id="S4.I4.i1.p1.1.m1.12.12.2.4" xref="S4.I4.i1.p1.1.m1.12.12.2.4.cmml"><mi id="S4.I4.i1.p1.1.m1.12.12.2.4.2.2" xref="S4.I4.i1.p1.1.m1.12.12.2.4.2.2.cmml">H</mi><mn id="S4.I4.i1.p1.1.m1.12.12.2.4.3" xref="S4.I4.i1.p1.1.m1.12.12.2.4.3.cmml">0</mn><mrow id="S4.I4.i1.p1.1.m1.2.2.2.4" xref="S4.I4.i1.p1.1.m1.2.2.2.3.cmml"><mi id="S4.I4.i1.p1.1.m1.1.1.1.1" xref="S4.I4.i1.p1.1.m1.1.1.1.1.cmml">s</mi><mo id="S4.I4.i1.p1.1.m1.2.2.2.4.1" xref="S4.I4.i1.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S4.I4.i1.p1.1.m1.2.2.2.2" xref="S4.I4.i1.p1.1.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S4.I4.i1.p1.1.m1.12.12.2.3" xref="S4.I4.i1.p1.1.m1.12.12.2.3.cmml">⁢</mo><mrow id="S4.I4.i1.p1.1.m1.12.12.2.2.2" xref="S4.I4.i1.p1.1.m1.12.12.2.2.3.cmml"><mo id="S4.I4.i1.p1.1.m1.12.12.2.2.2.3" stretchy="false" xref="S4.I4.i1.p1.1.m1.12.12.2.2.3.cmml">(</mo><msup id="S4.I4.i1.p1.1.m1.11.11.1.1.1.1" xref="S4.I4.i1.p1.1.m1.11.11.1.1.1.1.cmml"><mi id="S4.I4.i1.p1.1.m1.11.11.1.1.1.1.2" xref="S4.I4.i1.p1.1.m1.11.11.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.I4.i1.p1.1.m1.11.11.1.1.1.1.3" xref="S4.I4.i1.p1.1.m1.11.11.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.I4.i1.p1.1.m1.12.12.2.2.2.4" xref="S4.I4.i1.p1.1.m1.12.12.2.2.3.cmml">,</mo><msub id="S4.I4.i1.p1.1.m1.12.12.2.2.2.2" xref="S4.I4.i1.p1.1.m1.12.12.2.2.2.2.cmml"><mi 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xref="S4.I4.i1.p1.1.m1.4.4.2.4.2.3.cmml">c</mi><mi id="S4.I4.i1.p1.1.m1.4.4.2.4.3" mathvariant="normal" xref="S4.I4.i1.p1.1.m1.4.4.2.4.3.cmml">∞</mi></msubsup><mo id="S4.I4.i1.p1.1.m1.4.4.2.3" xref="S4.I4.i1.p1.1.m1.4.4.2.3.cmml">⁢</mo><mrow id="S4.I4.i1.p1.1.m1.4.4.2.2.1" xref="S4.I4.i1.p1.1.m1.4.4.2.2.2.cmml"><mo id="S4.I4.i1.p1.1.m1.4.4.2.2.1.2" stretchy="false" xref="S4.I4.i1.p1.1.m1.4.4.2.2.2.cmml">(</mo><msubsup id="S4.I4.i1.p1.1.m1.4.4.2.2.1.1" xref="S4.I4.i1.p1.1.m1.4.4.2.2.1.1.cmml"><mi id="S4.I4.i1.p1.1.m1.4.4.2.2.1.1.2.2" xref="S4.I4.i1.p1.1.m1.4.4.2.2.1.1.2.2.cmml">ℝ</mi><mo id="S4.I4.i1.p1.1.m1.4.4.2.2.1.1.2.3" xref="S4.I4.i1.p1.1.m1.4.4.2.2.1.1.2.3.cmml">∙</mo><mi id="S4.I4.i1.p1.1.m1.4.4.2.2.1.1.3" xref="S4.I4.i1.p1.1.m1.4.4.2.2.1.1.3.cmml">d</mi></msubsup><mo id="S4.I4.i1.p1.1.m1.4.4.2.2.1.3" xref="S4.I4.i1.p1.1.m1.4.4.2.2.2.cmml">;</mo><mi id="S4.I4.i1.p1.1.m1.3.3.1.1" xref="S4.I4.i1.p1.1.m1.3.3.1.1.cmml">X</mi><mo id="S4.I4.i1.p1.1.m1.4.4.2.2.1.4" stretchy="false" 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id="S4.I4.i1.p1.1.m1.12c">H^{s,p}_{0}(\mathbb{R}^{d},w_{\gamma};X)=\overline{C_{\mathrm{c}}^{\infty}(% \mathbb{R}_{\bullet}^{d};X)}^{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i1.p1.1.m1.12d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = over¯ start_ARG italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X ) end_ARG start_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I4.i1.p1.1.1">,</span></p> </div> </li> <li class="ltx_item" id="S4.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S4.I4.i2.p1"> <p class="ltx_p" id="S4.I4.i2.p1.1"><math alttext="H^{s,p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\overline{C_{\mathrm{c}}^{\infty}% (\mathbb{R}^{d}_{+};X)}^{H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.I4.i2.p1.1.m1.12"><semantics id="S4.I4.i2.p1.1.m1.12a"><mrow id="S4.I4.i2.p1.1.m1.12.12" xref="S4.I4.i2.p1.1.m1.12.12.cmml"><mrow id="S4.I4.i2.p1.1.m1.12.12.2" xref="S4.I4.i2.p1.1.m1.12.12.2.cmml"><msubsup id="S4.I4.i2.p1.1.m1.12.12.2.4" xref="S4.I4.i2.p1.1.m1.12.12.2.4.cmml"><mi id="S4.I4.i2.p1.1.m1.12.12.2.4.2.2" xref="S4.I4.i2.p1.1.m1.12.12.2.4.2.2.cmml">H</mi><mn id="S4.I4.i2.p1.1.m1.12.12.2.4.3" xref="S4.I4.i2.p1.1.m1.12.12.2.4.3.cmml">0</mn><mrow 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(\mathbb{R}^{d}_{+};X)}^{H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.I4.i2.p1.1.m1.12d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = over¯ start_ARG italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) end_ARG start_POSTSUPERSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUPERSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S4.I4.i2.p1.1.1">.</span></p> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S4.SS2.SSS1.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.SSS1.3.p1"> <p class="ltx_p" id="S4.SS2.SSS1.3.p1.6">Let <math alttext="m\in\mathbb{N}_{0}\cup\{-1\}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.3.p1.1.m1.1"><semantics id="S4.SS2.SSS1.3.p1.1.m1.1a"><mrow id="S4.SS2.SSS1.3.p1.1.m1.1.1" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS1.3.p1.1.m1.1.1.3" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.3.cmml">m</mi><mo id="S4.SS2.SSS1.3.p1.1.m1.1.1.2" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.2.cmml">∈</mo><mrow id="S4.SS2.SSS1.3.p1.1.m1.1.1.1" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.cmml"><msub id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.2" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.3" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.3.cmml">0</mn></msub><mo id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.2" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.2.cmml">∪</mo><mrow id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.2.cmml"><mo id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.2.cmml">{</mo><mrow id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1a" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.2.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.3.p1.1.m1.1b"><apply id="S4.SS2.SSS1.3.p1.1.m1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1"><in id="S4.SS2.SSS1.3.p1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.2"></in><ci id="S4.SS2.SSS1.3.p1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.3">𝑚</ci><apply id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1"><union id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.2"></union><apply id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.3.3">0</cn></apply><set id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1"><apply id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1"><minus id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1"></minus><cn id="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.2.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.1.m1.1.1.1.1.1.1.2">1</cn></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.3.p1.1.m1.1c">m\in\mathbb{N}_{0}\cup\{-1\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.3.p1.1.m1.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∪ { - 1 }</annotation></semantics></math> be such that <math alttext="m+\frac{\gamma+1}{p}<s<m+1+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.3.p1.2.m2.1"><semantics id="S4.SS2.SSS1.3.p1.2.m2.1a"><mrow id="S4.SS2.SSS1.3.p1.2.m2.1.1" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS1.3.p1.2.m2.1.1.2" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.cmml"><mi id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.2" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.2.cmml">m</mi><mo id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.1" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.1.cmml">+</mo><mfrac id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.cmml"><mrow id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.cmml"><mi id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.2" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.2.cmml">γ</mi><mo id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.1" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.1.cmml">+</mo><mn id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.3" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.3" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S4.SS2.SSS1.3.p1.2.m2.1.1.3" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.3.cmml"><</mo><mi id="S4.SS2.SSS1.3.p1.2.m2.1.1.4" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.4.cmml">s</mi><mo id="S4.SS2.SSS1.3.p1.2.m2.1.1.5" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.5.cmml"><</mo><mrow id="S4.SS2.SSS1.3.p1.2.m2.1.1.6" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.cmml"><mi id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.2" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.2.cmml">m</mi><mo id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.1" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.1.cmml">+</mo><mn id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.3" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.3.cmml">1</mn><mo id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.1a" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.1.cmml">+</mo><mfrac id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.cmml"><mrow id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.cmml"><mi id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.2" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.2.cmml">γ</mi><mo id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.1" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.1.cmml">+</mo><mn id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.3" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.3.cmml">1</mn></mrow><mi id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.3" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.3.p1.2.m2.1b"><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1"><and id="S4.SS2.SSS1.3.p1.2.m2.1.1a.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1"></and><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1b.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1"><lt id="S4.SS2.SSS1.3.p1.2.m2.1.1.3.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.3"></lt><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2"><plus id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.1.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.1"></plus><ci id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.2.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.2">𝑚</ci><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3"><divide id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.1.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3"></divide><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2"><plus id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.1.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.1"></plus><ci id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.2.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.2">𝛾</ci><cn id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.3.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.2.3">1</cn></apply><ci id="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.3.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.2.3.3">𝑝</ci></apply></apply><ci id="S4.SS2.SSS1.3.p1.2.m2.1.1.4.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.4">𝑠</ci></apply><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1c.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1"><lt id="S4.SS2.SSS1.3.p1.2.m2.1.1.5.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S4.SS2.SSS1.3.p1.2.m2.1.1.4.cmml" id="S4.SS2.SSS1.3.p1.2.m2.1.1d.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1"></share><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6"><plus id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.1.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.1"></plus><ci id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.2.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.2">𝑚</ci><cn id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.3.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.3">1</cn><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4"><divide id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.1.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4"></divide><apply id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2"><plus id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.1.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.1"></plus><ci id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.2.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.2">𝛾</ci><cn id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.3.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.2.3">1</cn></apply><ci id="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.3.cmml" xref="S4.SS2.SSS1.3.p1.2.m2.1.1.6.4.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.3.p1.2.m2.1c">m+\frac{\gamma+1}{p}<s<m+1+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.3.p1.2.m2.1d">italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_s < italic_m + 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Recall that for <math alttext="m=-1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.3.p1.3.m3.1"><semantics id="S4.SS2.SSS1.3.p1.3.m3.1a"><mrow id="S4.SS2.SSS1.3.p1.3.m3.1.1" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.cmml"><mi id="S4.SS2.SSS1.3.p1.3.m3.1.1.2" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS1.3.p1.3.m3.1.1.1" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.1.cmml">=</mo><mrow id="S4.SS2.SSS1.3.p1.3.m3.1.1.3" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.3.cmml"><mo id="S4.SS2.SSS1.3.p1.3.m3.1.1.3a" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.3.cmml">−</mo><mn id="S4.SS2.SSS1.3.p1.3.m3.1.1.3.2" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.3.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.3.p1.3.m3.1b"><apply id="S4.SS2.SSS1.3.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS1.3.p1.3.m3.1.1"><eq id="S4.SS2.SSS1.3.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.1"></eq><ci id="S4.SS2.SSS1.3.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.2">𝑚</ci><apply id="S4.SS2.SSS1.3.p1.3.m3.1.1.3.cmml" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.3"><minus id="S4.SS2.SSS1.3.p1.3.m3.1.1.3.1.cmml" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.3"></minus><cn id="S4.SS2.SSS1.3.p1.3.m3.1.1.3.2.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.3.m3.1.1.3.2">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.3.p1.3.m3.1c">m=-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.3.p1.3.m3.1d">italic_m = - 1</annotation></semantics></math> the trace does not exist and the arguments below can easily be adapted to this case. Moreover, we note that derivatives in the directions <math alttext="\widetilde{x}=(x_{2},\dots,x_{d})" class="ltx_Math" display="inline" id="S4.SS2.SSS1.3.p1.4.m4.3"><semantics id="S4.SS2.SSS1.3.p1.4.m4.3a"><mrow id="S4.SS2.SSS1.3.p1.4.m4.3.3" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.cmml"><mover accent="true" id="S4.SS2.SSS1.3.p1.4.m4.3.3.4" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.4.cmml"><mi id="S4.SS2.SSS1.3.p1.4.m4.3.3.4.2" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.4.2.cmml">x</mi><mo id="S4.SS2.SSS1.3.p1.4.m4.3.3.4.1" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.4.1.cmml">~</mo></mover><mo id="S4.SS2.SSS1.3.p1.4.m4.3.3.3" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.3.cmml">=</mo><mrow id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.3.cmml"><mo id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.3" stretchy="false" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.3.cmml">(</mo><msub id="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1" xref="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.2" xref="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.2.cmml">x</mi><mn id="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.3" xref="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.3.cmml">2</mn></msub><mo id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.4" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.3.p1.4.m4.1.1" mathvariant="normal" xref="S4.SS2.SSS1.3.p1.4.m4.1.1.cmml">…</mi><mo id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.5" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.cmml"><mi id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.2" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.2.cmml">x</mi><mi id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.3" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.3.cmml">d</mi></msub><mo id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.6" stretchy="false" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.3.p1.4.m4.3b"><apply id="S4.SS2.SSS1.3.p1.4.m4.3.3.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3"><eq id="S4.SS2.SSS1.3.p1.4.m4.3.3.3.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.3"></eq><apply id="S4.SS2.SSS1.3.p1.4.m4.3.3.4.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.4"><ci id="S4.SS2.SSS1.3.p1.4.m4.3.3.4.1.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.4.1">~</ci><ci id="S4.SS2.SSS1.3.p1.4.m4.3.3.4.2.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.4.2">𝑥</ci></apply><vector id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.3.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2"><apply id="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.2">𝑥</ci><cn id="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.4.m4.2.2.1.1.1.3">2</cn></apply><ci id="S4.SS2.SSS1.3.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.1.1">…</ci><apply id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.1.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.2.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.2">𝑥</ci><ci id="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.3.cmml" xref="S4.SS2.SSS1.3.p1.4.m4.3.3.2.2.2.3">𝑑</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.3.p1.4.m4.3c">\widetilde{x}=(x_{2},\dots,x_{d})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.3.p1.4.m4.3d">over~ start_ARG italic_x end_ARG = ( italic_x start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_x start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT )</annotation></semantics></math> commute with the trace operator and therefore for <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.3.p1.5.m5.1"><semantics id="S4.SS2.SSS1.3.p1.5.m5.1a"><mrow id="S4.SS2.SSS1.3.p1.5.m5.1.1" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.cmml"><mi id="S4.SS2.SSS1.3.p1.5.m5.1.1.2" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS1.3.p1.5.m5.1.1.1" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS1.3.p1.5.m5.1.1.3" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.3.cmml"><mi id="S4.SS2.SSS1.3.p1.5.m5.1.1.3.2" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS1.3.p1.5.m5.1.1.3.3" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.3.p1.5.m5.1b"><apply id="S4.SS2.SSS1.3.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS1.3.p1.5.m5.1.1"><in id="S4.SS2.SSS1.3.p1.5.m5.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.1"></in><ci id="S4.SS2.SSS1.3.p1.5.m5.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.2">𝑚</ci><apply id="S4.SS2.SSS1.3.p1.5.m5.1.1.3.cmml" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.3.p1.5.m5.1.1.3.1.cmml" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.3.p1.5.m5.1.1.3.2.cmml" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS1.3.p1.5.m5.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.3.p1.5.m5.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.3.p1.5.m5.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.3.p1.5.m5.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.3.p1.6.m6.2"><semantics id="S4.SS2.SSS1.3.p1.6.m6.2a"><mrow id="S4.SS2.SSS1.3.p1.6.m6.2.2" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.SSS1.3.p1.6.m6.2.2.4" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.4.cmml">𝒪</mi><mo id="S4.SS2.SSS1.3.p1.6.m6.2.2.3" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.3.cmml">∈</mo><mrow id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.3.cmml"><mo id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.3.cmml">{</mo><msup id="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1" xref="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.2" xref="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.3" xref="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.4" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.3.cmml">,</mo><msubsup id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.2" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.3" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.3.cmml">+</mo><mi id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.3" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.5" stretchy="false" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.3.p1.6.m6.2b"><apply id="S4.SS2.SSS1.3.p1.6.m6.2.2.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2"><in id="S4.SS2.SSS1.3.p1.6.m6.2.2.3.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.3"></in><ci id="S4.SS2.SSS1.3.p1.6.m6.2.2.4.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.4">𝒪</ci><set id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.3.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2"><apply id="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.1.1.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2">superscript</csymbol><ci id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.2">ℝ</ci><ci id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.3.p1.6.m6.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.3.p1.6.m6.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.3.p1.6.m6.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math> it holds that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{s,p}_{0}(\mathcal{O},w_{\gamma};X)=\{f\in H^{s,p}(\mathcal{O},w_{\gamma};X)% :\overline{\operatorname{Tr}}_{m}f=0\}." class="ltx_Math" display="block" id="S4.Ex33.m1.9"><semantics id="S4.Ex33.m1.9a"><mrow id="S4.Ex33.m1.9.9.1" xref="S4.Ex33.m1.9.9.1.1.cmml"><mrow id="S4.Ex33.m1.9.9.1.1" xref="S4.Ex33.m1.9.9.1.1.cmml"><mrow id="S4.Ex33.m1.9.9.1.1.1" xref="S4.Ex33.m1.9.9.1.1.1.cmml"><msubsup id="S4.Ex33.m1.9.9.1.1.1.3" xref="S4.Ex33.m1.9.9.1.1.1.3.cmml"><mi id="S4.Ex33.m1.9.9.1.1.1.3.2.2" xref="S4.Ex33.m1.9.9.1.1.1.3.2.2.cmml">H</mi><mn id="S4.Ex33.m1.9.9.1.1.1.3.3" xref="S4.Ex33.m1.9.9.1.1.1.3.3.cmml">0</mn><mrow id="S4.Ex33.m1.2.2.2.4" xref="S4.Ex33.m1.2.2.2.3.cmml"><mi id="S4.Ex33.m1.1.1.1.1" xref="S4.Ex33.m1.1.1.1.1.cmml">s</mi><mo id="S4.Ex33.m1.2.2.2.4.1" xref="S4.Ex33.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex33.m1.2.2.2.2" xref="S4.Ex33.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S4.Ex33.m1.9.9.1.1.1.2" xref="S4.Ex33.m1.9.9.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex33.m1.9.9.1.1.1.1.1" xref="S4.Ex33.m1.9.9.1.1.1.1.2.cmml"><mo id="S4.Ex33.m1.9.9.1.1.1.1.1.2" stretchy="false" xref="S4.Ex33.m1.9.9.1.1.1.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S4.Ex33.m1.5.5" xref="S4.Ex33.m1.5.5.cmml">𝒪</mi><mo id="S4.Ex33.m1.9.9.1.1.1.1.1.3" xref="S4.Ex33.m1.9.9.1.1.1.1.2.cmml">,</mo><msub id="S4.Ex33.m1.9.9.1.1.1.1.1.1" xref="S4.Ex33.m1.9.9.1.1.1.1.1.1.cmml"><mi id="S4.Ex33.m1.9.9.1.1.1.1.1.1.2" xref="S4.Ex33.m1.9.9.1.1.1.1.1.1.2.cmml">w</mi><mi id="S4.Ex33.m1.9.9.1.1.1.1.1.1.3" xref="S4.Ex33.m1.9.9.1.1.1.1.1.1.3.cmml">γ</mi></msub><mo id="S4.Ex33.m1.9.9.1.1.1.1.1.4" xref="S4.Ex33.m1.9.9.1.1.1.1.2.cmml">;</mo><mi 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xref="S4.Ex33.m1.9.9.1.1.3.2.2.2.3">𝑓</ci></apply><cn id="S4.Ex33.m1.9.9.1.1.3.2.2.3.cmml" type="integer" xref="S4.Ex33.m1.9.9.1.1.3.2.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex33.m1.9c">H^{s,p}_{0}(\mathcal{O},w_{\gamma};X)=\{f\in H^{s,p}(\mathcal{O},w_{\gamma};X)% :\overline{\operatorname{Tr}}_{m}f=0\}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex33.m1.9d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = { italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) : over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0 } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S4.SS2.SSS1.4.p2"> <p class="ltx_p" id="S4.SS2.SSS1.4.p2.6"><span class="ltx_text ltx_font_italic" id="S4.SS2.SSS1.4.p2.6.1">Step 1: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I4.i1" title="item i ‣ Proposition 4.7. ‣ 4.2.1. Bessel potential spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>. </span> For <math alttext="f\in C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.1.m1.2"><semantics id="S4.SS2.SSS1.4.p2.1.m1.2a"><mrow id="S4.SS2.SSS1.4.p2.1.m1.2.2" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.1.m1.2.2.3" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.3.cmml">f</mi><mo id="S4.SS2.SSS1.4.p2.1.m1.2.2.2" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS1.4.p2.1.m1.2.2.1" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.cmml"><msubsup id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.cmml"><mi id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.2" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.2" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.2.cmml"><mo id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.2" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.3" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.3.cmml">∙</mo><mi id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.3" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.3" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.4.p2.1.m1.1.1" xref="S4.SS2.SSS1.4.p2.1.m1.1.1.cmml">X</mi><mo id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.4" stretchy="false" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.1.m1.2b"><apply id="S4.SS2.SSS1.4.p2.1.m1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2"><in id="S4.SS2.SSS1.4.p2.1.m1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.2"></in><ci id="S4.SS2.SSS1.4.p2.1.m1.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.3">𝑓</ci><apply id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1"><times id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.2"></times><apply id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.3.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.3.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.3.3"></infinity></apply><list id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1"><apply id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.2.3">∙</ci></apply><ci id="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.2.2.1.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.4.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.1.m1.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.1.m1.2c">f\in C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.1.m1.2d">italic_f ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> it is clear that <math alttext="\overline{\operatorname{Tr}}_{m}f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.2.m2.1"><semantics id="S4.SS2.SSS1.4.p2.2.m2.1a"><mrow id="S4.SS2.SSS1.4.p2.2.m2.1.1" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS1.4.p2.2.m2.1.1.2" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.cmml"><msub id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.2" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.1" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.3" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.1" lspace="0.167em" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.3" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.3.cmml">f</mi></mrow><mo id="S4.SS2.SSS1.4.p2.2.m2.1.1.1" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.4.p2.2.m2.1.1.3" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.2.m2.1b"><apply id="S4.SS2.SSS1.4.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1"><eq id="S4.SS2.SSS1.4.p2.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.1"></eq><apply id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2"><times id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.1"></times><apply id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2"><ci id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS2.SSS1.4.p2.2.m2.1.1.2.3.cmml" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.2.3">𝑓</ci></apply><cn id="S4.SS2.SSS1.4.p2.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.2.m2.1c">\overline{\operatorname{Tr}}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.2.m2.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math> and by continuity (see Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem5" title="Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.5</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I3.i1" title="item i ‣ Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>) this extends to the closure of <math alttext="C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.3.m3.2"><semantics id="S4.SS2.SSS1.4.p2.3.m3.2a"><mrow id="S4.SS2.SSS1.4.p2.3.m3.2.2" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.cmml"><msubsup id="S4.SS2.SSS1.4.p2.3.m3.2.2.3" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3.cmml"><mi id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.2" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.4.p2.3.m3.2.2.2" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.2.cmml"><mo id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.2" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.3" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.3.cmml">∙</mo><mi id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.3" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.3" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.4.p2.3.m3.1.1" xref="S4.SS2.SSS1.4.p2.3.m3.1.1.cmml">X</mi><mo id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.4" stretchy="false" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.3.m3.2b"><apply id="S4.SS2.SSS1.4.p2.3.m3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2"><times id="S4.SS2.SSS1.4.p2.3.m3.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.2"></times><apply id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.1.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.3.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.4.p2.3.m3.2.2.3.3.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.3.3"></infinity></apply><list id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.2.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1"><apply id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.2.3">∙</ci></apply><ci id="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.2.2.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.4.p2.3.m3.1.1.cmml" xref="S4.SS2.SSS1.4.p2.3.m3.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.3.m3.2c">C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.3.m3.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>. Conversely, let <math alttext="f\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.4.m4.5"><semantics id="S4.SS2.SSS1.4.p2.4.m4.5a"><mrow id="S4.SS2.SSS1.4.p2.4.m4.5.5" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.cmml"><mi id="S4.SS2.SSS1.4.p2.4.m4.5.5.4" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.4.cmml">f</mi><mo id="S4.SS2.SSS1.4.p2.4.m4.5.5.3" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.3.cmml">∈</mo><mrow id="S4.SS2.SSS1.4.p2.4.m4.5.5.2" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.cmml"><msup id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4.cmml"><mi id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4.2" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.4.p2.4.m4.2.2.2.4" xref="S4.SS2.SSS1.4.p2.4.m4.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.4.p2.4.m4.1.1.1.1" xref="S4.SS2.SSS1.4.p2.4.m4.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.4.p2.4.m4.2.2.2.4.1" xref="S4.SS2.SSS1.4.p2.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.4.p2.4.m4.2.2.2.2" xref="S4.SS2.SSS1.4.p2.4.m4.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.3" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.3.cmml"><mo id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.3.cmml">(</mo><msup id="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1" xref="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.2" xref="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.3" xref="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.4" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.2" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.3" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.5" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.3.cmml">;</mo><mi id="S4.SS2.SSS1.4.p2.4.m4.3.3" xref="S4.SS2.SSS1.4.p2.4.m4.3.3.cmml">X</mi><mo id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.6" stretchy="false" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.4.m4.5b"><apply id="S4.SS2.SSS1.4.p2.4.m4.5.5.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5"><in id="S4.SS2.SSS1.4.p2.4.m4.5.5.3.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.3"></in><ci id="S4.SS2.SSS1.4.p2.4.m4.5.5.4.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.4">𝑓</ci><apply id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2"><times id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.3.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.3"></times><apply id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4.1.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4.2.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.4.2">𝐻</ci><list id="S4.SS2.SSS1.4.p2.4.m4.2.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.2.2.2.4"><ci id="S4.SS2.SSS1.4.p2.4.m4.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.1.1.1.1">𝑠</ci><ci id="S4.SS2.SSS1.4.p2.4.m4.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2"><apply id="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.4.p2.4.m4.3.3.cmml" xref="S4.SS2.SSS1.4.p2.4.m4.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.4.m4.5c">f\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.4.m4.5d">italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\overline{\operatorname{Tr}}_{m}f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.5.m5.1"><semantics id="S4.SS2.SSS1.4.p2.5.m5.1a"><mrow id="S4.SS2.SSS1.4.p2.5.m5.1.1" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.cmml"><mrow id="S4.SS2.SSS1.4.p2.5.m5.1.1.2" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.cmml"><msub id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.2" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.1" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.3" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.1" lspace="0.167em" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.3" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.3.cmml">f</mi></mrow><mo id="S4.SS2.SSS1.4.p2.5.m5.1.1.1" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.4.p2.5.m5.1.1.3" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.5.m5.1b"><apply id="S4.SS2.SSS1.4.p2.5.m5.1.1.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1"><eq id="S4.SS2.SSS1.4.p2.5.m5.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.1"></eq><apply id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2"><times id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.1"></times><apply id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2"><ci id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS2.SSS1.4.p2.5.m5.1.1.2.3.cmml" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.2.3">𝑓</ci></apply><cn id="S4.SS2.SSS1.4.p2.5.m5.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.5.m5.1c">\overline{\operatorname{Tr}}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.5.m5.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math>. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 3.4]</cite> there exists a sequence <math alttext="(f^{(1)}_{n})_{n\geq 1}\subseteq C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.6.m6.4"><semantics id="S4.SS2.SSS1.4.p2.6.m6.4a"><mrow id="S4.SS2.SSS1.4.p2.6.m6.4.4" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.cmml"><msub id="S4.SS2.SSS1.4.p2.6.m6.3.3.1" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.cmml"><mrow id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.cmml">(</mo><msubsup id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.2.2" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.3" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.6.m6.1.1.1.3" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.4.p2.6.m6.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.6.m6.1.1.1.1" xref="S4.SS2.SSS1.4.p2.6.m6.1.1.1.1.cmml">1</mn><mo id="S4.SS2.SSS1.4.p2.6.m6.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.1.1.1.cmml">)</mo></mrow><mrow id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.cmml"><mi id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.2" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.2.cmml">n</mi><mo id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.1" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.1.cmml">≥</mo><mn id="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.3" xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.4.p2.6.m6.4.4.3" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.3.cmml">⊆</mo><mrow id="S4.SS2.SSS1.4.p2.6.m6.4.4.2" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.cmml"><msubsup id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.cmml"><mi id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.2" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.2" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.2.cmml"><mo id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.2.cmml">(</mo><msup id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.2" 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xref="S4.SS2.SSS1.4.p2.6.m6.3.3.1.3.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2"><times id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.2.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.2"></times><apply id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.1.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.3.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.3.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.3.3"></infinity></apply><list id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.2.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1"><apply id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.4.4.2.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.4.p2.6.m6.2.2.cmml" xref="S4.SS2.SSS1.4.p2.6.m6.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.6.m6.4c">(f^{(1)}_{n})_{n\geq 1}\subseteq C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.6.m6.4d">( italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 1 end_POSTSUBSCRIPT ⊆ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f^{(1)}_{n}\to f\quad\text{ in }H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)\,\text{ % as }\,n\to\infty." class="ltx_Math" display="block" id="S4.E5.m1.5"><semantics id="S4.E5.m1.5a"><mrow id="S4.E5.m1.5.5.1"><mrow id="S4.E5.m1.5.5.1.1.2" xref="S4.E5.m1.5.5.1.1.3.cmml"><mrow id="S4.E5.m1.5.5.1.1.1.1" xref="S4.E5.m1.5.5.1.1.1.1.cmml"><msubsup id="S4.E5.m1.5.5.1.1.1.1.2" xref="S4.E5.m1.5.5.1.1.1.1.2.cmml"><mi id="S4.E5.m1.5.5.1.1.1.1.2.2.2" xref="S4.E5.m1.5.5.1.1.1.1.2.2.2.cmml">f</mi><mi id="S4.E5.m1.5.5.1.1.1.1.2.3" 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.4.p2.13">Let <math alttext="\overline{\operatorname{ext}}_{m}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.7.m1.1"><semantics id="S4.SS2.SSS1.4.p2.7.m1.1a"><msub id="S4.SS2.SSS1.4.p2.7.m1.1.1" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.cmml"><mover accent="true" id="S4.SS2.SSS1.4.p2.7.m1.1.1.2" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.2.cmml"><mi id="S4.SS2.SSS1.4.p2.7.m1.1.1.2.2" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.2.2.cmml">ext</mi><mo id="S4.SS2.SSS1.4.p2.7.m1.1.1.2.1" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.4.p2.7.m1.1.1.3" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.7.m1.1b"><apply id="S4.SS2.SSS1.4.p2.7.m1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.7.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.7.m1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.7.m1.1.1">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.7.m1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.2"><ci id="S4.SS2.SSS1.4.p2.7.m1.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.7.m1.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.2.2">ext</ci></apply><ci id="S4.SS2.SSS1.4.p2.7.m1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.7.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.7.m1.1c">\overline{\operatorname{ext}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.7.m1.1d">over¯ start_ARG roman_ext end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> be as in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem5" title="Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.5</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I3.i1" title="item i ‣ Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> and define <math alttext="f^{(2)}_{n}:=f^{(1)}_{n}-\overline{\operatorname{ext}}_{m}(\overline{% \operatorname{Tr}}_{m}f^{(1)}_{n})" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.8.m2.4"><semantics id="S4.SS2.SSS1.4.p2.8.m2.4a"><mrow id="S4.SS2.SSS1.4.p2.8.m2.4.4" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.cmml"><msubsup id="S4.SS2.SSS1.4.p2.8.m2.4.4.3" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3.cmml"><mi id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.2.2" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.3" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.8.m2.1.1.1.3" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3.cmml"><mo id="S4.SS2.SSS1.4.p2.8.m2.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.8.m2.1.1.1.1" 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xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.3" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.8.m2.3.3.1.3" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.cmml"><mo id="S4.SS2.SSS1.4.p2.8.m2.3.3.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.8.m2.3.3.1.1" xref="S4.SS2.SSS1.4.p2.8.m2.3.3.1.1.cmml">1</mn><mo id="S4.SS2.SSS1.4.p2.8.m2.3.3.1.3.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.8.m2.4b"><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4"><csymbol cd="latexml" id="S4.SS2.SSS1.4.p2.8.m2.4.4.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.2">assign</csymbol><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3.2.2">𝑓</ci><cn id="S4.SS2.SSS1.4.p2.8.m2.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.8.m2.1.1.1.1">2</cn></apply><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.3.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.3.3">𝑛</ci></apply><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1"><minus id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.2"></minus><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.2.2">𝑓</ci><cn id="S4.SS2.SSS1.4.p2.8.m2.2.2.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.8.m2.2.2.1.1">1</cn></apply><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.3.3">𝑛</ci></apply><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1"><times id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.2"></times><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.2"><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.2.2">ext</ci></apply><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.3.3">𝑚</ci></apply><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1"><times id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.1"></times><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.2"><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.2.3">𝑚</ci></apply><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.2.2">𝑓</ci><cn id="S4.SS2.SSS1.4.p2.8.m2.3.3.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.8.m2.3.3.1.1">1</cn></apply><ci id="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.4.p2.8.m2.4.4.1.1.1.1.1.3.3">𝑛</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.8.m2.4c">f^{(2)}_{n}:=f^{(1)}_{n}-\overline{\operatorname{ext}}_{m}(\overline{% \operatorname{Tr}}_{m}f^{(1)}_{n})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.8.m2.4d">italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT := italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT - over¯ start_ARG roman_ext end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> for <math alttext="n\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.9.m3.1"><semantics id="S4.SS2.SSS1.4.p2.9.m3.1a"><mrow id="S4.SS2.SSS1.4.p2.9.m3.1.1" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.9.m3.1.1.2" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.4.p2.9.m3.1.1.1" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS1.4.p2.9.m3.1.1.3" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.3.cmml"><mi id="S4.SS2.SSS1.4.p2.9.m3.1.1.3.2" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS1.4.p2.9.m3.1.1.3.3" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.9.m3.1b"><apply id="S4.SS2.SSS1.4.p2.9.m3.1.1.cmml" xref="S4.SS2.SSS1.4.p2.9.m3.1.1"><in id="S4.SS2.SSS1.4.p2.9.m3.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.1"></in><ci id="S4.SS2.SSS1.4.p2.9.m3.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.2">𝑛</ci><apply id="S4.SS2.SSS1.4.p2.9.m3.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.9.m3.1.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.9.m3.1.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS1.4.p2.9.m3.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.9.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.9.m3.1c">n\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.9.m3.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Then by Remark <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem3" title="Remark 4.3. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.3</span></a> we have <math alttext="f^{(2)}_{n}\in C^{\infty}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.10.m4.3"><semantics id="S4.SS2.SSS1.4.p2.10.m4.3a"><mrow id="S4.SS2.SSS1.4.p2.10.m4.3.3" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.cmml"><msubsup id="S4.SS2.SSS1.4.p2.10.m4.3.3.3" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.cmml"><mi id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.2.2" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.3" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.10.m4.1.1.1.3" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.cmml"><mo id="S4.SS2.SSS1.4.p2.10.m4.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.10.m4.1.1.1.1" xref="S4.SS2.SSS1.4.p2.10.m4.1.1.1.1.cmml">2</mn><mo id="S4.SS2.SSS1.4.p2.10.m4.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.cmml">)</mo></mrow></msubsup><mo id="S4.SS2.SSS1.4.p2.10.m4.3.3.2" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.2.cmml">∈</mo><mrow id="S4.SS2.SSS1.4.p2.10.m4.3.3.1" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.cmml"><msup id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.cmml"><mi id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.2" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.2.cmml">C</mi><mi id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.3.cmml">∞</mi></msup><mo id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.2" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.2.cmml"><mo id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.2.cmml">(</mo><msup id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.2" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.3" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.3" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.4.p2.10.m4.2.2" xref="S4.SS2.SSS1.4.p2.10.m4.2.2.cmml">X</mi><mo id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.4" stretchy="false" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.10.m4.3b"><apply id="S4.SS2.SSS1.4.p2.10.m4.3.3.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3"><in id="S4.SS2.SSS1.4.p2.10.m4.3.3.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.2"></in><apply id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.1.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.2.2">𝑓</ci><cn id="S4.SS2.SSS1.4.p2.10.m4.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.10.m4.1.1.1.1">2</cn></apply><ci id="S4.SS2.SSS1.4.p2.10.m4.3.3.3.3.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.3.3">𝑛</ci></apply><apply id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1"><times id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.2"></times><apply id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.2">𝐶</ci><infinity id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.3.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.3.3"></infinity></apply><list id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1"><apply id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.3.3.1.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.4.p2.10.m4.2.2.cmml" xref="S4.SS2.SSS1.4.p2.10.m4.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.10.m4.3c">f^{(2)}_{n}\in C^{\infty}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.10.m4.3d">italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> and from Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem5" title="Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.5</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I3.i1" title="item i ‣ Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> it follows that <math alttext="\overline{\operatorname{Tr}}_{m}f^{(2)}_{n}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.11.m5.1"><semantics id="S4.SS2.SSS1.4.p2.11.m5.1a"><mrow id="S4.SS2.SSS1.4.p2.11.m5.1.2" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.cmml"><mrow id="S4.SS2.SSS1.4.p2.11.m5.1.2.2" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.cmml"><msub id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.2" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.1" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.3" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.1" lspace="0.167em" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.1.cmml">⁢</mo><msubsup id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.cmml"><mi id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.2.2" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.3" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.11.m5.1.1.1.3" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.cmml"><mo id="S4.SS2.SSS1.4.p2.11.m5.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.11.m5.1.1.1.1" xref="S4.SS2.SSS1.4.p2.11.m5.1.1.1.1.cmml">2</mn><mo id="S4.SS2.SSS1.4.p2.11.m5.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.SS2.SSS1.4.p2.11.m5.1.2.1" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.1.cmml">=</mo><mn id="S4.SS2.SSS1.4.p2.11.m5.1.2.3" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.11.m5.1b"><apply id="S4.SS2.SSS1.4.p2.11.m5.1.2.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2"><eq id="S4.SS2.SSS1.4.p2.11.m5.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.1"></eq><apply id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2"><times id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.1"></times><apply id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2"><ci id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.2.3">𝑚</ci></apply><apply id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.1.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.2.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.2.2">𝑓</ci><cn id="S4.SS2.SSS1.4.p2.11.m5.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.11.m5.1.1.1.1">2</cn></apply><ci id="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.3.cmml" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.2.3.3">𝑛</ci></apply></apply><cn id="S4.SS2.SSS1.4.p2.11.m5.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.11.m5.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.11.m5.1c">\overline{\operatorname{Tr}}_{m}f^{(2)}_{n}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.11.m5.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT = 0</annotation></semantics></math> for all <math alttext="n\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.12.m6.1"><semantics id="S4.SS2.SSS1.4.p2.12.m6.1a"><mrow id="S4.SS2.SSS1.4.p2.12.m6.1.1" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.12.m6.1.1.2" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.4.p2.12.m6.1.1.1" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS1.4.p2.12.m6.1.1.3" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.3.cmml"><mi id="S4.SS2.SSS1.4.p2.12.m6.1.1.3.2" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS1.4.p2.12.m6.1.1.3.3" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.12.m6.1b"><apply id="S4.SS2.SSS1.4.p2.12.m6.1.1.cmml" xref="S4.SS2.SSS1.4.p2.12.m6.1.1"><in id="S4.SS2.SSS1.4.p2.12.m6.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.1"></in><ci id="S4.SS2.SSS1.4.p2.12.m6.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.2">𝑛</ci><apply id="S4.SS2.SSS1.4.p2.12.m6.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.12.m6.1.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.12.m6.1.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS1.4.p2.12.m6.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.12.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.12.m6.1c">n\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.12.m6.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, using <math alttext="\overline{\operatorname{Tr}}_{m}f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.13.m7.1"><semantics id="S4.SS2.SSS1.4.p2.13.m7.1a"><mrow id="S4.SS2.SSS1.4.p2.13.m7.1.1" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.cmml"><mrow id="S4.SS2.SSS1.4.p2.13.m7.1.1.2" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.cmml"><msub id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.2" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.1" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.3" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.1" lspace="0.167em" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.3" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.3.cmml">f</mi></mrow><mo id="S4.SS2.SSS1.4.p2.13.m7.1.1.1" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.4.p2.13.m7.1.1.3" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.13.m7.1b"><apply id="S4.SS2.SSS1.4.p2.13.m7.1.1.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1"><eq id="S4.SS2.SSS1.4.p2.13.m7.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.1"></eq><apply id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2"><times id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.1"></times><apply id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2"><ci id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS2.SSS1.4.p2.13.m7.1.1.2.3.cmml" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.2.3">𝑓</ci></apply><cn id="S4.SS2.SSS1.4.p2.13.m7.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.13.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.13.m7.1c">\overline{\operatorname{Tr}}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.13.m7.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math>, Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem5" title="Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.5</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I3.i1" title="item i ‣ Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> twice and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E5" title="In Proof. ‣ 4.2.1. Bessel potential spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.5</span></a>), it follows</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx22"> <tbody id="S4.Ex34"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\overline{\operatorname{ext}}_{m}(\overline{\operatorname{Tr}}_% {m}f_{n}^{(1)})\|_{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.Ex34.m1.7"><semantics id="S4.Ex34.m1.7a"><msub id="S4.Ex34.m1.7.7" xref="S4.Ex34.m1.7.7.cmml"><mrow id="S4.Ex34.m1.7.7.1.1" xref="S4.Ex34.m1.7.7.1.2.cmml"><mo id="S4.Ex34.m1.7.7.1.1.2" stretchy="false" xref="S4.Ex34.m1.7.7.1.2.1.cmml">‖</mo><mrow id="S4.Ex34.m1.7.7.1.1.1" xref="S4.Ex34.m1.7.7.1.1.1.cmml"><msub id="S4.Ex34.m1.7.7.1.1.1.3" xref="S4.Ex34.m1.7.7.1.1.1.3.cmml"><mover accent="true" id="S4.Ex34.m1.7.7.1.1.1.3.2" xref="S4.Ex34.m1.7.7.1.1.1.3.2.cmml"><mi id="S4.Ex34.m1.7.7.1.1.1.3.2.2" xref="S4.Ex34.m1.7.7.1.1.1.3.2.2.cmml">ext</mi><mo id="S4.Ex34.m1.7.7.1.1.1.3.2.1" xref="S4.Ex34.m1.7.7.1.1.1.3.2.1.cmml">¯</mo></mover><mi id="S4.Ex34.m1.7.7.1.1.1.3.3" xref="S4.Ex34.m1.7.7.1.1.1.3.3.cmml">m</mi></msub><mo id="S4.Ex34.m1.7.7.1.1.1.2" xref="S4.Ex34.m1.7.7.1.1.1.2.cmml">⁢</mo><mrow id="S4.Ex34.m1.7.7.1.1.1.1.1" xref="S4.Ex34.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S4.Ex34.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="S4.Ex34.m1.7.7.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Ex34.m1.7.7.1.1.1.1.1.1" xref="S4.Ex34.m1.7.7.1.1.1.1.1.1.cmml"><msub id="S4.Ex34.m1.7.7.1.1.1.1.1.1.2" xref="S4.Ex34.m1.7.7.1.1.1.1.1.1.2.cmml"><mover accent="true" id="S4.Ex34.m1.7.7.1.1.1.1.1.1.2.2" xref="S4.Ex34.m1.7.7.1.1.1.1.1.1.2.2.cmml"><mi id="S4.Ex34.m1.7.7.1.1.1.1.1.1.2.2.2" xref="S4.Ex34.m1.7.7.1.1.1.1.1.1.2.2.2.cmml">Tr</mi><mo 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xref="S4.Ex34.m1.7.7.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.Ex34.m1.7.7.1.1.3" stretchy="false" xref="S4.Ex34.m1.7.7.1.2.1.cmml">‖</mo></mrow><mrow id="S4.Ex34.m1.6.6.5" xref="S4.Ex34.m1.6.6.5.cmml"><msup id="S4.Ex34.m1.6.6.5.7" xref="S4.Ex34.m1.6.6.5.7.cmml"><mi id="S4.Ex34.m1.6.6.5.7.2" xref="S4.Ex34.m1.6.6.5.7.2.cmml">H</mi><mrow id="S4.Ex34.m1.3.3.2.2.2.4" xref="S4.Ex34.m1.3.3.2.2.2.3.cmml"><mi id="S4.Ex34.m1.2.2.1.1.1.1" xref="S4.Ex34.m1.2.2.1.1.1.1.cmml">s</mi><mo id="S4.Ex34.m1.3.3.2.2.2.4.1" xref="S4.Ex34.m1.3.3.2.2.2.3.cmml">,</mo><mi id="S4.Ex34.m1.3.3.2.2.2.2" xref="S4.Ex34.m1.3.3.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex34.m1.6.6.5.6" xref="S4.Ex34.m1.6.6.5.6.cmml">⁢</mo><mrow id="S4.Ex34.m1.6.6.5.5.2" xref="S4.Ex34.m1.6.6.5.5.3.cmml"><mo id="S4.Ex34.m1.6.6.5.5.2.3" stretchy="false" xref="S4.Ex34.m1.6.6.5.5.3.cmml">(</mo><msup id="S4.Ex34.m1.5.5.4.4.1.1" xref="S4.Ex34.m1.5.5.4.4.1.1.cmml"><mi id="S4.Ex34.m1.5.5.4.4.1.1.2" xref="S4.Ex34.m1.5.5.4.4.1.1.2.cmml">ℝ</mi><mi id="S4.Ex34.m1.5.5.4.4.1.1.3" xref="S4.Ex34.m1.5.5.4.4.1.1.3.cmml">d</mi></msup><mo id="S4.Ex34.m1.6.6.5.5.2.4" xref="S4.Ex34.m1.6.6.5.5.3.cmml">,</mo><msub id="S4.Ex34.m1.6.6.5.5.2.2" xref="S4.Ex34.m1.6.6.5.5.2.2.cmml"><mi id="S4.Ex34.m1.6.6.5.5.2.2.2" xref="S4.Ex34.m1.6.6.5.5.2.2.2.cmml">w</mi><mi id="S4.Ex34.m1.6.6.5.5.2.2.3" xref="S4.Ex34.m1.6.6.5.5.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex34.m1.6.6.5.5.2.5" xref="S4.Ex34.m1.6.6.5.5.3.cmml">;</mo><mi id="S4.Ex34.m1.4.4.3.3" xref="S4.Ex34.m1.4.4.3.3.cmml">X</mi><mo id="S4.Ex34.m1.6.6.5.5.2.6" stretchy="false" xref="S4.Ex34.m1.6.6.5.5.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.Ex34.m1.7b"><apply id="S4.Ex34.m1.7.7.cmml" xref="S4.Ex34.m1.7.7"><csymbol cd="ambiguous" id="S4.Ex34.m1.7.7.2.cmml" xref="S4.Ex34.m1.7.7">subscript</csymbol><apply id="S4.Ex34.m1.7.7.1.2.cmml" xref="S4.Ex34.m1.7.7.1.1"><csymbol cd="latexml" id="S4.Ex34.m1.7.7.1.2.1.cmml" 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xref="S4.Ex34.m1.5.5.4.4.1.1">superscript</csymbol><ci id="S4.Ex34.m1.5.5.4.4.1.1.2.cmml" xref="S4.Ex34.m1.5.5.4.4.1.1.2">ℝ</ci><ci id="S4.Ex34.m1.5.5.4.4.1.1.3.cmml" xref="S4.Ex34.m1.5.5.4.4.1.1.3">𝑑</ci></apply><apply id="S4.Ex34.m1.6.6.5.5.2.2.cmml" xref="S4.Ex34.m1.6.6.5.5.2.2"><csymbol cd="ambiguous" id="S4.Ex34.m1.6.6.5.5.2.2.1.cmml" xref="S4.Ex34.m1.6.6.5.5.2.2">subscript</csymbol><ci id="S4.Ex34.m1.6.6.5.5.2.2.2.cmml" xref="S4.Ex34.m1.6.6.5.5.2.2.2">𝑤</ci><ci id="S4.Ex34.m1.6.6.5.5.2.2.3.cmml" xref="S4.Ex34.m1.6.6.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex34.m1.4.4.3.3.cmml" xref="S4.Ex34.m1.4.4.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex34.m1.7c">\displaystyle\|\overline{\operatorname{ext}}_{m}(\overline{\operatorname{Tr}}_% {m}f_{n}^{(1)})\|_{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex34.m1.7d">∥ over¯ start_ARG roman_ext end_ARG start_POSTSUBSCRIPT italic_m 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id="S4.Ex34.m2.5.5.4.4.2.cmml" xref="S4.Ex34.m2.5.5.4.4.1"><apply id="S4.Ex34.m2.5.5.4.4.1.1.cmml" xref="S4.Ex34.m2.5.5.4.4.1.1"><csymbol cd="ambiguous" id="S4.Ex34.m2.5.5.4.4.1.1.1.cmml" xref="S4.Ex34.m2.5.5.4.4.1.1">superscript</csymbol><ci id="S4.Ex34.m2.5.5.4.4.1.1.2.cmml" xref="S4.Ex34.m2.5.5.4.4.1.1.2">ℝ</ci><apply id="S4.Ex34.m2.5.5.4.4.1.1.3.cmml" xref="S4.Ex34.m2.5.5.4.4.1.1.3"><minus id="S4.Ex34.m2.5.5.4.4.1.1.3.1.cmml" xref="S4.Ex34.m2.5.5.4.4.1.1.3.1"></minus><ci id="S4.Ex34.m2.5.5.4.4.1.1.3.2.cmml" xref="S4.Ex34.m2.5.5.4.4.1.1.3.2">𝑑</ci><cn id="S4.Ex34.m2.5.5.4.4.1.1.3.3.cmml" type="integer" xref="S4.Ex34.m2.5.5.4.4.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex34.m2.4.4.3.3.cmml" xref="S4.Ex34.m2.4.4.3.3">𝑋</ci></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex34.m2.6c">\displaystyle\leq C\sum_{j=0}^{m}\|\overline{\operatorname{Tr}}_{m}f-\overline% {\operatorname{Tr}}_{m}f^{(1)}_{n}\|_{B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb% {R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex34.m2.6d">≤ italic_C ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ∥ over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f - over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody 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xref="S4.Ex35.m1.6.6.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex35.m1.4.4.3.3.cmml" xref="S4.Ex35.m1.4.4.3.3">𝑋</ci></vector></apply></apply></apply></apply><apply id="S4.Ex35.m1.7.7.1.1c.cmml" xref="S4.Ex35.m1.7.7.1.1"><ci id="S4.Ex35.m1.7.7.1.1.5.cmml" xref="S4.Ex35.m1.7.7.1.1.5">→</ci><share href="https://arxiv.org/html/2503.14636v1#S4.Ex35.m1.7.7.1.1.1.cmml" id="S4.Ex35.m1.7.7.1.1d.cmml" xref="S4.Ex35.m1.7.7.1.1"></share><cn id="S4.Ex35.m1.7.7.1.1.6.cmml" type="integer" xref="S4.Ex35.m1.7.7.1.1.6">0</cn></apply></apply><apply id="S4.Ex35.m1.8.8.2.2.cmml" xref="S4.Ex35.m1.8.8.2.2"><ci id="S4.Ex35.m1.8.8.2.2.1.cmml" xref="S4.Ex35.m1.8.8.2.2.1">→</ci><apply id="S4.Ex35.m1.8.8.2.2.2.cmml" xref="S4.Ex35.m1.8.8.2.2.2"><times id="S4.Ex35.m1.8.8.2.2.2.1.cmml" xref="S4.Ex35.m1.8.8.2.2.2.1"></times><ci id="S4.Ex35.m1.8.8.2.2.2.2a.cmml" xref="S4.Ex35.m1.8.8.2.2.2.2"><mtext id="S4.Ex35.m1.8.8.2.2.2.2.cmml" xref="S4.Ex35.m1.8.8.2.2.2.2"> as </mtext></ci><ci id="S4.Ex35.m1.8.8.2.2.2.3.cmml" xref="S4.Ex35.m1.8.8.2.2.2.3">𝑛</ci></apply><infinity id="S4.Ex35.m1.8.8.2.2.3.cmml" xref="S4.Ex35.m1.8.8.2.2.3"></infinity></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex35.m1.8c">\displaystyle\leq C\|f-f_{n}^{(1)}\|_{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}\to 0% \quad\text{ as }n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S4.Ex35.m1.8d">≤ italic_C ∥ italic_f - italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT → 0 as italic_n → ∞</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.4.p2.21">and therefore</p> <table 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xref="S4.Ex36.m2.9.9.2.2.2.2">𝑝</ci></list></apply><vector id="S4.Ex36.m2.12.12.5.5.3.cmml" xref="S4.Ex36.m2.12.12.5.5.2"><apply id="S4.Ex36.m2.11.11.4.4.1.1.cmml" xref="S4.Ex36.m2.11.11.4.4.1.1"><csymbol cd="ambiguous" id="S4.Ex36.m2.11.11.4.4.1.1.1.cmml" xref="S4.Ex36.m2.11.11.4.4.1.1">superscript</csymbol><ci id="S4.Ex36.m2.11.11.4.4.1.1.2.cmml" xref="S4.Ex36.m2.11.11.4.4.1.1.2">ℝ</ci><ci id="S4.Ex36.m2.11.11.4.4.1.1.3.cmml" xref="S4.Ex36.m2.11.11.4.4.1.1.3">𝑑</ci></apply><apply id="S4.Ex36.m2.12.12.5.5.2.2.cmml" xref="S4.Ex36.m2.12.12.5.5.2.2"><csymbol cd="ambiguous" id="S4.Ex36.m2.12.12.5.5.2.2.1.cmml" xref="S4.Ex36.m2.12.12.5.5.2.2">subscript</csymbol><ci id="S4.Ex36.m2.12.12.5.5.2.2.2.cmml" xref="S4.Ex36.m2.12.12.5.5.2.2.2">𝑤</ci><ci id="S4.Ex36.m2.12.12.5.5.2.2.3.cmml" xref="S4.Ex36.m2.12.12.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex36.m2.10.10.3.3.cmml" xref="S4.Ex36.m2.10.10.3.3">𝑋</ci></vector></apply></apply></apply></apply><apply id="S4.Ex36.m2.14.14c.cmml" xref="S4.Ex36.m2.14.14"><ci id="S4.Ex36.m2.14.14.6.cmml" xref="S4.Ex36.m2.14.14.6">→</ci><share href="https://arxiv.org/html/2503.14636v1#S4.Ex36.m2.14.14.2.cmml" id="S4.Ex36.m2.14.14d.cmml" xref="S4.Ex36.m2.14.14"></share><cn id="S4.Ex36.m2.14.14.7.cmml" type="integer" xref="S4.Ex36.m2.14.14.7">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex36.m2.14c">\displaystyle\leq\|f-f_{n}^{(1)}\|_{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}+\|% \overline{\operatorname{ext}}_{m}(\overline{\operatorname{Tr}}_{m}f_{n}^{(1)})% \|_{H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)}\to 0</annotation><annotation encoding="application/x-llamapun" id="S4.Ex36.m2.14d">≤ ∥ italic_f - italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT + ∥ over¯ start_ARG roman_ext end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT → 0</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS1.4.p2.20">as <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.14.m1.1"><semantics id="S4.SS2.SSS1.4.p2.14.m1.1a"><mrow id="S4.SS2.SSS1.4.p2.14.m1.1.1" xref="S4.SS2.SSS1.4.p2.14.m1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.14.m1.1.1.2" xref="S4.SS2.SSS1.4.p2.14.m1.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.4.p2.14.m1.1.1.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.14.m1.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.4.p2.14.m1.1.1.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.14.m1.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.14.m1.1b"><apply id="S4.SS2.SSS1.4.p2.14.m1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.14.m1.1.1"><ci id="S4.SS2.SSS1.4.p2.14.m1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.14.m1.1.1.1">→</ci><ci id="S4.SS2.SSS1.4.p2.14.m1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.14.m1.1.1.2">𝑛</ci><infinity id="S4.SS2.SSS1.4.p2.14.m1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.14.m1.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.14.m1.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.14.m1.1d">italic_n → ∞</annotation></semantics></math>. By a standard cut-off argument we find a sequence <math alttext="(f^{(3)}_{n})_{n\geq 1}\subseteq C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.15.m2.4"><semantics id="S4.SS2.SSS1.4.p2.15.m2.4a"><mrow id="S4.SS2.SSS1.4.p2.15.m2.4.4" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.cmml"><msub id="S4.SS2.SSS1.4.p2.15.m2.3.3.1" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.cmml"><mrow id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml">(</mo><msubsup id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.2.2" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.3" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.15.m2.1.1.1.3" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.4.p2.15.m2.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.15.m2.1.1.1.1" xref="S4.SS2.SSS1.4.p2.15.m2.1.1.1.1.cmml">3</mn><mo id="S4.SS2.SSS1.4.p2.15.m2.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml">)</mo></mrow></msubsup><mo id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml">)</mo></mrow><mrow id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.cmml"><mi id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.2" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.2.cmml">n</mi><mo id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.1" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.1.cmml">≥</mo><mn id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.3" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.3.cmml">1</mn></mrow></msub><mo id="S4.SS2.SSS1.4.p2.15.m2.4.4.3" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.3.cmml">⊆</mo><mrow id="S4.SS2.SSS1.4.p2.15.m2.4.4.2" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.cmml"><msubsup id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.cmml"><mi id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.2" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.2" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.2.cmml"><mo id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.2.cmml">(</mo><msup id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.2" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.3" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.3" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.4.p2.15.m2.2.2" xref="S4.SS2.SSS1.4.p2.15.m2.2.2.cmml">X</mi><mo id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.4" stretchy="false" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.15.m2.4b"><apply id="S4.SS2.SSS1.4.p2.15.m2.4.4.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4"><subset id="S4.SS2.SSS1.4.p2.15.m2.4.4.3.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.3"></subset><apply id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.2.2">𝑓</ci><cn id="S4.SS2.SSS1.4.p2.15.m2.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.15.m2.1.1.1.1">3</cn></apply><ci id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.1.1.1.3">𝑛</ci></apply><apply id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3"><geq id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.1"></geq><ci id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.2">𝑛</ci><cn id="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.15.m2.3.3.1.3.3">1</cn></apply></apply><apply id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2"><times id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.2"></times><apply id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.3.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.3.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.3.3"></infinity></apply><list id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1"><apply id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.4.4.2.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.4.p2.15.m2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.15.m2.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.15.m2.4c">(f^{(3)}_{n})_{n\geq 1}\subseteq C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.15.m2.4d">( italic_f start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 1 end_POSTSUBSCRIPT ⊆ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\overline{\operatorname{Tr}}_{m}f_{n}^{(3)}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.16.m3.1"><semantics id="S4.SS2.SSS1.4.p2.16.m3.1a"><mrow id="S4.SS2.SSS1.4.p2.16.m3.1.2" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.cmml"><mrow id="S4.SS2.SSS1.4.p2.16.m3.1.2.2" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.cmml"><msub id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.2" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.1" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.3" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.1" lspace="0.167em" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.1.cmml">⁢</mo><msubsup id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.cmml"><mi id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.2" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.3" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.16.m3.1.1.1.3" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.cmml"><mo id="S4.SS2.SSS1.4.p2.16.m3.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.16.m3.1.1.1.1" xref="S4.SS2.SSS1.4.p2.16.m3.1.1.1.1.cmml">3</mn><mo id="S4.SS2.SSS1.4.p2.16.m3.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.cmml">)</mo></mrow></msubsup></mrow><mo id="S4.SS2.SSS1.4.p2.16.m3.1.2.1" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.1.cmml">=</mo><mn id="S4.SS2.SSS1.4.p2.16.m3.1.2.3" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.16.m3.1b"><apply id="S4.SS2.SSS1.4.p2.16.m3.1.2.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2"><eq id="S4.SS2.SSS1.4.p2.16.m3.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.1"></eq><apply id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2"><times id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.1"></times><apply id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2"><ci id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.2.3">𝑚</ci></apply><apply id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.1.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.1.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.2.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.2">𝑓</ci><ci id="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.3.cmml" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.2.3.2.3">𝑛</ci></apply><cn id="S4.SS2.SSS1.4.p2.16.m3.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.16.m3.1.1.1.1">3</cn></apply></apply><cn id="S4.SS2.SSS1.4.p2.16.m3.1.2.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.16.m3.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.16.m3.1c">\overline{\operatorname{Tr}}_{m}f_{n}^{(3)}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.16.m3.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT = 0</annotation></semantics></math> for all <math alttext="n\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.17.m4.1"><semantics id="S4.SS2.SSS1.4.p2.17.m4.1a"><mrow id="S4.SS2.SSS1.4.p2.17.m4.1.1" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.17.m4.1.1.2" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.4.p2.17.m4.1.1.1" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS1.4.p2.17.m4.1.1.3" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.3.cmml"><mi id="S4.SS2.SSS1.4.p2.17.m4.1.1.3.2" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS1.4.p2.17.m4.1.1.3.3" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.17.m4.1b"><apply id="S4.SS2.SSS1.4.p2.17.m4.1.1.cmml" xref="S4.SS2.SSS1.4.p2.17.m4.1.1"><in id="S4.SS2.SSS1.4.p2.17.m4.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.1"></in><ci id="S4.SS2.SSS1.4.p2.17.m4.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.2">𝑛</ci><apply id="S4.SS2.SSS1.4.p2.17.m4.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.17.m4.1.1.3.1.cmml" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.17.m4.1.1.3.2.cmml" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS1.4.p2.17.m4.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.17.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.17.m4.1c">n\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.17.m4.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="f_{n}^{(3)}\to f" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.18.m5.1"><semantics id="S4.SS2.SSS1.4.p2.18.m5.1a"><mrow id="S4.SS2.SSS1.4.p2.18.m5.1.2" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.cmml"><msubsup id="S4.SS2.SSS1.4.p2.18.m5.1.2.2" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.2" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.3" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.3.cmml">n</mi><mrow id="S4.SS2.SSS1.4.p2.18.m5.1.1.1.3" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.cmml"><mo id="S4.SS2.SSS1.4.p2.18.m5.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.cmml">(</mo><mn id="S4.SS2.SSS1.4.p2.18.m5.1.1.1.1" xref="S4.SS2.SSS1.4.p2.18.m5.1.1.1.1.cmml">3</mn><mo id="S4.SS2.SSS1.4.p2.18.m5.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.cmml">)</mo></mrow></msubsup><mo id="S4.SS2.SSS1.4.p2.18.m5.1.2.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.1.cmml">→</mo><mi id="S4.SS2.SSS1.4.p2.18.m5.1.2.3" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.18.m5.1b"><apply id="S4.SS2.SSS1.4.p2.18.m5.1.2.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2"><ci id="S4.SS2.SSS1.4.p2.18.m5.1.2.1.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.1">→</ci><apply id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2">superscript</csymbol><apply id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.2">𝑓</ci><ci id="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.2.2.3">𝑛</ci></apply><cn id="S4.SS2.SSS1.4.p2.18.m5.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS1.4.p2.18.m5.1.1.1.1">3</cn></apply><ci id="S4.SS2.SSS1.4.p2.18.m5.1.2.3.cmml" xref="S4.SS2.SSS1.4.p2.18.m5.1.2.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.18.m5.1c">f_{n}^{(3)}\to f</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.18.m5.1d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 3 ) end_POSTSUPERSCRIPT → italic_f</annotation></semantics></math> in <math alttext="H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.19.m6.5"><semantics id="S4.SS2.SSS1.4.p2.19.m6.5a"><mrow id="S4.SS2.SSS1.4.p2.19.m6.5.5" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.cmml"><msup id="S4.SS2.SSS1.4.p2.19.m6.5.5.4" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.4.cmml"><mi id="S4.SS2.SSS1.4.p2.19.m6.5.5.4.2" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.4.p2.19.m6.2.2.2.4" xref="S4.SS2.SSS1.4.p2.19.m6.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.4.p2.19.m6.1.1.1.1" xref="S4.SS2.SSS1.4.p2.19.m6.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.4.p2.19.m6.2.2.2.4.1" xref="S4.SS2.SSS1.4.p2.19.m6.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.4.p2.19.m6.2.2.2.2" xref="S4.SS2.SSS1.4.p2.19.m6.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS2.SSS1.4.p2.19.m6.5.5.3" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.3.cmml"><mo id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.3" stretchy="false" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.3.cmml">(</mo><msup id="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1" xref="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.2" xref="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.3" xref="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.4" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.cmml"><mi id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.2" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.3" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.5" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.3.cmml">;</mo><mi id="S4.SS2.SSS1.4.p2.19.m6.3.3" xref="S4.SS2.SSS1.4.p2.19.m6.3.3.cmml">X</mi><mo id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.6" stretchy="false" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.19.m6.5b"><apply id="S4.SS2.SSS1.4.p2.19.m6.5.5.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5"><times id="S4.SS2.SSS1.4.p2.19.m6.5.5.3.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.3"></times><apply id="S4.SS2.SSS1.4.p2.19.m6.5.5.4.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.19.m6.5.5.4.1.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.4">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.19.m6.5.5.4.2.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.4.2">𝐻</ci><list id="S4.SS2.SSS1.4.p2.19.m6.2.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.2.2.2.4"><ci id="S4.SS2.SSS1.4.p2.19.m6.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.1.1.1.1">𝑠</ci><ci id="S4.SS2.SSS1.4.p2.19.m6.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.3.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2"><apply id="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.4.4.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.1.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.3.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.4.p2.19.m6.3.3.cmml" xref="S4.SS2.SSS1.4.p2.19.m6.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.19.m6.5c">H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.19.m6.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> as <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S4.SS2.SSS1.4.p2.20.m7.1"><semantics id="S4.SS2.SSS1.4.p2.20.m7.1a"><mrow id="S4.SS2.SSS1.4.p2.20.m7.1.1" xref="S4.SS2.SSS1.4.p2.20.m7.1.1.cmml"><mi id="S4.SS2.SSS1.4.p2.20.m7.1.1.2" xref="S4.SS2.SSS1.4.p2.20.m7.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.4.p2.20.m7.1.1.1" stretchy="false" xref="S4.SS2.SSS1.4.p2.20.m7.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.4.p2.20.m7.1.1.3" mathvariant="normal" xref="S4.SS2.SSS1.4.p2.20.m7.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.4.p2.20.m7.1b"><apply id="S4.SS2.SSS1.4.p2.20.m7.1.1.cmml" xref="S4.SS2.SSS1.4.p2.20.m7.1.1"><ci id="S4.SS2.SSS1.4.p2.20.m7.1.1.1.cmml" xref="S4.SS2.SSS1.4.p2.20.m7.1.1.1">→</ci><ci id="S4.SS2.SSS1.4.p2.20.m7.1.1.2.cmml" xref="S4.SS2.SSS1.4.p2.20.m7.1.1.2">𝑛</ci><infinity id="S4.SS2.SSS1.4.p2.20.m7.1.1.3.cmml" xref="S4.SS2.SSS1.4.p2.20.m7.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.4.p2.20.m7.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.4.p2.20.m7.1d">italic_n → ∞</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.5.p3"> <p class="ltx_p" id="S4.SS2.SSS1.5.p3.17">It remains to show that any <math alttext="g\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.1.m1.2"><semantics id="S4.SS2.SSS1.5.p3.1.m1.2a"><mrow id="S4.SS2.SSS1.5.p3.1.m1.2.2" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.cmml"><mi id="S4.SS2.SSS1.5.p3.1.m1.2.2.3" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.3.cmml">g</mi><mo id="S4.SS2.SSS1.5.p3.1.m1.2.2.2" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS1.5.p3.1.m1.2.2.1" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.cmml"><msubsup id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.cmml"><mi id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.2" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.2" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.2.cmml"><mo id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.2.cmml">(</mo><msup id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.2" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.3" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.3" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.5.p3.1.m1.1.1" xref="S4.SS2.SSS1.5.p3.1.m1.1.1.cmml">X</mi><mo id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.4" stretchy="false" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.1.m1.2b"><apply id="S4.SS2.SSS1.5.p3.1.m1.2.2.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2"><in id="S4.SS2.SSS1.5.p3.1.m1.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.2"></in><ci id="S4.SS2.SSS1.5.p3.1.m1.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.3">𝑔</ci><apply id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1"><times id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.2.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.2"></times><apply id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.1.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3">superscript</csymbol><apply id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.1.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.2.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.3.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.3.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.3.3"></infinity></apply><list id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1"><apply id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.2.2.1.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.5.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.1.m1.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.1.m1.2c">g\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.1.m1.2d">italic_g ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> with <math alttext="\overline{\operatorname{Tr}}_{m}g=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.2.m2.1"><semantics id="S4.SS2.SSS1.5.p3.2.m2.1a"><mrow id="S4.SS2.SSS1.5.p3.2.m2.1.1" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.cmml"><mrow id="S4.SS2.SSS1.5.p3.2.m2.1.1.2" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.cmml"><msub id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.2" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.1" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.3" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.1" lspace="0.167em" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.3" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.3.cmml">g</mi></mrow><mo id="S4.SS2.SSS1.5.p3.2.m2.1.1.1" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.5.p3.2.m2.1.1.3" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.2.m2.1b"><apply id="S4.SS2.SSS1.5.p3.2.m2.1.1.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1"><eq id="S4.SS2.SSS1.5.p3.2.m2.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.1"></eq><apply id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2"><times id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.1.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.1"></times><apply id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2"><ci id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS2.SSS1.5.p3.2.m2.1.1.2.3.cmml" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.2.3">𝑔</ci></apply><cn id="S4.SS2.SSS1.5.p3.2.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.5.p3.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.2.m2.1c">\overline{\operatorname{Tr}}_{m}g=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.2.m2.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_g = 0</annotation></semantics></math> can be approximated by functions in <math alttext="C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.3.m3.2"><semantics id="S4.SS2.SSS1.5.p3.3.m3.2a"><mrow id="S4.SS2.SSS1.5.p3.3.m3.2.2" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.cmml"><msubsup id="S4.SS2.SSS1.5.p3.3.m3.2.2.3" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3.cmml"><mi id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.2" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.5.p3.3.m3.2.2.2" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.2.cmml"><mo id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.2" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.3" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.3.cmml">∙</mo><mi id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.3" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.3" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.5.p3.3.m3.1.1" xref="S4.SS2.SSS1.5.p3.3.m3.1.1.cmml">X</mi><mo id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.4" stretchy="false" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.3.m3.2b"><apply id="S4.SS2.SSS1.5.p3.3.m3.2.2.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2"><times id="S4.SS2.SSS1.5.p3.3.m3.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.2"></times><apply id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.1.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3">superscript</csymbol><apply id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.1.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.2.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.3.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.5.p3.3.m3.2.2.3.3.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.3.3"></infinity></apply><list id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.2.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1"><apply id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.2.3">∙</ci></apply><ci id="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.2.2.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.5.p3.3.m3.1.1.cmml" xref="S4.SS2.SSS1.5.p3.3.m3.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.3.m3.2c">C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.3.m3.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math>. By writing <math alttext="g=\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}g+\operatorname{\mathbf{1}}_{% \mathbb{R}^{d}_{-}}g=:g_{+}+g_{-}" class="ltx_math_unparsed" display="inline" id="S4.SS2.SSS1.5.p3.4.m4.1"><semantics id="S4.SS2.SSS1.5.p3.4.m4.1a"><mrow id="S4.SS2.SSS1.5.p3.4.m4.1b"><mi id="S4.SS2.SSS1.5.p3.4.m4.1.1">g</mi><mo id="S4.SS2.SSS1.5.p3.4.m4.1.2">=</mo><msub id="S4.SS2.SSS1.5.p3.4.m4.1.3"><mi id="S4.SS2.SSS1.5.p3.4.m4.1.3.2">𝟏</mi><msubsup id="S4.SS2.SSS1.5.p3.4.m4.1.3.3"><mi id="S4.SS2.SSS1.5.p3.4.m4.1.3.3.2.2">ℝ</mi><mo id="S4.SS2.SSS1.5.p3.4.m4.1.3.3.3">+</mo><mi id="S4.SS2.SSS1.5.p3.4.m4.1.3.3.2.3">d</mi></msubsup></msub><mi id="S4.SS2.SSS1.5.p3.4.m4.1.4">g</mi><mo id="S4.SS2.SSS1.5.p3.4.m4.1.5">+</mo><msub id="S4.SS2.SSS1.5.p3.4.m4.1.6"><mi id="S4.SS2.SSS1.5.p3.4.m4.1.6.2">𝟏</mi><msubsup id="S4.SS2.SSS1.5.p3.4.m4.1.6.3"><mi id="S4.SS2.SSS1.5.p3.4.m4.1.6.3.2.2">ℝ</mi><mo id="S4.SS2.SSS1.5.p3.4.m4.1.6.3.3">−</mo><mi id="S4.SS2.SSS1.5.p3.4.m4.1.6.3.2.3">d</mi></msubsup></msub><mi id="S4.SS2.SSS1.5.p3.4.m4.1.7">g</mi><mo id="S4.SS2.SSS1.5.p3.4.m4.1.8" rspace="0em">=</mo><mo id="S4.SS2.SSS1.5.p3.4.m4.1.9" rspace="0.278em">:</mo><msub id="S4.SS2.SSS1.5.p3.4.m4.1.10"><mi id="S4.SS2.SSS1.5.p3.4.m4.1.10.2">g</mi><mo id="S4.SS2.SSS1.5.p3.4.m4.1.10.3">+</mo></msub><mo id="S4.SS2.SSS1.5.p3.4.m4.1.11">+</mo><msub id="S4.SS2.SSS1.5.p3.4.m4.1.12"><mi id="S4.SS2.SSS1.5.p3.4.m4.1.12.2">g</mi><mo id="S4.SS2.SSS1.5.p3.4.m4.1.12.3">−</mo></msub></mrow><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.4.m4.1c">g=\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}g+\operatorname{\mathbf{1}}_{% \mathbb{R}^{d}_{-}}g=:g_{+}+g_{-}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.4.m4.1d">italic_g = bold_1 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_g + bold_1 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_g = : italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT + italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math>, we see that <math alttext="g_{\pm}\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.5.m5.5"><semantics id="S4.SS2.SSS1.5.p3.5.m5.5a"><mrow id="S4.SS2.SSS1.5.p3.5.m5.5.5" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.cmml"><msub id="S4.SS2.SSS1.5.p3.5.m5.5.5.4" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.4.cmml"><mi id="S4.SS2.SSS1.5.p3.5.m5.5.5.4.2" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.4.2.cmml">g</mi><mo id="S4.SS2.SSS1.5.p3.5.m5.5.5.4.3" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.4.3.cmml">±</mo></msub><mo id="S4.SS2.SSS1.5.p3.5.m5.5.5.3" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.3.cmml">∈</mo><mrow id="S4.SS2.SSS1.5.p3.5.m5.5.5.2" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.cmml"><msup id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4.cmml"><mi id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4.2" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.5.p3.5.m5.2.2.2.4" xref="S4.SS2.SSS1.5.p3.5.m5.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.5.p3.5.m5.1.1.1.1" xref="S4.SS2.SSS1.5.p3.5.m5.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.5.p3.5.m5.2.2.2.4.1" xref="S4.SS2.SSS1.5.p3.5.m5.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.5.p3.5.m5.2.2.2.2" xref="S4.SS2.SSS1.5.p3.5.m5.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.3" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.3.cmml"><mo id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.3.cmml">(</mo><msup id="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1" xref="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.2" xref="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.3" xref="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.4" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.2" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.3" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.5" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.3.cmml">;</mo><mi id="S4.SS2.SSS1.5.p3.5.m5.3.3" xref="S4.SS2.SSS1.5.p3.5.m5.3.3.cmml">X</mi><mo id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.6" stretchy="false" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.5.m5.5b"><apply id="S4.SS2.SSS1.5.p3.5.m5.5.5.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5"><in id="S4.SS2.SSS1.5.p3.5.m5.5.5.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.3"></in><apply id="S4.SS2.SSS1.5.p3.5.m5.5.5.4.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.5.m5.5.5.4.1.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.4">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.5.m5.5.5.4.2.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.4.2">𝑔</ci><csymbol cd="latexml" id="S4.SS2.SSS1.5.p3.5.m5.5.5.4.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.4.3">plus-or-minus</csymbol></apply><apply id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2"><times id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.3"></times><apply id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4.1.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4.2.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.4.2">𝐻</ci><list id="S4.SS2.SSS1.5.p3.5.m5.2.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.2.2.2.4"><ci id="S4.SS2.SSS1.5.p3.5.m5.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.1.1.1.1">𝑠</ci><ci id="S4.SS2.SSS1.5.p3.5.m5.2.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2"><apply id="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.5.p3.5.m5.3.3.cmml" xref="S4.SS2.SSS1.5.p3.5.m5.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.5.m5.5c">g_{\pm}\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.5.m5.5d">italic_g start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> by Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem6" title="Proposition 4.6. ‣ 4.2.1. Bessel potential spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.6</span></a> and thus it suffices to approximate <math alttext="g_{+}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.6.m6.1"><semantics id="S4.SS2.SSS1.5.p3.6.m6.1a"><msub id="S4.SS2.SSS1.5.p3.6.m6.1.1" xref="S4.SS2.SSS1.5.p3.6.m6.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.6.m6.1.1.2" xref="S4.SS2.SSS1.5.p3.6.m6.1.1.2.cmml">g</mi><mo id="S4.SS2.SSS1.5.p3.6.m6.1.1.3" xref="S4.SS2.SSS1.5.p3.6.m6.1.1.3.cmml">+</mo></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.6.m6.1b"><apply id="S4.SS2.SSS1.5.p3.6.m6.1.1.cmml" xref="S4.SS2.SSS1.5.p3.6.m6.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.6.m6.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.6.m6.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.6.m6.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.6.m6.1.1.2">𝑔</ci><plus id="S4.SS2.SSS1.5.p3.6.m6.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.6.m6.1.1.3"></plus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.6.m6.1c">g_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.6.m6.1d">italic_g start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="g_{-}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.7.m7.1"><semantics id="S4.SS2.SSS1.5.p3.7.m7.1a"><msub id="S4.SS2.SSS1.5.p3.7.m7.1.1" xref="S4.SS2.SSS1.5.p3.7.m7.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.7.m7.1.1.2" xref="S4.SS2.SSS1.5.p3.7.m7.1.1.2.cmml">g</mi><mo id="S4.SS2.SSS1.5.p3.7.m7.1.1.3" xref="S4.SS2.SSS1.5.p3.7.m7.1.1.3.cmml">−</mo></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.7.m7.1b"><apply id="S4.SS2.SSS1.5.p3.7.m7.1.1.cmml" xref="S4.SS2.SSS1.5.p3.7.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.7.m7.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.7.m7.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.7.m7.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.7.m7.1.1.2">𝑔</ci><minus id="S4.SS2.SSS1.5.p3.7.m7.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.7.m7.1.1.3"></minus></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.7.m7.1c">g_{-}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.7.m7.1d">italic_g start_POSTSUBSCRIPT - end_POSTSUBSCRIPT</annotation></semantics></math> separately. Let <math alttext="\phi\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d})" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.8.m8.1"><semantics id="S4.SS2.SSS1.5.p3.8.m8.1a"><mrow id="S4.SS2.SSS1.5.p3.8.m8.1.1" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.8.m8.1.1.3" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.3.cmml">ϕ</mi><mo id="S4.SS2.SSS1.5.p3.8.m8.1.1.2" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.2.cmml">∈</mo><mrow id="S4.SS2.SSS1.5.p3.8.m8.1.1.1" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.cmml"><msubsup id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.cmml"><mi id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.2" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.2" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.cmml">(</mo><msup id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.2" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.3" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.8.m8.1b"><apply id="S4.SS2.SSS1.5.p3.8.m8.1.1.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1"><in id="S4.SS2.SSS1.5.p3.8.m8.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.2"></in><ci id="S4.SS2.SSS1.5.p3.8.m8.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.3">italic-ϕ</ci><apply id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1"><times id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.2"></times><apply id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.1.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3">superscript</csymbol><apply id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.1.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.2.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.3.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.3.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.3.3"></infinity></apply><apply id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.8.m8.1.1.1.1.1.1.3">𝑑</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.8.m8.1c">\phi\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.8.m8.1d">italic_ϕ ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT )</annotation></semantics></math> such that <math alttext="\int_{\mathbb{R}^{d}}\phi\hskip 2.0pt\mathrm{d}x=1" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.9.m9.1"><semantics id="S4.SS2.SSS1.5.p3.9.m9.1a"><mrow id="S4.SS2.SSS1.5.p3.9.m9.1.1" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.cmml"><mrow id="S4.SS2.SSS1.5.p3.9.m9.1.1.2" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.cmml"><msub id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.cmml"><mo id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.2" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.2.cmml">∫</mo><msup id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.cmml"><mi id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.2" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.3" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.3.cmml">d</mi></msup></msub><mrow id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.cmml"><mi id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.2" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.2.cmml">ϕ</mi><mo id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.1" lspace="0.200em" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.cmml"><mo id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.1" rspace="0em" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.1.cmml">d</mo><mi id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.2" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.2.cmml">x</mi></mrow></mrow></mrow><mo id="S4.SS2.SSS1.5.p3.9.m9.1.1.1" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.5.p3.9.m9.1.1.3" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.9.m9.1b"><apply id="S4.SS2.SSS1.5.p3.9.m9.1.1.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1"><eq id="S4.SS2.SSS1.5.p3.9.m9.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.1"></eq><apply id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2"><apply id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.1.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1">subscript</csymbol><int id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.2.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.2"></int><apply id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.1.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.2.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.2">ℝ</ci><ci id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.3.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.1.3.3">𝑑</ci></apply></apply><apply id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2"><times id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.1"></times><ci id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.2">italic-ϕ</ci><apply id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3"><csymbol cd="latexml" id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.1.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.1">differential-d</csymbol><ci id="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.2.cmml" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.2.2.3.2">𝑥</ci></apply></apply></apply><cn id="S4.SS2.SSS1.5.p3.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.5.p3.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.9.m9.1c">\int_{\mathbb{R}^{d}}\phi\hskip 2.0pt\mathrm{d}x=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.9.m9.1d">∫ start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_ϕ roman_d italic_x = 1</annotation></semantics></math> and <math alttext="\text{\rm supp\,}\phi\subseteq[1,\infty)\times\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.10.m10.2"><semantics id="S4.SS2.SSS1.5.p3.10.m10.2a"><mrow id="S4.SS2.SSS1.5.p3.10.m10.2.3" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.cmml"><mrow id="S4.SS2.SSS1.5.p3.10.m10.2.3.2" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.cmml"><mtext id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.2" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.2a.cmml">supp </mtext><mo id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.1" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.3" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.3.cmml">ϕ</mi></mrow><mo id="S4.SS2.SSS1.5.p3.10.m10.2.3.1" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.1.cmml">⊆</mo><mrow id="S4.SS2.SSS1.5.p3.10.m10.2.3.3" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.cmml"><mrow id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.2" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.1.cmml"><mo id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.2.1" stretchy="false" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.1.cmml">[</mo><mn id="S4.SS2.SSS1.5.p3.10.m10.1.1" xref="S4.SS2.SSS1.5.p3.10.m10.1.1.cmml">1</mn><mo id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.2.2" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.1.cmml">,</mo><mi id="S4.SS2.SSS1.5.p3.10.m10.2.2" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.10.m10.2.2.cmml">∞</mi><mo id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.2.3" rspace="0.055em" stretchy="false" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.1" rspace="0.222em" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.1.cmml">×</mo><msup id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.cmml"><mi id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.2" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.2.cmml">ℝ</mi><mrow id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.cmml"><mi id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.2" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.2.cmml">d</mi><mo id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.1" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.1.cmml">−</mo><mn id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.3" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.3.cmml">1</mn></mrow></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.10.m10.2b"><apply id="S4.SS2.SSS1.5.p3.10.m10.2.3.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3"><subset id="S4.SS2.SSS1.5.p3.10.m10.2.3.1.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.1"></subset><apply id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2"><times id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.1.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.1"></times><ci id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.2a.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.2"><mtext id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.2.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.2">supp </mtext></ci><ci id="S4.SS2.SSS1.5.p3.10.m10.2.3.2.3.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.2.3">italic-ϕ</ci></apply><apply id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3"><times id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.1.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.1"></times><interval closure="closed-open" id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.1.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.2.2"><cn id="S4.SS2.SSS1.5.p3.10.m10.1.1.cmml" type="integer" xref="S4.SS2.SSS1.5.p3.10.m10.1.1">1</cn><infinity id="S4.SS2.SSS1.5.p3.10.m10.2.2.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.2"></infinity></interval><apply id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.1.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.2.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.2">ℝ</ci><apply id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3"><minus id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.1.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.1"></minus><ci id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.2.cmml" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.2">𝑑</ci><cn id="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.3.cmml" type="integer" xref="S4.SS2.SSS1.5.p3.10.m10.2.3.3.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.10.m10.2c">\text{\rm supp\,}\phi\subseteq[1,\infty)\times\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.10.m10.2d">supp italic_ϕ ⊆ [ 1 , ∞ ) × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. Define <math alttext="\phi_{n}(x):=n^{d}\phi(nx)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.11.m11.2"><semantics id="S4.SS2.SSS1.5.p3.11.m11.2a"><mrow id="S4.SS2.SSS1.5.p3.11.m11.2.2" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.cmml"><mrow id="S4.SS2.SSS1.5.p3.11.m11.2.2.3" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.cmml"><msub id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.cmml"><mi id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.2" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.2.cmml">ϕ</mi><mi id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.3" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.3.cmml">n</mi></msub><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.1" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.3.2" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.cmml"><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.cmml">(</mo><mi id="S4.SS2.SSS1.5.p3.11.m11.1.1" xref="S4.SS2.SSS1.5.p3.11.m11.1.1.cmml">x</mi><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.3.2.2" rspace="0.278em" stretchy="false" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.2" rspace="0.278em" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.2.cmml">:=</mo><mrow id="S4.SS2.SSS1.5.p3.11.m11.2.2.1" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.cmml"><msup id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.cmml"><mi id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.2" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.2.cmml">n</mi><mi id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.3" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.2" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.4" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.4.cmml">ϕ</mi><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.2a" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.2" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.1" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.3" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.3.cmml">x</mi></mrow><mo id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.3" stretchy="false" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.11.m11.2b"><apply id="S4.SS2.SSS1.5.p3.11.m11.2.2.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2"><csymbol cd="latexml" id="S4.SS2.SSS1.5.p3.11.m11.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.2">assign</csymbol><apply id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3"><times id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.1.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.1"></times><apply id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.1.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.2.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.2">italic-ϕ</ci><ci id="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.3.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.3.2.3">𝑛</ci></apply><ci id="S4.SS2.SSS1.5.p3.11.m11.1.1.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.1.1">𝑥</ci></apply><apply id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1"><times id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.2.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.2"></times><apply id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.1.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.2.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.2">𝑛</ci><ci id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.3.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.3.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.4.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.4">italic-ϕ</ci><apply id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1"><times id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.1"></times><ci id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.2">𝑛</ci><ci id="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.11.m11.2.2.1.1.1.1.3">𝑥</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.11.m11.2c">\phi_{n}(x):=n^{d}\phi(nx)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.11.m11.2d">italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) := italic_n start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT italic_ϕ ( italic_n italic_x )</annotation></semantics></math> for <math alttext="n\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.12.m12.1"><semantics id="S4.SS2.SSS1.5.p3.12.m12.1a"><mrow id="S4.SS2.SSS1.5.p3.12.m12.1.1" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.12.m12.1.1.2" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.5.p3.12.m12.1.1.1" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS1.5.p3.12.m12.1.1.3" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.3.cmml"><mi id="S4.SS2.SSS1.5.p3.12.m12.1.1.3.2" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS1.5.p3.12.m12.1.1.3.3" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.12.m12.1b"><apply id="S4.SS2.SSS1.5.p3.12.m12.1.1.cmml" xref="S4.SS2.SSS1.5.p3.12.m12.1.1"><in id="S4.SS2.SSS1.5.p3.12.m12.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.1"></in><ci id="S4.SS2.SSS1.5.p3.12.m12.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.2">𝑛</ci><apply id="S4.SS2.SSS1.5.p3.12.m12.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.12.m12.1.1.3.1.cmml" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.12.m12.1.1.3.2.cmml" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS1.5.p3.12.m12.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS1.5.p3.12.m12.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.12.m12.1c">n\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.12.m12.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="x\in\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.13.m13.1"><semantics id="S4.SS2.SSS1.5.p3.13.m13.1a"><mrow id="S4.SS2.SSS1.5.p3.13.m13.1.1" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.13.m13.1.1.2" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.2.cmml">x</mi><mo id="S4.SS2.SSS1.5.p3.13.m13.1.1.1" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.1.cmml">∈</mo><msup id="S4.SS2.SSS1.5.p3.13.m13.1.1.3" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.3.cmml"><mi id="S4.SS2.SSS1.5.p3.13.m13.1.1.3.2" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.3.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.5.p3.13.m13.1.1.3.3" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.13.m13.1b"><apply id="S4.SS2.SSS1.5.p3.13.m13.1.1.cmml" xref="S4.SS2.SSS1.5.p3.13.m13.1.1"><in id="S4.SS2.SSS1.5.p3.13.m13.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.1"></in><ci id="S4.SS2.SSS1.5.p3.13.m13.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.2">𝑥</ci><apply id="S4.SS2.SSS1.5.p3.13.m13.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.13.m13.1.1.3.1.cmml" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS1.5.p3.13.m13.1.1.3.2.cmml" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.3.2">ℝ</ci><ci id="S4.SS2.SSS1.5.p3.13.m13.1.1.3.3.cmml" xref="S4.SS2.SSS1.5.p3.13.m13.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.13.m13.1c">x\in\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.13.m13.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. Then <math alttext="\phi_{n}\ast g_{\pm}\in C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.14.m14.2"><semantics id="S4.SS2.SSS1.5.p3.14.m14.2a"><mrow id="S4.SS2.SSS1.5.p3.14.m14.2.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.cmml"><mrow id="S4.SS2.SSS1.5.p3.14.m14.2.2.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.cmml"><msub id="S4.SS2.SSS1.5.p3.14.m14.2.2.3.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.2.cmml"><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.3.2.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.2.2.cmml">ϕ</mi><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.3.2.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.2.3.cmml">n</mi></msub><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.3.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.1.cmml">∗</mo><msub id="S4.SS2.SSS1.5.p3.14.m14.2.2.3.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.3.cmml"><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.3.3.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.3.2.cmml">g</mi><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.3.3.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.3.3.3.cmml">±</mo></msub></mrow><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS1.5.p3.14.m14.2.2.1" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.cmml"><msubsup id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.cmml"><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.2.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.2.cmml"><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.2" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.3.cmml">∙</mo><mi id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.3" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.5.p3.14.m14.1.1" xref="S4.SS2.SSS1.5.p3.14.m14.1.1.cmml">X</mi><mo id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.4" stretchy="false" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.14.m14.2b"><apply id="S4.SS2.SSS1.5.p3.14.m14.2.2.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2"><in id="S4.SS2.SSS1.5.p3.14.m14.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.2"></in><apply 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xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.3.3"></infinity></apply><list id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1"><apply id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.2.3">∙</ci></apply><ci id="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.14.m14.2.2.1.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.5.p3.14.m14.1.1.cmml" 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id="S4.SS2.SSS1.5.p3.16.m16.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.5.p3.16.m16.5.5.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.5.p3.16.m16.5.5.2.2.2.3.cmml" xref="S4.SS2.SSS1.5.p3.16.m16.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.5.p3.16.m16.3.3.cmml" xref="S4.SS2.SSS1.5.p3.16.m16.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.16.m16.5c">H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.16.m16.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> as <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S4.SS2.SSS1.5.p3.17.m17.1"><semantics id="S4.SS2.SSS1.5.p3.17.m17.1a"><mrow id="S4.SS2.SSS1.5.p3.17.m17.1.1" xref="S4.SS2.SSS1.5.p3.17.m17.1.1.cmml"><mi id="S4.SS2.SSS1.5.p3.17.m17.1.1.2" xref="S4.SS2.SSS1.5.p3.17.m17.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.5.p3.17.m17.1.1.1" stretchy="false" xref="S4.SS2.SSS1.5.p3.17.m17.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.5.p3.17.m17.1.1.3" mathvariant="normal" xref="S4.SS2.SSS1.5.p3.17.m17.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.5.p3.17.m17.1b"><apply id="S4.SS2.SSS1.5.p3.17.m17.1.1.cmml" xref="S4.SS2.SSS1.5.p3.17.m17.1.1"><ci id="S4.SS2.SSS1.5.p3.17.m17.1.1.1.cmml" xref="S4.SS2.SSS1.5.p3.17.m17.1.1.1">→</ci><ci id="S4.SS2.SSS1.5.p3.17.m17.1.1.2.cmml" xref="S4.SS2.SSS1.5.p3.17.m17.1.1.2">𝑛</ci><infinity id="S4.SS2.SSS1.5.p3.17.m17.1.1.3.cmml" xref="S4.SS2.SSS1.5.p3.17.m17.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.5.p3.17.m17.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.5.p3.17.m17.1d">italic_n → ∞</annotation></semantics></math> by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 3.6]</cite>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS1.6.p4"> <p class="ltx_p" id="S4.SS2.SSS1.6.p4.14"><span class="ltx_text ltx_font_italic" id="S4.SS2.SSS1.6.p4.14.1">Step 2: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I4.i2" title="item ii ‣ Proposition 4.7. ‣ 4.2.1. Bessel potential spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>. </span>Again, the embedding “<math alttext="\hookleftarrow" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.1.m1.1"><semantics id="S4.SS2.SSS1.6.p4.1.m1.1a"><mo id="S4.SS2.SSS1.6.p4.1.m1.1.1" stretchy="false" xref="S4.SS2.SSS1.6.p4.1.m1.1.1.cmml">↩</mo><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.1.m1.1b"><ci id="S4.SS2.SSS1.6.p4.1.m1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.1.m1.1.1">↩</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.1.m1.1c">\hookleftarrow</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.1.m1.1d">↩</annotation></semantics></math>” is straightforward to prove. For the other embedding, let <math alttext="f\in H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.2.m2.5"><semantics id="S4.SS2.SSS1.6.p4.2.m2.5a"><mrow id="S4.SS2.SSS1.6.p4.2.m2.5.5" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.cmml"><mi id="S4.SS2.SSS1.6.p4.2.m2.5.5.4" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.4.cmml">f</mi><mo id="S4.SS2.SSS1.6.p4.2.m2.5.5.3" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.3.cmml">∈</mo><mrow id="S4.SS2.SSS1.6.p4.2.m2.5.5.2" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.cmml"><msup id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.4" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.4.cmml"><mi id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.4.2" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.6.p4.2.m2.2.2.2.4" xref="S4.SS2.SSS1.6.p4.2.m2.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.6.p4.2.m2.1.1.1.1" xref="S4.SS2.SSS1.6.p4.2.m2.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.6.p4.2.m2.2.2.2.4.1" xref="S4.SS2.SSS1.6.p4.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.6.p4.2.m2.2.2.2.2" xref="S4.SS2.SSS1.6.p4.2.m2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.3" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.3.cmml"><mo id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.3.cmml">(</mo><msubsup id="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1" xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.2" xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.3" xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.3.cmml">+</mo><mi id="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.3" xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.4" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.2" 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xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.4.4.1.1.1.1.3"></plus></apply><apply id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.6.p4.2.m2.3.3.cmml" xref="S4.SS2.SSS1.6.p4.2.m2.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.2.m2.5c">f\in H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.2.m2.5d">italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\overline{\operatorname{Tr}}_{m}f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.3.m3.1"><semantics id="S4.SS2.SSS1.6.p4.3.m3.1a"><mrow id="S4.SS2.SSS1.6.p4.3.m3.1.1" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.cmml"><mrow id="S4.SS2.SSS1.6.p4.3.m3.1.1.2" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.cmml"><msub id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.cmml"><mover accent="true" id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.cmml"><mi id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.2" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.1" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.3" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.1" lspace="0.167em" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.3" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.3.cmml">f</mi></mrow><mo id="S4.SS2.SSS1.6.p4.3.m3.1.1.1" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS1.6.p4.3.m3.1.1.3" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.3.m3.1b"><apply id="S4.SS2.SSS1.6.p4.3.m3.1.1.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1"><eq id="S4.SS2.SSS1.6.p4.3.m3.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.1"></eq><apply id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2"><times id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.1.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.1"></times><apply id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2"><ci id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.2.3">𝑚</ci></apply><ci id="S4.SS2.SSS1.6.p4.3.m3.1.1.2.3.cmml" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.2.3">𝑓</ci></apply><cn id="S4.SS2.SSS1.6.p4.3.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS1.6.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.3.m3.1c">\overline{\operatorname{Tr}}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.3.m3.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math>. Then there exists an <math alttext="\widetilde{f}\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.4.m4.5"><semantics id="S4.SS2.SSS1.6.p4.4.m4.5a"><mrow id="S4.SS2.SSS1.6.p4.4.m4.5.5" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.cmml"><mover accent="true" id="S4.SS2.SSS1.6.p4.4.m4.5.5.4" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.4.cmml"><mi id="S4.SS2.SSS1.6.p4.4.m4.5.5.4.2" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.4.2.cmml">f</mi><mo id="S4.SS2.SSS1.6.p4.4.m4.5.5.4.1" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.4.1.cmml">~</mo></mover><mo id="S4.SS2.SSS1.6.p4.4.m4.5.5.3" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.3.cmml">∈</mo><mrow id="S4.SS2.SSS1.6.p4.4.m4.5.5.2" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.cmml"><msup id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4.cmml"><mi id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4.2" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.6.p4.4.m4.2.2.2.4" xref="S4.SS2.SSS1.6.p4.4.m4.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.6.p4.4.m4.1.1.1.1" xref="S4.SS2.SSS1.6.p4.4.m4.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.6.p4.4.m4.2.2.2.4.1" xref="S4.SS2.SSS1.6.p4.4.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.6.p4.4.m4.2.2.2.2" xref="S4.SS2.SSS1.6.p4.4.m4.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.3" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.3.cmml"><mo id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.3" stretchy="false" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.3.cmml">(</mo><msup id="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1" xref="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.2" xref="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.3" xref="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.4" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.cmml"><mi id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.2" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.3" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.5" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.3.cmml">;</mo><mi id="S4.SS2.SSS1.6.p4.4.m4.3.3" xref="S4.SS2.SSS1.6.p4.4.m4.3.3.cmml">X</mi><mo id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.6" stretchy="false" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.4.m4.5b"><apply id="S4.SS2.SSS1.6.p4.4.m4.5.5.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5"><in id="S4.SS2.SSS1.6.p4.4.m4.5.5.3.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.3"></in><apply id="S4.SS2.SSS1.6.p4.4.m4.5.5.4.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.4"><ci id="S4.SS2.SSS1.6.p4.4.m4.5.5.4.1.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.4.1">~</ci><ci id="S4.SS2.SSS1.6.p4.4.m4.5.5.4.2.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.4.2">𝑓</ci></apply><apply id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2"><times id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.3.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.3"></times><apply id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4.1.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4">superscript</csymbol><ci id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4.2.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.4.2">𝐻</ci><list id="S4.SS2.SSS1.6.p4.4.m4.2.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.2.2.2.4"><ci id="S4.SS2.SSS1.6.p4.4.m4.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.1.1.1.1">𝑠</ci><ci id="S4.SS2.SSS1.6.p4.4.m4.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2"><apply id="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.6.p4.4.m4.3.3.cmml" xref="S4.SS2.SSS1.6.p4.4.m4.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.4.m4.5c">\widetilde{f}\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.4.m4.5d">over~ start_ARG italic_f end_ARG ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\widetilde{f}|_{\mathbb{R}^{d}_{+}}=f" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.5.m5.2"><semantics id="S4.SS2.SSS1.6.p4.5.m5.2a"><mrow id="S4.SS2.SSS1.6.p4.5.m5.2.3" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.cmml"><msub id="S4.SS2.SSS1.6.p4.5.m5.2.3.2.2" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.2.1.cmml"><mrow id="S4.SS2.SSS1.6.p4.5.m5.2.3.2.2.2" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.2.1.cmml"><mover accent="true" id="S4.SS2.SSS1.6.p4.5.m5.1.1" xref="S4.SS2.SSS1.6.p4.5.m5.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.5.m5.1.1.2" xref="S4.SS2.SSS1.6.p4.5.m5.1.1.2.cmml">f</mi><mo id="S4.SS2.SSS1.6.p4.5.m5.1.1.1" xref="S4.SS2.SSS1.6.p4.5.m5.1.1.1.cmml">~</mo></mover><mo id="S4.SS2.SSS1.6.p4.5.m5.2.3.2.2.2.1" stretchy="false" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.2.1.1.cmml">|</mo></mrow><msubsup id="S4.SS2.SSS1.6.p4.5.m5.2.2.1" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1.cmml"><mi id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.2" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.3" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1.3.cmml">+</mo><mi id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.3" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.3.cmml">d</mi></msubsup></msub><mo id="S4.SS2.SSS1.6.p4.5.m5.2.3.1" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.1.cmml">=</mo><mi id="S4.SS2.SSS1.6.p4.5.m5.2.3.3" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.5.m5.2b"><apply id="S4.SS2.SSS1.6.p4.5.m5.2.3.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.3"><eq id="S4.SS2.SSS1.6.p4.5.m5.2.3.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.1"></eq><apply id="S4.SS2.SSS1.6.p4.5.m5.2.3.2.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.2.2"><csymbol cd="latexml" id="S4.SS2.SSS1.6.p4.5.m5.2.3.2.1.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.2.2.2.1">evaluated-at</csymbol><apply id="S4.SS2.SSS1.6.p4.5.m5.1.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.1.1"><ci id="S4.SS2.SSS1.6.p4.5.m5.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.1.1.1">~</ci><ci id="S4.SS2.SSS1.6.p4.5.m5.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.1.1.2">𝑓</ci></apply><apply id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1">subscript</csymbol><apply id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.1.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1">superscript</csymbol><ci id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.2.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.3.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS1.6.p4.5.m5.2.2.1.3.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.2.1.3"></plus></apply></apply><ci id="S4.SS2.SSS1.6.p4.5.m5.2.3.3.cmml" xref="S4.SS2.SSS1.6.p4.5.m5.2.3.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.5.m5.2c">\widetilde{f}|_{\mathbb{R}^{d}_{+}}=f</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.5.m5.2d">over~ start_ARG italic_f end_ARG | start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_f</annotation></semantics></math> and <math alttext="\overline{\operatorname{Tr}}_{m}\widetilde{f}=\overline{\operatorname{Tr}}_{m}% f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.6.m6.1"><semantics id="S4.SS2.SSS1.6.p4.6.m6.1a"><mrow id="S4.SS2.SSS1.6.p4.6.m6.1.1" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.cmml"><mrow 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xref="S4.SS2.SSS1.6.p4.6.m6.1.1.3.cmml">=</mo><mrow id="S4.SS2.SSS1.6.p4.6.m6.1.1.4" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.cmml"><msub id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.cmml"><mover accent="true" id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.cmml"><mi id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.2" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.2.cmml">Tr</mi><mo id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.1" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.1.cmml">¯</mo></mover><mi id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.3" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.3.cmml">m</mi></msub><mo id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.1" lspace="0.167em" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.1.cmml">⁢</mo><mi id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.3" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.3.cmml">f</mi></mrow><mo id="S4.SS2.SSS1.6.p4.6.m6.1.1.5" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.5.cmml">=</mo><mn id="S4.SS2.SSS1.6.p4.6.m6.1.1.6" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.6.m6.1b"><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1"><and id="S4.SS2.SSS1.6.p4.6.m6.1.1a.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1"></and><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1b.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1"><eq id="S4.SS2.SSS1.6.p4.6.m6.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.3"></eq><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2"><times id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.1"></times><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2">subscript</csymbol><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.2"><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.2.1">¯</ci><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.2.3">𝑚</ci></apply><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.3.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.3"><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.3.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.3.1">~</ci><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.2.3.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.2.3.2">𝑓</ci></apply></apply><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4"><times id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.1"></times><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2">subscript</csymbol><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2"><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.1">¯</ci><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.2.2">Tr</ci></apply><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.3.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.2.3">𝑚</ci></apply><ci id="S4.SS2.SSS1.6.p4.6.m6.1.1.4.3.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.4.3">𝑓</ci></apply></apply><apply id="S4.SS2.SSS1.6.p4.6.m6.1.1c.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1"><eq id="S4.SS2.SSS1.6.p4.6.m6.1.1.5.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S4.SS2.SSS1.6.p4.6.m6.1.1.4.cmml" id="S4.SS2.SSS1.6.p4.6.m6.1.1d.cmml" xref="S4.SS2.SSS1.6.p4.6.m6.1.1"></share><cn id="S4.SS2.SSS1.6.p4.6.m6.1.1.6.cmml" type="integer" xref="S4.SS2.SSS1.6.p4.6.m6.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.6.m6.1c">\overline{\operatorname{Tr}}_{m}\widetilde{f}=\overline{\operatorname{Tr}}_{m}% f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.6.m6.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT over~ start_ARG italic_f end_ARG = over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math> (see Remark <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem2" title="Remark 4.2. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.2</span></a>). By Step 1 there exists a sequence <math alttext="\widetilde{f}_{n}\in C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.7.m7.2"><semantics id="S4.SS2.SSS1.6.p4.7.m7.2a"><mrow id="S4.SS2.SSS1.6.p4.7.m7.2.2" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.cmml"><msub id="S4.SS2.SSS1.6.p4.7.m7.2.2.3" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.cmml"><mover accent="true" id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.cmml"><mi id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.2" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.2.cmml">f</mi><mo id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.1" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.1.cmml">~</mo></mover><mi id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.3" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.3.cmml">n</mi></msub><mo id="S4.SS2.SSS1.6.p4.7.m7.2.2.2" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS1.6.p4.7.m7.2.2.1" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.cmml"><msubsup id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.cmml"><mi id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.2" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.3" mathvariant="normal" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.2" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.2.cmml"><mo id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.2" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.3" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.3.cmml">∙</mo><mi id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.3" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.3" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.6.p4.7.m7.1.1" xref="S4.SS2.SSS1.6.p4.7.m7.1.1.cmml">X</mi><mo id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.4" stretchy="false" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.7.m7.2b"><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2"><in id="S4.SS2.SSS1.6.p4.7.m7.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.2"></in><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3">subscript</csymbol><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2"><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.1">~</ci><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.2.2">𝑓</ci></apply><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.3.3.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.3.3">𝑛</ci></apply><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1"><times id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.2"></times><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3">superscript</csymbol><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3">subscript</csymbol><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.2">𝐶</ci><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.3.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.3.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.3.3"></infinity></apply><list id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1"><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.2.3">∙</ci></apply><ci id="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.2.2.1.1.1.1.3">𝑑</ci></apply><ci id="S4.SS2.SSS1.6.p4.7.m7.1.1.cmml" xref="S4.SS2.SSS1.6.p4.7.m7.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.7.m7.2c">\widetilde{f}_{n}\in C_{\mathrm{c}}^{\infty}(\mathbb{R}_{\bullet}^{d};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.7.m7.2d">over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT ∙ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\widetilde{f}_{n}\to\widetilde{f}" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.8.m8.1"><semantics id="S4.SS2.SSS1.6.p4.8.m8.1a"><mrow id="S4.SS2.SSS1.6.p4.8.m8.1.1" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.cmml"><msub id="S4.SS2.SSS1.6.p4.8.m8.1.1.2" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.cmml"><mover accent="true" id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.cmml"><mi id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.2" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.2.cmml">f</mi><mo id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.1" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.1.cmml">~</mo></mover><mi id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.3" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.3.cmml">n</mi></msub><mo id="S4.SS2.SSS1.6.p4.8.m8.1.1.1" stretchy="false" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.1.cmml">→</mo><mover accent="true" id="S4.SS2.SSS1.6.p4.8.m8.1.1.3" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.3.cmml"><mi id="S4.SS2.SSS1.6.p4.8.m8.1.1.3.2" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.3.2.cmml">f</mi><mo id="S4.SS2.SSS1.6.p4.8.m8.1.1.3.1" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.3.1.cmml">~</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.8.m8.1b"><apply id="S4.SS2.SSS1.6.p4.8.m8.1.1.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1"><ci id="S4.SS2.SSS1.6.p4.8.m8.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.1">→</ci><apply id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.1.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2">subscript</csymbol><apply id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2"><ci id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.1">~</ci><ci id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.2.2">𝑓</ci></apply><ci id="S4.SS2.SSS1.6.p4.8.m8.1.1.2.3.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.2.3">𝑛</ci></apply><apply id="S4.SS2.SSS1.6.p4.8.m8.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.3"><ci id="S4.SS2.SSS1.6.p4.8.m8.1.1.3.1.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.3.1">~</ci><ci id="S4.SS2.SSS1.6.p4.8.m8.1.1.3.2.cmml" xref="S4.SS2.SSS1.6.p4.8.m8.1.1.3.2">𝑓</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.8.m8.1c">\widetilde{f}_{n}\to\widetilde{f}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.8.m8.1d">over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT → over~ start_ARG italic_f end_ARG</annotation></semantics></math> in <math alttext="H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.9.m9.5"><semantics id="S4.SS2.SSS1.6.p4.9.m9.5a"><mrow id="S4.SS2.SSS1.6.p4.9.m9.5.5" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.cmml"><msup id="S4.SS2.SSS1.6.p4.9.m9.5.5.4" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.4.cmml"><mi id="S4.SS2.SSS1.6.p4.9.m9.5.5.4.2" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.4.2.cmml">H</mi><mrow id="S4.SS2.SSS1.6.p4.9.m9.2.2.2.4" xref="S4.SS2.SSS1.6.p4.9.m9.2.2.2.3.cmml"><mi id="S4.SS2.SSS1.6.p4.9.m9.1.1.1.1" xref="S4.SS2.SSS1.6.p4.9.m9.1.1.1.1.cmml">s</mi><mo id="S4.SS2.SSS1.6.p4.9.m9.2.2.2.4.1" xref="S4.SS2.SSS1.6.p4.9.m9.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS1.6.p4.9.m9.2.2.2.2" xref="S4.SS2.SSS1.6.p4.9.m9.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.SS2.SSS1.6.p4.9.m9.5.5.3" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.3.cmml">⁢</mo><mrow id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.3.cmml"><mo id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.3" stretchy="false" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.3.cmml">(</mo><msup id="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1" xref="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.2" xref="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.3" xref="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.4" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.3.cmml">,</mo><msub id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.cmml"><mi id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.2" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.2.cmml">w</mi><mi id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.3" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.5" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.3.cmml">;</mo><mi id="S4.SS2.SSS1.6.p4.9.m9.3.3" xref="S4.SS2.SSS1.6.p4.9.m9.3.3.cmml">X</mi><mo id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.6" stretchy="false" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.9.m9.5b"><apply id="S4.SS2.SSS1.6.p4.9.m9.5.5.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5"><times id="S4.SS2.SSS1.6.p4.9.m9.5.5.3.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.3"></times><apply id="S4.SS2.SSS1.6.p4.9.m9.5.5.4.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.9.m9.5.5.4.1.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.4">superscript</csymbol><ci id="S4.SS2.SSS1.6.p4.9.m9.5.5.4.2.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.4.2">𝐻</ci><list id="S4.SS2.SSS1.6.p4.9.m9.2.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.2.2.2.4"><ci id="S4.SS2.SSS1.6.p4.9.m9.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.1.1.1.1">𝑠</ci><ci id="S4.SS2.SSS1.6.p4.9.m9.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.3.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2"><apply id="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.2">ℝ</ci><ci id="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.4.4.1.1.1.3">𝑑</ci></apply><apply id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.6.p4.9.m9.3.3.cmml" xref="S4.SS2.SSS1.6.p4.9.m9.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.9.m9.5c">H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.9.m9.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> as <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.10.m10.1"><semantics id="S4.SS2.SSS1.6.p4.10.m10.1a"><mrow id="S4.SS2.SSS1.6.p4.10.m10.1.1" xref="S4.SS2.SSS1.6.p4.10.m10.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.10.m10.1.1.2" xref="S4.SS2.SSS1.6.p4.10.m10.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.6.p4.10.m10.1.1.1" stretchy="false" xref="S4.SS2.SSS1.6.p4.10.m10.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.6.p4.10.m10.1.1.3" mathvariant="normal" xref="S4.SS2.SSS1.6.p4.10.m10.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.10.m10.1b"><apply id="S4.SS2.SSS1.6.p4.10.m10.1.1.cmml" xref="S4.SS2.SSS1.6.p4.10.m10.1.1"><ci id="S4.SS2.SSS1.6.p4.10.m10.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.10.m10.1.1.1">→</ci><ci id="S4.SS2.SSS1.6.p4.10.m10.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.10.m10.1.1.2">𝑛</ci><infinity id="S4.SS2.SSS1.6.p4.10.m10.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.10.m10.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.10.m10.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.10.m10.1d">italic_n → ∞</annotation></semantics></math>. It follows that <math alttext="f_{n}:=\widetilde{f}_{n}|_{\mathbb{R}^{d}_{+}}\in C_{\mathrm{c}}^{\infty}(% \mathbb{R}^{d}_{+};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.11.m11.4"><semantics id="S4.SS2.SSS1.6.p4.11.m11.4a"><mrow id="S4.SS2.SSS1.6.p4.11.m11.4.4" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.cmml"><msub id="S4.SS2.SSS1.6.p4.11.m11.4.4.4" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.4.cmml"><mi id="S4.SS2.SSS1.6.p4.11.m11.4.4.4.2" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.4.2.cmml">f</mi><mi id="S4.SS2.SSS1.6.p4.11.m11.4.4.4.3" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.4.3.cmml">n</mi></msub><mo id="S4.SS2.SSS1.6.p4.11.m11.4.4.5" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.5.cmml">:=</mo><msub id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.2.cmml"><mrow id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.2.cmml"><msub id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.cmml"><mover accent="true" id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.2" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.2.cmml"><mi id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.2.2" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.2.2.cmml">f</mi><mo id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.2.1" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.2.1.cmml">~</mo></mover><mi id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.3" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.2" stretchy="false" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.2.1.cmml">|</mo></mrow><msubsup id="S4.SS2.SSS1.6.p4.11.m11.1.1.1" xref="S4.SS2.SSS1.6.p4.11.m11.1.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.11.m11.1.1.1.2.2" xref="S4.SS2.SSS1.6.p4.11.m11.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.6.p4.11.m11.1.1.1.3" xref="S4.SS2.SSS1.6.p4.11.m11.1.1.1.3.cmml">+</mo><mi id="S4.SS2.SSS1.6.p4.11.m11.1.1.1.2.3" xref="S4.SS2.SSS1.6.p4.11.m11.1.1.1.2.3.cmml">d</mi></msubsup></msub><mo id="S4.SS2.SSS1.6.p4.11.m11.4.4.6" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.6.cmml">∈</mo><mrow id="S4.SS2.SSS1.6.p4.11.m11.4.4.2" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.cmml"><msubsup id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3.cmml"><mi id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3.2.2" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3.2.3" mathvariant="normal" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3.3" mathvariant="normal" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.2" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.2.cmml"><mo id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.2" stretchy="false" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.2" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.3" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.3.cmml">+</mo><mi id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.3" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.3" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.2.cmml">;</mo><mi id="S4.SS2.SSS1.6.p4.11.m11.2.2" xref="S4.SS2.SSS1.6.p4.11.m11.2.2.cmml">X</mi><mo id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.4" stretchy="false" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.11.m11.4b"><apply id="S4.SS2.SSS1.6.p4.11.m11.4.4.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4"><and id="S4.SS2.SSS1.6.p4.11.m11.4.4a.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4"></and><apply id="S4.SS2.SSS1.6.p4.11.m11.4.4b.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4"><csymbol cd="latexml" id="S4.SS2.SSS1.6.p4.11.m11.4.4.5.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.5">assign</csymbol><apply id="S4.SS2.SSS1.6.p4.11.m11.4.4.4.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.4"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.11.m11.4.4.4.1.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.4">subscript</csymbol><ci id="S4.SS2.SSS1.6.p4.11.m11.4.4.4.2.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.4.2">𝑓</ci><ci id="S4.SS2.SSS1.6.p4.11.m11.4.4.4.3.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.4.3">𝑛</ci></apply><apply id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.2.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1"><csymbol cd="latexml" id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.2.1.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.2">evaluated-at</csymbol><apply id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS1.6.p4.11.m11.3.3.1.1.1.1.2.cmml" 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id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.2.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.3.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.4.4.2.1.1.1.3"></plus></apply><ci id="S4.SS2.SSS1.6.p4.11.m11.2.2.cmml" xref="S4.SS2.SSS1.6.p4.11.m11.2.2">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.11.m11.4c">f_{n}:=\widetilde{f}_{n}|_{\mathbb{R}^{d}_{+}}\in C_{\mathrm{c}}^{\infty}(% \mathbb{R}^{d}_{+};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.11.m11.4d">italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT := over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT | start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> and <math alttext="f_{n}\to f" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.12.m12.1"><semantics id="S4.SS2.SSS1.6.p4.12.m12.1a"><mrow id="S4.SS2.SSS1.6.p4.12.m12.1.1" xref="S4.SS2.SSS1.6.p4.12.m12.1.1.cmml"><msub id="S4.SS2.SSS1.6.p4.12.m12.1.1.2" xref="S4.SS2.SSS1.6.p4.12.m12.1.1.2.cmml"><mi id="S4.SS2.SSS1.6.p4.12.m12.1.1.2.2" xref="S4.SS2.SSS1.6.p4.12.m12.1.1.2.2.cmml">f</mi><mi id="S4.SS2.SSS1.6.p4.12.m12.1.1.2.3" xref="S4.SS2.SSS1.6.p4.12.m12.1.1.2.3.cmml">n</mi></msub><mo id="S4.SS2.SSS1.6.p4.12.m12.1.1.1" stretchy="false" xref="S4.SS2.SSS1.6.p4.12.m12.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.6.p4.12.m12.1.1.3" xref="S4.SS2.SSS1.6.p4.12.m12.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" 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xref="S4.SS2.SSS1.6.p4.13.m13.5.5.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS1.6.p4.13.m13.5.5.2.2.2.1.cmml" xref="S4.SS2.SSS1.6.p4.13.m13.5.5.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS1.6.p4.13.m13.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS1.6.p4.13.m13.5.5.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS1.6.p4.13.m13.5.5.2.2.2.3.cmml" xref="S4.SS2.SSS1.6.p4.13.m13.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS1.6.p4.13.m13.3.3.cmml" xref="S4.SS2.SSS1.6.p4.13.m13.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.13.m13.5c">H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.13.m13.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> as <math alttext="n\to\infty" class="ltx_Math" display="inline" id="S4.SS2.SSS1.6.p4.14.m14.1"><semantics id="S4.SS2.SSS1.6.p4.14.m14.1a"><mrow id="S4.SS2.SSS1.6.p4.14.m14.1.1" xref="S4.SS2.SSS1.6.p4.14.m14.1.1.cmml"><mi id="S4.SS2.SSS1.6.p4.14.m14.1.1.2" xref="S4.SS2.SSS1.6.p4.14.m14.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS1.6.p4.14.m14.1.1.1" stretchy="false" xref="S4.SS2.SSS1.6.p4.14.m14.1.1.1.cmml">→</mo><mi id="S4.SS2.SSS1.6.p4.14.m14.1.1.3" mathvariant="normal" xref="S4.SS2.SSS1.6.p4.14.m14.1.1.3.cmml">∞</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS1.6.p4.14.m14.1b"><apply id="S4.SS2.SSS1.6.p4.14.m14.1.1.cmml" xref="S4.SS2.SSS1.6.p4.14.m14.1.1"><ci id="S4.SS2.SSS1.6.p4.14.m14.1.1.1.cmml" xref="S4.SS2.SSS1.6.p4.14.m14.1.1.1">→</ci><ci id="S4.SS2.SSS1.6.p4.14.m14.1.1.2.cmml" xref="S4.SS2.SSS1.6.p4.14.m14.1.1.2">𝑛</ci><infinity id="S4.SS2.SSS1.6.p4.14.m14.1.1.3.cmml" xref="S4.SS2.SSS1.6.p4.14.m14.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS1.6.p4.14.m14.1c">n\to\infty</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS1.6.p4.14.m14.1d">italic_n → ∞</annotation></semantics></math>. ∎</p> </div> </div> </section> <section class="ltx_subsubsection" id="S4.SS2.SSS2"> <h4 class="ltx_title ltx_title_subsubsection"> <span class="ltx_tag ltx_tag_subsubsection">4.2.2. </span>Sobolev spaces</h4> <div class="ltx_para" id="S4.SS2.SSS2.p1"> <p class="ltx_p" id="S4.SS2.SSS2.p1.4">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.1.m1.2"><semantics id="S4.SS2.SSS2.p1.1.m1.2a"><mrow id="S4.SS2.SSS2.p1.1.m1.2.3" xref="S4.SS2.SSS2.p1.1.m1.2.3.cmml"><mi id="S4.SS2.SSS2.p1.1.m1.2.3.2" xref="S4.SS2.SSS2.p1.1.m1.2.3.2.cmml">p</mi><mo id="S4.SS2.SSS2.p1.1.m1.2.3.1" xref="S4.SS2.SSS2.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.p1.1.m1.2.3.3.2" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml"><mo id="S4.SS2.SSS2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.SS2.SSS2.p1.1.m1.1.1" xref="S4.SS2.SSS2.p1.1.m1.1.1.cmml">1</mn><mo id="S4.SS2.SSS2.p1.1.m1.2.3.3.2.2" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.p1.1.m1.2.2" mathvariant="normal" xref="S4.SS2.SSS2.p1.1.m1.2.2.cmml">∞</mi><mo id="S4.SS2.SSS2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.1.m1.2b"><apply id="S4.SS2.SSS2.p1.1.m1.2.3.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3"><in id="S4.SS2.SSS2.p1.1.m1.2.3.1.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3.1"></in><ci id="S4.SS2.SSS2.p1.1.m1.2.3.2.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.SS2.SSS2.p1.1.m1.2.3.3.1.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.3.3.2"><cn id="S4.SS2.SSS2.p1.1.m1.1.1.cmml" type="integer" xref="S4.SS2.SSS2.p1.1.m1.1.1">1</cn><infinity id="S4.SS2.SSS2.p1.1.m1.2.2.cmml" xref="S4.SS2.SSS2.p1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.2.m2.1"><semantics id="S4.SS2.SSS2.p1.2.m2.1a"><mrow id="S4.SS2.SSS2.p1.2.m2.1.1" xref="S4.SS2.SSS2.p1.2.m2.1.1.cmml"><mi id="S4.SS2.SSS2.p1.2.m2.1.1.2" xref="S4.SS2.SSS2.p1.2.m2.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS2.p1.2.m2.1.1.1" xref="S4.SS2.SSS2.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS2.p1.2.m2.1.1.3" xref="S4.SS2.SSS2.p1.2.m2.1.1.3.cmml"><mi id="S4.SS2.SSS2.p1.2.m2.1.1.3.2" xref="S4.SS2.SSS2.p1.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS2.p1.2.m2.1.1.3.3" xref="S4.SS2.SSS2.p1.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.2.m2.1b"><apply id="S4.SS2.SSS2.p1.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1"><in id="S4.SS2.SSS2.p1.2.m2.1.1.1.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1.1"></in><ci id="S4.SS2.SSS2.p1.2.m2.1.1.2.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1.2">𝑘</ci><apply id="S4.SS2.SSS2.p1.2.m2.1.1.3.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.2.m2.1.1.3.1.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.p1.2.m2.1.1.3.2.cmml" xref="S4.SS2.SSS2.p1.2.m2.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS2.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.2.m2.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.2.m2.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.3.m3.4"><semantics id="S4.SS2.SSS2.p1.3.m3.4a"><mrow id="S4.SS2.SSS2.p1.3.m3.4.4" xref="S4.SS2.SSS2.p1.3.m3.4.4.cmml"><mi id="S4.SS2.SSS2.p1.3.m3.4.4.5" xref="S4.SS2.SSS2.p1.3.m3.4.4.5.cmml">γ</mi><mo id="S4.SS2.SSS2.p1.3.m3.4.4.4" xref="S4.SS2.SSS2.p1.3.m3.4.4.4.cmml">∈</mo><mrow id="S4.SS2.SSS2.p1.3.m3.4.4.3" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.cmml"><mrow id="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1" xref="S4.SS2.SSS2.p1.3.m3.2.2.1.1.2.cmml"><mo id="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.p1.3.m3.2.2.1.1.2.cmml">(</mo><mrow id="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.1" xref="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.1a" xref="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.1.2" xref="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.3" xref="S4.SS2.SSS2.p1.3.m3.2.2.1.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.p1.3.m3.1.1" mathvariant="normal" xref="S4.SS2.SSS2.p1.3.m3.1.1.cmml">∞</mi><mo id="S4.SS2.SSS2.p1.3.m3.2.2.1.1.1.4" stretchy="false" xref="S4.SS2.SSS2.p1.3.m3.2.2.1.1.2.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.p1.3.m3.4.4.3.4" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.4.cmml">∖</mo><mrow id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.3.cmml"><mo id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.3.1.cmml">{</mo><mrow id="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1" xref="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.cmml"><mrow id="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2" xref="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2.cmml"><mi id="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2.2" xref="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2.1" xref="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2.3" xref="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.1" xref="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.3" xref="S4.SS2.SSS2.p1.3.m3.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.3.1.cmml">:</mo><mrow id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.cmml"><mi id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.2" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.2.cmml">j</mi><mo id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.1" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.3" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.3.cmml"><mi id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.3.2" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.3.3" xref="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S4.SS2.SSS2.p1.3.m3.4.4.3.3.2.5" stretchy="false" 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italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.4.m4.1"><semantics id="S4.SS2.SSS2.p1.4.m4.1a"><mi id="S4.SS2.SSS2.p1.4.m4.1.1" xref="S4.SS2.SSS2.p1.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.4.m4.1b"><ci id="S4.SS2.SSS2.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS2.p1.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.4.m4.1d">italic_X</annotation></semantics></math> a Banach space. Then we define</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex37"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="W^{k,p}_{0}(\mathbb{R}_{+}^{d},w_{\gamma};X):=\left\{f\in W^{k,p}(\mathbb{R}_{% +}^{d},w_{\gamma};X):\operatorname{Tr}(\partial^{\alpha}f)=0\text{ if }k-|% \alpha|>\tfrac{\gamma+1}{p}\right\}." class="ltx_Math" display="block" id="S4.Ex37.m1.9"><semantics id="S4.Ex37.m1.9a"><mrow id="S4.Ex37.m1.9.9.1" xref="S4.Ex37.m1.9.9.1.1.cmml"><mrow id="S4.Ex37.m1.9.9.1.1" xref="S4.Ex37.m1.9.9.1.1.cmml"><mrow id="S4.Ex37.m1.9.9.1.1.2" xref="S4.Ex37.m1.9.9.1.1.2.cmml"><msubsup id="S4.Ex37.m1.9.9.1.1.2.4" xref="S4.Ex37.m1.9.9.1.1.2.4.cmml"><mi id="S4.Ex37.m1.9.9.1.1.2.4.2.2" xref="S4.Ex37.m1.9.9.1.1.2.4.2.2.cmml">W</mi><mn id="S4.Ex37.m1.9.9.1.1.2.4.3" xref="S4.Ex37.m1.9.9.1.1.2.4.3.cmml">0</mn><mrow id="S4.Ex37.m1.2.2.2.4" 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xref="S4.Ex37.m1.9.9.1.1.4.2.2.4.2.4">𝑘</ci></apply><apply id="S4.Ex37.m1.9.9.1.1.4.2.2.4.3.1.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.4.3.2"><abs id="S4.Ex37.m1.9.9.1.1.4.2.2.4.3.1.1.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.4.3.2.1"></abs><ci id="S4.Ex37.m1.8.8.cmml" xref="S4.Ex37.m1.8.8">𝛼</ci></apply></apply></apply><apply id="S4.Ex37.m1.9.9.1.1.4.2.2c.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2"><gt id="S4.Ex37.m1.9.9.1.1.4.2.2.5.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.5"></gt><share href="https://arxiv.org/html/2503.14636v1#S4.Ex37.m1.9.9.1.1.4.2.2.4.cmml" id="S4.Ex37.m1.9.9.1.1.4.2.2d.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2"></share><apply id="S4.Ex37.m1.9.9.1.1.4.2.2.6.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.6"><divide id="S4.Ex37.m1.9.9.1.1.4.2.2.6.1.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.6"></divide><apply id="S4.Ex37.m1.9.9.1.1.4.2.2.6.2.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.6.2"><plus id="S4.Ex37.m1.9.9.1.1.4.2.2.6.2.1.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.6.2.1"></plus><ci id="S4.Ex37.m1.9.9.1.1.4.2.2.6.2.2.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.6.2.2">𝛾</ci><cn id="S4.Ex37.m1.9.9.1.1.4.2.2.6.2.3.cmml" type="integer" xref="S4.Ex37.m1.9.9.1.1.4.2.2.6.2.3">1</cn></apply><ci id="S4.Ex37.m1.9.9.1.1.4.2.2.6.3.cmml" xref="S4.Ex37.m1.9.9.1.1.4.2.2.6.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex37.m1.9c">W^{k,p}_{0}(\mathbb{R}_{+}^{d},w_{\gamma};X):=\left\{f\in W^{k,p}(\mathbb{R}_{% +}^{d},w_{\gamma};X):\operatorname{Tr}(\partial^{\alpha}f)=0\text{ if }k-|% \alpha|>\tfrac{\gamma+1}{p}\right\}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex37.m1.9d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) := { italic_f ∈ italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) : roman_Tr ( ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_f ) = 0 if italic_k - | italic_α | > divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.p1.5">Moreover, if <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.5.m1.1"><semantics id="S4.SS2.SSS2.p1.5.m1.1a"><mrow id="S4.SS2.SSS2.p1.5.m1.1.1" xref="S4.SS2.SSS2.p1.5.m1.1.1.cmml"><mi id="S4.SS2.SSS2.p1.5.m1.1.1.2" xref="S4.SS2.SSS2.p1.5.m1.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS2.p1.5.m1.1.1.1" xref="S4.SS2.SSS2.p1.5.m1.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS2.p1.5.m1.1.1.3" xref="S4.SS2.SSS2.p1.5.m1.1.1.3.cmml"><mi id="S4.SS2.SSS2.p1.5.m1.1.1.3.2" xref="S4.SS2.SSS2.p1.5.m1.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS2.p1.5.m1.1.1.3.3" xref="S4.SS2.SSS2.p1.5.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.5.m1.1b"><apply id="S4.SS2.SSS2.p1.5.m1.1.1.cmml" xref="S4.SS2.SSS2.p1.5.m1.1.1"><in id="S4.SS2.SSS2.p1.5.m1.1.1.1.cmml" xref="S4.SS2.SSS2.p1.5.m1.1.1.1"></in><ci id="S4.SS2.SSS2.p1.5.m1.1.1.2.cmml" xref="S4.SS2.SSS2.p1.5.m1.1.1.2">𝑚</ci><apply id="S4.SS2.SSS2.p1.5.m1.1.1.3.cmml" xref="S4.SS2.SSS2.p1.5.m1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.5.m1.1.1.3.1.cmml" xref="S4.SS2.SSS2.p1.5.m1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.p1.5.m1.1.1.3.2.cmml" xref="S4.SS2.SSS2.p1.5.m1.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS2.p1.5.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.p1.5.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.5.m1.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.5.m1.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, then we define</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx24"> <tbody id="S4.Ex38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W^{k,p}_{\operatorname{Tr}_{m}}(\mathbb{R}_{+}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.Ex38.m1.5"><semantics id="S4.Ex38.m1.5a"><mrow id="S4.Ex38.m1.5.5" xref="S4.Ex38.m1.5.5.cmml"><msubsup id="S4.Ex38.m1.5.5.4" xref="S4.Ex38.m1.5.5.4.cmml"><mi id="S4.Ex38.m1.5.5.4.2.2" xref="S4.Ex38.m1.5.5.4.2.2.cmml">W</mi><msub id="S4.Ex38.m1.5.5.4.3" xref="S4.Ex38.m1.5.5.4.3.cmml"><mi id="S4.Ex38.m1.5.5.4.3.2" xref="S4.Ex38.m1.5.5.4.3.2.cmml">Tr</mi><mi id="S4.Ex38.m1.5.5.4.3.3" xref="S4.Ex38.m1.5.5.4.3.3.cmml">m</mi></msub><mrow id="S4.Ex38.m1.2.2.2.4" xref="S4.Ex38.m1.2.2.2.3.cmml"><mi id="S4.Ex38.m1.1.1.1.1" xref="S4.Ex38.m1.1.1.1.1.cmml">k</mi><mo id="S4.Ex38.m1.2.2.2.4.1" xref="S4.Ex38.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex38.m1.2.2.2.2" xref="S4.Ex38.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S4.Ex38.m1.5.5.3" xref="S4.Ex38.m1.5.5.3.cmml">⁢</mo><mrow id="S4.Ex38.m1.5.5.2.2" xref="S4.Ex38.m1.5.5.2.3.cmml"><mo id="S4.Ex38.m1.5.5.2.2.3" stretchy="false" xref="S4.Ex38.m1.5.5.2.3.cmml">(</mo><msubsup id="S4.Ex38.m1.4.4.1.1.1" xref="S4.Ex38.m1.4.4.1.1.1.cmml"><mi id="S4.Ex38.m1.4.4.1.1.1.2.2" xref="S4.Ex38.m1.4.4.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex38.m1.4.4.1.1.1.2.3" xref="S4.Ex38.m1.4.4.1.1.1.2.3.cmml">+</mo><mi id="S4.Ex38.m1.4.4.1.1.1.3" xref="S4.Ex38.m1.4.4.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.Ex38.m1.5.5.2.2.4" xref="S4.Ex38.m1.5.5.2.3.cmml">,</mo><msub id="S4.Ex38.m1.5.5.2.2.2" xref="S4.Ex38.m1.5.5.2.2.2.cmml"><mi id="S4.Ex38.m1.5.5.2.2.2.2" xref="S4.Ex38.m1.5.5.2.2.2.2.cmml">w</mi><mi id="S4.Ex38.m1.5.5.2.2.2.3" xref="S4.Ex38.m1.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex38.m1.5.5.2.2.5" xref="S4.Ex38.m1.5.5.2.3.cmml">;</mo><mi id="S4.Ex38.m1.3.3" xref="S4.Ex38.m1.3.3.cmml">X</mi><mo id="S4.Ex38.m1.5.5.2.2.6" stretchy="false" xref="S4.Ex38.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Ex38.m1.5b"><apply id="S4.Ex38.m1.5.5.cmml" xref="S4.Ex38.m1.5.5"><times id="S4.Ex38.m1.5.5.3.cmml" xref="S4.Ex38.m1.5.5.3"></times><apply id="S4.Ex38.m1.5.5.4.cmml" xref="S4.Ex38.m1.5.5.4"><csymbol cd="ambiguous" id="S4.Ex38.m1.5.5.4.1.cmml" xref="S4.Ex38.m1.5.5.4">subscript</csymbol><apply id="S4.Ex38.m1.5.5.4.2.cmml" xref="S4.Ex38.m1.5.5.4"><csymbol cd="ambiguous" id="S4.Ex38.m1.5.5.4.2.1.cmml" xref="S4.Ex38.m1.5.5.4">superscript</csymbol><ci id="S4.Ex38.m1.5.5.4.2.2.cmml" xref="S4.Ex38.m1.5.5.4.2.2">𝑊</ci><list id="S4.Ex38.m1.2.2.2.3.cmml" xref="S4.Ex38.m1.2.2.2.4"><ci id="S4.Ex38.m1.1.1.1.1.cmml" xref="S4.Ex38.m1.1.1.1.1">𝑘</ci><ci id="S4.Ex38.m1.2.2.2.2.cmml" xref="S4.Ex38.m1.2.2.2.2">𝑝</ci></list></apply><apply id="S4.Ex38.m1.5.5.4.3.cmml" xref="S4.Ex38.m1.5.5.4.3"><csymbol cd="ambiguous" id="S4.Ex38.m1.5.5.4.3.1.cmml" xref="S4.Ex38.m1.5.5.4.3">subscript</csymbol><ci id="S4.Ex38.m1.5.5.4.3.2.cmml" xref="S4.Ex38.m1.5.5.4.3.2">Tr</ci><ci id="S4.Ex38.m1.5.5.4.3.3.cmml" xref="S4.Ex38.m1.5.5.4.3.3">𝑚</ci></apply></apply><vector id="S4.Ex38.m1.5.5.2.3.cmml" xref="S4.Ex38.m1.5.5.2.2"><apply id="S4.Ex38.m1.4.4.1.1.1.cmml" xref="S4.Ex38.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.Ex38.m1.4.4.1.1.1.1.cmml" xref="S4.Ex38.m1.4.4.1.1.1">superscript</csymbol><apply id="S4.Ex38.m1.4.4.1.1.1.2.cmml" xref="S4.Ex38.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S4.Ex38.m1.4.4.1.1.1.2.1.cmml" xref="S4.Ex38.m1.4.4.1.1.1">subscript</csymbol><ci id="S4.Ex38.m1.4.4.1.1.1.2.2.cmml" xref="S4.Ex38.m1.4.4.1.1.1.2.2">ℝ</ci><plus id="S4.Ex38.m1.4.4.1.1.1.2.3.cmml" xref="S4.Ex38.m1.4.4.1.1.1.2.3"></plus></apply><ci id="S4.Ex38.m1.4.4.1.1.1.3.cmml" xref="S4.Ex38.m1.4.4.1.1.1.3">𝑑</ci></apply><apply id="S4.Ex38.m1.5.5.2.2.2.cmml" xref="S4.Ex38.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S4.Ex38.m1.5.5.2.2.2.1.cmml" xref="S4.Ex38.m1.5.5.2.2.2">subscript</csymbol><ci id="S4.Ex38.m1.5.5.2.2.2.2.cmml" xref="S4.Ex38.m1.5.5.2.2.2.2">𝑤</ci><ci id="S4.Ex38.m1.5.5.2.2.2.3.cmml" xref="S4.Ex38.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.Ex38.m1.3.3.cmml" xref="S4.Ex38.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex38.m1.5c">\displaystyle W^{k,p}_{\operatorname{Tr}_{m}}(\mathbb{R}_{+}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex38.m1.5d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:=\left\{f\in W^{k,p}(\mathbb{R}_{+}^{d},w_{\gamma};X):% \operatorname{Tr}_{m}f=0\text{ if }k>m+\tfrac{\gamma+1}{p}\right\}." class="ltx_Math" display="inline" id="S4.Ex38.m2.4"><semantics id="S4.Ex38.m2.4a"><mrow id="S4.Ex38.m2.4.4.1" xref="S4.Ex38.m2.4.4.1.1.cmml"><mrow id="S4.Ex38.m2.4.4.1.1" xref="S4.Ex38.m2.4.4.1.1.cmml"><mi id="S4.Ex38.m2.4.4.1.1.4" xref="S4.Ex38.m2.4.4.1.1.4.cmml"></mi><mo id="S4.Ex38.m2.4.4.1.1.3" lspace="0.278em" rspace="0.278em" xref="S4.Ex38.m2.4.4.1.1.3.cmml">:=</mo><mrow id="S4.Ex38.m2.4.4.1.1.2.2" xref="S4.Ex38.m2.4.4.1.1.2.3.cmml"><mo id="S4.Ex38.m2.4.4.1.1.2.2.3" xref="S4.Ex38.m2.4.4.1.1.2.3.1.cmml">{</mo><mrow id="S4.Ex38.m2.4.4.1.1.1.1.1" xref="S4.Ex38.m2.4.4.1.1.1.1.1.cmml"><mi id="S4.Ex38.m2.4.4.1.1.1.1.1.4" 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id="S4.Ex38.m2.4.4.1.1.2.2.2d.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2"></share><apply id="S4.Ex38.m2.4.4.1.1.2.2.2.6.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6"><plus id="S4.Ex38.m2.4.4.1.1.2.2.2.6.1.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.1"></plus><ci id="S4.Ex38.m2.4.4.1.1.2.2.2.6.2.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.2">𝑚</ci><apply id="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.3"><divide id="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.1.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.3"></divide><apply id="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2"><plus id="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2.1.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2.1"></plus><ci id="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2.2.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2.2">𝛾</ci><cn id="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2.3.cmml" type="integer" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.2.3">1</cn></apply><ci id="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.3.cmml" xref="S4.Ex38.m2.4.4.1.1.2.2.2.6.3.3">𝑝</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex38.m2.4c">\displaystyle:=\left\{f\in W^{k,p}(\mathbb{R}_{+}^{d},w_{\gamma};X):% \operatorname{Tr}_{m}f=0\text{ if }k>m+\tfrac{\gamma+1}{p}\right\}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex38.m2.4d">:= { italic_f ∈ italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) : roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0 if italic_k > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.p1.10">All the traces in the above definitions are well defined by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4" title="Theorem 4.4. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a>. Note that for <math alttext="m=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.6.m1.1"><semantics id="S4.SS2.SSS2.p1.6.m1.1a"><mrow id="S4.SS2.SSS2.p1.6.m1.1.1" xref="S4.SS2.SSS2.p1.6.m1.1.1.cmml"><mi id="S4.SS2.SSS2.p1.6.m1.1.1.2" xref="S4.SS2.SSS2.p1.6.m1.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS2.p1.6.m1.1.1.1" xref="S4.SS2.SSS2.p1.6.m1.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS2.p1.6.m1.1.1.3" xref="S4.SS2.SSS2.p1.6.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.6.m1.1b"><apply id="S4.SS2.SSS2.p1.6.m1.1.1.cmml" xref="S4.SS2.SSS2.p1.6.m1.1.1"><eq id="S4.SS2.SSS2.p1.6.m1.1.1.1.cmml" xref="S4.SS2.SSS2.p1.6.m1.1.1.1"></eq><ci id="S4.SS2.SSS2.p1.6.m1.1.1.2.cmml" xref="S4.SS2.SSS2.p1.6.m1.1.1.2">𝑚</ci><cn id="S4.SS2.SSS2.p1.6.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p1.6.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.6.m1.1c">m=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.6.m1.1d">italic_m = 0</annotation></semantics></math> and <math alttext="m=1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.7.m2.1"><semantics id="S4.SS2.SSS2.p1.7.m2.1a"><mrow id="S4.SS2.SSS2.p1.7.m2.1.1" xref="S4.SS2.SSS2.p1.7.m2.1.1.cmml"><mi id="S4.SS2.SSS2.p1.7.m2.1.1.2" xref="S4.SS2.SSS2.p1.7.m2.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS2.p1.7.m2.1.1.1" xref="S4.SS2.SSS2.p1.7.m2.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS2.p1.7.m2.1.1.3" xref="S4.SS2.SSS2.p1.7.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.7.m2.1b"><apply id="S4.SS2.SSS2.p1.7.m2.1.1.cmml" xref="S4.SS2.SSS2.p1.7.m2.1.1"><eq id="S4.SS2.SSS2.p1.7.m2.1.1.1.cmml" xref="S4.SS2.SSS2.p1.7.m2.1.1.1"></eq><ci id="S4.SS2.SSS2.p1.7.m2.1.1.2.cmml" xref="S4.SS2.SSS2.p1.7.m2.1.1.2">𝑚</ci><cn id="S4.SS2.SSS2.p1.7.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p1.7.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.7.m2.1c">m=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.7.m2.1d">italic_m = 1</annotation></semantics></math> the above Sobolev spaces with zero boundary conditions coincide with the Dirichlet and Neumann spaces as defined <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>]</cite>. Similar to the Bessel potential spaces, it holds that <math alttext="C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.8.m3.2"><semantics id="S4.SS2.SSS2.p1.8.m3.2a"><mrow id="S4.SS2.SSS2.p1.8.m3.2.2" xref="S4.SS2.SSS2.p1.8.m3.2.2.cmml"><msubsup id="S4.SS2.SSS2.p1.8.m3.2.2.3" xref="S4.SS2.SSS2.p1.8.m3.2.2.3.cmml"><mi id="S4.SS2.SSS2.p1.8.m3.2.2.3.2.2" xref="S4.SS2.SSS2.p1.8.m3.2.2.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS2.p1.8.m3.2.2.3.2.3" mathvariant="normal" xref="S4.SS2.SSS2.p1.8.m3.2.2.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS2.p1.8.m3.2.2.3.3" mathvariant="normal" xref="S4.SS2.SSS2.p1.8.m3.2.2.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS2.p1.8.m3.2.2.2" xref="S4.SS2.SSS2.p1.8.m3.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.8.m3.2.2.1.1" xref="S4.SS2.SSS2.p1.8.m3.2.2.1.2.cmml"><mo id="S4.SS2.SSS2.p1.8.m3.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS2.p1.8.m3.2.2.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1" 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xref="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS2.p1.8.m3.2.2.1.1.1.3"></plus></apply><ci id="S4.SS2.SSS2.p1.8.m3.1.1.cmml" xref="S4.SS2.SSS2.p1.8.m3.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.8.m3.2c">C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.8.m3.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> is dense in <math alttext="W_{0}^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.9.m4.5"><semantics id="S4.SS2.SSS2.p1.9.m4.5a"><mrow id="S4.SS2.SSS2.p1.9.m4.5.5" xref="S4.SS2.SSS2.p1.9.m4.5.5.cmml"><msubsup id="S4.SS2.SSS2.p1.9.m4.5.5.4" xref="S4.SS2.SSS2.p1.9.m4.5.5.4.cmml"><mi id="S4.SS2.SSS2.p1.9.m4.5.5.4.2.2" xref="S4.SS2.SSS2.p1.9.m4.5.5.4.2.2.cmml">W</mi><mn id="S4.SS2.SSS2.p1.9.m4.5.5.4.2.3" xref="S4.SS2.SSS2.p1.9.m4.5.5.4.2.3.cmml">0</mn><mrow id="S4.SS2.SSS2.p1.9.m4.2.2.2.4" xref="S4.SS2.SSS2.p1.9.m4.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.p1.9.m4.1.1.1.1" xref="S4.SS2.SSS2.p1.9.m4.1.1.1.1.cmml">k</mi><mo id="S4.SS2.SSS2.p1.9.m4.2.2.2.4.1" xref="S4.SS2.SSS2.p1.9.m4.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS2.p1.9.m4.2.2.2.2" xref="S4.SS2.SSS2.p1.9.m4.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S4.SS2.SSS2.p1.9.m4.5.5.3" xref="S4.SS2.SSS2.p1.9.m4.5.5.3.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p1.9.m4.5.5.2.2" xref="S4.SS2.SSS2.p1.9.m4.5.5.2.3.cmml"><mo id="S4.SS2.SSS2.p1.9.m4.5.5.2.2.3" stretchy="false" 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xref="S4.SS2.SSS2.p1.9.m4.5.5.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.9.m4.5.5.2.2.2.1.cmml" xref="S4.SS2.SSS2.p1.9.m4.5.5.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.p1.9.m4.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS2.p1.9.m4.5.5.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS2.p1.9.m4.5.5.2.2.2.3.cmml" xref="S4.SS2.SSS2.p1.9.m4.5.5.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS2.p1.9.m4.3.3.cmml" xref="S4.SS2.SSS2.p1.9.m4.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.9.m4.5c">W_{0}^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.9.m4.5d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>, see, e.g., <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>, Proposition 3.8]</cite>. To deal with higher-order boundary conditions we need spaces of smooth functions where only higher-order derivatives are compactly supported. For <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.10.m5.1"><semantics id="S4.SS2.SSS2.p1.10.m5.1a"><mrow id="S4.SS2.SSS2.p1.10.m5.1.1" xref="S4.SS2.SSS2.p1.10.m5.1.1.cmml"><mi id="S4.SS2.SSS2.p1.10.m5.1.1.2" xref="S4.SS2.SSS2.p1.10.m5.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS2.p1.10.m5.1.1.1" xref="S4.SS2.SSS2.p1.10.m5.1.1.1.cmml">∈</mo><msub id="S4.SS2.SSS2.p1.10.m5.1.1.3" xref="S4.SS2.SSS2.p1.10.m5.1.1.3.cmml"><mi id="S4.SS2.SSS2.p1.10.m5.1.1.3.2" xref="S4.SS2.SSS2.p1.10.m5.1.1.3.2.cmml">ℕ</mi><mn id="S4.SS2.SSS2.p1.10.m5.1.1.3.3" xref="S4.SS2.SSS2.p1.10.m5.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.10.m5.1b"><apply id="S4.SS2.SSS2.p1.10.m5.1.1.cmml" xref="S4.SS2.SSS2.p1.10.m5.1.1"><in id="S4.SS2.SSS2.p1.10.m5.1.1.1.cmml" xref="S4.SS2.SSS2.p1.10.m5.1.1.1"></in><ci id="S4.SS2.SSS2.p1.10.m5.1.1.2.cmml" xref="S4.SS2.SSS2.p1.10.m5.1.1.2">𝑚</ci><apply id="S4.SS2.SSS2.p1.10.m5.1.1.3.cmml" xref="S4.SS2.SSS2.p1.10.m5.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.10.m5.1.1.3.1.cmml" xref="S4.SS2.SSS2.p1.10.m5.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.p1.10.m5.1.1.3.2.cmml" xref="S4.SS2.SSS2.p1.10.m5.1.1.3.2">ℕ</ci><cn id="S4.SS2.SSS2.p1.10.m5.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.p1.10.m5.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.10.m5.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.10.m5.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> we define</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex39"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="C^{\infty}_{{\rm c},m}(\overline{\mathbb{R}^{d}_{+}};X):=\{f\in C_{\mathrm{c}}% 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end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X ) := { italic_f ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X ) : ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_f ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.p1.14">In particular, note that <math alttext="f\in C^{\infty}_{{\rm c},m}(\overline{\mathbb{R}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.11.m1.4"><semantics 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end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X )</annotation></semantics></math> satisfies <math alttext="\partial^{\alpha}f\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.12.m2.2"><semantics id="S4.SS2.SSS2.p1.12.m2.2a"><mrow id="S4.SS2.SSS2.p1.12.m2.2.2" xref="S4.SS2.SSS2.p1.12.m2.2.2.cmml"><mrow id="S4.SS2.SSS2.p1.12.m2.2.2.3" xref="S4.SS2.SSS2.p1.12.m2.2.2.3.cmml"><msup id="S4.SS2.SSS2.p1.12.m2.2.2.3.1" xref="S4.SS2.SSS2.p1.12.m2.2.2.3.1.cmml"><mo id="S4.SS2.SSS2.p1.12.m2.2.2.3.1.2" xref="S4.SS2.SSS2.p1.12.m2.2.2.3.1.2.cmml">∂</mo><mi id="S4.SS2.SSS2.p1.12.m2.2.2.3.1.3" xref="S4.SS2.SSS2.p1.12.m2.2.2.3.1.3.cmml">α</mi></msup><mi id="S4.SS2.SSS2.p1.12.m2.2.2.3.2" xref="S4.SS2.SSS2.p1.12.m2.2.2.3.2.cmml">f</mi></mrow><mo id="S4.SS2.SSS2.p1.12.m2.2.2.2" xref="S4.SS2.SSS2.p1.12.m2.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS2.p1.12.m2.2.2.1" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.cmml"><msubsup 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xref="S4.SS2.SSS2.p1.12.m2.2.2.1.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS2.p1.12.m2.2.2.1.3.3.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.3.3"></infinity></apply><list id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1"><apply id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.p1.12.m2.2.2.1.1.1.1.3"></plus></apply><ci id="S4.SS2.SSS2.p1.12.m2.1.1.cmml" xref="S4.SS2.SSS2.p1.12.m2.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.12.m2.2c">\partial^{\alpha}f\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.12.m2.2d">∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_f ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for all <math alttext="\alpha=(\alpha_{1},\widetilde{\alpha})\in\mathbb{N}_{0}\times\mathbb{N}^{d-1}_% {0}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.13.m3.2"><semantics id="S4.SS2.SSS2.p1.13.m3.2a"><mrow id="S4.SS2.SSS2.p1.13.m3.2.2" xref="S4.SS2.SSS2.p1.13.m3.2.2.cmml"><mi 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end_POSTSUBSCRIPT × blackboard_N start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\alpha_{1}\geq m" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p1.14.m4.1"><semantics id="S4.SS2.SSS2.p1.14.m4.1a"><mrow id="S4.SS2.SSS2.p1.14.m4.1.1" xref="S4.SS2.SSS2.p1.14.m4.1.1.cmml"><msub id="S4.SS2.SSS2.p1.14.m4.1.1.2" xref="S4.SS2.SSS2.p1.14.m4.1.1.2.cmml"><mi id="S4.SS2.SSS2.p1.14.m4.1.1.2.2" xref="S4.SS2.SSS2.p1.14.m4.1.1.2.2.cmml">α</mi><mn id="S4.SS2.SSS2.p1.14.m4.1.1.2.3" xref="S4.SS2.SSS2.p1.14.m4.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.SSS2.p1.14.m4.1.1.1" xref="S4.SS2.SSS2.p1.14.m4.1.1.1.cmml">≥</mo><mi id="S4.SS2.SSS2.p1.14.m4.1.1.3" xref="S4.SS2.SSS2.p1.14.m4.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p1.14.m4.1b"><apply id="S4.SS2.SSS2.p1.14.m4.1.1.cmml" xref="S4.SS2.SSS2.p1.14.m4.1.1"><geq id="S4.SS2.SSS2.p1.14.m4.1.1.1.cmml" xref="S4.SS2.SSS2.p1.14.m4.1.1.1"></geq><apply id="S4.SS2.SSS2.p1.14.m4.1.1.2.cmml" xref="S4.SS2.SSS2.p1.14.m4.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p1.14.m4.1.1.2.1.cmml" xref="S4.SS2.SSS2.p1.14.m4.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS2.p1.14.m4.1.1.2.2.cmml" xref="S4.SS2.SSS2.p1.14.m4.1.1.2.2">𝛼</ci><cn id="S4.SS2.SSS2.p1.14.m4.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS2.p1.14.m4.1.1.2.3">1</cn></apply><ci id="S4.SS2.SSS2.p1.14.m4.1.1.3.cmml" xref="S4.SS2.SSS2.p1.14.m4.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p1.14.m4.1c">\alpha_{1}\geq m</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p1.14.m4.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_m</annotation></semantics></math>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S4.SS2.SSS2.p2"> <p class="ltx_p" id="S4.SS2.SSS2.p2.2">The following two density results will be required in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib48" title="">48</a>]</cite> to find trace characterisations for weighted Sobolev on domains with, e.g., <math alttext="C^{1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.1.m1.1"><semantics id="S4.SS2.SSS2.p2.1.m1.1a"><msup id="S4.SS2.SSS2.p2.1.m1.1.1" xref="S4.SS2.SSS2.p2.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.p2.1.m1.1.1.2" xref="S4.SS2.SSS2.p2.1.m1.1.1.2.cmml">C</mi><mn id="S4.SS2.SSS2.p2.1.m1.1.1.3" xref="S4.SS2.SSS2.p2.1.m1.1.1.3.cmml">1</mn></msup><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.1.m1.1b"><apply id="S4.SS2.SSS2.p2.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p2.1.m1.1.1.1.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1">superscript</csymbol><ci id="S4.SS2.SSS2.p2.1.m1.1.1.2.cmml" xref="S4.SS2.SSS2.p2.1.m1.1.1.2">𝐶</ci><cn id="S4.SS2.SSS2.p2.1.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.p2.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.1.m1.1c">C^{1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.1.m1.1d">italic_C start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math>-boundary. We first prove a density result for homogeneous boundary conditions of order <math alttext="m" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p2.2.m2.1"><semantics id="S4.SS2.SSS2.p2.2.m2.1a"><mi id="S4.SS2.SSS2.p2.2.m2.1.1" xref="S4.SS2.SSS2.p2.2.m2.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p2.2.m2.1b"><ci id="S4.SS2.SSS2.p2.2.m2.1.1.cmml" xref="S4.SS2.SSS2.p2.2.m2.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p2.2.m2.1c">m</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p2.2.m2.1d">italic_m</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.1.1.1">Proposition 4.8</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem8.p1"> <p class="ltx_p" id="S4.Thmtheorem8.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem8.p1.5.5">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.1.1.m1.2"><semantics id="S4.Thmtheorem8.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem8.p1.1.1.m1.2.3" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem8.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S4.Thmtheorem8.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem8.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem8.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem8.p1.1.1.m1.1.1" xref="S4.Thmtheorem8.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem8.p1.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem8.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem8.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem8.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.1.1.m1.2b"><apply id="S4.Thmtheorem8.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.2.3"><in id="S4.Thmtheorem8.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.1"></in><ci id="S4.Thmtheorem8.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.2.3.3.2"><cn id="S4.Thmtheorem8.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem8.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem8.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem8.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="k,m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.2.2.m2.2"><semantics id="S4.Thmtheorem8.p1.2.2.m2.2a"><mrow id="S4.Thmtheorem8.p1.2.2.m2.2.3" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem8.p1.2.2.m2.2.3.2.2" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.2.1.cmml"><mi id="S4.Thmtheorem8.p1.2.2.m2.1.1" xref="S4.Thmtheorem8.p1.2.2.m2.1.1.cmml">k</mi><mo id="S4.Thmtheorem8.p1.2.2.m2.2.3.2.2.1" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem8.p1.2.2.m2.2.2" xref="S4.Thmtheorem8.p1.2.2.m2.2.2.cmml">m</mi></mrow><mo id="S4.Thmtheorem8.p1.2.2.m2.2.3.1" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.1.cmml">∈</mo><msub id="S4.Thmtheorem8.p1.2.2.m2.2.3.3" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.3.cmml"><mi id="S4.Thmtheorem8.p1.2.2.m2.2.3.3.2" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.3.2.cmml">ℕ</mi><mn id="S4.Thmtheorem8.p1.2.2.m2.2.3.3.3" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.2.2.m2.2b"><apply id="S4.Thmtheorem8.p1.2.2.m2.2.3.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.2.3"><in id="S4.Thmtheorem8.p1.2.2.m2.2.3.1.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.1"></in><list id="S4.Thmtheorem8.p1.2.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.2.2"><ci id="S4.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.1.1">𝑘</ci><ci id="S4.Thmtheorem8.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.2.2">𝑚</ci></list><apply id="S4.Thmtheorem8.p1.2.2.m2.2.3.3.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem8.p1.2.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.3">subscript</csymbol><ci id="S4.Thmtheorem8.p1.2.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.3.2">ℕ</ci><cn id="S4.Thmtheorem8.p1.2.2.m2.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.2.2.m2.2.3.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.2.2.m2.2c">k,m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.2.2.m2.2d">italic_k , italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="k\geq m" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.3.3.m3.1"><semantics id="S4.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem8.p1.3.3.m3.1.1" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.1.cmml">≥</mo><mi id="S4.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.3.3.m3.1b"><apply id="S4.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1"><geq id="S4.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.1"></geq><ci id="S4.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.2">𝑘</ci><ci id="S4.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem8.p1.3.3.m3.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.3.3.m3.1c">k\geq m</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.3.3.m3.1d">italic_k ≥ italic_m</annotation></semantics></math>, <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\{1,\dots,k-m\}\}" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.4.4.m4.6"><semantics id="S4.Thmtheorem8.p1.4.4.m4.6a"><mrow id="S4.Thmtheorem8.p1.4.4.m4.6.6" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.cmml"><mi id="S4.Thmtheorem8.p1.4.4.m4.6.6.5" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.5.cmml">γ</mi><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.4" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.4.cmml">∈</mo><mrow id="S4.Thmtheorem8.p1.4.4.m4.6.6.3" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.cmml"><mrow id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.2.cmml"><mo id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.2" stretchy="false" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.2.cmml">(</mo><mrow id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.cmml"><mo id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1a" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.2" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.2.cmml">1</mn></mrow><mo id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.3" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.2.cmml">,</mo><mi id="S4.Thmtheorem8.p1.4.4.m4.1.1" mathvariant="normal" xref="S4.Thmtheorem8.p1.4.4.m4.1.1.cmml">∞</mi><mo id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.4" stretchy="false" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.2.cmml">)</mo></mrow><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.4" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.4.cmml">∖</mo><mrow id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.3.cmml"><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.3" stretchy="false" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.3.1.cmml">{</mo><mrow id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.cmml"><mrow id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.cmml"><mi id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.2" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.2.cmml">j</mi><mo id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.1" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.1.cmml">⁢</mo><mi id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.3" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.3" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.3.1.cmml">:</mo><mrow id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.cmml"><mi id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.3" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.3.cmml">j</mi><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.2" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.2.cmml">∈</mo><mrow id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.2.cmml"><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.2" stretchy="false" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.2.cmml">{</mo><mn id="S4.Thmtheorem8.p1.4.4.m4.2.2" xref="S4.Thmtheorem8.p1.4.4.m4.2.2.cmml">1</mn><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.3" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.2.cmml">,</mo><mi id="S4.Thmtheorem8.p1.4.4.m4.3.3" mathvariant="normal" xref="S4.Thmtheorem8.p1.4.4.m4.3.3.cmml">…</mi><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.4" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.2.cmml">,</mo><mrow id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.cmml"><mi id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.2" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.1" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.1.cmml">−</mo><mi id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.3" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.3.cmml">m</mi></mrow><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.5" stretchy="false" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.2.cmml">}</mo></mrow></mrow><mo id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.5" stretchy="false" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.4.4.m4.6b"><apply id="S4.Thmtheorem8.p1.4.4.m4.6.6.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6"><in id="S4.Thmtheorem8.p1.4.4.m4.6.6.4.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.4"></in><ci id="S4.Thmtheorem8.p1.4.4.m4.6.6.5.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.5">𝛾</ci><apply id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3"><setdiff id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.4.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.4"></setdiff><interval closure="open" id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1"><apply id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1"><minus id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1"></minus><cn id="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.2.cmml" type="integer" xref="S4.Thmtheorem8.p1.4.4.m4.4.4.1.1.1.1.2">1</cn></apply><infinity id="S4.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.1.1"></infinity></interval><apply id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.3.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2"><csymbol cd="latexml" id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.3.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.3">conditional-set</csymbol><apply id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1"><minus id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.1"></minus><apply id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2"><times id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.1"></times><ci id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.2">𝑗</ci><ci id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.3.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.2.3">𝑝</ci></apply><cn id="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.3.cmml" type="integer" xref="S4.Thmtheorem8.p1.4.4.m4.5.5.2.2.1.1.3">1</cn></apply><apply id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2"><in id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.2"></in><ci id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.3.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.3">𝑗</ci><set id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1"><cn id="S4.Thmtheorem8.p1.4.4.m4.2.2.cmml" type="integer" xref="S4.Thmtheorem8.p1.4.4.m4.2.2">1</cn><ci id="S4.Thmtheorem8.p1.4.4.m4.3.3.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.3.3">…</ci><apply id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1"><minus id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.1"></minus><ci id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.2.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.2">𝑘</ci><ci id="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.3.cmml" xref="S4.Thmtheorem8.p1.4.4.m4.6.6.3.3.2.2.1.1.1.3">𝑚</ci></apply></set></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.4.4.m4.6c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\{1,\dots,k-m\}\}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.4.4.m4.6d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ { 1 , … , italic_k - italic_m } }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem8.p1.5.5.m5.1"><semantics id="S4.Thmtheorem8.p1.5.5.m5.1a"><mi id="S4.Thmtheorem8.p1.5.5.m5.1.1" xref="S4.Thmtheorem8.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem8.p1.5.5.m5.1b"><ci id="S4.Thmtheorem8.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem8.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem8.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem8.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex40"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="W^{k,p}_{\operatorname{Tr}_{m}}(\mathbb{R}_{+}^{d},w_{\gamma};X)=\overline{% \bigl{\{}f\in C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X):(% \partial_{1}^{m}f)|_{\partial\mathbb{R}^{d}_{+}}=0\bigr{\}}}^{W^{k,p}(\mathbb{% R}^{d}_{+},w_{\gamma};X)}." class="ltx_Math" display="block" id="S4.Ex40.m1.14"><semantics id="S4.Ex40.m1.14a"><mrow id="S4.Ex40.m1.14.14.1" xref="S4.Ex40.m1.14.14.1.1.cmml"><mrow id="S4.Ex40.m1.14.14.1.1" xref="S4.Ex40.m1.14.14.1.1.cmml"><mrow id="S4.Ex40.m1.14.14.1.1.2" xref="S4.Ex40.m1.14.14.1.1.2.cmml"><msubsup id="S4.Ex40.m1.14.14.1.1.2.4" xref="S4.Ex40.m1.14.14.1.1.2.4.cmml"><mi id="S4.Ex40.m1.14.14.1.1.2.4.2.2" xref="S4.Ex40.m1.14.14.1.1.2.4.2.2.cmml">W</mi><msub id="S4.Ex40.m1.14.14.1.1.2.4.3" 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xref="S4.Ex40.m1.10.10.3.3">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex40.m1.14c">W^{k,p}_{\operatorname{Tr}_{m}}(\mathbb{R}_{+}^{d},w_{\gamma};X)=\overline{% \bigl{\{}f\in C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X):(% \partial_{1}^{m}f)|_{\partial\mathbb{R}^{d}_{+}}=0\bigr{\}}}^{W^{k,p}(\mathbb{% R}^{d}_{+},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex40.m1.14d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = over¯ start_ARG { italic_f ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X ) : ( ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_f ) | start_POSTSUBSCRIPT ∂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 } end_ARG start_POSTSUPERSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS2.SSS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.SSS2.1.p1"> <p class="ltx_p" id="S4.SS2.SSS2.1.p1.3"><span class="ltx_text ltx_font_italic" id="S4.SS2.SSS2.1.p1.1.1">Step 1: the case <math alttext="\gamma>(k-m)p-1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.1.p1.1.1.m1.1"><semantics id="S4.SS2.SSS2.1.p1.1.1.m1.1a"><mrow id="S4.SS2.SSS2.1.p1.1.1.m1.1.1" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.3" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.3.cmml">γ</mi><mo id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.2" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.2.cmml">></mo><mrow id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.cmml"><mrow id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.cmml"><mrow id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.cmml"><mi 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xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1"><gt id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.2"></gt><ci id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.3.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.3">𝛾</ci><apply id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1"><minus id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.2"></minus><apply id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1"><times id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.2"></times><apply id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1"><minus id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.2">𝑘</ci><ci id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.1.1.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.1.3">𝑝</ci></apply><cn id="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.1.p1.1.1.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.1.p1.1.1.m1.1c">\gamma>(k-m)p-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.1.p1.1.1.m1.1d">italic_γ > ( italic_k - italic_m ) italic_p - 1</annotation></semantics></math>.</span> Note that for <math alttext="\gamma>(k-m)p-1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.1.p1.2.m1.1"><semantics id="S4.SS2.SSS2.1.p1.2.m1.1a"><mrow id="S4.SS2.SSS2.1.p1.2.m1.1.1" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.cmml"><mi id="S4.SS2.SSS2.1.p1.2.m1.1.1.3" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.3.cmml">γ</mi><mo id="S4.SS2.SSS2.1.p1.2.m1.1.1.2" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.2.cmml">></mo><mrow id="S4.SS2.SSS2.1.p1.2.m1.1.1.1" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.cmml"><mrow id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.cmml"><mrow id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.3.cmml">m</mi></mrow><mo id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.2" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.3" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.2" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.2.cmml">−</mo><mn id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.3" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.1.p1.2.m1.1b"><apply id="S4.SS2.SSS2.1.p1.2.m1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1"><gt id="S4.SS2.SSS2.1.p1.2.m1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.2"></gt><ci id="S4.SS2.SSS2.1.p1.2.m1.1.1.3.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.3">𝛾</ci><apply id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1"><minus id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.2"></minus><apply id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1"><times id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.2"></times><apply id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1"><minus id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.2">𝑘</ci><ci id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.1.1.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.1.3">𝑝</ci></apply><cn id="S4.SS2.SSS2.1.p1.2.m1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.1.p1.2.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.1.p1.2.m1.1c">\gamma>(k-m)p-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.1.p1.2.m1.1d">italic_γ > ( italic_k - italic_m ) italic_p - 1</annotation></semantics></math> it holds that <math alttext="W^{k,p}_{\operatorname{Tr}_{m}}(\mathbb{R}_{+}^{d},w_{\gamma};X)=W^{k,p}(% \mathbb{R}_{+}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.1.p1.3.m2.10"><semantics id="S4.SS2.SSS2.1.p1.3.m2.10a"><mrow id="S4.SS2.SSS2.1.p1.3.m2.10.10" xref="S4.SS2.SSS2.1.p1.3.m2.10.10.cmml"><mrow id="S4.SS2.SSS2.1.p1.3.m2.8.8.2" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.cmml"><msubsup id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.cmml"><mi id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.2.2" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.2.2.cmml">W</mi><msub id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.3" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.3.cmml"><mi id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.3.2" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.3.2.cmml">Tr</mi><mi id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.3.3" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.4.3.3.cmml">m</mi></msub><mrow id="S4.SS2.SSS2.1.p1.3.m2.2.2.2.4" xref="S4.SS2.SSS2.1.p1.3.m2.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.1.p1.3.m2.1.1.1.1" xref="S4.SS2.SSS2.1.p1.3.m2.1.1.1.1.cmml">k</mi><mo id="S4.SS2.SSS2.1.p1.3.m2.2.2.2.4.1" xref="S4.SS2.SSS2.1.p1.3.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS2.1.p1.3.m2.2.2.2.2" xref="S4.SS2.SSS2.1.p1.3.m2.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.3" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.3.cmml">⁢</mo><mrow id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.2" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.3.cmml"><mo id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.2.3" stretchy="false" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.3.cmml">(</mo><msubsup id="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1" xref="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1.2.2" xref="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1.2.3" xref="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1.2.3.cmml">+</mo><mi id="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1.3" xref="S4.SS2.SSS2.1.p1.3.m2.7.7.1.1.1.1.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.2.4" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.3.cmml">,</mo><msub id="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.2.2" xref="S4.SS2.SSS2.1.p1.3.m2.8.8.2.2.2.2.cmml"><mi 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xref="S4.Ex41.m1.3.3.2"><csymbol cd="ambiguous" id="S4.Ex41.m1.3.3.2.1.cmml" xref="S4.Ex41.m1.3.3.2">subscript</csymbol><apply id="S4.Ex41.m1.3.3.2.2.cmml" xref="S4.Ex41.m1.3.3.2"><csymbol cd="ambiguous" id="S4.Ex41.m1.3.3.2.2.1.cmml" xref="S4.Ex41.m1.3.3.2">superscript</csymbol><ci id="S4.Ex41.m1.3.3.2.2.2.cmml" xref="S4.Ex41.m1.3.3.2.2.2">ℝ</ci><ci id="S4.Ex41.m1.3.3.2.2.3.cmml" xref="S4.Ex41.m1.3.3.2.2.3">𝑑</ci></apply><plus id="S4.Ex41.m1.3.3.2.3.cmml" xref="S4.Ex41.m1.3.3.2.3"></plus></apply></apply><ci id="S4.Ex41.m1.4.4.cmml" xref="S4.Ex41.m1.4.4">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex41.m1.4c">\displaystyle C_{\mathrm{c},m}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex41.m1.4d">italic_C start_POSTSUBSCRIPT roman_c , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT 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id="S4.Ex41.m2.5c">\displaystyle\subseteq\bigl{\{}f\in C_{\mathrm{c}}^{\infty}(\overline{\mathbb{% R}^{d}_{+}};X):(\partial_{1}^{m}f)|_{\partial\mathbb{R}^{d}_{+}}=0\bigr{\}}% \quad\text{and}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex41.m2.5d">⊆ { italic_f ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X ) : ( ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_f ) | start_POSTSUBSCRIPT ∂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 } and</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle C_{\mathrm{c},m}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.Ex42.m1.4"><semantics id="S4.Ex42.m1.4a"><mrow id="S4.Ex42.m1.4.5" xref="S4.Ex42.m1.4.5.cmml"><msubsup id="S4.Ex42.m1.4.5.2" xref="S4.Ex42.m1.4.5.2.cmml"><mi id="S4.Ex42.m1.4.5.2.2.2" xref="S4.Ex42.m1.4.5.2.2.2.cmml">C</mi><mrow id="S4.Ex42.m1.2.2.2.4" xref="S4.Ex42.m1.2.2.2.3.cmml"><mi id="S4.Ex42.m1.1.1.1.1" mathvariant="normal" xref="S4.Ex42.m1.1.1.1.1.cmml">c</mi><mo id="S4.Ex42.m1.2.2.2.4.1" xref="S4.Ex42.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex42.m1.2.2.2.2" xref="S4.Ex42.m1.2.2.2.2.cmml">m</mi></mrow><mi id="S4.Ex42.m1.4.5.2.3" mathvariant="normal" xref="S4.Ex42.m1.4.5.2.3.cmml">∞</mi></msubsup><mo id="S4.Ex42.m1.4.5.1" xref="S4.Ex42.m1.4.5.1.cmml">⁢</mo><mrow id="S4.Ex42.m1.4.5.3.2" xref="S4.Ex42.m1.4.5.3.1.cmml"><mo id="S4.Ex42.m1.4.5.3.2.1" stretchy="false" 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id="S4.Ex42.m1.3.3.2.2.1.cmml" xref="S4.Ex42.m1.3.3.2">superscript</csymbol><ci id="S4.Ex42.m1.3.3.2.2.2.cmml" xref="S4.Ex42.m1.3.3.2.2.2">ℝ</ci><ci id="S4.Ex42.m1.3.3.2.2.3.cmml" xref="S4.Ex42.m1.3.3.2.2.3">𝑑</ci></apply><plus id="S4.Ex42.m1.3.3.2.3.cmml" xref="S4.Ex42.m1.3.3.2.3"></plus></apply></apply><ci id="S4.Ex42.m1.4.4.cmml" xref="S4.Ex42.m1.4.4">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex42.m1.4c">\displaystyle C_{\mathrm{c},m}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.Ex42.m1.4d">italic_C start_POSTSUBSCRIPT roman_c , italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S4.Ex42.m2.4c">\displaystyle\stackrel{{\scriptstyle\text{dense}}}{{\hookrightarrow}}W^{k,p}(% \mathbb{R}^{d}_{+},w_{\gamma};X),</annotation><annotation encoding="application/x-llamapun" id="S4.Ex42.m2.4d">start_RELOP SUPERSCRIPTOP start_ARG ↪ end_ARG start_ARG dense end_ARG end_RELOP italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.1.p1.4">see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Lemma 3.4]</cite>.</p> </div> <div class="ltx_para" id="S4.SS2.SSS2.2.p2"> <p class="ltx_p" id="S4.SS2.SSS2.2.p2.5"><span class="ltx_text ltx_font_italic" id="S4.SS2.SSS2.2.p2.1.1">Step 2: the case <math alttext="\gamma<(k-m)p-1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.1.1.m1.1"><semantics id="S4.SS2.SSS2.2.p2.1.1.m1.1a"><mrow id="S4.SS2.SSS2.2.p2.1.1.m1.1.1" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.3" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.3.cmml">γ</mi><mo id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.2" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.2.cmml"><</mo><mrow id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.cmml"><mrow id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.cmml"><mrow id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.1.1" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.1.1.1.2" 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xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.1.3">𝑝</ci></apply><cn id="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.1.1.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.1.1.m1.1c">\gamma<(k-m)p-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.1.1.m1.1d">italic_γ < ( italic_k - italic_m ) italic_p - 1</annotation></semantics></math>.</span> Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.2.m1.1"><semantics id="S4.SS2.SSS2.2.p2.2.m1.1a"><mrow id="S4.SS2.SSS2.2.p2.2.m1.1.1" xref="S4.SS2.SSS2.2.p2.2.m1.1.1.cmml"><mi id="S4.SS2.SSS2.2.p2.2.m1.1.1.2" xref="S4.SS2.SSS2.2.p2.2.m1.1.1.2.cmml">ε</mi><mo id="S4.SS2.SSS2.2.p2.2.m1.1.1.1" xref="S4.SS2.SSS2.2.p2.2.m1.1.1.1.cmml">></mo><mn id="S4.SS2.SSS2.2.p2.2.m1.1.1.3" xref="S4.SS2.SSS2.2.p2.2.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" 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xref="S4.SS2.SSS2.2.p2.3.m2.4.4.1.1.1.1.3"></plus></apply><apply id="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2.3.cmml" xref="S4.SS2.SSS2.2.p2.3.m2.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS2.2.p2.3.m2.3.3.cmml" xref="S4.SS2.SSS2.2.p2.3.m2.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.3.m2.5c">f\in W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.3.m2.5d">italic_f ∈ italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\operatorname{Tr}_{m}f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.4.m3.1"><semantics id="S4.SS2.SSS2.2.p2.4.m3.1a"><mrow id="S4.SS2.SSS2.2.p2.4.m3.1.1" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.cmml"><mrow id="S4.SS2.SSS2.2.p2.4.m3.1.1.2" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.cmml"><msub id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.cmml"><mi id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.2" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.2.cmml">Tr</mi><mi id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.3" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.3.cmml">m</mi></msub><mo id="S4.SS2.SSS2.2.p2.4.m3.1.1.2a" lspace="0.167em" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.cmml">⁡</mo><mi id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.2" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.2.cmml">f</mi></mrow><mo id="S4.SS2.SSS2.2.p2.4.m3.1.1.1" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS2.2.p2.4.m3.1.1.3" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.2.p2.4.m3.1b"><apply id="S4.SS2.SSS2.2.p2.4.m3.1.1.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1"><eq id="S4.SS2.SSS2.2.p2.4.m3.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.1"></eq><apply id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2"><apply id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.1.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1">subscript</csymbol><ci id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.2.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.2">Tr</ci><ci id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.3.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.2.p2.4.m3.1.1.2.2.cmml" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.2.2">𝑓</ci></apply><cn id="S4.SS2.SSS2.2.p2.4.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.4.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.4.m3.1c">\operatorname{Tr}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.4.m3.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math>. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib42" title="">42</a>, Theorem 7.2 & Remark 11.12(iii)]</cite>, which also holds in the vector-valued case, and a standard cut-off argument, there exists an <math alttext="f^{(1)}\in C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.5.m4.3"><semantics id="S4.SS2.SSS2.2.p2.5.m4.3a"><mrow id="S4.SS2.SSS2.2.p2.5.m4.3.4" xref="S4.SS2.SSS2.2.p2.5.m4.3.4.cmml"><msup id="S4.SS2.SSS2.2.p2.5.m4.3.4.2" xref="S4.SS2.SSS2.2.p2.5.m4.3.4.2.cmml"><mi id="S4.SS2.SSS2.2.p2.5.m4.3.4.2.2" xref="S4.SS2.SSS2.2.p2.5.m4.3.4.2.2.cmml">f</mi><mrow id="S4.SS2.SSS2.2.p2.5.m4.1.1.1.3" xref="S4.SS2.SSS2.2.p2.5.m4.3.4.2.cmml"><mo id="S4.SS2.SSS2.2.p2.5.m4.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS2.2.p2.5.m4.3.4.2.cmml">(</mo><mn id="S4.SS2.SSS2.2.p2.5.m4.1.1.1.1" xref="S4.SS2.SSS2.2.p2.5.m4.1.1.1.1.cmml">1</mn><mo 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)</annotation></semantics></math> such that</p> <table class="ltx_equation ltx_eqn_table" id="S4.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f-f^{(1)}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}<\varepsilon." class="ltx_Math" display="block" id="S4.E6.m1.7"><semantics id="S4.E6.m1.7a"><mrow id="S4.E6.m1.7.7.1" xref="S4.E6.m1.7.7.1.1.cmml"><mrow id="S4.E6.m1.7.7.1.1" xref="S4.E6.m1.7.7.1.1.cmml"><msub id="S4.E6.m1.7.7.1.1.1" xref="S4.E6.m1.7.7.1.1.1.cmml"><mrow id="S4.E6.m1.7.7.1.1.1.1.1" xref="S4.E6.m1.7.7.1.1.1.1.2.cmml"><mo id="S4.E6.m1.7.7.1.1.1.1.1.2" stretchy="false" xref="S4.E6.m1.7.7.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E6.m1.7.7.1.1.1.1.1.1" xref="S4.E6.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S4.E6.m1.7.7.1.1.1.1.1.1.2" xref="S4.E6.m1.7.7.1.1.1.1.1.1.2.cmml">f</mi><mo id="S4.E6.m1.7.7.1.1.1.1.1.1.1" 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id="S4.E6.m1.4.4.3.3" xref="S4.E6.m1.4.4.3.3.cmml">X</mi><mo id="S4.E6.m1.6.6.5.5.2.6" stretchy="false" xref="S4.E6.m1.6.6.5.5.3.cmml">)</mo></mrow></mrow></msub><mo id="S4.E6.m1.7.7.1.1.2" xref="S4.E6.m1.7.7.1.1.2.cmml"><</mo><mi id="S4.E6.m1.7.7.1.1.3" xref="S4.E6.m1.7.7.1.1.3.cmml">ε</mi></mrow><mo id="S4.E6.m1.7.7.1.2" lspace="0em" xref="S4.E6.m1.7.7.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E6.m1.7b"><apply id="S4.E6.m1.7.7.1.1.cmml" xref="S4.E6.m1.7.7.1"><lt id="S4.E6.m1.7.7.1.1.2.cmml" xref="S4.E6.m1.7.7.1.1.2"></lt><apply id="S4.E6.m1.7.7.1.1.1.cmml" xref="S4.E6.m1.7.7.1.1.1"><csymbol cd="ambiguous" id="S4.E6.m1.7.7.1.1.1.2.cmml" xref="S4.E6.m1.7.7.1.1.1">subscript</csymbol><apply id="S4.E6.m1.7.7.1.1.1.1.2.cmml" xref="S4.E6.m1.7.7.1.1.1.1.1"><csymbol cd="latexml" id="S4.E6.m1.7.7.1.1.1.1.2.1.cmml" xref="S4.E6.m1.7.7.1.1.1.1.1.2">norm</csymbol><apply id="S4.E6.m1.7.7.1.1.1.1.1.1.cmml" xref="S4.E6.m1.7.7.1.1.1.1.1.1"><minus 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xref="S4.E6.m1.6.6.5.5.2.2.2">𝑤</ci><ci id="S4.E6.m1.6.6.5.5.2.2.3.cmml" xref="S4.E6.m1.6.6.5.5.2.2.3">𝛾</ci></apply><ci id="S4.E6.m1.4.4.3.3.cmml" xref="S4.E6.m1.4.4.3.3">𝑋</ci></vector></apply></apply><ci id="S4.E6.m1.7.7.1.1.3.cmml" xref="S4.E6.m1.7.7.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E6.m1.7c">\|f-f^{(1)}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}<\varepsilon.</annotation><annotation encoding="application/x-llamapun" id="S4.E6.m1.7d">∥ italic_f - italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT < italic_ε .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.2.p2.9">Let <math alttext="\operatorname{ext}_{m}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.6.m1.1"><semantics id="S4.SS2.SSS2.2.p2.6.m1.1a"><msub id="S4.SS2.SSS2.2.p2.6.m1.1.1" xref="S4.SS2.SSS2.2.p2.6.m1.1.1.cmml"><mi id="S4.SS2.SSS2.2.p2.6.m1.1.1.2" xref="S4.SS2.SSS2.2.p2.6.m1.1.1.2.cmml">ext</mi><mi id="S4.SS2.SSS2.2.p2.6.m1.1.1.3" xref="S4.SS2.SSS2.2.p2.6.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.2.p2.6.m1.1b"><apply id="S4.SS2.SSS2.2.p2.6.m1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.6.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.6.m1.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.6.m1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.2.p2.6.m1.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.6.m1.1.1.2">ext</ci><ci id="S4.SS2.SSS2.2.p2.6.m1.1.1.3.cmml" xref="S4.SS2.SSS2.2.p2.6.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.6.m1.1c">\operatorname{ext}_{m}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.6.m1.1d">roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> be as in Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4" title="Theorem 4.4. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a> and define <math alttext="f^{(2)}:=f^{(1)}-\operatorname{ext}_{m}(\operatorname{Tr}_{m}f^{(1)})" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.7.m2.5"><semantics id="S4.SS2.SSS2.2.p2.7.m2.5a"><mrow id="S4.SS2.SSS2.2.p2.7.m2.5.5" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.cmml"><msup id="S4.SS2.SSS2.2.p2.7.m2.5.5.4" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4.cmml"><mi id="S4.SS2.SSS2.2.p2.7.m2.5.5.4.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4.2.cmml">f</mi><mrow id="S4.SS2.SSS2.2.p2.7.m2.1.1.1.3" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4.cmml"><mo id="S4.SS2.SSS2.2.p2.7.m2.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4.cmml">(</mo><mn id="S4.SS2.SSS2.2.p2.7.m2.1.1.1.1" xref="S4.SS2.SSS2.2.p2.7.m2.1.1.1.1.cmml">2</mn><mo id="S4.SS2.SSS2.2.p2.7.m2.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4.cmml">)</mo></mrow></msup><mo id="S4.SS2.SSS2.2.p2.7.m2.5.5.3" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.3.cmml">:=</mo><mrow id="S4.SS2.SSS2.2.p2.7.m2.5.5.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.cmml"><msup id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.cmml"><mi id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.2.cmml">f</mi><mrow id="S4.SS2.SSS2.2.p2.7.m2.2.2.1.3" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.cmml"><mo id="S4.SS2.SSS2.2.p2.7.m2.2.2.1.3.1" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.cmml">(</mo><mn id="S4.SS2.SSS2.2.p2.7.m2.2.2.1.1" xref="S4.SS2.SSS2.2.p2.7.m2.2.2.1.1.cmml">1</mn><mo id="S4.SS2.SSS2.2.p2.7.m2.2.2.1.3.2" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.cmml">)</mo></mrow></msup><mo id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.3" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.3.cmml">−</mo><mrow id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.3.cmml"><msub id="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1" xref="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.2" xref="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.2.cmml">ext</mi><mi id="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.3" xref="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.3.cmml">m</mi></msub><mo id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2a" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.3.cmml">⁡</mo><mrow id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.3.cmml"><mo id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.2" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.3.cmml">(</mo><mrow id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.cmml"><msub id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.cmml"><mi id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.2.cmml">Tr</mi><mi id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.3" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.3.cmml">m</mi></msub><mo id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1a" lspace="0.167em" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.cmml">⁡</mo><msup id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.cmml"><mi id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.2" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.2.cmml">f</mi><mrow id="S4.SS2.SSS2.2.p2.7.m2.3.3.1.3" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.cmml"><mo id="S4.SS2.SSS2.2.p2.7.m2.3.3.1.3.1" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.cmml">(</mo><mn id="S4.SS2.SSS2.2.p2.7.m2.3.3.1.1" xref="S4.SS2.SSS2.2.p2.7.m2.3.3.1.1.cmml">1</mn><mo id="S4.SS2.SSS2.2.p2.7.m2.3.3.1.3.2" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.cmml">)</mo></mrow></msup></mrow><mo id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.3" stretchy="false" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.2.p2.7.m2.5b"><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5"><csymbol cd="latexml" id="S4.SS2.SSS2.2.p2.7.m2.5.5.3.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.3">assign</csymbol><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.4.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.7.m2.5.5.4.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.7.m2.5.5.4.2.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.4.2">𝑓</ci><cn id="S4.SS2.SSS2.2.p2.7.m2.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.7.m2.1.1.1.1">2</cn></apply><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2"><minus id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.3.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.3"></minus><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.2.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.4.2">𝑓</ci><cn id="S4.SS2.SSS2.2.p2.7.m2.2.2.1.1.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.7.m2.2.2.1.1">1</cn></apply><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.3.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2"><apply id="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.2">ext</ci><ci id="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.4.4.1.1.1.1.3">𝑚</ci></apply><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1"><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.2">Tr</ci><ci id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.3.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.1.3">𝑚</ci></apply><apply id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.1.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.2.cmml" xref="S4.SS2.SSS2.2.p2.7.m2.5.5.2.2.2.2.1.2.2">𝑓</ci><cn id="S4.SS2.SSS2.2.p2.7.m2.3.3.1.1.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.7.m2.3.3.1.1">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.7.m2.5c">f^{(2)}:=f^{(1)}-\operatorname{ext}_{m}(\operatorname{Tr}_{m}f^{(1)})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.7.m2.5d">italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT := italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT - roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT )</annotation></semantics></math>. Then by Remark <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem3" title="Remark 4.3. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.3</span></a> we have <math alttext="f^{(2)}\in C^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.8.m3.3"><semantics id="S4.SS2.SSS2.2.p2.8.m3.3a"><mrow id="S4.SS2.SSS2.2.p2.8.m3.3.4" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.cmml"><msup id="S4.SS2.SSS2.2.p2.8.m3.3.4.2" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2.cmml"><mi id="S4.SS2.SSS2.2.p2.8.m3.3.4.2.2" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2.2.cmml">f</mi><mrow id="S4.SS2.SSS2.2.p2.8.m3.1.1.1.3" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2.cmml"><mo id="S4.SS2.SSS2.2.p2.8.m3.1.1.1.3.1" stretchy="false" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2.cmml">(</mo><mn id="S4.SS2.SSS2.2.p2.8.m3.1.1.1.1" xref="S4.SS2.SSS2.2.p2.8.m3.1.1.1.1.cmml">2</mn><mo id="S4.SS2.SSS2.2.p2.8.m3.1.1.1.3.2" stretchy="false" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2.cmml">)</mo></mrow></msup><mo id="S4.SS2.SSS2.2.p2.8.m3.3.4.1" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.2.p2.8.m3.3.4.3" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.cmml"><msup id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.cmml"><mi id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.2" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.3" mathvariant="normal" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.3.cmml">∞</mi></msup><mo id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.1" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.2" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.1.cmml"><mo id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.1.cmml">(</mo><mover accent="true" id="S4.SS2.SSS2.2.p2.8.m3.2.2" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.cmml"><msubsup id="S4.SS2.SSS2.2.p2.8.m3.2.2.2" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2.cmml"><mi id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.2" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.3" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2.3.cmml">+</mo><mi id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.3" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS2.2.p2.8.m3.2.2.1" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.1.cmml">¯</mo></mover><mo id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.2.2" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.1.cmml">;</mo><mi id="S4.SS2.SSS2.2.p2.8.m3.3.3" xref="S4.SS2.SSS2.2.p2.8.m3.3.3.cmml">X</mi><mo id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.2.p2.8.m3.3b"><apply id="S4.SS2.SSS2.2.p2.8.m3.3.4.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4"><in id="S4.SS2.SSS2.2.p2.8.m3.3.4.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.1"></in><apply id="S4.SS2.SSS2.2.p2.8.m3.3.4.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.8.m3.3.4.2.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.8.m3.3.4.2.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.2.2">𝑓</ci><cn id="S4.SS2.SSS2.2.p2.8.m3.1.1.1.1.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.8.m3.1.1.1.1">2</cn></apply><apply id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3"><times id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.1"></times><apply id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.2">𝐶</ci><infinity id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.3.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.2.3"></infinity></apply><list id="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.4.3.3.2"><apply id="S4.SS2.SSS2.2.p2.8.m3.2.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2"><ci id="S4.SS2.SSS2.2.p2.8.m3.2.2.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.1">¯</ci><apply id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2">subscript</csymbol><apply id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.2">ℝ</ci><ci id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.3.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.2.p2.8.m3.2.2.2.3.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.2.2.2.3"></plus></apply></apply><ci id="S4.SS2.SSS2.2.p2.8.m3.3.3.cmml" xref="S4.SS2.SSS2.2.p2.8.m3.3.3">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.8.m3.3c">f^{(2)}\in C^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.8.m3.3d">italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X )</annotation></semantics></math> and from Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4" title="Theorem 4.4. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a> it follows that <math alttext="(\partial_{1}^{m}f)|_{\mathbb{R}^{d}_{+}}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.9.m4.2"><semantics id="S4.SS2.SSS2.2.p2.9.m4.2a"><mrow id="S4.SS2.SSS2.2.p2.9.m4.2.2" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.cmml"><msub id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.2.cmml"><mrow id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.2.cmml"><mrow id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1.cmml"><msubsup id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1.1" 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xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1.3.cmml">+</mo><mi id="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.3" xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.3.cmml">d</mi></msubsup></msub><mo id="S4.SS2.SSS2.2.p2.9.m4.2.2.2" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.2.cmml">=</mo><mn id="S4.SS2.SSS2.2.p2.9.m4.2.2.3" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.2.p2.9.m4.2b"><apply id="S4.SS2.SSS2.2.p2.9.m4.2.2.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.2.2"><eq id="S4.SS2.SSS2.2.p2.9.m4.2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.2"></eq><apply id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.2.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1"><csymbol cd="latexml" id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.2.1.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.2">evaluated-at</csymbol><apply id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1"><apply id="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" 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id="S4.SS2.SSS2.2.p2.9.m4.1.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.2.p2.9.m4.1.1.1.3.cmml" xref="S4.SS2.SSS2.2.p2.9.m4.1.1.1.3"></plus></apply></apply><cn id="S4.SS2.SSS2.2.p2.9.m4.2.2.3.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.9.m4.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.9.m4.2c">(\partial_{1}^{m}f)|_{\mathbb{R}^{d}_{+}}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.9.m4.2d">( ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_f ) | start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math> and</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx26"> <tbody id="S4.Ex43"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|f-f^{(2)}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}\leq" class="ltx_Math" display="inline" id="S4.Ex43.m1.7"><semantics id="S4.Ex43.m1.7a"><mrow id="S4.Ex43.m1.7.7" xref="S4.Ex43.m1.7.7.cmml"><msub id="S4.Ex43.m1.7.7.1" xref="S4.Ex43.m1.7.7.1.cmml"><mrow id="S4.Ex43.m1.7.7.1.1.1" xref="S4.Ex43.m1.7.7.1.1.2.cmml"><mo id="S4.Ex43.m1.7.7.1.1.1.2" stretchy="false" xref="S4.Ex43.m1.7.7.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex43.m1.7.7.1.1.1.1" xref="S4.Ex43.m1.7.7.1.1.1.1.cmml"><mi 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id="S4.Ex43.m2.12.12.5.5.2.2.2.cmml" xref="S4.Ex43.m2.12.12.5.5.2.2.2">𝑤</ci><ci id="S4.Ex43.m2.12.12.5.5.2.2.3.cmml" xref="S4.Ex43.m2.12.12.5.5.2.2.3">𝛾</ci></apply><ci id="S4.Ex43.m2.10.10.3.3.cmml" xref="S4.Ex43.m2.10.10.3.3">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex43.m2.13c">\displaystyle\;\|f-f^{(1)}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}+\|% \operatorname{ext}_{m}(\operatorname{Tr}_{m}f^{(1)})\|_{W^{k,p}(\mathbb{R}^{d}% _{+},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex43.m2.13d">∥ italic_f - italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT + ∥ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.2.p2.10">From <math alttext="\operatorname{Tr}_{m}f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.10.m1.1"><semantics id="S4.SS2.SSS2.2.p2.10.m1.1a"><mrow id="S4.SS2.SSS2.2.p2.10.m1.1.1" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.cmml"><mrow id="S4.SS2.SSS2.2.p2.10.m1.1.1.2" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.cmml"><msub id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.cmml"><mi id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.2" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.2.cmml">Tr</mi><mi id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.3" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.3.cmml">m</mi></msub><mo id="S4.SS2.SSS2.2.p2.10.m1.1.1.2a" lspace="0.167em" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.cmml">⁡</mo><mi id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.2" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.2.cmml">f</mi></mrow><mo id="S4.SS2.SSS2.2.p2.10.m1.1.1.1" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS2.2.p2.10.m1.1.1.3" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.2.p2.10.m1.1b"><apply id="S4.SS2.SSS2.2.p2.10.m1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1"><eq id="S4.SS2.SSS2.2.p2.10.m1.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.1"></eq><apply id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2"><apply id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.1.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1">subscript</csymbol><ci id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.2.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.2">Tr</ci><ci id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.3.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.2.p2.10.m1.1.1.2.2.cmml" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.2.2">𝑓</ci></apply><cn id="S4.SS2.SSS2.2.p2.10.m1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.10.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.10.m1.1c">\operatorname{Tr}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.10.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math>, Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4" title="Theorem 4.4. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a> twice and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E6" title="In Proof. ‣ 4.2.2. Sobolev spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.6</span></a>), it follows</p> <table class="ltx_equationgroup ltx_eqn_table" id="S4.Ex44"> <tbody id="S4.Ex44X"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|\operatorname{ext}_{m}(\operatorname{Tr}_{m}f^{(1)})\|_{W^{k,p}% (\mathbb{R}^{d}_{+},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S4.Ex44X.2.1.1.m1.7"><semantics id="S4.Ex44X.2.1.1.m1.7a"><msub id="S4.Ex44X.2.1.1.m1.7.7" xref="S4.Ex44X.2.1.1.m1.7.7.cmml"><mrow id="S4.Ex44X.2.1.1.m1.7.7.1.1" xref="S4.Ex44X.2.1.1.m1.7.7.1.2.cmml"><mo id="S4.Ex44X.2.1.1.m1.7.7.1.1.2" stretchy="false" xref="S4.Ex44X.2.1.1.m1.7.7.1.2.1.cmml">‖</mo><mrow id="S4.Ex44X.2.1.1.m1.7.7.1.1.1.2" xref="S4.Ex44X.2.1.1.m1.7.7.1.1.1.3.cmml"><msub 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id="S4.Ex44X.2.1.1.m1.7c">\displaystyle\|\operatorname{ext}_{m}(\operatorname{Tr}_{m}f^{(1)})\|_{W^{k,p}% (\mathbb{R}^{d}_{+},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex44X.2.1.1.m1.7d">∥ roman_ext start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ) ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq C\|\operatorname{Tr}_{m}f-\operatorname{Tr}_{m}f^{(1)}\|_{B^% {k-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)}" class="ltx_Math" display="inline" id="S4.Ex44X.3.2.2.m1.6"><semantics 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xref="S4.Ex44X.3.2.2.m1.5.5.4.4.1.1.3.1"></minus><ci id="S4.Ex44X.3.2.2.m1.5.5.4.4.1.1.3.2.cmml" xref="S4.Ex44X.3.2.2.m1.5.5.4.4.1.1.3.2">𝑑</ci><cn id="S4.Ex44X.3.2.2.m1.5.5.4.4.1.1.3.3.cmml" type="integer" xref="S4.Ex44X.3.2.2.m1.5.5.4.4.1.1.3.3">1</cn></apply></apply><ci id="S4.Ex44X.3.2.2.m1.4.4.3.3.cmml" xref="S4.Ex44X.3.2.2.m1.4.4.3.3">𝑋</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex44X.3.2.2.m1.6c">\displaystyle\leq C\|\operatorname{Tr}_{m}f-\operatorname{Tr}_{m}f^{(1)}\|_{B^% {k-m-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex44X.3.2.2.m1.6d">≤ italic_C ∥ roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f - roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_B start_POSTSUPERSCRIPT italic_k - italic_m - divide start_ARG italic_γ + 1 end_ARG 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xref="S4.Ex44Xa.2.1.1.m1.7.7.1.1.6.2">𝐶</ci><ci id="S4.Ex44Xa.2.1.1.m1.7.7.1.1.6.3.cmml" xref="S4.Ex44Xa.2.1.1.m1.7.7.1.1.6.3">𝜀</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex44Xa.2.1.1.m1.7c">\displaystyle\leq C\|f-f^{(1)}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}<C\varepsilon.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex44Xa.2.1.1.m1.7d">≤ italic_C ∥ italic_f - italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT < italic_C italic_ε .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.2.p2.12">This proves that <math 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id="S4.SS2.SSS2.2.p2.11.m1.8.8.2.1.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.11.m1.8.8.2.1.1.1.2">𝐶</ci><cn id="S4.SS2.SSS2.2.p2.11.m1.8.8.2.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.2.p2.11.m1.8.8.2.1.1.1.3">1</cn></apply><ci id="S4.SS2.SSS2.2.p2.11.m1.8.8.2.3.cmml" xref="S4.SS2.SSS2.2.p2.11.m1.8.8.2.3">𝜀</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.11.m1.8c">\|f-f^{(2)}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}<(C+1)\varepsilon</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.11.m1.8d">∥ italic_f - italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT < ( italic_C + 1 ) italic_ε</annotation></semantics></math>. Finally, a standard cut-off argument gives an approximating sequence in <math alttext="C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.2.p2.12.m2.2"><semantics id="S4.SS2.SSS2.2.p2.12.m2.2a"><mrow id="S4.SS2.SSS2.2.p2.12.m2.2.3" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.cmml"><msubsup id="S4.SS2.SSS2.2.p2.12.m2.2.3.2" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.2.cmml"><mi id="S4.SS2.SSS2.2.p2.12.m2.2.3.2.2.2" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.2.2.2.cmml">C</mi><mi id="S4.SS2.SSS2.2.p2.12.m2.2.3.2.2.3" mathvariant="normal" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.2.2.3.cmml">c</mi><mi id="S4.SS2.SSS2.2.p2.12.m2.2.3.2.3" mathvariant="normal" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.2.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS2.2.p2.12.m2.2.3.1" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.2.p2.12.m2.2.3.3.2" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.3.1.cmml"><mo id="S4.SS2.SSS2.2.p2.12.m2.2.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.3.1.cmml">(</mo><mover accent="true" id="S4.SS2.SSS2.2.p2.12.m2.1.1" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.cmml"><msubsup id="S4.SS2.SSS2.2.p2.12.m2.1.1.2" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2.cmml"><mi id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.2" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.3" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2.3.cmml">+</mo><mi id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.3" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS2.2.p2.12.m2.1.1.1" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.1.cmml">¯</mo></mover><mo id="S4.SS2.SSS2.2.p2.12.m2.2.3.3.2.2" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.3.1.cmml">;</mo><mi id="S4.SS2.SSS2.2.p2.12.m2.2.2" xref="S4.SS2.SSS2.2.p2.12.m2.2.2.cmml">X</mi><mo id="S4.SS2.SSS2.2.p2.12.m2.2.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.2.p2.12.m2.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.2.p2.12.m2.2b"><apply 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id="S4.SS2.SSS2.2.p2.12.m2.1.1.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1"><ci id="S4.SS2.SSS2.2.p2.12.m2.1.1.1.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.1">¯</ci><apply id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.1.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2">subscript</csymbol><apply id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.1.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2">superscript</csymbol><ci id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.2">ℝ</ci><ci id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.3.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.2.p2.12.m2.1.1.2.3.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.1.1.2.3"></plus></apply></apply><ci id="S4.SS2.SSS2.2.p2.12.m2.2.2.cmml" xref="S4.SS2.SSS2.2.p2.12.m2.2.2">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.2.p2.12.m2.2c">C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.2.p2.12.m2.2d">italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X )</annotation></semantics></math> satisfying the boundary condition. ∎</p> </div> </div> <div class="ltx_para" id="S4.SS2.SSS2.p3"> <p class="ltx_p" id="S4.SS2.SSS2.p3.2">Under suitable restrictions on <math alttext="\gamma" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p3.1.m1.1"><semantics id="S4.SS2.SSS2.p3.1.m1.1a"><mi id="S4.SS2.SSS2.p3.1.m1.1.1" xref="S4.SS2.SSS2.p3.1.m1.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p3.1.m1.1b"><ci id="S4.SS2.SSS2.p3.1.m1.1.1.cmml" xref="S4.SS2.SSS2.p3.1.m1.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p3.1.m1.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p3.1.m1.1d">italic_γ</annotation></semantics></math> we can prove the following density result involving <math alttext="C^{\infty}_{{\rm c},m}(\overline{\mathbb{R}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.p3.2.m2.4"><semantics id="S4.SS2.SSS2.p3.2.m2.4a"><mrow id="S4.SS2.SSS2.p3.2.m2.4.5" xref="S4.SS2.SSS2.p3.2.m2.4.5.cmml"><msubsup id="S4.SS2.SSS2.p3.2.m2.4.5.2" xref="S4.SS2.SSS2.p3.2.m2.4.5.2.cmml"><mi id="S4.SS2.SSS2.p3.2.m2.4.5.2.2.2" xref="S4.SS2.SSS2.p3.2.m2.4.5.2.2.2.cmml">C</mi><mrow id="S4.SS2.SSS2.p3.2.m2.2.2.2.4" xref="S4.SS2.SSS2.p3.2.m2.2.2.2.3.cmml"><mi id="S4.SS2.SSS2.p3.2.m2.1.1.1.1" mathvariant="normal" xref="S4.SS2.SSS2.p3.2.m2.1.1.1.1.cmml">c</mi><mo id="S4.SS2.SSS2.p3.2.m2.2.2.2.4.1" xref="S4.SS2.SSS2.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S4.SS2.SSS2.p3.2.m2.2.2.2.2" xref="S4.SS2.SSS2.p3.2.m2.2.2.2.2.cmml">m</mi></mrow><mi id="S4.SS2.SSS2.p3.2.m2.4.5.2.2.3" mathvariant="normal" xref="S4.SS2.SSS2.p3.2.m2.4.5.2.2.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS2.p3.2.m2.4.5.1" xref="S4.SS2.SSS2.p3.2.m2.4.5.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.p3.2.m2.4.5.3.2" xref="S4.SS2.SSS2.p3.2.m2.4.5.3.1.cmml"><mo id="S4.SS2.SSS2.p3.2.m2.4.5.3.2.1" stretchy="false" xref="S4.SS2.SSS2.p3.2.m2.4.5.3.1.cmml">(</mo><mover accent="true" id="S4.SS2.SSS2.p3.2.m2.3.3" xref="S4.SS2.SSS2.p3.2.m2.3.3.cmml"><msubsup id="S4.SS2.SSS2.p3.2.m2.3.3.2" xref="S4.SS2.SSS2.p3.2.m2.3.3.2.cmml"><mi id="S4.SS2.SSS2.p3.2.m2.3.3.2.2.2" xref="S4.SS2.SSS2.p3.2.m2.3.3.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.p3.2.m2.3.3.2.3" xref="S4.SS2.SSS2.p3.2.m2.3.3.2.3.cmml">+</mo><mi id="S4.SS2.SSS2.p3.2.m2.3.3.2.2.3" xref="S4.SS2.SSS2.p3.2.m2.3.3.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS2.p3.2.m2.3.3.1" xref="S4.SS2.SSS2.p3.2.m2.3.3.1.cmml">¯</mo></mover><mo id="S4.SS2.SSS2.p3.2.m2.4.5.3.2.2" xref="S4.SS2.SSS2.p3.2.m2.4.5.3.1.cmml">;</mo><mi id="S4.SS2.SSS2.p3.2.m2.4.4" xref="S4.SS2.SSS2.p3.2.m2.4.4.cmml">X</mi><mo id="S4.SS2.SSS2.p3.2.m2.4.5.3.2.3" stretchy="false" xref="S4.SS2.SSS2.p3.2.m2.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.p3.2.m2.4b"><apply id="S4.SS2.SSS2.p3.2.m2.4.5.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5"><times id="S4.SS2.SSS2.p3.2.m2.4.5.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.1"></times><apply id="S4.SS2.SSS2.p3.2.m2.4.5.2.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p3.2.m2.4.5.2.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.2">subscript</csymbol><apply id="S4.SS2.SSS2.p3.2.m2.4.5.2.2.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p3.2.m2.4.5.2.2.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.2">superscript</csymbol><ci id="S4.SS2.SSS2.p3.2.m2.4.5.2.2.2.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.2.2.2">𝐶</ci><infinity id="S4.SS2.SSS2.p3.2.m2.4.5.2.2.3.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.2.2.3"></infinity></apply><list id="S4.SS2.SSS2.p3.2.m2.2.2.2.3.cmml" xref="S4.SS2.SSS2.p3.2.m2.2.2.2.4"><ci id="S4.SS2.SSS2.p3.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.1.1.1.1">c</ci><ci id="S4.SS2.SSS2.p3.2.m2.2.2.2.2.cmml" xref="S4.SS2.SSS2.p3.2.m2.2.2.2.2">𝑚</ci></list></apply><list id="S4.SS2.SSS2.p3.2.m2.4.5.3.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.5.3.2"><apply id="S4.SS2.SSS2.p3.2.m2.3.3.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3"><ci id="S4.SS2.SSS2.p3.2.m2.3.3.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.1">¯</ci><apply id="S4.SS2.SSS2.p3.2.m2.3.3.2.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p3.2.m2.3.3.2.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.2">subscript</csymbol><apply id="S4.SS2.SSS2.p3.2.m2.3.3.2.2.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.p3.2.m2.3.3.2.2.1.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.2">superscript</csymbol><ci id="S4.SS2.SSS2.p3.2.m2.3.3.2.2.2.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.2.2.2">ℝ</ci><ci id="S4.SS2.SSS2.p3.2.m2.3.3.2.2.3.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.p3.2.m2.3.3.2.3.cmml" xref="S4.SS2.SSS2.p3.2.m2.3.3.2.3"></plus></apply></apply><ci id="S4.SS2.SSS2.p3.2.m2.4.4.cmml" xref="S4.SS2.SSS2.p3.2.m2.4.4">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.p3.2.m2.4c">C^{\infty}_{{\rm c},m}(\overline{\mathbb{R}^{d}_{+}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.p3.2.m2.4d">italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_c , italic_m end_POSTSUBSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X )</annotation></semantics></math>. The proof combines the construction of an approximating sequence from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Lemma 3.4]</cite> with the result of Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem8" title="Proposition 4.8. ‣ 4.2.2. Sobolev spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.8</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S4.Thmtheorem9"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.1.1.1">Proposition 4.9</span></span><span class="ltx_text ltx_font_bold" id="S4.Thmtheorem9.2.2">.</span> </h6> <div class="ltx_para" id="S4.Thmtheorem9.p1"> <p class="ltx_p" id="S4.Thmtheorem9.p1.5"><span class="ltx_text ltx_font_italic" id="S4.Thmtheorem9.p1.5.5">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.1.1.m1.2"><semantics id="S4.Thmtheorem9.p1.1.1.m1.2a"><mrow id="S4.Thmtheorem9.p1.1.1.m1.2.3" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.cmml"><mi id="S4.Thmtheorem9.p1.1.1.m1.2.3.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.1" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml"><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S4.Thmtheorem9.p1.1.1.m1.1.1" xref="S4.Thmtheorem9.p1.1.1.m1.1.1.cmml">1</mn><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2.2" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.1.1.m1.2.2" mathvariant="normal" xref="S4.Thmtheorem9.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.1.1.m1.2b"><apply id="S4.Thmtheorem9.p1.1.1.m1.2.3.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3"><in id="S4.Thmtheorem9.p1.1.1.m1.2.3.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.1"></in><ci id="S4.Thmtheorem9.p1.1.1.m1.2.3.2.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S4.Thmtheorem9.p1.1.1.m1.2.3.3.1.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.3.3.2"><cn id="S4.Thmtheorem9.p1.1.1.m1.1.1.cmml" type="integer" xref="S4.Thmtheorem9.p1.1.1.m1.1.1">1</cn><infinity id="S4.Thmtheorem9.p1.1.1.m1.2.2.cmml" xref="S4.Thmtheorem9.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="k,m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.2.2.m2.2"><semantics id="S4.Thmtheorem9.p1.2.2.m2.2a"><mrow id="S4.Thmtheorem9.p1.2.2.m2.2.3" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.cmml"><mrow id="S4.Thmtheorem9.p1.2.2.m2.2.3.2.2" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.2.1.cmml"><mi id="S4.Thmtheorem9.p1.2.2.m2.1.1" xref="S4.Thmtheorem9.p1.2.2.m2.1.1.cmml">k</mi><mo id="S4.Thmtheorem9.p1.2.2.m2.2.3.2.2.1" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.2.1.cmml">,</mo><mi id="S4.Thmtheorem9.p1.2.2.m2.2.2" xref="S4.Thmtheorem9.p1.2.2.m2.2.2.cmml">m</mi></mrow><mo id="S4.Thmtheorem9.p1.2.2.m2.2.3.1" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.1.cmml">∈</mo><msub id="S4.Thmtheorem9.p1.2.2.m2.2.3.3" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.3.cmml"><mi id="S4.Thmtheorem9.p1.2.2.m2.2.3.3.2" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.3.2.cmml">ℕ</mi><mn id="S4.Thmtheorem9.p1.2.2.m2.2.3.3.3" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.2.2.m2.2b"><apply id="S4.Thmtheorem9.p1.2.2.m2.2.3.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.3"><in id="S4.Thmtheorem9.p1.2.2.m2.2.3.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.1"></in><list id="S4.Thmtheorem9.p1.2.2.m2.2.3.2.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.2.2"><ci id="S4.Thmtheorem9.p1.2.2.m2.1.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.1.1">𝑘</ci><ci id="S4.Thmtheorem9.p1.2.2.m2.2.2.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.2">𝑚</ci></list><apply id="S4.Thmtheorem9.p1.2.2.m2.2.3.3.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.3"><csymbol cd="ambiguous" id="S4.Thmtheorem9.p1.2.2.m2.2.3.3.1.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.3">subscript</csymbol><ci id="S4.Thmtheorem9.p1.2.2.m2.2.3.3.2.cmml" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.3.2">ℕ</ci><cn id="S4.Thmtheorem9.p1.2.2.m2.2.3.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.2.2.m2.2.3.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.2.2.m2.2c">k,m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.2.2.m2.2d">italic_k , italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="k\geq m+1" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.3.3.m3.1"><semantics id="S4.Thmtheorem9.p1.3.3.m3.1a"><mrow id="S4.Thmtheorem9.p1.3.3.m3.1.1" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.cmml"><mi id="S4.Thmtheorem9.p1.3.3.m3.1.1.2" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem9.p1.3.3.m3.1.1.1" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.1.cmml">≥</mo><mrow id="S4.Thmtheorem9.p1.3.3.m3.1.1.3" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.cmml"><mi id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2.cmml">m</mi><mo id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.1" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.1.cmml">+</mo><mn id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.3.3.m3.1b"><apply id="S4.Thmtheorem9.p1.3.3.m3.1.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1"><geq id="S4.Thmtheorem9.p1.3.3.m3.1.1.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.1"></geq><ci id="S4.Thmtheorem9.p1.3.3.m3.1.1.2.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.2">𝑘</ci><apply id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3"><plus id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.1.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.1"></plus><ci id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2.cmml" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.2">𝑚</ci><cn id="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.3.3.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.3.3.m3.1c">k\geq m+1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.3.3.m3.1d">italic_k ≥ italic_m + 1</annotation></semantics></math>, <math alttext="(k-m-1)p-1<\gamma<(k-m)p-1" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.4.4.m4.2"><semantics id="S4.Thmtheorem9.p1.4.4.m4.2a"><mrow id="S4.Thmtheorem9.p1.4.4.m4.2.2" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.cmml"><mrow id="S4.Thmtheorem9.p1.4.4.m4.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.cmml"><mrow id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.cmml"><mrow id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.2" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.3" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.3.cmml">m</mi><mo id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.1a" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.4" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.4.cmml">1</mn></mrow><mo id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.2" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.2.cmml">⁢</mo><mi id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.3" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.3.cmml">p</mi></mrow><mo id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.2" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.2.cmml">−</mo><mn id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.3" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.3.cmml">1</mn></mrow><mo id="S4.Thmtheorem9.p1.4.4.m4.2.2.4" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.4.cmml"><</mo><mi id="S4.Thmtheorem9.p1.4.4.m4.2.2.5" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.5.cmml">γ</mi><mo id="S4.Thmtheorem9.p1.4.4.m4.2.2.6" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.6.cmml"><</mo><mrow id="S4.Thmtheorem9.p1.4.4.m4.2.2.2" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.cmml"><mrow id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.cmml"><mrow id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.cmml"><mo id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.2" stretchy="false" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.cmml"><mi id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.2" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.2.cmml">k</mi><mo id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.1" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.1.cmml">−</mo><mi id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.3" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.3.cmml">m</mi></mrow><mo id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.3" stretchy="false" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.2" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.2.cmml">⁢</mo><mi id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.3" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.3.cmml">p</mi></mrow><mo id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.2" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.2.cmml">−</mo><mn id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.3" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.4.4.m4.2b"><apply id="S4.Thmtheorem9.p1.4.4.m4.2.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2"><and id="S4.Thmtheorem9.p1.4.4.m4.2.2a.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2"></and><apply id="S4.Thmtheorem9.p1.4.4.m4.2.2b.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2"><lt id="S4.Thmtheorem9.p1.4.4.m4.2.2.4.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.4"></lt><apply id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1"><minus id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.2"></minus><apply id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1"><times id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.2"></times><apply id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1"><minus id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.1"></minus><ci id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.2">𝑘</ci><ci id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.3">𝑚</ci><cn id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.4.cmml" type="integer" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.1.1.1.4">1</cn></apply><ci id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.1.3">𝑝</ci></apply><cn id="S4.Thmtheorem9.p1.4.4.m4.1.1.1.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.4.4.m4.1.1.1.3">1</cn></apply><ci id="S4.Thmtheorem9.p1.4.4.m4.2.2.5.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.5">𝛾</ci></apply><apply id="S4.Thmtheorem9.p1.4.4.m4.2.2c.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2"><lt id="S4.Thmtheorem9.p1.4.4.m4.2.2.6.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.6"></lt><share href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem9.p1.4.4.m4.2.2.5.cmml" id="S4.Thmtheorem9.p1.4.4.m4.2.2d.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2"></share><apply id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2"><minus id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.2"></minus><apply id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1"><times id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.2"></times><apply id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1"><minus id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.1.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.1"></minus><ci id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.2.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.2">𝑘</ci><ci id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.3.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.1.1.1.3">𝑚</ci></apply><ci id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.3.cmml" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.1.3">𝑝</ci></apply><cn id="S4.Thmtheorem9.p1.4.4.m4.2.2.2.3.cmml" type="integer" xref="S4.Thmtheorem9.p1.4.4.m4.2.2.2.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.4.4.m4.2c">(k-m-1)p-1<\gamma<(k-m)p-1</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.4.4.m4.2d">( italic_k - italic_m - 1 ) italic_p - 1 < italic_γ < ( italic_k - italic_m ) italic_p - 1</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S4.Thmtheorem9.p1.5.5.m5.1"><semantics id="S4.Thmtheorem9.p1.5.5.m5.1a"><mi id="S4.Thmtheorem9.p1.5.5.m5.1.1" xref="S4.Thmtheorem9.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S4.Thmtheorem9.p1.5.5.m5.1b"><ci id="S4.Thmtheorem9.p1.5.5.m5.1.1.cmml" xref="S4.Thmtheorem9.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.Thmtheorem9.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S4.Thmtheorem9.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a Banach space. 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xref="S4.Ex45.m1.13.13.1.1.2.4.3.2.cmml">Tr</mi><mi id="S4.Ex45.m1.13.13.1.1.2.4.3.3" xref="S4.Ex45.m1.13.13.1.1.2.4.3.3.cmml">m</mi></msub><mrow id="S4.Ex45.m1.2.2.2.4" xref="S4.Ex45.m1.2.2.2.3.cmml"><mi id="S4.Ex45.m1.1.1.1.1" xref="S4.Ex45.m1.1.1.1.1.cmml">k</mi><mo id="S4.Ex45.m1.2.2.2.4.1" xref="S4.Ex45.m1.2.2.2.3.cmml">,</mo><mi id="S4.Ex45.m1.2.2.2.2" xref="S4.Ex45.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S4.Ex45.m1.13.13.1.1.2.3" xref="S4.Ex45.m1.13.13.1.1.2.3.cmml">⁢</mo><mrow id="S4.Ex45.m1.13.13.1.1.2.2.2" xref="S4.Ex45.m1.13.13.1.1.2.2.3.cmml"><mo id="S4.Ex45.m1.13.13.1.1.2.2.2.3" stretchy="false" xref="S4.Ex45.m1.13.13.1.1.2.2.3.cmml">(</mo><msubsup id="S4.Ex45.m1.13.13.1.1.1.1.1.1" xref="S4.Ex45.m1.13.13.1.1.1.1.1.1.cmml"><mi id="S4.Ex45.m1.13.13.1.1.1.1.1.1.2.2" xref="S4.Ex45.m1.13.13.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex45.m1.13.13.1.1.1.1.1.1.3" xref="S4.Ex45.m1.13.13.1.1.1.1.1.1.3.cmml">+</mo><mi id="S4.Ex45.m1.13.13.1.1.1.1.1.1.2.3" 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encoding="application/x-tex" id="S4.Ex45.m1.13c">W^{k,p}_{\operatorname{Tr}_{m}}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\overline{C^{% \infty}_{{\rm c},m}(\overline{\mathbb{R}^{d}_{+}};X)}^{W^{k,p}(\mathbb{R}^{d}_% {+},w_{\gamma};X)}.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex45.m1.13d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = over¯ start_ARG italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_c , italic_m end_POSTSUBSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X ) end_ARG start_POSTSUPERSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S4.SS2.SSS2.3"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S4.SS2.SSS2.3.p1"> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.10">Let <math alttext="\varepsilon>0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.1.m1.1"><semantics id="S4.SS2.SSS2.3.p1.1.m1.1a"><mrow id="S4.SS2.SSS2.3.p1.1.m1.1.1" xref="S4.SS2.SSS2.3.p1.1.m1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.1.m1.1.1.2" xref="S4.SS2.SSS2.3.p1.1.m1.1.1.2.cmml">ε</mi><mo id="S4.SS2.SSS2.3.p1.1.m1.1.1.1" xref="S4.SS2.SSS2.3.p1.1.m1.1.1.1.cmml">></mo><mn id="S4.SS2.SSS2.3.p1.1.m1.1.1.3" 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id="S4.SS2.SSS2.3.p1.2.m2.2.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.2.2.2.4"><ci id="S4.SS2.SSS2.3.p1.2.m2.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.1.1.1.1">𝑘</ci><ci id="S4.SS2.SSS2.3.p1.2.m2.2.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.2.2.2.2">𝑝</ci></list></apply><vector id="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2"><apply id="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.4.4.1.1.1.1.3"></plus></apply><apply id="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2.2">𝑤</ci><ci id="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S4.SS2.SSS2.3.p1.2.m2.3.3.cmml" xref="S4.SS2.SSS2.3.p1.2.m2.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.2.m2.5c">f\in W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.2.m2.5d">italic_f ∈ italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> such that <math alttext="\operatorname{Tr}_{m}f=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.3.m3.1"><semantics id="S4.SS2.SSS2.3.p1.3.m3.1a"><mrow id="S4.SS2.SSS2.3.p1.3.m3.1.1" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.cmml"><mrow id="S4.SS2.SSS2.3.p1.3.m3.1.1.2" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.cmml"><msub id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.cmml"><mi id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.2" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.2.cmml">Tr</mi><mi id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.3" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.3.cmml">m</mi></msub><mo id="S4.SS2.SSS2.3.p1.3.m3.1.1.2a" lspace="0.167em" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.cmml">⁡</mo><mi id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.2" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.2.cmml">f</mi></mrow><mo id="S4.SS2.SSS2.3.p1.3.m3.1.1.1" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS2.3.p1.3.m3.1.1.3" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.3.m3.1b"><apply id="S4.SS2.SSS2.3.p1.3.m3.1.1.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1"><eq id="S4.SS2.SSS2.3.p1.3.m3.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.1"></eq><apply id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2"><apply id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.1.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.2.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.2">Tr</ci><ci id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.3.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.3.p1.3.m3.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.2.2">𝑓</ci></apply><cn id="S4.SS2.SSS2.3.p1.3.m3.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.3.m3.1c">\operatorname{Tr}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.3.m3.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math>. The conditions on <math alttext="\gamma" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.4.m4.1"><semantics id="S4.SS2.SSS2.3.p1.4.m4.1a"><mi id="S4.SS2.SSS2.3.p1.4.m4.1.1" xref="S4.SS2.SSS2.3.p1.4.m4.1.1.cmml">γ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.4.m4.1b"><ci id="S4.SS2.SSS2.3.p1.4.m4.1.1.cmml" xref="S4.SS2.SSS2.3.p1.4.m4.1.1">𝛾</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.4.m4.1c">\gamma</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.4.m4.1d">italic_γ</annotation></semantics></math> guarantee that <math alttext="\operatorname{Tr}_{m}f" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.5.m5.1"><semantics id="S4.SS2.SSS2.3.p1.5.m5.1a"><mrow id="S4.SS2.SSS2.3.p1.5.m5.1.1" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.cmml"><msub id="S4.SS2.SSS2.3.p1.5.m5.1.1.1" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.5.m5.1.1.1.2" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.1.2.cmml">Tr</mi><mi id="S4.SS2.SSS2.3.p1.5.m5.1.1.1.3" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.1.3.cmml">m</mi></msub><mo id="S4.SS2.SSS2.3.p1.5.m5.1.1a" lspace="0.167em" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.cmml">⁡</mo><mi id="S4.SS2.SSS2.3.p1.5.m5.1.1.2" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.5.m5.1b"><apply id="S4.SS2.SSS2.3.p1.5.m5.1.1.cmml" xref="S4.SS2.SSS2.3.p1.5.m5.1.1"><apply id="S4.SS2.SSS2.3.p1.5.m5.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.5.m5.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.5.m5.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.1.2">Tr</ci><ci id="S4.SS2.SSS2.3.p1.5.m5.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.3.p1.5.m5.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.5.m5.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.5.m5.1c">\operatorname{Tr}_{m}f</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.5.m5.1d">roman_Tr start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f</annotation></semantics></math> exists and traces of order <math alttext="\geq m+1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.6.m6.1"><semantics id="S4.SS2.SSS2.3.p1.6.m6.1a"><mrow id="S4.SS2.SSS2.3.p1.6.m6.1.1" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.6.m6.1.1.2" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.2.cmml"></mi><mo id="S4.SS2.SSS2.3.p1.6.m6.1.1.1" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.1.cmml">≥</mo><mrow id="S4.SS2.SSS2.3.p1.6.m6.1.1.3" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.6.m6.1.1.3.2" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3.2.cmml">m</mi><mo id="S4.SS2.SSS2.3.p1.6.m6.1.1.3.1" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3.1.cmml">+</mo><mn id="S4.SS2.SSS2.3.p1.6.m6.1.1.3.3" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.6.m6.1b"><apply id="S4.SS2.SSS2.3.p1.6.m6.1.1.cmml" xref="S4.SS2.SSS2.3.p1.6.m6.1.1"><geq id="S4.SS2.SSS2.3.p1.6.m6.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.1"></geq><csymbol cd="latexml" id="S4.SS2.SSS2.3.p1.6.m6.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.2">absent</csymbol><apply id="S4.SS2.SSS2.3.p1.6.m6.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3"><plus id="S4.SS2.SSS2.3.p1.6.m6.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3.1"></plus><ci id="S4.SS2.SSS2.3.p1.6.m6.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3.2">𝑚</ci><cn id="S4.SS2.SSS2.3.p1.6.m6.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.6.m6.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.6.m6.1c">\geq m+1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.6.m6.1d">≥ italic_m + 1</annotation></semantics></math> do not exist. By Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem8" title="Proposition 4.8. ‣ 4.2.2. Sobolev spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.8</span></a> there exists a <math alttext="g\in C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.7.m7.2"><semantics id="S4.SS2.SSS2.3.p1.7.m7.2a"><mrow id="S4.SS2.SSS2.3.p1.7.m7.2.3" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.cmml"><mi id="S4.SS2.SSS2.3.p1.7.m7.2.3.2" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.2.cmml">g</mi><mo id="S4.SS2.SSS2.3.p1.7.m7.2.3.1" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.1.cmml">∈</mo><mrow id="S4.SS2.SSS2.3.p1.7.m7.2.3.3" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.cmml"><msubsup id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.cmml"><mi id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.2" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.2.cmml">C</mi><mi id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.3" mathvariant="normal" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.3.cmml">c</mi><mi id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.3" mathvariant="normal" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.1" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.2" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.1.cmml"><mo id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.1.cmml">(</mo><mover accent="true" id="S4.SS2.SSS2.3.p1.7.m7.1.1" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.cmml"><msubsup id="S4.SS2.SSS2.3.p1.7.m7.1.1.2" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2.cmml"><mi id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.2" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.3" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2.3.cmml">+</mo><mi id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.3" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS2.3.p1.7.m7.1.1.1" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.1.cmml">¯</mo></mover><mo id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.2.2" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.1.cmml">;</mo><mi id="S4.SS2.SSS2.3.p1.7.m7.2.2" xref="S4.SS2.SSS2.3.p1.7.m7.2.2.cmml">X</mi><mo id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.2.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.7.m7.2b"><apply id="S4.SS2.SSS2.3.p1.7.m7.2.3.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3"><in id="S4.SS2.SSS2.3.p1.7.m7.2.3.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.1"></in><ci id="S4.SS2.SSS2.3.p1.7.m7.2.3.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.2">𝑔</ci><apply id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3"><times id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.1"></times><apply id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2">superscript</csymbol><apply id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.2">𝐶</ci><ci id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.2.3">c</ci></apply><infinity id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.3.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.2.3"></infinity></apply><list id="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.3.3.3.2"><apply id="S4.SS2.SSS2.3.p1.7.m7.1.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1"><ci id="S4.SS2.SSS2.3.p1.7.m7.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.1">¯</ci><apply id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.2">ℝ</ci><ci id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.3.p1.7.m7.1.1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.1.1.2.3"></plus></apply></apply><ci id="S4.SS2.SSS2.3.p1.7.m7.2.2.cmml" xref="S4.SS2.SSS2.3.p1.7.m7.2.2">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.7.m7.2c">g\in C_{\mathrm{c}}^{\infty}(\overline{\mathbb{R}^{d}_{+}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.7.m7.2d">italic_g ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( over¯ start_ARG blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_ARG ; italic_X )</annotation></semantics></math> with its support in <math alttext="[0,R]\times[-R,R]^{d-1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.8.m8.4"><semantics id="S4.SS2.SSS2.3.p1.8.m8.4a"><mrow id="S4.SS2.SSS2.3.p1.8.m8.4.4" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.cmml"><mrow id="S4.SS2.SSS2.3.p1.8.m8.4.4.3.2" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.3.1.cmml"><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.3.2.1" stretchy="false" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.3.1.cmml">[</mo><mn id="S4.SS2.SSS2.3.p1.8.m8.1.1" xref="S4.SS2.SSS2.3.p1.8.m8.1.1.cmml">0</mn><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.3.2.2" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.3.p1.8.m8.2.2" xref="S4.SS2.SSS2.3.p1.8.m8.2.2.cmml">R</mi><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.3.2.3" rspace="0.055em" stretchy="false" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.3.1.cmml">]</mo></mrow><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.2" rspace="0.222em" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.2.cmml">×</mo><msup id="S4.SS2.SSS2.3.p1.8.m8.4.4.1" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.cmml"><mrow id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.2.cmml"><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.2.cmml">[</mo><mrow id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1a" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.2.cmml">R</mi></mrow><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.3" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.3.p1.8.m8.3.3" xref="S4.SS2.SSS2.3.p1.8.m8.3.3.cmml">R</mi><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.4" stretchy="false" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.2.cmml">]</mo></mrow><mrow id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.2" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.2.cmml">d</mi><mo id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.1" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.3" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.8.m8.4b"><apply id="S4.SS2.SSS2.3.p1.8.m8.4.4.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4"><times id="S4.SS2.SSS2.3.p1.8.m8.4.4.2.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.2"></times><interval closure="closed" id="S4.SS2.SSS2.3.p1.8.m8.4.4.3.1.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.3.2"><cn id="S4.SS2.SSS2.3.p1.8.m8.1.1.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.8.m8.1.1">0</cn><ci id="S4.SS2.SSS2.3.p1.8.m8.2.2.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.2.2">𝑅</ci></interval><apply id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.2.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1">superscript</csymbol><interval closure="closed" id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1"><apply id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1"><minus id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1"></minus><ci id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.1.1.1.2">𝑅</ci></apply><ci id="S4.SS2.SSS2.3.p1.8.m8.3.3.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.3.3">𝑅</ci></interval><apply id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3"><minus id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.1"></minus><ci id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.2">𝑑</ci><cn id="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.8.m8.4.4.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.8.m8.4c">[0,R]\times[-R,R]^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.8.m8.4d">[ 0 , italic_R ] × [ - italic_R , italic_R ] start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> for some <math alttext="R>0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.9.m9.1"><semantics id="S4.SS2.SSS2.3.p1.9.m9.1a"><mrow id="S4.SS2.SSS2.3.p1.9.m9.1.1" xref="S4.SS2.SSS2.3.p1.9.m9.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.9.m9.1.1.2" xref="S4.SS2.SSS2.3.p1.9.m9.1.1.2.cmml">R</mi><mo id="S4.SS2.SSS2.3.p1.9.m9.1.1.1" xref="S4.SS2.SSS2.3.p1.9.m9.1.1.1.cmml">></mo><mn id="S4.SS2.SSS2.3.p1.9.m9.1.1.3" xref="S4.SS2.SSS2.3.p1.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.9.m9.1b"><apply id="S4.SS2.SSS2.3.p1.9.m9.1.1.cmml" xref="S4.SS2.SSS2.3.p1.9.m9.1.1"><gt id="S4.SS2.SSS2.3.p1.9.m9.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.9.m9.1.1.1"></gt><ci id="S4.SS2.SSS2.3.p1.9.m9.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.9.m9.1.1.2">𝑅</ci><cn id="S4.SS2.SSS2.3.p1.9.m9.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.9.m9.1c">R>0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.9.m9.1d">italic_R > 0</annotation></semantics></math> such that <math alttext="(\partial_{1}^{m}g)|_{\partial\mathbb{R}^{d}_{+}}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.10.m10.2"><semantics id="S4.SS2.SSS2.3.p1.10.m10.2a"><mrow id="S4.SS2.SSS2.3.p1.10.m10.2.2" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.cmml"><msub id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.2.cmml"><mrow id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.2.cmml"><mrow id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.cmml"><msubsup id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.2" lspace="0em" rspace="0em" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.2.cmml">∂</mo><mn id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.3" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.3.cmml">m</mi></msubsup><mi id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.2.cmml">g</mi></mrow><mo id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.2.1.cmml">|</mo></mrow><mrow id="S4.SS2.SSS2.3.p1.10.m10.1.1.1" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.1" rspace="0em" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.1.cmml">∂</mo><msubsup id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.cmml"><mi id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.2" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.3" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.3.cmml">+</mo><mi id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.3" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.3.cmml">d</mi></msubsup></mrow></msub><mo id="S4.SS2.SSS2.3.p1.10.m10.2.2.2" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.2.cmml">=</mo><mn id="S4.SS2.SSS2.3.p1.10.m10.2.2.3" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.10.m10.2b"><apply id="S4.SS2.SSS2.3.p1.10.m10.2.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2"><eq id="S4.SS2.SSS2.3.p1.10.m10.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.2"></eq><apply id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1"><csymbol cd="latexml" id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.2">evaluated-at</csymbol><apply id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1"><apply id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1">subscript</csymbol><partialdiff id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.2"></partialdiff><cn id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.1.1.1.1.1.1.2">𝑔</ci></apply><apply id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1"><partialdiff id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.1"></partialdiff><apply id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.2">ℝ</ci><ci id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.10.m10.1.1.1.2.3"></plus></apply></apply></apply><cn id="S4.SS2.SSS2.3.p1.10.m10.2.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.10.m10.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.10.m10.2c">(\partial_{1}^{m}g)|_{\partial\mathbb{R}^{d}_{+}}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.10.m10.2d">( ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g ) | start_POSTSUBSCRIPT ∂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math> and</p> <table class="ltx_equation ltx_eqn_table" id="S4.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f-g\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}<\varepsilon." class="ltx_Math" display="block" id="S4.E7.m1.6"><semantics id="S4.E7.m1.6a"><mrow id="S4.E7.m1.6.6.1" xref="S4.E7.m1.6.6.1.1.cmml"><mrow id="S4.E7.m1.6.6.1.1" xref="S4.E7.m1.6.6.1.1.cmml"><msub id="S4.E7.m1.6.6.1.1.1" xref="S4.E7.m1.6.6.1.1.1.cmml"><mrow id="S4.E7.m1.6.6.1.1.1.1.1" xref="S4.E7.m1.6.6.1.1.1.1.2.cmml"><mo id="S4.E7.m1.6.6.1.1.1.1.1.2" stretchy="false" xref="S4.E7.m1.6.6.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.E7.m1.6.6.1.1.1.1.1.1" xref="S4.E7.m1.6.6.1.1.1.1.1.1.cmml"><mi id="S4.E7.m1.6.6.1.1.1.1.1.1.2" xref="S4.E7.m1.6.6.1.1.1.1.1.1.2.cmml">f</mi><mo id="S4.E7.m1.6.6.1.1.1.1.1.1.1" xref="S4.E7.m1.6.6.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.E7.m1.6.6.1.1.1.1.1.1.3" xref="S4.E7.m1.6.6.1.1.1.1.1.1.3.cmml">g</mi></mrow><mo id="S4.E7.m1.6.6.1.1.1.1.1.3" stretchy="false" xref="S4.E7.m1.6.6.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S4.E7.m1.5.5.5" xref="S4.E7.m1.5.5.5.cmml"><msup id="S4.E7.m1.5.5.5.7" xref="S4.E7.m1.5.5.5.7.cmml"><mi id="S4.E7.m1.5.5.5.7.2" xref="S4.E7.m1.5.5.5.7.2.cmml">W</mi><mrow id="S4.E7.m1.2.2.2.2.2.4" xref="S4.E7.m1.2.2.2.2.2.3.cmml"><mi id="S4.E7.m1.1.1.1.1.1.1" xref="S4.E7.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S4.E7.m1.2.2.2.2.2.4.1" xref="S4.E7.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.E7.m1.2.2.2.2.2.2" xref="S4.E7.m1.2.2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.E7.m1.5.5.5.6" xref="S4.E7.m1.5.5.5.6.cmml">⁢</mo><mrow id="S4.E7.m1.5.5.5.5.2" xref="S4.E7.m1.5.5.5.5.3.cmml"><mo id="S4.E7.m1.5.5.5.5.2.3" stretchy="false" xref="S4.E7.m1.5.5.5.5.3.cmml">(</mo><msubsup id="S4.E7.m1.4.4.4.4.1.1" xref="S4.E7.m1.4.4.4.4.1.1.cmml"><mi id="S4.E7.m1.4.4.4.4.1.1.2.2" xref="S4.E7.m1.4.4.4.4.1.1.2.2.cmml">ℝ</mi><mo id="S4.E7.m1.4.4.4.4.1.1.3" xref="S4.E7.m1.4.4.4.4.1.1.3.cmml">+</mo><mi id="S4.E7.m1.4.4.4.4.1.1.2.3" xref="S4.E7.m1.4.4.4.4.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.E7.m1.5.5.5.5.2.4" xref="S4.E7.m1.5.5.5.5.3.cmml">,</mo><msub id="S4.E7.m1.5.5.5.5.2.2" xref="S4.E7.m1.5.5.5.5.2.2.cmml"><mi id="S4.E7.m1.5.5.5.5.2.2.2" xref="S4.E7.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S4.E7.m1.5.5.5.5.2.2.3" xref="S4.E7.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S4.E7.m1.5.5.5.5.2.5" xref="S4.E7.m1.5.5.5.5.3.cmml">;</mo><mi id="S4.E7.m1.3.3.3.3" xref="S4.E7.m1.3.3.3.3.cmml">X</mi><mo id="S4.E7.m1.5.5.5.5.2.6" stretchy="false" xref="S4.E7.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><mo id="S4.E7.m1.6.6.1.1.2" xref="S4.E7.m1.6.6.1.1.2.cmml"><</mo><mi id="S4.E7.m1.6.6.1.1.3" xref="S4.E7.m1.6.6.1.1.3.cmml">ε</mi></mrow><mo id="S4.E7.m1.6.6.1.2" lspace="0em" xref="S4.E7.m1.6.6.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.E7.m1.6b"><apply id="S4.E7.m1.6.6.1.1.cmml" xref="S4.E7.m1.6.6.1"><lt id="S4.E7.m1.6.6.1.1.2.cmml" xref="S4.E7.m1.6.6.1.1.2"></lt><apply id="S4.E7.m1.6.6.1.1.1.cmml" xref="S4.E7.m1.6.6.1.1.1"><csymbol cd="ambiguous" id="S4.E7.m1.6.6.1.1.1.2.cmml" xref="S4.E7.m1.6.6.1.1.1">subscript</csymbol><apply id="S4.E7.m1.6.6.1.1.1.1.2.cmml" xref="S4.E7.m1.6.6.1.1.1.1.1"><csymbol cd="latexml" id="S4.E7.m1.6.6.1.1.1.1.2.1.cmml" xref="S4.E7.m1.6.6.1.1.1.1.1.2">norm</csymbol><apply id="S4.E7.m1.6.6.1.1.1.1.1.1.cmml" xref="S4.E7.m1.6.6.1.1.1.1.1.1"><minus id="S4.E7.m1.6.6.1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.6.6.1.1.1.1.1.1.1"></minus><ci id="S4.E7.m1.6.6.1.1.1.1.1.1.2.cmml" xref="S4.E7.m1.6.6.1.1.1.1.1.1.2">𝑓</ci><ci id="S4.E7.m1.6.6.1.1.1.1.1.1.3.cmml" xref="S4.E7.m1.6.6.1.1.1.1.1.1.3">𝑔</ci></apply></apply><apply id="S4.E7.m1.5.5.5.cmml" xref="S4.E7.m1.5.5.5"><times id="S4.E7.m1.5.5.5.6.cmml" xref="S4.E7.m1.5.5.5.6"></times><apply id="S4.E7.m1.5.5.5.7.cmml" xref="S4.E7.m1.5.5.5.7"><csymbol cd="ambiguous" id="S4.E7.m1.5.5.5.7.1.cmml" xref="S4.E7.m1.5.5.5.7">superscript</csymbol><ci id="S4.E7.m1.5.5.5.7.2.cmml" xref="S4.E7.m1.5.5.5.7.2">𝑊</ci><list id="S4.E7.m1.2.2.2.2.2.3.cmml" xref="S4.E7.m1.2.2.2.2.2.4"><ci id="S4.E7.m1.1.1.1.1.1.1.cmml" xref="S4.E7.m1.1.1.1.1.1.1">𝑘</ci><ci id="S4.E7.m1.2.2.2.2.2.2.cmml" xref="S4.E7.m1.2.2.2.2.2.2">𝑝</ci></list></apply><vector id="S4.E7.m1.5.5.5.5.3.cmml" xref="S4.E7.m1.5.5.5.5.2"><apply id="S4.E7.m1.4.4.4.4.1.1.cmml" xref="S4.E7.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S4.E7.m1.4.4.4.4.1.1.1.cmml" xref="S4.E7.m1.4.4.4.4.1.1">subscript</csymbol><apply id="S4.E7.m1.4.4.4.4.1.1.2.cmml" xref="S4.E7.m1.4.4.4.4.1.1"><csymbol cd="ambiguous" id="S4.E7.m1.4.4.4.4.1.1.2.1.cmml" xref="S4.E7.m1.4.4.4.4.1.1">superscript</csymbol><ci id="S4.E7.m1.4.4.4.4.1.1.2.2.cmml" xref="S4.E7.m1.4.4.4.4.1.1.2.2">ℝ</ci><ci id="S4.E7.m1.4.4.4.4.1.1.2.3.cmml" xref="S4.E7.m1.4.4.4.4.1.1.2.3">𝑑</ci></apply><plus id="S4.E7.m1.4.4.4.4.1.1.3.cmml" xref="S4.E7.m1.4.4.4.4.1.1.3"></plus></apply><apply id="S4.E7.m1.5.5.5.5.2.2.cmml" xref="S4.E7.m1.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S4.E7.m1.5.5.5.5.2.2.1.cmml" xref="S4.E7.m1.5.5.5.5.2.2">subscript</csymbol><ci id="S4.E7.m1.5.5.5.5.2.2.2.cmml" xref="S4.E7.m1.5.5.5.5.2.2.2">𝑤</ci><ci id="S4.E7.m1.5.5.5.5.2.2.3.cmml" xref="S4.E7.m1.5.5.5.5.2.2.3">𝛾</ci></apply><ci id="S4.E7.m1.3.3.3.3.cmml" xref="S4.E7.m1.3.3.3.3">𝑋</ci></vector></apply></apply><ci id="S4.E7.m1.6.6.1.1.3.cmml" xref="S4.E7.m1.6.6.1.1.3">𝜀</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E7.m1.6c">\|f-g\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}<\varepsilon.</annotation><annotation encoding="application/x-llamapun" id="S4.E7.m1.6d">∥ italic_f - italic_g ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT < italic_ε .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.17">Let <math alttext="\phi\in C^{\infty}({\mathbb{R}}_{+})" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.11.m1.1"><semantics id="S4.SS2.SSS2.3.p1.11.m1.1a"><mrow id="S4.SS2.SSS2.3.p1.11.m1.1.1" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.11.m1.1.1.3" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.3.cmml">ϕ</mi><mo id="S4.SS2.SSS2.3.p1.11.m1.1.1.2" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.2.cmml">∈</mo><mrow id="S4.SS2.SSS2.3.p1.11.m1.1.1.1" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.cmml"><msup id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.2" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.2.cmml">C</mi><mi id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.3" mathvariant="normal" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.3.cmml">∞</mi></msup><mo id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.3.cmml">+</mo></msub><mo id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.11.m1.1b"><apply id="S4.SS2.SSS2.3.p1.11.m1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1"><in id="S4.SS2.SSS2.3.p1.11.m1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.2"></in><ci id="S4.SS2.SSS2.3.p1.11.m1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.3">italic-ϕ</ci><apply id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1"><times id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.2"></times><apply id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.2">𝐶</ci><infinity id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.3.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.3.3"></infinity></apply><apply id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.2">ℝ</ci><plus id="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.11.m1.1.1.1.1.1.1.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.11.m1.1c">\phi\in C^{\infty}({\mathbb{R}}_{+})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.11.m1.1d">italic_ϕ ∈ italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT )</annotation></semantics></math> be such that <math alttext="\phi=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.12.m2.1"><semantics id="S4.SS2.SSS2.3.p1.12.m2.1a"><mrow id="S4.SS2.SSS2.3.p1.12.m2.1.1" xref="S4.SS2.SSS2.3.p1.12.m2.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.12.m2.1.1.2" xref="S4.SS2.SSS2.3.p1.12.m2.1.1.2.cmml">ϕ</mi><mo id="S4.SS2.SSS2.3.p1.12.m2.1.1.1" xref="S4.SS2.SSS2.3.p1.12.m2.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS2.3.p1.12.m2.1.1.3" xref="S4.SS2.SSS2.3.p1.12.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.12.m2.1b"><apply id="S4.SS2.SSS2.3.p1.12.m2.1.1.cmml" xref="S4.SS2.SSS2.3.p1.12.m2.1.1"><eq id="S4.SS2.SSS2.3.p1.12.m2.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.12.m2.1.1.1"></eq><ci id="S4.SS2.SSS2.3.p1.12.m2.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.12.m2.1.1.2">italic-ϕ</ci><cn id="S4.SS2.SSS2.3.p1.12.m2.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.12.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.12.m2.1c">\phi=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.12.m2.1d">italic_ϕ = 0</annotation></semantics></math> on <math alttext="[0,\tfrac{1}{2}]" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.13.m3.2"><semantics id="S4.SS2.SSS2.3.p1.13.m3.2a"><mrow id="S4.SS2.SSS2.3.p1.13.m3.2.3.2" xref="S4.SS2.SSS2.3.p1.13.m3.2.3.1.cmml"><mo id="S4.SS2.SSS2.3.p1.13.m3.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.3.p1.13.m3.2.3.1.cmml">[</mo><mn id="S4.SS2.SSS2.3.p1.13.m3.1.1" xref="S4.SS2.SSS2.3.p1.13.m3.1.1.cmml">0</mn><mo id="S4.SS2.SSS2.3.p1.13.m3.2.3.2.2" xref="S4.SS2.SSS2.3.p1.13.m3.2.3.1.cmml">,</mo><mfrac id="S4.SS2.SSS2.3.p1.13.m3.2.2" xref="S4.SS2.SSS2.3.p1.13.m3.2.2.cmml"><mn id="S4.SS2.SSS2.3.p1.13.m3.2.2.2" xref="S4.SS2.SSS2.3.p1.13.m3.2.2.2.cmml">1</mn><mn id="S4.SS2.SSS2.3.p1.13.m3.2.2.3" xref="S4.SS2.SSS2.3.p1.13.m3.2.2.3.cmml">2</mn></mfrac><mo id="S4.SS2.SSS2.3.p1.13.m3.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.13.m3.2.3.1.cmml">]</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.13.m3.2b"><interval closure="closed" id="S4.SS2.SSS2.3.p1.13.m3.2.3.1.cmml" xref="S4.SS2.SSS2.3.p1.13.m3.2.3.2"><cn id="S4.SS2.SSS2.3.p1.13.m3.1.1.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.13.m3.1.1">0</cn><apply id="S4.SS2.SSS2.3.p1.13.m3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.13.m3.2.2"><divide id="S4.SS2.SSS2.3.p1.13.m3.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.13.m3.2.2"></divide><cn id="S4.SS2.SSS2.3.p1.13.m3.2.2.2.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.13.m3.2.2.2">1</cn><cn id="S4.SS2.SSS2.3.p1.13.m3.2.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.13.m3.2.2.3">2</cn></apply></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.13.m3.2c">[0,\tfrac{1}{2}]</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.13.m3.2d">[ 0 , divide start_ARG 1 end_ARG start_ARG 2 end_ARG ]</annotation></semantics></math> and <math alttext="\phi=1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.14.m4.1"><semantics id="S4.SS2.SSS2.3.p1.14.m4.1a"><mrow id="S4.SS2.SSS2.3.p1.14.m4.1.1" xref="S4.SS2.SSS2.3.p1.14.m4.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.14.m4.1.1.2" xref="S4.SS2.SSS2.3.p1.14.m4.1.1.2.cmml">ϕ</mi><mo id="S4.SS2.SSS2.3.p1.14.m4.1.1.1" xref="S4.SS2.SSS2.3.p1.14.m4.1.1.1.cmml">=</mo><mn id="S4.SS2.SSS2.3.p1.14.m4.1.1.3" xref="S4.SS2.SSS2.3.p1.14.m4.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.14.m4.1b"><apply id="S4.SS2.SSS2.3.p1.14.m4.1.1.cmml" xref="S4.SS2.SSS2.3.p1.14.m4.1.1"><eq id="S4.SS2.SSS2.3.p1.14.m4.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.14.m4.1.1.1"></eq><ci id="S4.SS2.SSS2.3.p1.14.m4.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.14.m4.1.1.2">italic-ϕ</ci><cn id="S4.SS2.SSS2.3.p1.14.m4.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.14.m4.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.14.m4.1c">\phi=1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.14.m4.1d">italic_ϕ = 1</annotation></semantics></math> on <math alttext="[1,\infty)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.15.m5.2"><semantics id="S4.SS2.SSS2.3.p1.15.m5.2a"><mrow id="S4.SS2.SSS2.3.p1.15.m5.2.3.2" xref="S4.SS2.SSS2.3.p1.15.m5.2.3.1.cmml"><mo id="S4.SS2.SSS2.3.p1.15.m5.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.3.p1.15.m5.2.3.1.cmml">[</mo><mn id="S4.SS2.SSS2.3.p1.15.m5.1.1" xref="S4.SS2.SSS2.3.p1.15.m5.1.1.cmml">1</mn><mo id="S4.SS2.SSS2.3.p1.15.m5.2.3.2.2" xref="S4.SS2.SSS2.3.p1.15.m5.2.3.1.cmml">,</mo><mi id="S4.SS2.SSS2.3.p1.15.m5.2.2" mathvariant="normal" xref="S4.SS2.SSS2.3.p1.15.m5.2.2.cmml">∞</mi><mo id="S4.SS2.SSS2.3.p1.15.m5.2.3.2.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.15.m5.2.3.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.15.m5.2b"><interval closure="closed-open" id="S4.SS2.SSS2.3.p1.15.m5.2.3.1.cmml" xref="S4.SS2.SSS2.3.p1.15.m5.2.3.2"><cn id="S4.SS2.SSS2.3.p1.15.m5.1.1.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.15.m5.1.1">1</cn><infinity id="S4.SS2.SSS2.3.p1.15.m5.2.2.cmml" xref="S4.SS2.SSS2.3.p1.15.m5.2.2"></infinity></interval></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.15.m5.2c">[1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.15.m5.2d">[ 1 , ∞ )</annotation></semantics></math> and set <math alttext="\phi_{n}(x_{1}):=\phi(nx_{1})" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.16.m6.2"><semantics id="S4.SS2.SSS2.3.p1.16.m6.2a"><mrow id="S4.SS2.SSS2.3.p1.16.m6.2.2" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.cmml"><mrow id="S4.SS2.SSS2.3.p1.16.m6.1.1.1" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.cmml"><msub id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.2" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.2.cmml">ϕ</mi><mi id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.3" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.3.cmml">n</mi></msub><mo id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.2" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.2.cmml">x</mi><mn id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.3" rspace="0.278em" stretchy="false" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.3.p1.16.m6.2.2.3" rspace="0.278em" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.3.cmml">:=</mo><mrow id="S4.SS2.SSS2.3.p1.16.m6.2.2.2" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.cmml"><mi id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.3" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.3.cmml">ϕ</mi><mo id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.2" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.2" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.2.cmml">n</mi><mo id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.1" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.1.cmml">⁢</mo><msub id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.2" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.2.cmml">x</mi><mn id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.3" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.16.m6.2b"><apply id="S4.SS2.SSS2.3.p1.16.m6.2.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2"><csymbol cd="latexml" id="S4.SS2.SSS2.3.p1.16.m6.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.3">assign</csymbol><apply id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1"><times id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.2"></times><apply id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.2">italic-ϕ</ci><ci id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.3.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.3.3">𝑛</ci></apply><apply id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.2">𝑥</ci><cn id="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.16.m6.1.1.1.1.1.1.3">1</cn></apply></apply><apply id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2"><times id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.2"></times><ci id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.3">italic-ϕ</ci><apply id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1"><times id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.1"></times><ci id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.2">𝑛</ci><apply id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.2">𝑥</ci><cn id="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.16.m6.2.2.2.1.1.1.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.16.m6.2c">\phi_{n}(x_{1}):=\phi(nx_{1})</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.16.m6.2d">italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) := italic_ϕ ( italic_n italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT )</annotation></semantics></math>. We construct a sequence <math alttext="(g_{n})_{n\geq 1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.17.m7.1"><semantics id="S4.SS2.SSS2.3.p1.17.m7.1a"><msub id="S4.SS2.SSS2.3.p1.17.m7.1.1" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.cmml"><mrow id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.cmml">(</mo><msub id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.2.cmml">g</mi><mi id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.3.cmml">n</mi></msub><mo id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S4.SS2.SSS2.3.p1.17.m7.1.1.3" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.17.m7.1.1.3.2" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3.2.cmml">n</mi><mo id="S4.SS2.SSS2.3.p1.17.m7.1.1.3.1" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3.1.cmml">≥</mo><mn id="S4.SS2.SSS2.3.p1.17.m7.1.1.3.3" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3.3.cmml">1</mn></mrow></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.17.m7.1b"><apply id="S4.SS2.SSS2.3.p1.17.m7.1.1.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.17.m7.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.2">𝑔</ci><ci id="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.1.1.1.3">𝑛</ci></apply><apply id="S4.SS2.SSS2.3.p1.17.m7.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3"><geq id="S4.SS2.SSS2.3.p1.17.m7.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3.1"></geq><ci id="S4.SS2.SSS2.3.p1.17.m7.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3.2">𝑛</ci><cn id="S4.SS2.SSS2.3.p1.17.m7.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.17.m7.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.17.m7.1c">(g_{n})_{n\geq 1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.17.m7.1d">( italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_n ≥ 1 end_POSTSUBSCRIPT</annotation></semantics></math> as follows:</p> <ul class="ltx_itemize" id="S4.I5"> <li class="ltx_item" id="S4.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I5.i1.p1"> <p class="ltx_p" id="S4.I5.i1.p1.2">If <math alttext="m=0" class="ltx_Math" display="inline" id="S4.I5.i1.p1.1.m1.1"><semantics id="S4.I5.i1.p1.1.m1.1a"><mrow id="S4.I5.i1.p1.1.m1.1.1" xref="S4.I5.i1.p1.1.m1.1.1.cmml"><mi id="S4.I5.i1.p1.1.m1.1.1.2" xref="S4.I5.i1.p1.1.m1.1.1.2.cmml">m</mi><mo id="S4.I5.i1.p1.1.m1.1.1.1" xref="S4.I5.i1.p1.1.m1.1.1.1.cmml">=</mo><mn id="S4.I5.i1.p1.1.m1.1.1.3" xref="S4.I5.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I5.i1.p1.1.m1.1b"><apply id="S4.I5.i1.p1.1.m1.1.1.cmml" xref="S4.I5.i1.p1.1.m1.1.1"><eq id="S4.I5.i1.p1.1.m1.1.1.1.cmml" xref="S4.I5.i1.p1.1.m1.1.1.1"></eq><ci id="S4.I5.i1.p1.1.m1.1.1.2.cmml" xref="S4.I5.i1.p1.1.m1.1.1.2">𝑚</ci><cn id="S4.I5.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I5.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I5.i1.p1.1.m1.1c">m=0</annotation><annotation encoding="application/x-llamapun" id="S4.I5.i1.p1.1.m1.1d">italic_m = 0</annotation></semantics></math>, define <math alttext="g_{n}(x):=\phi_{n}(x_{1})g(x)" class="ltx_Math" display="inline" id="S4.I5.i1.p1.2.m2.3"><semantics id="S4.I5.i1.p1.2.m2.3a"><mrow id="S4.I5.i1.p1.2.m2.3.3" xref="S4.I5.i1.p1.2.m2.3.3.cmml"><mrow id="S4.I5.i1.p1.2.m2.3.3.3" xref="S4.I5.i1.p1.2.m2.3.3.3.cmml"><msub id="S4.I5.i1.p1.2.m2.3.3.3.2" xref="S4.I5.i1.p1.2.m2.3.3.3.2.cmml"><mi id="S4.I5.i1.p1.2.m2.3.3.3.2.2" xref="S4.I5.i1.p1.2.m2.3.3.3.2.2.cmml">g</mi><mi id="S4.I5.i1.p1.2.m2.3.3.3.2.3" xref="S4.I5.i1.p1.2.m2.3.3.3.2.3.cmml">n</mi></msub><mo id="S4.I5.i1.p1.2.m2.3.3.3.1" xref="S4.I5.i1.p1.2.m2.3.3.3.1.cmml">⁢</mo><mrow id="S4.I5.i1.p1.2.m2.3.3.3.3.2" xref="S4.I5.i1.p1.2.m2.3.3.3.cmml"><mo id="S4.I5.i1.p1.2.m2.3.3.3.3.2.1" stretchy="false" xref="S4.I5.i1.p1.2.m2.3.3.3.cmml">(</mo><mi id="S4.I5.i1.p1.2.m2.1.1" xref="S4.I5.i1.p1.2.m2.1.1.cmml">x</mi><mo id="S4.I5.i1.p1.2.m2.3.3.3.3.2.2" rspace="0.278em" stretchy="false" xref="S4.I5.i1.p1.2.m2.3.3.3.cmml">)</mo></mrow></mrow><mo id="S4.I5.i1.p1.2.m2.3.3.2" rspace="0.278em" xref="S4.I5.i1.p1.2.m2.3.3.2.cmml">:=</mo><mrow id="S4.I5.i1.p1.2.m2.3.3.1" xref="S4.I5.i1.p1.2.m2.3.3.1.cmml"><msub id="S4.I5.i1.p1.2.m2.3.3.1.3" xref="S4.I5.i1.p1.2.m2.3.3.1.3.cmml"><mi id="S4.I5.i1.p1.2.m2.3.3.1.3.2" xref="S4.I5.i1.p1.2.m2.3.3.1.3.2.cmml">ϕ</mi><mi id="S4.I5.i1.p1.2.m2.3.3.1.3.3" xref="S4.I5.i1.p1.2.m2.3.3.1.3.3.cmml">n</mi></msub><mo id="S4.I5.i1.p1.2.m2.3.3.1.2" xref="S4.I5.i1.p1.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S4.I5.i1.p1.2.m2.3.3.1.1.1" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.cmml"><mo id="S4.I5.i1.p1.2.m2.3.3.1.1.1.2" stretchy="false" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.cmml">(</mo><msub id="S4.I5.i1.p1.2.m2.3.3.1.1.1.1" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.cmml"><mi id="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.2" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.2.cmml">x</mi><mn id="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.3" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.3.cmml">1</mn></msub><mo id="S4.I5.i1.p1.2.m2.3.3.1.1.1.3" stretchy="false" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.cmml">)</mo></mrow><mo id="S4.I5.i1.p1.2.m2.3.3.1.2a" xref="S4.I5.i1.p1.2.m2.3.3.1.2.cmml">⁢</mo><mi id="S4.I5.i1.p1.2.m2.3.3.1.4" xref="S4.I5.i1.p1.2.m2.3.3.1.4.cmml">g</mi><mo id="S4.I5.i1.p1.2.m2.3.3.1.2b" xref="S4.I5.i1.p1.2.m2.3.3.1.2.cmml">⁢</mo><mrow id="S4.I5.i1.p1.2.m2.3.3.1.5.2" xref="S4.I5.i1.p1.2.m2.3.3.1.cmml"><mo id="S4.I5.i1.p1.2.m2.3.3.1.5.2.1" stretchy="false" xref="S4.I5.i1.p1.2.m2.3.3.1.cmml">(</mo><mi id="S4.I5.i1.p1.2.m2.2.2" xref="S4.I5.i1.p1.2.m2.2.2.cmml">x</mi><mo id="S4.I5.i1.p1.2.m2.3.3.1.5.2.2" stretchy="false" xref="S4.I5.i1.p1.2.m2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.I5.i1.p1.2.m2.3b"><apply id="S4.I5.i1.p1.2.m2.3.3.cmml" xref="S4.I5.i1.p1.2.m2.3.3"><csymbol cd="latexml" id="S4.I5.i1.p1.2.m2.3.3.2.cmml" xref="S4.I5.i1.p1.2.m2.3.3.2">assign</csymbol><apply id="S4.I5.i1.p1.2.m2.3.3.3.cmml" xref="S4.I5.i1.p1.2.m2.3.3.3"><times id="S4.I5.i1.p1.2.m2.3.3.3.1.cmml" xref="S4.I5.i1.p1.2.m2.3.3.3.1"></times><apply id="S4.I5.i1.p1.2.m2.3.3.3.2.cmml" xref="S4.I5.i1.p1.2.m2.3.3.3.2"><csymbol cd="ambiguous" id="S4.I5.i1.p1.2.m2.3.3.3.2.1.cmml" xref="S4.I5.i1.p1.2.m2.3.3.3.2">subscript</csymbol><ci id="S4.I5.i1.p1.2.m2.3.3.3.2.2.cmml" xref="S4.I5.i1.p1.2.m2.3.3.3.2.2">𝑔</ci><ci id="S4.I5.i1.p1.2.m2.3.3.3.2.3.cmml" xref="S4.I5.i1.p1.2.m2.3.3.3.2.3">𝑛</ci></apply><ci id="S4.I5.i1.p1.2.m2.1.1.cmml" xref="S4.I5.i1.p1.2.m2.1.1">𝑥</ci></apply><apply id="S4.I5.i1.p1.2.m2.3.3.1.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1"><times id="S4.I5.i1.p1.2.m2.3.3.1.2.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.2"></times><apply id="S4.I5.i1.p1.2.m2.3.3.1.3.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.3"><csymbol cd="ambiguous" id="S4.I5.i1.p1.2.m2.3.3.1.3.1.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.3">subscript</csymbol><ci id="S4.I5.i1.p1.2.m2.3.3.1.3.2.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.3.2">italic-ϕ</ci><ci id="S4.I5.i1.p1.2.m2.3.3.1.3.3.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.3.3">𝑛</ci></apply><apply id="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1">subscript</csymbol><ci id="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.2.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.2">𝑥</ci><cn id="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.3.cmml" type="integer" xref="S4.I5.i1.p1.2.m2.3.3.1.1.1.1.3">1</cn></apply><ci id="S4.I5.i1.p1.2.m2.3.3.1.4.cmml" xref="S4.I5.i1.p1.2.m2.3.3.1.4">𝑔</ci><ci id="S4.I5.i1.p1.2.m2.2.2.cmml" xref="S4.I5.i1.p1.2.m2.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I5.i1.p1.2.m2.3c">g_{n}(x):=\phi_{n}(x_{1})g(x)</annotation><annotation encoding="application/x-llamapun" id="S4.I5.i1.p1.2.m2.3d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) := italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) italic_g ( italic_x )</annotation></semantics></math>.</p> </div> </li> <li class="ltx_item" id="S4.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">•</span> <div class="ltx_para" id="S4.I5.i2.p1"> <p class="ltx_p" id="S4.I5.i2.p1.1">If <math alttext="m\geq 1" class="ltx_Math" display="inline" id="S4.I5.i2.p1.1.m1.1"><semantics id="S4.I5.i2.p1.1.m1.1a"><mrow id="S4.I5.i2.p1.1.m1.1.1" xref="S4.I5.i2.p1.1.m1.1.1.cmml"><mi id="S4.I5.i2.p1.1.m1.1.1.2" xref="S4.I5.i2.p1.1.m1.1.1.2.cmml">m</mi><mo id="S4.I5.i2.p1.1.m1.1.1.1" xref="S4.I5.i2.p1.1.m1.1.1.1.cmml">≥</mo><mn id="S4.I5.i2.p1.1.m1.1.1.3" xref="S4.I5.i2.p1.1.m1.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.I5.i2.p1.1.m1.1b"><apply id="S4.I5.i2.p1.1.m1.1.1.cmml" xref="S4.I5.i2.p1.1.m1.1.1"><geq id="S4.I5.i2.p1.1.m1.1.1.1.cmml" xref="S4.I5.i2.p1.1.m1.1.1.1"></geq><ci id="S4.I5.i2.p1.1.m1.1.1.2.cmml" xref="S4.I5.i2.p1.1.m1.1.1.2">𝑚</ci><cn id="S4.I5.i2.p1.1.m1.1.1.3.cmml" type="integer" xref="S4.I5.i2.p1.1.m1.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.I5.i2.p1.1.m1.1c">m\geq 1</annotation><annotation encoding="application/x-llamapun" id="S4.I5.i2.p1.1.m1.1d">italic_m ≥ 1</annotation></semantics></math>, define</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex46"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{n}(x):=\sum_{j=0}^{m-1}g(1,\widetilde{x})\frac{(x_{1}-1)^{j}}{j!}+\frac{1}{% (m-1)!}\int_{1}^{x_{1}}(x_{1}-t)^{m-1}\phi_{n}(t)\partial^{m}_{1}g(t,% \widetilde{x})\hskip 2.0pt\mathrm{d}t." class="ltx_Math" display="block" id="S4.Ex46.m1.9"><semantics id="S4.Ex46.m1.9a"><mrow id="S4.Ex46.m1.9.9.1" xref="S4.Ex46.m1.9.9.1.1.cmml"><mrow id="S4.Ex46.m1.9.9.1.1" xref="S4.Ex46.m1.9.9.1.1.cmml"><mrow id="S4.Ex46.m1.9.9.1.1.3" xref="S4.Ex46.m1.9.9.1.1.3.cmml"><msub id="S4.Ex46.m1.9.9.1.1.3.2" xref="S4.Ex46.m1.9.9.1.1.3.2.cmml"><mi id="S4.Ex46.m1.9.9.1.1.3.2.2" 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xref="S4.Ex46.m1.9.9.1.1.1.3.1.2.3.cmml"><mi id="S4.Ex46.m1.9.9.1.1.1.3.1.2.3.2" xref="S4.Ex46.m1.9.9.1.1.1.3.1.2.3.2.cmml">j</mi><mo id="S4.Ex46.m1.9.9.1.1.1.3.1.2.3.1" xref="S4.Ex46.m1.9.9.1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S4.Ex46.m1.9.9.1.1.1.3.1.2.3.3" xref="S4.Ex46.m1.9.9.1.1.1.3.1.2.3.3.cmml">0</mn></mrow><mrow id="S4.Ex46.m1.9.9.1.1.1.3.1.3" xref="S4.Ex46.m1.9.9.1.1.1.3.1.3.cmml"><mi id="S4.Ex46.m1.9.9.1.1.1.3.1.3.2" xref="S4.Ex46.m1.9.9.1.1.1.3.1.3.2.cmml">m</mi><mo id="S4.Ex46.m1.9.9.1.1.1.3.1.3.1" xref="S4.Ex46.m1.9.9.1.1.1.3.1.3.1.cmml">−</mo><mn id="S4.Ex46.m1.9.9.1.1.1.3.1.3.3" xref="S4.Ex46.m1.9.9.1.1.1.3.1.3.3.cmml">1</mn></mrow></munderover><mrow id="S4.Ex46.m1.9.9.1.1.1.3.2" xref="S4.Ex46.m1.9.9.1.1.1.3.2.cmml"><mi id="S4.Ex46.m1.9.9.1.1.1.3.2.2" xref="S4.Ex46.m1.9.9.1.1.1.3.2.2.cmml">g</mi><mo id="S4.Ex46.m1.9.9.1.1.1.3.2.1" xref="S4.Ex46.m1.9.9.1.1.1.3.2.1.cmml">⁢</mo><mrow id="S4.Ex46.m1.9.9.1.1.1.3.2.3.2" xref="S4.Ex46.m1.9.9.1.1.1.3.2.3.1.cmml"><mo 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2.0pt\mathrm{d}t.</annotation><annotation encoding="application/x-llamapun" id="S4.Ex46.m1.9d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) := ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT italic_g ( 1 , over~ start_ARG italic_x end_ARG ) divide start_ARG ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 ) start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_ARG start_ARG italic_j ! end_ARG + divide start_ARG 1 end_ARG start_ARG ( italic_m - 1 ) ! end_ARG ∫ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_t ) start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_t ) ∂ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_g ( italic_t , over~ start_ARG italic_x end_ARG ) roman_d italic_t .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> </ul> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.24">Note that by integration by parts <math alttext="g_{n}(x)=g(x)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.18.m1.2"><semantics id="S4.SS2.SSS2.3.p1.18.m1.2a"><mrow id="S4.SS2.SSS2.3.p1.18.m1.2.3" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.cmml"><mrow id="S4.SS2.SSS2.3.p1.18.m1.2.3.2" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.cmml"><msub id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.cmml"><mi id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.2" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.2.cmml">g</mi><mi id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.3" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.3.cmml">n</mi></msub><mo id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.1" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.3.2" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.cmml"><mo id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.3.2.1" stretchy="false" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.cmml">(</mo><mi id="S4.SS2.SSS2.3.p1.18.m1.1.1" xref="S4.SS2.SSS2.3.p1.18.m1.1.1.cmml">x</mi><mo id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.3.2.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.cmml">)</mo></mrow></mrow><mo id="S4.SS2.SSS2.3.p1.18.m1.2.3.1" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.1.cmml">=</mo><mrow id="S4.SS2.SSS2.3.p1.18.m1.2.3.3" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.cmml"><mi id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.2" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.2.cmml">g</mi><mo id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.1" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.1.cmml">⁢</mo><mrow id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.3.2" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.cmml"><mo id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.3.2.1" stretchy="false" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.cmml">(</mo><mi id="S4.SS2.SSS2.3.p1.18.m1.2.2" xref="S4.SS2.SSS2.3.p1.18.m1.2.2.cmml">x</mi><mo id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.3.2.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.18.m1.2b"><apply id="S4.SS2.SSS2.3.p1.18.m1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3"><eq id="S4.SS2.SSS2.3.p1.18.m1.2.3.1.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.1"></eq><apply id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2"><times id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.1"></times><apply id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.2">𝑔</ci><ci id="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.2.2.3">𝑛</ci></apply><ci id="S4.SS2.SSS2.3.p1.18.m1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.1.1">𝑥</ci></apply><apply id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3"><times id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.1.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.1"></times><ci id="S4.SS2.SSS2.3.p1.18.m1.2.3.3.2.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.3.3.2">𝑔</ci><ci id="S4.SS2.SSS2.3.p1.18.m1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.18.m1.2.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.18.m1.2c">g_{n}(x)=g(x)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.18.m1.2d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = italic_g ( italic_x )</annotation></semantics></math> for all <math alttext="x\in{\mathbb{R}}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.19.m2.1"><semantics id="S4.SS2.SSS2.3.p1.19.m2.1a"><mrow id="S4.SS2.SSS2.3.p1.19.m2.1.1" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.19.m2.1.1.2" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.2.cmml">x</mi><mo id="S4.SS2.SSS2.3.p1.19.m2.1.1.1" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.1.cmml">∈</mo><msubsup id="S4.SS2.SSS2.3.p1.19.m2.1.1.3" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.2" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.3" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3.3.cmml">+</mo><mi id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.3" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.19.m2.1b"><apply id="S4.SS2.SSS2.3.p1.19.m2.1.1.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1"><in id="S4.SS2.SSS2.3.p1.19.m2.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.1"></in><ci id="S4.SS2.SSS2.3.p1.19.m2.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.2">𝑥</ci><apply id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.2">ℝ</ci><ci id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.3.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.3.p1.19.m2.1.1.3.3.cmml" xref="S4.SS2.SSS2.3.p1.19.m2.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.19.m2.1c">x\in{\mathbb{R}}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.19.m2.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="x_{1}\geq\frac{1}{n}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.20.m3.1"><semantics id="S4.SS2.SSS2.3.p1.20.m3.1a"><mrow id="S4.SS2.SSS2.3.p1.20.m3.1.1" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.cmml"><msub id="S4.SS2.SSS2.3.p1.20.m3.1.1.2" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.2.cmml"><mi id="S4.SS2.SSS2.3.p1.20.m3.1.1.2.2" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.2.2.cmml">x</mi><mn id="S4.SS2.SSS2.3.p1.20.m3.1.1.2.3" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.SSS2.3.p1.20.m3.1.1.1" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.1.cmml">≥</mo><mfrac id="S4.SS2.SSS2.3.p1.20.m3.1.1.3" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.3.cmml"><mn id="S4.SS2.SSS2.3.p1.20.m3.1.1.3.2" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.3.2.cmml">1</mn><mi id="S4.SS2.SSS2.3.p1.20.m3.1.1.3.3" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.3.3.cmml">n</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.20.m3.1b"><apply id="S4.SS2.SSS2.3.p1.20.m3.1.1.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1"><geq id="S4.SS2.SSS2.3.p1.20.m3.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.1"></geq><apply id="S4.SS2.SSS2.3.p1.20.m3.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.20.m3.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.20.m3.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.2.2">𝑥</ci><cn id="S4.SS2.SSS2.3.p1.20.m3.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.2.3">1</cn></apply><apply id="S4.SS2.SSS2.3.p1.20.m3.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.3"><divide id="S4.SS2.SSS2.3.p1.20.m3.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.3"></divide><cn id="S4.SS2.SSS2.3.p1.20.m3.1.1.3.2.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.3.2">1</cn><ci id="S4.SS2.SSS2.3.p1.20.m3.1.1.3.3.cmml" xref="S4.SS2.SSS2.3.p1.20.m3.1.1.3.3">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.20.m3.1c">x_{1}\geq\frac{1}{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.20.m3.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ divide start_ARG 1 end_ARG start_ARG italic_n end_ARG</annotation></semantics></math>. For <math alttext="\lvert\alpha\rvert\leq k" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.21.m4.1"><semantics id="S4.SS2.SSS2.3.p1.21.m4.1a"><mrow id="S4.SS2.SSS2.3.p1.21.m4.1.2" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.cmml"><mrow id="S4.SS2.SSS2.3.p1.21.m4.1.2.2.2" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.2.1.cmml"><mo id="S4.SS2.SSS2.3.p1.21.m4.1.2.2.2.1" stretchy="false" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.2.1.1.cmml">|</mo><mi id="S4.SS2.SSS2.3.p1.21.m4.1.1" xref="S4.SS2.SSS2.3.p1.21.m4.1.1.cmml">α</mi><mo id="S4.SS2.SSS2.3.p1.21.m4.1.2.2.2.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="S4.SS2.SSS2.3.p1.21.m4.1.2.1" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.1.cmml">≤</mo><mi id="S4.SS2.SSS2.3.p1.21.m4.1.2.3" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.21.m4.1b"><apply id="S4.SS2.SSS2.3.p1.21.m4.1.2.cmml" xref="S4.SS2.SSS2.3.p1.21.m4.1.2"><leq id="S4.SS2.SSS2.3.p1.21.m4.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.1"></leq><apply id="S4.SS2.SSS2.3.p1.21.m4.1.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.2.2"><abs id="S4.SS2.SSS2.3.p1.21.m4.1.2.2.1.1.cmml" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.2.2.1"></abs><ci id="S4.SS2.SSS2.3.p1.21.m4.1.1.cmml" xref="S4.SS2.SSS2.3.p1.21.m4.1.1">𝛼</ci></apply><ci id="S4.SS2.SSS2.3.p1.21.m4.1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.21.m4.1.2.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.21.m4.1c">\lvert\alpha\rvert\leq k</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.21.m4.1d">| italic_α | ≤ italic_k</annotation></semantics></math> with <math alttext="\alpha_{1}\leq m" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.22.m5.1"><semantics id="S4.SS2.SSS2.3.p1.22.m5.1a"><mrow id="S4.SS2.SSS2.3.p1.22.m5.1.1" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.cmml"><msub id="S4.SS2.SSS2.3.p1.22.m5.1.1.2" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.2.cmml"><mi id="S4.SS2.SSS2.3.p1.22.m5.1.1.2.2" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.2.2.cmml">α</mi><mn id="S4.SS2.SSS2.3.p1.22.m5.1.1.2.3" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.SSS2.3.p1.22.m5.1.1.1" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.1.cmml">≤</mo><mi id="S4.SS2.SSS2.3.p1.22.m5.1.1.3" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.22.m5.1b"><apply id="S4.SS2.SSS2.3.p1.22.m5.1.1.cmml" xref="S4.SS2.SSS2.3.p1.22.m5.1.1"><leq id="S4.SS2.SSS2.3.p1.22.m5.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.1"></leq><apply id="S4.SS2.SSS2.3.p1.22.m5.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.22.m5.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.22.m5.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.2.2">𝛼</ci><cn id="S4.SS2.SSS2.3.p1.22.m5.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.2.3">1</cn></apply><ci id="S4.SS2.SSS2.3.p1.22.m5.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.22.m5.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.22.m5.1c">\alpha_{1}\leq m</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.22.m5.1d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_m</annotation></semantics></math> and <math alttext="x\in{\mathbb{R}}^{d}_{+}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.23.m6.1"><semantics id="S4.SS2.SSS2.3.p1.23.m6.1a"><mrow id="S4.SS2.SSS2.3.p1.23.m6.1.1" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.23.m6.1.1.2" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.2.cmml">x</mi><mo id="S4.SS2.SSS2.3.p1.23.m6.1.1.1" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.1.cmml">∈</mo><msubsup id="S4.SS2.SSS2.3.p1.23.m6.1.1.3" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.2" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.3" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3.3.cmml">+</mo><mi id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.3" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.23.m6.1b"><apply id="S4.SS2.SSS2.3.p1.23.m6.1.1.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1"><in id="S4.SS2.SSS2.3.p1.23.m6.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.1"></in><ci id="S4.SS2.SSS2.3.p1.23.m6.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.2">𝑥</ci><apply id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.2">ℝ</ci><ci id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.3.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.3.p1.23.m6.1.1.3.3.cmml" xref="S4.SS2.SSS2.3.p1.23.m6.1.1.3.3"></plus></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.23.m6.1c">x\in{\mathbb{R}}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.23.m6.1d">italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="x_{1}<1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.24.m7.1"><semantics id="S4.SS2.SSS2.3.p1.24.m7.1a"><mrow id="S4.SS2.SSS2.3.p1.24.m7.1.1" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.cmml"><msub id="S4.SS2.SSS2.3.p1.24.m7.1.1.2" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.2.cmml"><mi id="S4.SS2.SSS2.3.p1.24.m7.1.1.2.2" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.2.2.cmml">x</mi><mn id="S4.SS2.SSS2.3.p1.24.m7.1.1.2.3" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.2.3.cmml">1</mn></msub><mo id="S4.SS2.SSS2.3.p1.24.m7.1.1.1" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.1.cmml"><</mo><mn id="S4.SS2.SSS2.3.p1.24.m7.1.1.3" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.24.m7.1b"><apply id="S4.SS2.SSS2.3.p1.24.m7.1.1.cmml" xref="S4.SS2.SSS2.3.p1.24.m7.1.1"><lt id="S4.SS2.SSS2.3.p1.24.m7.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.1"></lt><apply id="S4.SS2.SSS2.3.p1.24.m7.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.24.m7.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.2">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.24.m7.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.2.2">𝑥</ci><cn id="S4.SS2.SSS2.3.p1.24.m7.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.2.3">1</cn></apply><cn id="S4.SS2.SSS2.3.p1.24.m7.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.24.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.24.m7.1c">x_{1}<1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.24.m7.1d">italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < 1</annotation></semantics></math> we have</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex47"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\lvert\partial^{\alpha}g_{n}(x)\rvert\leq C_{m}\lVert g\rVert_{C_{\rm b}^{k+m}% ({\mathbb{R}}^{d}_{+};X)}\lVert\phi\rVert_{L^{\infty}({\mathbb{R}}_{+})}." class="ltx_Math" display="block" id="S4.Ex47.m1.7"><semantics id="S4.Ex47.m1.7a"><mrow id="S4.Ex47.m1.7.7.1" xref="S4.Ex47.m1.7.7.1.1.cmml"><mrow id="S4.Ex47.m1.7.7.1.1" xref="S4.Ex47.m1.7.7.1.1.cmml"><mrow id="S4.Ex47.m1.7.7.1.1.1.1" xref="S4.Ex47.m1.7.7.1.1.1.2.cmml"><mo id="S4.Ex47.m1.7.7.1.1.1.1.2" stretchy="false" xref="S4.Ex47.m1.7.7.1.1.1.2.1.cmml">|</mo><mrow id="S4.Ex47.m1.7.7.1.1.1.1.1" xref="S4.Ex47.m1.7.7.1.1.1.1.1.cmml"><msup id="S4.Ex47.m1.7.7.1.1.1.1.1.1" xref="S4.Ex47.m1.7.7.1.1.1.1.1.1.cmml"><mo id="S4.Ex47.m1.7.7.1.1.1.1.1.1.2" lspace="0em" rspace="0em" xref="S4.Ex47.m1.7.7.1.1.1.1.1.1.2.cmml">∂</mo><mi id="S4.Ex47.m1.7.7.1.1.1.1.1.1.3" xref="S4.Ex47.m1.7.7.1.1.1.1.1.1.3.cmml">α</mi></msup><mrow id="S4.Ex47.m1.7.7.1.1.1.1.1.2" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.cmml"><msub id="S4.Ex47.m1.7.7.1.1.1.1.1.2.2" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.2.cmml"><mi id="S4.Ex47.m1.7.7.1.1.1.1.1.2.2.2" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.2.2.cmml">g</mi><mi id="S4.Ex47.m1.7.7.1.1.1.1.1.2.2.3" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.2.3.cmml">n</mi></msub><mo id="S4.Ex47.m1.7.7.1.1.1.1.1.2.1" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.1.cmml">⁢</mo><mrow id="S4.Ex47.m1.7.7.1.1.1.1.1.2.3.2" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.cmml"><mo id="S4.Ex47.m1.7.7.1.1.1.1.1.2.3.2.1" stretchy="false" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.cmml">(</mo><mi id="S4.Ex47.m1.4.4" xref="S4.Ex47.m1.4.4.cmml">x</mi><mo id="S4.Ex47.m1.7.7.1.1.1.1.1.2.3.2.2" stretchy="false" xref="S4.Ex47.m1.7.7.1.1.1.1.1.2.cmml">)</mo></mrow></mrow></mrow><mo id="S4.Ex47.m1.7.7.1.1.1.1.3" stretchy="false" xref="S4.Ex47.m1.7.7.1.1.1.2.1.cmml">|</mo></mrow><mo id="S4.Ex47.m1.7.7.1.1.2" xref="S4.Ex47.m1.7.7.1.1.2.cmml">≤</mo><mrow id="S4.Ex47.m1.7.7.1.1.3" xref="S4.Ex47.m1.7.7.1.1.3.cmml"><msub id="S4.Ex47.m1.7.7.1.1.3.2" xref="S4.Ex47.m1.7.7.1.1.3.2.cmml"><mi id="S4.Ex47.m1.7.7.1.1.3.2.2" xref="S4.Ex47.m1.7.7.1.1.3.2.2.cmml">C</mi><mi id="S4.Ex47.m1.7.7.1.1.3.2.3" xref="S4.Ex47.m1.7.7.1.1.3.2.3.cmml">m</mi></msub><mo id="S4.Ex47.m1.7.7.1.1.3.1" lspace="0em" xref="S4.Ex47.m1.7.7.1.1.3.1.cmml">⁢</mo><msub id="S4.Ex47.m1.7.7.1.1.3.3" xref="S4.Ex47.m1.7.7.1.1.3.3.cmml"><mrow id="S4.Ex47.m1.7.7.1.1.3.3.2.2" xref="S4.Ex47.m1.7.7.1.1.3.3.2.1.cmml"><mo fence="true" id="S4.Ex47.m1.7.7.1.1.3.3.2.2.1" rspace="0em" xref="S4.Ex47.m1.7.7.1.1.3.3.2.1.1.cmml">∥</mo><mi id="S4.Ex47.m1.5.5" xref="S4.Ex47.m1.5.5.cmml">g</mi><mo fence="true" id="S4.Ex47.m1.7.7.1.1.3.3.2.2.2" lspace="0em" xref="S4.Ex47.m1.7.7.1.1.3.3.2.1.1.cmml">∥</mo></mrow><mrow id="S4.Ex47.m1.2.2.2" xref="S4.Ex47.m1.2.2.2.cmml"><msubsup id="S4.Ex47.m1.2.2.2.4" xref="S4.Ex47.m1.2.2.2.4.cmml"><mi id="S4.Ex47.m1.2.2.2.4.2.2" xref="S4.Ex47.m1.2.2.2.4.2.2.cmml">C</mi><mi id="S4.Ex47.m1.2.2.2.4.2.3" mathvariant="normal" xref="S4.Ex47.m1.2.2.2.4.2.3.cmml">b</mi><mrow id="S4.Ex47.m1.2.2.2.4.3" xref="S4.Ex47.m1.2.2.2.4.3.cmml"><mi id="S4.Ex47.m1.2.2.2.4.3.2" xref="S4.Ex47.m1.2.2.2.4.3.2.cmml">k</mi><mo id="S4.Ex47.m1.2.2.2.4.3.1" xref="S4.Ex47.m1.2.2.2.4.3.1.cmml">+</mo><mi id="S4.Ex47.m1.2.2.2.4.3.3" xref="S4.Ex47.m1.2.2.2.4.3.3.cmml">m</mi></mrow></msubsup><mo id="S4.Ex47.m1.2.2.2.3" xref="S4.Ex47.m1.2.2.2.3.cmml">⁢</mo><mrow id="S4.Ex47.m1.2.2.2.2.1" xref="S4.Ex47.m1.2.2.2.2.2.cmml"><mo id="S4.Ex47.m1.2.2.2.2.1.2" stretchy="false" xref="S4.Ex47.m1.2.2.2.2.2.cmml">(</mo><msubsup id="S4.Ex47.m1.2.2.2.2.1.1" xref="S4.Ex47.m1.2.2.2.2.1.1.cmml"><mi id="S4.Ex47.m1.2.2.2.2.1.1.2.2" xref="S4.Ex47.m1.2.2.2.2.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex47.m1.2.2.2.2.1.1.3" xref="S4.Ex47.m1.2.2.2.2.1.1.3.cmml">+</mo><mi id="S4.Ex47.m1.2.2.2.2.1.1.2.3" xref="S4.Ex47.m1.2.2.2.2.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.Ex47.m1.2.2.2.2.1.3" xref="S4.Ex47.m1.2.2.2.2.2.cmml">;</mo><mi id="S4.Ex47.m1.1.1.1.1" xref="S4.Ex47.m1.1.1.1.1.cmml">X</mi><mo id="S4.Ex47.m1.2.2.2.2.1.4" stretchy="false" xref="S4.Ex47.m1.2.2.2.2.2.cmml">)</mo></mrow></mrow></msub><mo id="S4.Ex47.m1.7.7.1.1.3.1a" lspace="0em" xref="S4.Ex47.m1.7.7.1.1.3.1.cmml">⁢</mo><msub id="S4.Ex47.m1.7.7.1.1.3.4" xref="S4.Ex47.m1.7.7.1.1.3.4.cmml"><mrow id="S4.Ex47.m1.7.7.1.1.3.4.2.2" xref="S4.Ex47.m1.7.7.1.1.3.4.2.1.cmml"><mo fence="true" id="S4.Ex47.m1.7.7.1.1.3.4.2.2.1" 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start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) | ≤ italic_C start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ∥ italic_g ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_m end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ∥ italic_ϕ ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.34">Moreover, we have</p> <table class="ltx_equation ltx_eqn_table" id="S4.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td 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xref="S4.E8.m1.3.3.1.1.2.2.3.2.3">𝑑</ci></apply><plus id="S4.E8.m1.3.3.1.1.2.2.3.3.cmml" xref="S4.E8.m1.3.3.1.1.2.2.3.3"></plus></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E8.m1.3c">\partial_{1}^{m}g_{n}(x)=\phi_{n}(x_{1})\partial^{m}_{1}g(x),\qquad x\in{% \mathbb{R}}^{d}_{+},</annotation><annotation encoding="application/x-llamapun" id="S4.E8.m1.3d">∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) = italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∂ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_g ( italic_x ) , italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4.8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.27">so that in particular <math alttext="\partial_{1}^{m}g_{n}\in C_{\mathrm{c}}^{\infty}({{\mathbb{R}}^{d}_{+}};X)" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.25.m1.2"><semantics id="S4.SS2.SSS2.3.p1.25.m1.2a"><mrow id="S4.SS2.SSS2.3.p1.25.m1.2.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.cmml"><mrow id="S4.SS2.SSS2.3.p1.25.m1.2.2.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.cmml"><msubsup id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.cmml"><mo id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.2.cmml">∂</mo><mn id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.3.cmml">1</mn><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.3.cmml">m</mi></msubsup><msub id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.cmml"><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.2.cmml">g</mi><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.3.cmml">n</mi></msub></mrow><mo id="S4.SS2.SSS2.3.p1.25.m1.2.2.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.2.cmml">∈</mo><mrow id="S4.SS2.SSS2.3.p1.25.m1.2.2.1" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.cmml"><msubsup id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.2.cmml">C</mi><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.3" mathvariant="normal" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.3.cmml">c</mi><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.3" mathvariant="normal" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.3.cmml">∞</mi></msubsup><mo id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.2.cmml">⁢</mo><mrow id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.2.cmml"><mo id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.2.cmml">(</mo><msubsup id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.2" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.3.cmml">+</mo><mi id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.3" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.2.cmml">;</mo><mi id="S4.SS2.SSS2.3.p1.25.m1.1.1" xref="S4.SS2.SSS2.3.p1.25.m1.1.1.cmml">X</mi><mo id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.4" stretchy="false" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.25.m1.2b"><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2"><in id="S4.SS2.SSS2.3.p1.25.m1.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.2"></in><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3"><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1">superscript</csymbol><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1">subscript</csymbol><partialdiff id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.2"></partialdiff><cn id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.2.3">1</cn></apply><ci id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.1.3">𝑚</ci></apply><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.2">𝑔</ci><ci id="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.3.2.3">𝑛</ci></apply></apply><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1"><times id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.2"></times><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3">superscript</csymbol><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.2">𝐶</ci><ci id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.2.3">c</ci></apply><infinity id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.3.3"></infinity></apply><list id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1"><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.2">ℝ</ci><ci id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.2.2.1.1.1.1.3"></plus></apply><ci id="S4.SS2.SSS2.3.p1.25.m1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.25.m1.1.1">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.25.m1.2c">\partial_{1}^{m}g_{n}\in C_{\mathrm{c}}^{\infty}({{\mathbb{R}}^{d}_{+}};X)</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.25.m1.2d">∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. We write <math alttext="\alpha=(\alpha_{1},\widetilde{\alpha})\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{d% -1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.26.m2.2"><semantics id="S4.SS2.SSS2.3.p1.26.m2.2a"><mrow id="S4.SS2.SSS2.3.p1.26.m2.2.2" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.cmml"><mi id="S4.SS2.SSS2.3.p1.26.m2.2.2.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.3.cmml">α</mi><mo id="S4.SS2.SSS2.3.p1.26.m2.2.2.4" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.4.cmml">=</mo><mrow id="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.1.2.cmml"><mo id="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.1.2.cmml">(</mo><msub id="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.1" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.1.2" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.1.2.cmml">α</mi><mn id="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.1.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.1.2.cmml">,</mo><mover accent="true" id="S4.SS2.SSS2.3.p1.26.m2.1.1" xref="S4.SS2.SSS2.3.p1.26.m2.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.26.m2.1.1.2" xref="S4.SS2.SSS2.3.p1.26.m2.1.1.2.cmml">α</mi><mo id="S4.SS2.SSS2.3.p1.26.m2.1.1.1" xref="S4.SS2.SSS2.3.p1.26.m2.1.1.1.cmml">~</mo></mover><mo id="S4.SS2.SSS2.3.p1.26.m2.2.2.1.1.4" stretchy="false" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.1.2.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.3.p1.26.m2.2.2.5" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.5.cmml">∈</mo><mrow id="S4.SS2.SSS2.3.p1.26.m2.2.2.6" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.cmml"><msub id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.2" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.2.cmml"><mi id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.2.2" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.2.2.cmml">ℕ</mi><mn id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.2.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.2.3.cmml">0</mn></msub><mo id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.1" lspace="0.222em" rspace="0.222em" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.1.cmml">×</mo><msubsup id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.cmml"><mi id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.2" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.2.cmml">ℕ</mi><mn id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.3.cmml">0</mn><mrow id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.cmml"><mi id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.2" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.2.cmml">d</mi><mo id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.1" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.1.cmml">−</mo><mn id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.3" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.26.m2.2b"><apply id="S4.SS2.SSS2.3.p1.26.m2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2"><and id="S4.SS2.SSS2.3.p1.26.m2.2.2a.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2"></and><apply 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id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.1.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3">superscript</csymbol><apply id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.2">ℕ</ci><cn id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.2.3">0</cn></apply><apply id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3"><minus id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.1.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.1"></minus><ci id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.2.cmml" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.2">𝑑</ci><cn id="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.26.m2.2.2.6.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.26.m2.2c">\alpha=(\alpha_{1},\widetilde{\alpha})\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{d% -1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.26.m2.2d">italic_α = ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_α end_ARG ) ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. If <math alttext="\alpha_{1}\in\{m+1,\dots,k\}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.27.m3.3"><semantics id="S4.SS2.SSS2.3.p1.27.m3.3a"><mrow id="S4.SS2.SSS2.3.p1.27.m3.3.3" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.cmml"><msub id="S4.SS2.SSS2.3.p1.27.m3.3.3.3" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.3.cmml"><mi id="S4.SS2.SSS2.3.p1.27.m3.3.3.3.2" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.3.2.cmml">α</mi><mn id="S4.SS2.SSS2.3.p1.27.m3.3.3.3.3" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.3.3.cmml">1</mn></msub><mo id="S4.SS2.SSS2.3.p1.27.m3.3.3.2" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.2.cmml">∈</mo><mrow id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.2.cmml"><mo id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.2.cmml">{</mo><mrow id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.2" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.2.cmml">m</mi><mo id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.1" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.1.cmml">+</mo><mn id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.3" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.3" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.3.p1.27.m3.1.1" mathvariant="normal" xref="S4.SS2.SSS2.3.p1.27.m3.1.1.cmml">…</mi><mo id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.4" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.3.p1.27.m3.2.2" xref="S4.SS2.SSS2.3.p1.27.m3.2.2.cmml">k</mi><mo id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.5" stretchy="false" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.27.m3.3b"><apply id="S4.SS2.SSS2.3.p1.27.m3.3.3.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3"><in id="S4.SS2.SSS2.3.p1.27.m3.3.3.2.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.2"></in><apply id="S4.SS2.SSS2.3.p1.27.m3.3.3.3.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.27.m3.3.3.3.1.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.3">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.27.m3.3.3.3.2.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.3.2">𝛼</ci><cn id="S4.SS2.SSS2.3.p1.27.m3.3.3.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.3.3">1</cn></apply><set id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.2.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1"><apply id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1"><plus id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.1"></plus><ci id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.2">𝑚</ci><cn id="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.27.m3.3.3.1.1.1.3">1</cn></apply><ci id="S4.SS2.SSS2.3.p1.27.m3.1.1.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.1.1">…</ci><ci id="S4.SS2.SSS2.3.p1.27.m3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.27.m3.2.2">𝑘</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.27.m3.3c">\alpha_{1}\in\{m+1,\dots,k\}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.27.m3.3d">italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ { italic_m + 1 , … , italic_k }</annotation></semantics></math>, then it follows from (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E8" title="In Proof. ‣ 4.2.2. Sobolev spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.8</span></a>) and the product rule that</p> <table class="ltx_equationgroup ltx_eqn_table" id="S4.E9"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E9X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle|\partial_{1}^{\alpha_{1}}g_{n}(x)|" class="ltx_Math" display="inline" id="S4.E9X.2.1.1.m1.2"><semantics id="S4.E9X.2.1.1.m1.2a"><mrow id="S4.E9X.2.1.1.m1.2.2.1" xref="S4.E9X.2.1.1.m1.2.2.2.cmml"><mo id="S4.E9X.2.1.1.m1.2.2.1.2" stretchy="false" xref="S4.E9X.2.1.1.m1.2.2.2.1.cmml">|</mo><mrow id="S4.E9X.2.1.1.m1.2.2.1.1" xref="S4.E9X.2.1.1.m1.2.2.1.1.cmml"><msubsup id="S4.E9X.2.1.1.m1.2.2.1.1.1" xref="S4.E9X.2.1.1.m1.2.2.1.1.1.cmml"><mo id="S4.E9X.2.1.1.m1.2.2.1.1.1.2.2" lspace="0em" rspace="0em" 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id="S4.E9X.3.2.2.m1.2c">\displaystyle=\big{|}\partial_{1}^{\alpha_{1}-m}\big{(}\phi_{n}(x_{1})\partial% ^{m}_{1}g(x)\big{)}\big{|}</annotation><annotation encoding="application/x-llamapun" id="S4.E9X.3.2.2.m1.2d">= | ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_m end_POSTSUPERSCRIPT ( italic_ϕ start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∂ start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_g ( italic_x ) ) |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="3"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(4.9)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E9Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td 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id="S4.E9Xa.2.1.1.m1.8c">\displaystyle\leq n^{\alpha_{1}-m}\big{|}\phi^{(\alpha_{1}-m)}(nx_{1})\partial% _{1}^{m}g(x)\big{|}+C_{k,m}\sum_{j=0}^{\alpha_{1}-m-1}n^{j}\big{|}\phi^{(j)}(% nx_{1})\partial_{1}^{\alpha_{1}-j}g(x)\big{|}</annotation><annotation encoding="application/x-llamapun" id="S4.E9Xa.2.1.1.m1.8d">≤ italic_n start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_m end_POSTSUPERSCRIPT | italic_ϕ start_POSTSUPERSCRIPT ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_m ) end_POSTSUPERSCRIPT ( italic_n italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g ( italic_x ) | + italic_C start_POSTSUBSCRIPT italic_k , italic_m end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_m - 1 end_POSTSUPERSCRIPT italic_n start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT | italic_ϕ start_POSTSUPERSCRIPT ( italic_j ) end_POSTSUPERSCRIPT ( italic_n italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j end_POSTSUPERSCRIPT italic_g ( italic_x ) |</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S4.E9Xb"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq n^{k-m}\|\phi\|_{C_{{\rm b}}^{k-m}(\mathbb{R}_{+})}\big{|}% \partial_{1}^{m}g(x)\big{|}+C_{k,m}\,n^{k-m-1}\|\phi\|_{C_{{\rm b}}^{k-m-1}(% \mathbb{R}_{+})}\|g\|_{C_{{\rm b}}^{k}(\mathbb{R}^{d}_{+};X)}." class="ltx_Math" display="inline" id="S4.E9Xb.2.1.1.m1.11"><semantics id="S4.E9Xb.2.1.1.m1.11a"><mrow id="S4.E9Xb.2.1.1.m1.11.11.1" 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xref="S4.E9Xb.2.1.1.m1.6.6.2.4.2.2">𝐶</ci><ci id="S4.E9Xb.2.1.1.m1.6.6.2.4.2.3.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.4.2.3">b</ci></apply><ci id="S4.E9Xb.2.1.1.m1.6.6.2.4.3.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.4.3">𝑘</ci></apply><list id="S4.E9Xb.2.1.1.m1.6.6.2.2.2.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1"><apply id="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1"><csymbol cd="ambiguous" id="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.1.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1">subscript</csymbol><apply id="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.2.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1"><csymbol cd="ambiguous" id="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.2.1.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1">superscript</csymbol><ci id="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.2.2.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.2.2">ℝ</ci><ci id="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.2.3.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.2.3">𝑑</ci></apply><plus id="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.3.cmml" xref="S4.E9Xb.2.1.1.m1.6.6.2.2.1.1.3"></plus></apply><ci id="S4.E9Xb.2.1.1.m1.5.5.1.1.cmml" xref="S4.E9Xb.2.1.1.m1.5.5.1.1">𝑋</ci></list></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E9Xb.2.1.1.m1.11c">\displaystyle\leq n^{k-m}\|\phi\|_{C_{{\rm b}}^{k-m}(\mathbb{R}_{+})}\big{|}% \partial_{1}^{m}g(x)\big{|}+C_{k,m}\,n^{k-m-1}\|\phi\|_{C_{{\rm b}}^{k-m-1}(% \mathbb{R}_{+})}\|g\|_{C_{{\rm b}}^{k}(\mathbb{R}^{d}_{+};X)}.</annotation><annotation encoding="application/x-llamapun" id="S4.E9Xb.2.1.1.m1.11d">≤ italic_n start_POSTSUPERSCRIPT italic_k - italic_m end_POSTSUPERSCRIPT ∥ italic_ϕ ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT | ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g ( italic_x ) | + italic_C start_POSTSUBSCRIPT italic_k , italic_m end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT italic_k - italic_m - 1 end_POSTSUPERSCRIPT ∥ italic_ϕ ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m - 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ∥ italic_g ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.29">Let <math alttext="K_{R}:=[-R,R]^{d-1}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.28.m1.2"><semantics id="S4.SS2.SSS2.3.p1.28.m1.2a"><mrow id="S4.SS2.SSS2.3.p1.28.m1.2.2" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.cmml"><msub id="S4.SS2.SSS2.3.p1.28.m1.2.2.3" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.3.cmml"><mi id="S4.SS2.SSS2.3.p1.28.m1.2.2.3.2" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.3.2.cmml">K</mi><mi id="S4.SS2.SSS2.3.p1.28.m1.2.2.3.3" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.3.3.cmml">R</mi></msub><mo id="S4.SS2.SSS2.3.p1.28.m1.2.2.2" lspace="0.278em" rspace="0.278em" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.2.cmml">:=</mo><msup id="S4.SS2.SSS2.3.p1.28.m1.2.2.1" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.cmml"><mrow id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.2.cmml"><mo id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.2.cmml">[</mo><mrow id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1a" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.2.cmml">R</mi></mrow><mo id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.3" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.2.cmml">,</mo><mi id="S4.SS2.SSS2.3.p1.28.m1.1.1" xref="S4.SS2.SSS2.3.p1.28.m1.1.1.cmml">R</mi><mo id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.4" stretchy="false" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.2.cmml">]</mo></mrow><mrow id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.cmml"><mi id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.2" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.2.cmml">d</mi><mo id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.1" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.3" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.28.m1.2b"><apply id="S4.SS2.SSS2.3.p1.28.m1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2"><csymbol cd="latexml" id="S4.SS2.SSS2.3.p1.28.m1.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.2">assign</csymbol><apply id="S4.SS2.SSS2.3.p1.28.m1.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.3"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.28.m1.2.2.3.1.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.3">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.28.m1.2.2.3.2.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.3.2">𝐾</ci><ci id="S4.SS2.SSS2.3.p1.28.m1.2.2.3.3.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.3.3">𝑅</ci></apply><apply id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.2.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1">superscript</csymbol><interval closure="closed" id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1"><apply id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1"><minus id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1"></minus><ci id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.1.1.1.2">𝑅</ci></apply><ci id="S4.SS2.SSS2.3.p1.28.m1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.1.1">𝑅</ci></interval><apply id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3"><minus id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.1"></minus><ci id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.2">𝑑</ci><cn id="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.28.m1.2.2.1.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.28.m1.2c">K_{R}:=[-R,R]^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.28.m1.2d">italic_K start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT := [ - italic_R , italic_R ] start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>. The properties of <math alttext="g_{n}" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.29.m2.1"><semantics id="S4.SS2.SSS2.3.p1.29.m2.1a"><msub id="S4.SS2.SSS2.3.p1.29.m2.1.1" xref="S4.SS2.SSS2.3.p1.29.m2.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.29.m2.1.1.2" xref="S4.SS2.SSS2.3.p1.29.m2.1.1.2.cmml">g</mi><mi id="S4.SS2.SSS2.3.p1.29.m2.1.1.3" xref="S4.SS2.SSS2.3.p1.29.m2.1.1.3.cmml">n</mi></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.29.m2.1b"><apply id="S4.SS2.SSS2.3.p1.29.m2.1.1.cmml" xref="S4.SS2.SSS2.3.p1.29.m2.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.29.m2.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.29.m2.1.1">subscript</csymbol><ci id="S4.SS2.SSS2.3.p1.29.m2.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.29.m2.1.1.2">𝑔</ci><ci id="S4.SS2.SSS2.3.p1.29.m2.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.29.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.29.m2.1c">g_{n}</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.29.m2.1d">italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E9" title="In Proof. ‣ 4.2.2. Sobolev spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.9</span></a>) imply that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx27"> <tbody id="S4.Ex48"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|g-g_{n}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}\leq" class="ltx_Math" display="inline" id="S4.Ex48.m1.6"><semantics id="S4.Ex48.m1.6a"><mrow id="S4.Ex48.m1.6.6" xref="S4.Ex48.m1.6.6.cmml"><msub id="S4.Ex48.m1.6.6.1" xref="S4.Ex48.m1.6.6.1.cmml"><mrow id="S4.Ex48.m1.6.6.1.1.1" xref="S4.Ex48.m1.6.6.1.1.2.cmml"><mo id="S4.Ex48.m1.6.6.1.1.1.2" stretchy="false" xref="S4.Ex48.m1.6.6.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex48.m1.6.6.1.1.1.1" xref="S4.Ex48.m1.6.6.1.1.1.1.cmml"><mi id="S4.Ex48.m1.6.6.1.1.1.1.2" xref="S4.Ex48.m1.6.6.1.1.1.1.2.cmml">g</mi><mo id="S4.Ex48.m1.6.6.1.1.1.1.1" xref="S4.Ex48.m1.6.6.1.1.1.1.1.cmml">−</mo><msub id="S4.Ex48.m1.6.6.1.1.1.1.3" xref="S4.Ex48.m1.6.6.1.1.1.1.3.cmml"><mi id="S4.Ex48.m1.6.6.1.1.1.1.3.2" xref="S4.Ex48.m1.6.6.1.1.1.1.3.2.cmml">g</mi><mi id="S4.Ex48.m1.6.6.1.1.1.1.3.3" xref="S4.Ex48.m1.6.6.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.Ex48.m1.6.6.1.1.1.3" stretchy="false" xref="S4.Ex48.m1.6.6.1.1.2.1.cmml">‖</mo></mrow><mrow id="S4.Ex48.m1.5.5.5" xref="S4.Ex48.m1.5.5.5.cmml"><msup id="S4.Ex48.m1.5.5.5.7" xref="S4.Ex48.m1.5.5.5.7.cmml"><mi id="S4.Ex48.m1.5.5.5.7.2" xref="S4.Ex48.m1.5.5.5.7.2.cmml">W</mi><mrow id="S4.Ex48.m1.2.2.2.2.2.4" xref="S4.Ex48.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex48.m1.1.1.1.1.1.1" xref="S4.Ex48.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S4.Ex48.m1.2.2.2.2.2.4.1" xref="S4.Ex48.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.Ex48.m1.2.2.2.2.2.2" xref="S4.Ex48.m1.2.2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex48.m1.5.5.5.6" xref="S4.Ex48.m1.5.5.5.6.cmml">⁢</mo><mrow id="S4.Ex48.m1.5.5.5.5.2" xref="S4.Ex48.m1.5.5.5.5.3.cmml"><mo id="S4.Ex48.m1.5.5.5.5.2.3" stretchy="false" xref="S4.Ex48.m1.5.5.5.5.3.cmml">(</mo><msubsup id="S4.Ex48.m1.4.4.4.4.1.1" xref="S4.Ex48.m1.4.4.4.4.1.1.cmml"><mi id="S4.Ex48.m1.4.4.4.4.1.1.2.2" xref="S4.Ex48.m1.4.4.4.4.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex48.m1.4.4.4.4.1.1.3" xref="S4.Ex48.m1.4.4.4.4.1.1.3.cmml">+</mo><mi id="S4.Ex48.m1.4.4.4.4.1.1.2.3" xref="S4.Ex48.m1.4.4.4.4.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.Ex48.m1.5.5.5.5.2.4" xref="S4.Ex48.m1.5.5.5.5.3.cmml">,</mo><msub id="S4.Ex48.m1.5.5.5.5.2.2" xref="S4.Ex48.m1.5.5.5.5.2.2.cmml"><mi id="S4.Ex48.m1.5.5.5.5.2.2.2" xref="S4.Ex48.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S4.Ex48.m1.5.5.5.5.2.2.3" xref="S4.Ex48.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex48.m1.5.5.5.5.2.5" xref="S4.Ex48.m1.5.5.5.5.3.cmml">;</mo><mi id="S4.Ex48.m1.3.3.3.3" xref="S4.Ex48.m1.3.3.3.3.cmml">X</mi><mo id="S4.Ex48.m1.5.5.5.5.2.6" stretchy="false" 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id="S4.Ex48.m2.3.3.1.1.1.1.1.1.5.2.2.cmml" xref="S4.Ex48.m2.3.3.1.1.1.1.1.1.5.2.2">𝑥</ci></apply></apply></apply></apply></apply><apply id="S4.Ex48.m2.3.3.1.3.cmml" xref="S4.Ex48.m2.3.3.1.3"><divide id="S4.Ex48.m2.3.3.1.3.1.cmml" xref="S4.Ex48.m2.3.3.1.3"></divide><cn id="S4.Ex48.m2.3.3.1.3.2.cmml" type="integer" xref="S4.Ex48.m2.3.3.1.3.2">1</cn><ci id="S4.Ex48.m2.3.3.1.3.3.cmml" xref="S4.Ex48.m2.3.3.1.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex48.m2.3c">\displaystyle\;\sum_{|\alpha|\leq k}\Big{(}\int_{K_{R}}\int_{0}^{\frac{1}{n}}x% _{1}^{\gamma}\|\partial^{\alpha}g(x)\|_{X}^{p}\hskip 2.0pt\mathrm{d}x_{1}% \hskip 2.0pt\mathrm{d}\widetilde{x}\Big{)}^{\frac{1}{p}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex48.m2.3d">∑ start_POSTSUBSCRIPT | italic_α | ≤ italic_k end_POSTSUBSCRIPT ( ∫ start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_n end_ARG end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT ∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_g ( italic_x ) ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex49"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+\sum_{\begin{subarray}{c}\lvert\alpha\rvert\leq k\\ 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xref="S4.Ex49.m1.3.3.1.1.1.1.1.1.1.5.2.2">𝑥</ci></apply></apply></apply></apply></apply><apply id="S4.Ex49.m1.3.3.1.1.3.cmml" xref="S4.Ex49.m1.3.3.1.1.3"><divide id="S4.Ex49.m1.3.3.1.1.3.1.cmml" xref="S4.Ex49.m1.3.3.1.1.3"></divide><cn id="S4.Ex49.m1.3.3.1.1.3.2.cmml" type="integer" xref="S4.Ex49.m1.3.3.1.1.3.2">1</cn><ci id="S4.Ex49.m1.3.3.1.1.3.3.cmml" xref="S4.Ex49.m1.3.3.1.1.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex49.m1.3c">\displaystyle+\sum_{\begin{subarray}{c}\lvert\alpha\rvert\leq k\\ \alpha_{1}\leq m\end{subarray}}\Big{(}\int_{K_{R}}\int_{0}^{\frac{1}{n}}x_{1}^% {\gamma}\|\partial^{\alpha}g_{n}(x)\|_{X}^{p}\hskip 2.0pt\mathrm{d}x_{1}\hskip 2% .0pt\mathrm{d}\widetilde{x}\Big{)}^{\frac{1}{p}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex49.m1.3d">+ ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL | italic_α | ≤ italic_k end_CELL end_ROW start_ROW start_CELL italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_m end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ( ∫ start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_n end_ARG end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT ∥ ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_x ) ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex50"><tr class="ltx_equation 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K_{R}}\int_{0}^{\frac{1}{n}}x_{1}^{\gamma}\|\partial^{\widetilde{\alpha}}% \partial_{1}^{m}g(x)\|_{X}^{p}\hskip 2.0pt\mathrm{d}x_{1}\hskip 2.0pt\mathrm{d% }\widetilde{x}\Big{)}^{\frac{1}{p}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex50.m1.4d">+ italic_n start_POSTSUPERSCRIPT italic_k - italic_m end_POSTSUPERSCRIPT ∥ italic_ϕ ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( ∫ start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_n end_ARG end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT ∥ ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g ( italic_x ) ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex51"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+C_{k,m}\,n^{k-m-1}\|\phi\|_{C_{{\rm b}}^{k-m-1}(\mathbb{R}_{+})}% \|g\|_{C_{{\rm b}}^{k}(\mathbb{R}^{d}_{+};X)}\Big{(}\int_{K_{R}}\int_{0}^{% \frac{1}{n}}x_{1}^{\gamma}\hskip 2.0pt\mathrm{d}x_{1}\hskip 2.0pt\mathrm{d}% 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id="S4.Ex51.m1.8.8.1.1.3.2.cmml" type="integer" xref="S4.Ex51.m1.8.8.1.1.3.2">1</cn><ci id="S4.Ex51.m1.8.8.1.1.3.3.cmml" xref="S4.Ex51.m1.8.8.1.1.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex51.m1.8c">\displaystyle+C_{k,m}\,n^{k-m-1}\|\phi\|_{C_{{\rm b}}^{k-m-1}(\mathbb{R}_{+})}% \|g\|_{C_{{\rm b}}^{k}(\mathbb{R}^{d}_{+};X)}\Big{(}\int_{K_{R}}\int_{0}^{% \frac{1}{n}}x_{1}^{\gamma}\hskip 2.0pt\mathrm{d}x_{1}\hskip 2.0pt\mathrm{d}% \widetilde{x}\Big{)}^{\frac{1}{p}}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex51.m1.8d">+ italic_C start_POSTSUBSCRIPT italic_k , italic_m end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT italic_k - italic_m - 1 end_POSTSUPERSCRIPT ∥ italic_ϕ ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m - 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ∥ italic_g ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ( ∫ start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_n end_ARG end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td 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xref="S4.Ex52.m2.7.7.1.3">superscript</csymbol><apply id="S4.Ex52.m2.7.7.1.3.2.cmml" xref="S4.Ex52.m2.7.7.1.3"><csymbol cd="ambiguous" id="S4.Ex52.m2.7.7.1.3.2.1.cmml" xref="S4.Ex52.m2.7.7.1.3">subscript</csymbol><ci id="S4.Ex52.m2.7.7.1.3.2.2.cmml" xref="S4.Ex52.m2.7.7.1.3.2.2">𝐶</ci><ci id="S4.Ex52.m2.7.7.1.3.2.3.cmml" xref="S4.Ex52.m2.7.7.1.3.2.3">b</ci></apply><apply id="S4.Ex52.m2.7.7.1.3.3.cmml" xref="S4.Ex52.m2.7.7.1.3.3"><minus id="S4.Ex52.m2.7.7.1.3.3.1.cmml" xref="S4.Ex52.m2.7.7.1.3.3.1"></minus><ci id="S4.Ex52.m2.7.7.1.3.3.2.cmml" xref="S4.Ex52.m2.7.7.1.3.3.2">𝑘</ci><ci id="S4.Ex52.m2.7.7.1.3.3.3.cmml" xref="S4.Ex52.m2.7.7.1.3.3.3">𝑚</ci><cn id="S4.Ex52.m2.7.7.1.3.3.4.cmml" type="integer" xref="S4.Ex52.m2.7.7.1.3.3.4">1</cn></apply></apply><apply id="S4.Ex52.m2.7.7.1.1.1.1.cmml" xref="S4.Ex52.m2.7.7.1.1.1"><csymbol cd="ambiguous" id="S4.Ex52.m2.7.7.1.1.1.1.1.cmml" xref="S4.Ex52.m2.7.7.1.1.1">subscript</csymbol><ci id="S4.Ex52.m2.7.7.1.1.1.1.2.cmml" xref="S4.Ex52.m2.7.7.1.1.1.1.2">ℝ</ci><plus id="S4.Ex52.m2.7.7.1.1.1.1.3.cmml" xref="S4.Ex52.m2.7.7.1.1.1.1.3"></plus></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex52.m2.9c">\displaystyle\;C_{k,m}\,n^{\frac{1}{p}((k-m-1)p-1-\gamma)}\cdot\tfrac{(2R)^{% \frac{d-1}{p}}}{\gamma+1}\|g\|_{C_{\rm b}^{k+m}({\mathbb{R}}^{d}_{+};X)}\|\phi% \|_{C_{{\rm b}}^{k-m-1}(\mathbb{R}_{+})}</annotation><annotation encoding="application/x-llamapun" id="S4.Ex52.m2.9d">italic_C start_POSTSUBSCRIPT italic_k , italic_m end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG ( ( italic_k - italic_m - 1 ) italic_p - 1 - italic_γ ) end_POSTSUPERSCRIPT ⋅ divide start_ARG ( 2 italic_R ) start_POSTSUPERSCRIPT divide start_ARG italic_d - 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT end_ARG start_ARG italic_γ + 1 end_ARG ∥ italic_g ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_m end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ∥ italic_ϕ ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m - 1 end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex53"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle+n^{k-m-1}\|\phi\|_{C_{{\rm b}}^{k-m}(\mathbb{R}_{+})}\Big{(}\int% _{K_{R}}\int_{0}^{\frac{1}{n}}x_{1}^{\gamma-p}\|\partial^{\widetilde{\alpha}}% \partial_{1}^{m}g(x)\|_{X}^{p}\hskip 2.0pt\mathrm{d}x_{1}\hskip 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end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ( ∫ start_POSTSUBSCRIPT italic_K start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT end_POSTSUBSCRIPT ∫ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_n end_ARG end_POSTSUPERSCRIPT italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_γ - italic_p end_POSTSUPERSCRIPT ∥ ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g ( italic_x ) ∥ start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT roman_d italic_x start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT roman_d over~ start_ARG italic_x end_ARG ) start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S4.Ex54"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S4.Ex54.m1.1"><semantics id="S4.Ex54.m1.1a"><mo id="S4.Ex54.m1.1.1" xref="S4.Ex54.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S4.Ex54.m1.1b"><leq id="S4.Ex54.m1.1.1.cmml" xref="S4.Ex54.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex54.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S4.Ex54.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;C_{k,m,p}\,n^{\frac{1}{p}((k-m-1)p-1-\gamma)}\cdot\tfrac{(2R)^{% \frac{d-1}{p}}}{\gamma+1}\|g\|_{C_{\rm 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xref="S4.Ex54.m2.7.7.2.2.1.1.2.3">𝑑</ci></apply><plus id="S4.Ex54.m2.7.7.2.2.1.1.3.cmml" xref="S4.Ex54.m2.7.7.2.2.1.1.3"></plus></apply><ci id="S4.Ex54.m2.6.6.1.1.cmml" xref="S4.Ex54.m2.6.6.1.1">𝑋</ci></list></apply></apply><apply id="S4.Ex54.m2.11.11.1.1.4.cmml" xref="S4.Ex54.m2.11.11.1.1.4"><csymbol cd="ambiguous" id="S4.Ex54.m2.11.11.1.1.4.1.cmml" xref="S4.Ex54.m2.11.11.1.1.4">subscript</csymbol><apply id="S4.Ex54.m2.11.11.1.1.4.2.1.cmml" xref="S4.Ex54.m2.11.11.1.1.4.2.2"><csymbol cd="latexml" id="S4.Ex54.m2.11.11.1.1.4.2.1.1.cmml" xref="S4.Ex54.m2.11.11.1.1.4.2.2.1">norm</csymbol><ci id="S4.Ex54.m2.10.10.cmml" xref="S4.Ex54.m2.10.10">italic-ϕ</ci></apply><apply id="S4.Ex54.m2.8.8.1.cmml" xref="S4.Ex54.m2.8.8.1"><times id="S4.Ex54.m2.8.8.1.2.cmml" xref="S4.Ex54.m2.8.8.1.2"></times><apply id="S4.Ex54.m2.8.8.1.3.cmml" xref="S4.Ex54.m2.8.8.1.3"><csymbol cd="ambiguous" id="S4.Ex54.m2.8.8.1.3.1.cmml" xref="S4.Ex54.m2.8.8.1.3">superscript</csymbol><apply id="S4.Ex54.m2.8.8.1.3.2.cmml" 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encoding="application/x-tex" id="S4.Ex54.m2.11c">\displaystyle\;C_{k,m,p}\,n^{\frac{1}{p}((k-m-1)p-1-\gamma)}\cdot\tfrac{(2R)^{% \frac{d-1}{p}}}{\gamma+1}\|g\|_{C_{\rm b}^{k+m}({\mathbb{R}}^{d}_{+};X)}\|\phi% \|_{C_{{\rm b}}^{k-m}(\mathbb{R}_{+})},</annotation><annotation encoding="application/x-llamapun" id="S4.Ex54.m2.11d">italic_C start_POSTSUBSCRIPT italic_k , italic_m , italic_p end_POSTSUBSCRIPT italic_n start_POSTSUPERSCRIPT divide start_ARG 1 end_ARG start_ARG italic_p end_ARG ( ( italic_k - italic_m - 1 ) italic_p - 1 - italic_γ ) end_POSTSUPERSCRIPT ⋅ divide start_ARG ( 2 italic_R ) start_POSTSUPERSCRIPT divide start_ARG italic_d - 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT end_ARG start_ARG italic_γ + 1 end_ARG ∥ italic_g ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k + italic_m end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ∥ italic_ϕ ∥ start_POSTSUBSCRIPT italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_m end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.33">where in the last step we have applied Hardy’s inequality (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Lemma 3.1]</cite>, using that <math alttext="\gamma-p>-p-1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.30.m1.1"><semantics id="S4.SS2.SSS2.3.p1.30.m1.1a"><mrow id="S4.SS2.SSS2.3.p1.30.m1.1.1" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.cmml"><mrow id="S4.SS2.SSS2.3.p1.30.m1.1.1.2" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2.cmml"><mi id="S4.SS2.SSS2.3.p1.30.m1.1.1.2.2" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2.2.cmml">γ</mi><mo id="S4.SS2.SSS2.3.p1.30.m1.1.1.2.1" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2.1.cmml">−</mo><mi id="S4.SS2.SSS2.3.p1.30.m1.1.1.2.3" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2.3.cmml">p</mi></mrow><mo id="S4.SS2.SSS2.3.p1.30.m1.1.1.1" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.1.cmml">></mo><mrow id="S4.SS2.SSS2.3.p1.30.m1.1.1.3" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.cmml"><mrow id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.cmml"><mo id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2a" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.cmml">−</mo><mi id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.2" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.2.cmml">p</mi></mrow><mo id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.1" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.1.cmml">−</mo><mn id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.3" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.30.m1.1b"><apply id="S4.SS2.SSS2.3.p1.30.m1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1"><gt id="S4.SS2.SSS2.3.p1.30.m1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.1"></gt><apply id="S4.SS2.SSS2.3.p1.30.m1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2"><minus id="S4.SS2.SSS2.3.p1.30.m1.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2.1"></minus><ci id="S4.SS2.SSS2.3.p1.30.m1.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2.2">𝛾</ci><ci id="S4.SS2.SSS2.3.p1.30.m1.1.1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.2.3">𝑝</ci></apply><apply id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3"><minus id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.1.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.1"></minus><apply id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2"><minus id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.1.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2"></minus><ci id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.2.cmml" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.2.2">𝑝</ci></apply><cn id="S4.SS2.SSS2.3.p1.30.m1.1.1.3.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.30.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.30.m1.1c">\gamma-p>-p-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.30.m1.1d">italic_γ - italic_p > - italic_p - 1</annotation></semantics></math> and that <math alttext="(\partial_{1}^{m}g)|_{\partial\mathbb{R}^{d}_{+}}=0" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.31.m2.2"><semantics id="S4.SS2.SSS2.3.p1.31.m2.2a"><mrow id="S4.SS2.SSS2.3.p1.31.m2.2.2" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.cmml"><msub id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.2.cmml"><mrow id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.2.cmml"><mrow id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.cmml"><msubsup id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.2" lspace="0em" rspace="0em" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.2.cmml">∂</mo><mn id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.3" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.3.cmml">1</mn><mi id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.3.cmml">m</mi></msubsup><mi id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.2.cmml">g</mi></mrow><mo id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.2.1.cmml">|</mo></mrow><mrow id="S4.SS2.SSS2.3.p1.31.m2.1.1.1" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.1" rspace="0em" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.1.cmml">∂</mo><msubsup id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.cmml"><mi id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.2" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.2.cmml">ℝ</mi><mo id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.3" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.3.cmml">+</mo><mi id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.3" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.3.cmml">d</mi></msubsup></mrow></msub><mo id="S4.SS2.SSS2.3.p1.31.m2.2.2.2" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.2.cmml">=</mo><mn id="S4.SS2.SSS2.3.p1.31.m2.2.2.3" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.31.m2.2b"><apply id="S4.SS2.SSS2.3.p1.31.m2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2"><eq id="S4.SS2.SSS2.3.p1.31.m2.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.2"></eq><apply id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1"><csymbol cd="latexml" id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.2">evaluated-at</csymbol><apply id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1"><apply id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1">superscript</csymbol><apply id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1">subscript</csymbol><partialdiff id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.2"></partialdiff><cn id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.1.3">𝑚</ci></apply><ci id="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.1.1.1.1.1.1.2">𝑔</ci></apply><apply id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1"><partialdiff id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.1"></partialdiff><apply id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2">subscript</csymbol><apply id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2"><csymbol cd="ambiguous" id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.1.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2">superscript</csymbol><ci id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.2.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.2">ℝ</ci><ci id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.3.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.2.3">𝑑</ci></apply><plus id="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.3.cmml" xref="S4.SS2.SSS2.3.p1.31.m2.1.1.1.2.3"></plus></apply></apply></apply><cn id="S4.SS2.SSS2.3.p1.31.m2.2.2.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.31.m2.2.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.31.m2.2c">(\partial_{1}^{m}g)|_{\partial\mathbb{R}^{d}_{+}}=0</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.31.m2.2d">( ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_g ) | start_POSTSUBSCRIPT ∂ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math>). Hence, as <math alttext="\gamma>(k-m-1)p-1" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.32.m3.1"><semantics id="S4.SS2.SSS2.3.p1.32.m3.1a"><mrow id="S4.SS2.SSS2.3.p1.32.m3.1.1" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.32.m3.1.1.3" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.3.cmml">γ</mi><mo id="S4.SS2.SSS2.3.p1.32.m3.1.1.2" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.2.cmml">></mo><mrow id="S4.SS2.SSS2.3.p1.32.m3.1.1.1" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.cmml"><mrow id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.cmml"><mrow id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.cmml"><mo id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.2" stretchy="false" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.cmml"><mi id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.2.cmml">k</mi><mo id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.1" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.3.cmml">m</mi><mo id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.1a" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.1.cmml">−</mo><mn id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.4" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.4.cmml">1</mn></mrow><mo id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.3" stretchy="false" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.2" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.2.cmml">⁢</mo><mi id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.3" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.3.cmml">p</mi></mrow><mo id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.2" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.2.cmml">−</mo><mn id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.3" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.32.m3.1b"><apply id="S4.SS2.SSS2.3.p1.32.m3.1.1.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1"><gt id="S4.SS2.SSS2.3.p1.32.m3.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.2"></gt><ci id="S4.SS2.SSS2.3.p1.32.m3.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.3">𝛾</ci><apply id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1"><minus id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.2"></minus><apply id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1"><times id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.2"></times><apply id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1"><minus id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.1.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.1"></minus><ci id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.2.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.2">𝑘</ci><ci id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.3">𝑚</ci><cn id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.4.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.1.1.1.4">1</cn></apply><ci id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.3.cmml" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.1.3">𝑝</ci></apply><cn id="S4.SS2.SSS2.3.p1.32.m3.1.1.1.3.cmml" type="integer" xref="S4.SS2.SSS2.3.p1.32.m3.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.32.m3.1c">\gamma>(k-m-1)p-1</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.32.m3.1d">italic_γ > ( italic_k - italic_m - 1 ) italic_p - 1</annotation></semantics></math>, taking <math alttext="n" class="ltx_Math" display="inline" id="S4.SS2.SSS2.3.p1.33.m4.1"><semantics id="S4.SS2.SSS2.3.p1.33.m4.1a"><mi id="S4.SS2.SSS2.3.p1.33.m4.1.1" xref="S4.SS2.SSS2.3.p1.33.m4.1.1.cmml">n</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.SSS2.3.p1.33.m4.1b"><ci id="S4.SS2.SSS2.3.p1.33.m4.1.1.cmml" xref="S4.SS2.SSS2.3.p1.33.m4.1.1">𝑛</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.SSS2.3.p1.33.m4.1c">n</annotation><annotation encoding="application/x-llamapun" id="S4.SS2.SSS2.3.p1.33.m4.1d">italic_n</annotation></semantics></math> large enough gives with (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.E7" title="In Proof. ‣ 4.2.2. Sobolev spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.7</span></a>) that</p> <table class="ltx_equation ltx_eqn_table" id="S4.Ex55"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|f-g_{n}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}\leq\|f-g\|_{W^{k,p}(% \mathbb{R}^{d}_{+},w_{\gamma};X)}+\|g-g_{n}\|_{W^{k,p}(\mathbb{R}^{d}_{+},w_{% \gamma};X)}<2\varepsilon." class="ltx_Math" display="block" id="S4.Ex55.m1.16"><semantics id="S4.Ex55.m1.16a"><mrow id="S4.Ex55.m1.16.16.1" xref="S4.Ex55.m1.16.16.1.1.cmml"><mrow id="S4.Ex55.m1.16.16.1.1" xref="S4.Ex55.m1.16.16.1.1.cmml"><msub id="S4.Ex55.m1.16.16.1.1.1" xref="S4.Ex55.m1.16.16.1.1.1.cmml"><mrow id="S4.Ex55.m1.16.16.1.1.1.1.1" xref="S4.Ex55.m1.16.16.1.1.1.1.2.cmml"><mo id="S4.Ex55.m1.16.16.1.1.1.1.1.2" stretchy="false" xref="S4.Ex55.m1.16.16.1.1.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex55.m1.16.16.1.1.1.1.1.1" xref="S4.Ex55.m1.16.16.1.1.1.1.1.1.cmml"><mi id="S4.Ex55.m1.16.16.1.1.1.1.1.1.2" xref="S4.Ex55.m1.16.16.1.1.1.1.1.1.2.cmml">f</mi><mo id="S4.Ex55.m1.16.16.1.1.1.1.1.1.1" xref="S4.Ex55.m1.16.16.1.1.1.1.1.1.1.cmml">−</mo><msub id="S4.Ex55.m1.16.16.1.1.1.1.1.1.3" xref="S4.Ex55.m1.16.16.1.1.1.1.1.1.3.cmml"><mi id="S4.Ex55.m1.16.16.1.1.1.1.1.1.3.2" xref="S4.Ex55.m1.16.16.1.1.1.1.1.1.3.2.cmml">g</mi><mi id="S4.Ex55.m1.16.16.1.1.1.1.1.1.3.3" xref="S4.Ex55.m1.16.16.1.1.1.1.1.1.3.3.cmml">n</mi></msub></mrow><mo id="S4.Ex55.m1.16.16.1.1.1.1.1.3" stretchy="false" xref="S4.Ex55.m1.16.16.1.1.1.1.2.1.cmml">‖</mo></mrow><mrow id="S4.Ex55.m1.5.5.5" xref="S4.Ex55.m1.5.5.5.cmml"><msup id="S4.Ex55.m1.5.5.5.7" xref="S4.Ex55.m1.5.5.5.7.cmml"><mi id="S4.Ex55.m1.5.5.5.7.2" xref="S4.Ex55.m1.5.5.5.7.2.cmml">W</mi><mrow id="S4.Ex55.m1.2.2.2.2.2.4" xref="S4.Ex55.m1.2.2.2.2.2.3.cmml"><mi id="S4.Ex55.m1.1.1.1.1.1.1" xref="S4.Ex55.m1.1.1.1.1.1.1.cmml">k</mi><mo id="S4.Ex55.m1.2.2.2.2.2.4.1" xref="S4.Ex55.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S4.Ex55.m1.2.2.2.2.2.2" xref="S4.Ex55.m1.2.2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex55.m1.5.5.5.6" xref="S4.Ex55.m1.5.5.5.6.cmml">⁢</mo><mrow id="S4.Ex55.m1.5.5.5.5.2" xref="S4.Ex55.m1.5.5.5.5.3.cmml"><mo id="S4.Ex55.m1.5.5.5.5.2.3" stretchy="false" xref="S4.Ex55.m1.5.5.5.5.3.cmml">(</mo><msubsup id="S4.Ex55.m1.4.4.4.4.1.1" xref="S4.Ex55.m1.4.4.4.4.1.1.cmml"><mi id="S4.Ex55.m1.4.4.4.4.1.1.2.2" xref="S4.Ex55.m1.4.4.4.4.1.1.2.2.cmml">ℝ</mi><mo id="S4.Ex55.m1.4.4.4.4.1.1.3" xref="S4.Ex55.m1.4.4.4.4.1.1.3.cmml">+</mo><mi id="S4.Ex55.m1.4.4.4.4.1.1.2.3" xref="S4.Ex55.m1.4.4.4.4.1.1.2.3.cmml">d</mi></msubsup><mo id="S4.Ex55.m1.5.5.5.5.2.4" xref="S4.Ex55.m1.5.5.5.5.3.cmml">,</mo><msub id="S4.Ex55.m1.5.5.5.5.2.2" xref="S4.Ex55.m1.5.5.5.5.2.2.cmml"><mi id="S4.Ex55.m1.5.5.5.5.2.2.2" xref="S4.Ex55.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S4.Ex55.m1.5.5.5.5.2.2.3" xref="S4.Ex55.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S4.Ex55.m1.5.5.5.5.2.5" xref="S4.Ex55.m1.5.5.5.5.3.cmml">;</mo><mi id="S4.Ex55.m1.3.3.3.3" xref="S4.Ex55.m1.3.3.3.3.cmml">X</mi><mo id="S4.Ex55.m1.5.5.5.5.2.6" stretchy="false" xref="S4.Ex55.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><mo id="S4.Ex55.m1.16.16.1.1.5" xref="S4.Ex55.m1.16.16.1.1.5.cmml">≤</mo><mrow id="S4.Ex55.m1.16.16.1.1.3" xref="S4.Ex55.m1.16.16.1.1.3.cmml"><msub id="S4.Ex55.m1.16.16.1.1.2.1" xref="S4.Ex55.m1.16.16.1.1.2.1.cmml"><mrow id="S4.Ex55.m1.16.16.1.1.2.1.1.1" xref="S4.Ex55.m1.16.16.1.1.2.1.1.2.cmml"><mo id="S4.Ex55.m1.16.16.1.1.2.1.1.1.2" stretchy="false" xref="S4.Ex55.m1.16.16.1.1.2.1.1.2.1.cmml">‖</mo><mrow id="S4.Ex55.m1.16.16.1.1.2.1.1.1.1" xref="S4.Ex55.m1.16.16.1.1.2.1.1.1.1.cmml"><mi id="S4.Ex55.m1.16.16.1.1.2.1.1.1.1.2" xref="S4.Ex55.m1.16.16.1.1.2.1.1.1.1.2.cmml">f</mi><mo id="S4.Ex55.m1.16.16.1.1.2.1.1.1.1.1" xref="S4.Ex55.m1.16.16.1.1.2.1.1.1.1.1.cmml">−</mo><mi id="S4.Ex55.m1.16.16.1.1.2.1.1.1.1.3" xref="S4.Ex55.m1.16.16.1.1.2.1.1.1.1.3.cmml">g</mi></mrow><mo id="S4.Ex55.m1.16.16.1.1.2.1.1.1.3" stretchy="false" xref="S4.Ex55.m1.16.16.1.1.2.1.1.2.1.cmml">‖</mo></mrow><mrow id="S4.Ex55.m1.10.10.5" xref="S4.Ex55.m1.10.10.5.cmml"><msup id="S4.Ex55.m1.10.10.5.7" xref="S4.Ex55.m1.10.10.5.7.cmml"><mi id="S4.Ex55.m1.10.10.5.7.2" xref="S4.Ex55.m1.10.10.5.7.2.cmml">W</mi><mrow id="S4.Ex55.m1.7.7.2.2.2.4" xref="S4.Ex55.m1.7.7.2.2.2.3.cmml"><mi id="S4.Ex55.m1.6.6.1.1.1.1" xref="S4.Ex55.m1.6.6.1.1.1.1.cmml">k</mi><mo id="S4.Ex55.m1.7.7.2.2.2.4.1" xref="S4.Ex55.m1.7.7.2.2.2.3.cmml">,</mo><mi id="S4.Ex55.m1.7.7.2.2.2.2" xref="S4.Ex55.m1.7.7.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S4.Ex55.m1.10.10.5.6" xref="S4.Ex55.m1.10.10.5.6.cmml">⁢</mo><mrow id="S4.Ex55.m1.10.10.5.5.2" xref="S4.Ex55.m1.10.10.5.5.3.cmml"><mo id="S4.Ex55.m1.10.10.5.5.2.3" stretchy="false" 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encoding="application/x-llamapun" id="S4.Ex55.m1.16d">∥ italic_f - italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ ∥ italic_f - italic_g ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT + ∥ italic_g - italic_g start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT < 2 italic_ε .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S4.SS2.SSS2.3.p1.35">This completes the proof. ∎</p> </div> </div> </section> </section> </section> <section class="ltx_section" id="S5"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">5. </span>Trace theorem for boundary operators</h2> <div class="ltx_para" id="S5.p1"> <p class="ltx_p" id="S5.p1.1">In this section, we study differential operators on the boundary of the half-space and we prove a trace theorem for systems of normal boundary operators on weighted Bessel potential and Sobolev spaces. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S5.p2"> <p class="ltx_p" id="S5.p2.1">We start with the definition of a boundary operator which is in the same spirit as <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Section VIII.2]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.1.1.1">Definition 5.1</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.2.2"> </span>(Boundary operators)<span class="ltx_text ltx_font_bold" id="S5.Thmtheorem1.3.3">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem1.p1"> <p class="ltx_p" id="S5.Thmtheorem1.p1.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.1.m1.2"><semantics id="S5.Thmtheorem1.p1.1.m1.2a"><mrow id="S5.Thmtheorem1.p1.1.m1.2.3" xref="S5.Thmtheorem1.p1.1.m1.2.3.cmml"><mi id="S5.Thmtheorem1.p1.1.m1.2.3.2" xref="S5.Thmtheorem1.p1.1.m1.2.3.2.cmml">p</mi><mo id="S5.Thmtheorem1.p1.1.m1.2.3.1" 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xref="S5.Thmtheorem1.p1.3.m3.2.2.1.1.1.1.2">1</cn></apply><infinity id="S5.Thmtheorem1.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.1.1"></infinity></interval><apply id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.3.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2"><csymbol cd="latexml" id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.3.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.3">conditional-set</csymbol><apply id="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1"><minus id="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.1"></minus><apply id="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2"><times id="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2.1"></times><ci id="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2.3.cmml" xref="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.m3.3.3.2.2.1.1.3">1</cn></apply><apply id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2"><in id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.1"></in><ci id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.2">𝑗</ci><apply id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3.1.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3">subscript</csymbol><ci id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3.2.cmml" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.3.m3.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.3.m3.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.3.m3.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.4.m4.1"><semantics id="S5.Thmtheorem1.p1.4.m4.1a"><mrow id="S5.Thmtheorem1.p1.4.m4.1.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.cmml"><mi id="S5.Thmtheorem1.p1.4.m4.1.1.2" xref="S5.Thmtheorem1.p1.4.m4.1.1.2.cmml">m</mi><mo id="S5.Thmtheorem1.p1.4.m4.1.1.1" xref="S5.Thmtheorem1.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem1.p1.4.m4.1.1.3" xref="S5.Thmtheorem1.p1.4.m4.1.1.3.cmml"><mi id="S5.Thmtheorem1.p1.4.m4.1.1.3.2" xref="S5.Thmtheorem1.p1.4.m4.1.1.3.2.cmml">ℕ</mi><mn id="S5.Thmtheorem1.p1.4.m4.1.1.3.3" xref="S5.Thmtheorem1.p1.4.m4.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.4.m4.1b"><apply id="S5.Thmtheorem1.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1"><in id="S5.Thmtheorem1.p1.4.m4.1.1.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.1"></in><ci id="S5.Thmtheorem1.p1.4.m4.1.1.2.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.2">𝑚</ci><apply id="S5.Thmtheorem1.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem1.p1.4.m4.1.1.3.1.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem1.p1.4.m4.1.1.3.2.cmml" xref="S5.Thmtheorem1.p1.4.m4.1.1.3.2">ℕ</ci><cn id="S5.Thmtheorem1.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem1.p1.4.m4.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.4.m4.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.4.m4.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and let <math alttext="X,Y" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.5.m5.2"><semantics id="S5.Thmtheorem1.p1.5.m5.2a"><mrow id="S5.Thmtheorem1.p1.5.m5.2.3.2" xref="S5.Thmtheorem1.p1.5.m5.2.3.1.cmml"><mi id="S5.Thmtheorem1.p1.5.m5.1.1" xref="S5.Thmtheorem1.p1.5.m5.1.1.cmml">X</mi><mo id="S5.Thmtheorem1.p1.5.m5.2.3.2.1" xref="S5.Thmtheorem1.p1.5.m5.2.3.1.cmml">,</mo><mi id="S5.Thmtheorem1.p1.5.m5.2.2" xref="S5.Thmtheorem1.p1.5.m5.2.2.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.5.m5.2b"><list id="S5.Thmtheorem1.p1.5.m5.2.3.1.cmml" xref="S5.Thmtheorem1.p1.5.m5.2.3.2"><ci id="S5.Thmtheorem1.p1.5.m5.1.1.cmml" xref="S5.Thmtheorem1.p1.5.m5.1.1">𝑋</ci><ci id="S5.Thmtheorem1.p1.5.m5.2.2.cmml" xref="S5.Thmtheorem1.p1.5.m5.2.2">𝑌</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.5.m5.2c">X,Y</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.5.m5.2d">italic_X , italic_Y</annotation></semantics></math> be Banach spaces. Then the <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem1.p1.6.1">boundary operator of order <math alttext="m" class="ltx_Math" display="inline" id="S5.Thmtheorem1.p1.6.1.m1.1"><semantics id="S5.Thmtheorem1.p1.6.1.m1.1a"><mi id="S5.Thmtheorem1.p1.6.1.m1.1.1" xref="S5.Thmtheorem1.p1.6.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem1.p1.6.1.m1.1b"><ci id="S5.Thmtheorem1.p1.6.1.m1.1.1.cmml" xref="S5.Thmtheorem1.p1.6.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem1.p1.6.1.m1.1c">m</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem1.p1.6.1.m1.1d">italic_m</annotation></semantics></math></em>, given by</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}:=\sum_{|\alpha|\leq m}b_{\alpha}\operatorname{Tr}\partial^{\alpha}% \qquad\text{with }b_{\alpha}\neq 0\text{ for some }|\alpha|=m," class="ltx_Math" display="block" id="S5.Ex1.m1.3"><semantics id="S5.Ex1.m1.3a"><mrow id="S5.Ex1.m1.3.3.1"><mrow id="S5.Ex1.m1.3.3.1.1.2" xref="S5.Ex1.m1.3.3.1.1.3.cmml"><mrow id="S5.Ex1.m1.3.3.1.1.1.1" xref="S5.Ex1.m1.3.3.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex1.m1.3.3.1.1.1.1.2" xref="S5.Ex1.m1.3.3.1.1.1.1.2.cmml">ℬ</mi><mo id="S5.Ex1.m1.3.3.1.1.1.1.1" lspace="0.278em" rspace="0.111em" xref="S5.Ex1.m1.3.3.1.1.1.1.1.cmml">:=</mo><mrow id="S5.Ex1.m1.3.3.1.1.1.1.3" xref="S5.Ex1.m1.3.3.1.1.1.1.3.cmml"><munder id="S5.Ex1.m1.3.3.1.1.1.1.3.1" xref="S5.Ex1.m1.3.3.1.1.1.1.3.1.cmml"><mo id="S5.Ex1.m1.3.3.1.1.1.1.3.1.2" movablelimits="false" xref="S5.Ex1.m1.3.3.1.1.1.1.3.1.2.cmml">∑</mo><mrow id="S5.Ex1.m1.1.1.1" xref="S5.Ex1.m1.1.1.1.cmml"><mrow id="S5.Ex1.m1.1.1.1.3.2" xref="S5.Ex1.m1.1.1.1.3.1.cmml"><mo id="S5.Ex1.m1.1.1.1.3.2.1" stretchy="false" xref="S5.Ex1.m1.1.1.1.3.1.1.cmml">|</mo><mi id="S5.Ex1.m1.1.1.1.1" xref="S5.Ex1.m1.1.1.1.1.cmml">α</mi><mo id="S5.Ex1.m1.1.1.1.3.2.2" stretchy="false" xref="S5.Ex1.m1.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S5.Ex1.m1.1.1.1.2" xref="S5.Ex1.m1.1.1.1.2.cmml">≤</mo><mi id="S5.Ex1.m1.1.1.1.4" xref="S5.Ex1.m1.1.1.1.4.cmml">m</mi></mrow></munder><mrow id="S5.Ex1.m1.3.3.1.1.1.1.3.2" xref="S5.Ex1.m1.3.3.1.1.1.1.3.2.cmml"><msub id="S5.Ex1.m1.3.3.1.1.1.1.3.2.2" xref="S5.Ex1.m1.3.3.1.1.1.1.3.2.2.cmml"><mi id="S5.Ex1.m1.3.3.1.1.1.1.3.2.2.2" xref="S5.Ex1.m1.3.3.1.1.1.1.3.2.2.2.cmml">b</mi><mi id="S5.Ex1.m1.3.3.1.1.1.1.3.2.2.3" xref="S5.Ex1.m1.3.3.1.1.1.1.3.2.2.3.cmml">α</mi></msub><mo id="S5.Ex1.m1.3.3.1.1.1.1.3.2.1" lspace="0.167em" xref="S5.Ex1.m1.3.3.1.1.1.1.3.2.1.cmml">⁢</mo><mi id="S5.Ex1.m1.3.3.1.1.1.1.3.2.3" xref="S5.Ex1.m1.3.3.1.1.1.1.3.2.3.cmml">Tr</mi><mo id="S5.Ex1.m1.3.3.1.1.1.1.3.2.1a" lspace="0.167em" xref="S5.Ex1.m1.3.3.1.1.1.1.3.2.1.cmml">⁢</mo><msup id="S5.Ex1.m1.3.3.1.1.1.1.3.2.4" 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id="S5.I1.i1.p1.1.m1.1.1.3.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3"><plus id="S5.I1.i1.p1.1.m1.1.1.3.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.1"></plus><ci id="S5.I1.i1.p1.1.m1.1.1.3.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.2">𝑚</ci><apply id="S5.I1.i1.p1.1.m1.1.1.3.3.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.3"><divide id="S5.I1.i1.p1.1.m1.1.1.3.3.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.3"></divide><apply id="S5.I1.i1.p1.1.m1.1.1.3.3.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.3.2"><plus id="S5.I1.i1.p1.1.m1.1.1.3.3.2.1.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.3.2.1"></plus><ci id="S5.I1.i1.p1.1.m1.1.1.3.3.2.2.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S5.I1.i1.p1.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S5.I1.i1.p1.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S5.I1.i1.p1.1.m1.1.1.3.3.3.cmml" xref="S5.I1.i1.p1.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i1.p1.1.m1.1c">s>m+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i1.p1.1.m1.1d">italic_s > italic_m + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S5.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S5.I1.i2.p1"> <p class="ltx_p" id="S5.I1.i2.p1.4">for all <math alttext="|\alpha|\leq m" class="ltx_Math" display="inline" id="S5.I1.i2.p1.1.m1.1"><semantics id="S5.I1.i2.p1.1.m1.1a"><mrow id="S5.I1.i2.p1.1.m1.1.2" xref="S5.I1.i2.p1.1.m1.1.2.cmml"><mrow id="S5.I1.i2.p1.1.m1.1.2.2.2" xref="S5.I1.i2.p1.1.m1.1.2.2.1.cmml"><mo id="S5.I1.i2.p1.1.m1.1.2.2.2.1" stretchy="false" xref="S5.I1.i2.p1.1.m1.1.2.2.1.1.cmml">|</mo><mi id="S5.I1.i2.p1.1.m1.1.1" xref="S5.I1.i2.p1.1.m1.1.1.cmml">α</mi><mo id="S5.I1.i2.p1.1.m1.1.2.2.2.2" stretchy="false" xref="S5.I1.i2.p1.1.m1.1.2.2.1.1.cmml">|</mo></mrow><mo id="S5.I1.i2.p1.1.m1.1.2.1" xref="S5.I1.i2.p1.1.m1.1.2.1.cmml">≤</mo><mi id="S5.I1.i2.p1.1.m1.1.2.3" xref="S5.I1.i2.p1.1.m1.1.2.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.1.m1.1b"><apply id="S5.I1.i2.p1.1.m1.1.2.cmml" xref="S5.I1.i2.p1.1.m1.1.2"><leq id="S5.I1.i2.p1.1.m1.1.2.1.cmml" xref="S5.I1.i2.p1.1.m1.1.2.1"></leq><apply id="S5.I1.i2.p1.1.m1.1.2.2.1.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2.2"><abs id="S5.I1.i2.p1.1.m1.1.2.2.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.2.2.2.1"></abs><ci id="S5.I1.i2.p1.1.m1.1.1.cmml" xref="S5.I1.i2.p1.1.m1.1.1">𝛼</ci></apply><ci id="S5.I1.i2.p1.1.m1.1.2.3.cmml" xref="S5.I1.i2.p1.1.m1.1.2.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.1.m1.1c">|\alpha|\leq m</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.1.m1.1d">| italic_α | ≤ italic_m</annotation></semantics></math> we have <math alttext="b_{\alpha}\in C^{\ell}_{{\rm b}}(\mathbb{R}^{d-1};\mathcal{L}(X,Y))" class="ltx_Math" display="inline" id="S5.I1.i2.p1.2.m2.4"><semantics id="S5.I1.i2.p1.2.m2.4a"><mrow id="S5.I1.i2.p1.2.m2.4.4" xref="S5.I1.i2.p1.2.m2.4.4.cmml"><msub id="S5.I1.i2.p1.2.m2.4.4.4" xref="S5.I1.i2.p1.2.m2.4.4.4.cmml"><mi id="S5.I1.i2.p1.2.m2.4.4.4.2" xref="S5.I1.i2.p1.2.m2.4.4.4.2.cmml">b</mi><mi id="S5.I1.i2.p1.2.m2.4.4.4.3" xref="S5.I1.i2.p1.2.m2.4.4.4.3.cmml">α</mi></msub><mo id="S5.I1.i2.p1.2.m2.4.4.3" xref="S5.I1.i2.p1.2.m2.4.4.3.cmml">∈</mo><mrow id="S5.I1.i2.p1.2.m2.4.4.2" xref="S5.I1.i2.p1.2.m2.4.4.2.cmml"><msubsup id="S5.I1.i2.p1.2.m2.4.4.2.4" xref="S5.I1.i2.p1.2.m2.4.4.2.4.cmml"><mi id="S5.I1.i2.p1.2.m2.4.4.2.4.2.2" xref="S5.I1.i2.p1.2.m2.4.4.2.4.2.2.cmml">C</mi><mi id="S5.I1.i2.p1.2.m2.4.4.2.4.3" mathvariant="normal" xref="S5.I1.i2.p1.2.m2.4.4.2.4.3.cmml">b</mi><mi id="S5.I1.i2.p1.2.m2.4.4.2.4.2.3" mathvariant="normal" xref="S5.I1.i2.p1.2.m2.4.4.2.4.2.3.cmml">ℓ</mi></msubsup><mo id="S5.I1.i2.p1.2.m2.4.4.2.3" xref="S5.I1.i2.p1.2.m2.4.4.2.3.cmml">⁢</mo><mrow id="S5.I1.i2.p1.2.m2.4.4.2.2.2" xref="S5.I1.i2.p1.2.m2.4.4.2.2.3.cmml"><mo id="S5.I1.i2.p1.2.m2.4.4.2.2.2.3" stretchy="false" xref="S5.I1.i2.p1.2.m2.4.4.2.2.3.cmml">(</mo><msup id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.cmml"><mi id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.2" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.2.cmml">ℝ</mi><mrow id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.cmml"><mi id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.2" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.2.cmml">d</mi><mo id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.1" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.1.cmml">−</mo><mn id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.3" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S5.I1.i2.p1.2.m2.4.4.2.2.2.4" xref="S5.I1.i2.p1.2.m2.4.4.2.2.3.cmml">;</mo><mrow id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.2" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.2.cmml">ℒ</mi><mo id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.1" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.1.cmml">⁢</mo><mrow id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.2" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml"><mo id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.2.1" stretchy="false" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml">(</mo><mi id="S5.I1.i2.p1.2.m2.1.1" xref="S5.I1.i2.p1.2.m2.1.1.cmml">X</mi><mo id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.2.2" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml">,</mo><mi id="S5.I1.i2.p1.2.m2.2.2" xref="S5.I1.i2.p1.2.m2.2.2.cmml">Y</mi><mo id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.2.3" stretchy="false" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.I1.i2.p1.2.m2.4.4.2.2.2.5" stretchy="false" xref="S5.I1.i2.p1.2.m2.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.2.m2.4b"><apply id="S5.I1.i2.p1.2.m2.4.4.cmml" xref="S5.I1.i2.p1.2.m2.4.4"><in id="S5.I1.i2.p1.2.m2.4.4.3.cmml" xref="S5.I1.i2.p1.2.m2.4.4.3"></in><apply id="S5.I1.i2.p1.2.m2.4.4.4.cmml" xref="S5.I1.i2.p1.2.m2.4.4.4"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.4.4.4.1.cmml" xref="S5.I1.i2.p1.2.m2.4.4.4">subscript</csymbol><ci id="S5.I1.i2.p1.2.m2.4.4.4.2.cmml" xref="S5.I1.i2.p1.2.m2.4.4.4.2">𝑏</ci><ci id="S5.I1.i2.p1.2.m2.4.4.4.3.cmml" xref="S5.I1.i2.p1.2.m2.4.4.4.3">𝛼</ci></apply><apply id="S5.I1.i2.p1.2.m2.4.4.2.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2"><times id="S5.I1.i2.p1.2.m2.4.4.2.3.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.3"></times><apply id="S5.I1.i2.p1.2.m2.4.4.2.4.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.4"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.4.4.2.4.1.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.4">subscript</csymbol><apply id="S5.I1.i2.p1.2.m2.4.4.2.4.2.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.4"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.4.4.2.4.2.1.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.4">superscript</csymbol><ci id="S5.I1.i2.p1.2.m2.4.4.2.4.2.2.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.4.2.2">𝐶</ci><ci id="S5.I1.i2.p1.2.m2.4.4.2.4.2.3.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.4.2.3">ℓ</ci></apply><ci id="S5.I1.i2.p1.2.m2.4.4.2.4.3.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.4.3">b</ci></apply><list id="S5.I1.i2.p1.2.m2.4.4.2.2.3.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2"><apply id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.cmml" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1"><csymbol cd="ambiguous" id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.1.cmml" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1">superscript</csymbol><ci id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.2.cmml" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.2">ℝ</ci><apply id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.cmml" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3"><minus id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.1.cmml" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.1"></minus><ci id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.2.cmml" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.2">𝑑</ci><cn id="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.3.cmml" type="integer" xref="S5.I1.i2.p1.2.m2.3.3.1.1.1.1.3.3">1</cn></apply></apply><apply id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2"><times id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.1.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.1"></times><ci id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.2.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.2">ℒ</ci><interval closure="open" id="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml" xref="S5.I1.i2.p1.2.m2.4.4.2.2.2.2.3.2"><ci id="S5.I1.i2.p1.2.m2.1.1.cmml" xref="S5.I1.i2.p1.2.m2.1.1">𝑋</ci><ci id="S5.I1.i2.p1.2.m2.2.2.cmml" xref="S5.I1.i2.p1.2.m2.2.2">𝑌</ci></interval></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.2.m2.4c">b_{\alpha}\in C^{\ell}_{{\rm b}}(\mathbb{R}^{d-1};\mathcal{L}(X,Y))</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.2.m2.4d">italic_b start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT ∈ italic_C start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; caligraphic_L ( italic_X , italic_Y ) )</annotation></semantics></math> for some <math alttext="\ell\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.3.m3.1"><semantics id="S5.I1.i2.p1.3.m3.1a"><mrow id="S5.I1.i2.p1.3.m3.1.1" xref="S5.I1.i2.p1.3.m3.1.1.cmml"><mi id="S5.I1.i2.p1.3.m3.1.1.2" mathvariant="normal" xref="S5.I1.i2.p1.3.m3.1.1.2.cmml">ℓ</mi><mo id="S5.I1.i2.p1.3.m3.1.1.1" xref="S5.I1.i2.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S5.I1.i2.p1.3.m3.1.1.3" xref="S5.I1.i2.p1.3.m3.1.1.3.cmml"><mi id="S5.I1.i2.p1.3.m3.1.1.3.2" xref="S5.I1.i2.p1.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S5.I1.i2.p1.3.m3.1.1.3.3" xref="S5.I1.i2.p1.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.3.m3.1b"><apply id="S5.I1.i2.p1.3.m3.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1"><in id="S5.I1.i2.p1.3.m3.1.1.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.1"></in><ci id="S5.I1.i2.p1.3.m3.1.1.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.2">ℓ</ci><apply id="S5.I1.i2.p1.3.m3.1.1.3.cmml" xref="S5.I1.i2.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.I1.i2.p1.3.m3.1.1.3.1.cmml" xref="S5.I1.i2.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.I1.i2.p1.3.m3.1.1.3.2.cmml" xref="S5.I1.i2.p1.3.m3.1.1.3.2">ℕ</ci><cn id="S5.I1.i2.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S5.I1.i2.p1.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.3.m3.1c">\ell\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.3.m3.1d">roman_ℓ ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="\ell>s-m-\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S5.I1.i2.p1.4.m4.1"><semantics id="S5.I1.i2.p1.4.m4.1a"><mrow id="S5.I1.i2.p1.4.m4.1.1" xref="S5.I1.i2.p1.4.m4.1.1.cmml"><mi id="S5.I1.i2.p1.4.m4.1.1.2" mathvariant="normal" xref="S5.I1.i2.p1.4.m4.1.1.2.cmml">ℓ</mi><mo id="S5.I1.i2.p1.4.m4.1.1.1" xref="S5.I1.i2.p1.4.m4.1.1.1.cmml">></mo><mrow id="S5.I1.i2.p1.4.m4.1.1.3" xref="S5.I1.i2.p1.4.m4.1.1.3.cmml"><mi id="S5.I1.i2.p1.4.m4.1.1.3.2" xref="S5.I1.i2.p1.4.m4.1.1.3.2.cmml">s</mi><mo id="S5.I1.i2.p1.4.m4.1.1.3.1" xref="S5.I1.i2.p1.4.m4.1.1.3.1.cmml">−</mo><mi id="S5.I1.i2.p1.4.m4.1.1.3.3" xref="S5.I1.i2.p1.4.m4.1.1.3.3.cmml">m</mi><mo id="S5.I1.i2.p1.4.m4.1.1.3.1a" xref="S5.I1.i2.p1.4.m4.1.1.3.1.cmml">−</mo><mfrac id="S5.I1.i2.p1.4.m4.1.1.3.4" xref="S5.I1.i2.p1.4.m4.1.1.3.4.cmml"><mrow id="S5.I1.i2.p1.4.m4.1.1.3.4.2" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2.cmml"><mi id="S5.I1.i2.p1.4.m4.1.1.3.4.2.2" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2.2.cmml">γ</mi><mo id="S5.I1.i2.p1.4.m4.1.1.3.4.2.1" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2.1.cmml">+</mo><mn id="S5.I1.i2.p1.4.m4.1.1.3.4.2.3" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2.3.cmml">1</mn></mrow><mi id="S5.I1.i2.p1.4.m4.1.1.3.4.3" xref="S5.I1.i2.p1.4.m4.1.1.3.4.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I1.i2.p1.4.m4.1b"><apply id="S5.I1.i2.p1.4.m4.1.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1"><gt id="S5.I1.i2.p1.4.m4.1.1.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1.1"></gt><ci id="S5.I1.i2.p1.4.m4.1.1.2.cmml" xref="S5.I1.i2.p1.4.m4.1.1.2">ℓ</ci><apply id="S5.I1.i2.p1.4.m4.1.1.3.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3"><minus id="S5.I1.i2.p1.4.m4.1.1.3.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.1"></minus><ci id="S5.I1.i2.p1.4.m4.1.1.3.2.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.2">𝑠</ci><ci id="S5.I1.i2.p1.4.m4.1.1.3.3.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.3">𝑚</ci><apply id="S5.I1.i2.p1.4.m4.1.1.3.4.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.4"><divide id="S5.I1.i2.p1.4.m4.1.1.3.4.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.4"></divide><apply id="S5.I1.i2.p1.4.m4.1.1.3.4.2.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2"><plus id="S5.I1.i2.p1.4.m4.1.1.3.4.2.1.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2.1"></plus><ci id="S5.I1.i2.p1.4.m4.1.1.3.4.2.2.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2.2">𝛾</ci><cn id="S5.I1.i2.p1.4.m4.1.1.3.4.2.3.cmml" type="integer" xref="S5.I1.i2.p1.4.m4.1.1.3.4.2.3">1</cn></apply><ci id="S5.I1.i2.p1.4.m4.1.1.3.4.3.cmml" xref="S5.I1.i2.p1.4.m4.1.1.3.4.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I1.i2.p1.4.m4.1c">\ell>s-m-\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S5.I1.i2.p1.4.m4.1d">roman_ℓ > italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>.</p> </div> </li> </ol> </div> </div> <div class="ltx_para" id="S5.p3"> <p class="ltx_p" id="S5.p3.3">Note that for <math alttext="\alpha=(\alpha_{1},\widetilde{\alpha})\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{d% -1}" class="ltx_Math" display="inline" id="S5.p3.1.m1.2"><semantics id="S5.p3.1.m1.2a"><mrow id="S5.p3.1.m1.2.2" xref="S5.p3.1.m1.2.2.cmml"><mi id="S5.p3.1.m1.2.2.3" xref="S5.p3.1.m1.2.2.3.cmml">α</mi><mo id="S5.p3.1.m1.2.2.4" xref="S5.p3.1.m1.2.2.4.cmml">=</mo><mrow id="S5.p3.1.m1.2.2.1.1" xref="S5.p3.1.m1.2.2.1.2.cmml"><mo id="S5.p3.1.m1.2.2.1.1.2" stretchy="false" xref="S5.p3.1.m1.2.2.1.2.cmml">(</mo><msub id="S5.p3.1.m1.2.2.1.1.1" xref="S5.p3.1.m1.2.2.1.1.1.cmml"><mi id="S5.p3.1.m1.2.2.1.1.1.2" xref="S5.p3.1.m1.2.2.1.1.1.2.cmml">α</mi><mn id="S5.p3.1.m1.2.2.1.1.1.3" xref="S5.p3.1.m1.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S5.p3.1.m1.2.2.1.1.3" xref="S5.p3.1.m1.2.2.1.2.cmml">,</mo><mover accent="true" id="S5.p3.1.m1.1.1" xref="S5.p3.1.m1.1.1.cmml"><mi id="S5.p3.1.m1.1.1.2" xref="S5.p3.1.m1.1.1.2.cmml">α</mi><mo id="S5.p3.1.m1.1.1.1" xref="S5.p3.1.m1.1.1.1.cmml">~</mo></mover><mo id="S5.p3.1.m1.2.2.1.1.4" stretchy="false" xref="S5.p3.1.m1.2.2.1.2.cmml">)</mo></mrow><mo id="S5.p3.1.m1.2.2.5" xref="S5.p3.1.m1.2.2.5.cmml">∈</mo><mrow id="S5.p3.1.m1.2.2.6" xref="S5.p3.1.m1.2.2.6.cmml"><msub id="S5.p3.1.m1.2.2.6.2" xref="S5.p3.1.m1.2.2.6.2.cmml"><mi id="S5.p3.1.m1.2.2.6.2.2" xref="S5.p3.1.m1.2.2.6.2.2.cmml">ℕ</mi><mn id="S5.p3.1.m1.2.2.6.2.3" xref="S5.p3.1.m1.2.2.6.2.3.cmml">0</mn></msub><mo id="S5.p3.1.m1.2.2.6.1" lspace="0.222em" rspace="0.222em" xref="S5.p3.1.m1.2.2.6.1.cmml">×</mo><msubsup id="S5.p3.1.m1.2.2.6.3" xref="S5.p3.1.m1.2.2.6.3.cmml"><mi id="S5.p3.1.m1.2.2.6.3.2.2" xref="S5.p3.1.m1.2.2.6.3.2.2.cmml">ℕ</mi><mn id="S5.p3.1.m1.2.2.6.3.2.3" xref="S5.p3.1.m1.2.2.6.3.2.3.cmml">0</mn><mrow id="S5.p3.1.m1.2.2.6.3.3" xref="S5.p3.1.m1.2.2.6.3.3.cmml"><mi id="S5.p3.1.m1.2.2.6.3.3.2" xref="S5.p3.1.m1.2.2.6.3.3.2.cmml">d</mi><mo id="S5.p3.1.m1.2.2.6.3.3.1" xref="S5.p3.1.m1.2.2.6.3.3.1.cmml">−</mo><mn id="S5.p3.1.m1.2.2.6.3.3.3" xref="S5.p3.1.m1.2.2.6.3.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.1.m1.2b"><apply id="S5.p3.1.m1.2.2.cmml" xref="S5.p3.1.m1.2.2"><and id="S5.p3.1.m1.2.2a.cmml" xref="S5.p3.1.m1.2.2"></and><apply id="S5.p3.1.m1.2.2b.cmml" xref="S5.p3.1.m1.2.2"><eq id="S5.p3.1.m1.2.2.4.cmml" xref="S5.p3.1.m1.2.2.4"></eq><ci id="S5.p3.1.m1.2.2.3.cmml" xref="S5.p3.1.m1.2.2.3">𝛼</ci><interval closure="open" id="S5.p3.1.m1.2.2.1.2.cmml" xref="S5.p3.1.m1.2.2.1.1"><apply id="S5.p3.1.m1.2.2.1.1.1.cmml" xref="S5.p3.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.p3.1.m1.2.2.1.1.1.1.cmml" xref="S5.p3.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S5.p3.1.m1.2.2.1.1.1.2.cmml" xref="S5.p3.1.m1.2.2.1.1.1.2">𝛼</ci><cn id="S5.p3.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S5.p3.1.m1.2.2.1.1.1.3">1</cn></apply><apply id="S5.p3.1.m1.1.1.cmml" xref="S5.p3.1.m1.1.1"><ci id="S5.p3.1.m1.1.1.1.cmml" xref="S5.p3.1.m1.1.1.1">~</ci><ci id="S5.p3.1.m1.1.1.2.cmml" xref="S5.p3.1.m1.1.1.2">𝛼</ci></apply></interval></apply><apply id="S5.p3.1.m1.2.2c.cmml" xref="S5.p3.1.m1.2.2"><in id="S5.p3.1.m1.2.2.5.cmml" xref="S5.p3.1.m1.2.2.5"></in><share href="https://arxiv.org/html/2503.14636v1#S5.p3.1.m1.2.2.1.cmml" id="S5.p3.1.m1.2.2d.cmml" xref="S5.p3.1.m1.2.2"></share><apply id="S5.p3.1.m1.2.2.6.cmml" xref="S5.p3.1.m1.2.2.6"><times id="S5.p3.1.m1.2.2.6.1.cmml" xref="S5.p3.1.m1.2.2.6.1"></times><apply id="S5.p3.1.m1.2.2.6.2.cmml" xref="S5.p3.1.m1.2.2.6.2"><csymbol cd="ambiguous" id="S5.p3.1.m1.2.2.6.2.1.cmml" xref="S5.p3.1.m1.2.2.6.2">subscript</csymbol><ci id="S5.p3.1.m1.2.2.6.2.2.cmml" xref="S5.p3.1.m1.2.2.6.2.2">ℕ</ci><cn id="S5.p3.1.m1.2.2.6.2.3.cmml" type="integer" xref="S5.p3.1.m1.2.2.6.2.3">0</cn></apply><apply id="S5.p3.1.m1.2.2.6.3.cmml" xref="S5.p3.1.m1.2.2.6.3"><csymbol cd="ambiguous" id="S5.p3.1.m1.2.2.6.3.1.cmml" xref="S5.p3.1.m1.2.2.6.3">superscript</csymbol><apply id="S5.p3.1.m1.2.2.6.3.2.cmml" xref="S5.p3.1.m1.2.2.6.3"><csymbol cd="ambiguous" id="S5.p3.1.m1.2.2.6.3.2.1.cmml" xref="S5.p3.1.m1.2.2.6.3">subscript</csymbol><ci id="S5.p3.1.m1.2.2.6.3.2.2.cmml" xref="S5.p3.1.m1.2.2.6.3.2.2">ℕ</ci><cn id="S5.p3.1.m1.2.2.6.3.2.3.cmml" type="integer" xref="S5.p3.1.m1.2.2.6.3.2.3">0</cn></apply><apply id="S5.p3.1.m1.2.2.6.3.3.cmml" xref="S5.p3.1.m1.2.2.6.3.3"><minus id="S5.p3.1.m1.2.2.6.3.3.1.cmml" xref="S5.p3.1.m1.2.2.6.3.3.1"></minus><ci id="S5.p3.1.m1.2.2.6.3.3.2.cmml" xref="S5.p3.1.m1.2.2.6.3.3.2">𝑑</ci><cn id="S5.p3.1.m1.2.2.6.3.3.3.cmml" type="integer" xref="S5.p3.1.m1.2.2.6.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.1.m1.2c">\alpha=(\alpha_{1},\widetilde{\alpha})\in\mathbb{N}_{0}\times\mathbb{N}_{0}^{d% -1}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.1.m1.2d">italic_α = ( italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , over~ start_ARG italic_α end_ARG ) ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT × blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>, we have <math alttext="\operatorname{Tr}\partial^{\alpha}=\partial^{\widetilde{\alpha}}\operatorname{% Tr}_{\alpha_{1}}" class="ltx_Math" display="inline" id="S5.p3.2.m2.1"><semantics id="S5.p3.2.m2.1a"><mrow id="S5.p3.2.m2.1.1" xref="S5.p3.2.m2.1.1.cmml"><mrow id="S5.p3.2.m2.1.1.2" xref="S5.p3.2.m2.1.1.2.cmml"><mi id="S5.p3.2.m2.1.1.2.2" xref="S5.p3.2.m2.1.1.2.2.cmml">Tr</mi><mo id="S5.p3.2.m2.1.1.2.1" lspace="0.167em" xref="S5.p3.2.m2.1.1.2.1.cmml">⁢</mo><msup id="S5.p3.2.m2.1.1.2.3" xref="S5.p3.2.m2.1.1.2.3.cmml"><mo id="S5.p3.2.m2.1.1.2.3.2" rspace="0.1389em" xref="S5.p3.2.m2.1.1.2.3.2.cmml">∂</mo><mi id="S5.p3.2.m2.1.1.2.3.3" xref="S5.p3.2.m2.1.1.2.3.3.cmml">α</mi></msup></mrow><mo id="S5.p3.2.m2.1.1.1" lspace="0.1389em" rspace="0.1389em" xref="S5.p3.2.m2.1.1.1.cmml">=</mo><mrow id="S5.p3.2.m2.1.1.3" xref="S5.p3.2.m2.1.1.3.cmml"><msup id="S5.p3.2.m2.1.1.3.1" xref="S5.p3.2.m2.1.1.3.1.cmml"><mo id="S5.p3.2.m2.1.1.3.1.2" lspace="0.1389em" rspace="0.167em" xref="S5.p3.2.m2.1.1.3.1.2.cmml">∂</mo><mover accent="true" id="S5.p3.2.m2.1.1.3.1.3" xref="S5.p3.2.m2.1.1.3.1.3.cmml"><mi id="S5.p3.2.m2.1.1.3.1.3.2" xref="S5.p3.2.m2.1.1.3.1.3.2.cmml">α</mi><mo id="S5.p3.2.m2.1.1.3.1.3.1" xref="S5.p3.2.m2.1.1.3.1.3.1.cmml">~</mo></mover></msup><msub id="S5.p3.2.m2.1.1.3.2" xref="S5.p3.2.m2.1.1.3.2.cmml"><mi id="S5.p3.2.m2.1.1.3.2.2" xref="S5.p3.2.m2.1.1.3.2.2.cmml">Tr</mi><msub id="S5.p3.2.m2.1.1.3.2.3" xref="S5.p3.2.m2.1.1.3.2.3.cmml"><mi id="S5.p3.2.m2.1.1.3.2.3.2" xref="S5.p3.2.m2.1.1.3.2.3.2.cmml">α</mi><mn id="S5.p3.2.m2.1.1.3.2.3.3" xref="S5.p3.2.m2.1.1.3.2.3.3.cmml">1</mn></msub></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p3.2.m2.1b"><apply id="S5.p3.2.m2.1.1.cmml" xref="S5.p3.2.m2.1.1"><eq id="S5.p3.2.m2.1.1.1.cmml" xref="S5.p3.2.m2.1.1.1"></eq><apply id="S5.p3.2.m2.1.1.2.cmml" xref="S5.p3.2.m2.1.1.2"><times id="S5.p3.2.m2.1.1.2.1.cmml" xref="S5.p3.2.m2.1.1.2.1"></times><ci id="S5.p3.2.m2.1.1.2.2.cmml" xref="S5.p3.2.m2.1.1.2.2">Tr</ci><apply id="S5.p3.2.m2.1.1.2.3.cmml" xref="S5.p3.2.m2.1.1.2.3"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.2.3.1.cmml" xref="S5.p3.2.m2.1.1.2.3">superscript</csymbol><partialdiff id="S5.p3.2.m2.1.1.2.3.2.cmml" xref="S5.p3.2.m2.1.1.2.3.2"></partialdiff><ci id="S5.p3.2.m2.1.1.2.3.3.cmml" xref="S5.p3.2.m2.1.1.2.3.3">𝛼</ci></apply></apply><apply id="S5.p3.2.m2.1.1.3.cmml" xref="S5.p3.2.m2.1.1.3"><apply id="S5.p3.2.m2.1.1.3.1.cmml" xref="S5.p3.2.m2.1.1.3.1"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.3.1.1.cmml" xref="S5.p3.2.m2.1.1.3.1">superscript</csymbol><partialdiff id="S5.p3.2.m2.1.1.3.1.2.cmml" xref="S5.p3.2.m2.1.1.3.1.2"></partialdiff><apply id="S5.p3.2.m2.1.1.3.1.3.cmml" xref="S5.p3.2.m2.1.1.3.1.3"><ci id="S5.p3.2.m2.1.1.3.1.3.1.cmml" xref="S5.p3.2.m2.1.1.3.1.3.1">~</ci><ci id="S5.p3.2.m2.1.1.3.1.3.2.cmml" xref="S5.p3.2.m2.1.1.3.1.3.2">𝛼</ci></apply></apply><apply id="S5.p3.2.m2.1.1.3.2.cmml" xref="S5.p3.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.3.2.1.cmml" xref="S5.p3.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.p3.2.m2.1.1.3.2.2.cmml" xref="S5.p3.2.m2.1.1.3.2.2">Tr</ci><apply id="S5.p3.2.m2.1.1.3.2.3.cmml" xref="S5.p3.2.m2.1.1.3.2.3"><csymbol cd="ambiguous" id="S5.p3.2.m2.1.1.3.2.3.1.cmml" xref="S5.p3.2.m2.1.1.3.2.3">subscript</csymbol><ci id="S5.p3.2.m2.1.1.3.2.3.2.cmml" xref="S5.p3.2.m2.1.1.3.2.3.2">𝛼</ci><cn id="S5.p3.2.m2.1.1.3.2.3.3.cmml" type="integer" xref="S5.p3.2.m2.1.1.3.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.2.m2.1c">\operatorname{Tr}\partial^{\alpha}=\partial^{\widetilde{\alpha}}\operatorname{% Tr}_{\alpha_{1}}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.2.m2.1d">roman_Tr ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT = ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT roman_Tr start_POSTSUBSCRIPT italic_α start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> and therefore the boundary operator <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.p3.3.m3.1"><semantics id="S5.p3.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p3.3.m3.1.1" xref="S5.p3.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.p3.3.m3.1b"><ci id="S5.p3.3.m3.1.1.cmml" xref="S5.p3.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.3.m3.1d">caligraphic_B</annotation></semantics></math> can be rewritten as</p> <table class="ltx_equation ltx_eqn_table" id="S5.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}=\sum_{j=0}^{m}b_{j}\operatorname{Tr}_{j},\quad\text{ where }\quad b% _{j}(\cdot,\nabla_{\widetilde{x}}):=\sum_{|\widetilde{\alpha}|\leq m-j}b_{(0,% \widetilde{\alpha})}\partial^{\widetilde{\alpha}}." class="ltx_Math" display="block" id="S5.E1.m1.6"><semantics id="S5.E1.m1.6a"><mrow id="S5.E1.m1.6.6.1"><mrow id="S5.E1.m1.6.6.1.1.2" xref="S5.E1.m1.6.6.1.1.3.cmml"><mrow id="S5.E1.m1.6.6.1.1.1.1" xref="S5.E1.m1.6.6.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E1.m1.6.6.1.1.1.1.3" xref="S5.E1.m1.6.6.1.1.1.1.3.cmml">ℬ</mi><mo id="S5.E1.m1.6.6.1.1.1.1.2" rspace="0.111em" xref="S5.E1.m1.6.6.1.1.1.1.2.cmml">=</mo><mrow id="S5.E1.m1.6.6.1.1.1.1.1.1" xref="S5.E1.m1.6.6.1.1.1.1.1.2.cmml"><mrow id="S5.E1.m1.6.6.1.1.1.1.1.1.1" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.cmml"><munderover id="S5.E1.m1.6.6.1.1.1.1.1.1.1.1" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.cmml"><mo id="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.2" movablelimits="false" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.2.cmml">∑</mo><mrow id="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3.cmml"><mi id="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3.2" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3.2.cmml">j</mi><mo id="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3.1" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3.1.cmml">=</mo><mn id="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3.3" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.2.3.3.cmml">0</mn></mrow><mi id="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.3" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.1.3.cmml">m</mi></munderover><mrow id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.cmml"><msub id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.2" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.2.cmml"><mi id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.2.2" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.2.2.cmml">b</mi><mi id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.2.3" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.2.3.cmml">j</mi></msub><mo id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.1" lspace="0.167em" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.3" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.3.cmml"><mi id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.3.2" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.3.2.cmml">Tr</mi><mi id="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.3.3" xref="S5.E1.m1.6.6.1.1.1.1.1.1.1.2.3.3.cmml">j</mi></msub></mrow></mrow><mo id="S5.E1.m1.6.6.1.1.1.1.1.1.2" rspace="1.167em" xref="S5.E1.m1.6.6.1.1.1.1.1.2.cmml">,</mo><mtext id="S5.E1.m1.5.5" xref="S5.E1.m1.5.5a.cmml"> where </mtext></mrow></mrow><mspace id="S5.E1.m1.6.6.1.1.2.3" width="1em" xref="S5.E1.m1.6.6.1.1.3a.cmml"></mspace><mrow id="S5.E1.m1.6.6.1.1.2.2" xref="S5.E1.m1.6.6.1.1.2.2.cmml"><mrow id="S5.E1.m1.6.6.1.1.2.2.1" xref="S5.E1.m1.6.6.1.1.2.2.1.cmml"><msub id="S5.E1.m1.6.6.1.1.2.2.1.3" xref="S5.E1.m1.6.6.1.1.2.2.1.3.cmml"><mi id="S5.E1.m1.6.6.1.1.2.2.1.3.2" xref="S5.E1.m1.6.6.1.1.2.2.1.3.2.cmml">b</mi><mi id="S5.E1.m1.6.6.1.1.2.2.1.3.3" xref="S5.E1.m1.6.6.1.1.2.2.1.3.3.cmml">j</mi></msub><mo id="S5.E1.m1.6.6.1.1.2.2.1.2" xref="S5.E1.m1.6.6.1.1.2.2.1.2.cmml">⁢</mo><mrow id="S5.E1.m1.6.6.1.1.2.2.1.1.1" xref="S5.E1.m1.6.6.1.1.2.2.1.1.2.cmml"><mo id="S5.E1.m1.6.6.1.1.2.2.1.1.1.2" stretchy="false" xref="S5.E1.m1.6.6.1.1.2.2.1.1.2.cmml">(</mo><mo id="S5.E1.m1.4.4" lspace="0em" rspace="0em" xref="S5.E1.m1.4.4.cmml">⋅</mo><mo id="S5.E1.m1.6.6.1.1.2.2.1.1.1.3" xref="S5.E1.m1.6.6.1.1.2.2.1.1.2.cmml">,</mo><msub id="S5.E1.m1.6.6.1.1.2.2.1.1.1.1" xref="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.cmml"><mo id="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.2" xref="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.2.cmml">∇</mo><mover accent="true" id="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.3" xref="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.3.cmml"><mi id="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.3.2" xref="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.3.2.cmml">x</mi><mo id="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.3.1" xref="S5.E1.m1.6.6.1.1.2.2.1.1.1.1.3.1.cmml">~</mo></mover></msub><mo id="S5.E1.m1.6.6.1.1.2.2.1.1.1.4" rspace="0.278em" stretchy="false" xref="S5.E1.m1.6.6.1.1.2.2.1.1.2.cmml">)</mo></mrow></mrow><mo id="S5.E1.m1.6.6.1.1.2.2.2" rspace="0.111em" xref="S5.E1.m1.6.6.1.1.2.2.2.cmml">:=</mo><mrow id="S5.E1.m1.6.6.1.1.2.2.3" xref="S5.E1.m1.6.6.1.1.2.2.3.cmml"><munder id="S5.E1.m1.6.6.1.1.2.2.3.1" xref="S5.E1.m1.6.6.1.1.2.2.3.1.cmml"><mo id="S5.E1.m1.6.6.1.1.2.2.3.1.2" movablelimits="false" xref="S5.E1.m1.6.6.1.1.2.2.3.1.2.cmml">∑</mo><mrow 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start_POSTSUBSCRIPT ( 0 , over~ start_ARG italic_α end_ARG ) end_POSTSUBSCRIPT ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.1)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.p3.6">Note that the leading order coefficient <math alttext="b_{m}" class="ltx_Math" display="inline" id="S5.p3.4.m1.1"><semantics id="S5.p3.4.m1.1a"><msub id="S5.p3.4.m1.1.1" xref="S5.p3.4.m1.1.1.cmml"><mi id="S5.p3.4.m1.1.1.2" xref="S5.p3.4.m1.1.1.2.cmml">b</mi><mi id="S5.p3.4.m1.1.1.3" xref="S5.p3.4.m1.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p3.4.m1.1b"><apply id="S5.p3.4.m1.1.1.cmml" xref="S5.p3.4.m1.1.1"><csymbol cd="ambiguous" id="S5.p3.4.m1.1.1.1.cmml" xref="S5.p3.4.m1.1.1">subscript</csymbol><ci id="S5.p3.4.m1.1.1.2.cmml" xref="S5.p3.4.m1.1.1.2">𝑏</ci><ci id="S5.p3.4.m1.1.1.3.cmml" xref="S5.p3.4.m1.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.4.m1.1c">b_{m}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.4.m1.1d">italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> is independent of <math alttext="\nabla_{\widetilde{x}}" class="ltx_Math" display="inline" id="S5.p3.5.m2.1"><semantics id="S5.p3.5.m2.1a"><msub id="S5.p3.5.m2.1.1" xref="S5.p3.5.m2.1.1.cmml"><mo id="S5.p3.5.m2.1.1.2" xref="S5.p3.5.m2.1.1.2.cmml">∇</mo><mover accent="true" id="S5.p3.5.m2.1.1.3" xref="S5.p3.5.m2.1.1.3.cmml"><mi id="S5.p3.5.m2.1.1.3.2" xref="S5.p3.5.m2.1.1.3.2.cmml">x</mi><mo id="S5.p3.5.m2.1.1.3.1" xref="S5.p3.5.m2.1.1.3.1.cmml">~</mo></mover></msub><annotation-xml encoding="MathML-Content" id="S5.p3.5.m2.1b"><apply id="S5.p3.5.m2.1.1.cmml" xref="S5.p3.5.m2.1.1"><csymbol cd="ambiguous" id="S5.p3.5.m2.1.1.1.cmml" xref="S5.p3.5.m2.1.1">subscript</csymbol><ci id="S5.p3.5.m2.1.1.2.cmml" xref="S5.p3.5.m2.1.1.2">∇</ci><apply id="S5.p3.5.m2.1.1.3.cmml" xref="S5.p3.5.m2.1.1.3"><ci id="S5.p3.5.m2.1.1.3.1.cmml" xref="S5.p3.5.m2.1.1.3.1">~</ci><ci id="S5.p3.5.m2.1.1.3.2.cmml" xref="S5.p3.5.m2.1.1.3.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.5.m2.1c">\nabla_{\widetilde{x}}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.5.m2.1d">∇ start_POSTSUBSCRIPT over~ start_ARG italic_x end_ARG end_POSTSUBSCRIPT</annotation></semantics></math>. In the following lemma, we prove that <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.p3.6.m3.1"><semantics id="S5.p3.6.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.p3.6.m3.1.1" xref="S5.p3.6.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.p3.6.m3.1b"><ci id="S5.p3.6.m3.1.1.cmml" xref="S5.p3.6.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.p3.6.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.p3.6.m3.1d">caligraphic_B</annotation></semantics></math> is a bounded operator from a weighted Bessel potential or Sobolev space to its trace space.</p> </div> <div class="ltx_theorem ltx_theorem_lemma" id="S5.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.1.1.1">Lemma 5.2</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem2.p1"> <p class="ltx_p" id="S5.Thmtheorem2.p1.3"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem2.p1.3.3">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.1.1.m1.2"><semantics id="S5.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S5.Thmtheorem2.p1.1.1.m1.2.3" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.cmml"><mi id="S5.Thmtheorem2.p1.1.1.m1.2.3.2" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S5.Thmtheorem2.p1.1.1.m1.2.3.1" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem2.p1.1.1.m1.2.3.3.2" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml"><mo id="S5.Thmtheorem2.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem2.p1.1.1.m1.1.1" xref="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml">1</mn><mo id="S5.Thmtheorem2.p1.1.1.m1.2.3.3.2.2" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.1.1.m1.2.2" mathvariant="normal" xref="S5.Thmtheorem2.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S5.Thmtheorem2.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.1.1.m1.2b"><apply id="S5.Thmtheorem2.p1.1.1.m1.2.3.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.3"><in id="S5.Thmtheorem2.p1.1.1.m1.2.3.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.1"></in><ci id="S5.Thmtheorem2.p1.1.1.m1.2.3.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S5.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.3.3.2"><cn id="S5.Thmtheorem2.p1.1.1.m1.1.1.cmml" type="integer" xref="S5.Thmtheorem2.p1.1.1.m1.1.1">1</cn><infinity id="S5.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem2.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.2.2.m2.1"><semantics id="S5.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S5.Thmtheorem2.p1.2.2.m2.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml"><mi id="S5.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml">m</mi><mo id="S5.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><mi id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.3" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.2.2.m2.1b"><apply id="S5.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1"><in id="S5.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.1"></in><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.2">𝑚</ci><apply id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.2">ℕ</ci><cn id="S5.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem2.p1.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.2.2.m2.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.2.2.m2.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and let <math alttext="X,Y" class="ltx_Math" display="inline" id="S5.Thmtheorem2.p1.3.3.m3.2"><semantics id="S5.Thmtheorem2.p1.3.3.m3.2a"><mrow id="S5.Thmtheorem2.p1.3.3.m3.2.3.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.1.cmml"><mi id="S5.Thmtheorem2.p1.3.3.m3.1.1" xref="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml">X</mi><mo id="S5.Thmtheorem2.p1.3.3.m3.2.3.2.1" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.1.cmml">,</mo><mi id="S5.Thmtheorem2.p1.3.3.m3.2.2" xref="S5.Thmtheorem2.p1.3.3.m3.2.2.cmml">Y</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem2.p1.3.3.m3.2b"><list id="S5.Thmtheorem2.p1.3.3.m3.2.3.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.3.2"><ci id="S5.Thmtheorem2.p1.3.3.m3.1.1.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.1.1">𝑋</ci><ci id="S5.Thmtheorem2.p1.3.3.m3.2.2.cmml" xref="S5.Thmtheorem2.p1.3.3.m3.2.2">𝑌</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem2.p1.3.3.m3.2c">X,Y</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem2.p1.3.3.m3.2d">italic_X , italic_Y</annotation></semantics></math> be Banach spaces.</span></p> <ol class="ltx_enumerate" id="S5.I2"> <li class="ltx_item" id="S5.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S5.I2.i1.p1"> <p class="ltx_p" id="S5.I2.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.4.1">If </span><math alttext="s>0" class="ltx_Math" display="inline" id="S5.I2.i1.p1.1.m1.1"><semantics id="S5.I2.i1.p1.1.m1.1a"><mrow id="S5.I2.i1.p1.1.m1.1.1" xref="S5.I2.i1.p1.1.m1.1.1.cmml"><mi id="S5.I2.i1.p1.1.m1.1.1.2" xref="S5.I2.i1.p1.1.m1.1.1.2.cmml">s</mi><mo id="S5.I2.i1.p1.1.m1.1.1.1" xref="S5.I2.i1.p1.1.m1.1.1.1.cmml">></mo><mn id="S5.I2.i1.p1.1.m1.1.1.3" xref="S5.I2.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.1.m1.1b"><apply id="S5.I2.i1.p1.1.m1.1.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1"><gt id="S5.I2.i1.p1.1.m1.1.1.1.cmml" xref="S5.I2.i1.p1.1.m1.1.1.1"></gt><ci id="S5.I2.i1.p1.1.m1.1.1.2.cmml" xref="S5.I2.i1.p1.1.m1.1.1.2">𝑠</ci><cn id="S5.I2.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.I2.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.1.m1.1c">s>0</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.1.m1.1d">italic_s > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.4.2">, </span><math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S5.I2.i1.p1.2.m2.2"><semantics id="S5.I2.i1.p1.2.m2.2a"><mrow id="S5.I2.i1.p1.2.m2.2.2" xref="S5.I2.i1.p1.2.m2.2.2.cmml"><mi id="S5.I2.i1.p1.2.m2.2.2.4" xref="S5.I2.i1.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S5.I2.i1.p1.2.m2.2.2.3" xref="S5.I2.i1.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S5.I2.i1.p1.2.m2.2.2.2.2" xref="S5.I2.i1.p1.2.m2.2.2.2.3.cmml"><mo id="S5.I2.i1.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S5.I2.i1.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S5.I2.i1.p1.2.m2.1.1.1.1.1" xref="S5.I2.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S5.I2.i1.p1.2.m2.1.1.1.1.1a" xref="S5.I2.i1.p1.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S5.I2.i1.p1.2.m2.1.1.1.1.1.2" xref="S5.I2.i1.p1.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S5.I2.i1.p1.2.m2.2.2.2.2.4" xref="S5.I2.i1.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="S5.I2.i1.p1.2.m2.2.2.2.2.2" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2.cmml"><mi id="S5.I2.i1.p1.2.m2.2.2.2.2.2.2" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2.2.cmml">p</mi><mo id="S5.I2.i1.p1.2.m2.2.2.2.2.2.1" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2.1.cmml">−</mo><mn id="S5.I2.i1.p1.2.m2.2.2.2.2.2.3" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S5.I2.i1.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S5.I2.i1.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.2.m2.2b"><apply id="S5.I2.i1.p1.2.m2.2.2.cmml" xref="S5.I2.i1.p1.2.m2.2.2"><in id="S5.I2.i1.p1.2.m2.2.2.3.cmml" xref="S5.I2.i1.p1.2.m2.2.2.3"></in><ci id="S5.I2.i1.p1.2.m2.2.2.4.cmml" xref="S5.I2.i1.p1.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S5.I2.i1.p1.2.m2.2.2.2.3.cmml" xref="S5.I2.i1.p1.2.m2.2.2.2.2"><apply id="S5.I2.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S5.I2.i1.p1.2.m2.1.1.1.1.1"><minus id="S5.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.I2.i1.p1.2.m2.1.1.1.1.1"></minus><cn id="S5.I2.i1.p1.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S5.I2.i1.p1.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S5.I2.i1.p1.2.m2.2.2.2.2.2.cmml" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2"><minus id="S5.I2.i1.p1.2.m2.2.2.2.2.2.1.cmml" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2.1"></minus><ci id="S5.I2.i1.p1.2.m2.2.2.2.2.2.2.cmml" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2.2">𝑝</ci><cn id="S5.I2.i1.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S5.I2.i1.p1.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.2.m2.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.2.m2.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.4.3"> and </span><math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.I2.i1.p1.3.m3.1"><semantics id="S5.I2.i1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I2.i1.p1.3.m3.1.1" xref="S5.I2.i1.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.3.m3.1b"><ci id="S5.I2.i1.p1.3.m3.1.1.cmml" xref="S5.I2.i1.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.3.m3.1d">caligraphic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.4.4"> is of type </span><math alttext="(p,s,\gamma,m,Y)" class="ltx_Math" display="inline" id="S5.I2.i1.p1.4.m4.5"><semantics id="S5.I2.i1.p1.4.m4.5a"><mrow id="S5.I2.i1.p1.4.m4.5.6.2" xref="S5.I2.i1.p1.4.m4.5.6.1.cmml"><mo id="S5.I2.i1.p1.4.m4.5.6.2.1" stretchy="false" xref="S5.I2.i1.p1.4.m4.5.6.1.cmml">(</mo><mi id="S5.I2.i1.p1.4.m4.1.1" xref="S5.I2.i1.p1.4.m4.1.1.cmml">p</mi><mo id="S5.I2.i1.p1.4.m4.5.6.2.2" xref="S5.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i1.p1.4.m4.2.2" xref="S5.I2.i1.p1.4.m4.2.2.cmml">s</mi><mo id="S5.I2.i1.p1.4.m4.5.6.2.3" xref="S5.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i1.p1.4.m4.3.3" xref="S5.I2.i1.p1.4.m4.3.3.cmml">γ</mi><mo id="S5.I2.i1.p1.4.m4.5.6.2.4" xref="S5.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i1.p1.4.m4.4.4" xref="S5.I2.i1.p1.4.m4.4.4.cmml">m</mi><mo id="S5.I2.i1.p1.4.m4.5.6.2.5" xref="S5.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i1.p1.4.m4.5.5" xref="S5.I2.i1.p1.4.m4.5.5.cmml">Y</mi><mo id="S5.I2.i1.p1.4.m4.5.6.2.6" stretchy="false" xref="S5.I2.i1.p1.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i1.p1.4.m4.5b"><vector id="S5.I2.i1.p1.4.m4.5.6.1.cmml" xref="S5.I2.i1.p1.4.m4.5.6.2"><ci id="S5.I2.i1.p1.4.m4.1.1.cmml" xref="S5.I2.i1.p1.4.m4.1.1">𝑝</ci><ci id="S5.I2.i1.p1.4.m4.2.2.cmml" xref="S5.I2.i1.p1.4.m4.2.2">𝑠</ci><ci id="S5.I2.i1.p1.4.m4.3.3.cmml" xref="S5.I2.i1.p1.4.m4.3.3">𝛾</ci><ci id="S5.I2.i1.p1.4.m4.4.4.cmml" xref="S5.I2.i1.p1.4.m4.4.4">𝑚</ci><ci id="S5.I2.i1.p1.4.m4.5.5.cmml" xref="S5.I2.i1.p1.4.m4.5.5">𝑌</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i1.p1.4.m4.5c">(p,s,\gamma,m,Y)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i1.p1.4.m4.5d">( italic_p , italic_s , italic_γ , italic_m , italic_Y )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i1.p1.4.5">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^{s-m-\frac{\gamma+1}% {p}}_{p,p}(\mathbb{R}^{d-1};Y)\quad\text{ is bounded. }" class="ltx_Math" display="block" id="S5.Ex2.m1.10"><semantics id="S5.Ex2.m1.10a"><mrow 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id="S5.Ex2.m1.8.8.1.1.1.1.1" xref="S5.Ex2.m1.8.8.1.1.1.1.1.cmml"><mi id="S5.Ex2.m1.8.8.1.1.1.1.1.2.2" xref="S5.Ex2.m1.8.8.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S5.Ex2.m1.8.8.1.1.1.1.1.3" xref="S5.Ex2.m1.8.8.1.1.1.1.1.3.cmml">+</mo><mi id="S5.Ex2.m1.8.8.1.1.1.1.1.2.3" xref="S5.Ex2.m1.8.8.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S5.Ex2.m1.9.9.2.2.2.2.4" xref="S5.Ex2.m1.9.9.2.2.2.3.cmml">,</mo><msub id="S5.Ex2.m1.9.9.2.2.2.2.2" xref="S5.Ex2.m1.9.9.2.2.2.2.2.cmml"><mi id="S5.Ex2.m1.9.9.2.2.2.2.2.2" xref="S5.Ex2.m1.9.9.2.2.2.2.2.2.cmml">w</mi><mi id="S5.Ex2.m1.9.9.2.2.2.2.2.3" xref="S5.Ex2.m1.9.9.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S5.Ex2.m1.9.9.2.2.2.2.5" xref="S5.Ex2.m1.9.9.2.2.2.3.cmml">;</mo><mi id="S5.Ex2.m1.5.5" xref="S5.Ex2.m1.5.5.cmml">X</mi><mo id="S5.Ex2.m1.9.9.2.2.2.2.6" stretchy="false" xref="S5.Ex2.m1.9.9.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex2.m1.10.10.3.4" stretchy="false" xref="S5.Ex2.m1.10.10.3.4.cmml">→</mo><mrow id="S5.Ex2.m1.10.10.3.3.1" 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xref="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.2.cmml">ℝ</mi><mrow id="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3" xref="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3.cmml"><mi id="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3.2" xref="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3.2.cmml">d</mi><mo id="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3.1" xref="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3.1.cmml">−</mo><mn id="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3.3" xref="S5.Ex2.m1.10.10.3.3.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S5.Ex2.m1.10.10.3.3.1.1.1.1.3" xref="S5.Ex2.m1.10.10.3.3.1.1.1.2.cmml">;</mo><mi id="S5.Ex2.m1.6.6" xref="S5.Ex2.m1.6.6.cmml">Y</mi><mo id="S5.Ex2.m1.10.10.3.3.1.1.1.1.4" stretchy="false" xref="S5.Ex2.m1.10.10.3.3.1.1.1.2.cmml">)</mo></mrow></mrow><mspace id="S5.Ex2.m1.10.10.3.3.1.2" width="1em" xref="S5.Ex2.m1.10.10.3.3.2.cmml"></mspace><mtext class="ltx_mathvariant_italic" id="S5.Ex2.m1.7.7" xref="S5.Ex2.m1.7.7a.cmml"> is bounded. </mtext></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex2.m1.10b"><apply 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class="ltx_mathvariant_italic" id="S5.Ex2.m1.7.7.cmml" xref="S5.Ex2.m1.7.7"> is bounded. </mtext></ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex2.m1.10c">\mathcal{B}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^{s-m-\frac{\gamma+1}% {p}}_{p,p}(\mathbb{R}^{d-1};Y)\quad\text{ is bounded. }</annotation><annotation encoding="application/x-llamapun" id="S5.Ex2.m1.10d">caligraphic_B : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y ) is bounded.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S5.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S5.I2.i2.p1"> <p class="ltx_p" id="S5.I2.i2.p1.4"><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.4.1">If </span><math alttext="k\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.1.m1.1"><semantics id="S5.I2.i2.p1.1.m1.1a"><mrow id="S5.I2.i2.p1.1.m1.1.1" xref="S5.I2.i2.p1.1.m1.1.1.cmml"><mi id="S5.I2.i2.p1.1.m1.1.1.2" xref="S5.I2.i2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S5.I2.i2.p1.1.m1.1.1.1" xref="S5.I2.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S5.I2.i2.p1.1.m1.1.1.3" xref="S5.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S5.I2.i2.p1.1.m1.1.1.3.2" xref="S5.I2.i2.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S5.I2.i2.p1.1.m1.1.1.3.3" xref="S5.I2.i2.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml 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xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.3">conditional-set</csymbol><apply id="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.cmml" xref="S5.I2.i2.p1.2.m2.3.3.2.2.1.1"><minus id="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.1.cmml" xref="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.1"></minus><apply id="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2.cmml" xref="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2"><times id="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2.1.cmml" xref="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2.1"></times><ci id="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2.2.cmml" xref="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2.3.cmml" xref="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.3.cmml" type="integer" xref="S5.I2.i2.p1.2.m2.3.3.2.2.1.1.3">1</cn></apply><apply id="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.cmml" xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.2"><in id="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.1.cmml" xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.1"></in><ci id="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.2.cmml" xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.2">𝑗</ci><apply id="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3.cmml" xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3.1.cmml" xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3">subscript</csymbol><ci id="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3.2.cmml" xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S5.I2.i2.p1.2.m2.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.2.m2.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.2.m2.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.4.3"> and </span><math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.I2.i2.p1.3.m3.1"><semantics id="S5.I2.i2.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I2.i2.p1.3.m3.1.1" xref="S5.I2.i2.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.3.m3.1b"><ci id="S5.I2.i2.p1.3.m3.1.1.cmml" xref="S5.I2.i2.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.3.m3.1d">caligraphic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.4.4"> is of type </span><math alttext="(p,k,\gamma,m,Y)" class="ltx_Math" display="inline" id="S5.I2.i2.p1.4.m4.5"><semantics id="S5.I2.i2.p1.4.m4.5a"><mrow id="S5.I2.i2.p1.4.m4.5.6.2" xref="S5.I2.i2.p1.4.m4.5.6.1.cmml"><mo id="S5.I2.i2.p1.4.m4.5.6.2.1" stretchy="false" xref="S5.I2.i2.p1.4.m4.5.6.1.cmml">(</mo><mi id="S5.I2.i2.p1.4.m4.1.1" xref="S5.I2.i2.p1.4.m4.1.1.cmml">p</mi><mo id="S5.I2.i2.p1.4.m4.5.6.2.2" xref="S5.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i2.p1.4.m4.2.2" xref="S5.I2.i2.p1.4.m4.2.2.cmml">k</mi><mo id="S5.I2.i2.p1.4.m4.5.6.2.3" xref="S5.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i2.p1.4.m4.3.3" xref="S5.I2.i2.p1.4.m4.3.3.cmml">γ</mi><mo id="S5.I2.i2.p1.4.m4.5.6.2.4" xref="S5.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i2.p1.4.m4.4.4" xref="S5.I2.i2.p1.4.m4.4.4.cmml">m</mi><mo id="S5.I2.i2.p1.4.m4.5.6.2.5" xref="S5.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I2.i2.p1.4.m4.5.5" xref="S5.I2.i2.p1.4.m4.5.5.cmml">Y</mi><mo id="S5.I2.i2.p1.4.m4.5.6.2.6" stretchy="false" xref="S5.I2.i2.p1.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I2.i2.p1.4.m4.5b"><vector id="S5.I2.i2.p1.4.m4.5.6.1.cmml" xref="S5.I2.i2.p1.4.m4.5.6.2"><ci id="S5.I2.i2.p1.4.m4.1.1.cmml" xref="S5.I2.i2.p1.4.m4.1.1">𝑝</ci><ci id="S5.I2.i2.p1.4.m4.2.2.cmml" xref="S5.I2.i2.p1.4.m4.2.2">𝑘</ci><ci id="S5.I2.i2.p1.4.m4.3.3.cmml" xref="S5.I2.i2.p1.4.m4.3.3">𝛾</ci><ci id="S5.I2.i2.p1.4.m4.4.4.cmml" xref="S5.I2.i2.p1.4.m4.4.4">𝑚</ci><ci id="S5.I2.i2.p1.4.m4.5.5.cmml" xref="S5.I2.i2.p1.4.m4.5.5">𝑌</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.I2.i2.p1.4.m4.5c">(p,k,\gamma,m,Y)</annotation><annotation encoding="application/x-llamapun" id="S5.I2.i2.p1.4.m4.5d">( italic_p , italic_k , italic_γ , italic_m , italic_Y )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I2.i2.p1.4.5">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^{k-m-\frac{\gamma+1}% {p}}_{p,p}(\mathbb{R}^{d-1};Y)\quad\text{ is bounded. }" class="ltx_Math" display="block" id="S5.Ex3.m1.10"><semantics id="S5.Ex3.m1.10a"><mrow id="S5.Ex3.m1.10.10" xref="S5.Ex3.m1.10.10.cmml"><mi class="ltx_font_mathcaligraphic" 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id="S5.Ex3.m1.8.8.1.1.1.1.1.2.2" xref="S5.Ex3.m1.8.8.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S5.Ex3.m1.8.8.1.1.1.1.1.3" xref="S5.Ex3.m1.8.8.1.1.1.1.1.3.cmml">+</mo><mi id="S5.Ex3.m1.8.8.1.1.1.1.1.2.3" xref="S5.Ex3.m1.8.8.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S5.Ex3.m1.9.9.2.2.2.2.4" xref="S5.Ex3.m1.9.9.2.2.2.3.cmml">,</mo><msub id="S5.Ex3.m1.9.9.2.2.2.2.2" xref="S5.Ex3.m1.9.9.2.2.2.2.2.cmml"><mi id="S5.Ex3.m1.9.9.2.2.2.2.2.2" xref="S5.Ex3.m1.9.9.2.2.2.2.2.2.cmml">w</mi><mi id="S5.Ex3.m1.9.9.2.2.2.2.2.3" xref="S5.Ex3.m1.9.9.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S5.Ex3.m1.9.9.2.2.2.2.5" xref="S5.Ex3.m1.9.9.2.2.2.3.cmml">;</mo><mi id="S5.Ex3.m1.5.5" xref="S5.Ex3.m1.5.5.cmml">X</mi><mo id="S5.Ex3.m1.9.9.2.2.2.2.6" stretchy="false" xref="S5.Ex3.m1.9.9.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex3.m1.10.10.3.4" stretchy="false" xref="S5.Ex3.m1.10.10.3.4.cmml">→</mo><mrow id="S5.Ex3.m1.10.10.3.3.1" xref="S5.Ex3.m1.10.10.3.3.2.cmml"><mrow id="S5.Ex3.m1.10.10.3.3.1.1" 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encoding="application/x-tex" id="S5.Ex3.m1.10c">\mathcal{B}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^{k-m-\frac{\gamma+1}% {p}}_{p,p}(\mathbb{R}^{d-1};Y)\quad\text{ is bounded. }</annotation><annotation encoding="application/x-llamapun" id="S5.Ex3.m1.10d">caligraphic_B : italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_k - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y ) is bounded.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S5.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.1.p1"> <p class="ltx_p" id="S5.1.p1.3"><span class="ltx_text ltx_font_italic" id="S5.1.p1.3.1">Step 1: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.I2.i1" title="item i ‣ Lemma 5.2. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>.</span> By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> we have that</p> <table class="ltx_equation ltx_eqn_table" id="S5.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{j}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B_{p,p}^{k-j% -\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)\quad\text{ for all }j\in\{0,\dots,m\}," class="ltx_Math" display="block" id="S5.E2.m1.10"><semantics id="S5.E2.m1.10a"><mrow id="S5.E2.m1.10.10.1" xref="S5.E2.m1.10.10.1.1.cmml"><mrow id="S5.E2.m1.10.10.1.1" xref="S5.E2.m1.10.10.1.1.cmml"><msub id="S5.E2.m1.10.10.1.1.4" xref="S5.E2.m1.10.10.1.1.4.cmml"><mi id="S5.E2.m1.10.10.1.1.4.2" xref="S5.E2.m1.10.10.1.1.4.2.cmml">Tr</mi><mi id="S5.E2.m1.10.10.1.1.4.3" xref="S5.E2.m1.10.10.1.1.4.3.cmml">j</mi></msub><mo id="S5.E2.m1.10.10.1.1.3" lspace="0.278em" rspace="0.278em" xref="S5.E2.m1.10.10.1.1.3.cmml">:</mo><mrow id="S5.E2.m1.10.10.1.1.2.2" xref="S5.E2.m1.10.10.1.1.2.3.cmml"><mrow id="S5.E2.m1.10.10.1.1.1.1.1" xref="S5.E2.m1.10.10.1.1.1.1.1.cmml"><mrow id="S5.E2.m1.10.10.1.1.1.1.1.2" xref="S5.E2.m1.10.10.1.1.1.1.1.2.cmml"><msup id="S5.E2.m1.10.10.1.1.1.1.1.2.4" xref="S5.E2.m1.10.10.1.1.1.1.1.2.4.cmml"><mi id="S5.E2.m1.10.10.1.1.1.1.1.2.4.2" xref="S5.E2.m1.10.10.1.1.1.1.1.2.4.2.cmml">H</mi><mrow id="S5.E2.m1.2.2.2.4" xref="S5.E2.m1.2.2.2.3.cmml"><mi id="S5.E2.m1.1.1.1.1" xref="S5.E2.m1.1.1.1.1.cmml">s</mi><mo id="S5.E2.m1.2.2.2.4.1" xref="S5.E2.m1.2.2.2.3.cmml">,</mo><mi id="S5.E2.m1.2.2.2.2" xref="S5.E2.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S5.E2.m1.10.10.1.1.1.1.1.2.3" xref="S5.E2.m1.10.10.1.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S5.E2.m1.10.10.1.1.1.1.1.2.2.2" xref="S5.E2.m1.10.10.1.1.1.1.1.2.2.3.cmml"><mo 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id="S5.E2.m1.10.10.1.1.2.2.2.3.1.cmml" xref="S5.E2.m1.10.10.1.1.2.2.2.3.2"><cn id="S5.E2.m1.7.7.cmml" type="integer" xref="S5.E2.m1.7.7">0</cn><ci id="S5.E2.m1.8.8.cmml" xref="S5.E2.m1.8.8">…</ci><ci id="S5.E2.m1.9.9.cmml" xref="S5.E2.m1.9.9">𝑚</ci></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E2.m1.10c">\operatorname{Tr}_{j}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B_{p,p}^{k-j% -\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)\quad\text{ for all }j\in\{0,\dots,m\},</annotation><annotation encoding="application/x-llamapun" id="S5.E2.m1.10d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) for all italic_j ∈ { 0 , … , italic_m } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.4">is bounded and by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib57" title="">57</a>, Proposition 3.10]</cite> it follows that</p> <table class="ltx_equation ltx_eqn_table" id="S5.E3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\partial^{\widetilde{\alpha}}:B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1% };X)\to B_{p,p}^{s-j-|\widetilde{\alpha}|-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1}% ;X)\quad\text{ for all }|\widetilde{\alpha}|\leq m-j," class="ltx_Math" display="block" id="S5.E3.m1.9"><semantics id="S5.E3.m1.9a"><mrow id="S5.E3.m1.9.9.1" xref="S5.E3.m1.9.9.1.1.cmml"><mrow id="S5.E3.m1.9.9.1.1" xref="S5.E3.m1.9.9.1.1.cmml"><msup id="S5.E3.m1.9.9.1.1.4" xref="S5.E3.m1.9.9.1.1.4.cmml"><mo id="S5.E3.m1.9.9.1.1.4.2" xref="S5.E3.m1.9.9.1.1.4.2.cmml">∂</mo><mover accent="true" id="S5.E3.m1.9.9.1.1.4.3" xref="S5.E3.m1.9.9.1.1.4.3.cmml"><mi id="S5.E3.m1.9.9.1.1.4.3.2" xref="S5.E3.m1.9.9.1.1.4.3.2.cmml">α</mi><mo id="S5.E3.m1.9.9.1.1.4.3.1" xref="S5.E3.m1.9.9.1.1.4.3.1.cmml">~</mo></mover></msup><mo id="S5.E3.m1.9.9.1.1.3" rspace="0.278em" xref="S5.E3.m1.9.9.1.1.3.cmml">:</mo><mrow id="S5.E3.m1.9.9.1.1.2.2" xref="S5.E3.m1.9.9.1.1.2.3.cmml"><mrow id="S5.E3.m1.9.9.1.1.1.1.1" 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id="S5.E3.m1.9c">\partial^{\widetilde{\alpha}}:B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1% };X)\to B_{p,p}^{s-j-|\widetilde{\alpha}|-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1}% ;X)\quad\text{ for all }|\widetilde{\alpha}|\leq m-j,</annotation><annotation encoding="application/x-llamapun" id="S5.E3.m1.9d">∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT : italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_j - | over~ start_ARG italic_α end_ARG | - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) for all | over~ start_ARG italic_α end_ARG | ≤ italic_m - italic_j ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.3)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.2">is bounded. 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xref="S5.1.p1.1.m1.7.7.1.1.1.1.3.3">1</cn></apply></apply><ci id="S5.1.p1.1.m1.5.5.cmml" xref="S5.1.p1.1.m1.5.5">𝑋</ci></list></apply><apply id="S5.1.p1.1.m1.8.8.2.cmml" xref="S5.1.p1.1.m1.8.8.2"><times id="S5.1.p1.1.m1.8.8.2.2.cmml" xref="S5.1.p1.1.m1.8.8.2.2"></times><apply id="S5.1.p1.1.m1.8.8.2.3.cmml" xref="S5.1.p1.1.m1.8.8.2.3"><csymbol cd="ambiguous" id="S5.1.p1.1.m1.8.8.2.3.1.cmml" xref="S5.1.p1.1.m1.8.8.2.3">subscript</csymbol><apply id="S5.1.p1.1.m1.8.8.2.3.2.cmml" xref="S5.1.p1.1.m1.8.8.2.3"><csymbol cd="ambiguous" id="S5.1.p1.1.m1.8.8.2.3.2.1.cmml" xref="S5.1.p1.1.m1.8.8.2.3">superscript</csymbol><ci id="S5.1.p1.1.m1.8.8.2.3.2.2.cmml" xref="S5.1.p1.1.m1.8.8.2.3.2.2">𝐵</ci><apply id="S5.1.p1.1.m1.8.8.2.3.2.3.cmml" xref="S5.1.p1.1.m1.8.8.2.3.2.3"><csymbol cd="ambiguous" id="S5.1.p1.1.m1.8.8.2.3.2.3.1.cmml" xref="S5.1.p1.1.m1.8.8.2.3.2.3">subscript</csymbol><ci id="S5.1.p1.1.m1.8.8.2.3.2.3.2.cmml" xref="S5.1.p1.1.m1.8.8.2.3.2.3.2">𝑠</ci><cn id="S5.1.p1.1.m1.8.8.2.3.2.3.3.cmml" type="integer" xref="S5.1.p1.1.m1.8.8.2.3.2.3.3">2</cn></apply></apply><list id="S5.1.p1.1.m1.4.4.2.3.cmml" xref="S5.1.p1.1.m1.4.4.2.4"><ci id="S5.1.p1.1.m1.3.3.1.1.cmml" xref="S5.1.p1.1.m1.3.3.1.1">𝑝</ci><ci id="S5.1.p1.1.m1.4.4.2.2.cmml" xref="S5.1.p1.1.m1.4.4.2.2">𝑝</ci></list></apply><list id="S5.1.p1.1.m1.8.8.2.1.2.cmml" xref="S5.1.p1.1.m1.8.8.2.1.1"><apply id="S5.1.p1.1.m1.8.8.2.1.1.1.cmml" xref="S5.1.p1.1.m1.8.8.2.1.1.1"><csymbol cd="ambiguous" id="S5.1.p1.1.m1.8.8.2.1.1.1.1.cmml" xref="S5.1.p1.1.m1.8.8.2.1.1.1">superscript</csymbol><ci id="S5.1.p1.1.m1.8.8.2.1.1.1.2.cmml" xref="S5.1.p1.1.m1.8.8.2.1.1.1.2">ℝ</ci><apply id="S5.1.p1.1.m1.8.8.2.1.1.1.3.cmml" xref="S5.1.p1.1.m1.8.8.2.1.1.1.3"><minus id="S5.1.p1.1.m1.8.8.2.1.1.1.3.1.cmml" xref="S5.1.p1.1.m1.8.8.2.1.1.1.3.1"></minus><ci id="S5.1.p1.1.m1.8.8.2.1.1.1.3.2.cmml" xref="S5.1.p1.1.m1.8.8.2.1.1.1.3.2">𝑑</ci><cn id="S5.1.p1.1.m1.8.8.2.1.1.1.3.3.cmml" type="integer" xref="S5.1.p1.1.m1.8.8.2.1.1.1.3.3">1</cn></apply></apply><ci id="S5.1.p1.1.m1.6.6.cmml" xref="S5.1.p1.1.m1.6.6">𝑋</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.1.m1.8c">B^{s_{1}}_{p,p}(\mathbb{R}^{d-1};X)\hookrightarrow B^{s_{2}}_{p,p}(\mathbb{R}^% {d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.1.m1.8d">italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ↪ italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> for <math alttext="s_{1}\geq s_{2}" class="ltx_Math" display="inline" id="S5.1.p1.2.m2.1"><semantics id="S5.1.p1.2.m2.1a"><mrow id="S5.1.p1.2.m2.1.1" xref="S5.1.p1.2.m2.1.1.cmml"><msub id="S5.1.p1.2.m2.1.1.2" xref="S5.1.p1.2.m2.1.1.2.cmml"><mi id="S5.1.p1.2.m2.1.1.2.2" xref="S5.1.p1.2.m2.1.1.2.2.cmml">s</mi><mn id="S5.1.p1.2.m2.1.1.2.3" xref="S5.1.p1.2.m2.1.1.2.3.cmml">1</mn></msub><mo id="S5.1.p1.2.m2.1.1.1" xref="S5.1.p1.2.m2.1.1.1.cmml">≥</mo><msub id="S5.1.p1.2.m2.1.1.3" xref="S5.1.p1.2.m2.1.1.3.cmml"><mi id="S5.1.p1.2.m2.1.1.3.2" xref="S5.1.p1.2.m2.1.1.3.2.cmml">s</mi><mn id="S5.1.p1.2.m2.1.1.3.3" xref="S5.1.p1.2.m2.1.1.3.3.cmml">2</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.1.p1.2.m2.1b"><apply id="S5.1.p1.2.m2.1.1.cmml" xref="S5.1.p1.2.m2.1.1"><geq id="S5.1.p1.2.m2.1.1.1.cmml" xref="S5.1.p1.2.m2.1.1.1"></geq><apply id="S5.1.p1.2.m2.1.1.2.cmml" xref="S5.1.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.1.p1.2.m2.1.1.2.1.cmml" xref="S5.1.p1.2.m2.1.1.2">subscript</csymbol><ci id="S5.1.p1.2.m2.1.1.2.2.cmml" xref="S5.1.p1.2.m2.1.1.2.2">𝑠</ci><cn id="S5.1.p1.2.m2.1.1.2.3.cmml" type="integer" xref="S5.1.p1.2.m2.1.1.2.3">1</cn></apply><apply id="S5.1.p1.2.m2.1.1.3.cmml" xref="S5.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.1.p1.2.m2.1.1.3.1.cmml" xref="S5.1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.1.p1.2.m2.1.1.3.2.cmml" xref="S5.1.p1.2.m2.1.1.3.2">𝑠</ci><cn id="S5.1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S5.1.p1.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.1.p1.2.m2.1c">s_{1}\geq s_{2}</annotation><annotation encoding="application/x-llamapun" id="S5.1.p1.2.m2.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_s start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT</annotation></semantics></math>, imply that</p> <table class="ltx_equation ltx_eqn_table" id="S5.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\partial^{\widetilde{\alpha}}:B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1% };X)\to B_{p,p}^{s-m-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)\quad\text{ for % all }|\widetilde{\alpha}|\leq m-j," class="ltx_Math" display="block" id="S5.E4.m1.8"><semantics id="S5.E4.m1.8a"><mrow id="S5.E4.m1.8.8.1" xref="S5.E4.m1.8.8.1.1.cmml"><mrow id="S5.E4.m1.8.8.1.1" xref="S5.E4.m1.8.8.1.1.cmml"><msup id="S5.E4.m1.8.8.1.1.4" xref="S5.E4.m1.8.8.1.1.4.cmml"><mo id="S5.E4.m1.8.8.1.1.4.2" xref="S5.E4.m1.8.8.1.1.4.2.cmml">∂</mo><mover accent="true" id="S5.E4.m1.8.8.1.1.4.3" xref="S5.E4.m1.8.8.1.1.4.3.cmml"><mi id="S5.E4.m1.8.8.1.1.4.3.2" xref="S5.E4.m1.8.8.1.1.4.3.2.cmml">α</mi><mo id="S5.E4.m1.8.8.1.1.4.3.1" xref="S5.E4.m1.8.8.1.1.4.3.1.cmml">~</mo></mover></msup><mo id="S5.E4.m1.8.8.1.1.3" rspace="0.278em" xref="S5.E4.m1.8.8.1.1.3.cmml">:</mo><mrow id="S5.E4.m1.8.8.1.1.2.2" xref="S5.E4.m1.8.8.1.1.2.3.cmml"><mrow id="S5.E4.m1.8.8.1.1.1.1.1" xref="S5.E4.m1.8.8.1.1.1.1.1.cmml"><mrow 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xref="S5.E4.m1.8.8.1.1.2.2.2.2.2"> for all </mtext></ci><apply id="S5.E4.m1.8.8.1.1.2.2.2.2.3.1.cmml" xref="S5.E4.m1.8.8.1.1.2.2.2.2.3.2"><abs id="S5.E4.m1.8.8.1.1.2.2.2.2.3.1.1.cmml" xref="S5.E4.m1.8.8.1.1.2.2.2.2.3.2.1"></abs><apply id="S5.E4.m1.7.7.cmml" xref="S5.E4.m1.7.7"><ci id="S5.E4.m1.7.7.1.cmml" xref="S5.E4.m1.7.7.1">~</ci><ci id="S5.E4.m1.7.7.2.cmml" xref="S5.E4.m1.7.7.2">𝛼</ci></apply></apply></apply><apply id="S5.E4.m1.8.8.1.1.2.2.2.3.cmml" xref="S5.E4.m1.8.8.1.1.2.2.2.3"><minus id="S5.E4.m1.8.8.1.1.2.2.2.3.1.cmml" xref="S5.E4.m1.8.8.1.1.2.2.2.3.1"></minus><ci id="S5.E4.m1.8.8.1.1.2.2.2.3.2.cmml" xref="S5.E4.m1.8.8.1.1.2.2.2.3.2">𝑚</ci><ci id="S5.E4.m1.8.8.1.1.2.2.2.3.3.cmml" xref="S5.E4.m1.8.8.1.1.2.2.2.3.3">𝑗</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E4.m1.8c">\partial^{\widetilde{\alpha}}:B_{p,p}^{s-j-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1% };X)\to B_{p,p}^{s-m-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)\quad\text{ for % all }|\widetilde{\alpha}|\leq m-j,</annotation><annotation encoding="application/x-llamapun" id="S5.E4.m1.8d">∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT : italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) for all | over~ start_ARG italic_α end_ARG | ≤ italic_m - italic_j ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.5">is bounded as well. Moreover, by Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem1" title="Definition 5.1 (Boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.I1.i2" title="item ii ‣ Definition 5.1 (Boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Example 14.4.33]</cite>, the pointwise multiplication operator</p> <table class="ltx_equation ltx_eqn_table" id="S5.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="M_{b_{\alpha}}:B_{p,p}^{s-m-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};X)\to B_{p,p}% ^{s-m-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};Y)" class="ltx_Math" display="block" id="S5.E5.m1.8"><semantics id="S5.E5.m1.8a"><mrow id="S5.E5.m1.8.8" xref="S5.E5.m1.8.8.cmml"><msub id="S5.E5.m1.8.8.4" xref="S5.E5.m1.8.8.4.cmml"><mi id="S5.E5.m1.8.8.4.2" xref="S5.E5.m1.8.8.4.2.cmml">M</mi><msub id="S5.E5.m1.8.8.4.3" 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start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) → italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s - italic_m - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.1.p1.6">is bounded. Combining (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E1" title="In 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.1</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E2" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.2</span></a>), (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E4" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.4</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E5" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.5</span></a>), yields the result.</p> </div> <div class="ltx_para" id="S5.2.p2"> <p class="ltx_p" id="S5.2.p2.1"><span class="ltx_text ltx_font_italic" id="S5.2.p2.1.1">Step 2: proof of <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.I2.i2" title="item ii ‣ Lemma 5.2. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>.</span> This case is analogous to Step 1 using Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4" title="Theorem 4.4. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a> to obtain (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E2" title="In Proof. ‣ 5. 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xref="S5.2.p2.1.m1.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S5.2.p2.1.m1.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S5.2.p2.1.m1.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.2.p2.1.m1.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S5.2.p2.1.m1.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>. ∎</p> </div> </div> <div class="ltx_para" id="S5.p4"> <p class="ltx_p" id="S5.p4.1">We make the following definitions about (systems of) normal boundary operators. For the definition of a (co)retraction, we refer to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>, Section 1.2.4]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.1.1.1">Definition 5.3</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.2.2"> </span>(Normal boundary operators)<span class="ltx_text ltx_font_bold" id="S5.Thmtheorem3.3.3">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem3.p1"> <p class="ltx_p" id="S5.Thmtheorem3.p1.3">An operator <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.1.m1.1"><semantics id="S5.Thmtheorem3.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem3.p1.1.m1.1.1" xref="S5.Thmtheorem3.p1.1.m1.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.1.m1.1b"><ci id="S5.Thmtheorem3.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem3.p1.1.m1.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.1.m1.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.1.m1.1d">caligraphic_B</annotation></semantics></math> of type <math alttext="(p,s,\gamma,m,Y)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.2.m2.5"><semantics id="S5.Thmtheorem3.p1.2.m2.5a"><mrow id="S5.Thmtheorem3.p1.2.m2.5.6.2" xref="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml"><mo id="S5.Thmtheorem3.p1.2.m2.5.6.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.2.m2.1.1" xref="S5.Thmtheorem3.p1.2.m2.1.1.cmml">p</mi><mo id="S5.Thmtheorem3.p1.2.m2.5.6.2.2" xref="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.2.m2.2.2" xref="S5.Thmtheorem3.p1.2.m2.2.2.cmml">s</mi><mo id="S5.Thmtheorem3.p1.2.m2.5.6.2.3" xref="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.2.m2.3.3" xref="S5.Thmtheorem3.p1.2.m2.3.3.cmml">γ</mi><mo id="S5.Thmtheorem3.p1.2.m2.5.6.2.4" xref="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.2.m2.4.4" xref="S5.Thmtheorem3.p1.2.m2.4.4.cmml">m</mi><mo id="S5.Thmtheorem3.p1.2.m2.5.6.2.5" xref="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.2.m2.5.5" xref="S5.Thmtheorem3.p1.2.m2.5.5.cmml">Y</mi><mo id="S5.Thmtheorem3.p1.2.m2.5.6.2.6" stretchy="false" xref="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.2.m2.5b"><vector id="S5.Thmtheorem3.p1.2.m2.5.6.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.5.6.2"><ci id="S5.Thmtheorem3.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem3.p1.2.m2.1.1">𝑝</ci><ci id="S5.Thmtheorem3.p1.2.m2.2.2.cmml" xref="S5.Thmtheorem3.p1.2.m2.2.2">𝑠</ci><ci id="S5.Thmtheorem3.p1.2.m2.3.3.cmml" xref="S5.Thmtheorem3.p1.2.m2.3.3">𝛾</ci><ci id="S5.Thmtheorem3.p1.2.m2.4.4.cmml" xref="S5.Thmtheorem3.p1.2.m2.4.4">𝑚</ci><ci id="S5.Thmtheorem3.p1.2.m2.5.5.cmml" xref="S5.Thmtheorem3.p1.2.m2.5.5">𝑌</ci></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem3.p1.2.m2.5c">(p,s,\gamma,m,Y)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem3.p1.2.m2.5d">( italic_p , italic_s , italic_γ , italic_m , italic_Y )</annotation></semantics></math> is a <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem3.p1.3.1">normal boundary operator of type <math alttext="(p,s,\gamma,m,Y)" class="ltx_Math" display="inline" id="S5.Thmtheorem3.p1.3.1.m1.5"><semantics id="S5.Thmtheorem3.p1.3.1.m1.5a"><mrow id="S5.Thmtheorem3.p1.3.1.m1.5.6.2" xref="S5.Thmtheorem3.p1.3.1.m1.5.6.1.cmml"><mo id="S5.Thmtheorem3.p1.3.1.m1.5.6.2.1" stretchy="false" xref="S5.Thmtheorem3.p1.3.1.m1.5.6.1.cmml">(</mo><mi id="S5.Thmtheorem3.p1.3.1.m1.1.1" xref="S5.Thmtheorem3.p1.3.1.m1.1.1.cmml">p</mi><mo id="S5.Thmtheorem3.p1.3.1.m1.5.6.2.2" xref="S5.Thmtheorem3.p1.3.1.m1.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.3.1.m1.2.2" xref="S5.Thmtheorem3.p1.3.1.m1.2.2.cmml">s</mi><mo id="S5.Thmtheorem3.p1.3.1.m1.5.6.2.3" xref="S5.Thmtheorem3.p1.3.1.m1.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.3.1.m1.3.3" xref="S5.Thmtheorem3.p1.3.1.m1.3.3.cmml">γ</mi><mo id="S5.Thmtheorem3.p1.3.1.m1.5.6.2.4" xref="S5.Thmtheorem3.p1.3.1.m1.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.3.1.m1.4.4" xref="S5.Thmtheorem3.p1.3.1.m1.4.4.cmml">m</mi><mo id="S5.Thmtheorem3.p1.3.1.m1.5.6.2.5" xref="S5.Thmtheorem3.p1.3.1.m1.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem3.p1.3.1.m1.5.5" xref="S5.Thmtheorem3.p1.3.1.m1.5.5.cmml">Y</mi><mo id="S5.Thmtheorem3.p1.3.1.m1.5.6.2.6" stretchy="false" xref="S5.Thmtheorem3.p1.3.1.m1.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem3.p1.3.1.m1.5b"><vector 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id="S5.I3.i1.p1"> <p class="ltx_p" id="S5.I3.i1.p1.2">the leading order coefficient in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E1" title="In 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.1</span></a>), <math alttext="b_{m}(\widetilde{x})\in\mathcal{L}(X,Y)" class="ltx_Math" display="inline" id="S5.I3.i1.p1.1.m1.3"><semantics id="S5.I3.i1.p1.1.m1.3a"><mrow id="S5.I3.i1.p1.1.m1.3.4" xref="S5.I3.i1.p1.1.m1.3.4.cmml"><mrow id="S5.I3.i1.p1.1.m1.3.4.2" xref="S5.I3.i1.p1.1.m1.3.4.2.cmml"><msub id="S5.I3.i1.p1.1.m1.3.4.2.2" xref="S5.I3.i1.p1.1.m1.3.4.2.2.cmml"><mi id="S5.I3.i1.p1.1.m1.3.4.2.2.2" xref="S5.I3.i1.p1.1.m1.3.4.2.2.2.cmml">b</mi><mi id="S5.I3.i1.p1.1.m1.3.4.2.2.3" xref="S5.I3.i1.p1.1.m1.3.4.2.2.3.cmml">m</mi></msub><mo id="S5.I3.i1.p1.1.m1.3.4.2.1" xref="S5.I3.i1.p1.1.m1.3.4.2.1.cmml">⁢</mo><mrow id="S5.I3.i1.p1.1.m1.3.4.2.3.2" xref="S5.I3.i1.p1.1.m1.1.1.cmml"><mo id="S5.I3.i1.p1.1.m1.3.4.2.3.2.1" stretchy="false" xref="S5.I3.i1.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S5.I3.i1.p1.1.m1.1.1" xref="S5.I3.i1.p1.1.m1.1.1.cmml"><mi id="S5.I3.i1.p1.1.m1.1.1.2" xref="S5.I3.i1.p1.1.m1.1.1.2.cmml">x</mi><mo id="S5.I3.i1.p1.1.m1.1.1.1" xref="S5.I3.i1.p1.1.m1.1.1.1.cmml">~</mo></mover><mo id="S5.I3.i1.p1.1.m1.3.4.2.3.2.2" stretchy="false" xref="S5.I3.i1.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><mo id="S5.I3.i1.p1.1.m1.3.4.1" xref="S5.I3.i1.p1.1.m1.3.4.1.cmml">∈</mo><mrow id="S5.I3.i1.p1.1.m1.3.4.3" xref="S5.I3.i1.p1.1.m1.3.4.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I3.i1.p1.1.m1.3.4.3.2" xref="S5.I3.i1.p1.1.m1.3.4.3.2.cmml">ℒ</mi><mo id="S5.I3.i1.p1.1.m1.3.4.3.1" xref="S5.I3.i1.p1.1.m1.3.4.3.1.cmml">⁢</mo><mrow id="S5.I3.i1.p1.1.m1.3.4.3.3.2" xref="S5.I3.i1.p1.1.m1.3.4.3.3.1.cmml"><mo id="S5.I3.i1.p1.1.m1.3.4.3.3.2.1" stretchy="false" xref="S5.I3.i1.p1.1.m1.3.4.3.3.1.cmml">(</mo><mi id="S5.I3.i1.p1.1.m1.2.2" xref="S5.I3.i1.p1.1.m1.2.2.cmml">X</mi><mo id="S5.I3.i1.p1.1.m1.3.4.3.3.2.2" xref="S5.I3.i1.p1.1.m1.3.4.3.3.1.cmml">,</mo><mi id="S5.I3.i1.p1.1.m1.3.3" xref="S5.I3.i1.p1.1.m1.3.3.cmml">Y</mi><mo id="S5.I3.i1.p1.1.m1.3.4.3.3.2.3" stretchy="false" xref="S5.I3.i1.p1.1.m1.3.4.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.1.m1.3b"><apply id="S5.I3.i1.p1.1.m1.3.4.cmml" xref="S5.I3.i1.p1.1.m1.3.4"><in id="S5.I3.i1.p1.1.m1.3.4.1.cmml" xref="S5.I3.i1.p1.1.m1.3.4.1"></in><apply id="S5.I3.i1.p1.1.m1.3.4.2.cmml" xref="S5.I3.i1.p1.1.m1.3.4.2"><times id="S5.I3.i1.p1.1.m1.3.4.2.1.cmml" xref="S5.I3.i1.p1.1.m1.3.4.2.1"></times><apply id="S5.I3.i1.p1.1.m1.3.4.2.2.cmml" xref="S5.I3.i1.p1.1.m1.3.4.2.2"><csymbol cd="ambiguous" id="S5.I3.i1.p1.1.m1.3.4.2.2.1.cmml" xref="S5.I3.i1.p1.1.m1.3.4.2.2">subscript</csymbol><ci id="S5.I3.i1.p1.1.m1.3.4.2.2.2.cmml" xref="S5.I3.i1.p1.1.m1.3.4.2.2.2">𝑏</ci><ci id="S5.I3.i1.p1.1.m1.3.4.2.2.3.cmml" xref="S5.I3.i1.p1.1.m1.3.4.2.2.3">𝑚</ci></apply><apply id="S5.I3.i1.p1.1.m1.1.1.cmml" xref="S5.I3.i1.p1.1.m1.3.4.2.3.2"><ci id="S5.I3.i1.p1.1.m1.1.1.1.cmml" xref="S5.I3.i1.p1.1.m1.1.1.1">~</ci><ci id="S5.I3.i1.p1.1.m1.1.1.2.cmml" xref="S5.I3.i1.p1.1.m1.1.1.2">𝑥</ci></apply></apply><apply id="S5.I3.i1.p1.1.m1.3.4.3.cmml" xref="S5.I3.i1.p1.1.m1.3.4.3"><times id="S5.I3.i1.p1.1.m1.3.4.3.1.cmml" xref="S5.I3.i1.p1.1.m1.3.4.3.1"></times><ci id="S5.I3.i1.p1.1.m1.3.4.3.2.cmml" xref="S5.I3.i1.p1.1.m1.3.4.3.2">ℒ</ci><interval closure="open" id="S5.I3.i1.p1.1.m1.3.4.3.3.1.cmml" xref="S5.I3.i1.p1.1.m1.3.4.3.3.2"><ci id="S5.I3.i1.p1.1.m1.2.2.cmml" xref="S5.I3.i1.p1.1.m1.2.2">𝑋</ci><ci id="S5.I3.i1.p1.1.m1.3.3.cmml" xref="S5.I3.i1.p1.1.m1.3.3">𝑌</ci></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.1.m1.3c">b_{m}(\widetilde{x})\in\mathcal{L}(X,Y)</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.1.m1.3d">italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ( over~ start_ARG italic_x end_ARG ) ∈ caligraphic_L ( italic_X , italic_Y )</annotation></semantics></math> is a retraction for <math alttext="\widetilde{x}\in\mathbb{R}^{d-1}" class="ltx_Math" display="inline" id="S5.I3.i1.p1.2.m2.1"><semantics id="S5.I3.i1.p1.2.m2.1a"><mrow id="S5.I3.i1.p1.2.m2.1.1" xref="S5.I3.i1.p1.2.m2.1.1.cmml"><mover accent="true" id="S5.I3.i1.p1.2.m2.1.1.2" xref="S5.I3.i1.p1.2.m2.1.1.2.cmml"><mi id="S5.I3.i1.p1.2.m2.1.1.2.2" xref="S5.I3.i1.p1.2.m2.1.1.2.2.cmml">x</mi><mo id="S5.I3.i1.p1.2.m2.1.1.2.1" xref="S5.I3.i1.p1.2.m2.1.1.2.1.cmml">~</mo></mover><mo id="S5.I3.i1.p1.2.m2.1.1.1" xref="S5.I3.i1.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S5.I3.i1.p1.2.m2.1.1.3" xref="S5.I3.i1.p1.2.m2.1.1.3.cmml"><mi id="S5.I3.i1.p1.2.m2.1.1.3.2" xref="S5.I3.i1.p1.2.m2.1.1.3.2.cmml">ℝ</mi><mrow id="S5.I3.i1.p1.2.m2.1.1.3.3" xref="S5.I3.i1.p1.2.m2.1.1.3.3.cmml"><mi id="S5.I3.i1.p1.2.m2.1.1.3.3.2" xref="S5.I3.i1.p1.2.m2.1.1.3.3.2.cmml">d</mi><mo id="S5.I3.i1.p1.2.m2.1.1.3.3.1" xref="S5.I3.i1.p1.2.m2.1.1.3.3.1.cmml">−</mo><mn id="S5.I3.i1.p1.2.m2.1.1.3.3.3" xref="S5.I3.i1.p1.2.m2.1.1.3.3.3.cmml">1</mn></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i1.p1.2.m2.1b"><apply id="S5.I3.i1.p1.2.m2.1.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1"><in id="S5.I3.i1.p1.2.m2.1.1.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1.1"></in><apply id="S5.I3.i1.p1.2.m2.1.1.2.cmml" xref="S5.I3.i1.p1.2.m2.1.1.2"><ci id="S5.I3.i1.p1.2.m2.1.1.2.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1.2.1">~</ci><ci id="S5.I3.i1.p1.2.m2.1.1.2.2.cmml" xref="S5.I3.i1.p1.2.m2.1.1.2.2">𝑥</ci></apply><apply id="S5.I3.i1.p1.2.m2.1.1.3.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.I3.i1.p1.2.m2.1.1.3.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3">superscript</csymbol><ci id="S5.I3.i1.p1.2.m2.1.1.3.2.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3.2">ℝ</ci><apply id="S5.I3.i1.p1.2.m2.1.1.3.3.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3.3"><minus id="S5.I3.i1.p1.2.m2.1.1.3.3.1.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3.3.1"></minus><ci id="S5.I3.i1.p1.2.m2.1.1.3.3.2.cmml" xref="S5.I3.i1.p1.2.m2.1.1.3.3.2">𝑑</ci><cn id="S5.I3.i1.p1.2.m2.1.1.3.3.3.cmml" type="integer" xref="S5.I3.i1.p1.2.m2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i1.p1.2.m2.1c">\widetilde{x}\in\mathbb{R}^{d-1}</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i1.p1.2.m2.1d">over~ start_ARG italic_x end_ARG ∈ blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT</annotation></semantics></math>,</p> </div> </li> <li class="ltx_item" id="S5.I3.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S5.I3.i2.p1"> <p class="ltx_p" id="S5.I3.i2.p1.3">there exists a coretraction <math alttext="b_{m}^{{\rm c}}(\widetilde{x})" class="ltx_Math" display="inline" id="S5.I3.i2.p1.1.m1.1"><semantics id="S5.I3.i2.p1.1.m1.1a"><mrow id="S5.I3.i2.p1.1.m1.1.2" xref="S5.I3.i2.p1.1.m1.1.2.cmml"><msubsup id="S5.I3.i2.p1.1.m1.1.2.2" xref="S5.I3.i2.p1.1.m1.1.2.2.cmml"><mi id="S5.I3.i2.p1.1.m1.1.2.2.2.2" xref="S5.I3.i2.p1.1.m1.1.2.2.2.2.cmml">b</mi><mi id="S5.I3.i2.p1.1.m1.1.2.2.2.3" xref="S5.I3.i2.p1.1.m1.1.2.2.2.3.cmml">m</mi><mi id="S5.I3.i2.p1.1.m1.1.2.2.3" mathvariant="normal" xref="S5.I3.i2.p1.1.m1.1.2.2.3.cmml">c</mi></msubsup><mo id="S5.I3.i2.p1.1.m1.1.2.1" xref="S5.I3.i2.p1.1.m1.1.2.1.cmml">⁢</mo><mrow id="S5.I3.i2.p1.1.m1.1.2.3.2" xref="S5.I3.i2.p1.1.m1.1.1.cmml"><mo id="S5.I3.i2.p1.1.m1.1.2.3.2.1" stretchy="false" xref="S5.I3.i2.p1.1.m1.1.1.cmml">(</mo><mover accent="true" id="S5.I3.i2.p1.1.m1.1.1" xref="S5.I3.i2.p1.1.m1.1.1.cmml"><mi id="S5.I3.i2.p1.1.m1.1.1.2" xref="S5.I3.i2.p1.1.m1.1.1.2.cmml">x</mi><mo id="S5.I3.i2.p1.1.m1.1.1.1" xref="S5.I3.i2.p1.1.m1.1.1.1.cmml">~</mo></mover><mo id="S5.I3.i2.p1.1.m1.1.2.3.2.2" stretchy="false" xref="S5.I3.i2.p1.1.m1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.1.m1.1b"><apply id="S5.I3.i2.p1.1.m1.1.2.cmml" 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id="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.3.2.2" xref="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml">,</mo><mi id="S5.I3.i2.p1.2.m2.2.2" xref="S5.I3.i2.p1.2.m2.2.2.cmml">X</mi><mo id="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.3.2.3" stretchy="false" xref="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml">)</mo></mrow></mrow><mo id="S5.I3.i2.p1.2.m2.4.4.2.2.2.5" stretchy="false" xref="S5.I3.i2.p1.2.m2.4.4.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.2.m2.4b"><apply id="S5.I3.i2.p1.2.m2.4.4.cmml" xref="S5.I3.i2.p1.2.m2.4.4"><in id="S5.I3.i2.p1.2.m2.4.4.3.cmml" xref="S5.I3.i2.p1.2.m2.4.4.3"></in><apply id="S5.I3.i2.p1.2.m2.4.4.4.cmml" xref="S5.I3.i2.p1.2.m2.4.4.4"><csymbol cd="ambiguous" id="S5.I3.i2.p1.2.m2.4.4.4.1.cmml" xref="S5.I3.i2.p1.2.m2.4.4.4">superscript</csymbol><apply id="S5.I3.i2.p1.2.m2.4.4.4.2.cmml" xref="S5.I3.i2.p1.2.m2.4.4.4"><csymbol cd="ambiguous" id="S5.I3.i2.p1.2.m2.4.4.4.2.1.cmml" xref="S5.I3.i2.p1.2.m2.4.4.4">subscript</csymbol><ci 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id="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.2.cmml" xref="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.2">ℒ</ci><interval closure="open" id="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.3.1.cmml" xref="S5.I3.i2.p1.2.m2.4.4.2.2.2.2.3.2"><ci id="S5.I3.i2.p1.2.m2.1.1.cmml" xref="S5.I3.i2.p1.2.m2.1.1">𝑌</ci><ci id="S5.I3.i2.p1.2.m2.2.2.cmml" xref="S5.I3.i2.p1.2.m2.2.2">𝑋</ci></interval></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.2.m2.4c">b_{m}^{{\rm c}}\in C_{{\rm b}}^{\ell}(\mathbb{R}^{d-1};\mathcal{L}(Y,X))</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.2.m2.4d">italic_b start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT ∈ italic_C start_POSTSUBSCRIPT roman_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; caligraphic_L ( italic_Y , italic_X ) )</annotation></semantics></math>, where <math alttext="\ell" class="ltx_Math" display="inline" id="S5.I3.i2.p1.3.m3.1"><semantics id="S5.I3.i2.p1.3.m3.1a"><mi id="S5.I3.i2.p1.3.m3.1.1" mathvariant="normal" xref="S5.I3.i2.p1.3.m3.1.1.cmml">ℓ</mi><annotation-xml encoding="MathML-Content" id="S5.I3.i2.p1.3.m3.1b"><ci id="S5.I3.i2.p1.3.m3.1.1.cmml" xref="S5.I3.i2.p1.3.m3.1.1">ℓ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I3.i2.p1.3.m3.1c">\ell</annotation><annotation encoding="application/x-llamapun" id="S5.I3.i2.p1.3.m3.1d">roman_ℓ</annotation></semantics></math> is as in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem1" title="Definition 5.1 (Boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.1</span></a>.</p> </div> </li> </ol> </div> </div> <div class="ltx_theorem ltx_theorem_definition" id="S5.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.1.1.1">Definition 5.4</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.2.2"> </span>(System of normal boundary operators)<span class="ltx_text ltx_font_bold" id="S5.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem4.p1"> <p class="ltx_p" id="S5.Thmtheorem4.p1.1">Let <math alttext="n\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.1.m1.1"><semantics id="S5.Thmtheorem4.p1.1.m1.1a"><mrow id="S5.Thmtheorem4.p1.1.m1.1.1" xref="S5.Thmtheorem4.p1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem4.p1.1.m1.1.1.2" xref="S5.Thmtheorem4.p1.1.m1.1.1.2.cmml">n</mi><mo id="S5.Thmtheorem4.p1.1.m1.1.1.1" xref="S5.Thmtheorem4.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem4.p1.1.m1.1.1.3" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.cmml"><mi id="S5.Thmtheorem4.p1.1.m1.1.1.3.2" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S5.Thmtheorem4.p1.1.m1.1.1.3.3" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.1.m1.1b"><apply id="S5.Thmtheorem4.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1"><in id="S5.Thmtheorem4.p1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.1"></in><ci id="S5.Thmtheorem4.p1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.2">𝑛</ci><apply id="S5.Thmtheorem4.p1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem4.p1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.2">ℕ</ci><cn id="S5.Thmtheorem4.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.1.m1.1c">n\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.1.m1.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and</p> <ol class="ltx_enumerate" id="S5.I4"> <li class="ltx_item" id="S5.I4.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S5.I4.i1.p1"> <p class="ltx_p" id="S5.I4.i1.p1.1"><math alttext="0\leq m_{0}<m_{1}<\cdots<m_{n}" class="ltx_Math" display="inline" id="S5.I4.i1.p1.1.m1.1"><semantics id="S5.I4.i1.p1.1.m1.1a"><mrow id="S5.I4.i1.p1.1.m1.1.1" xref="S5.I4.i1.p1.1.m1.1.1.cmml"><mn id="S5.I4.i1.p1.1.m1.1.1.2" xref="S5.I4.i1.p1.1.m1.1.1.2.cmml">0</mn><mo id="S5.I4.i1.p1.1.m1.1.1.3" xref="S5.I4.i1.p1.1.m1.1.1.3.cmml">≤</mo><msub id="S5.I4.i1.p1.1.m1.1.1.4" xref="S5.I4.i1.p1.1.m1.1.1.4.cmml"><mi id="S5.I4.i1.p1.1.m1.1.1.4.2" xref="S5.I4.i1.p1.1.m1.1.1.4.2.cmml">m</mi><mn id="S5.I4.i1.p1.1.m1.1.1.4.3" xref="S5.I4.i1.p1.1.m1.1.1.4.3.cmml">0</mn></msub><mo id="S5.I4.i1.p1.1.m1.1.1.5" xref="S5.I4.i1.p1.1.m1.1.1.5.cmml"><</mo><msub id="S5.I4.i1.p1.1.m1.1.1.6" xref="S5.I4.i1.p1.1.m1.1.1.6.cmml"><mi id="S5.I4.i1.p1.1.m1.1.1.6.2" xref="S5.I4.i1.p1.1.m1.1.1.6.2.cmml">m</mi><mn id="S5.I4.i1.p1.1.m1.1.1.6.3" xref="S5.I4.i1.p1.1.m1.1.1.6.3.cmml">1</mn></msub><mo id="S5.I4.i1.p1.1.m1.1.1.7" xref="S5.I4.i1.p1.1.m1.1.1.7.cmml"><</mo><mi id="S5.I4.i1.p1.1.m1.1.1.8" mathvariant="normal" xref="S5.I4.i1.p1.1.m1.1.1.8.cmml">⋯</mi><mo id="S5.I4.i1.p1.1.m1.1.1.9" xref="S5.I4.i1.p1.1.m1.1.1.9.cmml"><</mo><msub id="S5.I4.i1.p1.1.m1.1.1.10" xref="S5.I4.i1.p1.1.m1.1.1.10.cmml"><mi id="S5.I4.i1.p1.1.m1.1.1.10.2" xref="S5.I4.i1.p1.1.m1.1.1.10.2.cmml">m</mi><mi id="S5.I4.i1.p1.1.m1.1.1.10.3" xref="S5.I4.i1.p1.1.m1.1.1.10.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i1.p1.1.m1.1b"><apply id="S5.I4.i1.p1.1.m1.1.1.cmml" xref="S5.I4.i1.p1.1.m1.1.1"><and id="S5.I4.i1.p1.1.m1.1.1a.cmml" xref="S5.I4.i1.p1.1.m1.1.1"></and><apply id="S5.I4.i1.p1.1.m1.1.1b.cmml" xref="S5.I4.i1.p1.1.m1.1.1"><leq id="S5.I4.i1.p1.1.m1.1.1.3.cmml" xref="S5.I4.i1.p1.1.m1.1.1.3"></leq><cn id="S5.I4.i1.p1.1.m1.1.1.2.cmml" type="integer" xref="S5.I4.i1.p1.1.m1.1.1.2">0</cn><apply id="S5.I4.i1.p1.1.m1.1.1.4.cmml" xref="S5.I4.i1.p1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S5.I4.i1.p1.1.m1.1.1.4.1.cmml" xref="S5.I4.i1.p1.1.m1.1.1.4">subscript</csymbol><ci id="S5.I4.i1.p1.1.m1.1.1.4.2.cmml" xref="S5.I4.i1.p1.1.m1.1.1.4.2">𝑚</ci><cn id="S5.I4.i1.p1.1.m1.1.1.4.3.cmml" type="integer" xref="S5.I4.i1.p1.1.m1.1.1.4.3">0</cn></apply></apply><apply id="S5.I4.i1.p1.1.m1.1.1c.cmml" xref="S5.I4.i1.p1.1.m1.1.1"><lt id="S5.I4.i1.p1.1.m1.1.1.5.cmml" xref="S5.I4.i1.p1.1.m1.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S5.I4.i1.p1.1.m1.1.1.4.cmml" id="S5.I4.i1.p1.1.m1.1.1d.cmml" xref="S5.I4.i1.p1.1.m1.1.1"></share><apply id="S5.I4.i1.p1.1.m1.1.1.6.cmml" xref="S5.I4.i1.p1.1.m1.1.1.6"><csymbol cd="ambiguous" id="S5.I4.i1.p1.1.m1.1.1.6.1.cmml" xref="S5.I4.i1.p1.1.m1.1.1.6">subscript</csymbol><ci id="S5.I4.i1.p1.1.m1.1.1.6.2.cmml" xref="S5.I4.i1.p1.1.m1.1.1.6.2">𝑚</ci><cn id="S5.I4.i1.p1.1.m1.1.1.6.3.cmml" type="integer" xref="S5.I4.i1.p1.1.m1.1.1.6.3">1</cn></apply></apply><apply id="S5.I4.i1.p1.1.m1.1.1e.cmml" xref="S5.I4.i1.p1.1.m1.1.1"><lt id="S5.I4.i1.p1.1.m1.1.1.7.cmml" xref="S5.I4.i1.p1.1.m1.1.1.7"></lt><share href="https://arxiv.org/html/2503.14636v1#S5.I4.i1.p1.1.m1.1.1.6.cmml" id="S5.I4.i1.p1.1.m1.1.1f.cmml" xref="S5.I4.i1.p1.1.m1.1.1"></share><ci id="S5.I4.i1.p1.1.m1.1.1.8.cmml" xref="S5.I4.i1.p1.1.m1.1.1.8">⋯</ci></apply><apply id="S5.I4.i1.p1.1.m1.1.1g.cmml" xref="S5.I4.i1.p1.1.m1.1.1"><lt id="S5.I4.i1.p1.1.m1.1.1.9.cmml" xref="S5.I4.i1.p1.1.m1.1.1.9"></lt><share href="https://arxiv.org/html/2503.14636v1#S5.I4.i1.p1.1.m1.1.1.8.cmml" id="S5.I4.i1.p1.1.m1.1.1h.cmml" xref="S5.I4.i1.p1.1.m1.1.1"></share><apply id="S5.I4.i1.p1.1.m1.1.1.10.cmml" xref="S5.I4.i1.p1.1.m1.1.1.10"><csymbol cd="ambiguous" id="S5.I4.i1.p1.1.m1.1.1.10.1.cmml" xref="S5.I4.i1.p1.1.m1.1.1.10">subscript</csymbol><ci id="S5.I4.i1.p1.1.m1.1.1.10.2.cmml" xref="S5.I4.i1.p1.1.m1.1.1.10.2">𝑚</ci><ci id="S5.I4.i1.p1.1.m1.1.1.10.3.cmml" xref="S5.I4.i1.p1.1.m1.1.1.10.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i1.p1.1.m1.1c">0\leq m_{0}<m_{1}<\cdots<m_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i1.p1.1.m1.1d">0 ≤ italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < italic_m start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < ⋯ < italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> be integers,</p> </div> </li> <li class="ltx_item" id="S5.I4.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S5.I4.i2.p1"> <p class="ltx_p" id="S5.I4.i2.p1.1"><math alttext="Y_{0},\dots,Y_{n}" class="ltx_Math" display="inline" id="S5.I4.i2.p1.1.m1.3"><semantics id="S5.I4.i2.p1.1.m1.3a"><mrow id="S5.I4.i2.p1.1.m1.3.3.2" xref="S5.I4.i2.p1.1.m1.3.3.3.cmml"><msub id="S5.I4.i2.p1.1.m1.2.2.1.1" xref="S5.I4.i2.p1.1.m1.2.2.1.1.cmml"><mi id="S5.I4.i2.p1.1.m1.2.2.1.1.2" xref="S5.I4.i2.p1.1.m1.2.2.1.1.2.cmml">Y</mi><mn id="S5.I4.i2.p1.1.m1.2.2.1.1.3" xref="S5.I4.i2.p1.1.m1.2.2.1.1.3.cmml">0</mn></msub><mo id="S5.I4.i2.p1.1.m1.3.3.2.3" xref="S5.I4.i2.p1.1.m1.3.3.3.cmml">,</mo><mi id="S5.I4.i2.p1.1.m1.1.1" mathvariant="normal" xref="S5.I4.i2.p1.1.m1.1.1.cmml">…</mi><mo id="S5.I4.i2.p1.1.m1.3.3.2.4" xref="S5.I4.i2.p1.1.m1.3.3.3.cmml">,</mo><msub id="S5.I4.i2.p1.1.m1.3.3.2.2" xref="S5.I4.i2.p1.1.m1.3.3.2.2.cmml"><mi id="S5.I4.i2.p1.1.m1.3.3.2.2.2" xref="S5.I4.i2.p1.1.m1.3.3.2.2.2.cmml">Y</mi><mi id="S5.I4.i2.p1.1.m1.3.3.2.2.3" xref="S5.I4.i2.p1.1.m1.3.3.2.2.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i2.p1.1.m1.3b"><list id="S5.I4.i2.p1.1.m1.3.3.3.cmml" xref="S5.I4.i2.p1.1.m1.3.3.2"><apply id="S5.I4.i2.p1.1.m1.2.2.1.1.cmml" xref="S5.I4.i2.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="S5.I4.i2.p1.1.m1.2.2.1.1.1.cmml" xref="S5.I4.i2.p1.1.m1.2.2.1.1">subscript</csymbol><ci id="S5.I4.i2.p1.1.m1.2.2.1.1.2.cmml" xref="S5.I4.i2.p1.1.m1.2.2.1.1.2">𝑌</ci><cn id="S5.I4.i2.p1.1.m1.2.2.1.1.3.cmml" type="integer" xref="S5.I4.i2.p1.1.m1.2.2.1.1.3">0</cn></apply><ci id="S5.I4.i2.p1.1.m1.1.1.cmml" xref="S5.I4.i2.p1.1.m1.1.1">…</ci><apply id="S5.I4.i2.p1.1.m1.3.3.2.2.cmml" xref="S5.I4.i2.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="S5.I4.i2.p1.1.m1.3.3.2.2.1.cmml" xref="S5.I4.i2.p1.1.m1.3.3.2.2">subscript</csymbol><ci id="S5.I4.i2.p1.1.m1.3.3.2.2.2.cmml" xref="S5.I4.i2.p1.1.m1.3.3.2.2.2">𝑌</ci><ci id="S5.I4.i2.p1.1.m1.3.3.2.2.3.cmml" xref="S5.I4.i2.p1.1.m1.3.3.2.2.3">𝑛</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i2.p1.1.m1.3c">Y_{0},\dots,Y_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i2.p1.1.m1.3d">italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_Y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> be Banach spaces,</p> </div> </li> <li class="ltx_item" id="S5.I4.i3" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(iii)</span> <div class="ltx_para" id="S5.I4.i3.p1"> <p class="ltx_p" id="S5.I4.i3.p1.2"><math alttext="\mathcal{B}^{m_{i}}" class="ltx_Math" display="inline" id="S5.I4.i3.p1.1.m1.1"><semantics id="S5.I4.i3.p1.1.m1.1a"><msup id="S5.I4.i3.p1.1.m1.1.1" xref="S5.I4.i3.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.I4.i3.p1.1.m1.1.1.2" xref="S5.I4.i3.p1.1.m1.1.1.2.cmml">ℬ</mi><msub id="S5.I4.i3.p1.1.m1.1.1.3" xref="S5.I4.i3.p1.1.m1.1.1.3.cmml"><mi id="S5.I4.i3.p1.1.m1.1.1.3.2" xref="S5.I4.i3.p1.1.m1.1.1.3.2.cmml">m</mi><mi id="S5.I4.i3.p1.1.m1.1.1.3.3" xref="S5.I4.i3.p1.1.m1.1.1.3.3.cmml">i</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.1.m1.1b"><apply id="S5.I4.i3.p1.1.m1.1.1.cmml" xref="S5.I4.i3.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.I4.i3.p1.1.m1.1.1.1.cmml" xref="S5.I4.i3.p1.1.m1.1.1">superscript</csymbol><ci id="S5.I4.i3.p1.1.m1.1.1.2.cmml" xref="S5.I4.i3.p1.1.m1.1.1.2">ℬ</ci><apply id="S5.I4.i3.p1.1.m1.1.1.3.cmml" xref="S5.I4.i3.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.I4.i3.p1.1.m1.1.1.3.1.cmml" xref="S5.I4.i3.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.I4.i3.p1.1.m1.1.1.3.2.cmml" xref="S5.I4.i3.p1.1.m1.1.1.3.2">𝑚</ci><ci id="S5.I4.i3.p1.1.m1.1.1.3.3.cmml" xref="S5.I4.i3.p1.1.m1.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.1.m1.1c">\mathcal{B}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.1.m1.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> be a normal boundary operator of type <math alttext="(p,s,\gamma,m_{i},Y_{i})" class="ltx_Math" display="inline" id="S5.I4.i3.p1.2.m2.5"><semantics id="S5.I4.i3.p1.2.m2.5a"><mrow id="S5.I4.i3.p1.2.m2.5.5.2" xref="S5.I4.i3.p1.2.m2.5.5.3.cmml"><mo id="S5.I4.i3.p1.2.m2.5.5.2.3" stretchy="false" xref="S5.I4.i3.p1.2.m2.5.5.3.cmml">(</mo><mi id="S5.I4.i3.p1.2.m2.1.1" xref="S5.I4.i3.p1.2.m2.1.1.cmml">p</mi><mo id="S5.I4.i3.p1.2.m2.5.5.2.4" xref="S5.I4.i3.p1.2.m2.5.5.3.cmml">,</mo><mi id="S5.I4.i3.p1.2.m2.2.2" xref="S5.I4.i3.p1.2.m2.2.2.cmml">s</mi><mo id="S5.I4.i3.p1.2.m2.5.5.2.5" xref="S5.I4.i3.p1.2.m2.5.5.3.cmml">,</mo><mi id="S5.I4.i3.p1.2.m2.3.3" xref="S5.I4.i3.p1.2.m2.3.3.cmml">γ</mi><mo id="S5.I4.i3.p1.2.m2.5.5.2.6" xref="S5.I4.i3.p1.2.m2.5.5.3.cmml">,</mo><msub id="S5.I4.i3.p1.2.m2.4.4.1.1" xref="S5.I4.i3.p1.2.m2.4.4.1.1.cmml"><mi id="S5.I4.i3.p1.2.m2.4.4.1.1.2" xref="S5.I4.i3.p1.2.m2.4.4.1.1.2.cmml">m</mi><mi id="S5.I4.i3.p1.2.m2.4.4.1.1.3" xref="S5.I4.i3.p1.2.m2.4.4.1.1.3.cmml">i</mi></msub><mo id="S5.I4.i3.p1.2.m2.5.5.2.7" xref="S5.I4.i3.p1.2.m2.5.5.3.cmml">,</mo><msub id="S5.I4.i3.p1.2.m2.5.5.2.2" xref="S5.I4.i3.p1.2.m2.5.5.2.2.cmml"><mi id="S5.I4.i3.p1.2.m2.5.5.2.2.2" xref="S5.I4.i3.p1.2.m2.5.5.2.2.2.cmml">Y</mi><mi id="S5.I4.i3.p1.2.m2.5.5.2.2.3" xref="S5.I4.i3.p1.2.m2.5.5.2.2.3.cmml">i</mi></msub><mo id="S5.I4.i3.p1.2.m2.5.5.2.8" stretchy="false" xref="S5.I4.i3.p1.2.m2.5.5.3.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I4.i3.p1.2.m2.5b"><vector id="S5.I4.i3.p1.2.m2.5.5.3.cmml" xref="S5.I4.i3.p1.2.m2.5.5.2"><ci id="S5.I4.i3.p1.2.m2.1.1.cmml" xref="S5.I4.i3.p1.2.m2.1.1">𝑝</ci><ci id="S5.I4.i3.p1.2.m2.2.2.cmml" xref="S5.I4.i3.p1.2.m2.2.2">𝑠</ci><ci id="S5.I4.i3.p1.2.m2.3.3.cmml" xref="S5.I4.i3.p1.2.m2.3.3">𝛾</ci><apply id="S5.I4.i3.p1.2.m2.4.4.1.1.cmml" xref="S5.I4.i3.p1.2.m2.4.4.1.1"><csymbol cd="ambiguous" id="S5.I4.i3.p1.2.m2.4.4.1.1.1.cmml" xref="S5.I4.i3.p1.2.m2.4.4.1.1">subscript</csymbol><ci id="S5.I4.i3.p1.2.m2.4.4.1.1.2.cmml" xref="S5.I4.i3.p1.2.m2.4.4.1.1.2">𝑚</ci><ci id="S5.I4.i3.p1.2.m2.4.4.1.1.3.cmml" xref="S5.I4.i3.p1.2.m2.4.4.1.1.3">𝑖</ci></apply><apply id="S5.I4.i3.p1.2.m2.5.5.2.2.cmml" xref="S5.I4.i3.p1.2.m2.5.5.2.2"><csymbol cd="ambiguous" id="S5.I4.i3.p1.2.m2.5.5.2.2.1.cmml" xref="S5.I4.i3.p1.2.m2.5.5.2.2">subscript</csymbol><ci id="S5.I4.i3.p1.2.m2.5.5.2.2.2.cmml" xref="S5.I4.i3.p1.2.m2.5.5.2.2.2">𝑌</ci><ci id="S5.I4.i3.p1.2.m2.5.5.2.2.3.cmml" xref="S5.I4.i3.p1.2.m2.5.5.2.2.3">𝑖</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.I4.i3.p1.2.m2.5c">(p,s,\gamma,m_{i},Y_{i})</annotation><annotation encoding="application/x-llamapun" id="S5.I4.i3.p1.2.m2.5d">( italic_p , italic_s , italic_γ , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math>.</p> </div> </li> </ol> <p class="ltx_p" id="S5.Thmtheorem4.p1.3">Let <math alttext="\overline{m}:=(m_{0},\dots,m_{n})" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.2.m1.3"><semantics id="S5.Thmtheorem4.p1.2.m1.3a"><mrow id="S5.Thmtheorem4.p1.2.m1.3.3" xref="S5.Thmtheorem4.p1.2.m1.3.3.cmml"><mover accent="true" id="S5.Thmtheorem4.p1.2.m1.3.3.4" xref="S5.Thmtheorem4.p1.2.m1.3.3.4.cmml"><mi id="S5.Thmtheorem4.p1.2.m1.3.3.4.2" xref="S5.Thmtheorem4.p1.2.m1.3.3.4.2.cmml">m</mi><mo id="S5.Thmtheorem4.p1.2.m1.3.3.4.1" xref="S5.Thmtheorem4.p1.2.m1.3.3.4.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem4.p1.2.m1.3.3.3" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem4.p1.2.m1.3.3.3.cmml">:=</mo><mrow id="S5.Thmtheorem4.p1.2.m1.3.3.2.2" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.3.cmml"><mo id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.3" stretchy="false" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.3.cmml">(</mo><msub id="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1" xref="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.2" xref="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.3" xref="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.4" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.3.cmml">,</mo><mi id="S5.Thmtheorem4.p1.2.m1.1.1" mathvariant="normal" xref="S5.Thmtheorem4.p1.2.m1.1.1.cmml">…</mi><mo id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.5" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.3.cmml">,</mo><msub id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.cmml"><mi id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.2" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.2.cmml">m</mi><mi id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.3" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.6" stretchy="false" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.2.m1.3b"><apply id="S5.Thmtheorem4.p1.2.m1.3.3.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3"><csymbol cd="latexml" id="S5.Thmtheorem4.p1.2.m1.3.3.3.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.3">assign</csymbol><apply id="S5.Thmtheorem4.p1.2.m1.3.3.4.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.4"><ci id="S5.Thmtheorem4.p1.2.m1.3.3.4.1.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.4.1">¯</ci><ci id="S5.Thmtheorem4.p1.2.m1.3.3.4.2.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.4.2">𝑚</ci></apply><vector id="S5.Thmtheorem4.p1.2.m1.3.3.2.3.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2"><apply id="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.cmml" xref="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.2">𝑚</ci><cn id="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.2.m1.2.2.1.1.1.3">0</cn></apply><ci id="S5.Thmtheorem4.p1.2.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.2.m1.1.1">…</ci><apply id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.1.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.2.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.2">𝑚</ci><ci id="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.3.cmml" xref="S5.Thmtheorem4.p1.2.m1.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.2.m1.3c">\overline{m}:=(m_{0},\dots,m_{n})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.2.m1.3d">over¯ start_ARG italic_m end_ARG := ( italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math> and <math alttext="\overline{Y}:=(Y_{0},\dots,Y_{n})" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.3.m2.3"><semantics id="S5.Thmtheorem4.p1.3.m2.3a"><mrow id="S5.Thmtheorem4.p1.3.m2.3.3" xref="S5.Thmtheorem4.p1.3.m2.3.3.cmml"><mover accent="true" id="S5.Thmtheorem4.p1.3.m2.3.3.4" xref="S5.Thmtheorem4.p1.3.m2.3.3.4.cmml"><mi id="S5.Thmtheorem4.p1.3.m2.3.3.4.2" xref="S5.Thmtheorem4.p1.3.m2.3.3.4.2.cmml">Y</mi><mo id="S5.Thmtheorem4.p1.3.m2.3.3.4.1" xref="S5.Thmtheorem4.p1.3.m2.3.3.4.1.cmml">¯</mo></mover><mo id="S5.Thmtheorem4.p1.3.m2.3.3.3" lspace="0.278em" rspace="0.278em" xref="S5.Thmtheorem4.p1.3.m2.3.3.3.cmml">:=</mo><mrow id="S5.Thmtheorem4.p1.3.m2.3.3.2.2" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.3.cmml"><mo id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.3" stretchy="false" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.3.cmml">(</mo><msub id="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1" xref="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.2" xref="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.2.cmml">Y</mi><mn id="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.3" xref="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.4" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.3.cmml">,</mo><mi id="S5.Thmtheorem4.p1.3.m2.1.1" mathvariant="normal" xref="S5.Thmtheorem4.p1.3.m2.1.1.cmml">…</mi><mo id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.5" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.3.cmml">,</mo><msub id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.cmml"><mi id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.2" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.2.cmml">Y</mi><mi id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.3" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.6" stretchy="false" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem4.p1.3.m2.3b"><apply id="S5.Thmtheorem4.p1.3.m2.3.3.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3"><csymbol cd="latexml" id="S5.Thmtheorem4.p1.3.m2.3.3.3.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.3">assign</csymbol><apply id="S5.Thmtheorem4.p1.3.m2.3.3.4.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.4"><ci id="S5.Thmtheorem4.p1.3.m2.3.3.4.1.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.4.1">¯</ci><ci id="S5.Thmtheorem4.p1.3.m2.3.3.4.2.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.4.2">𝑌</ci></apply><vector id="S5.Thmtheorem4.p1.3.m2.3.3.2.3.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2"><apply id="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.2">𝑌</ci><cn id="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem4.p1.3.m2.2.2.1.1.1.3">0</cn></apply><ci id="S5.Thmtheorem4.p1.3.m2.1.1.cmml" xref="S5.Thmtheorem4.p1.3.m2.1.1">…</ci><apply id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.1.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.2.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.2">𝑌</ci><ci id="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.3.cmml" xref="S5.Thmtheorem4.p1.3.m2.3.3.2.2.2.3">𝑛</ci></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.3.m2.3c">\overline{Y}:=(Y_{0},\dots,Y_{n})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.3.m2.3d">over¯ start_ARG italic_Y end_ARG := ( italic_Y start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_Y start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT )</annotation></semantics></math>. Then the operator</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}:=(\mathcal{B}^{m_{0}},\dots,\mathcal{B}^{m_{n}})" class="ltx_Math" display="block" id="S5.Ex4.m1.3"><semantics id="S5.Ex4.m1.3a"><mrow id="S5.Ex4.m1.3.3" xref="S5.Ex4.m1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex4.m1.3.3.4" xref="S5.Ex4.m1.3.3.4.cmml">ℬ</mi><mo id="S5.Ex4.m1.3.3.3" lspace="0.278em" rspace="0.278em" xref="S5.Ex4.m1.3.3.3.cmml">:=</mo><mrow id="S5.Ex4.m1.3.3.2.2" xref="S5.Ex4.m1.3.3.2.3.cmml"><mo id="S5.Ex4.m1.3.3.2.2.3" stretchy="false" xref="S5.Ex4.m1.3.3.2.3.cmml">(</mo><msup id="S5.Ex4.m1.2.2.1.1.1" xref="S5.Ex4.m1.2.2.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex4.m1.2.2.1.1.1.2" xref="S5.Ex4.m1.2.2.1.1.1.2.cmml">ℬ</mi><msub id="S5.Ex4.m1.2.2.1.1.1.3" xref="S5.Ex4.m1.2.2.1.1.1.3.cmml"><mi id="S5.Ex4.m1.2.2.1.1.1.3.2" xref="S5.Ex4.m1.2.2.1.1.1.3.2.cmml">m</mi><mn id="S5.Ex4.m1.2.2.1.1.1.3.3" xref="S5.Ex4.m1.2.2.1.1.1.3.3.cmml">0</mn></msub></msup><mo id="S5.Ex4.m1.3.3.2.2.4" xref="S5.Ex4.m1.3.3.2.3.cmml">,</mo><mi id="S5.Ex4.m1.1.1" mathvariant="normal" xref="S5.Ex4.m1.1.1.cmml">…</mi><mo id="S5.Ex4.m1.3.3.2.2.5" xref="S5.Ex4.m1.3.3.2.3.cmml">,</mo><msup id="S5.Ex4.m1.3.3.2.2.2" xref="S5.Ex4.m1.3.3.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex4.m1.3.3.2.2.2.2" xref="S5.Ex4.m1.3.3.2.2.2.2.cmml">ℬ</mi><msub id="S5.Ex4.m1.3.3.2.2.2.3" xref="S5.Ex4.m1.3.3.2.2.2.3.cmml"><mi id="S5.Ex4.m1.3.3.2.2.2.3.2" xref="S5.Ex4.m1.3.3.2.2.2.3.2.cmml">m</mi><mi id="S5.Ex4.m1.3.3.2.2.2.3.3" xref="S5.Ex4.m1.3.3.2.2.2.3.3.cmml">n</mi></msub></msup><mo id="S5.Ex4.m1.3.3.2.2.6" stretchy="false" xref="S5.Ex4.m1.3.3.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex4.m1.3b"><apply id="S5.Ex4.m1.3.3.cmml" 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xref="S5.Ex4.m1.3.3.2.2.2">superscript</csymbol><ci id="S5.Ex4.m1.3.3.2.2.2.2.cmml" xref="S5.Ex4.m1.3.3.2.2.2.2">ℬ</ci><apply id="S5.Ex4.m1.3.3.2.2.2.3.cmml" xref="S5.Ex4.m1.3.3.2.2.2.3"><csymbol cd="ambiguous" id="S5.Ex4.m1.3.3.2.2.2.3.1.cmml" xref="S5.Ex4.m1.3.3.2.2.2.3">subscript</csymbol><ci id="S5.Ex4.m1.3.3.2.2.2.3.2.cmml" xref="S5.Ex4.m1.3.3.2.2.2.3.2">𝑚</ci><ci id="S5.Ex4.m1.3.3.2.2.2.3.3.cmml" xref="S5.Ex4.m1.3.3.2.2.2.3.3">𝑛</ci></apply></apply></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex4.m1.3c">\mathcal{B}:=(\mathcal{B}^{m_{0}},\dots,\mathcal{B}^{m_{n}})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex4.m1.3d">caligraphic_B := ( caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT , … , caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.Thmtheorem4.p1.4">is a <em class="ltx_emph ltx_font_italic" id="S5.Thmtheorem4.p1.4.1">normal boundary operator of type <math alttext="(p,s,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S5.Thmtheorem4.p1.4.1.m1.5"><semantics id="S5.Thmtheorem4.p1.4.1.m1.5a"><mrow id="S5.Thmtheorem4.p1.4.1.m1.5.6.2" xref="S5.Thmtheorem4.p1.4.1.m1.5.6.1.cmml"><mo id="S5.Thmtheorem4.p1.4.1.m1.5.6.2.1" stretchy="false" xref="S5.Thmtheorem4.p1.4.1.m1.5.6.1.cmml">(</mo><mi id="S5.Thmtheorem4.p1.4.1.m1.1.1" xref="S5.Thmtheorem4.p1.4.1.m1.1.1.cmml">p</mi><mo id="S5.Thmtheorem4.p1.4.1.m1.5.6.2.2" xref="S5.Thmtheorem4.p1.4.1.m1.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem4.p1.4.1.m1.2.2" xref="S5.Thmtheorem4.p1.4.1.m1.2.2.cmml">s</mi><mo id="S5.Thmtheorem4.p1.4.1.m1.5.6.2.3" xref="S5.Thmtheorem4.p1.4.1.m1.5.6.1.cmml">,</mo><mi id="S5.Thmtheorem4.p1.4.1.m1.3.3" xref="S5.Thmtheorem4.p1.4.1.m1.3.3.cmml">γ</mi><mo 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xref="S5.Thmtheorem4.p1.4.1.m1.5.6.2"><ci id="S5.Thmtheorem4.p1.4.1.m1.1.1.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.1.1">𝑝</ci><ci id="S5.Thmtheorem4.p1.4.1.m1.2.2.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.2.2">𝑠</ci><ci id="S5.Thmtheorem4.p1.4.1.m1.3.3.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.3.3">𝛾</ci><apply id="S5.Thmtheorem4.p1.4.1.m1.4.4.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.4.4"><ci id="S5.Thmtheorem4.p1.4.1.m1.4.4.1.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.4.4.1">¯</ci><ci id="S5.Thmtheorem4.p1.4.1.m1.4.4.2.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.4.4.2">𝑚</ci></apply><apply id="S5.Thmtheorem4.p1.4.1.m1.5.5.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.5.5"><ci id="S5.Thmtheorem4.p1.4.1.m1.5.5.1.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.5.5.1">¯</ci><ci id="S5.Thmtheorem4.p1.4.1.m1.5.5.2.cmml" xref="S5.Thmtheorem4.p1.4.1.m1.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem4.p1.4.1.m1.5c">(p,s,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem4.p1.4.1.m1.5d">( italic_p , italic_s , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math></em>.</p> </div> </div> <div class="ltx_para" id="S5.p5"> <p class="ltx_p" id="S5.p5.9">Note that a normal boundary operator of type <math alttext="(p,s,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S5.p5.1.m1.5"><semantics id="S5.p5.1.m1.5a"><mrow id="S5.p5.1.m1.5.6.2" xref="S5.p5.1.m1.5.6.1.cmml"><mo id="S5.p5.1.m1.5.6.2.1" stretchy="false" xref="S5.p5.1.m1.5.6.1.cmml">(</mo><mi id="S5.p5.1.m1.1.1" xref="S5.p5.1.m1.1.1.cmml">p</mi><mo id="S5.p5.1.m1.5.6.2.2" xref="S5.p5.1.m1.5.6.1.cmml">,</mo><mi id="S5.p5.1.m1.2.2" xref="S5.p5.1.m1.2.2.cmml">s</mi><mo id="S5.p5.1.m1.5.6.2.3" xref="S5.p5.1.m1.5.6.1.cmml">,</mo><mi id="S5.p5.1.m1.3.3" xref="S5.p5.1.m1.3.3.cmml">γ</mi><mo id="S5.p5.1.m1.5.6.2.4" xref="S5.p5.1.m1.5.6.1.cmml">,</mo><mover accent="true" 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xref="S5.p5.1.m1.4.4.2">𝑚</ci></apply><apply id="S5.p5.1.m1.5.5.cmml" xref="S5.p5.1.m1.5.5"><ci id="S5.p5.1.m1.5.5.1.cmml" xref="S5.p5.1.m1.5.5.1">¯</ci><ci id="S5.p5.1.m1.5.5.2.cmml" xref="S5.p5.1.m1.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.1.m1.5c">(p,s,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S5.p5.1.m1.5d">( italic_p , italic_s , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math> satisfies <math alttext="s>m_{n}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S5.p5.2.m2.1"><semantics id="S5.p5.2.m2.1a"><mrow id="S5.p5.2.m2.1.1" xref="S5.p5.2.m2.1.1.cmml"><mi id="S5.p5.2.m2.1.1.2" xref="S5.p5.2.m2.1.1.2.cmml">s</mi><mo id="S5.p5.2.m2.1.1.1" xref="S5.p5.2.m2.1.1.1.cmml">></mo><mrow id="S5.p5.2.m2.1.1.3" xref="S5.p5.2.m2.1.1.3.cmml"><msub id="S5.p5.2.m2.1.1.3.2" xref="S5.p5.2.m2.1.1.3.2.cmml"><mi id="S5.p5.2.m2.1.1.3.2.2" xref="S5.p5.2.m2.1.1.3.2.2.cmml">m</mi><mi id="S5.p5.2.m2.1.1.3.2.3" xref="S5.p5.2.m2.1.1.3.2.3.cmml">n</mi></msub><mo id="S5.p5.2.m2.1.1.3.1" xref="S5.p5.2.m2.1.1.3.1.cmml">+</mo><mfrac id="S5.p5.2.m2.1.1.3.3" xref="S5.p5.2.m2.1.1.3.3.cmml"><mrow id="S5.p5.2.m2.1.1.3.3.2" xref="S5.p5.2.m2.1.1.3.3.2.cmml"><mi id="S5.p5.2.m2.1.1.3.3.2.2" xref="S5.p5.2.m2.1.1.3.3.2.2.cmml">γ</mi><mo id="S5.p5.2.m2.1.1.3.3.2.1" xref="S5.p5.2.m2.1.1.3.3.2.1.cmml">+</mo><mn id="S5.p5.2.m2.1.1.3.3.2.3" xref="S5.p5.2.m2.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S5.p5.2.m2.1.1.3.3.3" xref="S5.p5.2.m2.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.2.m2.1b"><apply id="S5.p5.2.m2.1.1.cmml" xref="S5.p5.2.m2.1.1"><gt id="S5.p5.2.m2.1.1.1.cmml" xref="S5.p5.2.m2.1.1.1"></gt><ci id="S5.p5.2.m2.1.1.2.cmml" xref="S5.p5.2.m2.1.1.2">𝑠</ci><apply id="S5.p5.2.m2.1.1.3.cmml" xref="S5.p5.2.m2.1.1.3"><plus id="S5.p5.2.m2.1.1.3.1.cmml" xref="S5.p5.2.m2.1.1.3.1"></plus><apply id="S5.p5.2.m2.1.1.3.2.cmml" xref="S5.p5.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S5.p5.2.m2.1.1.3.2.1.cmml" xref="S5.p5.2.m2.1.1.3.2">subscript</csymbol><ci id="S5.p5.2.m2.1.1.3.2.2.cmml" xref="S5.p5.2.m2.1.1.3.2.2">𝑚</ci><ci id="S5.p5.2.m2.1.1.3.2.3.cmml" xref="S5.p5.2.m2.1.1.3.2.3">𝑛</ci></apply><apply id="S5.p5.2.m2.1.1.3.3.cmml" xref="S5.p5.2.m2.1.1.3.3"><divide id="S5.p5.2.m2.1.1.3.3.1.cmml" xref="S5.p5.2.m2.1.1.3.3"></divide><apply id="S5.p5.2.m2.1.1.3.3.2.cmml" xref="S5.p5.2.m2.1.1.3.3.2"><plus id="S5.p5.2.m2.1.1.3.3.2.1.cmml" xref="S5.p5.2.m2.1.1.3.3.2.1"></plus><ci id="S5.p5.2.m2.1.1.3.3.2.2.cmml" xref="S5.p5.2.m2.1.1.3.3.2.2">𝛾</ci><cn id="S5.p5.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S5.p5.2.m2.1.1.3.3.2.3">1</cn></apply><ci id="S5.p5.2.m2.1.1.3.3.3.cmml" xref="S5.p5.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.2.m2.1c">s>m_{n}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S5.p5.2.m2.1d">italic_s > italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. In particular, <math alttext="(\operatorname{Tr}_{m_{i}})_{i=0}^{n}" class="ltx_Math" display="inline" id="S5.p5.3.m3.1"><semantics id="S5.p5.3.m3.1a"><msubsup id="S5.p5.3.m3.1.1" xref="S5.p5.3.m3.1.1.cmml"><mrow id="S5.p5.3.m3.1.1.1.1.1" xref="S5.p5.3.m3.1.1.1.1.1.1.cmml"><mo id="S5.p5.3.m3.1.1.1.1.1.2" stretchy="false" xref="S5.p5.3.m3.1.1.1.1.1.1.cmml">(</mo><msub id="S5.p5.3.m3.1.1.1.1.1.1" xref="S5.p5.3.m3.1.1.1.1.1.1.cmml"><mi id="S5.p5.3.m3.1.1.1.1.1.1.2" xref="S5.p5.3.m3.1.1.1.1.1.1.2.cmml">Tr</mi><msub id="S5.p5.3.m3.1.1.1.1.1.1.3" xref="S5.p5.3.m3.1.1.1.1.1.1.3.cmml"><mi id="S5.p5.3.m3.1.1.1.1.1.1.3.2" xref="S5.p5.3.m3.1.1.1.1.1.1.3.2.cmml">m</mi><mi id="S5.p5.3.m3.1.1.1.1.1.1.3.3" xref="S5.p5.3.m3.1.1.1.1.1.1.3.3.cmml">i</mi></msub></msub><mo id="S5.p5.3.m3.1.1.1.1.1.3" stretchy="false" xref="S5.p5.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.p5.3.m3.1.1.1.3" xref="S5.p5.3.m3.1.1.1.3.cmml"><mi id="S5.p5.3.m3.1.1.1.3.2" xref="S5.p5.3.m3.1.1.1.3.2.cmml">i</mi><mo id="S5.p5.3.m3.1.1.1.3.1" xref="S5.p5.3.m3.1.1.1.3.1.cmml">=</mo><mn id="S5.p5.3.m3.1.1.1.3.3" xref="S5.p5.3.m3.1.1.1.3.3.cmml">0</mn></mrow><mi id="S5.p5.3.m3.1.1.3" xref="S5.p5.3.m3.1.1.3.cmml">n</mi></msubsup><annotation-xml encoding="MathML-Content" id="S5.p5.3.m3.1b"><apply id="S5.p5.3.m3.1.1.cmml" xref="S5.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p5.3.m3.1.1.2.cmml" xref="S5.p5.3.m3.1.1">superscript</csymbol><apply id="S5.p5.3.m3.1.1.1.cmml" xref="S5.p5.3.m3.1.1"><csymbol cd="ambiguous" id="S5.p5.3.m3.1.1.1.2.cmml" xref="S5.p5.3.m3.1.1">subscript</csymbol><apply id="S5.p5.3.m3.1.1.1.1.1.1.cmml" xref="S5.p5.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.p5.3.m3.1.1.1.1.1.1.1.cmml" xref="S5.p5.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S5.p5.3.m3.1.1.1.1.1.1.2.cmml" xref="S5.p5.3.m3.1.1.1.1.1.1.2">Tr</ci><apply id="S5.p5.3.m3.1.1.1.1.1.1.3.cmml" xref="S5.p5.3.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.p5.3.m3.1.1.1.1.1.1.3.1.cmml" xref="S5.p5.3.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.p5.3.m3.1.1.1.1.1.1.3.2.cmml" xref="S5.p5.3.m3.1.1.1.1.1.1.3.2">𝑚</ci><ci id="S5.p5.3.m3.1.1.1.1.1.1.3.3.cmml" xref="S5.p5.3.m3.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="S5.p5.3.m3.1.1.1.3.cmml" xref="S5.p5.3.m3.1.1.1.3"><eq id="S5.p5.3.m3.1.1.1.3.1.cmml" xref="S5.p5.3.m3.1.1.1.3.1"></eq><ci id="S5.p5.3.m3.1.1.1.3.2.cmml" xref="S5.p5.3.m3.1.1.1.3.2">𝑖</ci><cn id="S5.p5.3.m3.1.1.1.3.3.cmml" type="integer" xref="S5.p5.3.m3.1.1.1.3.3">0</cn></apply></apply><ci id="S5.p5.3.m3.1.1.3.cmml" xref="S5.p5.3.m3.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.3.m3.1c">(\operatorname{Tr}_{m_{i}})_{i=0}^{n}</annotation><annotation encoding="application/x-llamapun" id="S5.p5.3.m3.1d">( roman_Tr start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT</annotation></semantics></math> is a normal boundary operator of type <math alttext="(p,s,\gamma,\overline{m},\overline{X})" class="ltx_Math" display="inline" id="S5.p5.4.m4.5"><semantics id="S5.p5.4.m4.5a"><mrow id="S5.p5.4.m4.5.6.2" xref="S5.p5.4.m4.5.6.1.cmml"><mo id="S5.p5.4.m4.5.6.2.1" stretchy="false" xref="S5.p5.4.m4.5.6.1.cmml">(</mo><mi id="S5.p5.4.m4.1.1" xref="S5.p5.4.m4.1.1.cmml">p</mi><mo id="S5.p5.4.m4.5.6.2.2" xref="S5.p5.4.m4.5.6.1.cmml">,</mo><mi id="S5.p5.4.m4.2.2" xref="S5.p5.4.m4.2.2.cmml">s</mi><mo id="S5.p5.4.m4.5.6.2.3" xref="S5.p5.4.m4.5.6.1.cmml">,</mo><mi id="S5.p5.4.m4.3.3" xref="S5.p5.4.m4.3.3.cmml">γ</mi><mo id="S5.p5.4.m4.5.6.2.4" xref="S5.p5.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S5.p5.4.m4.4.4" xref="S5.p5.4.m4.4.4.cmml"><mi id="S5.p5.4.m4.4.4.2" xref="S5.p5.4.m4.4.4.2.cmml">m</mi><mo id="S5.p5.4.m4.4.4.1" xref="S5.p5.4.m4.4.4.1.cmml">¯</mo></mover><mo id="S5.p5.4.m4.5.6.2.5" xref="S5.p5.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S5.p5.4.m4.5.5" xref="S5.p5.4.m4.5.5.cmml"><mi id="S5.p5.4.m4.5.5.2" xref="S5.p5.4.m4.5.5.2.cmml">X</mi><mo id="S5.p5.4.m4.5.5.1" xref="S5.p5.4.m4.5.5.1.cmml">¯</mo></mover><mo id="S5.p5.4.m4.5.6.2.6" stretchy="false" xref="S5.p5.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.4.m4.5b"><vector id="S5.p5.4.m4.5.6.1.cmml" xref="S5.p5.4.m4.5.6.2"><ci id="S5.p5.4.m4.1.1.cmml" xref="S5.p5.4.m4.1.1">𝑝</ci><ci id="S5.p5.4.m4.2.2.cmml" xref="S5.p5.4.m4.2.2">𝑠</ci><ci id="S5.p5.4.m4.3.3.cmml" xref="S5.p5.4.m4.3.3">𝛾</ci><apply id="S5.p5.4.m4.4.4.cmml" xref="S5.p5.4.m4.4.4"><ci id="S5.p5.4.m4.4.4.1.cmml" xref="S5.p5.4.m4.4.4.1">¯</ci><ci id="S5.p5.4.m4.4.4.2.cmml" xref="S5.p5.4.m4.4.4.2">𝑚</ci></apply><apply id="S5.p5.4.m4.5.5.cmml" xref="S5.p5.4.m4.5.5"><ci id="S5.p5.4.m4.5.5.1.cmml" xref="S5.p5.4.m4.5.5.1">¯</ci><ci id="S5.p5.4.m4.5.5.2.cmml" xref="S5.p5.4.m4.5.5.2">𝑋</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.4.m4.5c">(p,s,\gamma,\overline{m},\overline{X})</annotation><annotation encoding="application/x-llamapun" id="S5.p5.4.m4.5d">( italic_p , italic_s , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_X end_ARG )</annotation></semantics></math> for <math alttext="s>m_{n}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S5.p5.5.m5.1"><semantics id="S5.p5.5.m5.1a"><mrow id="S5.p5.5.m5.1.1" xref="S5.p5.5.m5.1.1.cmml"><mi id="S5.p5.5.m5.1.1.2" xref="S5.p5.5.m5.1.1.2.cmml">s</mi><mo id="S5.p5.5.m5.1.1.1" xref="S5.p5.5.m5.1.1.1.cmml">></mo><mrow id="S5.p5.5.m5.1.1.3" xref="S5.p5.5.m5.1.1.3.cmml"><msub id="S5.p5.5.m5.1.1.3.2" xref="S5.p5.5.m5.1.1.3.2.cmml"><mi id="S5.p5.5.m5.1.1.3.2.2" xref="S5.p5.5.m5.1.1.3.2.2.cmml">m</mi><mi id="S5.p5.5.m5.1.1.3.2.3" xref="S5.p5.5.m5.1.1.3.2.3.cmml">n</mi></msub><mo id="S5.p5.5.m5.1.1.3.1" xref="S5.p5.5.m5.1.1.3.1.cmml">+</mo><mfrac id="S5.p5.5.m5.1.1.3.3" xref="S5.p5.5.m5.1.1.3.3.cmml"><mrow id="S5.p5.5.m5.1.1.3.3.2" xref="S5.p5.5.m5.1.1.3.3.2.cmml"><mi id="S5.p5.5.m5.1.1.3.3.2.2" xref="S5.p5.5.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S5.p5.5.m5.1.1.3.3.2.1" xref="S5.p5.5.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S5.p5.5.m5.1.1.3.3.2.3" xref="S5.p5.5.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S5.p5.5.m5.1.1.3.3.3" xref="S5.p5.5.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.5.m5.1b"><apply id="S5.p5.5.m5.1.1.cmml" xref="S5.p5.5.m5.1.1"><gt id="S5.p5.5.m5.1.1.1.cmml" xref="S5.p5.5.m5.1.1.1"></gt><ci id="S5.p5.5.m5.1.1.2.cmml" xref="S5.p5.5.m5.1.1.2">𝑠</ci><apply id="S5.p5.5.m5.1.1.3.cmml" xref="S5.p5.5.m5.1.1.3"><plus id="S5.p5.5.m5.1.1.3.1.cmml" xref="S5.p5.5.m5.1.1.3.1"></plus><apply id="S5.p5.5.m5.1.1.3.2.cmml" xref="S5.p5.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S5.p5.5.m5.1.1.3.2.1.cmml" xref="S5.p5.5.m5.1.1.3.2">subscript</csymbol><ci id="S5.p5.5.m5.1.1.3.2.2.cmml" xref="S5.p5.5.m5.1.1.3.2.2">𝑚</ci><ci id="S5.p5.5.m5.1.1.3.2.3.cmml" xref="S5.p5.5.m5.1.1.3.2.3">𝑛</ci></apply><apply id="S5.p5.5.m5.1.1.3.3.cmml" xref="S5.p5.5.m5.1.1.3.3"><divide id="S5.p5.5.m5.1.1.3.3.1.cmml" xref="S5.p5.5.m5.1.1.3.3"></divide><apply id="S5.p5.5.m5.1.1.3.3.2.cmml" xref="S5.p5.5.m5.1.1.3.3.2"><plus id="S5.p5.5.m5.1.1.3.3.2.1.cmml" xref="S5.p5.5.m5.1.1.3.3.2.1"></plus><ci id="S5.p5.5.m5.1.1.3.3.2.2.cmml" xref="S5.p5.5.m5.1.1.3.3.2.2">𝛾</ci><cn id="S5.p5.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S5.p5.5.m5.1.1.3.3.2.3">1</cn></apply><ci id="S5.p5.5.m5.1.1.3.3.3.cmml" xref="S5.p5.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.5.m5.1c">s>m_{n}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S5.p5.5.m5.1d">italic_s > italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. This includes the important cases for Dirichlet (<math alttext="n=0" class="ltx_Math" display="inline" id="S5.p5.6.m6.1"><semantics id="S5.p5.6.m6.1a"><mrow id="S5.p5.6.m6.1.1" xref="S5.p5.6.m6.1.1.cmml"><mi id="S5.p5.6.m6.1.1.2" xref="S5.p5.6.m6.1.1.2.cmml">n</mi><mo id="S5.p5.6.m6.1.1.1" xref="S5.p5.6.m6.1.1.1.cmml">=</mo><mn id="S5.p5.6.m6.1.1.3" xref="S5.p5.6.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.6.m6.1b"><apply id="S5.p5.6.m6.1.1.cmml" xref="S5.p5.6.m6.1.1"><eq id="S5.p5.6.m6.1.1.1.cmml" xref="S5.p5.6.m6.1.1.1"></eq><ci id="S5.p5.6.m6.1.1.2.cmml" xref="S5.p5.6.m6.1.1.2">𝑛</ci><cn id="S5.p5.6.m6.1.1.3.cmml" type="integer" xref="S5.p5.6.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.6.m6.1c">n=0</annotation><annotation encoding="application/x-llamapun" id="S5.p5.6.m6.1d">italic_n = 0</annotation></semantics></math> and <math alttext="m_{0}=0" class="ltx_Math" display="inline" id="S5.p5.7.m7.1"><semantics id="S5.p5.7.m7.1a"><mrow id="S5.p5.7.m7.1.1" xref="S5.p5.7.m7.1.1.cmml"><msub id="S5.p5.7.m7.1.1.2" xref="S5.p5.7.m7.1.1.2.cmml"><mi id="S5.p5.7.m7.1.1.2.2" xref="S5.p5.7.m7.1.1.2.2.cmml">m</mi><mn id="S5.p5.7.m7.1.1.2.3" xref="S5.p5.7.m7.1.1.2.3.cmml">0</mn></msub><mo id="S5.p5.7.m7.1.1.1" xref="S5.p5.7.m7.1.1.1.cmml">=</mo><mn id="S5.p5.7.m7.1.1.3" xref="S5.p5.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.7.m7.1b"><apply id="S5.p5.7.m7.1.1.cmml" xref="S5.p5.7.m7.1.1"><eq id="S5.p5.7.m7.1.1.1.cmml" xref="S5.p5.7.m7.1.1.1"></eq><apply id="S5.p5.7.m7.1.1.2.cmml" xref="S5.p5.7.m7.1.1.2"><csymbol cd="ambiguous" id="S5.p5.7.m7.1.1.2.1.cmml" xref="S5.p5.7.m7.1.1.2">subscript</csymbol><ci id="S5.p5.7.m7.1.1.2.2.cmml" xref="S5.p5.7.m7.1.1.2.2">𝑚</ci><cn id="S5.p5.7.m7.1.1.2.3.cmml" type="integer" xref="S5.p5.7.m7.1.1.2.3">0</cn></apply><cn id="S5.p5.7.m7.1.1.3.cmml" type="integer" xref="S5.p5.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.7.m7.1c">m_{0}=0</annotation><annotation encoding="application/x-llamapun" id="S5.p5.7.m7.1d">italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math>) and Neumann (<math alttext="n=0" class="ltx_Math" display="inline" id="S5.p5.8.m8.1"><semantics id="S5.p5.8.m8.1a"><mrow id="S5.p5.8.m8.1.1" xref="S5.p5.8.m8.1.1.cmml"><mi id="S5.p5.8.m8.1.1.2" xref="S5.p5.8.m8.1.1.2.cmml">n</mi><mo id="S5.p5.8.m8.1.1.1" xref="S5.p5.8.m8.1.1.1.cmml">=</mo><mn id="S5.p5.8.m8.1.1.3" xref="S5.p5.8.m8.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.8.m8.1b"><apply id="S5.p5.8.m8.1.1.cmml" xref="S5.p5.8.m8.1.1"><eq id="S5.p5.8.m8.1.1.1.cmml" xref="S5.p5.8.m8.1.1.1"></eq><ci id="S5.p5.8.m8.1.1.2.cmml" xref="S5.p5.8.m8.1.1.2">𝑛</ci><cn id="S5.p5.8.m8.1.1.3.cmml" type="integer" xref="S5.p5.8.m8.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.8.m8.1c">n=0</annotation><annotation encoding="application/x-llamapun" id="S5.p5.8.m8.1d">italic_n = 0</annotation></semantics></math> and <math alttext="m_{0}=1" class="ltx_Math" display="inline" id="S5.p5.9.m9.1"><semantics id="S5.p5.9.m9.1a"><mrow id="S5.p5.9.m9.1.1" xref="S5.p5.9.m9.1.1.cmml"><msub id="S5.p5.9.m9.1.1.2" xref="S5.p5.9.m9.1.1.2.cmml"><mi id="S5.p5.9.m9.1.1.2.2" xref="S5.p5.9.m9.1.1.2.2.cmml">m</mi><mn id="S5.p5.9.m9.1.1.2.3" xref="S5.p5.9.m9.1.1.2.3.cmml">0</mn></msub><mo id="S5.p5.9.m9.1.1.1" xref="S5.p5.9.m9.1.1.1.cmml">=</mo><mn id="S5.p5.9.m9.1.1.3" xref="S5.p5.9.m9.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.p5.9.m9.1b"><apply id="S5.p5.9.m9.1.1.cmml" xref="S5.p5.9.m9.1.1"><eq id="S5.p5.9.m9.1.1.1.cmml" xref="S5.p5.9.m9.1.1.1"></eq><apply id="S5.p5.9.m9.1.1.2.cmml" xref="S5.p5.9.m9.1.1.2"><csymbol cd="ambiguous" id="S5.p5.9.m9.1.1.2.1.cmml" xref="S5.p5.9.m9.1.1.2">subscript</csymbol><ci id="S5.p5.9.m9.1.1.2.2.cmml" xref="S5.p5.9.m9.1.1.2.2">𝑚</ci><cn id="S5.p5.9.m9.1.1.2.3.cmml" type="integer" xref="S5.p5.9.m9.1.1.2.3">0</cn></apply><cn id="S5.p5.9.m9.1.1.3.cmml" type="integer" xref="S5.p5.9.m9.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p5.9.m9.1c">m_{0}=1</annotation><annotation encoding="application/x-llamapun" id="S5.p5.9.m9.1d">italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 1</annotation></semantics></math>) boundary conditions.</p> </div> <div class="ltx_theorem ltx_theorem_remark" id="S5.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem5.1.1.1">Remark 5.5</span></span><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem5.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem5.p1"> <p class="ltx_p" id="S5.Thmtheorem5.p1.9">In the case that <math alttext="X=\mathbb{C}^{q}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.1.m1.1"><semantics id="S5.Thmtheorem5.p1.1.m1.1a"><mrow id="S5.Thmtheorem5.p1.1.m1.1.1" xref="S5.Thmtheorem5.p1.1.m1.1.1.cmml"><mi id="S5.Thmtheorem5.p1.1.m1.1.1.2" xref="S5.Thmtheorem5.p1.1.m1.1.1.2.cmml">X</mi><mo id="S5.Thmtheorem5.p1.1.m1.1.1.1" xref="S5.Thmtheorem5.p1.1.m1.1.1.1.cmml">=</mo><msup id="S5.Thmtheorem5.p1.1.m1.1.1.3" xref="S5.Thmtheorem5.p1.1.m1.1.1.3.cmml"><mi id="S5.Thmtheorem5.p1.1.m1.1.1.3.2" xref="S5.Thmtheorem5.p1.1.m1.1.1.3.2.cmml">ℂ</mi><mi id="S5.Thmtheorem5.p1.1.m1.1.1.3.3" xref="S5.Thmtheorem5.p1.1.m1.1.1.3.3.cmml">q</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.1.m1.1b"><apply id="S5.Thmtheorem5.p1.1.m1.1.1.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1"><eq id="S5.Thmtheorem5.p1.1.m1.1.1.1.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1.1"></eq><ci id="S5.Thmtheorem5.p1.1.m1.1.1.2.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1.2">𝑋</ci><apply id="S5.Thmtheorem5.p1.1.m1.1.1.3.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.1.m1.1.1.3.1.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1.3">superscript</csymbol><ci id="S5.Thmtheorem5.p1.1.m1.1.1.3.2.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1.3.2">ℂ</ci><ci id="S5.Thmtheorem5.p1.1.m1.1.1.3.3.cmml" xref="S5.Thmtheorem5.p1.1.m1.1.1.3.3">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.1.m1.1c">X=\mathbb{C}^{q}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.1.m1.1d">italic_X = blackboard_C start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT</annotation></semantics></math> for some <math alttext="q\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.2.m2.1"><semantics id="S5.Thmtheorem5.p1.2.m2.1a"><mrow id="S5.Thmtheorem5.p1.2.m2.1.1" xref="S5.Thmtheorem5.p1.2.m2.1.1.cmml"><mi id="S5.Thmtheorem5.p1.2.m2.1.1.2" xref="S5.Thmtheorem5.p1.2.m2.1.1.2.cmml">q</mi><mo id="S5.Thmtheorem5.p1.2.m2.1.1.1" xref="S5.Thmtheorem5.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S5.Thmtheorem5.p1.2.m2.1.1.3" xref="S5.Thmtheorem5.p1.2.m2.1.1.3.cmml"><mi id="S5.Thmtheorem5.p1.2.m2.1.1.3.2" xref="S5.Thmtheorem5.p1.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S5.Thmtheorem5.p1.2.m2.1.1.3.3" xref="S5.Thmtheorem5.p1.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.2.m2.1b"><apply id="S5.Thmtheorem5.p1.2.m2.1.1.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1"><in id="S5.Thmtheorem5.p1.2.m2.1.1.1.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1.1"></in><ci id="S5.Thmtheorem5.p1.2.m2.1.1.2.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1.2">𝑞</ci><apply id="S5.Thmtheorem5.p1.2.m2.1.1.3.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.2.m2.1.1.3.1.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem5.p1.2.m2.1.1.3.2.cmml" xref="S5.Thmtheorem5.p1.2.m2.1.1.3.2">ℕ</ci><cn id="S5.Thmtheorem5.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S5.Thmtheorem5.p1.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.2.m2.1c">q\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.2.m2.1d">italic_q ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, the definition of a normal boundary system as given in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib65" title="">65</a>, Definition 3.1]</cite> coincides with our definition above. The definition in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib65" title="">65</a>]</cite> requires that the coefficient matrix for the highest-order derivative in <math alttext="\mathcal{B}^{m_{i}}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.3.m3.1"><semantics id="S5.Thmtheorem5.p1.3.m3.1a"><msup id="S5.Thmtheorem5.p1.3.m3.1.1" xref="S5.Thmtheorem5.p1.3.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem5.p1.3.m3.1.1.2" xref="S5.Thmtheorem5.p1.3.m3.1.1.2.cmml">ℬ</mi><msub id="S5.Thmtheorem5.p1.3.m3.1.1.3" xref="S5.Thmtheorem5.p1.3.m3.1.1.3.cmml"><mi id="S5.Thmtheorem5.p1.3.m3.1.1.3.2" xref="S5.Thmtheorem5.p1.3.m3.1.1.3.2.cmml">m</mi><mi id="S5.Thmtheorem5.p1.3.m3.1.1.3.3" xref="S5.Thmtheorem5.p1.3.m3.1.1.3.3.cmml">i</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.3.m3.1b"><apply id="S5.Thmtheorem5.p1.3.m3.1.1.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.3.m3.1.1.1.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1">superscript</csymbol><ci id="S5.Thmtheorem5.p1.3.m3.1.1.2.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1.2">ℬ</ci><apply id="S5.Thmtheorem5.p1.3.m3.1.1.3.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.3.m3.1.1.3.1.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem5.p1.3.m3.1.1.3.2.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1.3.2">𝑚</ci><ci id="S5.Thmtheorem5.p1.3.m3.1.1.3.3.cmml" xref="S5.Thmtheorem5.p1.3.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.3.m3.1c">\mathcal{B}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.3.m3.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is surjective, which implies that the boundary operator can be replaced by a <math alttext="q\times q" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.4.m4.1"><semantics id="S5.Thmtheorem5.p1.4.m4.1a"><mrow id="S5.Thmtheorem5.p1.4.m4.1.1" xref="S5.Thmtheorem5.p1.4.m4.1.1.cmml"><mi id="S5.Thmtheorem5.p1.4.m4.1.1.2" xref="S5.Thmtheorem5.p1.4.m4.1.1.2.cmml">q</mi><mo id="S5.Thmtheorem5.p1.4.m4.1.1.1" lspace="0.222em" rspace="0.222em" xref="S5.Thmtheorem5.p1.4.m4.1.1.1.cmml">×</mo><mi id="S5.Thmtheorem5.p1.4.m4.1.1.3" xref="S5.Thmtheorem5.p1.4.m4.1.1.3.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.4.m4.1b"><apply id="S5.Thmtheorem5.p1.4.m4.1.1.cmml" xref="S5.Thmtheorem5.p1.4.m4.1.1"><times id="S5.Thmtheorem5.p1.4.m4.1.1.1.cmml" xref="S5.Thmtheorem5.p1.4.m4.1.1.1"></times><ci id="S5.Thmtheorem5.p1.4.m4.1.1.2.cmml" xref="S5.Thmtheorem5.p1.4.m4.1.1.2">𝑞</ci><ci id="S5.Thmtheorem5.p1.4.m4.1.1.3.cmml" xref="S5.Thmtheorem5.p1.4.m4.1.1.3">𝑞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.4.m4.1c">q\times q</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.4.m4.1d">italic_q × italic_q</annotation></semantics></math> matrix operator (leaving the kernel of <math alttext="\mathcal{B}^{m_{i}}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.5.m5.1"><semantics id="S5.Thmtheorem5.p1.5.m5.1a"><msup id="S5.Thmtheorem5.p1.5.m5.1.1" xref="S5.Thmtheorem5.p1.5.m5.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem5.p1.5.m5.1.1.2" xref="S5.Thmtheorem5.p1.5.m5.1.1.2.cmml">ℬ</mi><msub id="S5.Thmtheorem5.p1.5.m5.1.1.3" xref="S5.Thmtheorem5.p1.5.m5.1.1.3.cmml"><mi id="S5.Thmtheorem5.p1.5.m5.1.1.3.2" xref="S5.Thmtheorem5.p1.5.m5.1.1.3.2.cmml">m</mi><mi id="S5.Thmtheorem5.p1.5.m5.1.1.3.3" xref="S5.Thmtheorem5.p1.5.m5.1.1.3.3.cmml">i</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.5.m5.1b"><apply id="S5.Thmtheorem5.p1.5.m5.1.1.cmml" xref="S5.Thmtheorem5.p1.5.m5.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.5.m5.1.1.1.cmml" xref="S5.Thmtheorem5.p1.5.m5.1.1">superscript</csymbol><ci id="S5.Thmtheorem5.p1.5.m5.1.1.2.cmml" xref="S5.Thmtheorem5.p1.5.m5.1.1.2">ℬ</ci><apply id="S5.Thmtheorem5.p1.5.m5.1.1.3.cmml" xref="S5.Thmtheorem5.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.5.m5.1.1.3.1.cmml" xref="S5.Thmtheorem5.p1.5.m5.1.1.3">subscript</csymbol><ci id="S5.Thmtheorem5.p1.5.m5.1.1.3.2.cmml" xref="S5.Thmtheorem5.p1.5.m5.1.1.3.2">𝑚</ci><ci id="S5.Thmtheorem5.p1.5.m5.1.1.3.3.cmml" xref="S5.Thmtheorem5.p1.5.m5.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.5.m5.1c">\mathcal{B}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.5.m5.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> invariant) such that the coefficient for the highest order derivative is a projection-valued matrix, see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib65" title="">65</a>, Section 3]</cite>. This is a key ingredient for characterising the complex interpolation spaces in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib65" title="">65</a>]</cite>. For general Banach spaces <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.6.m6.1"><semantics id="S5.Thmtheorem5.p1.6.m6.1a"><mi id="S5.Thmtheorem5.p1.6.m6.1.1" xref="S5.Thmtheorem5.p1.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.6.m6.1b"><ci id="S5.Thmtheorem5.p1.6.m6.1.1.cmml" xref="S5.Thmtheorem5.p1.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.6.m6.1d">italic_X</annotation></semantics></math> we replace these conditions by the retraction and coretraction in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem3" title="Definition 5.3 (Normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.3</span></a>. In the case that <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.7.m7.1"><semantics id="S5.Thmtheorem5.p1.7.m7.1a"><mi id="S5.Thmtheorem5.p1.7.m7.1.1" xref="S5.Thmtheorem5.p1.7.m7.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.7.m7.1b"><ci id="S5.Thmtheorem5.p1.7.m7.1.1.cmml" xref="S5.Thmtheorem5.p1.7.m7.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.7.m7.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.7.m7.1d">italic_X</annotation></semantics></math> and <math alttext="Y_{i}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.8.m8.1"><semantics id="S5.Thmtheorem5.p1.8.m8.1a"><msub id="S5.Thmtheorem5.p1.8.m8.1.1" xref="S5.Thmtheorem5.p1.8.m8.1.1.cmml"><mi id="S5.Thmtheorem5.p1.8.m8.1.1.2" xref="S5.Thmtheorem5.p1.8.m8.1.1.2.cmml">Y</mi><mi id="S5.Thmtheorem5.p1.8.m8.1.1.3" xref="S5.Thmtheorem5.p1.8.m8.1.1.3.cmml">i</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.8.m8.1b"><apply id="S5.Thmtheorem5.p1.8.m8.1.1.cmml" xref="S5.Thmtheorem5.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem5.p1.8.m8.1.1.1.cmml" xref="S5.Thmtheorem5.p1.8.m8.1.1">subscript</csymbol><ci id="S5.Thmtheorem5.p1.8.m8.1.1.2.cmml" xref="S5.Thmtheorem5.p1.8.m8.1.1.2">𝑌</ci><ci id="S5.Thmtheorem5.p1.8.m8.1.1.3.cmml" xref="S5.Thmtheorem5.p1.8.m8.1.1.3">𝑖</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.8.m8.1c">Y_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.8.m8.1d">italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> are Hilbert spaces we refer to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Lemma VIII.2.1.3]</cite> for a sufficient condition for <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.Thmtheorem5.p1.9.m9.1"><semantics id="S5.Thmtheorem5.p1.9.m9.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem5.p1.9.m9.1.1" xref="S5.Thmtheorem5.p1.9.m9.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem5.p1.9.m9.1b"><ci id="S5.Thmtheorem5.p1.9.m9.1.1.cmml" xref="S5.Thmtheorem5.p1.9.m9.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem5.p1.9.m9.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem5.p1.9.m9.1d">caligraphic_B</annotation></semantics></math> to be normal.</p> </div> <div class="ltx_para" id="S5.Thmtheorem5.p2"> <p class="ltx_p" id="S5.Thmtheorem5.p2.1">We note that <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib65" title="">65</a>, Definition 3.1]</cite> arises from Agmon’s condition <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib3" title="">3</a>]</cite> about minimal growth of the resolvent of an elliptic operator.</p> </div> </div> <div class="ltx_para" id="S5.p6"> <p class="ltx_p" id="S5.p6.1">We can now prove the following trace theorem for normal boundary operators analogous to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorem VIII.2.2.1]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S5.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem6.1.1.1">Theorem 5.6</span></span><span class="ltx_text ltx_font_bold" id="S5.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S5.Thmtheorem6.p1"> <p class="ltx_p" id="S5.Thmtheorem6.p1.2"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem6.p1.2.2">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.1.1.m1.2"><semantics id="S5.Thmtheorem6.p1.1.1.m1.2a"><mrow id="S5.Thmtheorem6.p1.1.1.m1.2.3" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.cmml"><mi id="S5.Thmtheorem6.p1.1.1.m1.2.3.2" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S5.Thmtheorem6.p1.1.1.m1.2.3.1" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S5.Thmtheorem6.p1.1.1.m1.2.3.3.2" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml"><mo id="S5.Thmtheorem6.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S5.Thmtheorem6.p1.1.1.m1.1.1" xref="S5.Thmtheorem6.p1.1.1.m1.1.1.cmml">1</mn><mo id="S5.Thmtheorem6.p1.1.1.m1.2.3.3.2.2" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S5.Thmtheorem6.p1.1.1.m1.2.2" mathvariant="normal" xref="S5.Thmtheorem6.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S5.Thmtheorem6.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.1.1.m1.2b"><apply id="S5.Thmtheorem6.p1.1.1.m1.2.3.cmml" xref="S5.Thmtheorem6.p1.1.1.m1.2.3"><in id="S5.Thmtheorem6.p1.1.1.m1.2.3.1.cmml" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.1"></in><ci id="S5.Thmtheorem6.p1.1.1.m1.2.3.2.cmml" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S5.Thmtheorem6.p1.1.1.m1.2.3.3.1.cmml" xref="S5.Thmtheorem6.p1.1.1.m1.2.3.3.2"><cn id="S5.Thmtheorem6.p1.1.1.m1.1.1.cmml" type="integer" xref="S5.Thmtheorem6.p1.1.1.m1.1.1">1</cn><infinity id="S5.Thmtheorem6.p1.1.1.m1.2.2.cmml" xref="S5.Thmtheorem6.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.2.2.m2.1"><semantics id="S5.Thmtheorem6.p1.2.2.m2.1a"><mi id="S5.Thmtheorem6.p1.2.2.m2.1.1" xref="S5.Thmtheorem6.p1.2.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.2.2.m2.1b"><ci id="S5.Thmtheorem6.p1.2.2.m2.1.1.cmml" xref="S5.Thmtheorem6.p1.2.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.2.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.2.2.m2.1d">italic_X</annotation></semantics></math> be a Banach space. Assume that the conditions <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.I4.i1" title="item i ‣ Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.I4.i2" title="item ii ‣ Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> of Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem4" title="Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.4</span></a> hold.</span></p> <ol class="ltx_enumerate" id="S5.I5"> <li class="ltx_item" id="S5.I5.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S5.I5.i1.p1"> <p class="ltx_p" id="S5.I5.i1.p1.4"><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.4.1">If </span><math alttext="s>0" class="ltx_Math" display="inline" id="S5.I5.i1.p1.1.m1.1"><semantics id="S5.I5.i1.p1.1.m1.1a"><mrow id="S5.I5.i1.p1.1.m1.1.1" xref="S5.I5.i1.p1.1.m1.1.1.cmml"><mi id="S5.I5.i1.p1.1.m1.1.1.2" xref="S5.I5.i1.p1.1.m1.1.1.2.cmml">s</mi><mo id="S5.I5.i1.p1.1.m1.1.1.1" xref="S5.I5.i1.p1.1.m1.1.1.1.cmml">></mo><mn id="S5.I5.i1.p1.1.m1.1.1.3" xref="S5.I5.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.1.m1.1b"><apply id="S5.I5.i1.p1.1.m1.1.1.cmml" xref="S5.I5.i1.p1.1.m1.1.1"><gt id="S5.I5.i1.p1.1.m1.1.1.1.cmml" xref="S5.I5.i1.p1.1.m1.1.1.1"></gt><ci id="S5.I5.i1.p1.1.m1.1.1.2.cmml" xref="S5.I5.i1.p1.1.m1.1.1.2">𝑠</ci><cn id="S5.I5.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S5.I5.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.1.m1.1c">s>0</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.1.m1.1d">italic_s > 0</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.4.2">, </span><math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S5.I5.i1.p1.2.m2.2"><semantics id="S5.I5.i1.p1.2.m2.2a"><mrow id="S5.I5.i1.p1.2.m2.2.2" xref="S5.I5.i1.p1.2.m2.2.2.cmml"><mi id="S5.I5.i1.p1.2.m2.2.2.4" xref="S5.I5.i1.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S5.I5.i1.p1.2.m2.2.2.3" xref="S5.I5.i1.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S5.I5.i1.p1.2.m2.2.2.2.2" xref="S5.I5.i1.p1.2.m2.2.2.2.3.cmml"><mo id="S5.I5.i1.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S5.I5.i1.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S5.I5.i1.p1.2.m2.1.1.1.1.1" xref="S5.I5.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S5.I5.i1.p1.2.m2.1.1.1.1.1a" xref="S5.I5.i1.p1.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S5.I5.i1.p1.2.m2.1.1.1.1.1.2" xref="S5.I5.i1.p1.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S5.I5.i1.p1.2.m2.2.2.2.2.4" xref="S5.I5.i1.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="S5.I5.i1.p1.2.m2.2.2.2.2.2" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2.cmml"><mi id="S5.I5.i1.p1.2.m2.2.2.2.2.2.2" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2.2.cmml">p</mi><mo id="S5.I5.i1.p1.2.m2.2.2.2.2.2.1" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2.1.cmml">−</mo><mn id="S5.I5.i1.p1.2.m2.2.2.2.2.2.3" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S5.I5.i1.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S5.I5.i1.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.2.m2.2b"><apply id="S5.I5.i1.p1.2.m2.2.2.cmml" xref="S5.I5.i1.p1.2.m2.2.2"><in id="S5.I5.i1.p1.2.m2.2.2.3.cmml" xref="S5.I5.i1.p1.2.m2.2.2.3"></in><ci id="S5.I5.i1.p1.2.m2.2.2.4.cmml" xref="S5.I5.i1.p1.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S5.I5.i1.p1.2.m2.2.2.2.3.cmml" xref="S5.I5.i1.p1.2.m2.2.2.2.2"><apply id="S5.I5.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S5.I5.i1.p1.2.m2.1.1.1.1.1"><minus id="S5.I5.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S5.I5.i1.p1.2.m2.1.1.1.1.1"></minus><cn id="S5.I5.i1.p1.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S5.I5.i1.p1.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S5.I5.i1.p1.2.m2.2.2.2.2.2.cmml" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2"><minus id="S5.I5.i1.p1.2.m2.2.2.2.2.2.1.cmml" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2.1"></minus><ci id="S5.I5.i1.p1.2.m2.2.2.2.2.2.2.cmml" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2.2">𝑝</ci><cn id="S5.I5.i1.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S5.I5.i1.p1.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.2.m2.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.2.m2.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.4.3"> and </span><math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.I5.i1.p1.3.m3.1"><semantics id="S5.I5.i1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i1.p1.3.m3.1.1" xref="S5.I5.i1.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.3.m3.1b"><ci id="S5.I5.i1.p1.3.m3.1.1.cmml" xref="S5.I5.i1.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.3.m3.1d">caligraphic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.4.4"> is of type </span><math alttext="(p,s,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S5.I5.i1.p1.4.m4.5"><semantics id="S5.I5.i1.p1.4.m4.5a"><mrow id="S5.I5.i1.p1.4.m4.5.6.2" xref="S5.I5.i1.p1.4.m4.5.6.1.cmml"><mo id="S5.I5.i1.p1.4.m4.5.6.2.1" stretchy="false" xref="S5.I5.i1.p1.4.m4.5.6.1.cmml">(</mo><mi id="S5.I5.i1.p1.4.m4.1.1" xref="S5.I5.i1.p1.4.m4.1.1.cmml">p</mi><mo id="S5.I5.i1.p1.4.m4.5.6.2.2" xref="S5.I5.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I5.i1.p1.4.m4.2.2" xref="S5.I5.i1.p1.4.m4.2.2.cmml">s</mi><mo id="S5.I5.i1.p1.4.m4.5.6.2.3" xref="S5.I5.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I5.i1.p1.4.m4.3.3" xref="S5.I5.i1.p1.4.m4.3.3.cmml">γ</mi><mo id="S5.I5.i1.p1.4.m4.5.6.2.4" xref="S5.I5.i1.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S5.I5.i1.p1.4.m4.4.4" xref="S5.I5.i1.p1.4.m4.4.4.cmml"><mi id="S5.I5.i1.p1.4.m4.4.4.2" xref="S5.I5.i1.p1.4.m4.4.4.2.cmml">m</mi><mo id="S5.I5.i1.p1.4.m4.4.4.1" xref="S5.I5.i1.p1.4.m4.4.4.1.cmml">¯</mo></mover><mo id="S5.I5.i1.p1.4.m4.5.6.2.5" xref="S5.I5.i1.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S5.I5.i1.p1.4.m4.5.5" xref="S5.I5.i1.p1.4.m4.5.5.cmml"><mi id="S5.I5.i1.p1.4.m4.5.5.2" xref="S5.I5.i1.p1.4.m4.5.5.2.cmml">Y</mi><mo id="S5.I5.i1.p1.4.m4.5.5.1" xref="S5.I5.i1.p1.4.m4.5.5.1.cmml">¯</mo></mover><mo id="S5.I5.i1.p1.4.m4.5.6.2.6" stretchy="false" xref="S5.I5.i1.p1.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i1.p1.4.m4.5b"><vector id="S5.I5.i1.p1.4.m4.5.6.1.cmml" xref="S5.I5.i1.p1.4.m4.5.6.2"><ci id="S5.I5.i1.p1.4.m4.1.1.cmml" xref="S5.I5.i1.p1.4.m4.1.1">𝑝</ci><ci id="S5.I5.i1.p1.4.m4.2.2.cmml" xref="S5.I5.i1.p1.4.m4.2.2">𝑠</ci><ci id="S5.I5.i1.p1.4.m4.3.3.cmml" xref="S5.I5.i1.p1.4.m4.3.3">𝛾</ci><apply id="S5.I5.i1.p1.4.m4.4.4.cmml" xref="S5.I5.i1.p1.4.m4.4.4"><ci id="S5.I5.i1.p1.4.m4.4.4.1.cmml" xref="S5.I5.i1.p1.4.m4.4.4.1">¯</ci><ci id="S5.I5.i1.p1.4.m4.4.4.2.cmml" xref="S5.I5.i1.p1.4.m4.4.4.2">𝑚</ci></apply><apply id="S5.I5.i1.p1.4.m4.5.5.cmml" xref="S5.I5.i1.p1.4.m4.5.5"><ci id="S5.I5.i1.p1.4.m4.5.5.1.cmml" xref="S5.I5.i1.p1.4.m4.5.5.1">¯</ci><ci id="S5.I5.i1.p1.4.m4.5.5.2.cmml" xref="S5.I5.i1.p1.4.m4.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i1.p1.4.m4.5c">(p,s,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i1.p1.4.m4.5d">( italic_p , italic_s , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.4.5">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to\prod_{j=0}^{n}B^{s-m_{% j}-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};Y_{j})" class="ltx_Math" display="block" id="S5.Ex5.m1.9"><semantics id="S5.Ex5.m1.9a"><mrow id="S5.Ex5.m1.9.9" xref="S5.Ex5.m1.9.9.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex5.m1.9.9.6" xref="S5.Ex5.m1.9.9.6.cmml">ℬ</mi><mo id="S5.Ex5.m1.9.9.5" lspace="0.278em" rspace="0.278em" xref="S5.Ex5.m1.9.9.5.cmml">:</mo><mrow id="S5.Ex5.m1.9.9.4" xref="S5.Ex5.m1.9.9.4.cmml"><mrow id="S5.Ex5.m1.7.7.2.2" xref="S5.Ex5.m1.7.7.2.2.cmml"><msup id="S5.Ex5.m1.7.7.2.2.4" xref="S5.Ex5.m1.7.7.2.2.4.cmml"><mi id="S5.Ex5.m1.7.7.2.2.4.2" xref="S5.Ex5.m1.7.7.2.2.4.2.cmml">H</mi><mrow id="S5.Ex5.m1.2.2.2.4" xref="S5.Ex5.m1.2.2.2.3.cmml"><mi id="S5.Ex5.m1.1.1.1.1" xref="S5.Ex5.m1.1.1.1.1.cmml">s</mi><mo id="S5.Ex5.m1.2.2.2.4.1" xref="S5.Ex5.m1.2.2.2.3.cmml">,</mo><mi id="S5.Ex5.m1.2.2.2.2" xref="S5.Ex5.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S5.Ex5.m1.7.7.2.2.3" xref="S5.Ex5.m1.7.7.2.2.3.cmml">⁢</mo><mrow id="S5.Ex5.m1.7.7.2.2.2.2" xref="S5.Ex5.m1.7.7.2.2.2.3.cmml"><mo id="S5.Ex5.m1.7.7.2.2.2.2.3" stretchy="false" xref="S5.Ex5.m1.7.7.2.2.2.3.cmml">(</mo><msubsup id="S5.Ex5.m1.6.6.1.1.1.1.1" xref="S5.Ex5.m1.6.6.1.1.1.1.1.cmml"><mi id="S5.Ex5.m1.6.6.1.1.1.1.1.2.2" xref="S5.Ex5.m1.6.6.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S5.Ex5.m1.6.6.1.1.1.1.1.3" xref="S5.Ex5.m1.6.6.1.1.1.1.1.3.cmml">+</mo><mi id="S5.Ex5.m1.6.6.1.1.1.1.1.2.3" xref="S5.Ex5.m1.6.6.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S5.Ex5.m1.7.7.2.2.2.2.4" xref="S5.Ex5.m1.7.7.2.2.2.3.cmml">,</mo><msub id="S5.Ex5.m1.7.7.2.2.2.2.2" xref="S5.Ex5.m1.7.7.2.2.2.2.2.cmml"><mi id="S5.Ex5.m1.7.7.2.2.2.2.2.2" xref="S5.Ex5.m1.7.7.2.2.2.2.2.2.cmml">w</mi><mi id="S5.Ex5.m1.7.7.2.2.2.2.2.3" xref="S5.Ex5.m1.7.7.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S5.Ex5.m1.7.7.2.2.2.2.5" xref="S5.Ex5.m1.7.7.2.2.2.3.cmml">;</mo><mi id="S5.Ex5.m1.5.5" xref="S5.Ex5.m1.5.5.cmml">X</mi><mo id="S5.Ex5.m1.7.7.2.2.2.2.6" stretchy="false" xref="S5.Ex5.m1.7.7.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex5.m1.9.9.4.5" rspace="0.111em" stretchy="false" xref="S5.Ex5.m1.9.9.4.5.cmml">→</mo><mrow id="S5.Ex5.m1.9.9.4.4" xref="S5.Ex5.m1.9.9.4.4.cmml"><munderover id="S5.Ex5.m1.9.9.4.4.3" xref="S5.Ex5.m1.9.9.4.4.3.cmml"><mo id="S5.Ex5.m1.9.9.4.4.3.2.2" movablelimits="false" xref="S5.Ex5.m1.9.9.4.4.3.2.2.cmml">∏</mo><mrow id="S5.Ex5.m1.9.9.4.4.3.2.3" xref="S5.Ex5.m1.9.9.4.4.3.2.3.cmml"><mi id="S5.Ex5.m1.9.9.4.4.3.2.3.2" xref="S5.Ex5.m1.9.9.4.4.3.2.3.2.cmml">j</mi><mo id="S5.Ex5.m1.9.9.4.4.3.2.3.1" xref="S5.Ex5.m1.9.9.4.4.3.2.3.1.cmml">=</mo><mn id="S5.Ex5.m1.9.9.4.4.3.2.3.3" xref="S5.Ex5.m1.9.9.4.4.3.2.3.3.cmml">0</mn></mrow><mi id="S5.Ex5.m1.9.9.4.4.3.3" xref="S5.Ex5.m1.9.9.4.4.3.3.cmml">n</mi></munderover><mrow id="S5.Ex5.m1.9.9.4.4.2" xref="S5.Ex5.m1.9.9.4.4.2.cmml"><msubsup id="S5.Ex5.m1.9.9.4.4.2.4" xref="S5.Ex5.m1.9.9.4.4.2.4.cmml"><mi id="S5.Ex5.m1.9.9.4.4.2.4.2.2" xref="S5.Ex5.m1.9.9.4.4.2.4.2.2.cmml">B</mi><mrow id="S5.Ex5.m1.4.4.2.4" xref="S5.Ex5.m1.4.4.2.3.cmml"><mi id="S5.Ex5.m1.3.3.1.1" xref="S5.Ex5.m1.3.3.1.1.cmml">p</mi><mo id="S5.Ex5.m1.4.4.2.4.1" xref="S5.Ex5.m1.4.4.2.3.cmml">,</mo><mi 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italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.I5.i1.p1.5"><span class="ltx_text ltx_font_italic" id="S5.I5.i1.p1.5.1">is a continuous and surjective operator.</span></p> </div> </li> <li class="ltx_item" id="S5.I5.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S5.I5.i2.p1"> <p class="ltx_p" id="S5.I5.i2.p1.4"><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.4.1">If </span><math alttext="k\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S5.I5.i2.p1.1.m1.1"><semantics id="S5.I5.i2.p1.1.m1.1a"><mrow id="S5.I5.i2.p1.1.m1.1.1" xref="S5.I5.i2.p1.1.m1.1.1.cmml"><mi id="S5.I5.i2.p1.1.m1.1.1.2" xref="S5.I5.i2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S5.I5.i2.p1.1.m1.1.1.1" xref="S5.I5.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S5.I5.i2.p1.1.m1.1.1.3" xref="S5.I5.i2.p1.1.m1.1.1.3.cmml"><mi id="S5.I5.i2.p1.1.m1.1.1.3.2" xref="S5.I5.i2.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S5.I5.i2.p1.1.m1.1.1.3.3" xref="S5.I5.i2.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.1.m1.1b"><apply id="S5.I5.i2.p1.1.m1.1.1.cmml" xref="S5.I5.i2.p1.1.m1.1.1"><in id="S5.I5.i2.p1.1.m1.1.1.1.cmml" xref="S5.I5.i2.p1.1.m1.1.1.1"></in><ci id="S5.I5.i2.p1.1.m1.1.1.2.cmml" xref="S5.I5.i2.p1.1.m1.1.1.2">𝑘</ci><apply id="S5.I5.i2.p1.1.m1.1.1.3.cmml" xref="S5.I5.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.I5.i2.p1.1.m1.1.1.3.1.cmml" xref="S5.I5.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S5.I5.i2.p1.1.m1.1.1.3.2.cmml" xref="S5.I5.i2.p1.1.m1.1.1.3.2">ℕ</ci><cn id="S5.I5.i2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S5.I5.i2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.1.m1.1c">k\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.1.m1.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.4.2">, </span><math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S5.I5.i2.p1.2.m2.4"><semantics id="S5.I5.i2.p1.2.m2.4a"><mrow id="S5.I5.i2.p1.2.m2.4.4" xref="S5.I5.i2.p1.2.m2.4.4.cmml"><mi id="S5.I5.i2.p1.2.m2.4.4.5" xref="S5.I5.i2.p1.2.m2.4.4.5.cmml">γ</mi><mo id="S5.I5.i2.p1.2.m2.4.4.4" xref="S5.I5.i2.p1.2.m2.4.4.4.cmml">∈</mo><mrow id="S5.I5.i2.p1.2.m2.4.4.3" xref="S5.I5.i2.p1.2.m2.4.4.3.cmml"><mrow id="S5.I5.i2.p1.2.m2.2.2.1.1.1" xref="S5.I5.i2.p1.2.m2.2.2.1.1.2.cmml"><mo id="S5.I5.i2.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S5.I5.i2.p1.2.m2.2.2.1.1.2.cmml">(</mo><mrow id="S5.I5.i2.p1.2.m2.2.2.1.1.1.1" xref="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.cmml"><mo id="S5.I5.i2.p1.2.m2.2.2.1.1.1.1a" xref="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.cmml">−</mo><mn id="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.2" xref="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S5.I5.i2.p1.2.m2.2.2.1.1.1.3" xref="S5.I5.i2.p1.2.m2.2.2.1.1.2.cmml">,</mo><mi id="S5.I5.i2.p1.2.m2.1.1" mathvariant="normal" xref="S5.I5.i2.p1.2.m2.1.1.cmml">∞</mi><mo id="S5.I5.i2.p1.2.m2.2.2.1.1.1.4" stretchy="false" xref="S5.I5.i2.p1.2.m2.2.2.1.1.2.cmml">)</mo></mrow><mo id="S5.I5.i2.p1.2.m2.4.4.3.4" xref="S5.I5.i2.p1.2.m2.4.4.3.4.cmml">∖</mo><mrow id="S5.I5.i2.p1.2.m2.4.4.3.3.2" xref="S5.I5.i2.p1.2.m2.4.4.3.3.3.cmml"><mo id="S5.I5.i2.p1.2.m2.4.4.3.3.2.3" stretchy="false" xref="S5.I5.i2.p1.2.m2.4.4.3.3.3.1.cmml">{</mo><mrow id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.cmml"><mrow id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.cmml"><mi id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.2" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.1" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.3" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.1" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.1.cmml">−</mo><mn id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.3" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S5.I5.i2.p1.2.m2.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S5.I5.i2.p1.2.m2.4.4.3.3.3.1.cmml">:</mo><mrow id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.cmml"><mi id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.2" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.2.cmml">j</mi><mo id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.1" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.cmml"><mi id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.2" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.3" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S5.I5.i2.p1.2.m2.4.4.3.3.2.5" stretchy="false" xref="S5.I5.i2.p1.2.m2.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.2.m2.4b"><apply id="S5.I5.i2.p1.2.m2.4.4.cmml" xref="S5.I5.i2.p1.2.m2.4.4"><in id="S5.I5.i2.p1.2.m2.4.4.4.cmml" xref="S5.I5.i2.p1.2.m2.4.4.4"></in><ci id="S5.I5.i2.p1.2.m2.4.4.5.cmml" xref="S5.I5.i2.p1.2.m2.4.4.5">𝛾</ci><apply id="S5.I5.i2.p1.2.m2.4.4.3.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3"><setdiff id="S5.I5.i2.p1.2.m2.4.4.3.4.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.4"></setdiff><interval closure="open" id="S5.I5.i2.p1.2.m2.2.2.1.1.2.cmml" xref="S5.I5.i2.p1.2.m2.2.2.1.1.1"><apply id="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.cmml" xref="S5.I5.i2.p1.2.m2.2.2.1.1.1.1"><minus id="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S5.I5.i2.p1.2.m2.2.2.1.1.1.1"></minus><cn id="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.2.cmml" type="integer" xref="S5.I5.i2.p1.2.m2.2.2.1.1.1.1.2">1</cn></apply><infinity id="S5.I5.i2.p1.2.m2.1.1.cmml" xref="S5.I5.i2.p1.2.m2.1.1"></infinity></interval><apply id="S5.I5.i2.p1.2.m2.4.4.3.3.3.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2"><csymbol cd="latexml" id="S5.I5.i2.p1.2.m2.4.4.3.3.3.1.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.3">conditional-set</csymbol><apply id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.cmml" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1"><minus id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.1.cmml" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.1"></minus><apply id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.cmml" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2"><times id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.1.cmml" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.1"></times><ci id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.2.cmml" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.3.cmml" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.3.cmml" type="integer" xref="S5.I5.i2.p1.2.m2.3.3.2.2.1.1.3">1</cn></apply><apply id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2"><in id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.1.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.1"></in><ci id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.2.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.2">𝑗</ci><apply id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.1.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3">subscript</csymbol><ci id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.2.cmml" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S5.I5.i2.p1.2.m2.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.2.m2.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.2.m2.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.4.3"> and </span><math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.I5.i2.p1.3.m3.1"><semantics id="S5.I5.i2.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S5.I5.i2.p1.3.m3.1.1" xref="S5.I5.i2.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.3.m3.1b"><ci id="S5.I5.i2.p1.3.m3.1.1.cmml" xref="S5.I5.i2.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.3.m3.1d">caligraphic_B</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.4.4"> is of type </span><math alttext="(p,k,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S5.I5.i2.p1.4.m4.5"><semantics id="S5.I5.i2.p1.4.m4.5a"><mrow id="S5.I5.i2.p1.4.m4.5.6.2" xref="S5.I5.i2.p1.4.m4.5.6.1.cmml"><mo id="S5.I5.i2.p1.4.m4.5.6.2.1" stretchy="false" xref="S5.I5.i2.p1.4.m4.5.6.1.cmml">(</mo><mi id="S5.I5.i2.p1.4.m4.1.1" xref="S5.I5.i2.p1.4.m4.1.1.cmml">p</mi><mo id="S5.I5.i2.p1.4.m4.5.6.2.2" xref="S5.I5.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I5.i2.p1.4.m4.2.2" xref="S5.I5.i2.p1.4.m4.2.2.cmml">k</mi><mo id="S5.I5.i2.p1.4.m4.5.6.2.3" xref="S5.I5.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S5.I5.i2.p1.4.m4.3.3" xref="S5.I5.i2.p1.4.m4.3.3.cmml">γ</mi><mo id="S5.I5.i2.p1.4.m4.5.6.2.4" xref="S5.I5.i2.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S5.I5.i2.p1.4.m4.4.4" xref="S5.I5.i2.p1.4.m4.4.4.cmml"><mi id="S5.I5.i2.p1.4.m4.4.4.2" xref="S5.I5.i2.p1.4.m4.4.4.2.cmml">m</mi><mo id="S5.I5.i2.p1.4.m4.4.4.1" xref="S5.I5.i2.p1.4.m4.4.4.1.cmml">¯</mo></mover><mo id="S5.I5.i2.p1.4.m4.5.6.2.5" xref="S5.I5.i2.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S5.I5.i2.p1.4.m4.5.5" xref="S5.I5.i2.p1.4.m4.5.5.cmml"><mi id="S5.I5.i2.p1.4.m4.5.5.2" xref="S5.I5.i2.p1.4.m4.5.5.2.cmml">Y</mi><mo id="S5.I5.i2.p1.4.m4.5.5.1" xref="S5.I5.i2.p1.4.m4.5.5.1.cmml">¯</mo></mover><mo id="S5.I5.i2.p1.4.m4.5.6.2.6" stretchy="false" xref="S5.I5.i2.p1.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.I5.i2.p1.4.m4.5b"><vector id="S5.I5.i2.p1.4.m4.5.6.1.cmml" xref="S5.I5.i2.p1.4.m4.5.6.2"><ci id="S5.I5.i2.p1.4.m4.1.1.cmml" xref="S5.I5.i2.p1.4.m4.1.1">𝑝</ci><ci id="S5.I5.i2.p1.4.m4.2.2.cmml" xref="S5.I5.i2.p1.4.m4.2.2">𝑘</ci><ci id="S5.I5.i2.p1.4.m4.3.3.cmml" xref="S5.I5.i2.p1.4.m4.3.3">𝛾</ci><apply id="S5.I5.i2.p1.4.m4.4.4.cmml" xref="S5.I5.i2.p1.4.m4.4.4"><ci id="S5.I5.i2.p1.4.m4.4.4.1.cmml" xref="S5.I5.i2.p1.4.m4.4.4.1">¯</ci><ci id="S5.I5.i2.p1.4.m4.4.4.2.cmml" xref="S5.I5.i2.p1.4.m4.4.4.2">𝑚</ci></apply><apply id="S5.I5.i2.p1.4.m4.5.5.cmml" xref="S5.I5.i2.p1.4.m4.5.5"><ci id="S5.I5.i2.p1.4.m4.5.5.1.cmml" xref="S5.I5.i2.p1.4.m4.5.5.1">¯</ci><ci id="S5.I5.i2.p1.4.m4.5.5.2.cmml" xref="S5.I5.i2.p1.4.m4.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S5.I5.i2.p1.4.m4.5c">(p,k,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S5.I5.i2.p1.4.m4.5d">( italic_p , italic_k , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.4.5">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to\prod_{j=0}^{n}B^{k-m_{% j}-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};Y_{j})" class="ltx_Math" display="block" id="S5.Ex6.m1.9"><semantics id="S5.Ex6.m1.9a"><mrow id="S5.Ex6.m1.9.9" xref="S5.Ex6.m1.9.9.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex6.m1.9.9.6" xref="S5.Ex6.m1.9.9.6.cmml">ℬ</mi><mo id="S5.Ex6.m1.9.9.5" lspace="0.278em" rspace="0.278em" xref="S5.Ex6.m1.9.9.5.cmml">:</mo><mrow id="S5.Ex6.m1.9.9.4" xref="S5.Ex6.m1.9.9.4.cmml"><mrow id="S5.Ex6.m1.7.7.2.2" xref="S5.Ex6.m1.7.7.2.2.cmml"><msup id="S5.Ex6.m1.7.7.2.2.4" xref="S5.Ex6.m1.7.7.2.2.4.cmml"><mi id="S5.Ex6.m1.7.7.2.2.4.2" xref="S5.Ex6.m1.7.7.2.2.4.2.cmml">W</mi><mrow id="S5.Ex6.m1.2.2.2.4" xref="S5.Ex6.m1.2.2.2.3.cmml"><mi id="S5.Ex6.m1.1.1.1.1" xref="S5.Ex6.m1.1.1.1.1.cmml">k</mi><mo id="S5.Ex6.m1.2.2.2.4.1" xref="S5.Ex6.m1.2.2.2.3.cmml">,</mo><mi id="S5.Ex6.m1.2.2.2.2" xref="S5.Ex6.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S5.Ex6.m1.7.7.2.2.3" xref="S5.Ex6.m1.7.7.2.2.3.cmml">⁢</mo><mrow id="S5.Ex6.m1.7.7.2.2.2.2" xref="S5.Ex6.m1.7.7.2.2.2.3.cmml"><mo id="S5.Ex6.m1.7.7.2.2.2.2.3" stretchy="false" xref="S5.Ex6.m1.7.7.2.2.2.3.cmml">(</mo><msubsup id="S5.Ex6.m1.6.6.1.1.1.1.1" xref="S5.Ex6.m1.6.6.1.1.1.1.1.cmml"><mi id="S5.Ex6.m1.6.6.1.1.1.1.1.2.2" xref="S5.Ex6.m1.6.6.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S5.Ex6.m1.6.6.1.1.1.1.1.3" xref="S5.Ex6.m1.6.6.1.1.1.1.1.3.cmml">+</mo><mi id="S5.Ex6.m1.6.6.1.1.1.1.1.2.3" xref="S5.Ex6.m1.6.6.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S5.Ex6.m1.7.7.2.2.2.2.4" xref="S5.Ex6.m1.7.7.2.2.2.3.cmml">,</mo><msub id="S5.Ex6.m1.7.7.2.2.2.2.2" xref="S5.Ex6.m1.7.7.2.2.2.2.2.cmml"><mi id="S5.Ex6.m1.7.7.2.2.2.2.2.2" xref="S5.Ex6.m1.7.7.2.2.2.2.2.2.cmml">w</mi><mi id="S5.Ex6.m1.7.7.2.2.2.2.2.3" xref="S5.Ex6.m1.7.7.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S5.Ex6.m1.7.7.2.2.2.2.5" xref="S5.Ex6.m1.7.7.2.2.2.3.cmml">;</mo><mi id="S5.Ex6.m1.5.5" xref="S5.Ex6.m1.5.5.cmml">X</mi><mo id="S5.Ex6.m1.7.7.2.2.2.2.6" stretchy="false" xref="S5.Ex6.m1.7.7.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex6.m1.9.9.4.5" rspace="0.111em" stretchy="false" xref="S5.Ex6.m1.9.9.4.5.cmml">→</mo><mrow id="S5.Ex6.m1.9.9.4.4" xref="S5.Ex6.m1.9.9.4.4.cmml"><munderover id="S5.Ex6.m1.9.9.4.4.3" xref="S5.Ex6.m1.9.9.4.4.3.cmml"><mo id="S5.Ex6.m1.9.9.4.4.3.2.2" movablelimits="false" xref="S5.Ex6.m1.9.9.4.4.3.2.2.cmml">∏</mo><mrow id="S5.Ex6.m1.9.9.4.4.3.2.3" xref="S5.Ex6.m1.9.9.4.4.3.2.3.cmml"><mi id="S5.Ex6.m1.9.9.4.4.3.2.3.2" xref="S5.Ex6.m1.9.9.4.4.3.2.3.2.cmml">j</mi><mo id="S5.Ex6.m1.9.9.4.4.3.2.3.1" xref="S5.Ex6.m1.9.9.4.4.3.2.3.1.cmml">=</mo><mn id="S5.Ex6.m1.9.9.4.4.3.2.3.3" xref="S5.Ex6.m1.9.9.4.4.3.2.3.3.cmml">0</mn></mrow><mi id="S5.Ex6.m1.9.9.4.4.3.3" xref="S5.Ex6.m1.9.9.4.4.3.3.cmml">n</mi></munderover><mrow id="S5.Ex6.m1.9.9.4.4.2" xref="S5.Ex6.m1.9.9.4.4.2.cmml"><msubsup id="S5.Ex6.m1.9.9.4.4.2.4" xref="S5.Ex6.m1.9.9.4.4.2.4.cmml"><mi id="S5.Ex6.m1.9.9.4.4.2.4.2.2" xref="S5.Ex6.m1.9.9.4.4.2.4.2.2.cmml">B</mi><mrow id="S5.Ex6.m1.4.4.2.4" xref="S5.Ex6.m1.4.4.2.3.cmml"><mi id="S5.Ex6.m1.3.3.1.1" xref="S5.Ex6.m1.3.3.1.1.cmml">p</mi><mo id="S5.Ex6.m1.4.4.2.4.1" xref="S5.Ex6.m1.4.4.2.3.cmml">,</mo><mi id="S5.Ex6.m1.4.4.2.2" xref="S5.Ex6.m1.4.4.2.2.cmml">p</mi></mrow><mrow id="S5.Ex6.m1.9.9.4.4.2.4.2.3" xref="S5.Ex6.m1.9.9.4.4.2.4.2.3.cmml"><mi id="S5.Ex6.m1.9.9.4.4.2.4.2.3.2" xref="S5.Ex6.m1.9.9.4.4.2.4.2.3.2.cmml">k</mi><mo id="S5.Ex6.m1.9.9.4.4.2.4.2.3.1" xref="S5.Ex6.m1.9.9.4.4.2.4.2.3.1.cmml">−</mo><msub id="S5.Ex6.m1.9.9.4.4.2.4.2.3.3" xref="S5.Ex6.m1.9.9.4.4.2.4.2.3.3.cmml"><mi id="S5.Ex6.m1.9.9.4.4.2.4.2.3.3.2" 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id="S5.Ex6.m1.9.9.4.4.2.2.2.2.cmml" xref="S5.Ex6.m1.9.9.4.4.2.2.2.2"><csymbol cd="ambiguous" id="S5.Ex6.m1.9.9.4.4.2.2.2.2.1.cmml" xref="S5.Ex6.m1.9.9.4.4.2.2.2.2">subscript</csymbol><ci id="S5.Ex6.m1.9.9.4.4.2.2.2.2.2.cmml" xref="S5.Ex6.m1.9.9.4.4.2.2.2.2.2">𝑌</ci><ci id="S5.Ex6.m1.9.9.4.4.2.2.2.2.3.cmml" xref="S5.Ex6.m1.9.9.4.4.2.2.2.2.3">𝑗</ci></apply></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex6.m1.9c">\mathcal{B}:W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to\prod_{j=0}^{n}B^{k-m_{% j}-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};Y_{j})</annotation><annotation encoding="application/x-llamapun" id="S5.Ex6.m1.9d">caligraphic_B : italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_k - italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.I5.i2.p1.5"><span class="ltx_text ltx_font_italic" id="S5.I5.i2.p1.5.1">is a continuous and surjective operator.</span></p> </div> </li> </ol> <p class="ltx_p" id="S5.Thmtheorem6.p1.9"><span class="ltx_text ltx_font_italic" id="S5.Thmtheorem6.p1.9.7">In both cases, there exists a continuous right inverse <math alttext="\operatorname{ext}_{\mathcal{B}}" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.3.1.m1.1"><semantics id="S5.Thmtheorem6.p1.3.1.m1.1a"><msub id="S5.Thmtheorem6.p1.3.1.m1.1.1" xref="S5.Thmtheorem6.p1.3.1.m1.1.1.cmml"><mi id="S5.Thmtheorem6.p1.3.1.m1.1.1.2" xref="S5.Thmtheorem6.p1.3.1.m1.1.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem6.p1.3.1.m1.1.1.3" xref="S5.Thmtheorem6.p1.3.1.m1.1.1.3.cmml">ℬ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.3.1.m1.1b"><apply id="S5.Thmtheorem6.p1.3.1.m1.1.1.cmml" xref="S5.Thmtheorem6.p1.3.1.m1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem6.p1.3.1.m1.1.1.1.cmml" xref="S5.Thmtheorem6.p1.3.1.m1.1.1">subscript</csymbol><ci id="S5.Thmtheorem6.p1.3.1.m1.1.1.2.cmml" xref="S5.Thmtheorem6.p1.3.1.m1.1.1.2">ext</ci><ci id="S5.Thmtheorem6.p1.3.1.m1.1.1.3.cmml" xref="S5.Thmtheorem6.p1.3.1.m1.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.3.1.m1.1c">\operatorname{ext}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.3.1.m1.1d">roman_ext start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> of <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.4.2.m2.1"><semantics id="S5.Thmtheorem6.p1.4.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem6.p1.4.2.m2.1.1" xref="S5.Thmtheorem6.p1.4.2.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.4.2.m2.1b"><ci id="S5.Thmtheorem6.p1.4.2.m2.1.1.cmml" xref="S5.Thmtheorem6.p1.4.2.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.4.2.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.4.2.m2.1d">caligraphic_B</annotation></semantics></math> which is independent of <math alttext="k,s,p,\gamma,\overline{Y}" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.5.3.m3.5"><semantics id="S5.Thmtheorem6.p1.5.3.m3.5a"><mrow 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id="S5.Thmtheorem6.p1.5.3.m3.5.5.1" xref="S5.Thmtheorem6.p1.5.3.m3.5.5.1.cmml">¯</mo></mover></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.5.3.m3.5b"><list id="S5.Thmtheorem6.p1.5.3.m3.5.6.1.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.5.6.2"><ci id="S5.Thmtheorem6.p1.5.3.m3.1.1.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.1.1">𝑘</ci><ci id="S5.Thmtheorem6.p1.5.3.m3.2.2.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.2.2">𝑠</ci><ci id="S5.Thmtheorem6.p1.5.3.m3.3.3.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.3.3">𝑝</ci><ci id="S5.Thmtheorem6.p1.5.3.m3.4.4.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.4.4">𝛾</ci><apply id="S5.Thmtheorem6.p1.5.3.m3.5.5.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.5.5"><ci id="S5.Thmtheorem6.p1.5.3.m3.5.5.1.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.5.5.1">¯</ci><ci id="S5.Thmtheorem6.p1.5.3.m3.5.5.2.cmml" xref="S5.Thmtheorem6.p1.5.3.m3.5.5.2">𝑌</ci></apply></list></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.5.3.m3.5c">k,s,p,\gamma,\overline{Y}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.5.3.m3.5d">italic_k , italic_s , italic_p , italic_γ , over¯ start_ARG italic_Y end_ARG</annotation></semantics></math> and <math alttext="X" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.6.4.m4.1"><semantics id="S5.Thmtheorem6.p1.6.4.m4.1a"><mi id="S5.Thmtheorem6.p1.6.4.m4.1.1" xref="S5.Thmtheorem6.p1.6.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.6.4.m4.1b"><ci id="S5.Thmtheorem6.p1.6.4.m4.1.1.cmml" xref="S5.Thmtheorem6.p1.6.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.6.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.6.4.m4.1d">italic_X</annotation></semantics></math>. For any <math alttext="0\leq j<m_{n}" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.7.5.m5.1"><semantics id="S5.Thmtheorem6.p1.7.5.m5.1a"><mrow id="S5.Thmtheorem6.p1.7.5.m5.1.1" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.cmml"><mn id="S5.Thmtheorem6.p1.7.5.m5.1.1.2" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.2.cmml">0</mn><mo id="S5.Thmtheorem6.p1.7.5.m5.1.1.3" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.3.cmml">≤</mo><mi id="S5.Thmtheorem6.p1.7.5.m5.1.1.4" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.4.cmml">j</mi><mo id="S5.Thmtheorem6.p1.7.5.m5.1.1.5" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.5.cmml"><</mo><msub id="S5.Thmtheorem6.p1.7.5.m5.1.1.6" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.6.cmml"><mi id="S5.Thmtheorem6.p1.7.5.m5.1.1.6.2" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.6.2.cmml">m</mi><mi id="S5.Thmtheorem6.p1.7.5.m5.1.1.6.3" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.6.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.7.5.m5.1b"><apply id="S5.Thmtheorem6.p1.7.5.m5.1.1.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1"><and id="S5.Thmtheorem6.p1.7.5.m5.1.1a.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1"></and><apply id="S5.Thmtheorem6.p1.7.5.m5.1.1b.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1"><leq id="S5.Thmtheorem6.p1.7.5.m5.1.1.3.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.3"></leq><cn id="S5.Thmtheorem6.p1.7.5.m5.1.1.2.cmml" type="integer" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.2">0</cn><ci id="S5.Thmtheorem6.p1.7.5.m5.1.1.4.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.4">𝑗</ci></apply><apply id="S5.Thmtheorem6.p1.7.5.m5.1.1c.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1"><lt id="S5.Thmtheorem6.p1.7.5.m5.1.1.5.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem6.p1.7.5.m5.1.1.4.cmml" id="S5.Thmtheorem6.p1.7.5.m5.1.1d.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1"></share><apply id="S5.Thmtheorem6.p1.7.5.m5.1.1.6.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.6"><csymbol cd="ambiguous" id="S5.Thmtheorem6.p1.7.5.m5.1.1.6.1.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.6">subscript</csymbol><ci id="S5.Thmtheorem6.p1.7.5.m5.1.1.6.2.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.6.2">𝑚</ci><ci id="S5.Thmtheorem6.p1.7.5.m5.1.1.6.3.cmml" xref="S5.Thmtheorem6.p1.7.5.m5.1.1.6.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.7.5.m5.1c">0\leq j<m_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.7.5.m5.1d">0 ≤ italic_j < italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="j\notin\{m_{0},\dots,m_{n-1}\}" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.8.6.m6.3"><semantics id="S5.Thmtheorem6.p1.8.6.m6.3a"><mrow id="S5.Thmtheorem6.p1.8.6.m6.3.3" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.cmml"><mi id="S5.Thmtheorem6.p1.8.6.m6.3.3.4" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.4.cmml">j</mi><mo id="S5.Thmtheorem6.p1.8.6.m6.3.3.3" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.3.cmml">∉</mo><mrow id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.3.cmml"><mo id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.3" stretchy="false" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.3.cmml">{</mo><msub id="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1" xref="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.cmml"><mi id="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.2" xref="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.2.cmml">m</mi><mn id="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.3" xref="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.4" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.3.cmml">,</mo><mi id="S5.Thmtheorem6.p1.8.6.m6.1.1" mathvariant="normal" xref="S5.Thmtheorem6.p1.8.6.m6.1.1.cmml">…</mi><mo id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.5" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.3.cmml">,</mo><msub id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.cmml"><mi id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.2" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.2.cmml">m</mi><mrow id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.cmml"><mi id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.2" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.2.cmml">n</mi><mo id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.1" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.1.cmml">−</mo><mn id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.3" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.3.cmml">1</mn></mrow></msub><mo id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.6" stretchy="false" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.8.6.m6.3b"><apply id="S5.Thmtheorem6.p1.8.6.m6.3.3.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3"><notin id="S5.Thmtheorem6.p1.8.6.m6.3.3.3.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.3"></notin><ci id="S5.Thmtheorem6.p1.8.6.m6.3.3.4.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.4">𝑗</ci><set id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.3.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2"><apply id="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.1.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1">subscript</csymbol><ci id="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.2.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.2">𝑚</ci><cn id="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.3.cmml" type="integer" xref="S5.Thmtheorem6.p1.8.6.m6.2.2.1.1.1.3">0</cn></apply><ci id="S5.Thmtheorem6.p1.8.6.m6.1.1.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.1.1">…</ci><apply id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.1.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2">subscript</csymbol><ci id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.2.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.2">𝑚</ci><apply id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3"><minus id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.1.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.1"></minus><ci id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.2.cmml" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.2">𝑛</ci><cn id="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.3.cmml" type="integer" xref="S5.Thmtheorem6.p1.8.6.m6.3.3.2.2.2.3.3">1</cn></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.8.6.m6.3c">j\notin\{m_{0},\dots,m_{n-1}\}</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.8.6.m6.3d">italic_j ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, we have <math alttext="\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{\mathcal{B}}=0" class="ltx_Math" display="inline" id="S5.Thmtheorem6.p1.9.7.m7.1"><semantics id="S5.Thmtheorem6.p1.9.7.m7.1a"><mrow id="S5.Thmtheorem6.p1.9.7.m7.1.1" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.cmml"><mrow id="S5.Thmtheorem6.p1.9.7.m7.1.1.2" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.cmml"><msub id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.cmml"><mi id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.2" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.2.cmml">Tr</mi><mi id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.3" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.3.cmml">j</mi></msub><mo id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.1" lspace="0.167em" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.1.cmml">⁢</mo><mrow id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.cmml"><mo id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.1" rspace="0.167em" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.1.cmml">∘</mo><msub id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.cmml"><mi id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.2" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.3" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.3.cmml">ℬ</mi></msub></mrow></mrow><mo id="S5.Thmtheorem6.p1.9.7.m7.1.1.1" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.1.cmml">=</mo><mn id="S5.Thmtheorem6.p1.9.7.m7.1.1.3" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.Thmtheorem6.p1.9.7.m7.1b"><apply id="S5.Thmtheorem6.p1.9.7.m7.1.1.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1"><eq id="S5.Thmtheorem6.p1.9.7.m7.1.1.1.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.1"></eq><apply id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2"><times id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.1.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.1"></times><apply id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2"><csymbol cd="ambiguous" id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.1.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2">subscript</csymbol><ci id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.2.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.2">Tr</ci><ci id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.3.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.2.3">𝑗</ci></apply><apply id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3"><compose id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.1.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.1"></compose><apply id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2"><csymbol cd="ambiguous" id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.1.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2">subscript</csymbol><ci id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.2.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.2">ext</ci><ci id="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.3.cmml" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.2.3.2.3">ℬ</ci></apply></apply></apply><cn id="S5.Thmtheorem6.p1.9.7.m7.1.1.3.cmml" type="integer" xref="S5.Thmtheorem6.p1.9.7.m7.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Thmtheorem6.p1.9.7.m7.1c">\operatorname{Tr}_{j}\mathop{\circ}\nolimits\operatorname{ext}_{\mathcal{B}}=0</annotation><annotation encoding="application/x-llamapun" id="S5.Thmtheorem6.p1.9.7.m7.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∘ roman_ext start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT = 0</annotation></semantics></math>.</span></p> </div> </div> <div class="ltx_proof" id="S5.6"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S5.3.p1"> <p class="ltx_p" id="S5.3.p1.1">The proof is similar to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorem VIII.2.2.1]</cite> and for the convenience of the reader, we provide the proof.</p> </div> <div class="ltx_para" id="S5.4.p2"> <p class="ltx_p" id="S5.4.p2.2">The continuity of <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.4.p2.1.m1.1"><semantics id="S5.4.p2.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.1.m1.1.1" xref="S5.4.p2.1.m1.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.4.p2.1.m1.1b"><ci id="S5.4.p2.1.m1.1.1.cmml" xref="S5.4.p2.1.m1.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.1.m1.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.1.m1.1d">caligraphic_B</annotation></semantics></math> follows from Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem2" title="Lemma 5.2. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.2</span></a>. We construct the right inverse <math alttext="\operatorname{ext}_{\mathcal{B}}" class="ltx_Math" display="inline" id="S5.4.p2.2.m2.1"><semantics id="S5.4.p2.2.m2.1a"><msub id="S5.4.p2.2.m2.1.1" xref="S5.4.p2.2.m2.1.1.cmml"><mi id="S5.4.p2.2.m2.1.1.2" xref="S5.4.p2.2.m2.1.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.2.m2.1.1.3" xref="S5.4.p2.2.m2.1.1.3.cmml">ℬ</mi></msub><annotation-xml encoding="MathML-Content" id="S5.4.p2.2.m2.1b"><apply id="S5.4.p2.2.m2.1.1.cmml" xref="S5.4.p2.2.m2.1.1"><csymbol cd="ambiguous" id="S5.4.p2.2.m2.1.1.1.cmml" xref="S5.4.p2.2.m2.1.1">subscript</csymbol><ci id="S5.4.p2.2.m2.1.1.2.cmml" xref="S5.4.p2.2.m2.1.1.2">ext</ci><ci id="S5.4.p2.2.m2.1.1.3.cmml" xref="S5.4.p2.2.m2.1.1.3">ℬ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.2.m2.1c">\operatorname{ext}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.2.m2.1d">roman_ext start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> in the case of Bessel potential spaces. The case of Sobolev spaces is the same if one invokes Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem4" title="Theorem 4.4. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.4</span></a> instead of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> in the construction below. Let</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}^{m_{i}}=\sum_{j=0}^{m_{i}}b_{i,j}\operatorname{Tr}_{j},\quad b_{i,% j}=\sum_{|\widetilde{\alpha}|\leq m_{i}-j}b^{m_{i}}_{(0,\widetilde{\alpha})}% \partial^{\widetilde{\alpha}}\quad\text{ for }i\in\{0,\dots,n\}," class="ltx_Math" display="block" id="S5.Ex7.m1.11"><semantics id="S5.Ex7.m1.11a"><mrow id="S5.Ex7.m1.11.11.1"><mrow id="S5.Ex7.m1.11.11.1.1.2" xref="S5.Ex7.m1.11.11.1.1.3.cmml"><mrow id="S5.Ex7.m1.11.11.1.1.1.1" xref="S5.Ex7.m1.11.11.1.1.1.1.cmml"><msup id="S5.Ex7.m1.11.11.1.1.1.1.2" xref="S5.Ex7.m1.11.11.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex7.m1.11.11.1.1.1.1.2.2" xref="S5.Ex7.m1.11.11.1.1.1.1.2.2.cmml">ℬ</mi><msub id="S5.Ex7.m1.11.11.1.1.1.1.2.3" xref="S5.Ex7.m1.11.11.1.1.1.1.2.3.cmml"><mi id="S5.Ex7.m1.11.11.1.1.1.1.2.3.2" xref="S5.Ex7.m1.11.11.1.1.1.1.2.3.2.cmml">m</mi><mi id="S5.Ex7.m1.11.11.1.1.1.1.2.3.3" xref="S5.Ex7.m1.11.11.1.1.1.1.2.3.3.cmml">i</mi></msub></msup><mo id="S5.Ex7.m1.11.11.1.1.1.1.1" rspace="0.111em" xref="S5.Ex7.m1.11.11.1.1.1.1.1.cmml">=</mo><mrow id="S5.Ex7.m1.11.11.1.1.1.1.3" xref="S5.Ex7.m1.11.11.1.1.1.1.3.cmml"><munderover id="S5.Ex7.m1.11.11.1.1.1.1.3.1" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.cmml"><mo id="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.2" movablelimits="false" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.2.cmml">∑</mo><mrow id="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3.cmml"><mi id="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3.2" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3.2.cmml">j</mi><mo id="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3.1" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3.1.cmml">=</mo><mn id="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3.3" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.2.3.3.cmml">0</mn></mrow><msub id="S5.Ex7.m1.11.11.1.1.1.1.3.1.3" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.3.cmml"><mi id="S5.Ex7.m1.11.11.1.1.1.1.3.1.3.2" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.3.2.cmml">m</mi><mi id="S5.Ex7.m1.11.11.1.1.1.1.3.1.3.3" xref="S5.Ex7.m1.11.11.1.1.1.1.3.1.3.3.cmml">i</mi></msub></munderover><mrow id="S5.Ex7.m1.11.11.1.1.1.1.3.2" xref="S5.Ex7.m1.11.11.1.1.1.1.3.2.cmml"><msub id="S5.Ex7.m1.11.11.1.1.1.1.3.2.2" xref="S5.Ex7.m1.11.11.1.1.1.1.3.2.2.cmml"><mi id="S5.Ex7.m1.11.11.1.1.1.1.3.2.2.2" xref="S5.Ex7.m1.11.11.1.1.1.1.3.2.2.2.cmml">b</mi><mrow id="S5.Ex7.m1.2.2.2.4" xref="S5.Ex7.m1.2.2.2.3.cmml"><mi id="S5.Ex7.m1.1.1.1.1" xref="S5.Ex7.m1.1.1.1.1.cmml">i</mi><mo id="S5.Ex7.m1.2.2.2.4.1" xref="S5.Ex7.m1.2.2.2.3.cmml">,</mo><mi id="S5.Ex7.m1.2.2.2.2" xref="S5.Ex7.m1.2.2.2.2.cmml">j</mi></mrow></msub><mo id="S5.Ex7.m1.11.11.1.1.1.1.3.2.1" lspace="0.167em" xref="S5.Ex7.m1.11.11.1.1.1.1.3.2.1.cmml">⁢</mo><msub id="S5.Ex7.m1.11.11.1.1.1.1.3.2.3" xref="S5.Ex7.m1.11.11.1.1.1.1.3.2.3.cmml"><mi id="S5.Ex7.m1.11.11.1.1.1.1.3.2.3.2" xref="S5.Ex7.m1.11.11.1.1.1.1.3.2.3.2.cmml">Tr</mi><mi id="S5.Ex7.m1.11.11.1.1.1.1.3.2.3.3" xref="S5.Ex7.m1.11.11.1.1.1.1.3.2.3.3.cmml">j</mi></msub></mrow></mrow></mrow><mo id="S5.Ex7.m1.11.11.1.1.2.3" rspace="1.167em" xref="S5.Ex7.m1.11.11.1.1.3a.cmml">,</mo><mrow id="S5.Ex7.m1.11.11.1.1.2.2.2" xref="S5.Ex7.m1.11.11.1.1.2.2.3.cmml"><mrow id="S5.Ex7.m1.11.11.1.1.2.2.1.1" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.cmml"><msub id="S5.Ex7.m1.11.11.1.1.2.2.1.1.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.2.cmml"><mi id="S5.Ex7.m1.11.11.1.1.2.2.1.1.2.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.2.2.cmml">b</mi><mrow id="S5.Ex7.m1.4.4.2.4" xref="S5.Ex7.m1.4.4.2.3.cmml"><mi id="S5.Ex7.m1.3.3.1.1" xref="S5.Ex7.m1.3.3.1.1.cmml">i</mi><mo id="S5.Ex7.m1.4.4.2.4.1" xref="S5.Ex7.m1.4.4.2.3.cmml">,</mo><mi id="S5.Ex7.m1.4.4.2.2" xref="S5.Ex7.m1.4.4.2.2.cmml">j</mi></mrow></msub><mo id="S5.Ex7.m1.11.11.1.1.2.2.1.1.1" rspace="0.111em" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.1.cmml">=</mo><mrow id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.cmml"><munder id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.1" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.1.cmml"><mo id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.1.2" movablelimits="false" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.1.2.cmml">∑</mo><mrow id="S5.Ex7.m1.5.5.1" xref="S5.Ex7.m1.5.5.1.cmml"><mrow id="S5.Ex7.m1.5.5.1.3.2" xref="S5.Ex7.m1.5.5.1.3.1.cmml"><mo id="S5.Ex7.m1.5.5.1.3.2.1" stretchy="false" xref="S5.Ex7.m1.5.5.1.3.1.1.cmml">|</mo><mover accent="true" id="S5.Ex7.m1.5.5.1.1" xref="S5.Ex7.m1.5.5.1.1.cmml"><mi id="S5.Ex7.m1.5.5.1.1.2" xref="S5.Ex7.m1.5.5.1.1.2.cmml">α</mi><mo id="S5.Ex7.m1.5.5.1.1.1" xref="S5.Ex7.m1.5.5.1.1.1.cmml">~</mo></mover><mo id="S5.Ex7.m1.5.5.1.3.2.2" stretchy="false" xref="S5.Ex7.m1.5.5.1.3.1.1.cmml">|</mo></mrow><mo id="S5.Ex7.m1.5.5.1.2" xref="S5.Ex7.m1.5.5.1.2.cmml">≤</mo><mrow id="S5.Ex7.m1.5.5.1.4" xref="S5.Ex7.m1.5.5.1.4.cmml"><msub id="S5.Ex7.m1.5.5.1.4.2" xref="S5.Ex7.m1.5.5.1.4.2.cmml"><mi id="S5.Ex7.m1.5.5.1.4.2.2" xref="S5.Ex7.m1.5.5.1.4.2.2.cmml">m</mi><mi id="S5.Ex7.m1.5.5.1.4.2.3" xref="S5.Ex7.m1.5.5.1.4.2.3.cmml">i</mi></msub><mo id="S5.Ex7.m1.5.5.1.4.1" xref="S5.Ex7.m1.5.5.1.4.1.cmml">−</mo><mi id="S5.Ex7.m1.5.5.1.4.3" xref="S5.Ex7.m1.5.5.1.4.3.cmml">j</mi></mrow></mrow></munder><mrow id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.cmml"><msubsup id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.cmml"><mi id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.2.cmml">b</mi><mrow id="S5.Ex7.m1.7.7.2.4" xref="S5.Ex7.m1.7.7.2.3.cmml"><mo id="S5.Ex7.m1.7.7.2.4.1" stretchy="false" xref="S5.Ex7.m1.7.7.2.3.cmml">(</mo><mn id="S5.Ex7.m1.6.6.1.1" xref="S5.Ex7.m1.6.6.1.1.cmml">0</mn><mo id="S5.Ex7.m1.7.7.2.4.2" xref="S5.Ex7.m1.7.7.2.3.cmml">,</mo><mover accent="true" id="S5.Ex7.m1.7.7.2.2" xref="S5.Ex7.m1.7.7.2.2.cmml"><mi id="S5.Ex7.m1.7.7.2.2.2" xref="S5.Ex7.m1.7.7.2.2.2.cmml">α</mi><mo id="S5.Ex7.m1.7.7.2.2.1" xref="S5.Ex7.m1.7.7.2.2.1.cmml">~</mo></mover><mo id="S5.Ex7.m1.7.7.2.4.3" stretchy="false" xref="S5.Ex7.m1.7.7.2.3.cmml">)</mo></mrow><msub id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.3" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.3.cmml"><mi id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.3.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.3.2.cmml">m</mi><mi id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.3.3" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.2.2.3.3.cmml">i</mi></msub></msubsup><mo id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.1" lspace="0em" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.1.cmml">⁢</mo><msup id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3.cmml"><mo id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3.2.cmml">∂</mo><mover accent="true" id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3.3" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3.3.cmml"><mi id="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3.3.2" xref="S5.Ex7.m1.11.11.1.1.2.2.1.1.3.2.3.3.2.cmml">α</mi><mo 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}i\in\{0,\dots,n\},</annotation><annotation encoding="application/x-llamapun" id="S5.Ex7.m1.11d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_b start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , italic_b start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT | over~ start_ARG italic_α end_ARG | ≤ italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - italic_j end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ( 0 , over~ start_ARG italic_α end_ARG ) end_POSTSUBSCRIPT ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_α end_ARG end_POSTSUPERSCRIPT for italic_i ∈ { 0 , … , italic_n } ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.4.p2.6">where <math alttext="b^{m_{i}}_{\alpha}" class="ltx_Math" display="inline" id="S5.4.p2.3.m1.1"><semantics id="S5.4.p2.3.m1.1a"><msubsup id="S5.4.p2.3.m1.1.1" xref="S5.4.p2.3.m1.1.1.cmml"><mi id="S5.4.p2.3.m1.1.1.2.2" xref="S5.4.p2.3.m1.1.1.2.2.cmml">b</mi><mi id="S5.4.p2.3.m1.1.1.3" xref="S5.4.p2.3.m1.1.1.3.cmml">α</mi><msub id="S5.4.p2.3.m1.1.1.2.3" xref="S5.4.p2.3.m1.1.1.2.3.cmml"><mi id="S5.4.p2.3.m1.1.1.2.3.2" xref="S5.4.p2.3.m1.1.1.2.3.2.cmml">m</mi><mi id="S5.4.p2.3.m1.1.1.2.3.3" xref="S5.4.p2.3.m1.1.1.2.3.3.cmml">i</mi></msub></msubsup><annotation-xml encoding="MathML-Content" id="S5.4.p2.3.m1.1b"><apply id="S5.4.p2.3.m1.1.1.cmml" xref="S5.4.p2.3.m1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.3.m1.1.1.1.cmml" xref="S5.4.p2.3.m1.1.1">subscript</csymbol><apply id="S5.4.p2.3.m1.1.1.2.cmml" xref="S5.4.p2.3.m1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.3.m1.1.1.2.1.cmml" xref="S5.4.p2.3.m1.1.1">superscript</csymbol><ci id="S5.4.p2.3.m1.1.1.2.2.cmml" xref="S5.4.p2.3.m1.1.1.2.2">𝑏</ci><apply id="S5.4.p2.3.m1.1.1.2.3.cmml" xref="S5.4.p2.3.m1.1.1.2.3"><csymbol cd="ambiguous" id="S5.4.p2.3.m1.1.1.2.3.1.cmml" xref="S5.4.p2.3.m1.1.1.2.3">subscript</csymbol><ci id="S5.4.p2.3.m1.1.1.2.3.2.cmml" xref="S5.4.p2.3.m1.1.1.2.3.2">𝑚</ci><ci id="S5.4.p2.3.m1.1.1.2.3.3.cmml" xref="S5.4.p2.3.m1.1.1.2.3.3">𝑖</ci></apply></apply><ci id="S5.4.p2.3.m1.1.1.3.cmml" xref="S5.4.p2.3.m1.1.1.3">𝛼</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.3.m1.1c">b^{m_{i}}_{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.3.m1.1d">italic_b start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_α end_POSTSUBSCRIPT</annotation></semantics></math> are the coefficients of the operator <math alttext="\mathcal{B}^{m_{i}}" class="ltx_Math" display="inline" id="S5.4.p2.4.m2.1"><semantics id="S5.4.p2.4.m2.1a"><msup id="S5.4.p2.4.m2.1.1" xref="S5.4.p2.4.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.4.m2.1.1.2" xref="S5.4.p2.4.m2.1.1.2.cmml">ℬ</mi><msub id="S5.4.p2.4.m2.1.1.3" xref="S5.4.p2.4.m2.1.1.3.cmml"><mi id="S5.4.p2.4.m2.1.1.3.2" xref="S5.4.p2.4.m2.1.1.3.2.cmml">m</mi><mi id="S5.4.p2.4.m2.1.1.3.3" xref="S5.4.p2.4.m2.1.1.3.3.cmml">i</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S5.4.p2.4.m2.1b"><apply id="S5.4.p2.4.m2.1.1.cmml" xref="S5.4.p2.4.m2.1.1"><csymbol cd="ambiguous" id="S5.4.p2.4.m2.1.1.1.cmml" xref="S5.4.p2.4.m2.1.1">superscript</csymbol><ci id="S5.4.p2.4.m2.1.1.2.cmml" xref="S5.4.p2.4.m2.1.1.2">ℬ</ci><apply id="S5.4.p2.4.m2.1.1.3.cmml" xref="S5.4.p2.4.m2.1.1.3"><csymbol cd="ambiguous" id="S5.4.p2.4.m2.1.1.3.1.cmml" xref="S5.4.p2.4.m2.1.1.3">subscript</csymbol><ci id="S5.4.p2.4.m2.1.1.3.2.cmml" xref="S5.4.p2.4.m2.1.1.3.2">𝑚</ci><ci id="S5.4.p2.4.m2.1.1.3.3.cmml" xref="S5.4.p2.4.m2.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.4.m2.1c">\mathcal{B}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.4.m2.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math>. According to Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem3" title="Definition 5.3 (Normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.3</span></a> there exists a coretraction <math alttext="b^{{\rm c}}_{i,m_{i}}(\widetilde{x})" class="ltx_Math" display="inline" id="S5.4.p2.5.m3.3"><semantics id="S5.4.p2.5.m3.3a"><mrow id="S5.4.p2.5.m3.3.4" xref="S5.4.p2.5.m3.3.4.cmml"><msubsup id="S5.4.p2.5.m3.3.4.2" xref="S5.4.p2.5.m3.3.4.2.cmml"><mi id="S5.4.p2.5.m3.3.4.2.2.2" xref="S5.4.p2.5.m3.3.4.2.2.2.cmml">b</mi><mrow id="S5.4.p2.5.m3.2.2.2.2" xref="S5.4.p2.5.m3.2.2.2.3.cmml"><mi id="S5.4.p2.5.m3.1.1.1.1" xref="S5.4.p2.5.m3.1.1.1.1.cmml">i</mi><mo id="S5.4.p2.5.m3.2.2.2.2.2" xref="S5.4.p2.5.m3.2.2.2.3.cmml">,</mo><msub id="S5.4.p2.5.m3.2.2.2.2.1" xref="S5.4.p2.5.m3.2.2.2.2.1.cmml"><mi id="S5.4.p2.5.m3.2.2.2.2.1.2" xref="S5.4.p2.5.m3.2.2.2.2.1.2.cmml">m</mi><mi id="S5.4.p2.5.m3.2.2.2.2.1.3" xref="S5.4.p2.5.m3.2.2.2.2.1.3.cmml">i</mi></msub></mrow><mi id="S5.4.p2.5.m3.3.4.2.2.3" mathvariant="normal" xref="S5.4.p2.5.m3.3.4.2.2.3.cmml">c</mi></msubsup><mo id="S5.4.p2.5.m3.3.4.1" xref="S5.4.p2.5.m3.3.4.1.cmml">⁢</mo><mrow id="S5.4.p2.5.m3.3.4.3.2" xref="S5.4.p2.5.m3.3.3.cmml"><mo id="S5.4.p2.5.m3.3.4.3.2.1" stretchy="false" xref="S5.4.p2.5.m3.3.3.cmml">(</mo><mover accent="true" id="S5.4.p2.5.m3.3.3" xref="S5.4.p2.5.m3.3.3.cmml"><mi id="S5.4.p2.5.m3.3.3.2" xref="S5.4.p2.5.m3.3.3.2.cmml">x</mi><mo id="S5.4.p2.5.m3.3.3.1" xref="S5.4.p2.5.m3.3.3.1.cmml">~</mo></mover><mo id="S5.4.p2.5.m3.3.4.3.2.2" stretchy="false" xref="S5.4.p2.5.m3.3.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.5.m3.3b"><apply id="S5.4.p2.5.m3.3.4.cmml" xref="S5.4.p2.5.m3.3.4"><times id="S5.4.p2.5.m3.3.4.1.cmml" xref="S5.4.p2.5.m3.3.4.1"></times><apply id="S5.4.p2.5.m3.3.4.2.cmml" xref="S5.4.p2.5.m3.3.4.2"><csymbol cd="ambiguous" id="S5.4.p2.5.m3.3.4.2.1.cmml" xref="S5.4.p2.5.m3.3.4.2">subscript</csymbol><apply id="S5.4.p2.5.m3.3.4.2.2.cmml" xref="S5.4.p2.5.m3.3.4.2"><csymbol cd="ambiguous" id="S5.4.p2.5.m3.3.4.2.2.1.cmml" xref="S5.4.p2.5.m3.3.4.2">superscript</csymbol><ci id="S5.4.p2.5.m3.3.4.2.2.2.cmml" xref="S5.4.p2.5.m3.3.4.2.2.2">𝑏</ci><ci id="S5.4.p2.5.m3.3.4.2.2.3.cmml" xref="S5.4.p2.5.m3.3.4.2.2.3">c</ci></apply><list id="S5.4.p2.5.m3.2.2.2.3.cmml" xref="S5.4.p2.5.m3.2.2.2.2"><ci id="S5.4.p2.5.m3.1.1.1.1.cmml" xref="S5.4.p2.5.m3.1.1.1.1">𝑖</ci><apply id="S5.4.p2.5.m3.2.2.2.2.1.cmml" xref="S5.4.p2.5.m3.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.4.p2.5.m3.2.2.2.2.1.1.cmml" xref="S5.4.p2.5.m3.2.2.2.2.1">subscript</csymbol><ci id="S5.4.p2.5.m3.2.2.2.2.1.2.cmml" xref="S5.4.p2.5.m3.2.2.2.2.1.2">𝑚</ci><ci id="S5.4.p2.5.m3.2.2.2.2.1.3.cmml" xref="S5.4.p2.5.m3.2.2.2.2.1.3">𝑖</ci></apply></list></apply><apply id="S5.4.p2.5.m3.3.3.cmml" xref="S5.4.p2.5.m3.3.4.3.2"><ci id="S5.4.p2.5.m3.3.3.1.cmml" xref="S5.4.p2.5.m3.3.3.1">~</ci><ci id="S5.4.p2.5.m3.3.3.2.cmml" xref="S5.4.p2.5.m3.3.3.2">𝑥</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.5.m3.3c">b^{{\rm c}}_{i,m_{i}}(\widetilde{x})</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.5.m3.3d">italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( over~ start_ARG italic_x end_ARG )</annotation></semantics></math> such that <math alttext="b^{{\rm c}}_{i,m_{i}}\in C_{{\rm b}}^{\ell}(\mathbb{R}^{d-1};\mathcal{L}(Y_{i}% ,X))" class="ltx_Math" display="inline" id="S5.4.p2.6.m4.5"><semantics id="S5.4.p2.6.m4.5a"><mrow id="S5.4.p2.6.m4.5.5" xref="S5.4.p2.6.m4.5.5.cmml"><msubsup id="S5.4.p2.6.m4.5.5.4" xref="S5.4.p2.6.m4.5.5.4.cmml"><mi id="S5.4.p2.6.m4.5.5.4.2.2" xref="S5.4.p2.6.m4.5.5.4.2.2.cmml">b</mi><mrow id="S5.4.p2.6.m4.2.2.2.2" xref="S5.4.p2.6.m4.2.2.2.3.cmml"><mi id="S5.4.p2.6.m4.1.1.1.1" xref="S5.4.p2.6.m4.1.1.1.1.cmml">i</mi><mo id="S5.4.p2.6.m4.2.2.2.2.2" xref="S5.4.p2.6.m4.2.2.2.3.cmml">,</mo><msub id="S5.4.p2.6.m4.2.2.2.2.1" xref="S5.4.p2.6.m4.2.2.2.2.1.cmml"><mi id="S5.4.p2.6.m4.2.2.2.2.1.2" xref="S5.4.p2.6.m4.2.2.2.2.1.2.cmml">m</mi><mi id="S5.4.p2.6.m4.2.2.2.2.1.3" xref="S5.4.p2.6.m4.2.2.2.2.1.3.cmml">i</mi></msub></mrow><mi id="S5.4.p2.6.m4.5.5.4.2.3" mathvariant="normal" xref="S5.4.p2.6.m4.5.5.4.2.3.cmml">c</mi></msubsup><mo id="S5.4.p2.6.m4.5.5.3" xref="S5.4.p2.6.m4.5.5.3.cmml">∈</mo><mrow id="S5.4.p2.6.m4.5.5.2" xref="S5.4.p2.6.m4.5.5.2.cmml"><msubsup id="S5.4.p2.6.m4.5.5.2.4" xref="S5.4.p2.6.m4.5.5.2.4.cmml"><mi id="S5.4.p2.6.m4.5.5.2.4.2.2" xref="S5.4.p2.6.m4.5.5.2.4.2.2.cmml">C</mi><mi id="S5.4.p2.6.m4.5.5.2.4.2.3" mathvariant="normal" xref="S5.4.p2.6.m4.5.5.2.4.2.3.cmml">b</mi><mi id="S5.4.p2.6.m4.5.5.2.4.3" mathvariant="normal" xref="S5.4.p2.6.m4.5.5.2.4.3.cmml">ℓ</mi></msubsup><mo id="S5.4.p2.6.m4.5.5.2.3" xref="S5.4.p2.6.m4.5.5.2.3.cmml">⁢</mo><mrow 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end_POSTSUBSCRIPT , italic_X ) )</annotation></semantics></math>. Define the operator</p> <table class="ltx_equation ltx_eqn_table" id="S5.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\mathcal{C}}^{m_{i}}=-\sum_{j=0}^{m_{i}-1}b^{{\rm c}}_{i,m_{i}}b_{i% ,j}\operatorname{Tr}_{j},\quad\widetilde{\mathcal{C}}^{0}:=0." class="ltx_Math" display="block" id="S5.E6.m1.5"><semantics id="S5.E6.m1.5a"><mrow id="S5.E6.m1.5.5.1"><mrow id="S5.E6.m1.5.5.1.1.2" xref="S5.E6.m1.5.5.1.1.3.cmml"><mrow id="S5.E6.m1.5.5.1.1.1.1" xref="S5.E6.m1.5.5.1.1.1.1.cmml"><msup id="S5.E6.m1.5.5.1.1.1.1.2" xref="S5.E6.m1.5.5.1.1.1.1.2.cmml"><mover accent="true" id="S5.E6.m1.5.5.1.1.1.1.2.2" xref="S5.E6.m1.5.5.1.1.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E6.m1.5.5.1.1.1.1.2.2.2" xref="S5.E6.m1.5.5.1.1.1.1.2.2.2.cmml">𝒞</mi><mo id="S5.E6.m1.5.5.1.1.1.1.2.2.1" xref="S5.E6.m1.5.5.1.1.1.1.2.2.1.cmml">~</mo></mover><msub 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xref="S5.E6.m1.5.5.1.1.2.2.2.2.2">𝒞</ci></apply><cn id="S5.E6.m1.5.5.1.1.2.2.2.3.cmml" type="integer" xref="S5.E6.m1.5.5.1.1.2.2.2.3">0</cn></apply><cn id="S5.E6.m1.5.5.1.1.2.2.3.cmml" type="integer" xref="S5.E6.m1.5.5.1.1.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E6.m1.5c">\widetilde{\mathcal{C}}^{m_{i}}=-\sum_{j=0}^{m_{i}-1}b^{{\rm c}}_{i,m_{i}}b_{i% ,j}\operatorname{Tr}_{j},\quad\widetilde{\mathcal{C}}^{0}:=0.</annotation><annotation encoding="application/x-llamapun" id="S5.E6.m1.5d">over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = - ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT , over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT := 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.4.p2.25">Reasoning as in the proof of Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem2" title="Lemma 5.2. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.2</span></a> gives that</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\mathcal{C}}^{m_{i}}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^% {s-m_{i}-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S5.Ex8.m1.9"><semantics id="S5.Ex8.m1.9a"><mrow id="S5.Ex8.m1.9.9" xref="S5.Ex8.m1.9.9.cmml"><msup id="S5.Ex8.m1.9.9.5" xref="S5.Ex8.m1.9.9.5.cmml"><mover accent="true" id="S5.Ex8.m1.9.9.5.2" xref="S5.Ex8.m1.9.9.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex8.m1.9.9.5.2.2" xref="S5.Ex8.m1.9.9.5.2.2.cmml">𝒞</mi><mo id="S5.Ex8.m1.9.9.5.2.1" xref="S5.Ex8.m1.9.9.5.2.1.cmml">~</mo></mover><msub 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encoding="application/x-tex" id="S5.Ex8.m1.9c">\widetilde{\mathcal{C}}^{m_{i}}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to B^% {s-m_{i}-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex8.m1.9d">over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_B start_POSTSUPERSCRIPT italic_s - italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.4.p2.12">is a bounded linear operator. Now, define the operator <math alttext="\widetilde{\mathcal{C}}:=(\widetilde{\mathcal{C}}^{j})_{j=0}^{m_{n}}" class="ltx_Math" display="inline" id="S5.4.p2.7.m1.1"><semantics id="S5.4.p2.7.m1.1a"><mrow id="S5.4.p2.7.m1.1.1" xref="S5.4.p2.7.m1.1.1.cmml"><mover accent="true" id="S5.4.p2.7.m1.1.1.3" xref="S5.4.p2.7.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.7.m1.1.1.3.2" xref="S5.4.p2.7.m1.1.1.3.2.cmml">𝒞</mi><mo id="S5.4.p2.7.m1.1.1.3.1" xref="S5.4.p2.7.m1.1.1.3.1.cmml">~</mo></mover><mo id="S5.4.p2.7.m1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S5.4.p2.7.m1.1.1.2.cmml">:=</mo><msubsup id="S5.4.p2.7.m1.1.1.1" xref="S5.4.p2.7.m1.1.1.1.cmml"><mrow id="S5.4.p2.7.m1.1.1.1.1.1.1" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.cmml"><mo id="S5.4.p2.7.m1.1.1.1.1.1.1.2" stretchy="false" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.cmml">(</mo><msup id="S5.4.p2.7.m1.1.1.1.1.1.1.1" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.cmml"><mover accent="true" id="S5.4.p2.7.m1.1.1.1.1.1.1.1.2" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.2" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.2.cmml">𝒞</mi><mo id="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.1" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.1.cmml">~</mo></mover><mi id="S5.4.p2.7.m1.1.1.1.1.1.1.1.3" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.3.cmml">j</mi></msup><mo id="S5.4.p2.7.m1.1.1.1.1.1.1.3" stretchy="false" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mrow id="S5.4.p2.7.m1.1.1.1.1.3" xref="S5.4.p2.7.m1.1.1.1.1.3.cmml"><mi id="S5.4.p2.7.m1.1.1.1.1.3.2" xref="S5.4.p2.7.m1.1.1.1.1.3.2.cmml">j</mi><mo id="S5.4.p2.7.m1.1.1.1.1.3.1" xref="S5.4.p2.7.m1.1.1.1.1.3.1.cmml">=</mo><mn id="S5.4.p2.7.m1.1.1.1.1.3.3" xref="S5.4.p2.7.m1.1.1.1.1.3.3.cmml">0</mn></mrow><msub id="S5.4.p2.7.m1.1.1.1.3" xref="S5.4.p2.7.m1.1.1.1.3.cmml"><mi id="S5.4.p2.7.m1.1.1.1.3.2" xref="S5.4.p2.7.m1.1.1.1.3.2.cmml">m</mi><mi id="S5.4.p2.7.m1.1.1.1.3.3" xref="S5.4.p2.7.m1.1.1.1.3.3.cmml">n</mi></msub></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.7.m1.1b"><apply id="S5.4.p2.7.m1.1.1.cmml" xref="S5.4.p2.7.m1.1.1"><csymbol cd="latexml" id="S5.4.p2.7.m1.1.1.2.cmml" xref="S5.4.p2.7.m1.1.1.2">assign</csymbol><apply id="S5.4.p2.7.m1.1.1.3.cmml" xref="S5.4.p2.7.m1.1.1.3"><ci id="S5.4.p2.7.m1.1.1.3.1.cmml" xref="S5.4.p2.7.m1.1.1.3.1">~</ci><ci id="S5.4.p2.7.m1.1.1.3.2.cmml" xref="S5.4.p2.7.m1.1.1.3.2">𝒞</ci></apply><apply id="S5.4.p2.7.m1.1.1.1.cmml" xref="S5.4.p2.7.m1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.7.m1.1.1.1.2.cmml" xref="S5.4.p2.7.m1.1.1.1">superscript</csymbol><apply id="S5.4.p2.7.m1.1.1.1.1.cmml" xref="S5.4.p2.7.m1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.7.m1.1.1.1.1.2.cmml" xref="S5.4.p2.7.m1.1.1.1">subscript</csymbol><apply id="S5.4.p2.7.m1.1.1.1.1.1.1.1.cmml" xref="S5.4.p2.7.m1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.7.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.4.p2.7.m1.1.1.1.1.1.1">superscript</csymbol><apply id="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.2"><ci id="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.1">~</ci><ci id="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.2.2">𝒞</ci></apply><ci id="S5.4.p2.7.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.4.p2.7.m1.1.1.1.1.1.1.1.3">𝑗</ci></apply><apply id="S5.4.p2.7.m1.1.1.1.1.3.cmml" xref="S5.4.p2.7.m1.1.1.1.1.3"><eq id="S5.4.p2.7.m1.1.1.1.1.3.1.cmml" xref="S5.4.p2.7.m1.1.1.1.1.3.1"></eq><ci id="S5.4.p2.7.m1.1.1.1.1.3.2.cmml" xref="S5.4.p2.7.m1.1.1.1.1.3.2">𝑗</ci><cn id="S5.4.p2.7.m1.1.1.1.1.3.3.cmml" type="integer" xref="S5.4.p2.7.m1.1.1.1.1.3.3">0</cn></apply></apply><apply id="S5.4.p2.7.m1.1.1.1.3.cmml" xref="S5.4.p2.7.m1.1.1.1.3"><csymbol cd="ambiguous" id="S5.4.p2.7.m1.1.1.1.3.1.cmml" xref="S5.4.p2.7.m1.1.1.1.3">subscript</csymbol><ci id="S5.4.p2.7.m1.1.1.1.3.2.cmml" xref="S5.4.p2.7.m1.1.1.1.3.2">𝑚</ci><ci id="S5.4.p2.7.m1.1.1.1.3.3.cmml" xref="S5.4.p2.7.m1.1.1.1.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.7.m1.1c">\widetilde{\mathcal{C}}:=(\widetilde{\mathcal{C}}^{j})_{j=0}^{m_{n}}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.7.m1.1d">over~ start_ARG caligraphic_C end_ARG := ( over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ) start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> by setting <math alttext="\widetilde{\mathcal{C}}^{j}=\widetilde{\mathcal{C}}^{m_{i}}" class="ltx_Math" display="inline" id="S5.4.p2.8.m2.1"><semantics id="S5.4.p2.8.m2.1a"><mrow id="S5.4.p2.8.m2.1.1" xref="S5.4.p2.8.m2.1.1.cmml"><msup id="S5.4.p2.8.m2.1.1.2" xref="S5.4.p2.8.m2.1.1.2.cmml"><mover accent="true" id="S5.4.p2.8.m2.1.1.2.2" xref="S5.4.p2.8.m2.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.8.m2.1.1.2.2.2" xref="S5.4.p2.8.m2.1.1.2.2.2.cmml">𝒞</mi><mo id="S5.4.p2.8.m2.1.1.2.2.1" xref="S5.4.p2.8.m2.1.1.2.2.1.cmml">~</mo></mover><mi id="S5.4.p2.8.m2.1.1.2.3" xref="S5.4.p2.8.m2.1.1.2.3.cmml">j</mi></msup><mo id="S5.4.p2.8.m2.1.1.1" xref="S5.4.p2.8.m2.1.1.1.cmml">=</mo><msup id="S5.4.p2.8.m2.1.1.3" xref="S5.4.p2.8.m2.1.1.3.cmml"><mover accent="true" id="S5.4.p2.8.m2.1.1.3.2" xref="S5.4.p2.8.m2.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.8.m2.1.1.3.2.2" xref="S5.4.p2.8.m2.1.1.3.2.2.cmml">𝒞</mi><mo id="S5.4.p2.8.m2.1.1.3.2.1" xref="S5.4.p2.8.m2.1.1.3.2.1.cmml">~</mo></mover><msub id="S5.4.p2.8.m2.1.1.3.3" xref="S5.4.p2.8.m2.1.1.3.3.cmml"><mi id="S5.4.p2.8.m2.1.1.3.3.2" xref="S5.4.p2.8.m2.1.1.3.3.2.cmml">m</mi><mi id="S5.4.p2.8.m2.1.1.3.3.3" xref="S5.4.p2.8.m2.1.1.3.3.3.cmml">i</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.8.m2.1b"><apply id="S5.4.p2.8.m2.1.1.cmml" xref="S5.4.p2.8.m2.1.1"><eq id="S5.4.p2.8.m2.1.1.1.cmml" xref="S5.4.p2.8.m2.1.1.1"></eq><apply id="S5.4.p2.8.m2.1.1.2.cmml" xref="S5.4.p2.8.m2.1.1.2"><csymbol cd="ambiguous" id="S5.4.p2.8.m2.1.1.2.1.cmml" xref="S5.4.p2.8.m2.1.1.2">superscript</csymbol><apply id="S5.4.p2.8.m2.1.1.2.2.cmml" xref="S5.4.p2.8.m2.1.1.2.2"><ci id="S5.4.p2.8.m2.1.1.2.2.1.cmml" xref="S5.4.p2.8.m2.1.1.2.2.1">~</ci><ci id="S5.4.p2.8.m2.1.1.2.2.2.cmml" xref="S5.4.p2.8.m2.1.1.2.2.2">𝒞</ci></apply><ci id="S5.4.p2.8.m2.1.1.2.3.cmml" xref="S5.4.p2.8.m2.1.1.2.3">𝑗</ci></apply><apply id="S5.4.p2.8.m2.1.1.3.cmml" xref="S5.4.p2.8.m2.1.1.3"><csymbol cd="ambiguous" id="S5.4.p2.8.m2.1.1.3.1.cmml" xref="S5.4.p2.8.m2.1.1.3">superscript</csymbol><apply id="S5.4.p2.8.m2.1.1.3.2.cmml" xref="S5.4.p2.8.m2.1.1.3.2"><ci id="S5.4.p2.8.m2.1.1.3.2.1.cmml" xref="S5.4.p2.8.m2.1.1.3.2.1">~</ci><ci id="S5.4.p2.8.m2.1.1.3.2.2.cmml" xref="S5.4.p2.8.m2.1.1.3.2.2">𝒞</ci></apply><apply id="S5.4.p2.8.m2.1.1.3.3.cmml" xref="S5.4.p2.8.m2.1.1.3.3"><csymbol cd="ambiguous" id="S5.4.p2.8.m2.1.1.3.3.1.cmml" xref="S5.4.p2.8.m2.1.1.3.3">subscript</csymbol><ci id="S5.4.p2.8.m2.1.1.3.3.2.cmml" xref="S5.4.p2.8.m2.1.1.3.3.2">𝑚</ci><ci id="S5.4.p2.8.m2.1.1.3.3.3.cmml" xref="S5.4.p2.8.m2.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.8.m2.1c">\widetilde{\mathcal{C}}^{j}=\widetilde{\mathcal{C}}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.8.m2.1d">over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> if <math alttext="j=m_{i}" class="ltx_Math" display="inline" id="S5.4.p2.9.m3.1"><semantics id="S5.4.p2.9.m3.1a"><mrow id="S5.4.p2.9.m3.1.1" xref="S5.4.p2.9.m3.1.1.cmml"><mi id="S5.4.p2.9.m3.1.1.2" xref="S5.4.p2.9.m3.1.1.2.cmml">j</mi><mo id="S5.4.p2.9.m3.1.1.1" xref="S5.4.p2.9.m3.1.1.1.cmml">=</mo><msub id="S5.4.p2.9.m3.1.1.3" xref="S5.4.p2.9.m3.1.1.3.cmml"><mi id="S5.4.p2.9.m3.1.1.3.2" xref="S5.4.p2.9.m3.1.1.3.2.cmml">m</mi><mi id="S5.4.p2.9.m3.1.1.3.3" xref="S5.4.p2.9.m3.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.9.m3.1b"><apply id="S5.4.p2.9.m3.1.1.cmml" xref="S5.4.p2.9.m3.1.1"><eq id="S5.4.p2.9.m3.1.1.1.cmml" xref="S5.4.p2.9.m3.1.1.1"></eq><ci id="S5.4.p2.9.m3.1.1.2.cmml" xref="S5.4.p2.9.m3.1.1.2">𝑗</ci><apply id="S5.4.p2.9.m3.1.1.3.cmml" xref="S5.4.p2.9.m3.1.1.3"><csymbol cd="ambiguous" id="S5.4.p2.9.m3.1.1.3.1.cmml" xref="S5.4.p2.9.m3.1.1.3">subscript</csymbol><ci id="S5.4.p2.9.m3.1.1.3.2.cmml" xref="S5.4.p2.9.m3.1.1.3.2">𝑚</ci><ci id="S5.4.p2.9.m3.1.1.3.3.cmml" xref="S5.4.p2.9.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.9.m3.1c">j=m_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.9.m3.1d">italic_j = italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="i\in\{0,\dots,n\}" class="ltx_Math" display="inline" id="S5.4.p2.10.m4.3"><semantics id="S5.4.p2.10.m4.3a"><mrow id="S5.4.p2.10.m4.3.4" xref="S5.4.p2.10.m4.3.4.cmml"><mi id="S5.4.p2.10.m4.3.4.2" xref="S5.4.p2.10.m4.3.4.2.cmml">i</mi><mo id="S5.4.p2.10.m4.3.4.1" xref="S5.4.p2.10.m4.3.4.1.cmml">∈</mo><mrow id="S5.4.p2.10.m4.3.4.3.2" xref="S5.4.p2.10.m4.3.4.3.1.cmml"><mo id="S5.4.p2.10.m4.3.4.3.2.1" stretchy="false" xref="S5.4.p2.10.m4.3.4.3.1.cmml">{</mo><mn id="S5.4.p2.10.m4.1.1" xref="S5.4.p2.10.m4.1.1.cmml">0</mn><mo id="S5.4.p2.10.m4.3.4.3.2.2" xref="S5.4.p2.10.m4.3.4.3.1.cmml">,</mo><mi id="S5.4.p2.10.m4.2.2" mathvariant="normal" xref="S5.4.p2.10.m4.2.2.cmml">…</mi><mo id="S5.4.p2.10.m4.3.4.3.2.3" xref="S5.4.p2.10.m4.3.4.3.1.cmml">,</mo><mi id="S5.4.p2.10.m4.3.3" xref="S5.4.p2.10.m4.3.3.cmml">n</mi><mo id="S5.4.p2.10.m4.3.4.3.2.4" stretchy="false" xref="S5.4.p2.10.m4.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.10.m4.3b"><apply id="S5.4.p2.10.m4.3.4.cmml" xref="S5.4.p2.10.m4.3.4"><in id="S5.4.p2.10.m4.3.4.1.cmml" xref="S5.4.p2.10.m4.3.4.1"></in><ci id="S5.4.p2.10.m4.3.4.2.cmml" xref="S5.4.p2.10.m4.3.4.2">𝑖</ci><set id="S5.4.p2.10.m4.3.4.3.1.cmml" xref="S5.4.p2.10.m4.3.4.3.2"><cn id="S5.4.p2.10.m4.1.1.cmml" type="integer" xref="S5.4.p2.10.m4.1.1">0</cn><ci id="S5.4.p2.10.m4.2.2.cmml" xref="S5.4.p2.10.m4.2.2">…</ci><ci id="S5.4.p2.10.m4.3.3.cmml" xref="S5.4.p2.10.m4.3.3">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.10.m4.3c">i\in\{0,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.10.m4.3d">italic_i ∈ { 0 , … , italic_n }</annotation></semantics></math>, and <math alttext="\widetilde{\mathcal{C}}^{j}=0" class="ltx_Math" display="inline" id="S5.4.p2.11.m5.1"><semantics id="S5.4.p2.11.m5.1a"><mrow id="S5.4.p2.11.m5.1.1" xref="S5.4.p2.11.m5.1.1.cmml"><msup id="S5.4.p2.11.m5.1.1.2" xref="S5.4.p2.11.m5.1.1.2.cmml"><mover accent="true" id="S5.4.p2.11.m5.1.1.2.2" xref="S5.4.p2.11.m5.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.11.m5.1.1.2.2.2" xref="S5.4.p2.11.m5.1.1.2.2.2.cmml">𝒞</mi><mo id="S5.4.p2.11.m5.1.1.2.2.1" xref="S5.4.p2.11.m5.1.1.2.2.1.cmml">~</mo></mover><mi id="S5.4.p2.11.m5.1.1.2.3" xref="S5.4.p2.11.m5.1.1.2.3.cmml">j</mi></msup><mo id="S5.4.p2.11.m5.1.1.1" xref="S5.4.p2.11.m5.1.1.1.cmml">=</mo><mn id="S5.4.p2.11.m5.1.1.3" xref="S5.4.p2.11.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.11.m5.1b"><apply id="S5.4.p2.11.m5.1.1.cmml" xref="S5.4.p2.11.m5.1.1"><eq id="S5.4.p2.11.m5.1.1.1.cmml" xref="S5.4.p2.11.m5.1.1.1"></eq><apply id="S5.4.p2.11.m5.1.1.2.cmml" xref="S5.4.p2.11.m5.1.1.2"><csymbol cd="ambiguous" id="S5.4.p2.11.m5.1.1.2.1.cmml" xref="S5.4.p2.11.m5.1.1.2">superscript</csymbol><apply id="S5.4.p2.11.m5.1.1.2.2.cmml" xref="S5.4.p2.11.m5.1.1.2.2"><ci id="S5.4.p2.11.m5.1.1.2.2.1.cmml" xref="S5.4.p2.11.m5.1.1.2.2.1">~</ci><ci id="S5.4.p2.11.m5.1.1.2.2.2.cmml" xref="S5.4.p2.11.m5.1.1.2.2.2">𝒞</ci></apply><ci id="S5.4.p2.11.m5.1.1.2.3.cmml" xref="S5.4.p2.11.m5.1.1.2.3">𝑗</ci></apply><cn id="S5.4.p2.11.m5.1.1.3.cmml" type="integer" xref="S5.4.p2.11.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.11.m5.1c">\widetilde{\mathcal{C}}^{j}=0</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.11.m5.1d">over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = 0</annotation></semantics></math> if <math alttext="j\notin\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S5.4.p2.12.m6.3"><semantics id="S5.4.p2.12.m6.3a"><mrow id="S5.4.p2.12.m6.3.3" xref="S5.4.p2.12.m6.3.3.cmml"><mi id="S5.4.p2.12.m6.3.3.4" xref="S5.4.p2.12.m6.3.3.4.cmml">j</mi><mo id="S5.4.p2.12.m6.3.3.3" xref="S5.4.p2.12.m6.3.3.3.cmml">∉</mo><mrow id="S5.4.p2.12.m6.3.3.2.2" xref="S5.4.p2.12.m6.3.3.2.3.cmml"><mo id="S5.4.p2.12.m6.3.3.2.2.3" stretchy="false" xref="S5.4.p2.12.m6.3.3.2.3.cmml">{</mo><msub id="S5.4.p2.12.m6.2.2.1.1.1" xref="S5.4.p2.12.m6.2.2.1.1.1.cmml"><mi id="S5.4.p2.12.m6.2.2.1.1.1.2" xref="S5.4.p2.12.m6.2.2.1.1.1.2.cmml">m</mi><mn id="S5.4.p2.12.m6.2.2.1.1.1.3" xref="S5.4.p2.12.m6.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.4.p2.12.m6.3.3.2.2.4" xref="S5.4.p2.12.m6.3.3.2.3.cmml">,</mo><mi id="S5.4.p2.12.m6.1.1" mathvariant="normal" xref="S5.4.p2.12.m6.1.1.cmml">…</mi><mo id="S5.4.p2.12.m6.3.3.2.2.5" xref="S5.4.p2.12.m6.3.3.2.3.cmml">,</mo><msub id="S5.4.p2.12.m6.3.3.2.2.2" xref="S5.4.p2.12.m6.3.3.2.2.2.cmml"><mi id="S5.4.p2.12.m6.3.3.2.2.2.2" xref="S5.4.p2.12.m6.3.3.2.2.2.2.cmml">m</mi><mi id="S5.4.p2.12.m6.3.3.2.2.2.3" xref="S5.4.p2.12.m6.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S5.4.p2.12.m6.3.3.2.2.6" stretchy="false" xref="S5.4.p2.12.m6.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.12.m6.3b"><apply id="S5.4.p2.12.m6.3.3.cmml" xref="S5.4.p2.12.m6.3.3"><notin id="S5.4.p2.12.m6.3.3.3.cmml" xref="S5.4.p2.12.m6.3.3.3"></notin><ci id="S5.4.p2.12.m6.3.3.4.cmml" xref="S5.4.p2.12.m6.3.3.4">𝑗</ci><set id="S5.4.p2.12.m6.3.3.2.3.cmml" xref="S5.4.p2.12.m6.3.3.2.2"><apply id="S5.4.p2.12.m6.2.2.1.1.1.cmml" xref="S5.4.p2.12.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.12.m6.2.2.1.1.1.1.cmml" xref="S5.4.p2.12.m6.2.2.1.1.1">subscript</csymbol><ci id="S5.4.p2.12.m6.2.2.1.1.1.2.cmml" xref="S5.4.p2.12.m6.2.2.1.1.1.2">𝑚</ci><cn id="S5.4.p2.12.m6.2.2.1.1.1.3.cmml" type="integer" xref="S5.4.p2.12.m6.2.2.1.1.1.3">0</cn></apply><ci id="S5.4.p2.12.m6.1.1.cmml" xref="S5.4.p2.12.m6.1.1">…</ci><apply id="S5.4.p2.12.m6.3.3.2.2.2.cmml" xref="S5.4.p2.12.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.4.p2.12.m6.3.3.2.2.2.1.cmml" xref="S5.4.p2.12.m6.3.3.2.2.2">subscript</csymbol><ci id="S5.4.p2.12.m6.3.3.2.2.2.2.cmml" xref="S5.4.p2.12.m6.3.3.2.2.2.2">𝑚</ci><ci id="S5.4.p2.12.m6.3.3.2.2.2.3.cmml" xref="S5.4.p2.12.m6.3.3.2.2.2.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.12.m6.3c">j\notin\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.12.m6.3d">italic_j ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\mathcal{C}}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to\prod_{j=0}% ^{m_{n}}B^{s-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)" class="ltx_Math" display="block" id="S5.Ex9.m1.9"><semantics id="S5.Ex9.m1.9a"><mrow id="S5.Ex9.m1.9.9" xref="S5.Ex9.m1.9.9.cmml"><mover accent="true" id="S5.Ex9.m1.9.9.5" xref="S5.Ex9.m1.9.9.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex9.m1.9.9.5.2" xref="S5.Ex9.m1.9.9.5.2.cmml">𝒞</mi><mo id="S5.Ex9.m1.9.9.5.1" xref="S5.Ex9.m1.9.9.5.1.cmml">~</mo></mover><mo id="S5.Ex9.m1.9.9.4" lspace="0.278em" rspace="0.278em" xref="S5.Ex9.m1.9.9.4.cmml">:</mo><mrow id="S5.Ex9.m1.9.9.3" xref="S5.Ex9.m1.9.9.3.cmml"><mrow id="S5.Ex9.m1.8.8.2.2" xref="S5.Ex9.m1.8.8.2.2.cmml"><msup id="S5.Ex9.m1.8.8.2.2.4" xref="S5.Ex9.m1.8.8.2.2.4.cmml"><mi id="S5.Ex9.m1.8.8.2.2.4.2" xref="S5.Ex9.m1.8.8.2.2.4.2.cmml">H</mi><mrow id="S5.Ex9.m1.2.2.2.4" xref="S5.Ex9.m1.2.2.2.3.cmml"><mi id="S5.Ex9.m1.1.1.1.1" xref="S5.Ex9.m1.1.1.1.1.cmml">s</mi><mo id="S5.Ex9.m1.2.2.2.4.1" xref="S5.Ex9.m1.2.2.2.3.cmml">,</mo><mi id="S5.Ex9.m1.2.2.2.2" xref="S5.Ex9.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S5.Ex9.m1.8.8.2.2.3" xref="S5.Ex9.m1.8.8.2.2.3.cmml">⁢</mo><mrow id="S5.Ex9.m1.8.8.2.2.2.2" xref="S5.Ex9.m1.8.8.2.2.2.3.cmml"><mo id="S5.Ex9.m1.8.8.2.2.2.2.3" stretchy="false" xref="S5.Ex9.m1.8.8.2.2.2.3.cmml">(</mo><msubsup id="S5.Ex9.m1.7.7.1.1.1.1.1" xref="S5.Ex9.m1.7.7.1.1.1.1.1.cmml"><mi id="S5.Ex9.m1.7.7.1.1.1.1.1.2.2" xref="S5.Ex9.m1.7.7.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S5.Ex9.m1.7.7.1.1.1.1.1.3" xref="S5.Ex9.m1.7.7.1.1.1.1.1.3.cmml">+</mo><mi id="S5.Ex9.m1.7.7.1.1.1.1.1.2.3" xref="S5.Ex9.m1.7.7.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S5.Ex9.m1.8.8.2.2.2.2.4" xref="S5.Ex9.m1.8.8.2.2.2.3.cmml">,</mo><msub id="S5.Ex9.m1.8.8.2.2.2.2.2" xref="S5.Ex9.m1.8.8.2.2.2.2.2.cmml"><mi id="S5.Ex9.m1.8.8.2.2.2.2.2.2" xref="S5.Ex9.m1.8.8.2.2.2.2.2.2.cmml">w</mi><mi id="S5.Ex9.m1.8.8.2.2.2.2.2.3" xref="S5.Ex9.m1.8.8.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S5.Ex9.m1.8.8.2.2.2.2.5" xref="S5.Ex9.m1.8.8.2.2.2.3.cmml">;</mo><mi id="S5.Ex9.m1.5.5" xref="S5.Ex9.m1.5.5.cmml">X</mi><mo id="S5.Ex9.m1.8.8.2.2.2.2.6" stretchy="false" xref="S5.Ex9.m1.8.8.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex9.m1.9.9.3.4" rspace="0.111em" stretchy="false" xref="S5.Ex9.m1.9.9.3.4.cmml">→</mo><mrow id="S5.Ex9.m1.9.9.3.3" xref="S5.Ex9.m1.9.9.3.3.cmml"><munderover id="S5.Ex9.m1.9.9.3.3.2" xref="S5.Ex9.m1.9.9.3.3.2.cmml"><mo id="S5.Ex9.m1.9.9.3.3.2.2.2" movablelimits="false" xref="S5.Ex9.m1.9.9.3.3.2.2.2.cmml">∏</mo><mrow id="S5.Ex9.m1.9.9.3.3.2.2.3" xref="S5.Ex9.m1.9.9.3.3.2.2.3.cmml"><mi id="S5.Ex9.m1.9.9.3.3.2.2.3.2" xref="S5.Ex9.m1.9.9.3.3.2.2.3.2.cmml">j</mi><mo id="S5.Ex9.m1.9.9.3.3.2.2.3.1" xref="S5.Ex9.m1.9.9.3.3.2.2.3.1.cmml">=</mo><mn id="S5.Ex9.m1.9.9.3.3.2.2.3.3" xref="S5.Ex9.m1.9.9.3.3.2.2.3.3.cmml">0</mn></mrow><msub id="S5.Ex9.m1.9.9.3.3.2.3" xref="S5.Ex9.m1.9.9.3.3.2.3.cmml"><mi id="S5.Ex9.m1.9.9.3.3.2.3.2" xref="S5.Ex9.m1.9.9.3.3.2.3.2.cmml">m</mi><mi id="S5.Ex9.m1.9.9.3.3.2.3.3" xref="S5.Ex9.m1.9.9.3.3.2.3.3.cmml">n</mi></msub></munderover><mrow id="S5.Ex9.m1.9.9.3.3.1" xref="S5.Ex9.m1.9.9.3.3.1.cmml"><msubsup id="S5.Ex9.m1.9.9.3.3.1.3" xref="S5.Ex9.m1.9.9.3.3.1.3.cmml"><mi id="S5.Ex9.m1.9.9.3.3.1.3.2.2" xref="S5.Ex9.m1.9.9.3.3.1.3.2.2.cmml">B</mi><mrow id="S5.Ex9.m1.4.4.2.4" xref="S5.Ex9.m1.4.4.2.3.cmml"><mi id="S5.Ex9.m1.3.3.1.1" xref="S5.Ex9.m1.3.3.1.1.cmml">p</mi><mo id="S5.Ex9.m1.4.4.2.4.1" xref="S5.Ex9.m1.4.4.2.3.cmml">,</mo><mi id="S5.Ex9.m1.4.4.2.2" xref="S5.Ex9.m1.4.4.2.2.cmml">p</mi></mrow><mrow id="S5.Ex9.m1.9.9.3.3.1.3.2.3" xref="S5.Ex9.m1.9.9.3.3.1.3.2.3.cmml"><mi id="S5.Ex9.m1.9.9.3.3.1.3.2.3.2" xref="S5.Ex9.m1.9.9.3.3.1.3.2.3.2.cmml">s</mi><mo 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id="S5.Ex9.m1.9c">\widetilde{\mathcal{C}}:H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to\prod_{j=0}% ^{m_{n}}B^{s-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex9.m1.9d">over~ start_ARG caligraphic_C end_ARG : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.4.p2.21">is linear and bounded as well. 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xref="S5.4.p2.13.m1.5.5.2.3.cmml">,</mo><msub id="S5.4.p2.13.m1.5.5.2.2.2" xref="S5.4.p2.13.m1.5.5.2.2.2.cmml"><mi id="S5.4.p2.13.m1.5.5.2.2.2.2" xref="S5.4.p2.13.m1.5.5.2.2.2.2.cmml">g</mi><msub id="S5.4.p2.13.m1.5.5.2.2.2.3" xref="S5.4.p2.13.m1.5.5.2.2.2.3.cmml"><mi id="S5.4.p2.13.m1.5.5.2.2.2.3.2" xref="S5.4.p2.13.m1.5.5.2.2.2.3.2.cmml">m</mi><mi id="S5.4.p2.13.m1.5.5.2.2.2.3.3" xref="S5.4.p2.13.m1.5.5.2.2.2.3.3.cmml">n</mi></msub></msub><mo id="S5.4.p2.13.m1.5.5.2.2.6" stretchy="false" xref="S5.4.p2.13.m1.5.5.2.3.cmml">)</mo></mrow><mo id="S5.4.p2.13.m1.7.7.8" rspace="0.111em" xref="S5.4.p2.13.m1.7.7.8.cmml">∈</mo><mrow id="S5.4.p2.13.m1.7.7.4" xref="S5.4.p2.13.m1.7.7.4.cmml"><msubsup id="S5.4.p2.13.m1.7.7.4.3" xref="S5.4.p2.13.m1.7.7.4.3.cmml"><mo id="S5.4.p2.13.m1.7.7.4.3.2.2" xref="S5.4.p2.13.m1.7.7.4.3.2.2.cmml">∏</mo><mrow id="S5.4.p2.13.m1.7.7.4.3.2.3" xref="S5.4.p2.13.m1.7.7.4.3.2.3.cmml"><mi id="S5.4.p2.13.m1.7.7.4.3.2.3.2" xref="S5.4.p2.13.m1.7.7.4.3.2.3.2.cmml">i</mi><mo 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end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ∈ ∏ start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT )</annotation></semantics></math> and define <math alttext="h=(h_{j})_{j=0}^{m_{n}}\in\prod_{j=0}^{m_{n}}B^{s-j-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};X)" class="ltx_Math" display="inline" id="S5.4.p2.14.m2.5"><semantics id="S5.4.p2.14.m2.5a"><mrow id="S5.4.p2.14.m2.5.5" xref="S5.4.p2.14.m2.5.5.cmml"><mi id="S5.4.p2.14.m2.5.5.4" xref="S5.4.p2.14.m2.5.5.4.cmml">h</mi><mo id="S5.4.p2.14.m2.5.5.5" xref="S5.4.p2.14.m2.5.5.5.cmml">=</mo><msubsup id="S5.4.p2.14.m2.4.4.1" 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id="S5.4.p2.14.m2.5c">h=(h_{j})_{j=0}^{m_{n}}\in\prod_{j=0}^{m_{n}}B^{s-j-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};X)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.14.m2.5d">italic_h = ( italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ∈ ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X )</annotation></semantics></math> by setting <math alttext="h_{j}=h_{m_{i}}:=b^{{\rm c}}_{i,m_{i}}g_{m_{i}}" class="ltx_Math" display="inline" 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xref="S5.4.p2.15.m3.2.3.4.2">ℎ</ci><apply id="S5.4.p2.15.m3.2.3.4.3.cmml" xref="S5.4.p2.15.m3.2.3.4.3"><csymbol cd="ambiguous" id="S5.4.p2.15.m3.2.3.4.3.1.cmml" xref="S5.4.p2.15.m3.2.3.4.3">subscript</csymbol><ci id="S5.4.p2.15.m3.2.3.4.3.2.cmml" xref="S5.4.p2.15.m3.2.3.4.3.2">𝑚</ci><ci id="S5.4.p2.15.m3.2.3.4.3.3.cmml" xref="S5.4.p2.15.m3.2.3.4.3.3">𝑖</ci></apply></apply></apply><apply id="S5.4.p2.15.m3.2.3c.cmml" xref="S5.4.p2.15.m3.2.3"><csymbol cd="latexml" id="S5.4.p2.15.m3.2.3.5.cmml" xref="S5.4.p2.15.m3.2.3.5">assign</csymbol><share href="https://arxiv.org/html/2503.14636v1#S5.4.p2.15.m3.2.3.4.cmml" id="S5.4.p2.15.m3.2.3d.cmml" xref="S5.4.p2.15.m3.2.3"></share><apply id="S5.4.p2.15.m3.2.3.6.cmml" xref="S5.4.p2.15.m3.2.3.6"><times id="S5.4.p2.15.m3.2.3.6.1.cmml" xref="S5.4.p2.15.m3.2.3.6.1"></times><apply id="S5.4.p2.15.m3.2.3.6.2.cmml" xref="S5.4.p2.15.m3.2.3.6.2"><csymbol cd="ambiguous" id="S5.4.p2.15.m3.2.3.6.2.1.cmml" xref="S5.4.p2.15.m3.2.3.6.2">subscript</csymbol><apply 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xref="S5.4.p2.15.m3.2.3.6.3">subscript</csymbol><ci id="S5.4.p2.15.m3.2.3.6.3.2.cmml" xref="S5.4.p2.15.m3.2.3.6.3.2">𝑔</ci><apply id="S5.4.p2.15.m3.2.3.6.3.3.cmml" xref="S5.4.p2.15.m3.2.3.6.3.3"><csymbol cd="ambiguous" id="S5.4.p2.15.m3.2.3.6.3.3.1.cmml" xref="S5.4.p2.15.m3.2.3.6.3.3">subscript</csymbol><ci id="S5.4.p2.15.m3.2.3.6.3.3.2.cmml" xref="S5.4.p2.15.m3.2.3.6.3.3.2">𝑚</ci><ci id="S5.4.p2.15.m3.2.3.6.3.3.3.cmml" xref="S5.4.p2.15.m3.2.3.6.3.3.3">𝑖</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.15.m3.2c">h_{j}=h_{m_{i}}:=b^{{\rm c}}_{i,m_{i}}g_{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.15.m3.2d">italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT := italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_g start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> if <math alttext="j=m_{i}" class="ltx_Math" display="inline" id="S5.4.p2.16.m4.1"><semantics id="S5.4.p2.16.m4.1a"><mrow id="S5.4.p2.16.m4.1.1" xref="S5.4.p2.16.m4.1.1.cmml"><mi id="S5.4.p2.16.m4.1.1.2" xref="S5.4.p2.16.m4.1.1.2.cmml">j</mi><mo id="S5.4.p2.16.m4.1.1.1" xref="S5.4.p2.16.m4.1.1.1.cmml">=</mo><msub id="S5.4.p2.16.m4.1.1.3" xref="S5.4.p2.16.m4.1.1.3.cmml"><mi id="S5.4.p2.16.m4.1.1.3.2" xref="S5.4.p2.16.m4.1.1.3.2.cmml">m</mi><mi id="S5.4.p2.16.m4.1.1.3.3" xref="S5.4.p2.16.m4.1.1.3.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.16.m4.1b"><apply id="S5.4.p2.16.m4.1.1.cmml" xref="S5.4.p2.16.m4.1.1"><eq id="S5.4.p2.16.m4.1.1.1.cmml" xref="S5.4.p2.16.m4.1.1.1"></eq><ci id="S5.4.p2.16.m4.1.1.2.cmml" xref="S5.4.p2.16.m4.1.1.2">𝑗</ci><apply id="S5.4.p2.16.m4.1.1.3.cmml" xref="S5.4.p2.16.m4.1.1.3"><csymbol cd="ambiguous" id="S5.4.p2.16.m4.1.1.3.1.cmml" xref="S5.4.p2.16.m4.1.1.3">subscript</csymbol><ci id="S5.4.p2.16.m4.1.1.3.2.cmml" xref="S5.4.p2.16.m4.1.1.3.2">𝑚</ci><ci id="S5.4.p2.16.m4.1.1.3.3.cmml" xref="S5.4.p2.16.m4.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.16.m4.1c">j=m_{i}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.16.m4.1d">italic_j = italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math> for some <math alttext="i\in\{0,\dots,n\}" class="ltx_Math" display="inline" id="S5.4.p2.17.m5.3"><semantics id="S5.4.p2.17.m5.3a"><mrow id="S5.4.p2.17.m5.3.4" xref="S5.4.p2.17.m5.3.4.cmml"><mi id="S5.4.p2.17.m5.3.4.2" xref="S5.4.p2.17.m5.3.4.2.cmml">i</mi><mo id="S5.4.p2.17.m5.3.4.1" xref="S5.4.p2.17.m5.3.4.1.cmml">∈</mo><mrow id="S5.4.p2.17.m5.3.4.3.2" xref="S5.4.p2.17.m5.3.4.3.1.cmml"><mo id="S5.4.p2.17.m5.3.4.3.2.1" stretchy="false" xref="S5.4.p2.17.m5.3.4.3.1.cmml">{</mo><mn id="S5.4.p2.17.m5.1.1" xref="S5.4.p2.17.m5.1.1.cmml">0</mn><mo id="S5.4.p2.17.m5.3.4.3.2.2" xref="S5.4.p2.17.m5.3.4.3.1.cmml">,</mo><mi id="S5.4.p2.17.m5.2.2" mathvariant="normal" xref="S5.4.p2.17.m5.2.2.cmml">…</mi><mo id="S5.4.p2.17.m5.3.4.3.2.3" xref="S5.4.p2.17.m5.3.4.3.1.cmml">,</mo><mi id="S5.4.p2.17.m5.3.3" xref="S5.4.p2.17.m5.3.3.cmml">n</mi><mo id="S5.4.p2.17.m5.3.4.3.2.4" stretchy="false" xref="S5.4.p2.17.m5.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.17.m5.3b"><apply id="S5.4.p2.17.m5.3.4.cmml" xref="S5.4.p2.17.m5.3.4"><in id="S5.4.p2.17.m5.3.4.1.cmml" xref="S5.4.p2.17.m5.3.4.1"></in><ci id="S5.4.p2.17.m5.3.4.2.cmml" xref="S5.4.p2.17.m5.3.4.2">𝑖</ci><set id="S5.4.p2.17.m5.3.4.3.1.cmml" xref="S5.4.p2.17.m5.3.4.3.2"><cn id="S5.4.p2.17.m5.1.1.cmml" type="integer" xref="S5.4.p2.17.m5.1.1">0</cn><ci id="S5.4.p2.17.m5.2.2.cmml" xref="S5.4.p2.17.m5.2.2">…</ci><ci id="S5.4.p2.17.m5.3.3.cmml" xref="S5.4.p2.17.m5.3.3">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.17.m5.3c">i\in\{0,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.17.m5.3d">italic_i ∈ { 0 , … , italic_n }</annotation></semantics></math>, and <math alttext="h_{j}=0" class="ltx_Math" display="inline" id="S5.4.p2.18.m6.1"><semantics id="S5.4.p2.18.m6.1a"><mrow id="S5.4.p2.18.m6.1.1" xref="S5.4.p2.18.m6.1.1.cmml"><msub id="S5.4.p2.18.m6.1.1.2" xref="S5.4.p2.18.m6.1.1.2.cmml"><mi id="S5.4.p2.18.m6.1.1.2.2" xref="S5.4.p2.18.m6.1.1.2.2.cmml">h</mi><mi id="S5.4.p2.18.m6.1.1.2.3" xref="S5.4.p2.18.m6.1.1.2.3.cmml">j</mi></msub><mo id="S5.4.p2.18.m6.1.1.1" xref="S5.4.p2.18.m6.1.1.1.cmml">=</mo><mn id="S5.4.p2.18.m6.1.1.3" xref="S5.4.p2.18.m6.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.18.m6.1b"><apply id="S5.4.p2.18.m6.1.1.cmml" xref="S5.4.p2.18.m6.1.1"><eq id="S5.4.p2.18.m6.1.1.1.cmml" xref="S5.4.p2.18.m6.1.1.1"></eq><apply id="S5.4.p2.18.m6.1.1.2.cmml" xref="S5.4.p2.18.m6.1.1.2"><csymbol cd="ambiguous" id="S5.4.p2.18.m6.1.1.2.1.cmml" xref="S5.4.p2.18.m6.1.1.2">subscript</csymbol><ci id="S5.4.p2.18.m6.1.1.2.2.cmml" xref="S5.4.p2.18.m6.1.1.2.2">ℎ</ci><ci id="S5.4.p2.18.m6.1.1.2.3.cmml" xref="S5.4.p2.18.m6.1.1.2.3">𝑗</ci></apply><cn id="S5.4.p2.18.m6.1.1.3.cmml" type="integer" xref="S5.4.p2.18.m6.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.18.m6.1c">h_{j}=0</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.18.m6.1d">italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = 0</annotation></semantics></math> if <math alttext="j\notin\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S5.4.p2.19.m7.3"><semantics id="S5.4.p2.19.m7.3a"><mrow id="S5.4.p2.19.m7.3.3" xref="S5.4.p2.19.m7.3.3.cmml"><mi id="S5.4.p2.19.m7.3.3.4" xref="S5.4.p2.19.m7.3.3.4.cmml">j</mi><mo id="S5.4.p2.19.m7.3.3.3" xref="S5.4.p2.19.m7.3.3.3.cmml">∉</mo><mrow id="S5.4.p2.19.m7.3.3.2.2" xref="S5.4.p2.19.m7.3.3.2.3.cmml"><mo id="S5.4.p2.19.m7.3.3.2.2.3" stretchy="false" xref="S5.4.p2.19.m7.3.3.2.3.cmml">{</mo><msub id="S5.4.p2.19.m7.2.2.1.1.1" xref="S5.4.p2.19.m7.2.2.1.1.1.cmml"><mi id="S5.4.p2.19.m7.2.2.1.1.1.2" xref="S5.4.p2.19.m7.2.2.1.1.1.2.cmml">m</mi><mn id="S5.4.p2.19.m7.2.2.1.1.1.3" xref="S5.4.p2.19.m7.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.4.p2.19.m7.3.3.2.2.4" xref="S5.4.p2.19.m7.3.3.2.3.cmml">,</mo><mi id="S5.4.p2.19.m7.1.1" mathvariant="normal" xref="S5.4.p2.19.m7.1.1.cmml">…</mi><mo id="S5.4.p2.19.m7.3.3.2.2.5" xref="S5.4.p2.19.m7.3.3.2.3.cmml">,</mo><msub id="S5.4.p2.19.m7.3.3.2.2.2" xref="S5.4.p2.19.m7.3.3.2.2.2.cmml"><mi id="S5.4.p2.19.m7.3.3.2.2.2.2" xref="S5.4.p2.19.m7.3.3.2.2.2.2.cmml">m</mi><mi id="S5.4.p2.19.m7.3.3.2.2.2.3" xref="S5.4.p2.19.m7.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S5.4.p2.19.m7.3.3.2.2.6" stretchy="false" xref="S5.4.p2.19.m7.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.19.m7.3b"><apply id="S5.4.p2.19.m7.3.3.cmml" xref="S5.4.p2.19.m7.3.3"><notin id="S5.4.p2.19.m7.3.3.3.cmml" xref="S5.4.p2.19.m7.3.3.3"></notin><ci id="S5.4.p2.19.m7.3.3.4.cmml" xref="S5.4.p2.19.m7.3.3.4">𝑗</ci><set id="S5.4.p2.19.m7.3.3.2.3.cmml" xref="S5.4.p2.19.m7.3.3.2.2"><apply id="S5.4.p2.19.m7.2.2.1.1.1.cmml" xref="S5.4.p2.19.m7.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.19.m7.2.2.1.1.1.1.cmml" xref="S5.4.p2.19.m7.2.2.1.1.1">subscript</csymbol><ci id="S5.4.p2.19.m7.2.2.1.1.1.2.cmml" xref="S5.4.p2.19.m7.2.2.1.1.1.2">𝑚</ci><cn id="S5.4.p2.19.m7.2.2.1.1.1.3.cmml" type="integer" xref="S5.4.p2.19.m7.2.2.1.1.1.3">0</cn></apply><ci id="S5.4.p2.19.m7.1.1.cmml" xref="S5.4.p2.19.m7.1.1">…</ci><apply id="S5.4.p2.19.m7.3.3.2.2.2.cmml" xref="S5.4.p2.19.m7.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.4.p2.19.m7.3.3.2.2.2.1.cmml" xref="S5.4.p2.19.m7.3.3.2.2.2">subscript</csymbol><ci id="S5.4.p2.19.m7.3.3.2.2.2.2.cmml" xref="S5.4.p2.19.m7.3.3.2.2.2.2">𝑚</ci><ci id="S5.4.p2.19.m7.3.3.2.2.2.3.cmml" xref="S5.4.p2.19.m7.3.3.2.2.2.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.19.m7.3c">j\notin\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.19.m7.3d">italic_j ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math>. Furthermore, let <math alttext="f_{0}:=\operatorname{ext}_{0}h_{0}" class="ltx_Math" display="inline" id="S5.4.p2.20.m8.1"><semantics id="S5.4.p2.20.m8.1a"><mrow id="S5.4.p2.20.m8.1.1" xref="S5.4.p2.20.m8.1.1.cmml"><msub id="S5.4.p2.20.m8.1.1.2" xref="S5.4.p2.20.m8.1.1.2.cmml"><mi id="S5.4.p2.20.m8.1.1.2.2" xref="S5.4.p2.20.m8.1.1.2.2.cmml">f</mi><mn id="S5.4.p2.20.m8.1.1.2.3" xref="S5.4.p2.20.m8.1.1.2.3.cmml">0</mn></msub><mo id="S5.4.p2.20.m8.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.4.p2.20.m8.1.1.1.cmml">:=</mo><mrow id="S5.4.p2.20.m8.1.1.3" xref="S5.4.p2.20.m8.1.1.3.cmml"><msub id="S5.4.p2.20.m8.1.1.3.1" xref="S5.4.p2.20.m8.1.1.3.1.cmml"><mi id="S5.4.p2.20.m8.1.1.3.1.2" xref="S5.4.p2.20.m8.1.1.3.1.2.cmml">ext</mi><mn id="S5.4.p2.20.m8.1.1.3.1.3" xref="S5.4.p2.20.m8.1.1.3.1.3.cmml">0</mn></msub><mo id="S5.4.p2.20.m8.1.1.3a" lspace="0.167em" xref="S5.4.p2.20.m8.1.1.3.cmml">⁡</mo><msub id="S5.4.p2.20.m8.1.1.3.2" xref="S5.4.p2.20.m8.1.1.3.2.cmml"><mi id="S5.4.p2.20.m8.1.1.3.2.2" xref="S5.4.p2.20.m8.1.1.3.2.2.cmml">h</mi><mn id="S5.4.p2.20.m8.1.1.3.2.3" xref="S5.4.p2.20.m8.1.1.3.2.3.cmml">0</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.20.m8.1b"><apply id="S5.4.p2.20.m8.1.1.cmml" xref="S5.4.p2.20.m8.1.1"><csymbol cd="latexml" id="S5.4.p2.20.m8.1.1.1.cmml" xref="S5.4.p2.20.m8.1.1.1">assign</csymbol><apply id="S5.4.p2.20.m8.1.1.2.cmml" xref="S5.4.p2.20.m8.1.1.2"><csymbol cd="ambiguous" id="S5.4.p2.20.m8.1.1.2.1.cmml" xref="S5.4.p2.20.m8.1.1.2">subscript</csymbol><ci id="S5.4.p2.20.m8.1.1.2.2.cmml" xref="S5.4.p2.20.m8.1.1.2.2">𝑓</ci><cn id="S5.4.p2.20.m8.1.1.2.3.cmml" type="integer" xref="S5.4.p2.20.m8.1.1.2.3">0</cn></apply><apply id="S5.4.p2.20.m8.1.1.3.cmml" xref="S5.4.p2.20.m8.1.1.3"><apply id="S5.4.p2.20.m8.1.1.3.1.cmml" xref="S5.4.p2.20.m8.1.1.3.1"><csymbol cd="ambiguous" id="S5.4.p2.20.m8.1.1.3.1.1.cmml" xref="S5.4.p2.20.m8.1.1.3.1">subscript</csymbol><ci id="S5.4.p2.20.m8.1.1.3.1.2.cmml" xref="S5.4.p2.20.m8.1.1.3.1.2">ext</ci><cn id="S5.4.p2.20.m8.1.1.3.1.3.cmml" type="integer" xref="S5.4.p2.20.m8.1.1.3.1.3">0</cn></apply><apply id="S5.4.p2.20.m8.1.1.3.2.cmml" xref="S5.4.p2.20.m8.1.1.3.2"><csymbol cd="ambiguous" id="S5.4.p2.20.m8.1.1.3.2.1.cmml" xref="S5.4.p2.20.m8.1.1.3.2">subscript</csymbol><ci id="S5.4.p2.20.m8.1.1.3.2.2.cmml" xref="S5.4.p2.20.m8.1.1.3.2.2">ℎ</ci><cn id="S5.4.p2.20.m8.1.1.3.2.3.cmml" type="integer" xref="S5.4.p2.20.m8.1.1.3.2.3">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.20.m8.1c">f_{0}:=\operatorname{ext}_{0}h_{0}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.20.m8.1d">italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT := roman_ext start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_h start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and for <math alttext="j\in\{1,\dots,m_{n}\}" class="ltx_Math" display="inline" id="S5.4.p2.21.m9.3"><semantics id="S5.4.p2.21.m9.3a"><mrow id="S5.4.p2.21.m9.3.3" xref="S5.4.p2.21.m9.3.3.cmml"><mi id="S5.4.p2.21.m9.3.3.3" xref="S5.4.p2.21.m9.3.3.3.cmml">j</mi><mo id="S5.4.p2.21.m9.3.3.2" xref="S5.4.p2.21.m9.3.3.2.cmml">∈</mo><mrow id="S5.4.p2.21.m9.3.3.1.1" xref="S5.4.p2.21.m9.3.3.1.2.cmml"><mo id="S5.4.p2.21.m9.3.3.1.1.2" stretchy="false" xref="S5.4.p2.21.m9.3.3.1.2.cmml">{</mo><mn id="S5.4.p2.21.m9.1.1" xref="S5.4.p2.21.m9.1.1.cmml">1</mn><mo id="S5.4.p2.21.m9.3.3.1.1.3" xref="S5.4.p2.21.m9.3.3.1.2.cmml">,</mo><mi id="S5.4.p2.21.m9.2.2" mathvariant="normal" xref="S5.4.p2.21.m9.2.2.cmml">…</mi><mo id="S5.4.p2.21.m9.3.3.1.1.4" xref="S5.4.p2.21.m9.3.3.1.2.cmml">,</mo><msub id="S5.4.p2.21.m9.3.3.1.1.1" xref="S5.4.p2.21.m9.3.3.1.1.1.cmml"><mi id="S5.4.p2.21.m9.3.3.1.1.1.2" xref="S5.4.p2.21.m9.3.3.1.1.1.2.cmml">m</mi><mi id="S5.4.p2.21.m9.3.3.1.1.1.3" xref="S5.4.p2.21.m9.3.3.1.1.1.3.cmml">n</mi></msub><mo id="S5.4.p2.21.m9.3.3.1.1.5" stretchy="false" xref="S5.4.p2.21.m9.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.21.m9.3b"><apply id="S5.4.p2.21.m9.3.3.cmml" xref="S5.4.p2.21.m9.3.3"><in id="S5.4.p2.21.m9.3.3.2.cmml" xref="S5.4.p2.21.m9.3.3.2"></in><ci id="S5.4.p2.21.m9.3.3.3.cmml" xref="S5.4.p2.21.m9.3.3.3">𝑗</ci><set id="S5.4.p2.21.m9.3.3.1.2.cmml" xref="S5.4.p2.21.m9.3.3.1.1"><cn id="S5.4.p2.21.m9.1.1.cmml" type="integer" xref="S5.4.p2.21.m9.1.1">1</cn><ci id="S5.4.p2.21.m9.2.2.cmml" xref="S5.4.p2.21.m9.2.2">…</ci><apply id="S5.4.p2.21.m9.3.3.1.1.1.cmml" xref="S5.4.p2.21.m9.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.21.m9.3.3.1.1.1.1.cmml" xref="S5.4.p2.21.m9.3.3.1.1.1">subscript</csymbol><ci id="S5.4.p2.21.m9.3.3.1.1.1.2.cmml" xref="S5.4.p2.21.m9.3.3.1.1.1.2">𝑚</ci><ci id="S5.4.p2.21.m9.3.3.1.1.1.3.cmml" xref="S5.4.p2.21.m9.3.3.1.1.1.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.21.m9.3c">j\in\{1,\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.21.m9.3d">italic_j ∈ { 1 , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math> we recursively define</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f_{j}:=f_{j-1}+\operatorname{ext}_{j}\big{(}h_{j}+\widetilde{\mathcal{C}}^{j}f% _{j-1}-\operatorname{Tr}_{j}f_{j-1}\big{)}." class="ltx_Math" display="block" id="S5.Ex10.m1.1"><semantics id="S5.Ex10.m1.1a"><mrow id="S5.Ex10.m1.1.1.1" xref="S5.Ex10.m1.1.1.1.1.cmml"><mrow id="S5.Ex10.m1.1.1.1.1" xref="S5.Ex10.m1.1.1.1.1.cmml"><msub id="S5.Ex10.m1.1.1.1.1.4" xref="S5.Ex10.m1.1.1.1.1.4.cmml"><mi id="S5.Ex10.m1.1.1.1.1.4.2" xref="S5.Ex10.m1.1.1.1.1.4.2.cmml">f</mi><mi id="S5.Ex10.m1.1.1.1.1.4.3" xref="S5.Ex10.m1.1.1.1.1.4.3.cmml">j</mi></msub><mo id="S5.Ex10.m1.1.1.1.1.3" lspace="0.278em" rspace="0.278em" xref="S5.Ex10.m1.1.1.1.1.3.cmml">:=</mo><mrow id="S5.Ex10.m1.1.1.1.1.2" xref="S5.Ex10.m1.1.1.1.1.2.cmml"><msub id="S5.Ex10.m1.1.1.1.1.2.4" xref="S5.Ex10.m1.1.1.1.1.2.4.cmml"><mi id="S5.Ex10.m1.1.1.1.1.2.4.2" xref="S5.Ex10.m1.1.1.1.1.2.4.2.cmml">f</mi><mrow id="S5.Ex10.m1.1.1.1.1.2.4.3" xref="S5.Ex10.m1.1.1.1.1.2.4.3.cmml"><mi id="S5.Ex10.m1.1.1.1.1.2.4.3.2" xref="S5.Ex10.m1.1.1.1.1.2.4.3.2.cmml">j</mi><mo id="S5.Ex10.m1.1.1.1.1.2.4.3.1" xref="S5.Ex10.m1.1.1.1.1.2.4.3.1.cmml">−</mo><mn id="S5.Ex10.m1.1.1.1.1.2.4.3.3" xref="S5.Ex10.m1.1.1.1.1.2.4.3.3.cmml">1</mn></mrow></msub><mo id="S5.Ex10.m1.1.1.1.1.2.3" xref="S5.Ex10.m1.1.1.1.1.2.3.cmml">+</mo><mrow id="S5.Ex10.m1.1.1.1.1.2.2.2" xref="S5.Ex10.m1.1.1.1.1.2.2.3.cmml"><msub id="S5.Ex10.m1.1.1.1.1.1.1.1.1" xref="S5.Ex10.m1.1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex10.m1.1.1.1.1.1.1.1.1.2" xref="S5.Ex10.m1.1.1.1.1.1.1.1.1.2.cmml">ext</mi><mi id="S5.Ex10.m1.1.1.1.1.1.1.1.1.3" xref="S5.Ex10.m1.1.1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S5.Ex10.m1.1.1.1.1.2.2.2a" xref="S5.Ex10.m1.1.1.1.1.2.2.3.cmml">⁡</mo><mrow id="S5.Ex10.m1.1.1.1.1.2.2.2.2" xref="S5.Ex10.m1.1.1.1.1.2.2.3.cmml"><mo id="S5.Ex10.m1.1.1.1.1.2.2.2.2.2" maxsize="120%" minsize="120%" xref="S5.Ex10.m1.1.1.1.1.2.2.3.cmml">(</mo><mrow id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.cmml"><mrow id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.cmml"><msub id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.2.cmml"><mi id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.2.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.2.2.cmml">h</mi><mi id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.2.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.2.3.cmml">j</mi></msub><mo id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.1" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.1.cmml">+</mo><mrow id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.cmml"><msup id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.cmml"><mover accent="true" id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.2.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.2.2.cmml">𝒞</mi><mo id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.2.1" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.2.1.cmml">~</mo></mover><mi id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.2.3.cmml">j</mi></msup><mo id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.1" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.1.cmml">⁢</mo><msub id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.cmml"><mi id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.2.cmml">f</mi><mrow id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3.cmml"><mi id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3.2.cmml">j</mi><mo id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3.1" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3.1.cmml">−</mo><mn id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.2.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mo id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.1" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.1.cmml">−</mo><mrow id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.cmml"><msub id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.1" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.1.cmml"><mi id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.1.2" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.1.2.cmml">Tr</mi><mi id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.1.3" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.1.3.cmml">j</mi></msub><mo id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3a" lspace="0.167em" xref="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.cmml">⁡</mo><msub id="S5.Ex10.m1.1.1.1.1.2.2.2.2.1.3.2" 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italic_j end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT - roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.4.p2.24">Note that <math alttext="f_{j}\in H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S5.4.p2.22.m1.5"><semantics id="S5.4.p2.22.m1.5a"><mrow id="S5.4.p2.22.m1.5.5" xref="S5.4.p2.22.m1.5.5.cmml"><msub id="S5.4.p2.22.m1.5.5.4" xref="S5.4.p2.22.m1.5.5.4.cmml"><mi id="S5.4.p2.22.m1.5.5.4.2" xref="S5.4.p2.22.m1.5.5.4.2.cmml">f</mi><mi id="S5.4.p2.22.m1.5.5.4.3" xref="S5.4.p2.22.m1.5.5.4.3.cmml">j</mi></msub><mo id="S5.4.p2.22.m1.5.5.3" xref="S5.4.p2.22.m1.5.5.3.cmml">∈</mo><mrow id="S5.4.p2.22.m1.5.5.2" xref="S5.4.p2.22.m1.5.5.2.cmml"><msup id="S5.4.p2.22.m1.5.5.2.4" xref="S5.4.p2.22.m1.5.5.2.4.cmml"><mi 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xref="S5.4.p2.22.m1.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.22.m1.4.4.1.1.1.1.2.1.cmml" xref="S5.4.p2.22.m1.4.4.1.1.1.1">superscript</csymbol><ci id="S5.4.p2.22.m1.4.4.1.1.1.1.2.2.cmml" xref="S5.4.p2.22.m1.4.4.1.1.1.1.2.2">ℝ</ci><ci id="S5.4.p2.22.m1.4.4.1.1.1.1.2.3.cmml" xref="S5.4.p2.22.m1.4.4.1.1.1.1.2.3">𝑑</ci></apply><plus id="S5.4.p2.22.m1.4.4.1.1.1.1.3.cmml" xref="S5.4.p2.22.m1.4.4.1.1.1.1.3"></plus></apply><apply id="S5.4.p2.22.m1.5.5.2.2.2.2.cmml" xref="S5.4.p2.22.m1.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S5.4.p2.22.m1.5.5.2.2.2.2.1.cmml" xref="S5.4.p2.22.m1.5.5.2.2.2.2">subscript</csymbol><ci id="S5.4.p2.22.m1.5.5.2.2.2.2.2.cmml" xref="S5.4.p2.22.m1.5.5.2.2.2.2.2">𝑤</ci><ci id="S5.4.p2.22.m1.5.5.2.2.2.2.3.cmml" xref="S5.4.p2.22.m1.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S5.4.p2.22.m1.3.3.cmml" xref="S5.4.p2.22.m1.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.22.m1.5c">f_{j}\in H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.22.m1.5d">italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for all <math alttext="j\in\{0,\dots,m_{n}\}" class="ltx_Math" display="inline" id="S5.4.p2.23.m2.3"><semantics id="S5.4.p2.23.m2.3a"><mrow id="S5.4.p2.23.m2.3.3" xref="S5.4.p2.23.m2.3.3.cmml"><mi id="S5.4.p2.23.m2.3.3.3" xref="S5.4.p2.23.m2.3.3.3.cmml">j</mi><mo id="S5.4.p2.23.m2.3.3.2" xref="S5.4.p2.23.m2.3.3.2.cmml">∈</mo><mrow id="S5.4.p2.23.m2.3.3.1.1" xref="S5.4.p2.23.m2.3.3.1.2.cmml"><mo id="S5.4.p2.23.m2.3.3.1.1.2" stretchy="false" xref="S5.4.p2.23.m2.3.3.1.2.cmml">{</mo><mn id="S5.4.p2.23.m2.1.1" xref="S5.4.p2.23.m2.1.1.cmml">0</mn><mo id="S5.4.p2.23.m2.3.3.1.1.3" xref="S5.4.p2.23.m2.3.3.1.2.cmml">,</mo><mi id="S5.4.p2.23.m2.2.2" mathvariant="normal" xref="S5.4.p2.23.m2.2.2.cmml">…</mi><mo id="S5.4.p2.23.m2.3.3.1.1.4" xref="S5.4.p2.23.m2.3.3.1.2.cmml">,</mo><msub id="S5.4.p2.23.m2.3.3.1.1.1" xref="S5.4.p2.23.m2.3.3.1.1.1.cmml"><mi id="S5.4.p2.23.m2.3.3.1.1.1.2" xref="S5.4.p2.23.m2.3.3.1.1.1.2.cmml">m</mi><mi id="S5.4.p2.23.m2.3.3.1.1.1.3" xref="S5.4.p2.23.m2.3.3.1.1.1.3.cmml">n</mi></msub><mo id="S5.4.p2.23.m2.3.3.1.1.5" stretchy="false" xref="S5.4.p2.23.m2.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.23.m2.3b"><apply id="S5.4.p2.23.m2.3.3.cmml" xref="S5.4.p2.23.m2.3.3"><in id="S5.4.p2.23.m2.3.3.2.cmml" xref="S5.4.p2.23.m2.3.3.2"></in><ci id="S5.4.p2.23.m2.3.3.3.cmml" xref="S5.4.p2.23.m2.3.3.3">𝑗</ci><set id="S5.4.p2.23.m2.3.3.1.2.cmml" xref="S5.4.p2.23.m2.3.3.1.1"><cn id="S5.4.p2.23.m2.1.1.cmml" type="integer" xref="S5.4.p2.23.m2.1.1">0</cn><ci id="S5.4.p2.23.m2.2.2.cmml" xref="S5.4.p2.23.m2.2.2">…</ci><apply id="S5.4.p2.23.m2.3.3.1.1.1.cmml" xref="S5.4.p2.23.m2.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.4.p2.23.m2.3.3.1.1.1.1.cmml" xref="S5.4.p2.23.m2.3.3.1.1.1">subscript</csymbol><ci id="S5.4.p2.23.m2.3.3.1.1.1.2.cmml" xref="S5.4.p2.23.m2.3.3.1.1.1.2">𝑚</ci><ci id="S5.4.p2.23.m2.3.3.1.1.1.3.cmml" xref="S5.4.p2.23.m2.3.3.1.1.1.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.23.m2.3c">j\in\{0,\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.23.m2.3d">italic_j ∈ { 0 , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math> by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>. We will prove that <math alttext="\operatorname{ext}_{\mathcal{B}}g:=f_{m_{n}}" class="ltx_Math" display="inline" id="S5.4.p2.24.m3.1"><semantics id="S5.4.p2.24.m3.1a"><mrow id="S5.4.p2.24.m3.1.1" xref="S5.4.p2.24.m3.1.1.cmml"><mrow id="S5.4.p2.24.m3.1.1.2" xref="S5.4.p2.24.m3.1.1.2.cmml"><msub id="S5.4.p2.24.m3.1.1.2.1" xref="S5.4.p2.24.m3.1.1.2.1.cmml"><mi id="S5.4.p2.24.m3.1.1.2.1.2" xref="S5.4.p2.24.m3.1.1.2.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S5.4.p2.24.m3.1.1.2.1.3" xref="S5.4.p2.24.m3.1.1.2.1.3.cmml">ℬ</mi></msub><mo id="S5.4.p2.24.m3.1.1.2a" lspace="0.167em" xref="S5.4.p2.24.m3.1.1.2.cmml">⁡</mo><mi id="S5.4.p2.24.m3.1.1.2.2" xref="S5.4.p2.24.m3.1.1.2.2.cmml">g</mi></mrow><mo id="S5.4.p2.24.m3.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.4.p2.24.m3.1.1.1.cmml">:=</mo><msub id="S5.4.p2.24.m3.1.1.3" xref="S5.4.p2.24.m3.1.1.3.cmml"><mi id="S5.4.p2.24.m3.1.1.3.2" xref="S5.4.p2.24.m3.1.1.3.2.cmml">f</mi><msub id="S5.4.p2.24.m3.1.1.3.3" xref="S5.4.p2.24.m3.1.1.3.3.cmml"><mi id="S5.4.p2.24.m3.1.1.3.3.2" xref="S5.4.p2.24.m3.1.1.3.3.2.cmml">m</mi><mi id="S5.4.p2.24.m3.1.1.3.3.3" xref="S5.4.p2.24.m3.1.1.3.3.3.cmml">n</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.4.p2.24.m3.1b"><apply id="S5.4.p2.24.m3.1.1.cmml" xref="S5.4.p2.24.m3.1.1"><csymbol cd="latexml" id="S5.4.p2.24.m3.1.1.1.cmml" xref="S5.4.p2.24.m3.1.1.1">assign</csymbol><apply id="S5.4.p2.24.m3.1.1.2.cmml" xref="S5.4.p2.24.m3.1.1.2"><apply id="S5.4.p2.24.m3.1.1.2.1.cmml" xref="S5.4.p2.24.m3.1.1.2.1"><csymbol cd="ambiguous" id="S5.4.p2.24.m3.1.1.2.1.1.cmml" xref="S5.4.p2.24.m3.1.1.2.1">subscript</csymbol><ci id="S5.4.p2.24.m3.1.1.2.1.2.cmml" xref="S5.4.p2.24.m3.1.1.2.1.2">ext</ci><ci id="S5.4.p2.24.m3.1.1.2.1.3.cmml" xref="S5.4.p2.24.m3.1.1.2.1.3">ℬ</ci></apply><ci id="S5.4.p2.24.m3.1.1.2.2.cmml" xref="S5.4.p2.24.m3.1.1.2.2">𝑔</ci></apply><apply id="S5.4.p2.24.m3.1.1.3.cmml" xref="S5.4.p2.24.m3.1.1.3"><csymbol cd="ambiguous" id="S5.4.p2.24.m3.1.1.3.1.cmml" xref="S5.4.p2.24.m3.1.1.3">subscript</csymbol><ci id="S5.4.p2.24.m3.1.1.3.2.cmml" xref="S5.4.p2.24.m3.1.1.3.2">𝑓</ci><apply id="S5.4.p2.24.m3.1.1.3.3.cmml" xref="S5.4.p2.24.m3.1.1.3.3"><csymbol cd="ambiguous" id="S5.4.p2.24.m3.1.1.3.3.1.cmml" xref="S5.4.p2.24.m3.1.1.3.3">subscript</csymbol><ci id="S5.4.p2.24.m3.1.1.3.3.2.cmml" xref="S5.4.p2.24.m3.1.1.3.3.2">𝑚</ci><ci id="S5.4.p2.24.m3.1.1.3.3.3.cmml" xref="S5.4.p2.24.m3.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.4.p2.24.m3.1c">\operatorname{ext}_{\mathcal{B}}g:=f_{m_{n}}</annotation><annotation encoding="application/x-llamapun" id="S5.4.p2.24.m3.1d">roman_ext start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT italic_g := italic_f start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> defines the right inverse we are looking for.</p> </div> <div class="ltx_para" id="S5.5.p3"> <p class="ltx_p" id="S5.5.p3.2">By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem1" title="Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.1</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I2.i1" title="item i ‣ Theorem 4.1. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> we have that <math alttext="\operatorname{Tr}_{\ell}\mathop{\circ}\nolimits\operatorname{ext}_{j}=\delta_{% j\ell}" class="ltx_Math" display="inline" id="S5.5.p3.1.m1.1"><semantics id="S5.5.p3.1.m1.1a"><mrow id="S5.5.p3.1.m1.1.1" xref="S5.5.p3.1.m1.1.1.cmml"><mrow id="S5.5.p3.1.m1.1.1.2" xref="S5.5.p3.1.m1.1.1.2.cmml"><msub id="S5.5.p3.1.m1.1.1.2.2" xref="S5.5.p3.1.m1.1.1.2.2.cmml"><mi id="S5.5.p3.1.m1.1.1.2.2.2" xref="S5.5.p3.1.m1.1.1.2.2.2.cmml">Tr</mi><mi id="S5.5.p3.1.m1.1.1.2.2.3" mathvariant="normal" xref="S5.5.p3.1.m1.1.1.2.2.3.cmml">ℓ</mi></msub><mo id="S5.5.p3.1.m1.1.1.2.1" lspace="0.167em" xref="S5.5.p3.1.m1.1.1.2.1.cmml">⁢</mo><mrow id="S5.5.p3.1.m1.1.1.2.3" xref="S5.5.p3.1.m1.1.1.2.3.cmml"><mo id="S5.5.p3.1.m1.1.1.2.3.1" rspace="0.167em" xref="S5.5.p3.1.m1.1.1.2.3.1.cmml">∘</mo><msub id="S5.5.p3.1.m1.1.1.2.3.2" xref="S5.5.p3.1.m1.1.1.2.3.2.cmml"><mi id="S5.5.p3.1.m1.1.1.2.3.2.2" xref="S5.5.p3.1.m1.1.1.2.3.2.2.cmml">ext</mi><mi id="S5.5.p3.1.m1.1.1.2.3.2.3" xref="S5.5.p3.1.m1.1.1.2.3.2.3.cmml">j</mi></msub></mrow></mrow><mo id="S5.5.p3.1.m1.1.1.1" xref="S5.5.p3.1.m1.1.1.1.cmml">=</mo><msub id="S5.5.p3.1.m1.1.1.3" xref="S5.5.p3.1.m1.1.1.3.cmml"><mi id="S5.5.p3.1.m1.1.1.3.2" xref="S5.5.p3.1.m1.1.1.3.2.cmml">δ</mi><mrow id="S5.5.p3.1.m1.1.1.3.3" xref="S5.5.p3.1.m1.1.1.3.3.cmml"><mi id="S5.5.p3.1.m1.1.1.3.3.2" xref="S5.5.p3.1.m1.1.1.3.3.2.cmml">j</mi><mo id="S5.5.p3.1.m1.1.1.3.3.1" xref="S5.5.p3.1.m1.1.1.3.3.1.cmml">⁢</mo><mi id="S5.5.p3.1.m1.1.1.3.3.3" mathvariant="normal" xref="S5.5.p3.1.m1.1.1.3.3.3.cmml">ℓ</mi></mrow></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.1.m1.1b"><apply id="S5.5.p3.1.m1.1.1.cmml" xref="S5.5.p3.1.m1.1.1"><eq id="S5.5.p3.1.m1.1.1.1.cmml" xref="S5.5.p3.1.m1.1.1.1"></eq><apply id="S5.5.p3.1.m1.1.1.2.cmml" 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xref="S5.5.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S5.5.p3.1.m1.1.1.3.1.cmml" xref="S5.5.p3.1.m1.1.1.3">subscript</csymbol><ci id="S5.5.p3.1.m1.1.1.3.2.cmml" xref="S5.5.p3.1.m1.1.1.3.2">𝛿</ci><apply id="S5.5.p3.1.m1.1.1.3.3.cmml" xref="S5.5.p3.1.m1.1.1.3.3"><times id="S5.5.p3.1.m1.1.1.3.3.1.cmml" xref="S5.5.p3.1.m1.1.1.3.3.1"></times><ci id="S5.5.p3.1.m1.1.1.3.3.2.cmml" xref="S5.5.p3.1.m1.1.1.3.3.2">𝑗</ci><ci id="S5.5.p3.1.m1.1.1.3.3.3.cmml" xref="S5.5.p3.1.m1.1.1.3.3.3">ℓ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.1.m1.1c">\operatorname{Tr}_{\ell}\mathop{\circ}\nolimits\operatorname{ext}_{j}=\delta_{% j\ell}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.1.m1.1d">roman_Tr start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT ∘ roman_ext start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_j roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="\ell\leq j" class="ltx_Math" display="inline" id="S5.5.p3.2.m2.1"><semantics id="S5.5.p3.2.m2.1a"><mrow id="S5.5.p3.2.m2.1.1" xref="S5.5.p3.2.m2.1.1.cmml"><mi id="S5.5.p3.2.m2.1.1.2" mathvariant="normal" xref="S5.5.p3.2.m2.1.1.2.cmml">ℓ</mi><mo id="S5.5.p3.2.m2.1.1.1" xref="S5.5.p3.2.m2.1.1.1.cmml">≤</mo><mi id="S5.5.p3.2.m2.1.1.3" xref="S5.5.p3.2.m2.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.2.m2.1b"><apply id="S5.5.p3.2.m2.1.1.cmml" xref="S5.5.p3.2.m2.1.1"><leq id="S5.5.p3.2.m2.1.1.1.cmml" xref="S5.5.p3.2.m2.1.1.1"></leq><ci id="S5.5.p3.2.m2.1.1.2.cmml" xref="S5.5.p3.2.m2.1.1.2">ℓ</ci><ci id="S5.5.p3.2.m2.1.1.3.cmml" xref="S5.5.p3.2.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.2.m2.1c">\ell\leq j</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.2.m2.1d">roman_ℓ ≤ italic_j</annotation></semantics></math> and therefore</p> <table class="ltx_equationgroup ltx_eqn_table" id="S5.E7"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S5.E7X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{Tr}_{j}f_{j}" class="ltx_Math" display="inline" id="S5.E7X.2.1.1.m1.1"><semantics id="S5.E7X.2.1.1.m1.1a"><mrow id="S5.E7X.2.1.1.m1.1.1" xref="S5.E7X.2.1.1.m1.1.1.cmml"><msub id="S5.E7X.2.1.1.m1.1.1.1" xref="S5.E7X.2.1.1.m1.1.1.1.cmml"><mi id="S5.E7X.2.1.1.m1.1.1.1.2" xref="S5.E7X.2.1.1.m1.1.1.1.2.cmml">Tr</mi><mi id="S5.E7X.2.1.1.m1.1.1.1.3" xref="S5.E7X.2.1.1.m1.1.1.1.3.cmml">j</mi></msub><mo id="S5.E7X.2.1.1.m1.1.1a" lspace="0.167em" xref="S5.E7X.2.1.1.m1.1.1.cmml">⁡</mo><msub id="S5.E7X.2.1.1.m1.1.1.2" xref="S5.E7X.2.1.1.m1.1.1.2.cmml"><mi id="S5.E7X.2.1.1.m1.1.1.2.2" xref="S5.E7X.2.1.1.m1.1.1.2.2.cmml">f</mi><mi id="S5.E7X.2.1.1.m1.1.1.2.3" xref="S5.E7X.2.1.1.m1.1.1.2.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.E7X.2.1.1.m1.1b"><apply id="S5.E7X.2.1.1.m1.1.1.cmml" xref="S5.E7X.2.1.1.m1.1.1"><apply id="S5.E7X.2.1.1.m1.1.1.1.cmml" xref="S5.E7X.2.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S5.E7X.2.1.1.m1.1.1.1.1.cmml" xref="S5.E7X.2.1.1.m1.1.1.1">subscript</csymbol><ci id="S5.E7X.2.1.1.m1.1.1.1.2.cmml" xref="S5.E7X.2.1.1.m1.1.1.1.2">Tr</ci><ci id="S5.E7X.2.1.1.m1.1.1.1.3.cmml" xref="S5.E7X.2.1.1.m1.1.1.1.3">𝑗</ci></apply><apply id="S5.E7X.2.1.1.m1.1.1.2.cmml" xref="S5.E7X.2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.E7X.2.1.1.m1.1.1.2.1.cmml" xref="S5.E7X.2.1.1.m1.1.1.2">subscript</csymbol><ci id="S5.E7X.2.1.1.m1.1.1.2.2.cmml" xref="S5.E7X.2.1.1.m1.1.1.2.2">𝑓</ci><ci id="S5.E7X.2.1.1.m1.1.1.2.3.cmml" xref="S5.E7X.2.1.1.m1.1.1.2.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7X.2.1.1.m1.1c">\displaystyle\operatorname{Tr}_{j}f_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.E7X.2.1.1.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=h_{j}+\widetilde{\mathcal{C}}^{j}f_{j-1}\qquad" class="ltx_Math" display="inline" id="S5.E7X.3.2.2.m1.1"><semantics id="S5.E7X.3.2.2.m1.1a"><mrow id="S5.E7X.3.2.2.m1.1.1.1" xref="S5.E7X.3.2.2.m1.1.1.1.1.cmml"><mrow id="S5.E7X.3.2.2.m1.1.1.1.1" xref="S5.E7X.3.2.2.m1.1.1.1.1.cmml"><mi id="S5.E7X.3.2.2.m1.1.1.1.1.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.2.cmml"></mi><mo id="S5.E7X.3.2.2.m1.1.1.1.1.1" xref="S5.E7X.3.2.2.m1.1.1.1.1.1.cmml">=</mo><mrow id="S5.E7X.3.2.2.m1.1.1.1.1.3" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.cmml"><msub id="S5.E7X.3.2.2.m1.1.1.1.1.3.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.2.cmml"><mi id="S5.E7X.3.2.2.m1.1.1.1.1.3.2.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.2.2.cmml">h</mi><mi id="S5.E7X.3.2.2.m1.1.1.1.1.3.2.3" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.2.3.cmml">j</mi></msub><mo id="S5.E7X.3.2.2.m1.1.1.1.1.3.1" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.1.cmml">+</mo><mrow id="S5.E7X.3.2.2.m1.1.1.1.1.3.3" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.cmml"><msup id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.cmml"><mover accent="true" id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.2.cmml">𝒞</mi><mo id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.1" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.1.cmml">~</mo></mover><mi id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.3" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.3.cmml">j</mi></msup><mo id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.1" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.1.cmml">⁢</mo><msub id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.cmml"><mi id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.2.cmml">f</mi><mrow id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.cmml"><mi id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.2" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.2.cmml">j</mi><mo id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.1" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.1.cmml">−</mo><mn id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.3" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow></mrow><mspace id="S5.E7X.3.2.2.m1.1.1.1.2" width="2em" xref="S5.E7X.3.2.2.m1.1.1.1.1.cmml"></mspace></mrow><annotation-xml encoding="MathML-Content" id="S5.E7X.3.2.2.m1.1b"><apply id="S5.E7X.3.2.2.m1.1.1.1.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1"><eq id="S5.E7X.3.2.2.m1.1.1.1.1.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.1"></eq><csymbol cd="latexml" id="S5.E7X.3.2.2.m1.1.1.1.1.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.2">absent</csymbol><apply id="S5.E7X.3.2.2.m1.1.1.1.1.3.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3"><plus id="S5.E7X.3.2.2.m1.1.1.1.1.3.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.1"></plus><apply id="S5.E7X.3.2.2.m1.1.1.1.1.3.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.E7X.3.2.2.m1.1.1.1.1.3.2.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.2">subscript</csymbol><ci id="S5.E7X.3.2.2.m1.1.1.1.1.3.2.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.2.2">ℎ</ci><ci id="S5.E7X.3.2.2.m1.1.1.1.1.3.2.3.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.2.3">𝑗</ci></apply><apply id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3"><times id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.1"></times><apply id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2"><csymbol cd="ambiguous" id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2">superscript</csymbol><apply id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2"><ci id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.1">~</ci><ci id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.2.2">𝒞</ci></apply><ci id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.3.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.2.3">𝑗</ci></apply><apply id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3">subscript</csymbol><ci id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.2">𝑓</ci><apply id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3"><minus id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.1.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.1"></minus><ci id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.2.cmml" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.2">𝑗</ci><cn id="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.3.cmml" type="integer" xref="S5.E7X.3.2.2.m1.1.1.1.1.3.3.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7X.3.2.2.m1.1c">\displaystyle=h_{j}+\widetilde{\mathcal{C}}^{j}f_{j-1}\qquad</annotation><annotation encoding="application/x-llamapun" id="S5.E7X.3.2.2.m1.1d">= italic_h start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\text{ for }0\leq j\leq m_{n}," class="ltx_Math" display="inline" id="S5.E7X.5.1.1.m1.1"><semantics id="S5.E7X.5.1.1.m1.1a"><mrow id="S5.E7X.5.1.1.m1.1.1.1" xref="S5.E7X.5.1.1.m1.1.1.1.1.cmml"><mrow id="S5.E7X.5.1.1.m1.1.1.1.1" xref="S5.E7X.5.1.1.m1.1.1.1.1.cmml"><mrow id="S5.E7X.5.1.1.m1.1.1.1.1.2" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.cmml"><mtext id="S5.E7X.5.1.1.m1.1.1.1.1.2.2" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.2a.cmml">for </mtext><mo id="S5.E7X.5.1.1.m1.1.1.1.1.2.1" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.1.cmml">⁢</mo><mn id="S5.E7X.5.1.1.m1.1.1.1.1.2.3" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.3.cmml">0</mn></mrow><mo id="S5.E7X.5.1.1.m1.1.1.1.1.3" xref="S5.E7X.5.1.1.m1.1.1.1.1.3.cmml">≤</mo><mi id="S5.E7X.5.1.1.m1.1.1.1.1.4" xref="S5.E7X.5.1.1.m1.1.1.1.1.4.cmml">j</mi><mo id="S5.E7X.5.1.1.m1.1.1.1.1.5" xref="S5.E7X.5.1.1.m1.1.1.1.1.5.cmml">≤</mo><msub id="S5.E7X.5.1.1.m1.1.1.1.1.6" xref="S5.E7X.5.1.1.m1.1.1.1.1.6.cmml"><mi id="S5.E7X.5.1.1.m1.1.1.1.1.6.2" xref="S5.E7X.5.1.1.m1.1.1.1.1.6.2.cmml">m</mi><mi id="S5.E7X.5.1.1.m1.1.1.1.1.6.3" xref="S5.E7X.5.1.1.m1.1.1.1.1.6.3.cmml">n</mi></msub></mrow><mo id="S5.E7X.5.1.1.m1.1.1.1.2" xref="S5.E7X.5.1.1.m1.1.1.1.1.cmml">,</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E7X.5.1.1.m1.1b"><apply id="S5.E7X.5.1.1.m1.1.1.1.1.cmml" xref="S5.E7X.5.1.1.m1.1.1.1"><and id="S5.E7X.5.1.1.m1.1.1.1.1a.cmml" xref="S5.E7X.5.1.1.m1.1.1.1"></and><apply id="S5.E7X.5.1.1.m1.1.1.1.1b.cmml" xref="S5.E7X.5.1.1.m1.1.1.1"><leq id="S5.E7X.5.1.1.m1.1.1.1.1.3.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.3"></leq><apply id="S5.E7X.5.1.1.m1.1.1.1.1.2.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.2"><times id="S5.E7X.5.1.1.m1.1.1.1.1.2.1.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.1"></times><ci id="S5.E7X.5.1.1.m1.1.1.1.1.2.2a.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.2"><mtext id="S5.E7X.5.1.1.m1.1.1.1.1.2.2.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.2">for </mtext></ci><cn id="S5.E7X.5.1.1.m1.1.1.1.1.2.3.cmml" type="integer" xref="S5.E7X.5.1.1.m1.1.1.1.1.2.3">0</cn></apply><ci id="S5.E7X.5.1.1.m1.1.1.1.1.4.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.4">𝑗</ci></apply><apply id="S5.E7X.5.1.1.m1.1.1.1.1c.cmml" xref="S5.E7X.5.1.1.m1.1.1.1"><leq id="S5.E7X.5.1.1.m1.1.1.1.1.5.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.E7X.5.1.1.m1.1.1.1.1.4.cmml" id="S5.E7X.5.1.1.m1.1.1.1.1d.cmml" xref="S5.E7X.5.1.1.m1.1.1.1"></share><apply id="S5.E7X.5.1.1.m1.1.1.1.1.6.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.6"><csymbol cd="ambiguous" id="S5.E7X.5.1.1.m1.1.1.1.1.6.1.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.6">subscript</csymbol><ci id="S5.E7X.5.1.1.m1.1.1.1.1.6.2.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.6.2">𝑚</ci><ci id="S5.E7X.5.1.1.m1.1.1.1.1.6.3.cmml" xref="S5.E7X.5.1.1.m1.1.1.1.1.6.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7X.5.1.1.m1.1c">\displaystyle\text{ for }0\leq j\leq m_{n},</annotation><annotation encoding="application/x-llamapun" id="S5.E7X.5.1.1.m1.1d">for 0 ≤ italic_j ≤ italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(5.7)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S5.E7Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\operatorname{Tr}_{\ell}f_{j}" class="ltx_Math" display="inline" id="S5.E7Xa.2.1.1.m1.1"><semantics id="S5.E7Xa.2.1.1.m1.1a"><mrow id="S5.E7Xa.2.1.1.m1.1.1" xref="S5.E7Xa.2.1.1.m1.1.1.cmml"><msub id="S5.E7Xa.2.1.1.m1.1.1.1" xref="S5.E7Xa.2.1.1.m1.1.1.1.cmml"><mi id="S5.E7Xa.2.1.1.m1.1.1.1.2" xref="S5.E7Xa.2.1.1.m1.1.1.1.2.cmml">Tr</mi><mi id="S5.E7Xa.2.1.1.m1.1.1.1.3" mathvariant="normal" xref="S5.E7Xa.2.1.1.m1.1.1.1.3.cmml">ℓ</mi></msub><mo id="S5.E7Xa.2.1.1.m1.1.1a" lspace="0.167em" xref="S5.E7Xa.2.1.1.m1.1.1.cmml">⁡</mo><msub id="S5.E7Xa.2.1.1.m1.1.1.2" xref="S5.E7Xa.2.1.1.m1.1.1.2.cmml"><mi id="S5.E7Xa.2.1.1.m1.1.1.2.2" xref="S5.E7Xa.2.1.1.m1.1.1.2.2.cmml">f</mi><mi id="S5.E7Xa.2.1.1.m1.1.1.2.3" xref="S5.E7Xa.2.1.1.m1.1.1.2.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.E7Xa.2.1.1.m1.1b"><apply id="S5.E7Xa.2.1.1.m1.1.1.cmml" xref="S5.E7Xa.2.1.1.m1.1.1"><apply id="S5.E7Xa.2.1.1.m1.1.1.1.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.1"><csymbol cd="ambiguous" id="S5.E7Xa.2.1.1.m1.1.1.1.1.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.1">subscript</csymbol><ci id="S5.E7Xa.2.1.1.m1.1.1.1.2.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.1.2">Tr</ci><ci id="S5.E7Xa.2.1.1.m1.1.1.1.3.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.1.3">ℓ</ci></apply><apply id="S5.E7Xa.2.1.1.m1.1.1.2.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S5.E7Xa.2.1.1.m1.1.1.2.1.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.2">subscript</csymbol><ci id="S5.E7Xa.2.1.1.m1.1.1.2.2.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.2.2">𝑓</ci><ci id="S5.E7Xa.2.1.1.m1.1.1.2.3.cmml" xref="S5.E7Xa.2.1.1.m1.1.1.2.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7Xa.2.1.1.m1.1c">\displaystyle\operatorname{Tr}_{\ell}f_{j}</annotation><annotation encoding="application/x-llamapun" id="S5.E7Xa.2.1.1.m1.1d">roman_Tr start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{Tr}_{\ell}f_{j-1}\qquad" class="ltx_Math" display="inline" id="S5.E7Xa.3.2.2.m1.1"><semantics id="S5.E7Xa.3.2.2.m1.1a"><mrow id="S5.E7Xa.3.2.2.m1.1.1.1" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.cmml"><mrow id="S5.E7Xa.3.2.2.m1.1.1.1.1" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.cmml"><mi id="S5.E7Xa.3.2.2.m1.1.1.1.1.2" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.2.cmml"></mi><mo id="S5.E7Xa.3.2.2.m1.1.1.1.1.1" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.1.cmml">=</mo><mrow id="S5.E7Xa.3.2.2.m1.1.1.1.1.3" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.cmml"><msub id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.cmml"><mi id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.2" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.2.cmml">Tr</mi><mi id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.3" mathvariant="normal" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.3.cmml">ℓ</mi></msub><mo id="S5.E7Xa.3.2.2.m1.1.1.1.1.3a" lspace="0.167em" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.cmml">⁡</mo><msub id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.cmml"><mi id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.2" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.2.cmml">f</mi><mrow id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.cmml"><mi id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.2" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.2.cmml">j</mi><mo id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.1" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.1.cmml">−</mo><mn id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.3" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.3.cmml">1</mn></mrow></msub></mrow></mrow><mspace id="S5.E7Xa.3.2.2.m1.1.1.1.2" width="2em" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.cmml"></mspace></mrow><annotation-xml encoding="MathML-Content" id="S5.E7Xa.3.2.2.m1.1b"><apply id="S5.E7Xa.3.2.2.m1.1.1.1.1.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1"><eq id="S5.E7Xa.3.2.2.m1.1.1.1.1.1.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.1"></eq><csymbol cd="latexml" id="S5.E7Xa.3.2.2.m1.1.1.1.1.2.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.2">absent</csymbol><apply id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3"><apply id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.1.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1">subscript</csymbol><ci id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.2.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.2">Tr</ci><ci id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.3.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.1.3">ℓ</ci></apply><apply id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2"><csymbol cd="ambiguous" id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.1.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2">subscript</csymbol><ci id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.2.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.2">𝑓</ci><apply id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3"><minus id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.1.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.1"></minus><ci id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.2.cmml" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.2">𝑗</ci><cn id="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.3.cmml" type="integer" xref="S5.E7Xa.3.2.2.m1.1.1.1.1.3.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7Xa.3.2.2.m1.1c">\displaystyle=\operatorname{Tr}_{\ell}f_{j-1}\qquad</annotation><annotation encoding="application/x-llamapun" id="S5.E7Xa.3.2.2.m1.1d">= roman_Tr start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\text{ for }0\leq\ell\leq j\leq m_{n}." class="ltx_Math" display="inline" id="S5.E7Xa.5.1.1.m1.1"><semantics id="S5.E7Xa.5.1.1.m1.1a"><mrow id="S5.E7Xa.5.1.1.m1.1.1.1" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.cmml"><mrow id="S5.E7Xa.5.1.1.m1.1.1.1.1" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.cmml"><mrow id="S5.E7Xa.5.1.1.m1.1.1.1.1.2" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.cmml"><mtext id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.2" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.2a.cmml">for </mtext><mo id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.1" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.1.cmml">⁢</mo><mn id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.3" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.3.cmml">0</mn></mrow><mo id="S5.E7Xa.5.1.1.m1.1.1.1.1.3" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.3.cmml">≤</mo><mi id="S5.E7Xa.5.1.1.m1.1.1.1.1.4" mathvariant="normal" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.4.cmml">ℓ</mi><mo id="S5.E7Xa.5.1.1.m1.1.1.1.1.5" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.5.cmml">≤</mo><mi id="S5.E7Xa.5.1.1.m1.1.1.1.1.6" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.6.cmml">j</mi><mo id="S5.E7Xa.5.1.1.m1.1.1.1.1.7" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.7.cmml">≤</mo><msub id="S5.E7Xa.5.1.1.m1.1.1.1.1.8" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.8.cmml"><mi id="S5.E7Xa.5.1.1.m1.1.1.1.1.8.2" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.8.2.cmml">m</mi><mi id="S5.E7Xa.5.1.1.m1.1.1.1.1.8.3" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.8.3.cmml">n</mi></msub></mrow><mo id="S5.E7Xa.5.1.1.m1.1.1.1.2" lspace="0em" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E7Xa.5.1.1.m1.1b"><apply id="S5.E7Xa.5.1.1.m1.1.1.1.1.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1"><and id="S5.E7Xa.5.1.1.m1.1.1.1.1a.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1"></and><apply id="S5.E7Xa.5.1.1.m1.1.1.1.1b.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1"><leq id="S5.E7Xa.5.1.1.m1.1.1.1.1.3.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.3"></leq><apply id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2"><times id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.1.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.1"></times><ci id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.2a.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.2"><mtext id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.2.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.2">for </mtext></ci><cn id="S5.E7Xa.5.1.1.m1.1.1.1.1.2.3.cmml" type="integer" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.2.3">0</cn></apply><ci id="S5.E7Xa.5.1.1.m1.1.1.1.1.4.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.4">ℓ</ci></apply><apply id="S5.E7Xa.5.1.1.m1.1.1.1.1c.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1"><leq id="S5.E7Xa.5.1.1.m1.1.1.1.1.5.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.E7Xa.5.1.1.m1.1.1.1.1.4.cmml" id="S5.E7Xa.5.1.1.m1.1.1.1.1d.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1"></share><ci id="S5.E7Xa.5.1.1.m1.1.1.1.1.6.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.6">𝑗</ci></apply><apply id="S5.E7Xa.5.1.1.m1.1.1.1.1e.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1"><leq id="S5.E7Xa.5.1.1.m1.1.1.1.1.7.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.7"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.E7Xa.5.1.1.m1.1.1.1.1.6.cmml" id="S5.E7Xa.5.1.1.m1.1.1.1.1f.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1"></share><apply id="S5.E7Xa.5.1.1.m1.1.1.1.1.8.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.8"><csymbol cd="ambiguous" id="S5.E7Xa.5.1.1.m1.1.1.1.1.8.1.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.8">subscript</csymbol><ci id="S5.E7Xa.5.1.1.m1.1.1.1.1.8.2.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.8.2">𝑚</ci><ci id="S5.E7Xa.5.1.1.m1.1.1.1.1.8.3.cmml" xref="S5.E7Xa.5.1.1.m1.1.1.1.1.8.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E7Xa.5.1.1.m1.1c">\displaystyle\text{ for }0\leq\ell\leq j\leq m_{n}.</annotation><annotation encoding="application/x-llamapun" id="S5.E7Xa.5.1.1.m1.1d">for 0 ≤ roman_ℓ ≤ italic_j ≤ italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S5.5.p3.3">Suppose that <math alttext="0\leq\ell\leq j\leq m_{n}" class="ltx_Math" display="inline" id="S5.5.p3.3.m1.1"><semantics id="S5.5.p3.3.m1.1a"><mrow id="S5.5.p3.3.m1.1.1" xref="S5.5.p3.3.m1.1.1.cmml"><mn id="S5.5.p3.3.m1.1.1.2" xref="S5.5.p3.3.m1.1.1.2.cmml">0</mn><mo id="S5.5.p3.3.m1.1.1.3" xref="S5.5.p3.3.m1.1.1.3.cmml">≤</mo><mi id="S5.5.p3.3.m1.1.1.4" mathvariant="normal" xref="S5.5.p3.3.m1.1.1.4.cmml">ℓ</mi><mo id="S5.5.p3.3.m1.1.1.5" xref="S5.5.p3.3.m1.1.1.5.cmml">≤</mo><mi id="S5.5.p3.3.m1.1.1.6" xref="S5.5.p3.3.m1.1.1.6.cmml">j</mi><mo id="S5.5.p3.3.m1.1.1.7" xref="S5.5.p3.3.m1.1.1.7.cmml">≤</mo><msub id="S5.5.p3.3.m1.1.1.8" xref="S5.5.p3.3.m1.1.1.8.cmml"><mi id="S5.5.p3.3.m1.1.1.8.2" xref="S5.5.p3.3.m1.1.1.8.2.cmml">m</mi><mi id="S5.5.p3.3.m1.1.1.8.3" xref="S5.5.p3.3.m1.1.1.8.3.cmml">n</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.3.m1.1b"><apply id="S5.5.p3.3.m1.1.1.cmml" xref="S5.5.p3.3.m1.1.1"><and id="S5.5.p3.3.m1.1.1a.cmml" xref="S5.5.p3.3.m1.1.1"></and><apply id="S5.5.p3.3.m1.1.1b.cmml" xref="S5.5.p3.3.m1.1.1"><leq id="S5.5.p3.3.m1.1.1.3.cmml" xref="S5.5.p3.3.m1.1.1.3"></leq><cn id="S5.5.p3.3.m1.1.1.2.cmml" type="integer" xref="S5.5.p3.3.m1.1.1.2">0</cn><ci id="S5.5.p3.3.m1.1.1.4.cmml" xref="S5.5.p3.3.m1.1.1.4">ℓ</ci></apply><apply id="S5.5.p3.3.m1.1.1c.cmml" xref="S5.5.p3.3.m1.1.1"><leq id="S5.5.p3.3.m1.1.1.5.cmml" xref="S5.5.p3.3.m1.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.5.p3.3.m1.1.1.4.cmml" id="S5.5.p3.3.m1.1.1d.cmml" xref="S5.5.p3.3.m1.1.1"></share><ci id="S5.5.p3.3.m1.1.1.6.cmml" xref="S5.5.p3.3.m1.1.1.6">𝑗</ci></apply><apply id="S5.5.p3.3.m1.1.1e.cmml" xref="S5.5.p3.3.m1.1.1"><leq id="S5.5.p3.3.m1.1.1.7.cmml" xref="S5.5.p3.3.m1.1.1.7"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.5.p3.3.m1.1.1.6.cmml" id="S5.5.p3.3.m1.1.1f.cmml" xref="S5.5.p3.3.m1.1.1"></share><apply id="S5.5.p3.3.m1.1.1.8.cmml" xref="S5.5.p3.3.m1.1.1.8"><csymbol cd="ambiguous" id="S5.5.p3.3.m1.1.1.8.1.cmml" xref="S5.5.p3.3.m1.1.1.8">subscript</csymbol><ci id="S5.5.p3.3.m1.1.1.8.2.cmml" xref="S5.5.p3.3.m1.1.1.8.2">𝑚</ci><ci id="S5.5.p3.3.m1.1.1.8.3.cmml" xref="S5.5.p3.3.m1.1.1.8.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.3.m1.1c">0\leq\ell\leq j\leq m_{n}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.3.m1.1d">0 ≤ roman_ℓ ≤ italic_j ≤ italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT</annotation></semantics></math>, then by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E7" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.7</span></a>) we have</p> <table class="ltx_equation ltx_eqn_table" id="S5.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{\ell}f_{j}=\operatorname{Tr}_{\ell}f_{j-1}=\cdots=% \operatorname{Tr}_{\ell}f_{\ell}=h_{\ell}+\widetilde{\mathcal{C}}^{\ell}f_{% \ell-1}." class="ltx_Math" display="block" id="S5.E8.m1.1"><semantics id="S5.E8.m1.1a"><mrow id="S5.E8.m1.1.1.1" xref="S5.E8.m1.1.1.1.1.cmml"><mrow id="S5.E8.m1.1.1.1.1" xref="S5.E8.m1.1.1.1.1.cmml"><mrow id="S5.E8.m1.1.1.1.1.2" xref="S5.E8.m1.1.1.1.1.2.cmml"><msub id="S5.E8.m1.1.1.1.1.2.1" xref="S5.E8.m1.1.1.1.1.2.1.cmml"><mi id="S5.E8.m1.1.1.1.1.2.1.2" xref="S5.E8.m1.1.1.1.1.2.1.2.cmml">Tr</mi><mi id="S5.E8.m1.1.1.1.1.2.1.3" mathvariant="normal" xref="S5.E8.m1.1.1.1.1.2.1.3.cmml">ℓ</mi></msub><mo id="S5.E8.m1.1.1.1.1.2a" lspace="0.167em" xref="S5.E8.m1.1.1.1.1.2.cmml">⁡</mo><msub id="S5.E8.m1.1.1.1.1.2.2" xref="S5.E8.m1.1.1.1.1.2.2.cmml"><mi id="S5.E8.m1.1.1.1.1.2.2.2" xref="S5.E8.m1.1.1.1.1.2.2.2.cmml">f</mi><mi id="S5.E8.m1.1.1.1.1.2.2.3" xref="S5.E8.m1.1.1.1.1.2.2.3.cmml">j</mi></msub></mrow><mo id="S5.E8.m1.1.1.1.1.3" xref="S5.E8.m1.1.1.1.1.3.cmml">=</mo><mrow id="S5.E8.m1.1.1.1.1.4" xref="S5.E8.m1.1.1.1.1.4.cmml"><msub id="S5.E8.m1.1.1.1.1.4.1" xref="S5.E8.m1.1.1.1.1.4.1.cmml"><mi id="S5.E8.m1.1.1.1.1.4.1.2" xref="S5.E8.m1.1.1.1.1.4.1.2.cmml">Tr</mi><mi id="S5.E8.m1.1.1.1.1.4.1.3" mathvariant="normal" xref="S5.E8.m1.1.1.1.1.4.1.3.cmml">ℓ</mi></msub><mo id="S5.E8.m1.1.1.1.1.4a" lspace="0.167em" xref="S5.E8.m1.1.1.1.1.4.cmml">⁡</mo><msub id="S5.E8.m1.1.1.1.1.4.2" xref="S5.E8.m1.1.1.1.1.4.2.cmml"><mi id="S5.E8.m1.1.1.1.1.4.2.2" xref="S5.E8.m1.1.1.1.1.4.2.2.cmml">f</mi><mrow id="S5.E8.m1.1.1.1.1.4.2.3" xref="S5.E8.m1.1.1.1.1.4.2.3.cmml"><mi id="S5.E8.m1.1.1.1.1.4.2.3.2" xref="S5.E8.m1.1.1.1.1.4.2.3.2.cmml">j</mi><mo id="S5.E8.m1.1.1.1.1.4.2.3.1" xref="S5.E8.m1.1.1.1.1.4.2.3.1.cmml">−</mo><mn id="S5.E8.m1.1.1.1.1.4.2.3.3" xref="S5.E8.m1.1.1.1.1.4.2.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S5.E8.m1.1.1.1.1.5" xref="S5.E8.m1.1.1.1.1.5.cmml">=</mo><mi id="S5.E8.m1.1.1.1.1.6" mathvariant="normal" xref="S5.E8.m1.1.1.1.1.6.cmml">⋯</mi><mo id="S5.E8.m1.1.1.1.1.7" xref="S5.E8.m1.1.1.1.1.7.cmml">=</mo><mrow id="S5.E8.m1.1.1.1.1.8" xref="S5.E8.m1.1.1.1.1.8.cmml"><msub id="S5.E8.m1.1.1.1.1.8.1" xref="S5.E8.m1.1.1.1.1.8.1.cmml"><mi id="S5.E8.m1.1.1.1.1.8.1.2" xref="S5.E8.m1.1.1.1.1.8.1.2.cmml">Tr</mi><mi id="S5.E8.m1.1.1.1.1.8.1.3" mathvariant="normal" xref="S5.E8.m1.1.1.1.1.8.1.3.cmml">ℓ</mi></msub><mo id="S5.E8.m1.1.1.1.1.8a" lspace="0.167em" xref="S5.E8.m1.1.1.1.1.8.cmml">⁡</mo><msub id="S5.E8.m1.1.1.1.1.8.2" xref="S5.E8.m1.1.1.1.1.8.2.cmml"><mi id="S5.E8.m1.1.1.1.1.8.2.2" 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xref="S5.E8.m1.1.1.1.1.10.3.2.2.1.cmml">~</mo></mover><mi id="S5.E8.m1.1.1.1.1.10.3.2.3" mathvariant="normal" xref="S5.E8.m1.1.1.1.1.10.3.2.3.cmml">ℓ</mi></msup><mo id="S5.E8.m1.1.1.1.1.10.3.1" xref="S5.E8.m1.1.1.1.1.10.3.1.cmml">⁢</mo><msub id="S5.E8.m1.1.1.1.1.10.3.3" xref="S5.E8.m1.1.1.1.1.10.3.3.cmml"><mi id="S5.E8.m1.1.1.1.1.10.3.3.2" xref="S5.E8.m1.1.1.1.1.10.3.3.2.cmml">f</mi><mrow id="S5.E8.m1.1.1.1.1.10.3.3.3" xref="S5.E8.m1.1.1.1.1.10.3.3.3.cmml"><mi id="S5.E8.m1.1.1.1.1.10.3.3.3.2" mathvariant="normal" xref="S5.E8.m1.1.1.1.1.10.3.3.3.2.cmml">ℓ</mi><mo id="S5.E8.m1.1.1.1.1.10.3.3.3.1" xref="S5.E8.m1.1.1.1.1.10.3.3.3.1.cmml">−</mo><mn id="S5.E8.m1.1.1.1.1.10.3.3.3.3" xref="S5.E8.m1.1.1.1.1.10.3.3.3.3.cmml">1</mn></mrow></msub></mrow></mrow></mrow><mo id="S5.E8.m1.1.1.1.2" lspace="0em" xref="S5.E8.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E8.m1.1b"><apply id="S5.E8.m1.1.1.1.1.cmml" xref="S5.E8.m1.1.1.1"><and id="S5.E8.m1.1.1.1.1a.cmml" xref="S5.E8.m1.1.1.1"></and><apply id="S5.E8.m1.1.1.1.1b.cmml" xref="S5.E8.m1.1.1.1"><eq id="S5.E8.m1.1.1.1.1.3.cmml" xref="S5.E8.m1.1.1.1.1.3"></eq><apply id="S5.E8.m1.1.1.1.1.2.cmml" xref="S5.E8.m1.1.1.1.1.2"><apply id="S5.E8.m1.1.1.1.1.2.1.cmml" xref="S5.E8.m1.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.2.1.1.cmml" xref="S5.E8.m1.1.1.1.1.2.1">subscript</csymbol><ci id="S5.E8.m1.1.1.1.1.2.1.2.cmml" xref="S5.E8.m1.1.1.1.1.2.1.2">Tr</ci><ci id="S5.E8.m1.1.1.1.1.2.1.3.cmml" xref="S5.E8.m1.1.1.1.1.2.1.3">ℓ</ci></apply><apply id="S5.E8.m1.1.1.1.1.2.2.cmml" xref="S5.E8.m1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.2.2.1.cmml" xref="S5.E8.m1.1.1.1.1.2.2">subscript</csymbol><ci id="S5.E8.m1.1.1.1.1.2.2.2.cmml" xref="S5.E8.m1.1.1.1.1.2.2.2">𝑓</ci><ci id="S5.E8.m1.1.1.1.1.2.2.3.cmml" xref="S5.E8.m1.1.1.1.1.2.2.3">𝑗</ci></apply></apply><apply id="S5.E8.m1.1.1.1.1.4.cmml" xref="S5.E8.m1.1.1.1.1.4"><apply id="S5.E8.m1.1.1.1.1.4.1.cmml" xref="S5.E8.m1.1.1.1.1.4.1"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.4.1.1.cmml" xref="S5.E8.m1.1.1.1.1.4.1">subscript</csymbol><ci id="S5.E8.m1.1.1.1.1.4.1.2.cmml" xref="S5.E8.m1.1.1.1.1.4.1.2">Tr</ci><ci id="S5.E8.m1.1.1.1.1.4.1.3.cmml" xref="S5.E8.m1.1.1.1.1.4.1.3">ℓ</ci></apply><apply id="S5.E8.m1.1.1.1.1.4.2.cmml" xref="S5.E8.m1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.4.2.1.cmml" xref="S5.E8.m1.1.1.1.1.4.2">subscript</csymbol><ci id="S5.E8.m1.1.1.1.1.4.2.2.cmml" xref="S5.E8.m1.1.1.1.1.4.2.2">𝑓</ci><apply id="S5.E8.m1.1.1.1.1.4.2.3.cmml" xref="S5.E8.m1.1.1.1.1.4.2.3"><minus id="S5.E8.m1.1.1.1.1.4.2.3.1.cmml" xref="S5.E8.m1.1.1.1.1.4.2.3.1"></minus><ci id="S5.E8.m1.1.1.1.1.4.2.3.2.cmml" xref="S5.E8.m1.1.1.1.1.4.2.3.2">𝑗</ci><cn id="S5.E8.m1.1.1.1.1.4.2.3.3.cmml" type="integer" xref="S5.E8.m1.1.1.1.1.4.2.3.3">1</cn></apply></apply></apply></apply><apply id="S5.E8.m1.1.1.1.1c.cmml" xref="S5.E8.m1.1.1.1"><eq id="S5.E8.m1.1.1.1.1.5.cmml" xref="S5.E8.m1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S5.E8.m1.1.1.1.1.4.cmml" id="S5.E8.m1.1.1.1.1d.cmml" xref="S5.E8.m1.1.1.1"></share><ci id="S5.E8.m1.1.1.1.1.6.cmml" xref="S5.E8.m1.1.1.1.1.6">⋯</ci></apply><apply id="S5.E8.m1.1.1.1.1e.cmml" xref="S5.E8.m1.1.1.1"><eq id="S5.E8.m1.1.1.1.1.7.cmml" xref="S5.E8.m1.1.1.1.1.7"></eq><share href="https://arxiv.org/html/2503.14636v1#S5.E8.m1.1.1.1.1.6.cmml" id="S5.E8.m1.1.1.1.1f.cmml" xref="S5.E8.m1.1.1.1"></share><apply id="S5.E8.m1.1.1.1.1.8.cmml" xref="S5.E8.m1.1.1.1.1.8"><apply id="S5.E8.m1.1.1.1.1.8.1.cmml" xref="S5.E8.m1.1.1.1.1.8.1"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.8.1.1.cmml" xref="S5.E8.m1.1.1.1.1.8.1">subscript</csymbol><ci id="S5.E8.m1.1.1.1.1.8.1.2.cmml" xref="S5.E8.m1.1.1.1.1.8.1.2">Tr</ci><ci id="S5.E8.m1.1.1.1.1.8.1.3.cmml" xref="S5.E8.m1.1.1.1.1.8.1.3">ℓ</ci></apply><apply id="S5.E8.m1.1.1.1.1.8.2.cmml" xref="S5.E8.m1.1.1.1.1.8.2"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.8.2.1.cmml" 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xref="S5.E8.m1.1.1.1.1.10.3"><times id="S5.E8.m1.1.1.1.1.10.3.1.cmml" xref="S5.E8.m1.1.1.1.1.10.3.1"></times><apply id="S5.E8.m1.1.1.1.1.10.3.2.cmml" xref="S5.E8.m1.1.1.1.1.10.3.2"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.10.3.2.1.cmml" xref="S5.E8.m1.1.1.1.1.10.3.2">superscript</csymbol><apply id="S5.E8.m1.1.1.1.1.10.3.2.2.cmml" xref="S5.E8.m1.1.1.1.1.10.3.2.2"><ci id="S5.E8.m1.1.1.1.1.10.3.2.2.1.cmml" xref="S5.E8.m1.1.1.1.1.10.3.2.2.1">~</ci><ci id="S5.E8.m1.1.1.1.1.10.3.2.2.2.cmml" xref="S5.E8.m1.1.1.1.1.10.3.2.2.2">𝒞</ci></apply><ci id="S5.E8.m1.1.1.1.1.10.3.2.3.cmml" xref="S5.E8.m1.1.1.1.1.10.3.2.3">ℓ</ci></apply><apply id="S5.E8.m1.1.1.1.1.10.3.3.cmml" xref="S5.E8.m1.1.1.1.1.10.3.3"><csymbol cd="ambiguous" id="S5.E8.m1.1.1.1.1.10.3.3.1.cmml" xref="S5.E8.m1.1.1.1.1.10.3.3">subscript</csymbol><ci id="S5.E8.m1.1.1.1.1.10.3.3.2.cmml" xref="S5.E8.m1.1.1.1.1.10.3.3.2">𝑓</ci><apply id="S5.E8.m1.1.1.1.1.10.3.3.3.cmml" xref="S5.E8.m1.1.1.1.1.10.3.3.3"><minus id="S5.E8.m1.1.1.1.1.10.3.3.3.1.cmml" xref="S5.E8.m1.1.1.1.1.10.3.3.3.1"></minus><ci id="S5.E8.m1.1.1.1.1.10.3.3.3.2.cmml" xref="S5.E8.m1.1.1.1.1.10.3.3.3.2">ℓ</ci><cn id="S5.E8.m1.1.1.1.1.10.3.3.3.3.cmml" type="integer" xref="S5.E8.m1.1.1.1.1.10.3.3.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E8.m1.1c">\operatorname{Tr}_{\ell}f_{j}=\operatorname{Tr}_{\ell}f_{j-1}=\cdots=% \operatorname{Tr}_{\ell}f_{\ell}=h_{\ell}+\widetilde{\mathcal{C}}^{\ell}f_{% \ell-1}.</annotation><annotation encoding="application/x-llamapun" id="S5.E8.m1.1d">roman_Tr start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = roman_Tr start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j - 1 end_POSTSUBSCRIPT = ⋯ = roman_Tr start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT + over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT roman_ℓ - 1 end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.8)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p3.4">Moreover, if <math alttext="\ell=m_{i}\in\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S5.5.p3.4.m1.3"><semantics id="S5.5.p3.4.m1.3a"><mrow id="S5.5.p3.4.m1.3.3" xref="S5.5.p3.4.m1.3.3.cmml"><mi id="S5.5.p3.4.m1.3.3.4" mathvariant="normal" xref="S5.5.p3.4.m1.3.3.4.cmml">ℓ</mi><mo id="S5.5.p3.4.m1.3.3.5" xref="S5.5.p3.4.m1.3.3.5.cmml">=</mo><msub id="S5.5.p3.4.m1.3.3.6" xref="S5.5.p3.4.m1.3.3.6.cmml"><mi id="S5.5.p3.4.m1.3.3.6.2" xref="S5.5.p3.4.m1.3.3.6.2.cmml">m</mi><mi id="S5.5.p3.4.m1.3.3.6.3" xref="S5.5.p3.4.m1.3.3.6.3.cmml">i</mi></msub><mo id="S5.5.p3.4.m1.3.3.7" xref="S5.5.p3.4.m1.3.3.7.cmml">∈</mo><mrow id="S5.5.p3.4.m1.3.3.2.2" xref="S5.5.p3.4.m1.3.3.2.3.cmml"><mo id="S5.5.p3.4.m1.3.3.2.2.3" stretchy="false" xref="S5.5.p3.4.m1.3.3.2.3.cmml">{</mo><msub id="S5.5.p3.4.m1.2.2.1.1.1" xref="S5.5.p3.4.m1.2.2.1.1.1.cmml"><mi id="S5.5.p3.4.m1.2.2.1.1.1.2" xref="S5.5.p3.4.m1.2.2.1.1.1.2.cmml">m</mi><mn id="S5.5.p3.4.m1.2.2.1.1.1.3" xref="S5.5.p3.4.m1.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.5.p3.4.m1.3.3.2.2.4" xref="S5.5.p3.4.m1.3.3.2.3.cmml">,</mo><mi id="S5.5.p3.4.m1.1.1" mathvariant="normal" xref="S5.5.p3.4.m1.1.1.cmml">…</mi><mo id="S5.5.p3.4.m1.3.3.2.2.5" xref="S5.5.p3.4.m1.3.3.2.3.cmml">,</mo><msub id="S5.5.p3.4.m1.3.3.2.2.2" xref="S5.5.p3.4.m1.3.3.2.2.2.cmml"><mi id="S5.5.p3.4.m1.3.3.2.2.2.2" xref="S5.5.p3.4.m1.3.3.2.2.2.2.cmml">m</mi><mi id="S5.5.p3.4.m1.3.3.2.2.2.3" xref="S5.5.p3.4.m1.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S5.5.p3.4.m1.3.3.2.2.6" stretchy="false" xref="S5.5.p3.4.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.4.m1.3b"><apply id="S5.5.p3.4.m1.3.3.cmml" xref="S5.5.p3.4.m1.3.3"><and id="S5.5.p3.4.m1.3.3a.cmml" xref="S5.5.p3.4.m1.3.3"></and><apply id="S5.5.p3.4.m1.3.3b.cmml" xref="S5.5.p3.4.m1.3.3"><eq id="S5.5.p3.4.m1.3.3.5.cmml" xref="S5.5.p3.4.m1.3.3.5"></eq><ci id="S5.5.p3.4.m1.3.3.4.cmml" xref="S5.5.p3.4.m1.3.3.4">ℓ</ci><apply id="S5.5.p3.4.m1.3.3.6.cmml" xref="S5.5.p3.4.m1.3.3.6"><csymbol cd="ambiguous" id="S5.5.p3.4.m1.3.3.6.1.cmml" xref="S5.5.p3.4.m1.3.3.6">subscript</csymbol><ci id="S5.5.p3.4.m1.3.3.6.2.cmml" xref="S5.5.p3.4.m1.3.3.6.2">𝑚</ci><ci id="S5.5.p3.4.m1.3.3.6.3.cmml" xref="S5.5.p3.4.m1.3.3.6.3">𝑖</ci></apply></apply><apply id="S5.5.p3.4.m1.3.3c.cmml" xref="S5.5.p3.4.m1.3.3"><in id="S5.5.p3.4.m1.3.3.7.cmml" xref="S5.5.p3.4.m1.3.3.7"></in><share href="https://arxiv.org/html/2503.14636v1#S5.5.p3.4.m1.3.3.6.cmml" id="S5.5.p3.4.m1.3.3d.cmml" xref="S5.5.p3.4.m1.3.3"></share><set id="S5.5.p3.4.m1.3.3.2.3.cmml" xref="S5.5.p3.4.m1.3.3.2.2"><apply id="S5.5.p3.4.m1.2.2.1.1.1.cmml" xref="S5.5.p3.4.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.5.p3.4.m1.2.2.1.1.1.1.cmml" xref="S5.5.p3.4.m1.2.2.1.1.1">subscript</csymbol><ci id="S5.5.p3.4.m1.2.2.1.1.1.2.cmml" xref="S5.5.p3.4.m1.2.2.1.1.1.2">𝑚</ci><cn id="S5.5.p3.4.m1.2.2.1.1.1.3.cmml" type="integer" xref="S5.5.p3.4.m1.2.2.1.1.1.3">0</cn></apply><ci id="S5.5.p3.4.m1.1.1.cmml" xref="S5.5.p3.4.m1.1.1">…</ci><apply id="S5.5.p3.4.m1.3.3.2.2.2.cmml" xref="S5.5.p3.4.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.5.p3.4.m1.3.3.2.2.2.1.cmml" xref="S5.5.p3.4.m1.3.3.2.2.2">subscript</csymbol><ci id="S5.5.p3.4.m1.3.3.2.2.2.2.cmml" xref="S5.5.p3.4.m1.3.3.2.2.2.2">𝑚</ci><ci id="S5.5.p3.4.m1.3.3.2.2.2.3.cmml" xref="S5.5.p3.4.m1.3.3.2.2.2.3">𝑛</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.4.m1.3c">\ell=m_{i}\in\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.4.m1.3d">roman_ℓ = italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math>, then by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E6" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.6</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E7" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.7</span></a>)</p> <table class="ltx_equation ltx_eqn_table" id="S5.E9"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\mathcal{C}}^{\ell}f_{\ell-1}=\widetilde{\mathcal{C}}^{\ell}f_{\ell% }=\dots=\widetilde{\mathcal{C}}^{\ell}f_{j}." class="ltx_Math" display="block" id="S5.E9.m1.1"><semantics id="S5.E9.m1.1a"><mrow id="S5.E9.m1.1.1.1" xref="S5.E9.m1.1.1.1.1.cmml"><mrow id="S5.E9.m1.1.1.1.1" xref="S5.E9.m1.1.1.1.1.cmml"><mrow id="S5.E9.m1.1.1.1.1.2" xref="S5.E9.m1.1.1.1.1.2.cmml"><msup id="S5.E9.m1.1.1.1.1.2.2" xref="S5.E9.m1.1.1.1.1.2.2.cmml"><mover accent="true" id="S5.E9.m1.1.1.1.1.2.2.2" xref="S5.E9.m1.1.1.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.E9.m1.1.1.1.1.2.2.2.2" xref="S5.E9.m1.1.1.1.1.2.2.2.2.cmml">𝒞</mi><mo id="S5.E9.m1.1.1.1.1.2.2.2.1" xref="S5.E9.m1.1.1.1.1.2.2.2.1.cmml">~</mo></mover><mi id="S5.E9.m1.1.1.1.1.2.2.3" mathvariant="normal" xref="S5.E9.m1.1.1.1.1.2.2.3.cmml">ℓ</mi></msup><mo id="S5.E9.m1.1.1.1.1.2.1" xref="S5.E9.m1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S5.E9.m1.1.1.1.1.2.3" xref="S5.E9.m1.1.1.1.1.2.3.cmml"><mi id="S5.E9.m1.1.1.1.1.2.3.2" xref="S5.E9.m1.1.1.1.1.2.3.2.cmml">f</mi><mrow id="S5.E9.m1.1.1.1.1.2.3.3" xref="S5.E9.m1.1.1.1.1.2.3.3.cmml"><mi id="S5.E9.m1.1.1.1.1.2.3.3.2" mathvariant="normal" xref="S5.E9.m1.1.1.1.1.2.3.3.2.cmml">ℓ</mi><mo id="S5.E9.m1.1.1.1.1.2.3.3.1" xref="S5.E9.m1.1.1.1.1.2.3.3.1.cmml">−</mo><mn id="S5.E9.m1.1.1.1.1.2.3.3.3" xref="S5.E9.m1.1.1.1.1.2.3.3.3.cmml">1</mn></mrow></msub></mrow><mo id="S5.E9.m1.1.1.1.1.3" xref="S5.E9.m1.1.1.1.1.3.cmml">=</mo><mrow id="S5.E9.m1.1.1.1.1.4" xref="S5.E9.m1.1.1.1.1.4.cmml"><msup id="S5.E9.m1.1.1.1.1.4.2" xref="S5.E9.m1.1.1.1.1.4.2.cmml"><mover accent="true" 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id="S5.E9.m1.1c">\widetilde{\mathcal{C}}^{\ell}f_{\ell-1}=\widetilde{\mathcal{C}}^{\ell}f_{\ell% }=\dots=\widetilde{\mathcal{C}}^{\ell}f_{j}.</annotation><annotation encoding="application/x-llamapun" id="S5.E9.m1.1d">over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT roman_ℓ - 1 end_POSTSUBSCRIPT = over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = ⋯ = over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.9)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p3.11">Hence, (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E6" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.6</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E9" title="In Proof. ‣ 5. 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xref="S5.Ex11.m1.1.1.1.1.2.2.6">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex11.m1.1c">\operatorname{Tr}_{m_{i}}f_{j}=h_{m_{i}}+\widetilde{\mathcal{C}}^{m_{i}}f_{j}% \quad\text{ for }m_{i}\leq j\leq m_{n},\quad 1\leq i\leq n.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex11.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_h start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT + over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT for italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_j ≤ italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , 1 ≤ italic_i ≤ italic_n .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p3.5">Multiplying from the left with <math alttext="b_{i,m_{i}}" class="ltx_Math" display="inline" id="S5.5.p3.5.m1.2"><semantics id="S5.5.p3.5.m1.2a"><msub id="S5.5.p3.5.m1.2.3" xref="S5.5.p3.5.m1.2.3.cmml"><mi id="S5.5.p3.5.m1.2.3.2" xref="S5.5.p3.5.m1.2.3.2.cmml">b</mi><mrow id="S5.5.p3.5.m1.2.2.2.2" xref="S5.5.p3.5.m1.2.2.2.3.cmml"><mi id="S5.5.p3.5.m1.1.1.1.1" xref="S5.5.p3.5.m1.1.1.1.1.cmml">i</mi><mo id="S5.5.p3.5.m1.2.2.2.2.2" xref="S5.5.p3.5.m1.2.2.2.3.cmml">,</mo><msub id="S5.5.p3.5.m1.2.2.2.2.1" xref="S5.5.p3.5.m1.2.2.2.2.1.cmml"><mi id="S5.5.p3.5.m1.2.2.2.2.1.2" xref="S5.5.p3.5.m1.2.2.2.2.1.2.cmml">m</mi><mi id="S5.5.p3.5.m1.2.2.2.2.1.3" xref="S5.5.p3.5.m1.2.2.2.2.1.3.cmml">i</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S5.5.p3.5.m1.2b"><apply id="S5.5.p3.5.m1.2.3.cmml" xref="S5.5.p3.5.m1.2.3"><csymbol cd="ambiguous" id="S5.5.p3.5.m1.2.3.1.cmml" xref="S5.5.p3.5.m1.2.3">subscript</csymbol><ci id="S5.5.p3.5.m1.2.3.2.cmml" xref="S5.5.p3.5.m1.2.3.2">𝑏</ci><list id="S5.5.p3.5.m1.2.2.2.3.cmml" xref="S5.5.p3.5.m1.2.2.2.2"><ci id="S5.5.p3.5.m1.1.1.1.1.cmml" xref="S5.5.p3.5.m1.1.1.1.1">𝑖</ci><apply id="S5.5.p3.5.m1.2.2.2.2.1.cmml" xref="S5.5.p3.5.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S5.5.p3.5.m1.2.2.2.2.1.1.cmml" xref="S5.5.p3.5.m1.2.2.2.2.1">subscript</csymbol><ci id="S5.5.p3.5.m1.2.2.2.2.1.2.cmml" xref="S5.5.p3.5.m1.2.2.2.2.1.2">𝑚</ci><ci id="S5.5.p3.5.m1.2.2.2.2.1.3.cmml" xref="S5.5.p3.5.m1.2.2.2.2.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.5.m1.2c">b_{i,m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.5.m1.2d">italic_b start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> gives</p> <table class="ltx_equation 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xref="S5.E10.m1.1.1.1.1.1.1.2.2.3.3.cmml">i</mi></msub></msup><mo id="S5.E10.m1.1.1.1.1.1.1.2.1" xref="S5.E10.m1.1.1.1.1.1.1.2.1.cmml">⁢</mo><msub id="S5.E10.m1.1.1.1.1.1.1.2.3" xref="S5.E10.m1.1.1.1.1.1.1.2.3.cmml"><mi id="S5.E10.m1.1.1.1.1.1.1.2.3.2" xref="S5.E10.m1.1.1.1.1.1.1.2.3.2.cmml">f</mi><mi id="S5.E10.m1.1.1.1.1.1.1.2.3.3" xref="S5.E10.m1.1.1.1.1.1.1.2.3.3.cmml">j</mi></msub></mrow><mo id="S5.E10.m1.1.1.1.1.1.1.1" xref="S5.E10.m1.1.1.1.1.1.1.1.cmml">=</mo><msub id="S5.E10.m1.1.1.1.1.1.1.3" xref="S5.E10.m1.1.1.1.1.1.1.3.cmml"><mi id="S5.E10.m1.1.1.1.1.1.1.3.2" xref="S5.E10.m1.1.1.1.1.1.1.3.2.cmml">g</mi><msub id="S5.E10.m1.1.1.1.1.1.1.3.3" xref="S5.E10.m1.1.1.1.1.1.1.3.3.cmml"><mi id="S5.E10.m1.1.1.1.1.1.1.3.3.2" xref="S5.E10.m1.1.1.1.1.1.1.3.3.2.cmml">m</mi><mi id="S5.E10.m1.1.1.1.1.1.1.3.3.3" xref="S5.E10.m1.1.1.1.1.1.1.3.3.3.cmml">i</mi></msub></msub></mrow><mspace id="S5.E10.m1.1.1.1.1.2.3" width="1em" xref="S5.E10.m1.1.1.1.1.3a.cmml"></mspace><mrow id="S5.E10.m1.1.1.1.1.2.2" xref="S5.E10.m1.1.1.1.1.2.2.cmml"><mrow id="S5.E10.m1.1.1.1.1.2.2.2" xref="S5.E10.m1.1.1.1.1.2.2.2.cmml"><mtext id="S5.E10.m1.1.1.1.1.2.2.2.2" xref="S5.E10.m1.1.1.1.1.2.2.2.2a.cmml"> for </mtext><mo id="S5.E10.m1.1.1.1.1.2.2.2.1" xref="S5.E10.m1.1.1.1.1.2.2.2.1.cmml">⁢</mo><msub id="S5.E10.m1.1.1.1.1.2.2.2.3" xref="S5.E10.m1.1.1.1.1.2.2.2.3.cmml"><mi id="S5.E10.m1.1.1.1.1.2.2.2.3.2" xref="S5.E10.m1.1.1.1.1.2.2.2.3.2.cmml">m</mi><mi id="S5.E10.m1.1.1.1.1.2.2.2.3.3" xref="S5.E10.m1.1.1.1.1.2.2.2.3.3.cmml">i</mi></msub></mrow><mo id="S5.E10.m1.1.1.1.1.2.2.1" xref="S5.E10.m1.1.1.1.1.2.2.1.cmml">≤</mo><msub id="S5.E10.m1.1.1.1.1.2.2.3" xref="S5.E10.m1.1.1.1.1.2.2.3.cmml"><mi id="S5.E10.m1.1.1.1.1.2.2.3.2" xref="S5.E10.m1.1.1.1.1.2.2.3.2.cmml">m</mi><mi id="S5.E10.m1.1.1.1.1.2.2.3.3" xref="S5.E10.m1.1.1.1.1.2.2.3.3.cmml">n</mi></msub></mrow></mrow><mo id="S5.E10.m1.1.1.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.E10.m1.1b"><apply 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xref="S5.E10.m1.1.1.1.1.1.1.2.2.3.2">𝑚</ci><ci id="S5.E10.m1.1.1.1.1.1.1.2.2.3.3.cmml" xref="S5.E10.m1.1.1.1.1.1.1.2.2.3.3">𝑖</ci></apply></apply><apply id="S5.E10.m1.1.1.1.1.1.1.2.3.cmml" xref="S5.E10.m1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S5.E10.m1.1.1.1.1.1.1.2.3.1.cmml" xref="S5.E10.m1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="S5.E10.m1.1.1.1.1.1.1.2.3.2.cmml" xref="S5.E10.m1.1.1.1.1.1.1.2.3.2">𝑓</ci><ci id="S5.E10.m1.1.1.1.1.1.1.2.3.3.cmml" xref="S5.E10.m1.1.1.1.1.1.1.2.3.3">𝑗</ci></apply></apply><apply id="S5.E10.m1.1.1.1.1.1.1.3.cmml" xref="S5.E10.m1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.E10.m1.1.1.1.1.1.1.3.1.cmml" xref="S5.E10.m1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S5.E10.m1.1.1.1.1.1.1.3.2.cmml" xref="S5.E10.m1.1.1.1.1.1.1.3.2">𝑔</ci><apply id="S5.E10.m1.1.1.1.1.1.1.3.3.cmml" xref="S5.E10.m1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S5.E10.m1.1.1.1.1.1.1.3.3.1.cmml" xref="S5.E10.m1.1.1.1.1.1.1.3.3">subscript</csymbol><ci 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xref="S5.E10.m1.1.1.1.1.2.2.2.3.3">𝑖</ci></apply></apply><apply id="S5.E10.m1.1.1.1.1.2.2.3.cmml" xref="S5.E10.m1.1.1.1.1.2.2.3"><csymbol cd="ambiguous" id="S5.E10.m1.1.1.1.1.2.2.3.1.cmml" xref="S5.E10.m1.1.1.1.1.2.2.3">subscript</csymbol><ci id="S5.E10.m1.1.1.1.1.2.2.3.2.cmml" xref="S5.E10.m1.1.1.1.1.2.2.3.2">𝑚</ci><ci id="S5.E10.m1.1.1.1.1.2.2.3.3.cmml" xref="S5.E10.m1.1.1.1.1.2.2.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.E10.m1.1c">\mathcal{B}^{m_{i}}f_{j}=g_{m_{i}}\quad\text{ for }m_{i}\leq m_{n}.</annotation><annotation encoding="application/x-llamapun" id="S5.E10.m1.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT = italic_g start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT for italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ≤ italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5.10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p3.12">Define</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{C}^{m_{i}}:\prod_{j=0}^{i}B^{s-m_{j}-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};Y_{j})\to H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="block" id="S5.Ex12.m1.9"><semantics id="S5.Ex12.m1.9a"><mrow id="S5.Ex12.m1.9.9" xref="S5.Ex12.m1.9.9.cmml"><msup id="S5.Ex12.m1.9.9.6" xref="S5.Ex12.m1.9.9.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex12.m1.9.9.6.2" 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id="S5.Ex12.m1.9.9.4.4.4.1.cmml" xref="S5.Ex12.m1.9.9.4.4.4">superscript</csymbol><ci id="S5.Ex12.m1.9.9.4.4.4.2.cmml" xref="S5.Ex12.m1.9.9.4.4.4.2">𝐻</ci><list id="S5.Ex12.m1.4.4.2.3.cmml" xref="S5.Ex12.m1.4.4.2.4"><ci id="S5.Ex12.m1.3.3.1.1.cmml" xref="S5.Ex12.m1.3.3.1.1">𝑠</ci><ci id="S5.Ex12.m1.4.4.2.2.cmml" xref="S5.Ex12.m1.4.4.2.2">𝑝</ci></list></apply><vector id="S5.Ex12.m1.9.9.4.4.2.3.cmml" xref="S5.Ex12.m1.9.9.4.4.2.2"><apply id="S5.Ex12.m1.8.8.3.3.1.1.1.cmml" xref="S5.Ex12.m1.8.8.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.Ex12.m1.8.8.3.3.1.1.1.1.cmml" xref="S5.Ex12.m1.8.8.3.3.1.1.1">subscript</csymbol><apply id="S5.Ex12.m1.8.8.3.3.1.1.1.2.cmml" xref="S5.Ex12.m1.8.8.3.3.1.1.1"><csymbol cd="ambiguous" id="S5.Ex12.m1.8.8.3.3.1.1.1.2.1.cmml" xref="S5.Ex12.m1.8.8.3.3.1.1.1">superscript</csymbol><ci id="S5.Ex12.m1.8.8.3.3.1.1.1.2.2.cmml" xref="S5.Ex12.m1.8.8.3.3.1.1.1.2.2">ℝ</ci><ci id="S5.Ex12.m1.8.8.3.3.1.1.1.2.3.cmml" xref="S5.Ex12.m1.8.8.3.3.1.1.1.2.3">𝑑</ci></apply><plus id="S5.Ex12.m1.8.8.3.3.1.1.1.3.cmml" xref="S5.Ex12.m1.8.8.3.3.1.1.1.3"></plus></apply><apply id="S5.Ex12.m1.9.9.4.4.2.2.2.cmml" xref="S5.Ex12.m1.9.9.4.4.2.2.2"><csymbol cd="ambiguous" id="S5.Ex12.m1.9.9.4.4.2.2.2.1.cmml" xref="S5.Ex12.m1.9.9.4.4.2.2.2">subscript</csymbol><ci id="S5.Ex12.m1.9.9.4.4.2.2.2.2.cmml" xref="S5.Ex12.m1.9.9.4.4.2.2.2.2">𝑤</ci><ci id="S5.Ex12.m1.9.9.4.4.2.2.2.3.cmml" xref="S5.Ex12.m1.9.9.4.4.2.2.2.3">𝛾</ci></apply><ci id="S5.Ex12.m1.5.5.cmml" xref="S5.Ex12.m1.5.5">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex12.m1.9c">\mathcal{C}^{m_{i}}:\prod_{j=0}^{i}B^{s-m_{j}-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};Y_{j})\to H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S5.Ex12.m1.9d">caligraphic_C start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT : ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_s - italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) → italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p3.8">by <math alttext="\mathcal{C}^{m_{i}}(g_{m_{0}},\dots,g_{m_{n}}):=f_{m_{i}}" class="ltx_Math" display="inline" id="S5.5.p3.6.m1.3"><semantics id="S5.5.p3.6.m1.3a"><mrow id="S5.5.p3.6.m1.3.3" xref="S5.5.p3.6.m1.3.3.cmml"><mrow id="S5.5.p3.6.m1.3.3.2" xref="S5.5.p3.6.m1.3.3.2.cmml"><msup id="S5.5.p3.6.m1.3.3.2.4" xref="S5.5.p3.6.m1.3.3.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p3.6.m1.3.3.2.4.2" xref="S5.5.p3.6.m1.3.3.2.4.2.cmml">𝒞</mi><msub id="S5.5.p3.6.m1.3.3.2.4.3" xref="S5.5.p3.6.m1.3.3.2.4.3.cmml"><mi id="S5.5.p3.6.m1.3.3.2.4.3.2" xref="S5.5.p3.6.m1.3.3.2.4.3.2.cmml">m</mi><mi id="S5.5.p3.6.m1.3.3.2.4.3.3" xref="S5.5.p3.6.m1.3.3.2.4.3.3.cmml">i</mi></msub></msup><mo id="S5.5.p3.6.m1.3.3.2.3" xref="S5.5.p3.6.m1.3.3.2.3.cmml">⁢</mo><mrow id="S5.5.p3.6.m1.3.3.2.2.2" xref="S5.5.p3.6.m1.3.3.2.2.3.cmml"><mo id="S5.5.p3.6.m1.3.3.2.2.2.3" stretchy="false" xref="S5.5.p3.6.m1.3.3.2.2.3.cmml">(</mo><msub id="S5.5.p3.6.m1.2.2.1.1.1.1" xref="S5.5.p3.6.m1.2.2.1.1.1.1.cmml"><mi id="S5.5.p3.6.m1.2.2.1.1.1.1.2" xref="S5.5.p3.6.m1.2.2.1.1.1.1.2.cmml">g</mi><msub id="S5.5.p3.6.m1.2.2.1.1.1.1.3" xref="S5.5.p3.6.m1.2.2.1.1.1.1.3.cmml"><mi id="S5.5.p3.6.m1.2.2.1.1.1.1.3.2" xref="S5.5.p3.6.m1.2.2.1.1.1.1.3.2.cmml">m</mi><mn id="S5.5.p3.6.m1.2.2.1.1.1.1.3.3" xref="S5.5.p3.6.m1.2.2.1.1.1.1.3.3.cmml">0</mn></msub></msub><mo id="S5.5.p3.6.m1.3.3.2.2.2.4" xref="S5.5.p3.6.m1.3.3.2.2.3.cmml">,</mo><mi id="S5.5.p3.6.m1.1.1" mathvariant="normal" xref="S5.5.p3.6.m1.1.1.cmml">…</mi><mo id="S5.5.p3.6.m1.3.3.2.2.2.5" xref="S5.5.p3.6.m1.3.3.2.2.3.cmml">,</mo><msub id="S5.5.p3.6.m1.3.3.2.2.2.2" xref="S5.5.p3.6.m1.3.3.2.2.2.2.cmml"><mi id="S5.5.p3.6.m1.3.3.2.2.2.2.2" xref="S5.5.p3.6.m1.3.3.2.2.2.2.2.cmml">g</mi><msub id="S5.5.p3.6.m1.3.3.2.2.2.2.3" xref="S5.5.p3.6.m1.3.3.2.2.2.2.3.cmml"><mi id="S5.5.p3.6.m1.3.3.2.2.2.2.3.2" xref="S5.5.p3.6.m1.3.3.2.2.2.2.3.2.cmml">m</mi><mi id="S5.5.p3.6.m1.3.3.2.2.2.2.3.3" xref="S5.5.p3.6.m1.3.3.2.2.2.2.3.3.cmml">n</mi></msub></msub><mo id="S5.5.p3.6.m1.3.3.2.2.2.6" rspace="0.278em" stretchy="false" xref="S5.5.p3.6.m1.3.3.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.5.p3.6.m1.3.3.3" rspace="0.278em" xref="S5.5.p3.6.m1.3.3.3.cmml">:=</mo><msub id="S5.5.p3.6.m1.3.3.4" xref="S5.5.p3.6.m1.3.3.4.cmml"><mi id="S5.5.p3.6.m1.3.3.4.2" xref="S5.5.p3.6.m1.3.3.4.2.cmml">f</mi><msub id="S5.5.p3.6.m1.3.3.4.3" xref="S5.5.p3.6.m1.3.3.4.3.cmml"><mi id="S5.5.p3.6.m1.3.3.4.3.2" xref="S5.5.p3.6.m1.3.3.4.3.2.cmml">m</mi><mi id="S5.5.p3.6.m1.3.3.4.3.3" xref="S5.5.p3.6.m1.3.3.4.3.3.cmml">i</mi></msub></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.6.m1.3b"><apply id="S5.5.p3.6.m1.3.3.cmml" xref="S5.5.p3.6.m1.3.3"><csymbol cd="latexml" id="S5.5.p3.6.m1.3.3.3.cmml" xref="S5.5.p3.6.m1.3.3.3">assign</csymbol><apply id="S5.5.p3.6.m1.3.3.2.cmml" xref="S5.5.p3.6.m1.3.3.2"><times id="S5.5.p3.6.m1.3.3.2.3.cmml" xref="S5.5.p3.6.m1.3.3.2.3"></times><apply id="S5.5.p3.6.m1.3.3.2.4.cmml" xref="S5.5.p3.6.m1.3.3.2.4"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.3.3.2.4.1.cmml" xref="S5.5.p3.6.m1.3.3.2.4">superscript</csymbol><ci id="S5.5.p3.6.m1.3.3.2.4.2.cmml" xref="S5.5.p3.6.m1.3.3.2.4.2">𝒞</ci><apply id="S5.5.p3.6.m1.3.3.2.4.3.cmml" xref="S5.5.p3.6.m1.3.3.2.4.3"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.3.3.2.4.3.1.cmml" xref="S5.5.p3.6.m1.3.3.2.4.3">subscript</csymbol><ci id="S5.5.p3.6.m1.3.3.2.4.3.2.cmml" xref="S5.5.p3.6.m1.3.3.2.4.3.2">𝑚</ci><ci id="S5.5.p3.6.m1.3.3.2.4.3.3.cmml" xref="S5.5.p3.6.m1.3.3.2.4.3.3">𝑖</ci></apply></apply><vector id="S5.5.p3.6.m1.3.3.2.2.3.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2"><apply id="S5.5.p3.6.m1.2.2.1.1.1.1.cmml" xref="S5.5.p3.6.m1.2.2.1.1.1.1"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.2.2.1.1.1.1.1.cmml" xref="S5.5.p3.6.m1.2.2.1.1.1.1">subscript</csymbol><ci id="S5.5.p3.6.m1.2.2.1.1.1.1.2.cmml" xref="S5.5.p3.6.m1.2.2.1.1.1.1.2">𝑔</ci><apply id="S5.5.p3.6.m1.2.2.1.1.1.1.3.cmml" xref="S5.5.p3.6.m1.2.2.1.1.1.1.3"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.2.2.1.1.1.1.3.1.cmml" xref="S5.5.p3.6.m1.2.2.1.1.1.1.3">subscript</csymbol><ci id="S5.5.p3.6.m1.2.2.1.1.1.1.3.2.cmml" xref="S5.5.p3.6.m1.2.2.1.1.1.1.3.2">𝑚</ci><cn id="S5.5.p3.6.m1.2.2.1.1.1.1.3.3.cmml" type="integer" xref="S5.5.p3.6.m1.2.2.1.1.1.1.3.3">0</cn></apply></apply><ci id="S5.5.p3.6.m1.1.1.cmml" xref="S5.5.p3.6.m1.1.1">…</ci><apply id="S5.5.p3.6.m1.3.3.2.2.2.2.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.3.3.2.2.2.2.1.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2.2">subscript</csymbol><ci id="S5.5.p3.6.m1.3.3.2.2.2.2.2.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2.2.2">𝑔</ci><apply id="S5.5.p3.6.m1.3.3.2.2.2.2.3.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2.2.3"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.3.3.2.2.2.2.3.1.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2.2.3">subscript</csymbol><ci id="S5.5.p3.6.m1.3.3.2.2.2.2.3.2.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2.2.3.2">𝑚</ci><ci id="S5.5.p3.6.m1.3.3.2.2.2.2.3.3.cmml" xref="S5.5.p3.6.m1.3.3.2.2.2.2.3.3">𝑛</ci></apply></apply></vector></apply><apply id="S5.5.p3.6.m1.3.3.4.cmml" xref="S5.5.p3.6.m1.3.3.4"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.3.3.4.1.cmml" xref="S5.5.p3.6.m1.3.3.4">subscript</csymbol><ci id="S5.5.p3.6.m1.3.3.4.2.cmml" xref="S5.5.p3.6.m1.3.3.4.2">𝑓</ci><apply id="S5.5.p3.6.m1.3.3.4.3.cmml" xref="S5.5.p3.6.m1.3.3.4.3"><csymbol cd="ambiguous" id="S5.5.p3.6.m1.3.3.4.3.1.cmml" xref="S5.5.p3.6.m1.3.3.4.3">subscript</csymbol><ci id="S5.5.p3.6.m1.3.3.4.3.2.cmml" xref="S5.5.p3.6.m1.3.3.4.3.2">𝑚</ci><ci id="S5.5.p3.6.m1.3.3.4.3.3.cmml" xref="S5.5.p3.6.m1.3.3.4.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.6.m1.3c">\mathcal{C}^{m_{i}}(g_{m_{0}},\dots,g_{m_{n}}):=f_{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.6.m1.3d">caligraphic_C start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_g start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , … , italic_g start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) := italic_f start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="i\in\{1,\dots,n\}" class="ltx_Math" display="inline" id="S5.5.p3.7.m2.3"><semantics id="S5.5.p3.7.m2.3a"><mrow id="S5.5.p3.7.m2.3.4" xref="S5.5.p3.7.m2.3.4.cmml"><mi id="S5.5.p3.7.m2.3.4.2" xref="S5.5.p3.7.m2.3.4.2.cmml">i</mi><mo id="S5.5.p3.7.m2.3.4.1" xref="S5.5.p3.7.m2.3.4.1.cmml">∈</mo><mrow id="S5.5.p3.7.m2.3.4.3.2" xref="S5.5.p3.7.m2.3.4.3.1.cmml"><mo id="S5.5.p3.7.m2.3.4.3.2.1" stretchy="false" xref="S5.5.p3.7.m2.3.4.3.1.cmml">{</mo><mn id="S5.5.p3.7.m2.1.1" xref="S5.5.p3.7.m2.1.1.cmml">1</mn><mo id="S5.5.p3.7.m2.3.4.3.2.2" xref="S5.5.p3.7.m2.3.4.3.1.cmml">,</mo><mi id="S5.5.p3.7.m2.2.2" mathvariant="normal" xref="S5.5.p3.7.m2.2.2.cmml">…</mi><mo id="S5.5.p3.7.m2.3.4.3.2.3" xref="S5.5.p3.7.m2.3.4.3.1.cmml">,</mo><mi id="S5.5.p3.7.m2.3.3" xref="S5.5.p3.7.m2.3.3.cmml">n</mi><mo id="S5.5.p3.7.m2.3.4.3.2.4" stretchy="false" xref="S5.5.p3.7.m2.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.7.m2.3b"><apply id="S5.5.p3.7.m2.3.4.cmml" xref="S5.5.p3.7.m2.3.4"><in id="S5.5.p3.7.m2.3.4.1.cmml" xref="S5.5.p3.7.m2.3.4.1"></in><ci id="S5.5.p3.7.m2.3.4.2.cmml" xref="S5.5.p3.7.m2.3.4.2">𝑖</ci><set id="S5.5.p3.7.m2.3.4.3.1.cmml" xref="S5.5.p3.7.m2.3.4.3.2"><cn id="S5.5.p3.7.m2.1.1.cmml" type="integer" xref="S5.5.p3.7.m2.1.1">1</cn><ci id="S5.5.p3.7.m2.2.2.cmml" xref="S5.5.p3.7.m2.2.2">…</ci><ci id="S5.5.p3.7.m2.3.3.cmml" xref="S5.5.p3.7.m2.3.3">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.7.m2.3c">i\in\{1,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.7.m2.3d">italic_i ∈ { 1 , … , italic_n }</annotation></semantics></math>. Then from (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E10" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.10</span></a>) we obtain for <math alttext="0\leq\ell\leq i\leq j\leq n" class="ltx_Math" display="inline" id="S5.5.p3.8.m3.1"><semantics id="S5.5.p3.8.m3.1a"><mrow id="S5.5.p3.8.m3.1.1" xref="S5.5.p3.8.m3.1.1.cmml"><mn id="S5.5.p3.8.m3.1.1.2" xref="S5.5.p3.8.m3.1.1.2.cmml">0</mn><mo id="S5.5.p3.8.m3.1.1.3" xref="S5.5.p3.8.m3.1.1.3.cmml">≤</mo><mi id="S5.5.p3.8.m3.1.1.4" mathvariant="normal" xref="S5.5.p3.8.m3.1.1.4.cmml">ℓ</mi><mo id="S5.5.p3.8.m3.1.1.5" xref="S5.5.p3.8.m3.1.1.5.cmml">≤</mo><mi id="S5.5.p3.8.m3.1.1.6" xref="S5.5.p3.8.m3.1.1.6.cmml">i</mi><mo id="S5.5.p3.8.m3.1.1.7" xref="S5.5.p3.8.m3.1.1.7.cmml">≤</mo><mi id="S5.5.p3.8.m3.1.1.8" xref="S5.5.p3.8.m3.1.1.8.cmml">j</mi><mo id="S5.5.p3.8.m3.1.1.9" xref="S5.5.p3.8.m3.1.1.9.cmml">≤</mo><mi id="S5.5.p3.8.m3.1.1.10" xref="S5.5.p3.8.m3.1.1.10.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.8.m3.1b"><apply id="S5.5.p3.8.m3.1.1.cmml" xref="S5.5.p3.8.m3.1.1"><and id="S5.5.p3.8.m3.1.1a.cmml" xref="S5.5.p3.8.m3.1.1"></and><apply id="S5.5.p3.8.m3.1.1b.cmml" xref="S5.5.p3.8.m3.1.1"><leq id="S5.5.p3.8.m3.1.1.3.cmml" xref="S5.5.p3.8.m3.1.1.3"></leq><cn id="S5.5.p3.8.m3.1.1.2.cmml" type="integer" xref="S5.5.p3.8.m3.1.1.2">0</cn><ci id="S5.5.p3.8.m3.1.1.4.cmml" xref="S5.5.p3.8.m3.1.1.4">ℓ</ci></apply><apply id="S5.5.p3.8.m3.1.1c.cmml" xref="S5.5.p3.8.m3.1.1"><leq id="S5.5.p3.8.m3.1.1.5.cmml" xref="S5.5.p3.8.m3.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.5.p3.8.m3.1.1.4.cmml" id="S5.5.p3.8.m3.1.1d.cmml" xref="S5.5.p3.8.m3.1.1"></share><ci id="S5.5.p3.8.m3.1.1.6.cmml" xref="S5.5.p3.8.m3.1.1.6">𝑖</ci></apply><apply id="S5.5.p3.8.m3.1.1e.cmml" xref="S5.5.p3.8.m3.1.1"><leq id="S5.5.p3.8.m3.1.1.7.cmml" xref="S5.5.p3.8.m3.1.1.7"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.5.p3.8.m3.1.1.6.cmml" id="S5.5.p3.8.m3.1.1f.cmml" xref="S5.5.p3.8.m3.1.1"></share><ci id="S5.5.p3.8.m3.1.1.8.cmml" xref="S5.5.p3.8.m3.1.1.8">𝑗</ci></apply><apply id="S5.5.p3.8.m3.1.1g.cmml" xref="S5.5.p3.8.m3.1.1"><leq id="S5.5.p3.8.m3.1.1.9.cmml" xref="S5.5.p3.8.m3.1.1.9"></leq><share href="https://arxiv.org/html/2503.14636v1#S5.5.p3.8.m3.1.1.8.cmml" id="S5.5.p3.8.m3.1.1h.cmml" xref="S5.5.p3.8.m3.1.1"></share><ci id="S5.5.p3.8.m3.1.1.10.cmml" xref="S5.5.p3.8.m3.1.1.10">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.8.m3.1c">0\leq\ell\leq i\leq j\leq n</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.8.m3.1d">0 ≤ roman_ℓ ≤ italic_i ≤ italic_j ≤ italic_n</annotation></semantics></math> that</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}^{m_{\ell}}\mathcal{C}^{m_{i}}(g_{0},\dots,g_{m_{i}})=\mathcal{B}^{% m_{\ell}}f_{m_{i}}=g_{m_{\ell}}." class="ltx_Math" display="block" id="S5.Ex13.m1.2"><semantics id="S5.Ex13.m1.2a"><mrow id="S5.Ex13.m1.2.2.1" xref="S5.Ex13.m1.2.2.1.1.cmml"><mrow id="S5.Ex13.m1.2.2.1.1" xref="S5.Ex13.m1.2.2.1.1.cmml"><mrow id="S5.Ex13.m1.2.2.1.1.2" xref="S5.Ex13.m1.2.2.1.1.2.cmml"><msup id="S5.Ex13.m1.2.2.1.1.2.4" xref="S5.Ex13.m1.2.2.1.1.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex13.m1.2.2.1.1.2.4.2" xref="S5.Ex13.m1.2.2.1.1.2.4.2.cmml">ℬ</mi><msub id="S5.Ex13.m1.2.2.1.1.2.4.3" xref="S5.Ex13.m1.2.2.1.1.2.4.3.cmml"><mi id="S5.Ex13.m1.2.2.1.1.2.4.3.2" xref="S5.Ex13.m1.2.2.1.1.2.4.3.2.cmml">m</mi><mi id="S5.Ex13.m1.2.2.1.1.2.4.3.3" mathvariant="normal" xref="S5.Ex13.m1.2.2.1.1.2.4.3.3.cmml">ℓ</mi></msub></msup><mo id="S5.Ex13.m1.2.2.1.1.2.3" xref="S5.Ex13.m1.2.2.1.1.2.3.cmml">⁢</mo><msup id="S5.Ex13.m1.2.2.1.1.2.5" xref="S5.Ex13.m1.2.2.1.1.2.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex13.m1.2.2.1.1.2.5.2" xref="S5.Ex13.m1.2.2.1.1.2.5.2.cmml">𝒞</mi><msub id="S5.Ex13.m1.2.2.1.1.2.5.3" xref="S5.Ex13.m1.2.2.1.1.2.5.3.cmml"><mi id="S5.Ex13.m1.2.2.1.1.2.5.3.2" xref="S5.Ex13.m1.2.2.1.1.2.5.3.2.cmml">m</mi><mi id="S5.Ex13.m1.2.2.1.1.2.5.3.3" xref="S5.Ex13.m1.2.2.1.1.2.5.3.3.cmml">i</mi></msub></msup><mo id="S5.Ex13.m1.2.2.1.1.2.3a" xref="S5.Ex13.m1.2.2.1.1.2.3.cmml">⁢</mo><mrow id="S5.Ex13.m1.2.2.1.1.2.2.2" xref="S5.Ex13.m1.2.2.1.1.2.2.3.cmml"><mo id="S5.Ex13.m1.2.2.1.1.2.2.2.3" stretchy="false" xref="S5.Ex13.m1.2.2.1.1.2.2.3.cmml">(</mo><msub id="S5.Ex13.m1.2.2.1.1.1.1.1.1" xref="S5.Ex13.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S5.Ex13.m1.2.2.1.1.1.1.1.1.2" xref="S5.Ex13.m1.2.2.1.1.1.1.1.1.2.cmml">g</mi><mn id="S5.Ex13.m1.2.2.1.1.1.1.1.1.3" xref="S5.Ex13.m1.2.2.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S5.Ex13.m1.2.2.1.1.2.2.2.4" xref="S5.Ex13.m1.2.2.1.1.2.2.3.cmml">,</mo><mi id="S5.Ex13.m1.1.1" mathvariant="normal" xref="S5.Ex13.m1.1.1.cmml">…</mi><mo id="S5.Ex13.m1.2.2.1.1.2.2.2.5" xref="S5.Ex13.m1.2.2.1.1.2.2.3.cmml">,</mo><msub id="S5.Ex13.m1.2.2.1.1.2.2.2.2" xref="S5.Ex13.m1.2.2.1.1.2.2.2.2.cmml"><mi id="S5.Ex13.m1.2.2.1.1.2.2.2.2.2" xref="S5.Ex13.m1.2.2.1.1.2.2.2.2.2.cmml">g</mi><msub id="S5.Ex13.m1.2.2.1.1.2.2.2.2.3" xref="S5.Ex13.m1.2.2.1.1.2.2.2.2.3.cmml"><mi id="S5.Ex13.m1.2.2.1.1.2.2.2.2.3.2" xref="S5.Ex13.m1.2.2.1.1.2.2.2.2.3.2.cmml">m</mi><mi id="S5.Ex13.m1.2.2.1.1.2.2.2.2.3.3" xref="S5.Ex13.m1.2.2.1.1.2.2.2.2.3.3.cmml">i</mi></msub></msub><mo id="S5.Ex13.m1.2.2.1.1.2.2.2.6" stretchy="false" xref="S5.Ex13.m1.2.2.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex13.m1.2.2.1.1.4" xref="S5.Ex13.m1.2.2.1.1.4.cmml">=</mo><mrow id="S5.Ex13.m1.2.2.1.1.5" xref="S5.Ex13.m1.2.2.1.1.5.cmml"><msup id="S5.Ex13.m1.2.2.1.1.5.2" xref="S5.Ex13.m1.2.2.1.1.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.Ex13.m1.2.2.1.1.5.2.2" xref="S5.Ex13.m1.2.2.1.1.5.2.2.cmml">ℬ</mi><msub id="S5.Ex13.m1.2.2.1.1.5.2.3" xref="S5.Ex13.m1.2.2.1.1.5.2.3.cmml"><mi id="S5.Ex13.m1.2.2.1.1.5.2.3.2" xref="S5.Ex13.m1.2.2.1.1.5.2.3.2.cmml">m</mi><mi id="S5.Ex13.m1.2.2.1.1.5.2.3.3" mathvariant="normal" xref="S5.Ex13.m1.2.2.1.1.5.2.3.3.cmml">ℓ</mi></msub></msup><mo id="S5.Ex13.m1.2.2.1.1.5.1" xref="S5.Ex13.m1.2.2.1.1.5.1.cmml">⁢</mo><msub id="S5.Ex13.m1.2.2.1.1.5.3" xref="S5.Ex13.m1.2.2.1.1.5.3.cmml"><mi id="S5.Ex13.m1.2.2.1.1.5.3.2" xref="S5.Ex13.m1.2.2.1.1.5.3.2.cmml">f</mi><msub id="S5.Ex13.m1.2.2.1.1.5.3.3" xref="S5.Ex13.m1.2.2.1.1.5.3.3.cmml"><mi id="S5.Ex13.m1.2.2.1.1.5.3.3.2" xref="S5.Ex13.m1.2.2.1.1.5.3.3.2.cmml">m</mi><mi id="S5.Ex13.m1.2.2.1.1.5.3.3.3" xref="S5.Ex13.m1.2.2.1.1.5.3.3.3.cmml">i</mi></msub></msub></mrow><mo id="S5.Ex13.m1.2.2.1.1.6" xref="S5.Ex13.m1.2.2.1.1.6.cmml">=</mo><msub id="S5.Ex13.m1.2.2.1.1.7" xref="S5.Ex13.m1.2.2.1.1.7.cmml"><mi id="S5.Ex13.m1.2.2.1.1.7.2" xref="S5.Ex13.m1.2.2.1.1.7.2.cmml">g</mi><msub id="S5.Ex13.m1.2.2.1.1.7.3" 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xref="S5.Ex13.m1.2.2.1.1.2.4.2">ℬ</ci><apply id="S5.Ex13.m1.2.2.1.1.2.4.3.cmml" xref="S5.Ex13.m1.2.2.1.1.2.4.3"><csymbol cd="ambiguous" id="S5.Ex13.m1.2.2.1.1.2.4.3.1.cmml" xref="S5.Ex13.m1.2.2.1.1.2.4.3">subscript</csymbol><ci id="S5.Ex13.m1.2.2.1.1.2.4.3.2.cmml" xref="S5.Ex13.m1.2.2.1.1.2.4.3.2">𝑚</ci><ci id="S5.Ex13.m1.2.2.1.1.2.4.3.3.cmml" xref="S5.Ex13.m1.2.2.1.1.2.4.3.3">ℓ</ci></apply></apply><apply id="S5.Ex13.m1.2.2.1.1.2.5.cmml" xref="S5.Ex13.m1.2.2.1.1.2.5"><csymbol cd="ambiguous" id="S5.Ex13.m1.2.2.1.1.2.5.1.cmml" xref="S5.Ex13.m1.2.2.1.1.2.5">superscript</csymbol><ci id="S5.Ex13.m1.2.2.1.1.2.5.2.cmml" xref="S5.Ex13.m1.2.2.1.1.2.5.2">𝒞</ci><apply id="S5.Ex13.m1.2.2.1.1.2.5.3.cmml" xref="S5.Ex13.m1.2.2.1.1.2.5.3"><csymbol cd="ambiguous" id="S5.Ex13.m1.2.2.1.1.2.5.3.1.cmml" xref="S5.Ex13.m1.2.2.1.1.2.5.3">subscript</csymbol><ci id="S5.Ex13.m1.2.2.1.1.2.5.3.2.cmml" xref="S5.Ex13.m1.2.2.1.1.2.5.3.2">𝑚</ci><ci id="S5.Ex13.m1.2.2.1.1.2.5.3.3.cmml" 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xref="S5.Ex13.m1.2.2.1.1.5.2.3.3">ℓ</ci></apply></apply><apply id="S5.Ex13.m1.2.2.1.1.5.3.cmml" xref="S5.Ex13.m1.2.2.1.1.5.3"><csymbol cd="ambiguous" id="S5.Ex13.m1.2.2.1.1.5.3.1.cmml" xref="S5.Ex13.m1.2.2.1.1.5.3">subscript</csymbol><ci id="S5.Ex13.m1.2.2.1.1.5.3.2.cmml" xref="S5.Ex13.m1.2.2.1.1.5.3.2">𝑓</ci><apply id="S5.Ex13.m1.2.2.1.1.5.3.3.cmml" xref="S5.Ex13.m1.2.2.1.1.5.3.3"><csymbol cd="ambiguous" id="S5.Ex13.m1.2.2.1.1.5.3.3.1.cmml" xref="S5.Ex13.m1.2.2.1.1.5.3.3">subscript</csymbol><ci id="S5.Ex13.m1.2.2.1.1.5.3.3.2.cmml" xref="S5.Ex13.m1.2.2.1.1.5.3.3.2">𝑚</ci><ci id="S5.Ex13.m1.2.2.1.1.5.3.3.3.cmml" xref="S5.Ex13.m1.2.2.1.1.5.3.3.3">𝑖</ci></apply></apply></apply></apply><apply id="S5.Ex13.m1.2.2.1.1c.cmml" xref="S5.Ex13.m1.2.2.1"><eq id="S5.Ex13.m1.2.2.1.1.6.cmml" xref="S5.Ex13.m1.2.2.1.1.6"></eq><share href="https://arxiv.org/html/2503.14636v1#S5.Ex13.m1.2.2.1.1.5.cmml" id="S5.Ex13.m1.2.2.1.1d.cmml" xref="S5.Ex13.m1.2.2.1"></share><apply id="S5.Ex13.m1.2.2.1.1.7.cmml" xref="S5.Ex13.m1.2.2.1.1.7"><csymbol cd="ambiguous" id="S5.Ex13.m1.2.2.1.1.7.1.cmml" xref="S5.Ex13.m1.2.2.1.1.7">subscript</csymbol><ci id="S5.Ex13.m1.2.2.1.1.7.2.cmml" xref="S5.Ex13.m1.2.2.1.1.7.2">𝑔</ci><apply id="S5.Ex13.m1.2.2.1.1.7.3.cmml" xref="S5.Ex13.m1.2.2.1.1.7.3"><csymbol cd="ambiguous" id="S5.Ex13.m1.2.2.1.1.7.3.1.cmml" xref="S5.Ex13.m1.2.2.1.1.7.3">subscript</csymbol><ci id="S5.Ex13.m1.2.2.1.1.7.3.2.cmml" xref="S5.Ex13.m1.2.2.1.1.7.3.2">𝑚</ci><ci id="S5.Ex13.m1.2.2.1.1.7.3.3.cmml" xref="S5.Ex13.m1.2.2.1.1.7.3.3">ℓ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex13.m1.2c">\mathcal{B}^{m_{\ell}}\mathcal{C}^{m_{i}}(g_{0},\dots,g_{m_{i}})=\mathcal{B}^{% m_{\ell}}f_{m_{i}}=g_{m_{\ell}}.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex13.m1.2d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT caligraphic_C start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_g start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) = caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_g start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.5.p3.10">So indeed, <math alttext="\operatorname{ext}_{\mathcal{B}}:=\mathcal{C}^{m_{n}}" class="ltx_Math" display="inline" id="S5.5.p3.9.m1.1"><semantics id="S5.5.p3.9.m1.1a"><mrow id="S5.5.p3.9.m1.1.1" xref="S5.5.p3.9.m1.1.1.cmml"><msub id="S5.5.p3.9.m1.1.1.2" xref="S5.5.p3.9.m1.1.1.2.cmml"><mi id="S5.5.p3.9.m1.1.1.2.2" xref="S5.5.p3.9.m1.1.1.2.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S5.5.p3.9.m1.1.1.2.3" xref="S5.5.p3.9.m1.1.1.2.3.cmml">ℬ</mi></msub><mo id="S5.5.p3.9.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S5.5.p3.9.m1.1.1.1.cmml">:=</mo><msup id="S5.5.p3.9.m1.1.1.3" xref="S5.5.p3.9.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.5.p3.9.m1.1.1.3.2" xref="S5.5.p3.9.m1.1.1.3.2.cmml">𝒞</mi><msub id="S5.5.p3.9.m1.1.1.3.3" xref="S5.5.p3.9.m1.1.1.3.3.cmml"><mi id="S5.5.p3.9.m1.1.1.3.3.2" xref="S5.5.p3.9.m1.1.1.3.3.2.cmml">m</mi><mi id="S5.5.p3.9.m1.1.1.3.3.3" xref="S5.5.p3.9.m1.1.1.3.3.3.cmml">n</mi></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S5.5.p3.9.m1.1b"><apply id="S5.5.p3.9.m1.1.1.cmml" xref="S5.5.p3.9.m1.1.1"><csymbol cd="latexml" id="S5.5.p3.9.m1.1.1.1.cmml" xref="S5.5.p3.9.m1.1.1.1">assign</csymbol><apply id="S5.5.p3.9.m1.1.1.2.cmml" xref="S5.5.p3.9.m1.1.1.2"><csymbol cd="ambiguous" id="S5.5.p3.9.m1.1.1.2.1.cmml" xref="S5.5.p3.9.m1.1.1.2">subscript</csymbol><ci id="S5.5.p3.9.m1.1.1.2.2.cmml" xref="S5.5.p3.9.m1.1.1.2.2">ext</ci><ci id="S5.5.p3.9.m1.1.1.2.3.cmml" xref="S5.5.p3.9.m1.1.1.2.3">ℬ</ci></apply><apply id="S5.5.p3.9.m1.1.1.3.cmml" xref="S5.5.p3.9.m1.1.1.3"><csymbol cd="ambiguous" id="S5.5.p3.9.m1.1.1.3.1.cmml" xref="S5.5.p3.9.m1.1.1.3">superscript</csymbol><ci id="S5.5.p3.9.m1.1.1.3.2.cmml" xref="S5.5.p3.9.m1.1.1.3.2">𝒞</ci><apply id="S5.5.p3.9.m1.1.1.3.3.cmml" xref="S5.5.p3.9.m1.1.1.3.3"><csymbol cd="ambiguous" id="S5.5.p3.9.m1.1.1.3.3.1.cmml" xref="S5.5.p3.9.m1.1.1.3.3">subscript</csymbol><ci id="S5.5.p3.9.m1.1.1.3.3.2.cmml" xref="S5.5.p3.9.m1.1.1.3.3.2">𝑚</ci><ci id="S5.5.p3.9.m1.1.1.3.3.3.cmml" xref="S5.5.p3.9.m1.1.1.3.3.3">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.9.m1.1c">\operatorname{ext}_{\mathcal{B}}:=\mathcal{C}^{m_{n}}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.9.m1.1d">roman_ext start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT := caligraphic_C start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> is the right inverse for <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S5.5.p3.10.m2.1"><semantics id="S5.5.p3.10.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S5.5.p3.10.m2.1.1" xref="S5.5.p3.10.m2.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S5.5.p3.10.m2.1b"><ci id="S5.5.p3.10.m2.1.1.cmml" xref="S5.5.p3.10.m2.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S5.5.p3.10.m2.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S5.5.p3.10.m2.1d">caligraphic_B</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S5.6.p4"> <p class="ltx_p" id="S5.6.p4.3">Finally, if <math alttext="\ell\notin\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S5.6.p4.1.m1.3"><semantics id="S5.6.p4.1.m1.3a"><mrow id="S5.6.p4.1.m1.3.3" xref="S5.6.p4.1.m1.3.3.cmml"><mi id="S5.6.p4.1.m1.3.3.4" mathvariant="normal" xref="S5.6.p4.1.m1.3.3.4.cmml">ℓ</mi><mo id="S5.6.p4.1.m1.3.3.3" xref="S5.6.p4.1.m1.3.3.3.cmml">∉</mo><mrow id="S5.6.p4.1.m1.3.3.2.2" xref="S5.6.p4.1.m1.3.3.2.3.cmml"><mo id="S5.6.p4.1.m1.3.3.2.2.3" stretchy="false" xref="S5.6.p4.1.m1.3.3.2.3.cmml">{</mo><msub id="S5.6.p4.1.m1.2.2.1.1.1" xref="S5.6.p4.1.m1.2.2.1.1.1.cmml"><mi id="S5.6.p4.1.m1.2.2.1.1.1.2" xref="S5.6.p4.1.m1.2.2.1.1.1.2.cmml">m</mi><mn id="S5.6.p4.1.m1.2.2.1.1.1.3" xref="S5.6.p4.1.m1.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S5.6.p4.1.m1.3.3.2.2.4" xref="S5.6.p4.1.m1.3.3.2.3.cmml">,</mo><mi id="S5.6.p4.1.m1.1.1" mathvariant="normal" xref="S5.6.p4.1.m1.1.1.cmml">…</mi><mo id="S5.6.p4.1.m1.3.3.2.2.5" xref="S5.6.p4.1.m1.3.3.2.3.cmml">,</mo><msub id="S5.6.p4.1.m1.3.3.2.2.2" xref="S5.6.p4.1.m1.3.3.2.2.2.cmml"><mi id="S5.6.p4.1.m1.3.3.2.2.2.2" xref="S5.6.p4.1.m1.3.3.2.2.2.2.cmml">m</mi><mi id="S5.6.p4.1.m1.3.3.2.2.2.3" xref="S5.6.p4.1.m1.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S5.6.p4.1.m1.3.3.2.2.6" stretchy="false" xref="S5.6.p4.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p4.1.m1.3b"><apply id="S5.6.p4.1.m1.3.3.cmml" xref="S5.6.p4.1.m1.3.3"><notin id="S5.6.p4.1.m1.3.3.3.cmml" xref="S5.6.p4.1.m1.3.3.3"></notin><ci id="S5.6.p4.1.m1.3.3.4.cmml" xref="S5.6.p4.1.m1.3.3.4">ℓ</ci><set id="S5.6.p4.1.m1.3.3.2.3.cmml" xref="S5.6.p4.1.m1.3.3.2.2"><apply id="S5.6.p4.1.m1.2.2.1.1.1.cmml" xref="S5.6.p4.1.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S5.6.p4.1.m1.2.2.1.1.1.1.cmml" xref="S5.6.p4.1.m1.2.2.1.1.1">subscript</csymbol><ci id="S5.6.p4.1.m1.2.2.1.1.1.2.cmml" xref="S5.6.p4.1.m1.2.2.1.1.1.2">𝑚</ci><cn id="S5.6.p4.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S5.6.p4.1.m1.2.2.1.1.1.3">0</cn></apply><ci id="S5.6.p4.1.m1.1.1.cmml" xref="S5.6.p4.1.m1.1.1">…</ci><apply id="S5.6.p4.1.m1.3.3.2.2.2.cmml" xref="S5.6.p4.1.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S5.6.p4.1.m1.3.3.2.2.2.1.cmml" xref="S5.6.p4.1.m1.3.3.2.2.2">subscript</csymbol><ci id="S5.6.p4.1.m1.3.3.2.2.2.2.cmml" xref="S5.6.p4.1.m1.3.3.2.2.2.2">𝑚</ci><ci id="S5.6.p4.1.m1.3.3.2.2.2.3.cmml" xref="S5.6.p4.1.m1.3.3.2.2.2.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p4.1.m1.3c">\ell\notin\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S5.6.p4.1.m1.3d">roman_ℓ ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math>, then <math alttext="h_{\ell}=0" class="ltx_Math" display="inline" id="S5.6.p4.2.m2.1"><semantics id="S5.6.p4.2.m2.1a"><mrow id="S5.6.p4.2.m2.1.1" xref="S5.6.p4.2.m2.1.1.cmml"><msub id="S5.6.p4.2.m2.1.1.2" xref="S5.6.p4.2.m2.1.1.2.cmml"><mi id="S5.6.p4.2.m2.1.1.2.2" xref="S5.6.p4.2.m2.1.1.2.2.cmml">h</mi><mi id="S5.6.p4.2.m2.1.1.2.3" mathvariant="normal" xref="S5.6.p4.2.m2.1.1.2.3.cmml">ℓ</mi></msub><mo id="S5.6.p4.2.m2.1.1.1" xref="S5.6.p4.2.m2.1.1.1.cmml">=</mo><mn id="S5.6.p4.2.m2.1.1.3" xref="S5.6.p4.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p4.2.m2.1b"><apply id="S5.6.p4.2.m2.1.1.cmml" xref="S5.6.p4.2.m2.1.1"><eq id="S5.6.p4.2.m2.1.1.1.cmml" xref="S5.6.p4.2.m2.1.1.1"></eq><apply id="S5.6.p4.2.m2.1.1.2.cmml" xref="S5.6.p4.2.m2.1.1.2"><csymbol cd="ambiguous" id="S5.6.p4.2.m2.1.1.2.1.cmml" xref="S5.6.p4.2.m2.1.1.2">subscript</csymbol><ci id="S5.6.p4.2.m2.1.1.2.2.cmml" xref="S5.6.p4.2.m2.1.1.2.2">ℎ</ci><ci id="S5.6.p4.2.m2.1.1.2.3.cmml" xref="S5.6.p4.2.m2.1.1.2.3">ℓ</ci></apply><cn id="S5.6.p4.2.m2.1.1.3.cmml" type="integer" xref="S5.6.p4.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p4.2.m2.1c">h_{\ell}=0</annotation><annotation encoding="application/x-llamapun" id="S5.6.p4.2.m2.1d">italic_h start_POSTSUBSCRIPT roman_ℓ end_POSTSUBSCRIPT = 0</annotation></semantics></math> and <math alttext="\widetilde{\mathcal{C}}^{\ell}=0" class="ltx_Math" display="inline" id="S5.6.p4.3.m3.1"><semantics id="S5.6.p4.3.m3.1a"><mrow id="S5.6.p4.3.m3.1.1" xref="S5.6.p4.3.m3.1.1.cmml"><msup id="S5.6.p4.3.m3.1.1.2" xref="S5.6.p4.3.m3.1.1.2.cmml"><mover accent="true" id="S5.6.p4.3.m3.1.1.2.2" xref="S5.6.p4.3.m3.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.6.p4.3.m3.1.1.2.2.2" xref="S5.6.p4.3.m3.1.1.2.2.2.cmml">𝒞</mi><mo id="S5.6.p4.3.m3.1.1.2.2.1" xref="S5.6.p4.3.m3.1.1.2.2.1.cmml">~</mo></mover><mi id="S5.6.p4.3.m3.1.1.2.3" mathvariant="normal" xref="S5.6.p4.3.m3.1.1.2.3.cmml">ℓ</mi></msup><mo id="S5.6.p4.3.m3.1.1.1" xref="S5.6.p4.3.m3.1.1.1.cmml">=</mo><mn id="S5.6.p4.3.m3.1.1.3" xref="S5.6.p4.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S5.6.p4.3.m3.1b"><apply id="S5.6.p4.3.m3.1.1.cmml" xref="S5.6.p4.3.m3.1.1"><eq id="S5.6.p4.3.m3.1.1.1.cmml" xref="S5.6.p4.3.m3.1.1.1"></eq><apply id="S5.6.p4.3.m3.1.1.2.cmml" xref="S5.6.p4.3.m3.1.1.2"><csymbol cd="ambiguous" id="S5.6.p4.3.m3.1.1.2.1.cmml" xref="S5.6.p4.3.m3.1.1.2">superscript</csymbol><apply id="S5.6.p4.3.m3.1.1.2.2.cmml" xref="S5.6.p4.3.m3.1.1.2.2"><ci id="S5.6.p4.3.m3.1.1.2.2.1.cmml" xref="S5.6.p4.3.m3.1.1.2.2.1">~</ci><ci id="S5.6.p4.3.m3.1.1.2.2.2.cmml" xref="S5.6.p4.3.m3.1.1.2.2.2">𝒞</ci></apply><ci id="S5.6.p4.3.m3.1.1.2.3.cmml" xref="S5.6.p4.3.m3.1.1.2.3">ℓ</ci></apply><cn id="S5.6.p4.3.m3.1.1.3.cmml" type="integer" xref="S5.6.p4.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.6.p4.3.m3.1c">\widetilde{\mathcal{C}}^{\ell}=0</annotation><annotation encoding="application/x-llamapun" id="S5.6.p4.3.m3.1d">over~ start_ARG caligraphic_C end_ARG start_POSTSUPERSCRIPT roman_ℓ end_POSTSUPERSCRIPT = 0</annotation></semantics></math>, so that (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E8" title="In Proof. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.8</span></a>) implies</p> <table class="ltx_equation ltx_eqn_table" id="S5.Ex14"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{j}(\operatorname{ext}_{\mathcal{B}}g)=\operatorname{Tr}_{j}% f_{m_{n}}=0." class="ltx_Math" display="block" id="S5.Ex14.m1.1"><semantics id="S5.Ex14.m1.1a"><mrow id="S5.Ex14.m1.1.1.1" xref="S5.Ex14.m1.1.1.1.1.cmml"><mrow id="S5.Ex14.m1.1.1.1.1" xref="S5.Ex14.m1.1.1.1.1.cmml"><mrow id="S5.Ex14.m1.1.1.1.1.2.2" xref="S5.Ex14.m1.1.1.1.1.2.3.cmml"><msub id="S5.Ex14.m1.1.1.1.1.1.1.1" xref="S5.Ex14.m1.1.1.1.1.1.1.1.cmml"><mi id="S5.Ex14.m1.1.1.1.1.1.1.1.2" xref="S5.Ex14.m1.1.1.1.1.1.1.1.2.cmml">Tr</mi><mi id="S5.Ex14.m1.1.1.1.1.1.1.1.3" xref="S5.Ex14.m1.1.1.1.1.1.1.1.3.cmml">j</mi></msub><mo id="S5.Ex14.m1.1.1.1.1.2.2a" xref="S5.Ex14.m1.1.1.1.1.2.3.cmml">⁡</mo><mrow id="S5.Ex14.m1.1.1.1.1.2.2.2" xref="S5.Ex14.m1.1.1.1.1.2.3.cmml"><mo id="S5.Ex14.m1.1.1.1.1.2.2.2.2" stretchy="false" xref="S5.Ex14.m1.1.1.1.1.2.3.cmml">(</mo><mrow id="S5.Ex14.m1.1.1.1.1.2.2.2.1" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.cmml"><msub id="S5.Ex14.m1.1.1.1.1.2.2.2.1.1" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.cmml"><mi id="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.2" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.3" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.3.cmml">ℬ</mi></msub><mo id="S5.Ex14.m1.1.1.1.1.2.2.2.1a" lspace="0.167em" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.cmml">⁡</mo><mi id="S5.Ex14.m1.1.1.1.1.2.2.2.1.2" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.2.cmml">g</mi></mrow><mo id="S5.Ex14.m1.1.1.1.1.2.2.2.3" stretchy="false" xref="S5.Ex14.m1.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S5.Ex14.m1.1.1.1.1.4" xref="S5.Ex14.m1.1.1.1.1.4.cmml">=</mo><mrow id="S5.Ex14.m1.1.1.1.1.5" xref="S5.Ex14.m1.1.1.1.1.5.cmml"><msub id="S5.Ex14.m1.1.1.1.1.5.1" xref="S5.Ex14.m1.1.1.1.1.5.1.cmml"><mi id="S5.Ex14.m1.1.1.1.1.5.1.2" xref="S5.Ex14.m1.1.1.1.1.5.1.2.cmml">Tr</mi><mi id="S5.Ex14.m1.1.1.1.1.5.1.3" xref="S5.Ex14.m1.1.1.1.1.5.1.3.cmml">j</mi></msub><mo id="S5.Ex14.m1.1.1.1.1.5a" lspace="0.167em" xref="S5.Ex14.m1.1.1.1.1.5.cmml">⁡</mo><msub id="S5.Ex14.m1.1.1.1.1.5.2" xref="S5.Ex14.m1.1.1.1.1.5.2.cmml"><mi id="S5.Ex14.m1.1.1.1.1.5.2.2" xref="S5.Ex14.m1.1.1.1.1.5.2.2.cmml">f</mi><msub id="S5.Ex14.m1.1.1.1.1.5.2.3" xref="S5.Ex14.m1.1.1.1.1.5.2.3.cmml"><mi id="S5.Ex14.m1.1.1.1.1.5.2.3.2" xref="S5.Ex14.m1.1.1.1.1.5.2.3.2.cmml">m</mi><mi id="S5.Ex14.m1.1.1.1.1.5.2.3.3" xref="S5.Ex14.m1.1.1.1.1.5.2.3.3.cmml">n</mi></msub></msub></mrow><mo id="S5.Ex14.m1.1.1.1.1.6" xref="S5.Ex14.m1.1.1.1.1.6.cmml">=</mo><mn id="S5.Ex14.m1.1.1.1.1.7" xref="S5.Ex14.m1.1.1.1.1.7.cmml">0</mn></mrow><mo id="S5.Ex14.m1.1.1.1.2" lspace="0em" xref="S5.Ex14.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S5.Ex14.m1.1b"><apply id="S5.Ex14.m1.1.1.1.1.cmml" xref="S5.Ex14.m1.1.1.1"><and id="S5.Ex14.m1.1.1.1.1a.cmml" xref="S5.Ex14.m1.1.1.1"></and><apply id="S5.Ex14.m1.1.1.1.1b.cmml" xref="S5.Ex14.m1.1.1.1"><eq id="S5.Ex14.m1.1.1.1.1.4.cmml" xref="S5.Ex14.m1.1.1.1.1.4"></eq><apply id="S5.Ex14.m1.1.1.1.1.2.3.cmml" xref="S5.Ex14.m1.1.1.1.1.2.2"><apply id="S5.Ex14.m1.1.1.1.1.1.1.1.cmml" xref="S5.Ex14.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S5.Ex14.m1.1.1.1.1.1.1.1.1.cmml" xref="S5.Ex14.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="S5.Ex14.m1.1.1.1.1.1.1.1.2.cmml" xref="S5.Ex14.m1.1.1.1.1.1.1.1.2">Tr</ci><ci id="S5.Ex14.m1.1.1.1.1.1.1.1.3.cmml" xref="S5.Ex14.m1.1.1.1.1.1.1.1.3">𝑗</ci></apply><apply id="S5.Ex14.m1.1.1.1.1.2.2.2.1.cmml" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1"><apply id="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.cmml" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.1"><csymbol cd="ambiguous" id="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.1.cmml" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.1">subscript</csymbol><ci id="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.2.cmml" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.2">ext</ci><ci id="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.3.cmml" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.1.3">ℬ</ci></apply><ci id="S5.Ex14.m1.1.1.1.1.2.2.2.1.2.cmml" xref="S5.Ex14.m1.1.1.1.1.2.2.2.1.2">𝑔</ci></apply></apply><apply id="S5.Ex14.m1.1.1.1.1.5.cmml" xref="S5.Ex14.m1.1.1.1.1.5"><apply id="S5.Ex14.m1.1.1.1.1.5.1.cmml" xref="S5.Ex14.m1.1.1.1.1.5.1"><csymbol cd="ambiguous" id="S5.Ex14.m1.1.1.1.1.5.1.1.cmml" xref="S5.Ex14.m1.1.1.1.1.5.1">subscript</csymbol><ci id="S5.Ex14.m1.1.1.1.1.5.1.2.cmml" xref="S5.Ex14.m1.1.1.1.1.5.1.2">Tr</ci><ci id="S5.Ex14.m1.1.1.1.1.5.1.3.cmml" xref="S5.Ex14.m1.1.1.1.1.5.1.3">𝑗</ci></apply><apply id="S5.Ex14.m1.1.1.1.1.5.2.cmml" xref="S5.Ex14.m1.1.1.1.1.5.2"><csymbol cd="ambiguous" id="S5.Ex14.m1.1.1.1.1.5.2.1.cmml" xref="S5.Ex14.m1.1.1.1.1.5.2">subscript</csymbol><ci id="S5.Ex14.m1.1.1.1.1.5.2.2.cmml" xref="S5.Ex14.m1.1.1.1.1.5.2.2">𝑓</ci><apply id="S5.Ex14.m1.1.1.1.1.5.2.3.cmml" xref="S5.Ex14.m1.1.1.1.1.5.2.3"><csymbol cd="ambiguous" id="S5.Ex14.m1.1.1.1.1.5.2.3.1.cmml" xref="S5.Ex14.m1.1.1.1.1.5.2.3">subscript</csymbol><ci id="S5.Ex14.m1.1.1.1.1.5.2.3.2.cmml" xref="S5.Ex14.m1.1.1.1.1.5.2.3.2">𝑚</ci><ci id="S5.Ex14.m1.1.1.1.1.5.2.3.3.cmml" xref="S5.Ex14.m1.1.1.1.1.5.2.3.3">𝑛</ci></apply></apply></apply></apply><apply id="S5.Ex14.m1.1.1.1.1c.cmml" xref="S5.Ex14.m1.1.1.1"><eq id="S5.Ex14.m1.1.1.1.1.6.cmml" xref="S5.Ex14.m1.1.1.1.1.6"></eq><share href="https://arxiv.org/html/2503.14636v1#S5.Ex14.m1.1.1.1.1.5.cmml" id="S5.Ex14.m1.1.1.1.1d.cmml" xref="S5.Ex14.m1.1.1.1"></share><cn id="S5.Ex14.m1.1.1.1.1.7.cmml" type="integer" xref="S5.Ex14.m1.1.1.1.1.7">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.Ex14.m1.1c">\operatorname{Tr}_{j}(\operatorname{ext}_{\mathcal{B}}g)=\operatorname{Tr}_{j}% f_{m_{n}}=0.</annotation><annotation encoding="application/x-llamapun" id="S5.Ex14.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( roman_ext start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT italic_g ) = roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S5.6.p4.4">This completes the proof. ∎</p> </div> </div> </section> <section class="ltx_section" id="S6"> <h2 class="ltx_title ltx_title_section"> <span class="ltx_tag ltx_tag_section">6. </span>Complex interpolation</h2> <div class="ltx_para" id="S6.p1"> <p class="ltx_p" id="S6.p1.1">In this final section, we prove the characterisations for the complex interpolation spaces of weighted Bessel potential and Sobolev spaces. In Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS1" title="6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.1</span></a> we start with some preliminary interpolation results on the half-space required for our main results which are stated in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS2" title="6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a>. The proofs of the main results are given in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS3" title="6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a>.</p> </div> <section class="ltx_subsection" id="S6.SS1"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.1. </span>Interpolation results for weighted spaces</h3> <div class="ltx_para" id="S6.SS1.p1"> <p class="ltx_p" id="S6.SS1.p1.1">We start with an interpolation result for the weighted Bessel potential spaces <math alttext="H_{0}^{s,p}" class="ltx_Math" display="inline" id="S6.SS1.p1.1.m1.2"><semantics id="S6.SS1.p1.1.m1.2a"><msubsup id="S6.SS1.p1.1.m1.2.3" xref="S6.SS1.p1.1.m1.2.3.cmml"><mi id="S6.SS1.p1.1.m1.2.3.2.2" xref="S6.SS1.p1.1.m1.2.3.2.2.cmml">H</mi><mn id="S6.SS1.p1.1.m1.2.3.2.3" xref="S6.SS1.p1.1.m1.2.3.2.3.cmml">0</mn><mrow id="S6.SS1.p1.1.m1.2.2.2.4" xref="S6.SS1.p1.1.m1.2.2.2.3.cmml"><mi id="S6.SS1.p1.1.m1.1.1.1.1" xref="S6.SS1.p1.1.m1.1.1.1.1.cmml">s</mi><mo id="S6.SS1.p1.1.m1.2.2.2.4.1" xref="S6.SS1.p1.1.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS1.p1.1.m1.2.2.2.2" xref="S6.SS1.p1.1.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S6.SS1.p1.1.m1.2b"><apply id="S6.SS1.p1.1.m1.2.3.cmml" xref="S6.SS1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S6.SS1.p1.1.m1.2.3.1.cmml" xref="S6.SS1.p1.1.m1.2.3">superscript</csymbol><apply id="S6.SS1.p1.1.m1.2.3.2.cmml" xref="S6.SS1.p1.1.m1.2.3"><csymbol cd="ambiguous" id="S6.SS1.p1.1.m1.2.3.2.1.cmml" xref="S6.SS1.p1.1.m1.2.3">subscript</csymbol><ci id="S6.SS1.p1.1.m1.2.3.2.2.cmml" xref="S6.SS1.p1.1.m1.2.3.2.2">𝐻</ci><cn id="S6.SS1.p1.1.m1.2.3.2.3.cmml" type="integer" xref="S6.SS1.p1.1.m1.2.3.2.3">0</cn></apply><list id="S6.SS1.p1.1.m1.2.2.2.3.cmml" xref="S6.SS1.p1.1.m1.2.2.2.4"><ci id="S6.SS1.p1.1.m1.1.1.1.1.cmml" xref="S6.SS1.p1.1.m1.1.1.1.1">𝑠</ci><ci id="S6.SS1.p1.1.m1.2.2.2.2.cmml" xref="S6.SS1.p1.1.m1.2.2.2.2">𝑝</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p1.1.m1.2c">H_{0}^{s,p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p1.1.m1.2d">italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>, see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS2.SSS1" title="4.2.1. Bessel potential spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.2.1</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem1"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.1.1.1">Proposition 6.1</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem1.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem1.p1"> <p class="ltx_p" id="S6.Thmtheorem1.p1.9"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem1.p1.9.9">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.1.1.m1.2"><semantics id="S6.Thmtheorem1.p1.1.1.m1.2a"><mrow id="S6.Thmtheorem1.p1.1.1.m1.2.3" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.cmml"><mi id="S6.Thmtheorem1.p1.1.1.m1.2.3.2" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S6.Thmtheorem1.p1.1.1.m1.2.3.1" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S6.Thmtheorem1.p1.1.1.m1.2.3.3.2" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml"><mo id="S6.Thmtheorem1.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S6.Thmtheorem1.p1.1.1.m1.1.1" xref="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml">1</mn><mo id="S6.Thmtheorem1.p1.1.1.m1.2.3.3.2.2" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem1.p1.1.1.m1.2.2" mathvariant="normal" xref="S6.Thmtheorem1.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S6.Thmtheorem1.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.1.1.m1.2b"><apply id="S6.Thmtheorem1.p1.1.1.m1.2.3.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.2.3"><in id="S6.Thmtheorem1.p1.1.1.m1.2.3.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.1"></in><ci id="S6.Thmtheorem1.p1.1.1.m1.2.3.2.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S6.Thmtheorem1.p1.1.1.m1.2.3.3.1.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.2.3.3.2"><cn id="S6.Thmtheorem1.p1.1.1.m1.1.1.cmml" type="integer" xref="S6.Thmtheorem1.p1.1.1.m1.1.1">1</cn><infinity id="S6.Thmtheorem1.p1.1.1.m1.2.2.cmml" xref="S6.Thmtheorem1.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.2.2.m2.2"><semantics id="S6.Thmtheorem1.p1.2.2.m2.2a"><mrow id="S6.Thmtheorem1.p1.2.2.m2.2.2" xref="S6.Thmtheorem1.p1.2.2.m2.2.2.cmml"><mi id="S6.Thmtheorem1.p1.2.2.m2.2.2.4" xref="S6.Thmtheorem1.p1.2.2.m2.2.2.4.cmml">γ</mi><mo id="S6.Thmtheorem1.p1.2.2.m2.2.2.3" 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id="S6.Thmtheorem1.p1.3.3.m3.2.2.4" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.4.cmml">𝒪</mi><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.3" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.3.cmml">∈</mo><mrow id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml"><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">{</mo><msup id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml">ℝ</mi><mi id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml">d</mi></msup><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.4" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">,</mo><msubsup id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.cmml"><mi id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.2" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.2.cmml">ℝ</mi><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3.cmml">+</mo><mi id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.3" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.3.cmml">d</mi></msubsup><mo id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.5" stretchy="false" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.3.3.m3.2b"><apply id="S6.Thmtheorem1.p1.3.3.m3.2.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2"><in id="S6.Thmtheorem1.p1.3.3.m3.2.2.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.3"></in><ci id="S6.Thmtheorem1.p1.3.3.m3.2.2.4.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.4">𝒪</ci><set id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2"><apply id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1">superscript</csymbol><ci id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.2">ℝ</ci><ci id="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.1.1.1.1.1.3">𝑑</ci></apply><apply id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2">subscript</csymbol><apply id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2">superscript</csymbol><ci id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.2">ℝ</ci><ci id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.2.3">𝑑</ci></apply><plus id="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.3.3.m3.2.2.2.2.2.3"></plus></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.3.3.m3.2c">\mathcal{O}\in\{\mathbb{R}^{d},\mathbb{R}^{d}_{+}\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.3.3.m3.2d">caligraphic_O ∈ { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.4.4.m4.1"><semantics id="S6.Thmtheorem1.p1.4.4.m4.1a"><mi id="S6.Thmtheorem1.p1.4.4.m4.1.1" xref="S6.Thmtheorem1.p1.4.4.m4.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.4.4.m4.1b"><ci id="S6.Thmtheorem1.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem1.p1.4.4.m4.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.4.4.m4.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.4.4.m4.1d">italic_X</annotation></semantics></math> be a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.5.5.m5.1"><semantics id="S6.Thmtheorem1.p1.5.5.m5.1a"><mi id="S6.Thmtheorem1.p1.5.5.m5.1.1" xref="S6.Thmtheorem1.p1.5.5.m5.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.5.5.m5.1b"><ci id="S6.Thmtheorem1.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem1.p1.5.5.m5.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.5.5.m5.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.5.5.m5.1d">roman_UMD</annotation></semantics></math> Banach space. Assume that <math alttext="\theta\in(0,1)" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.6.6.m6.2"><semantics id="S6.Thmtheorem1.p1.6.6.m6.2a"><mrow id="S6.Thmtheorem1.p1.6.6.m6.2.3" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.cmml"><mi id="S6.Thmtheorem1.p1.6.6.m6.2.3.2" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.2.cmml">θ</mi><mo id="S6.Thmtheorem1.p1.6.6.m6.2.3.1" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.1.cmml">∈</mo><mrow id="S6.Thmtheorem1.p1.6.6.m6.2.3.3.2" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml"><mo id="S6.Thmtheorem1.p1.6.6.m6.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml">(</mo><mn id="S6.Thmtheorem1.p1.6.6.m6.1.1" xref="S6.Thmtheorem1.p1.6.6.m6.1.1.cmml">0</mn><mo id="S6.Thmtheorem1.p1.6.6.m6.2.3.3.2.2" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml">,</mo><mn id="S6.Thmtheorem1.p1.6.6.m6.2.2" xref="S6.Thmtheorem1.p1.6.6.m6.2.2.cmml">1</mn><mo id="S6.Thmtheorem1.p1.6.6.m6.2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.6.6.m6.2b"><apply id="S6.Thmtheorem1.p1.6.6.m6.2.3.cmml" xref="S6.Thmtheorem1.p1.6.6.m6.2.3"><in id="S6.Thmtheorem1.p1.6.6.m6.2.3.1.cmml" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.1"></in><ci id="S6.Thmtheorem1.p1.6.6.m6.2.3.2.cmml" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.2">𝜃</ci><interval closure="open" id="S6.Thmtheorem1.p1.6.6.m6.2.3.3.1.cmml" xref="S6.Thmtheorem1.p1.6.6.m6.2.3.3.2"><cn id="S6.Thmtheorem1.p1.6.6.m6.1.1.cmml" type="integer" xref="S6.Thmtheorem1.p1.6.6.m6.1.1">0</cn><cn id="S6.Thmtheorem1.p1.6.6.m6.2.2.cmml" type="integer" xref="S6.Thmtheorem1.p1.6.6.m6.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.6.6.m6.2c">\theta\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.6.6.m6.2d">italic_θ ∈ ( 0 , 1 )</annotation></semantics></math>, <math alttext="-1+\frac{\gamma+1}{p}<s_{0}<s_{\theta}<s_{1}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.7.7.m7.1"><semantics id="S6.Thmtheorem1.p1.7.7.m7.1a"><mrow id="S6.Thmtheorem1.p1.7.7.m7.1.1" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.cmml"><mrow id="S6.Thmtheorem1.p1.7.7.m7.1.1.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.cmml"><mrow id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.cmml"><mo id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2a" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.cmml">−</mo><mn id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.2.cmml">1</mn></mrow><mo id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.1" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.1.cmml">+</mo><mfrac id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.cmml"><mrow id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.cmml"><mi id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.2.cmml">γ</mi><mo id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.1" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.1.cmml">+</mo><mn id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.3" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.3" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.Thmtheorem1.p1.7.7.m7.1.1.3" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.3.cmml"><</mo><msub id="S6.Thmtheorem1.p1.7.7.m7.1.1.4" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.4.cmml"><mi id="S6.Thmtheorem1.p1.7.7.m7.1.1.4.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.4.2.cmml">s</mi><mn id="S6.Thmtheorem1.p1.7.7.m7.1.1.4.3" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.4.3.cmml">0</mn></msub><mo id="S6.Thmtheorem1.p1.7.7.m7.1.1.5" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.5.cmml"><</mo><msub id="S6.Thmtheorem1.p1.7.7.m7.1.1.6" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.6.cmml"><mi id="S6.Thmtheorem1.p1.7.7.m7.1.1.6.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.6.2.cmml">s</mi><mi id="S6.Thmtheorem1.p1.7.7.m7.1.1.6.3" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.6.3.cmml">θ</mi></msub><mo id="S6.Thmtheorem1.p1.7.7.m7.1.1.7" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.7.cmml"><</mo><msub id="S6.Thmtheorem1.p1.7.7.m7.1.1.8" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.8.cmml"><mi id="S6.Thmtheorem1.p1.7.7.m7.1.1.8.2" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.8.2.cmml">s</mi><mn id="S6.Thmtheorem1.p1.7.7.m7.1.1.8.3" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.8.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.7.7.m7.1b"><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1"><and id="S6.Thmtheorem1.p1.7.7.m7.1.1a.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1"></and><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1b.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1"><lt id="S6.Thmtheorem1.p1.7.7.m7.1.1.3.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.3"></lt><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2"><plus id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.1"></plus><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2"><minus id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2"></minus><cn id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.2.cmml" type="integer" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.2.2">1</cn></apply><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3"><divide id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3"></divide><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2"><plus id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.1"></plus><ci id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.2.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.2">𝛾</ci><cn id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.2.3">1</cn></apply><ci id="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.3.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.2.3.3">𝑝</ci></apply></apply><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.4.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.4"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.7.7.m7.1.1.4.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.4">subscript</csymbol><ci id="S6.Thmtheorem1.p1.7.7.m7.1.1.4.2.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.4.2">𝑠</ci><cn id="S6.Thmtheorem1.p1.7.7.m7.1.1.4.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.4.3">0</cn></apply></apply><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1c.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1"><lt id="S6.Thmtheorem1.p1.7.7.m7.1.1.5.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem1.p1.7.7.m7.1.1.4.cmml" id="S6.Thmtheorem1.p1.7.7.m7.1.1d.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1"></share><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.6.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.6"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.7.7.m7.1.1.6.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.6">subscript</csymbol><ci id="S6.Thmtheorem1.p1.7.7.m7.1.1.6.2.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.6.2">𝑠</ci><ci id="S6.Thmtheorem1.p1.7.7.m7.1.1.6.3.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.6.3">𝜃</ci></apply></apply><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1e.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1"><lt id="S6.Thmtheorem1.p1.7.7.m7.1.1.7.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.7"></lt><share href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem1.p1.7.7.m7.1.1.6.cmml" id="S6.Thmtheorem1.p1.7.7.m7.1.1f.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1"></share><apply id="S6.Thmtheorem1.p1.7.7.m7.1.1.8.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.8"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.7.7.m7.1.1.8.1.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.8">subscript</csymbol><ci id="S6.Thmtheorem1.p1.7.7.m7.1.1.8.2.cmml" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.8.2">𝑠</ci><cn id="S6.Thmtheorem1.p1.7.7.m7.1.1.8.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.7.7.m7.1.1.8.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.7.7.m7.1c">-1+\frac{\gamma+1}{p}<s_{0}<s_{\theta}<s_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.7.7.m7.1d">- 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="s_{\theta}=(1-\theta)s_{0}+\theta s_{1}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.8.8.m8.1"><semantics id="S6.Thmtheorem1.p1.8.8.m8.1a"><mrow id="S6.Thmtheorem1.p1.8.8.m8.1.1" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.cmml"><msub id="S6.Thmtheorem1.p1.8.8.m8.1.1.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.3.cmml"><mi id="S6.Thmtheorem1.p1.8.8.m8.1.1.3.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.3.2.cmml">s</mi><mi id="S6.Thmtheorem1.p1.8.8.m8.1.1.3.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.3.3.cmml">θ</mi></msub><mo id="S6.Thmtheorem1.p1.8.8.m8.1.1.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.2.cmml">=</mo><mrow id="S6.Thmtheorem1.p1.8.8.m8.1.1.1" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.cmml"><mrow id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.cmml"><mrow id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.cmml"><mn id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.1" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.3.cmml">θ</mi></mrow><mo id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.2.cmml">⁢</mo><msub id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.2.cmml">s</mi><mn id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.2.cmml">+</mo><mrow id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.cmml"><mi id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.2.cmml">θ</mi><mo id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.1" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.1.cmml">⁢</mo><msub id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.cmml"><mi id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.2" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.2.cmml">s</mi><mn id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.3" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.8.8.m8.1b"><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1"><eq id="S6.Thmtheorem1.p1.8.8.m8.1.1.2.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.2"></eq><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.3.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.8.8.m8.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.8.8.m8.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.3.2">𝑠</ci><ci id="S6.Thmtheorem1.p1.8.8.m8.1.1.3.3.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.3.3">𝜃</ci></apply><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1"><plus id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.2"></plus><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1"><times id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.2"></times><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1"><minus id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.1"></minus><cn id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.2">1</cn><ci id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.1.1.1.3">𝜃</ci></apply><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.2">𝑠</ci><cn id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.1.3.3">0</cn></apply></apply><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3"><times id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.1"></times><ci id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.2.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.2">𝜃</ci><apply id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.1.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.2.cmml" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.2">𝑠</ci><cn id="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.8.8.m8.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.8.8.m8.1c">s_{\theta}=(1-\theta)s_{0}+\theta s_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.8.8.m8.1d">italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = ( 1 - italic_θ ) italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_θ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> satisfy <math alttext="s_{0},s_{\theta},s_{1}\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.Thmtheorem1.p1.9.9.m9.3"><semantics id="S6.Thmtheorem1.p1.9.9.m9.3a"><mrow id="S6.Thmtheorem1.p1.9.9.m9.3.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.cmml"><mrow id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.4.cmml"><msub id="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1" xref="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.2" xref="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.2.cmml">s</mi><mn id="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.3" xref="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.4" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.4.cmml">,</mo><msub id="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2" xref="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.cmml"><mi id="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.2" xref="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.2.cmml">s</mi><mi id="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.3" xref="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.3.cmml">θ</mi></msub><mo id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.5" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.4.cmml">,</mo><msub id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.cmml"><mi id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.2" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.2.cmml">s</mi><mn id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.3.cmml">1</mn></msub></mrow><mo id="S6.Thmtheorem1.p1.9.9.m9.3.3.4" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.4.cmml">∉</mo><mrow id="S6.Thmtheorem1.p1.9.9.m9.3.3.5" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.cmml"><msub id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.2" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.2.cmml"><mi id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.2.2" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.2.2.cmml">ℕ</mi><mn id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.2.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.2.3.cmml">0</mn></msub><mo id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.1" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.1.cmml">+</mo><mfrac id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.cmml"><mrow id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.cmml"><mi id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.2" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.2.cmml">γ</mi><mo id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.1" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.1.cmml">+</mo><mn id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.3.cmml">1</mn></mrow><mi id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.3" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem1.p1.9.9.m9.3b"><apply id="S6.Thmtheorem1.p1.9.9.m9.3.3.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3"><notin id="S6.Thmtheorem1.p1.9.9.m9.3.3.4.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.4"></notin><list id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.4.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3"><apply id="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1">subscript</csymbol><ci id="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.2.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.2">𝑠</ci><cn id="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.9.9.m9.1.1.1.1.1.3">0</cn></apply><apply id="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2">subscript</csymbol><ci id="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.2">𝑠</ci><ci id="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.3.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.2.2.2.2.2.3">𝜃</ci></apply><apply id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.1.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3">subscript</csymbol><ci id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.2.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.2">𝑠</ci><cn id="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.3.3.3.3">1</cn></apply></list><apply 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xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.1"></plus><ci id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.2.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.2">𝛾</ci><cn id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.3.cmml" type="integer" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.2.3">1</cn></apply><ci id="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.3.cmml" xref="S6.Thmtheorem1.p1.9.9.m9.3.3.5.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem1.p1.9.9.m9.3c">s_{0},s_{\theta},s_{1}\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem1.p1.9.9.m9.3d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∉ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{s_{\theta},p}_{0}(\mathcal{O},w_{\gamma};X)=\big{[}H^{s_{0},p}_{0}(\mathcal% {O},w_{\gamma};X),H^{s_{1},p}_{0}(\mathcal{O},w_{\gamma};X)\big{]}_{\theta}." class="ltx_Math" display="block" id="S6.Ex1.m1.13"><semantics id="S6.Ex1.m1.13a"><mrow id="S6.Ex1.m1.13.13.1" xref="S6.Ex1.m1.13.13.1.1.cmml"><mrow id="S6.Ex1.m1.13.13.1.1" xref="S6.Ex1.m1.13.13.1.1.cmml"><mrow id="S6.Ex1.m1.13.13.1.1.1" xref="S6.Ex1.m1.13.13.1.1.1.cmml"><msubsup id="S6.Ex1.m1.13.13.1.1.1.3" xref="S6.Ex1.m1.13.13.1.1.1.3.cmml"><mi id="S6.Ex1.m1.13.13.1.1.1.3.2.2" xref="S6.Ex1.m1.13.13.1.1.1.3.2.2.cmml">H</mi><mn id="S6.Ex1.m1.13.13.1.1.1.3.3" xref="S6.Ex1.m1.13.13.1.1.1.3.3.cmml">0</mn><mrow id="S6.Ex1.m1.2.2.2.2" xref="S6.Ex1.m1.2.2.2.3.cmml"><msub 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id="S6.Ex1.m1.13c">H^{s_{\theta},p}_{0}(\mathcal{O},w_{\gamma};X)=\big{[}H^{s_{0},p}_{0}(\mathcal% {O},w_{\gamma};X),H^{s_{1},p}_{0}(\mathcal{O},w_{\gamma};X)\big{]}_{\theta}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex1.m1.13d">italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( caligraphic_O , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S6.SS1.4"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS1.1.p1"> <p class="ltx_p" id="S6.SS1.1.p1.4"><span class="ltx_text ltx_font_italic" id="S6.SS1.1.p1.1.1">Step 1: the case <math alttext="\mathcal{O}=\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S6.SS1.1.p1.1.1.m1.1"><semantics id="S6.SS1.1.p1.1.1.m1.1a"><mrow id="S6.SS1.1.p1.1.1.m1.1.1" xref="S6.SS1.1.p1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS1.1.p1.1.1.m1.1.1.2" xref="S6.SS1.1.p1.1.1.m1.1.1.2.cmml">𝒪</mi><mo id="S6.SS1.1.p1.1.1.m1.1.1.1" xref="S6.SS1.1.p1.1.1.m1.1.1.1.cmml">=</mo><msup id="S6.SS1.1.p1.1.1.m1.1.1.3" xref="S6.SS1.1.p1.1.1.m1.1.1.3.cmml"><mi id="S6.SS1.1.p1.1.1.m1.1.1.3.2" xref="S6.SS1.1.p1.1.1.m1.1.1.3.2.cmml">ℝ</mi><mi id="S6.SS1.1.p1.1.1.m1.1.1.3.3" xref="S6.SS1.1.p1.1.1.m1.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.1.1.m1.1b"><apply id="S6.SS1.1.p1.1.1.m1.1.1.cmml" xref="S6.SS1.1.p1.1.1.m1.1.1"><eq id="S6.SS1.1.p1.1.1.m1.1.1.1.cmml" xref="S6.SS1.1.p1.1.1.m1.1.1.1"></eq><ci id="S6.SS1.1.p1.1.1.m1.1.1.2.cmml" xref="S6.SS1.1.p1.1.1.m1.1.1.2">𝒪</ci><apply id="S6.SS1.1.p1.1.1.m1.1.1.3.cmml" xref="S6.SS1.1.p1.1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.1.1.m1.1.1.3.1.cmml" xref="S6.SS1.1.p1.1.1.m1.1.1.3">superscript</csymbol><ci id="S6.SS1.1.p1.1.1.m1.1.1.3.2.cmml" xref="S6.SS1.1.p1.1.1.m1.1.1.3.2">ℝ</ci><ci id="S6.SS1.1.p1.1.1.m1.1.1.3.3.cmml" xref="S6.SS1.1.p1.1.1.m1.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.1.1.m1.1c">\mathcal{O}=\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.1.1.m1.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>. </span>We define <math alttext="H^{s,p}_{\mathbb{R}^{d}_{\pm}}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS1.1.p1.2.m1.5"><semantics id="S6.SS1.1.p1.2.m1.5a"><mrow id="S6.SS1.1.p1.2.m1.5.5" xref="S6.SS1.1.p1.2.m1.5.5.cmml"><msubsup id="S6.SS1.1.p1.2.m1.5.5.4" xref="S6.SS1.1.p1.2.m1.5.5.4.cmml"><mi id="S6.SS1.1.p1.2.m1.5.5.4.2.2" xref="S6.SS1.1.p1.2.m1.5.5.4.2.2.cmml">H</mi><msubsup id="S6.SS1.1.p1.2.m1.5.5.4.3" xref="S6.SS1.1.p1.2.m1.5.5.4.3.cmml"><mi id="S6.SS1.1.p1.2.m1.5.5.4.3.2.2" xref="S6.SS1.1.p1.2.m1.5.5.4.3.2.2.cmml">ℝ</mi><mo id="S6.SS1.1.p1.2.m1.5.5.4.3.3" xref="S6.SS1.1.p1.2.m1.5.5.4.3.3.cmml">±</mo><mi id="S6.SS1.1.p1.2.m1.5.5.4.3.2.3" xref="S6.SS1.1.p1.2.m1.5.5.4.3.2.3.cmml">d</mi></msubsup><mrow id="S6.SS1.1.p1.2.m1.2.2.2.4" xref="S6.SS1.1.p1.2.m1.2.2.2.3.cmml"><mi id="S6.SS1.1.p1.2.m1.1.1.1.1" xref="S6.SS1.1.p1.2.m1.1.1.1.1.cmml">s</mi><mo id="S6.SS1.1.p1.2.m1.2.2.2.4.1" xref="S6.SS1.1.p1.2.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS1.1.p1.2.m1.2.2.2.2" xref="S6.SS1.1.p1.2.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S6.SS1.1.p1.2.m1.5.5.3" xref="S6.SS1.1.p1.2.m1.5.5.3.cmml">⁢</mo><mrow id="S6.SS1.1.p1.2.m1.5.5.2.2" xref="S6.SS1.1.p1.2.m1.5.5.2.3.cmml"><mo id="S6.SS1.1.p1.2.m1.5.5.2.2.3" stretchy="false" xref="S6.SS1.1.p1.2.m1.5.5.2.3.cmml">(</mo><msup id="S6.SS1.1.p1.2.m1.4.4.1.1.1" xref="S6.SS1.1.p1.2.m1.4.4.1.1.1.cmml"><mi id="S6.SS1.1.p1.2.m1.4.4.1.1.1.2" xref="S6.SS1.1.p1.2.m1.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S6.SS1.1.p1.2.m1.4.4.1.1.1.3" xref="S6.SS1.1.p1.2.m1.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S6.SS1.1.p1.2.m1.5.5.2.2.4" xref="S6.SS1.1.p1.2.m1.5.5.2.3.cmml">,</mo><msub id="S6.SS1.1.p1.2.m1.5.5.2.2.2" xref="S6.SS1.1.p1.2.m1.5.5.2.2.2.cmml"><mi id="S6.SS1.1.p1.2.m1.5.5.2.2.2.2" xref="S6.SS1.1.p1.2.m1.5.5.2.2.2.2.cmml">w</mi><mi id="S6.SS1.1.p1.2.m1.5.5.2.2.2.3" xref="S6.SS1.1.p1.2.m1.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S6.SS1.1.p1.2.m1.5.5.2.2.5" xref="S6.SS1.1.p1.2.m1.5.5.2.3.cmml">;</mo><mi id="S6.SS1.1.p1.2.m1.3.3" xref="S6.SS1.1.p1.2.m1.3.3.cmml">X</mi><mo id="S6.SS1.1.p1.2.m1.5.5.2.2.6" stretchy="false" xref="S6.SS1.1.p1.2.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.2.m1.5b"><apply id="S6.SS1.1.p1.2.m1.5.5.cmml" xref="S6.SS1.1.p1.2.m1.5.5"><times id="S6.SS1.1.p1.2.m1.5.5.3.cmml" xref="S6.SS1.1.p1.2.m1.5.5.3"></times><apply id="S6.SS1.1.p1.2.m1.5.5.4.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4"><csymbol cd="ambiguous" id="S6.SS1.1.p1.2.m1.5.5.4.1.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4">subscript</csymbol><apply id="S6.SS1.1.p1.2.m1.5.5.4.2.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4"><csymbol cd="ambiguous" id="S6.SS1.1.p1.2.m1.5.5.4.2.1.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4">superscript</csymbol><ci id="S6.SS1.1.p1.2.m1.5.5.4.2.2.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.2.2">𝐻</ci><list id="S6.SS1.1.p1.2.m1.2.2.2.3.cmml" xref="S6.SS1.1.p1.2.m1.2.2.2.4"><ci id="S6.SS1.1.p1.2.m1.1.1.1.1.cmml" xref="S6.SS1.1.p1.2.m1.1.1.1.1">𝑠</ci><ci id="S6.SS1.1.p1.2.m1.2.2.2.2.cmml" xref="S6.SS1.1.p1.2.m1.2.2.2.2">𝑝</ci></list></apply><apply id="S6.SS1.1.p1.2.m1.5.5.4.3.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.2.m1.5.5.4.3.1.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.3">subscript</csymbol><apply id="S6.SS1.1.p1.2.m1.5.5.4.3.2.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.3"><csymbol cd="ambiguous" id="S6.SS1.1.p1.2.m1.5.5.4.3.2.1.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.3">superscript</csymbol><ci id="S6.SS1.1.p1.2.m1.5.5.4.3.2.2.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.3.2.2">ℝ</ci><ci id="S6.SS1.1.p1.2.m1.5.5.4.3.2.3.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.3.2.3">𝑑</ci></apply><csymbol cd="latexml" id="S6.SS1.1.p1.2.m1.5.5.4.3.3.cmml" xref="S6.SS1.1.p1.2.m1.5.5.4.3.3">plus-or-minus</csymbol></apply></apply><vector id="S6.SS1.1.p1.2.m1.5.5.2.3.cmml" xref="S6.SS1.1.p1.2.m1.5.5.2.2"><apply id="S6.SS1.1.p1.2.m1.4.4.1.1.1.cmml" xref="S6.SS1.1.p1.2.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.1.p1.2.m1.4.4.1.1.1.1.cmml" xref="S6.SS1.1.p1.2.m1.4.4.1.1.1">superscript</csymbol><ci id="S6.SS1.1.p1.2.m1.4.4.1.1.1.2.cmml" xref="S6.SS1.1.p1.2.m1.4.4.1.1.1.2">ℝ</ci><ci id="S6.SS1.1.p1.2.m1.4.4.1.1.1.3.cmml" xref="S6.SS1.1.p1.2.m1.4.4.1.1.1.3">𝑑</ci></apply><apply id="S6.SS1.1.p1.2.m1.5.5.2.2.2.cmml" xref="S6.SS1.1.p1.2.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.1.p1.2.m1.5.5.2.2.2.1.cmml" xref="S6.SS1.1.p1.2.m1.5.5.2.2.2">subscript</csymbol><ci id="S6.SS1.1.p1.2.m1.5.5.2.2.2.2.cmml" xref="S6.SS1.1.p1.2.m1.5.5.2.2.2.2">𝑤</ci><ci id="S6.SS1.1.p1.2.m1.5.5.2.2.2.3.cmml" xref="S6.SS1.1.p1.2.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S6.SS1.1.p1.2.m1.3.3.cmml" xref="S6.SS1.1.p1.2.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.2.m1.5c">H^{s,p}_{\mathbb{R}^{d}_{\pm}}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.2.m1.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> as the closed subspace of <math alttext="H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS1.1.p1.3.m2.5"><semantics id="S6.SS1.1.p1.3.m2.5a"><mrow id="S6.SS1.1.p1.3.m2.5.5" xref="S6.SS1.1.p1.3.m2.5.5.cmml"><msup id="S6.SS1.1.p1.3.m2.5.5.4" xref="S6.SS1.1.p1.3.m2.5.5.4.cmml"><mi id="S6.SS1.1.p1.3.m2.5.5.4.2" xref="S6.SS1.1.p1.3.m2.5.5.4.2.cmml">H</mi><mrow id="S6.SS1.1.p1.3.m2.2.2.2.4" xref="S6.SS1.1.p1.3.m2.2.2.2.3.cmml"><mi id="S6.SS1.1.p1.3.m2.1.1.1.1" xref="S6.SS1.1.p1.3.m2.1.1.1.1.cmml">s</mi><mo id="S6.SS1.1.p1.3.m2.2.2.2.4.1" xref="S6.SS1.1.p1.3.m2.2.2.2.3.cmml">,</mo><mi id="S6.SS1.1.p1.3.m2.2.2.2.2" xref="S6.SS1.1.p1.3.m2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S6.SS1.1.p1.3.m2.5.5.3" xref="S6.SS1.1.p1.3.m2.5.5.3.cmml">⁢</mo><mrow id="S6.SS1.1.p1.3.m2.5.5.2.2" xref="S6.SS1.1.p1.3.m2.5.5.2.3.cmml"><mo id="S6.SS1.1.p1.3.m2.5.5.2.2.3" stretchy="false" xref="S6.SS1.1.p1.3.m2.5.5.2.3.cmml">(</mo><msup id="S6.SS1.1.p1.3.m2.4.4.1.1.1" xref="S6.SS1.1.p1.3.m2.4.4.1.1.1.cmml"><mi id="S6.SS1.1.p1.3.m2.4.4.1.1.1.2" xref="S6.SS1.1.p1.3.m2.4.4.1.1.1.2.cmml">ℝ</mi><mi id="S6.SS1.1.p1.3.m2.4.4.1.1.1.3" xref="S6.SS1.1.p1.3.m2.4.4.1.1.1.3.cmml">d</mi></msup><mo id="S6.SS1.1.p1.3.m2.5.5.2.2.4" xref="S6.SS1.1.p1.3.m2.5.5.2.3.cmml">,</mo><msub id="S6.SS1.1.p1.3.m2.5.5.2.2.2" xref="S6.SS1.1.p1.3.m2.5.5.2.2.2.cmml"><mi id="S6.SS1.1.p1.3.m2.5.5.2.2.2.2" xref="S6.SS1.1.p1.3.m2.5.5.2.2.2.2.cmml">w</mi><mi id="S6.SS1.1.p1.3.m2.5.5.2.2.2.3" xref="S6.SS1.1.p1.3.m2.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S6.SS1.1.p1.3.m2.5.5.2.2.5" xref="S6.SS1.1.p1.3.m2.5.5.2.3.cmml">;</mo><mi id="S6.SS1.1.p1.3.m2.3.3" xref="S6.SS1.1.p1.3.m2.3.3.cmml">X</mi><mo id="S6.SS1.1.p1.3.m2.5.5.2.2.6" stretchy="false" xref="S6.SS1.1.p1.3.m2.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.3.m2.5b"><apply id="S6.SS1.1.p1.3.m2.5.5.cmml" xref="S6.SS1.1.p1.3.m2.5.5"><times id="S6.SS1.1.p1.3.m2.5.5.3.cmml" xref="S6.SS1.1.p1.3.m2.5.5.3"></times><apply id="S6.SS1.1.p1.3.m2.5.5.4.cmml" xref="S6.SS1.1.p1.3.m2.5.5.4"><csymbol cd="ambiguous" id="S6.SS1.1.p1.3.m2.5.5.4.1.cmml" xref="S6.SS1.1.p1.3.m2.5.5.4">superscript</csymbol><ci id="S6.SS1.1.p1.3.m2.5.5.4.2.cmml" xref="S6.SS1.1.p1.3.m2.5.5.4.2">𝐻</ci><list id="S6.SS1.1.p1.3.m2.2.2.2.3.cmml" xref="S6.SS1.1.p1.3.m2.2.2.2.4"><ci id="S6.SS1.1.p1.3.m2.1.1.1.1.cmml" xref="S6.SS1.1.p1.3.m2.1.1.1.1">𝑠</ci><ci id="S6.SS1.1.p1.3.m2.2.2.2.2.cmml" xref="S6.SS1.1.p1.3.m2.2.2.2.2">𝑝</ci></list></apply><vector id="S6.SS1.1.p1.3.m2.5.5.2.3.cmml" xref="S6.SS1.1.p1.3.m2.5.5.2.2"><apply id="S6.SS1.1.p1.3.m2.4.4.1.1.1.cmml" xref="S6.SS1.1.p1.3.m2.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.1.p1.3.m2.4.4.1.1.1.1.cmml" xref="S6.SS1.1.p1.3.m2.4.4.1.1.1">superscript</csymbol><ci id="S6.SS1.1.p1.3.m2.4.4.1.1.1.2.cmml" xref="S6.SS1.1.p1.3.m2.4.4.1.1.1.2">ℝ</ci><ci id="S6.SS1.1.p1.3.m2.4.4.1.1.1.3.cmml" xref="S6.SS1.1.p1.3.m2.4.4.1.1.1.3">𝑑</ci></apply><apply id="S6.SS1.1.p1.3.m2.5.5.2.2.2.cmml" xref="S6.SS1.1.p1.3.m2.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.1.p1.3.m2.5.5.2.2.2.1.cmml" xref="S6.SS1.1.p1.3.m2.5.5.2.2.2">subscript</csymbol><ci id="S6.SS1.1.p1.3.m2.5.5.2.2.2.2.cmml" xref="S6.SS1.1.p1.3.m2.5.5.2.2.2.2">𝑤</ci><ci id="S6.SS1.1.p1.3.m2.5.5.2.2.2.3.cmml" xref="S6.SS1.1.p1.3.m2.5.5.2.2.2.3">𝛾</ci></apply><ci id="S6.SS1.1.p1.3.m2.3.3.cmml" xref="S6.SS1.1.p1.3.m2.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.3.m2.5c">H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.3.m2.5d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> consisting of all functions with its support in <math alttext="\mathbb{R}^{d}_{\pm}" class="ltx_Math" display="inline" id="S6.SS1.1.p1.4.m3.1"><semantics id="S6.SS1.1.p1.4.m3.1a"><msubsup id="S6.SS1.1.p1.4.m3.1.1" xref="S6.SS1.1.p1.4.m3.1.1.cmml"><mi id="S6.SS1.1.p1.4.m3.1.1.2.2" xref="S6.SS1.1.p1.4.m3.1.1.2.2.cmml">ℝ</mi><mo id="S6.SS1.1.p1.4.m3.1.1.3" xref="S6.SS1.1.p1.4.m3.1.1.3.cmml">±</mo><mi id="S6.SS1.1.p1.4.m3.1.1.2.3" xref="S6.SS1.1.p1.4.m3.1.1.2.3.cmml">d</mi></msubsup><annotation-xml encoding="MathML-Content" id="S6.SS1.1.p1.4.m3.1b"><apply id="S6.SS1.1.p1.4.m3.1.1.cmml" xref="S6.SS1.1.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.1.p1.4.m3.1.1.1.cmml" xref="S6.SS1.1.p1.4.m3.1.1">subscript</csymbol><apply id="S6.SS1.1.p1.4.m3.1.1.2.cmml" xref="S6.SS1.1.p1.4.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.1.p1.4.m3.1.1.2.1.cmml" xref="S6.SS1.1.p1.4.m3.1.1">superscript</csymbol><ci id="S6.SS1.1.p1.4.m3.1.1.2.2.cmml" xref="S6.SS1.1.p1.4.m3.1.1.2.2">ℝ</ci><ci id="S6.SS1.1.p1.4.m3.1.1.2.3.cmml" xref="S6.SS1.1.p1.4.m3.1.1.2.3">𝑑</ci></apply><csymbol cd="latexml" id="S6.SS1.1.p1.4.m3.1.1.3.cmml" xref="S6.SS1.1.p1.4.m3.1.1.3">plus-or-minus</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.1.p1.4.m3.1c">\mathbb{R}^{d}_{\pm}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.1.p1.4.m3.1d">blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT</annotation></semantics></math>. These spaces form a complex interpolation scale (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.7]</cite>) and below we construct an isomorphism to reduce to this known interpolation result.</p> </div> <div class="ltx_para" id="S6.SS1.2.p2"> <p class="ltx_p" id="S6.SS1.2.p2.11">Let <math alttext="s>-1+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.1.m1.1"><semantics id="S6.SS1.2.p2.1.m1.1a"><mrow id="S6.SS1.2.p2.1.m1.1.1" xref="S6.SS1.2.p2.1.m1.1.1.cmml"><mi id="S6.SS1.2.p2.1.m1.1.1.2" xref="S6.SS1.2.p2.1.m1.1.1.2.cmml">s</mi><mo id="S6.SS1.2.p2.1.m1.1.1.1" xref="S6.SS1.2.p2.1.m1.1.1.1.cmml">></mo><mrow id="S6.SS1.2.p2.1.m1.1.1.3" xref="S6.SS1.2.p2.1.m1.1.1.3.cmml"><mrow id="S6.SS1.2.p2.1.m1.1.1.3.2" xref="S6.SS1.2.p2.1.m1.1.1.3.2.cmml"><mo id="S6.SS1.2.p2.1.m1.1.1.3.2a" xref="S6.SS1.2.p2.1.m1.1.1.3.2.cmml">−</mo><mn id="S6.SS1.2.p2.1.m1.1.1.3.2.2" xref="S6.SS1.2.p2.1.m1.1.1.3.2.2.cmml">1</mn></mrow><mo id="S6.SS1.2.p2.1.m1.1.1.3.1" xref="S6.SS1.2.p2.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS1.2.p2.1.m1.1.1.3.3" xref="S6.SS1.2.p2.1.m1.1.1.3.3.cmml"><mrow id="S6.SS1.2.p2.1.m1.1.1.3.3.2" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS1.2.p2.1.m1.1.1.3.3.2.2" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS1.2.p2.1.m1.1.1.3.3.2.1" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS1.2.p2.1.m1.1.1.3.3.2.3" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS1.2.p2.1.m1.1.1.3.3.3" xref="S6.SS1.2.p2.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.1.m1.1b"><apply id="S6.SS1.2.p2.1.m1.1.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1"><gt id="S6.SS1.2.p2.1.m1.1.1.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1.1"></gt><ci id="S6.SS1.2.p2.1.m1.1.1.2.cmml" xref="S6.SS1.2.p2.1.m1.1.1.2">𝑠</ci><apply id="S6.SS1.2.p2.1.m1.1.1.3.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3"><plus id="S6.SS1.2.p2.1.m1.1.1.3.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.1"></plus><apply id="S6.SS1.2.p2.1.m1.1.1.3.2.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.2"><minus id="S6.SS1.2.p2.1.m1.1.1.3.2.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.2"></minus><cn id="S6.SS1.2.p2.1.m1.1.1.3.2.2.cmml" type="integer" xref="S6.SS1.2.p2.1.m1.1.1.3.2.2">1</cn></apply><apply id="S6.SS1.2.p2.1.m1.1.1.3.3.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.3"><divide id="S6.SS1.2.p2.1.m1.1.1.3.3.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.3"></divide><apply id="S6.SS1.2.p2.1.m1.1.1.3.3.2.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2"><plus id="S6.SS1.2.p2.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS1.2.p2.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS1.2.p2.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS1.2.p2.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS1.2.p2.1.m1.1.1.3.3.3.cmml" xref="S6.SS1.2.p2.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.1.m1.1c">s>-1+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.1.m1.1d">italic_s > - 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> be such that <math alttext="s\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.2.m2.1"><semantics id="S6.SS1.2.p2.2.m2.1a"><mrow id="S6.SS1.2.p2.2.m2.1.1" xref="S6.SS1.2.p2.2.m2.1.1.cmml"><mi id="S6.SS1.2.p2.2.m2.1.1.2" xref="S6.SS1.2.p2.2.m2.1.1.2.cmml">s</mi><mo id="S6.SS1.2.p2.2.m2.1.1.1" xref="S6.SS1.2.p2.2.m2.1.1.1.cmml">∉</mo><mrow id="S6.SS1.2.p2.2.m2.1.1.3" xref="S6.SS1.2.p2.2.m2.1.1.3.cmml"><msub id="S6.SS1.2.p2.2.m2.1.1.3.2" xref="S6.SS1.2.p2.2.m2.1.1.3.2.cmml"><mi id="S6.SS1.2.p2.2.m2.1.1.3.2.2" xref="S6.SS1.2.p2.2.m2.1.1.3.2.2.cmml">ℕ</mi><mn id="S6.SS1.2.p2.2.m2.1.1.3.2.3" xref="S6.SS1.2.p2.2.m2.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS1.2.p2.2.m2.1.1.3.1" xref="S6.SS1.2.p2.2.m2.1.1.3.1.cmml">+</mo><mfrac id="S6.SS1.2.p2.2.m2.1.1.3.3" xref="S6.SS1.2.p2.2.m2.1.1.3.3.cmml"><mrow id="S6.SS1.2.p2.2.m2.1.1.3.3.2" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2.cmml"><mi id="S6.SS1.2.p2.2.m2.1.1.3.3.2.2" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS1.2.p2.2.m2.1.1.3.3.2.1" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS1.2.p2.2.m2.1.1.3.3.2.3" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS1.2.p2.2.m2.1.1.3.3.3" xref="S6.SS1.2.p2.2.m2.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.2.m2.1b"><apply id="S6.SS1.2.p2.2.m2.1.1.cmml" xref="S6.SS1.2.p2.2.m2.1.1"><notin id="S6.SS1.2.p2.2.m2.1.1.1.cmml" xref="S6.SS1.2.p2.2.m2.1.1.1"></notin><ci id="S6.SS1.2.p2.2.m2.1.1.2.cmml" xref="S6.SS1.2.p2.2.m2.1.1.2">𝑠</ci><apply id="S6.SS1.2.p2.2.m2.1.1.3.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3"><plus id="S6.SS1.2.p2.2.m2.1.1.3.1.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.1"></plus><apply id="S6.SS1.2.p2.2.m2.1.1.3.2.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.2.m2.1.1.3.2.1.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.2">subscript</csymbol><ci id="S6.SS1.2.p2.2.m2.1.1.3.2.2.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.2.2">ℕ</ci><cn id="S6.SS1.2.p2.2.m2.1.1.3.2.3.cmml" type="integer" xref="S6.SS1.2.p2.2.m2.1.1.3.2.3">0</cn></apply><apply id="S6.SS1.2.p2.2.m2.1.1.3.3.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.3"><divide id="S6.SS1.2.p2.2.m2.1.1.3.3.1.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.3"></divide><apply id="S6.SS1.2.p2.2.m2.1.1.3.3.2.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2"><plus id="S6.SS1.2.p2.2.m2.1.1.3.3.2.1.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2.1"></plus><ci id="S6.SS1.2.p2.2.m2.1.1.3.3.2.2.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS1.2.p2.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS1.2.p2.2.m2.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS1.2.p2.2.m2.1.1.3.3.3.cmml" xref="S6.SS1.2.p2.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.2.m2.1c">s\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.2.m2.1d">italic_s ∉ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Then we claim that <math alttext="\overline{\operatorname{Tr}}_{m}f=0" class="ltx_Math" display="inline" id="S6.SS1.2.p2.3.m3.1"><semantics id="S6.SS1.2.p2.3.m3.1a"><mrow id="S6.SS1.2.p2.3.m3.1.1" xref="S6.SS1.2.p2.3.m3.1.1.cmml"><mrow id="S6.SS1.2.p2.3.m3.1.1.2" xref="S6.SS1.2.p2.3.m3.1.1.2.cmml"><msub id="S6.SS1.2.p2.3.m3.1.1.2.2" xref="S6.SS1.2.p2.3.m3.1.1.2.2.cmml"><mover accent="true" id="S6.SS1.2.p2.3.m3.1.1.2.2.2" xref="S6.SS1.2.p2.3.m3.1.1.2.2.2.cmml"><mi id="S6.SS1.2.p2.3.m3.1.1.2.2.2.2" xref="S6.SS1.2.p2.3.m3.1.1.2.2.2.2.cmml">Tr</mi><mo id="S6.SS1.2.p2.3.m3.1.1.2.2.2.1" xref="S6.SS1.2.p2.3.m3.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S6.SS1.2.p2.3.m3.1.1.2.2.3" xref="S6.SS1.2.p2.3.m3.1.1.2.2.3.cmml">m</mi></msub><mo id="S6.SS1.2.p2.3.m3.1.1.2.1" lspace="0.167em" xref="S6.SS1.2.p2.3.m3.1.1.2.1.cmml">⁢</mo><mi id="S6.SS1.2.p2.3.m3.1.1.2.3" xref="S6.SS1.2.p2.3.m3.1.1.2.3.cmml">f</mi></mrow><mo id="S6.SS1.2.p2.3.m3.1.1.1" xref="S6.SS1.2.p2.3.m3.1.1.1.cmml">=</mo><mn id="S6.SS1.2.p2.3.m3.1.1.3" xref="S6.SS1.2.p2.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.3.m3.1b"><apply id="S6.SS1.2.p2.3.m3.1.1.cmml" xref="S6.SS1.2.p2.3.m3.1.1"><eq id="S6.SS1.2.p2.3.m3.1.1.1.cmml" xref="S6.SS1.2.p2.3.m3.1.1.1"></eq><apply id="S6.SS1.2.p2.3.m3.1.1.2.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2"><times id="S6.SS1.2.p2.3.m3.1.1.2.1.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.1"></times><apply id="S6.SS1.2.p2.3.m3.1.1.2.2.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.3.m3.1.1.2.2.1.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.2">subscript</csymbol><apply id="S6.SS1.2.p2.3.m3.1.1.2.2.2.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.2.2"><ci id="S6.SS1.2.p2.3.m3.1.1.2.2.2.1.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.2.2.1">¯</ci><ci id="S6.SS1.2.p2.3.m3.1.1.2.2.2.2.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.2.2.2">Tr</ci></apply><ci id="S6.SS1.2.p2.3.m3.1.1.2.2.3.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.2.3">𝑚</ci></apply><ci id="S6.SS1.2.p2.3.m3.1.1.2.3.cmml" xref="S6.SS1.2.p2.3.m3.1.1.2.3">𝑓</ci></apply><cn id="S6.SS1.2.p2.3.m3.1.1.3.cmml" type="integer" xref="S6.SS1.2.p2.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.3.m3.1c">\overline{\operatorname{Tr}}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.3.m3.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math> for integers <math alttext="m\in[0,s-\frac{\gamma+1}{p})" class="ltx_Math" display="inline" id="S6.SS1.2.p2.4.m4.2"><semantics id="S6.SS1.2.p2.4.m4.2a"><mrow id="S6.SS1.2.p2.4.m4.2.2" xref="S6.SS1.2.p2.4.m4.2.2.cmml"><mi id="S6.SS1.2.p2.4.m4.2.2.3" xref="S6.SS1.2.p2.4.m4.2.2.3.cmml">m</mi><mo id="S6.SS1.2.p2.4.m4.2.2.2" xref="S6.SS1.2.p2.4.m4.2.2.2.cmml">∈</mo><mrow id="S6.SS1.2.p2.4.m4.2.2.1.1" xref="S6.SS1.2.p2.4.m4.2.2.1.2.cmml"><mo id="S6.SS1.2.p2.4.m4.2.2.1.1.2" stretchy="false" xref="S6.SS1.2.p2.4.m4.2.2.1.2.cmml">[</mo><mn id="S6.SS1.2.p2.4.m4.1.1" xref="S6.SS1.2.p2.4.m4.1.1.cmml">0</mn><mo id="S6.SS1.2.p2.4.m4.2.2.1.1.3" xref="S6.SS1.2.p2.4.m4.2.2.1.2.cmml">,</mo><mrow id="S6.SS1.2.p2.4.m4.2.2.1.1.1" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.cmml"><mi id="S6.SS1.2.p2.4.m4.2.2.1.1.1.2" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.2.cmml">s</mi><mo id="S6.SS1.2.p2.4.m4.2.2.1.1.1.1" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.1.cmml">−</mo><mfrac id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.cmml"><mrow id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.cmml"><mi id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.2" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.2.cmml">γ</mi><mo id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.1" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.1.cmml">+</mo><mn id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.3" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.3.cmml">1</mn></mrow><mi id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.3" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS1.2.p2.4.m4.2.2.1.1.4" stretchy="false" xref="S6.SS1.2.p2.4.m4.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.4.m4.2b"><apply id="S6.SS1.2.p2.4.m4.2.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2"><in id="S6.SS1.2.p2.4.m4.2.2.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2.2"></in><ci id="S6.SS1.2.p2.4.m4.2.2.3.cmml" xref="S6.SS1.2.p2.4.m4.2.2.3">𝑚</ci><interval closure="closed-open" id="S6.SS1.2.p2.4.m4.2.2.1.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1"><cn id="S6.SS1.2.p2.4.m4.1.1.cmml" type="integer" xref="S6.SS1.2.p2.4.m4.1.1">0</cn><apply id="S6.SS1.2.p2.4.m4.2.2.1.1.1.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1"><minus id="S6.SS1.2.p2.4.m4.2.2.1.1.1.1.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.1"></minus><ci id="S6.SS1.2.p2.4.m4.2.2.1.1.1.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.2">𝑠</ci><apply id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3"><divide id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.1.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3"></divide><apply id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2"><plus id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.1.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.1"></plus><ci id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.2.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.2">𝛾</ci><cn id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.3.cmml" type="integer" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.2.3">1</cn></apply><ci id="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.3.cmml" xref="S6.SS1.2.p2.4.m4.2.2.1.1.1.3.3">𝑝</ci></apply></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.4.m4.2c">m\in[0,s-\frac{\gamma+1}{p})</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.4.m4.2d">italic_m ∈ [ 0 , italic_s - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG )</annotation></semantics></math> and <math alttext="f\in H^{s,p}_{\mathbb{R}^{d}_{\pm}}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS1.2.p2.5.m5.5"><semantics id="S6.SS1.2.p2.5.m5.5a"><mrow id="S6.SS1.2.p2.5.m5.5.5" xref="S6.SS1.2.p2.5.m5.5.5.cmml"><mi id="S6.SS1.2.p2.5.m5.5.5.4" xref="S6.SS1.2.p2.5.m5.5.5.4.cmml">f</mi><mo id="S6.SS1.2.p2.5.m5.5.5.3" xref="S6.SS1.2.p2.5.m5.5.5.3.cmml">∈</mo><mrow id="S6.SS1.2.p2.5.m5.5.5.2" xref="S6.SS1.2.p2.5.m5.5.5.2.cmml"><msubsup id="S6.SS1.2.p2.5.m5.5.5.2.4" xref="S6.SS1.2.p2.5.m5.5.5.2.4.cmml"><mi id="S6.SS1.2.p2.5.m5.5.5.2.4.2.2" xref="S6.SS1.2.p2.5.m5.5.5.2.4.2.2.cmml">H</mi><msubsup id="S6.SS1.2.p2.5.m5.5.5.2.4.3" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3.cmml"><mi id="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.2" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.2.cmml">ℝ</mi><mo id="S6.SS1.2.p2.5.m5.5.5.2.4.3.3" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3.3.cmml">±</mo><mi id="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.3" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.3.cmml">d</mi></msubsup><mrow id="S6.SS1.2.p2.5.m5.2.2.2.4" xref="S6.SS1.2.p2.5.m5.2.2.2.3.cmml"><mi id="S6.SS1.2.p2.5.m5.1.1.1.1" xref="S6.SS1.2.p2.5.m5.1.1.1.1.cmml">s</mi><mo id="S6.SS1.2.p2.5.m5.2.2.2.4.1" xref="S6.SS1.2.p2.5.m5.2.2.2.3.cmml">,</mo><mi id="S6.SS1.2.p2.5.m5.2.2.2.2" xref="S6.SS1.2.p2.5.m5.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S6.SS1.2.p2.5.m5.5.5.2.3" xref="S6.SS1.2.p2.5.m5.5.5.2.3.cmml">⁢</mo><mrow id="S6.SS1.2.p2.5.m5.5.5.2.2.2" xref="S6.SS1.2.p2.5.m5.5.5.2.2.3.cmml"><mo id="S6.SS1.2.p2.5.m5.5.5.2.2.2.3" stretchy="false" xref="S6.SS1.2.p2.5.m5.5.5.2.2.3.cmml">(</mo><msup id="S6.SS1.2.p2.5.m5.4.4.1.1.1.1" xref="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.cmml"><mi id="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.2" xref="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.3" xref="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S6.SS1.2.p2.5.m5.5.5.2.2.2.4" xref="S6.SS1.2.p2.5.m5.5.5.2.2.3.cmml">,</mo><msub id="S6.SS1.2.p2.5.m5.5.5.2.2.2.2" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.cmml"><mi id="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.2" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.2.cmml">w</mi><mi id="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.3" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.SS1.2.p2.5.m5.5.5.2.2.2.5" xref="S6.SS1.2.p2.5.m5.5.5.2.2.3.cmml">;</mo><mi id="S6.SS1.2.p2.5.m5.3.3" xref="S6.SS1.2.p2.5.m5.3.3.cmml">X</mi><mo id="S6.SS1.2.p2.5.m5.5.5.2.2.2.6" stretchy="false" xref="S6.SS1.2.p2.5.m5.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.5.m5.5b"><apply id="S6.SS1.2.p2.5.m5.5.5.cmml" xref="S6.SS1.2.p2.5.m5.5.5"><in id="S6.SS1.2.p2.5.m5.5.5.3.cmml" xref="S6.SS1.2.p2.5.m5.5.5.3"></in><ci id="S6.SS1.2.p2.5.m5.5.5.4.cmml" xref="S6.SS1.2.p2.5.m5.5.5.4">𝑓</ci><apply id="S6.SS1.2.p2.5.m5.5.5.2.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2"><times id="S6.SS1.2.p2.5.m5.5.5.2.3.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.3"></times><apply id="S6.SS1.2.p2.5.m5.5.5.2.4.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4"><csymbol cd="ambiguous" id="S6.SS1.2.p2.5.m5.5.5.2.4.1.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4">subscript</csymbol><apply id="S6.SS1.2.p2.5.m5.5.5.2.4.2.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4"><csymbol cd="ambiguous" id="S6.SS1.2.p2.5.m5.5.5.2.4.2.1.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4">superscript</csymbol><ci id="S6.SS1.2.p2.5.m5.5.5.2.4.2.2.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.2.2">𝐻</ci><list id="S6.SS1.2.p2.5.m5.2.2.2.3.cmml" xref="S6.SS1.2.p2.5.m5.2.2.2.4"><ci id="S6.SS1.2.p2.5.m5.1.1.1.1.cmml" xref="S6.SS1.2.p2.5.m5.1.1.1.1">𝑠</ci><ci id="S6.SS1.2.p2.5.m5.2.2.2.2.cmml" xref="S6.SS1.2.p2.5.m5.2.2.2.2">𝑝</ci></list></apply><apply id="S6.SS1.2.p2.5.m5.5.5.2.4.3.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3"><csymbol cd="ambiguous" id="S6.SS1.2.p2.5.m5.5.5.2.4.3.1.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3">subscript</csymbol><apply id="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3"><csymbol cd="ambiguous" id="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.1.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3">superscript</csymbol><ci id="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.2.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.2">ℝ</ci><ci id="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.3.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3.2.3">𝑑</ci></apply><csymbol cd="latexml" id="S6.SS1.2.p2.5.m5.5.5.2.4.3.3.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.4.3.3">plus-or-minus</csymbol></apply></apply><vector id="S6.SS1.2.p2.5.m5.5.5.2.2.3.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2"><apply id="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.cmml" xref="S6.SS1.2.p2.5.m5.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.1.cmml" xref="S6.SS1.2.p2.5.m5.4.4.1.1.1.1">superscript</csymbol><ci id="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.2.cmml" xref="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.2">ℝ</ci><ci id="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.3.cmml" xref="S6.SS1.2.p2.5.m5.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.1.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2.2">subscript</csymbol><ci id="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.2">𝑤</ci><ci id="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.3.cmml" xref="S6.SS1.2.p2.5.m5.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S6.SS1.2.p2.5.m5.3.3.cmml" xref="S6.SS1.2.p2.5.m5.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.5.m5.5c">f\in H^{s,p}_{\mathbb{R}^{d}_{\pm}}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.5.m5.5d">italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT ± end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>. Indeed, this follows from the fact that if <math alttext="f\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS1.2.p2.6.m6.5"><semantics id="S6.SS1.2.p2.6.m6.5a"><mrow id="S6.SS1.2.p2.6.m6.5.5" xref="S6.SS1.2.p2.6.m6.5.5.cmml"><mi id="S6.SS1.2.p2.6.m6.5.5.4" xref="S6.SS1.2.p2.6.m6.5.5.4.cmml">f</mi><mo id="S6.SS1.2.p2.6.m6.5.5.3" xref="S6.SS1.2.p2.6.m6.5.5.3.cmml">∈</mo><mrow id="S6.SS1.2.p2.6.m6.5.5.2" xref="S6.SS1.2.p2.6.m6.5.5.2.cmml"><msup id="S6.SS1.2.p2.6.m6.5.5.2.4" xref="S6.SS1.2.p2.6.m6.5.5.2.4.cmml"><mi id="S6.SS1.2.p2.6.m6.5.5.2.4.2" xref="S6.SS1.2.p2.6.m6.5.5.2.4.2.cmml">H</mi><mrow id="S6.SS1.2.p2.6.m6.2.2.2.4" xref="S6.SS1.2.p2.6.m6.2.2.2.3.cmml"><mi id="S6.SS1.2.p2.6.m6.1.1.1.1" xref="S6.SS1.2.p2.6.m6.1.1.1.1.cmml">s</mi><mo id="S6.SS1.2.p2.6.m6.2.2.2.4.1" xref="S6.SS1.2.p2.6.m6.2.2.2.3.cmml">,</mo><mi id="S6.SS1.2.p2.6.m6.2.2.2.2" xref="S6.SS1.2.p2.6.m6.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S6.SS1.2.p2.6.m6.5.5.2.3" xref="S6.SS1.2.p2.6.m6.5.5.2.3.cmml">⁢</mo><mrow id="S6.SS1.2.p2.6.m6.5.5.2.2.2" xref="S6.SS1.2.p2.6.m6.5.5.2.2.3.cmml"><mo id="S6.SS1.2.p2.6.m6.5.5.2.2.2.3" stretchy="false" xref="S6.SS1.2.p2.6.m6.5.5.2.2.3.cmml">(</mo><msup id="S6.SS1.2.p2.6.m6.4.4.1.1.1.1" xref="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.cmml"><mi id="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.2" xref="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.2.cmml">ℝ</mi><mi id="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.3" xref="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.3.cmml">d</mi></msup><mo id="S6.SS1.2.p2.6.m6.5.5.2.2.2.4" xref="S6.SS1.2.p2.6.m6.5.5.2.2.3.cmml">,</mo><msub id="S6.SS1.2.p2.6.m6.5.5.2.2.2.2" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.cmml"><mi id="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.2" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.2.cmml">w</mi><mi id="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.3" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.SS1.2.p2.6.m6.5.5.2.2.2.5" xref="S6.SS1.2.p2.6.m6.5.5.2.2.3.cmml">;</mo><mi id="S6.SS1.2.p2.6.m6.3.3" xref="S6.SS1.2.p2.6.m6.3.3.cmml">X</mi><mo id="S6.SS1.2.p2.6.m6.5.5.2.2.2.6" stretchy="false" xref="S6.SS1.2.p2.6.m6.5.5.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.6.m6.5b"><apply id="S6.SS1.2.p2.6.m6.5.5.cmml" xref="S6.SS1.2.p2.6.m6.5.5"><in id="S6.SS1.2.p2.6.m6.5.5.3.cmml" xref="S6.SS1.2.p2.6.m6.5.5.3"></in><ci id="S6.SS1.2.p2.6.m6.5.5.4.cmml" xref="S6.SS1.2.p2.6.m6.5.5.4">𝑓</ci><apply id="S6.SS1.2.p2.6.m6.5.5.2.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2"><times id="S6.SS1.2.p2.6.m6.5.5.2.3.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.3"></times><apply id="S6.SS1.2.p2.6.m6.5.5.2.4.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.4"><csymbol cd="ambiguous" id="S6.SS1.2.p2.6.m6.5.5.2.4.1.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.4">superscript</csymbol><ci id="S6.SS1.2.p2.6.m6.5.5.2.4.2.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.4.2">𝐻</ci><list id="S6.SS1.2.p2.6.m6.2.2.2.3.cmml" xref="S6.SS1.2.p2.6.m6.2.2.2.4"><ci id="S6.SS1.2.p2.6.m6.1.1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.1.1.1.1">𝑠</ci><ci id="S6.SS1.2.p2.6.m6.2.2.2.2.cmml" xref="S6.SS1.2.p2.6.m6.2.2.2.2">𝑝</ci></list></apply><vector id="S6.SS1.2.p2.6.m6.5.5.2.2.3.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2"><apply id="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.4.4.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.1.cmml" xref="S6.SS1.2.p2.6.m6.4.4.1.1.1.1">superscript</csymbol><ci id="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.2.cmml" xref="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.2">ℝ</ci><ci id="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.3.cmml" xref="S6.SS1.2.p2.6.m6.4.4.1.1.1.1.3">𝑑</ci></apply><apply id="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.1.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2.2">subscript</csymbol><ci id="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.2">𝑤</ci><ci id="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.3.cmml" xref="S6.SS1.2.p2.6.m6.5.5.2.2.2.2.3">𝛾</ci></apply><ci id="S6.SS1.2.p2.6.m6.3.3.cmml" xref="S6.SS1.2.p2.6.m6.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.6.m6.5c">f\in H^{s,p}(\mathbb{R}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.6.m6.5d">italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> satisfies <math alttext="f|_{(0,\delta)\times\mathbb{R}^{d-1}}=0" class="ltx_Math" display="inline" id="S6.SS1.2.p2.7.m7.3"><semantics id="S6.SS1.2.p2.7.m7.3a"><mrow id="S6.SS1.2.p2.7.m7.3.4" xref="S6.SS1.2.p2.7.m7.3.4.cmml"><msub id="S6.SS1.2.p2.7.m7.3.4.2.2" xref="S6.SS1.2.p2.7.m7.3.4.2.1.cmml"><mrow id="S6.SS1.2.p2.7.m7.3.4.2.2.2" xref="S6.SS1.2.p2.7.m7.3.4.2.1.cmml"><mi id="S6.SS1.2.p2.7.m7.3.3" xref="S6.SS1.2.p2.7.m7.3.3.cmml">f</mi><mo id="S6.SS1.2.p2.7.m7.3.4.2.2.2.1" stretchy="false" xref="S6.SS1.2.p2.7.m7.3.4.2.1.1.cmml">|</mo></mrow><mrow id="S6.SS1.2.p2.7.m7.2.2.2" xref="S6.SS1.2.p2.7.m7.2.2.2.cmml"><mrow id="S6.SS1.2.p2.7.m7.2.2.2.4.2" xref="S6.SS1.2.p2.7.m7.2.2.2.4.1.cmml"><mo id="S6.SS1.2.p2.7.m7.2.2.2.4.2.1" stretchy="false" xref="S6.SS1.2.p2.7.m7.2.2.2.4.1.cmml">(</mo><mn id="S6.SS1.2.p2.7.m7.1.1.1.1" xref="S6.SS1.2.p2.7.m7.1.1.1.1.cmml">0</mn><mo id="S6.SS1.2.p2.7.m7.2.2.2.4.2.2" xref="S6.SS1.2.p2.7.m7.2.2.2.4.1.cmml">,</mo><mi id="S6.SS1.2.p2.7.m7.2.2.2.2" xref="S6.SS1.2.p2.7.m7.2.2.2.2.cmml">δ</mi><mo id="S6.SS1.2.p2.7.m7.2.2.2.4.2.3" rspace="0.055em" stretchy="false" xref="S6.SS1.2.p2.7.m7.2.2.2.4.1.cmml">)</mo></mrow><mo id="S6.SS1.2.p2.7.m7.2.2.2.3" rspace="0.222em" xref="S6.SS1.2.p2.7.m7.2.2.2.3.cmml">×</mo><msup id="S6.SS1.2.p2.7.m7.2.2.2.5" xref="S6.SS1.2.p2.7.m7.2.2.2.5.cmml"><mi id="S6.SS1.2.p2.7.m7.2.2.2.5.2" xref="S6.SS1.2.p2.7.m7.2.2.2.5.2.cmml">ℝ</mi><mrow id="S6.SS1.2.p2.7.m7.2.2.2.5.3" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3.cmml"><mi id="S6.SS1.2.p2.7.m7.2.2.2.5.3.2" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3.2.cmml">d</mi><mo id="S6.SS1.2.p2.7.m7.2.2.2.5.3.1" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3.1.cmml">−</mo><mn id="S6.SS1.2.p2.7.m7.2.2.2.5.3.3" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3.3.cmml">1</mn></mrow></msup></mrow></msub><mo id="S6.SS1.2.p2.7.m7.3.4.1" xref="S6.SS1.2.p2.7.m7.3.4.1.cmml">=</mo><mn id="S6.SS1.2.p2.7.m7.3.4.3" xref="S6.SS1.2.p2.7.m7.3.4.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.7.m7.3b"><apply id="S6.SS1.2.p2.7.m7.3.4.cmml" xref="S6.SS1.2.p2.7.m7.3.4"><eq id="S6.SS1.2.p2.7.m7.3.4.1.cmml" xref="S6.SS1.2.p2.7.m7.3.4.1"></eq><apply id="S6.SS1.2.p2.7.m7.3.4.2.1.cmml" xref="S6.SS1.2.p2.7.m7.3.4.2.2"><csymbol cd="latexml" id="S6.SS1.2.p2.7.m7.3.4.2.1.1.cmml" xref="S6.SS1.2.p2.7.m7.3.4.2.2.2.1">evaluated-at</csymbol><ci id="S6.SS1.2.p2.7.m7.3.3.cmml" xref="S6.SS1.2.p2.7.m7.3.3">𝑓</ci><apply id="S6.SS1.2.p2.7.m7.2.2.2.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2"><times id="S6.SS1.2.p2.7.m7.2.2.2.3.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.3"></times><interval closure="open" id="S6.SS1.2.p2.7.m7.2.2.2.4.1.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.4.2"><cn id="S6.SS1.2.p2.7.m7.1.1.1.1.cmml" type="integer" xref="S6.SS1.2.p2.7.m7.1.1.1.1">0</cn><ci id="S6.SS1.2.p2.7.m7.2.2.2.2.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.2">𝛿</ci></interval><apply id="S6.SS1.2.p2.7.m7.2.2.2.5.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.5"><csymbol cd="ambiguous" id="S6.SS1.2.p2.7.m7.2.2.2.5.1.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.5">superscript</csymbol><ci id="S6.SS1.2.p2.7.m7.2.2.2.5.2.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.5.2">ℝ</ci><apply id="S6.SS1.2.p2.7.m7.2.2.2.5.3.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3"><minus id="S6.SS1.2.p2.7.m7.2.2.2.5.3.1.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3.1"></minus><ci id="S6.SS1.2.p2.7.m7.2.2.2.5.3.2.cmml" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3.2">𝑑</ci><cn id="S6.SS1.2.p2.7.m7.2.2.2.5.3.3.cmml" type="integer" xref="S6.SS1.2.p2.7.m7.2.2.2.5.3.3">1</cn></apply></apply></apply></apply><cn id="S6.SS1.2.p2.7.m7.3.4.3.cmml" type="integer" xref="S6.SS1.2.p2.7.m7.3.4.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.7.m7.3c">f|_{(0,\delta)\times\mathbb{R}^{d-1}}=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.7.m7.3d">italic_f | start_POSTSUBSCRIPT ( 0 , italic_δ ) × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math> or <math alttext="f|_{(-\delta,0)\times\mathbb{R}^{d-1}}=0" class="ltx_Math" display="inline" id="S6.SS1.2.p2.8.m8.3"><semantics id="S6.SS1.2.p2.8.m8.3a"><mrow id="S6.SS1.2.p2.8.m8.3.4" xref="S6.SS1.2.p2.8.m8.3.4.cmml"><msub id="S6.SS1.2.p2.8.m8.3.4.2.2" xref="S6.SS1.2.p2.8.m8.3.4.2.1.cmml"><mrow id="S6.SS1.2.p2.8.m8.3.4.2.2.2" xref="S6.SS1.2.p2.8.m8.3.4.2.1.cmml"><mi id="S6.SS1.2.p2.8.m8.3.3" xref="S6.SS1.2.p2.8.m8.3.3.cmml">f</mi><mo id="S6.SS1.2.p2.8.m8.3.4.2.2.2.1" stretchy="false" xref="S6.SS1.2.p2.8.m8.3.4.2.1.1.cmml">|</mo></mrow><mrow id="S6.SS1.2.p2.8.m8.2.2.2" xref="S6.SS1.2.p2.8.m8.2.2.2.cmml"><mrow id="S6.SS1.2.p2.8.m8.2.2.2.2.1" xref="S6.SS1.2.p2.8.m8.2.2.2.2.2.cmml"><mo id="S6.SS1.2.p2.8.m8.2.2.2.2.1.2" stretchy="false" xref="S6.SS1.2.p2.8.m8.2.2.2.2.2.cmml">(</mo><mrow id="S6.SS1.2.p2.8.m8.2.2.2.2.1.1" xref="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.cmml"><mo id="S6.SS1.2.p2.8.m8.2.2.2.2.1.1a" xref="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.cmml">−</mo><mi id="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.2" xref="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.2.cmml">δ</mi></mrow><mo id="S6.SS1.2.p2.8.m8.2.2.2.2.1.3" xref="S6.SS1.2.p2.8.m8.2.2.2.2.2.cmml">,</mo><mn id="S6.SS1.2.p2.8.m8.1.1.1.1" xref="S6.SS1.2.p2.8.m8.1.1.1.1.cmml">0</mn><mo id="S6.SS1.2.p2.8.m8.2.2.2.2.1.4" rspace="0.055em" stretchy="false" xref="S6.SS1.2.p2.8.m8.2.2.2.2.2.cmml">)</mo></mrow><mo id="S6.SS1.2.p2.8.m8.2.2.2.3" rspace="0.222em" xref="S6.SS1.2.p2.8.m8.2.2.2.3.cmml">×</mo><msup id="S6.SS1.2.p2.8.m8.2.2.2.4" xref="S6.SS1.2.p2.8.m8.2.2.2.4.cmml"><mi id="S6.SS1.2.p2.8.m8.2.2.2.4.2" xref="S6.SS1.2.p2.8.m8.2.2.2.4.2.cmml">ℝ</mi><mrow id="S6.SS1.2.p2.8.m8.2.2.2.4.3" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3.cmml"><mi id="S6.SS1.2.p2.8.m8.2.2.2.4.3.2" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3.2.cmml">d</mi><mo id="S6.SS1.2.p2.8.m8.2.2.2.4.3.1" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3.1.cmml">−</mo><mn id="S6.SS1.2.p2.8.m8.2.2.2.4.3.3" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3.3.cmml">1</mn></mrow></msup></mrow></msub><mo id="S6.SS1.2.p2.8.m8.3.4.1" xref="S6.SS1.2.p2.8.m8.3.4.1.cmml">=</mo><mn id="S6.SS1.2.p2.8.m8.3.4.3" xref="S6.SS1.2.p2.8.m8.3.4.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.8.m8.3b"><apply id="S6.SS1.2.p2.8.m8.3.4.cmml" xref="S6.SS1.2.p2.8.m8.3.4"><eq id="S6.SS1.2.p2.8.m8.3.4.1.cmml" xref="S6.SS1.2.p2.8.m8.3.4.1"></eq><apply id="S6.SS1.2.p2.8.m8.3.4.2.1.cmml" xref="S6.SS1.2.p2.8.m8.3.4.2.2"><csymbol cd="latexml" id="S6.SS1.2.p2.8.m8.3.4.2.1.1.cmml" xref="S6.SS1.2.p2.8.m8.3.4.2.2.2.1">evaluated-at</csymbol><ci id="S6.SS1.2.p2.8.m8.3.3.cmml" xref="S6.SS1.2.p2.8.m8.3.3">𝑓</ci><apply id="S6.SS1.2.p2.8.m8.2.2.2.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2"><times id="S6.SS1.2.p2.8.m8.2.2.2.3.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.3"></times><interval closure="open" id="S6.SS1.2.p2.8.m8.2.2.2.2.2.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.2.1"><apply id="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.2.1.1"><minus id="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.1.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.2.1.1"></minus><ci id="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.2.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.2.1.1.2">𝛿</ci></apply><cn id="S6.SS1.2.p2.8.m8.1.1.1.1.cmml" type="integer" xref="S6.SS1.2.p2.8.m8.1.1.1.1">0</cn></interval><apply id="S6.SS1.2.p2.8.m8.2.2.2.4.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.4"><csymbol cd="ambiguous" id="S6.SS1.2.p2.8.m8.2.2.2.4.1.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.4">superscript</csymbol><ci id="S6.SS1.2.p2.8.m8.2.2.2.4.2.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.4.2">ℝ</ci><apply id="S6.SS1.2.p2.8.m8.2.2.2.4.3.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3"><minus id="S6.SS1.2.p2.8.m8.2.2.2.4.3.1.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3.1"></minus><ci id="S6.SS1.2.p2.8.m8.2.2.2.4.3.2.cmml" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3.2">𝑑</ci><cn id="S6.SS1.2.p2.8.m8.2.2.2.4.3.3.cmml" type="integer" xref="S6.SS1.2.p2.8.m8.2.2.2.4.3.3">1</cn></apply></apply></apply></apply><cn id="S6.SS1.2.p2.8.m8.3.4.3.cmml" type="integer" xref="S6.SS1.2.p2.8.m8.3.4.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.8.m8.3c">f|_{(-\delta,0)\times\mathbb{R}^{d-1}}=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.8.m8.3d">italic_f | start_POSTSUBSCRIPT ( - italic_δ , 0 ) × blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT end_POSTSUBSCRIPT = 0</annotation></semantics></math> for some <math alttext="\delta>0" class="ltx_Math" display="inline" id="S6.SS1.2.p2.9.m9.1"><semantics id="S6.SS1.2.p2.9.m9.1a"><mrow id="S6.SS1.2.p2.9.m9.1.1" xref="S6.SS1.2.p2.9.m9.1.1.cmml"><mi id="S6.SS1.2.p2.9.m9.1.1.2" xref="S6.SS1.2.p2.9.m9.1.1.2.cmml">δ</mi><mo id="S6.SS1.2.p2.9.m9.1.1.1" xref="S6.SS1.2.p2.9.m9.1.1.1.cmml">></mo><mn id="S6.SS1.2.p2.9.m9.1.1.3" xref="S6.SS1.2.p2.9.m9.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.9.m9.1b"><apply id="S6.SS1.2.p2.9.m9.1.1.cmml" xref="S6.SS1.2.p2.9.m9.1.1"><gt id="S6.SS1.2.p2.9.m9.1.1.1.cmml" xref="S6.SS1.2.p2.9.m9.1.1.1"></gt><ci id="S6.SS1.2.p2.9.m9.1.1.2.cmml" xref="S6.SS1.2.p2.9.m9.1.1.2">𝛿</ci><cn id="S6.SS1.2.p2.9.m9.1.1.3.cmml" type="integer" xref="S6.SS1.2.p2.9.m9.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.9.m9.1c">\delta>0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.9.m9.1d">italic_δ > 0</annotation></semantics></math>, then <math alttext="\overline{\operatorname{Tr}}_{m}f=0" class="ltx_Math" display="inline" id="S6.SS1.2.p2.10.m10.1"><semantics id="S6.SS1.2.p2.10.m10.1a"><mrow id="S6.SS1.2.p2.10.m10.1.1" xref="S6.SS1.2.p2.10.m10.1.1.cmml"><mrow id="S6.SS1.2.p2.10.m10.1.1.2" xref="S6.SS1.2.p2.10.m10.1.1.2.cmml"><msub id="S6.SS1.2.p2.10.m10.1.1.2.2" xref="S6.SS1.2.p2.10.m10.1.1.2.2.cmml"><mover accent="true" id="S6.SS1.2.p2.10.m10.1.1.2.2.2" xref="S6.SS1.2.p2.10.m10.1.1.2.2.2.cmml"><mi id="S6.SS1.2.p2.10.m10.1.1.2.2.2.2" xref="S6.SS1.2.p2.10.m10.1.1.2.2.2.2.cmml">Tr</mi><mo id="S6.SS1.2.p2.10.m10.1.1.2.2.2.1" xref="S6.SS1.2.p2.10.m10.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S6.SS1.2.p2.10.m10.1.1.2.2.3" xref="S6.SS1.2.p2.10.m10.1.1.2.2.3.cmml">m</mi></msub><mo id="S6.SS1.2.p2.10.m10.1.1.2.1" lspace="0.167em" xref="S6.SS1.2.p2.10.m10.1.1.2.1.cmml">⁢</mo><mi id="S6.SS1.2.p2.10.m10.1.1.2.3" xref="S6.SS1.2.p2.10.m10.1.1.2.3.cmml">f</mi></mrow><mo id="S6.SS1.2.p2.10.m10.1.1.1" xref="S6.SS1.2.p2.10.m10.1.1.1.cmml">=</mo><mn id="S6.SS1.2.p2.10.m10.1.1.3" xref="S6.SS1.2.p2.10.m10.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.10.m10.1b"><apply id="S6.SS1.2.p2.10.m10.1.1.cmml" xref="S6.SS1.2.p2.10.m10.1.1"><eq id="S6.SS1.2.p2.10.m10.1.1.1.cmml" xref="S6.SS1.2.p2.10.m10.1.1.1"></eq><apply id="S6.SS1.2.p2.10.m10.1.1.2.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2"><times id="S6.SS1.2.p2.10.m10.1.1.2.1.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.1"></times><apply id="S6.SS1.2.p2.10.m10.1.1.2.2.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS1.2.p2.10.m10.1.1.2.2.1.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.2">subscript</csymbol><apply id="S6.SS1.2.p2.10.m10.1.1.2.2.2.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.2.2"><ci id="S6.SS1.2.p2.10.m10.1.1.2.2.2.1.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.2.2.1">¯</ci><ci id="S6.SS1.2.p2.10.m10.1.1.2.2.2.2.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.2.2.2">Tr</ci></apply><ci id="S6.SS1.2.p2.10.m10.1.1.2.2.3.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.2.3">𝑚</ci></apply><ci id="S6.SS1.2.p2.10.m10.1.1.2.3.cmml" xref="S6.SS1.2.p2.10.m10.1.1.2.3">𝑓</ci></apply><cn id="S6.SS1.2.p2.10.m10.1.1.3.cmml" type="integer" xref="S6.SS1.2.p2.10.m10.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.10.m10.1c">\overline{\operatorname{Tr}}_{m}f=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.10.m10.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT italic_f = 0</annotation></semantics></math>. This observation can be proved similarly as in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 6.3(2)]</cite> using the continuity of <math alttext="\overline{\operatorname{Tr}}_{m}" class="ltx_Math" display="inline" id="S6.SS1.2.p2.11.m11.1"><semantics id="S6.SS1.2.p2.11.m11.1a"><msub id="S6.SS1.2.p2.11.m11.1.1" xref="S6.SS1.2.p2.11.m11.1.1.cmml"><mover accent="true" id="S6.SS1.2.p2.11.m11.1.1.2" xref="S6.SS1.2.p2.11.m11.1.1.2.cmml"><mi id="S6.SS1.2.p2.11.m11.1.1.2.2" xref="S6.SS1.2.p2.11.m11.1.1.2.2.cmml">Tr</mi><mo id="S6.SS1.2.p2.11.m11.1.1.2.1" xref="S6.SS1.2.p2.11.m11.1.1.2.1.cmml">¯</mo></mover><mi id="S6.SS1.2.p2.11.m11.1.1.3" xref="S6.SS1.2.p2.11.m11.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS1.2.p2.11.m11.1b"><apply id="S6.SS1.2.p2.11.m11.1.1.cmml" xref="S6.SS1.2.p2.11.m11.1.1"><csymbol cd="ambiguous" id="S6.SS1.2.p2.11.m11.1.1.1.cmml" xref="S6.SS1.2.p2.11.m11.1.1">subscript</csymbol><apply id="S6.SS1.2.p2.11.m11.1.1.2.cmml" xref="S6.SS1.2.p2.11.m11.1.1.2"><ci id="S6.SS1.2.p2.11.m11.1.1.2.1.cmml" xref="S6.SS1.2.p2.11.m11.1.1.2.1">¯</ci><ci id="S6.SS1.2.p2.11.m11.1.1.2.2.cmml" xref="S6.SS1.2.p2.11.m11.1.1.2.2">Tr</ci></apply><ci id="S6.SS1.2.p2.11.m11.1.1.3.cmml" xref="S6.SS1.2.p2.11.m11.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.2.p2.11.m11.1c">\overline{\operatorname{Tr}}_{m}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.2.p2.11.m11.1d">over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT</annotation></semantics></math> from Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem5" title="Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.5</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.I3.i1" title="item i ‣ Proposition 4.5. ‣ 4.1. Trace spaces of weighted spaces on the half-space ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>.</p> </div> <div class="ltx_para" id="S6.SS1.3.p3"> <p class="ltx_p" id="S6.SS1.3.p3.2">The claim proves that the mapping</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="R:H^{s,p}_{\mathbb{R}^{d}_{+}}(\mathbb{R}^{d},w_{\gamma};X)\times H^{s,p}_{% \mathbb{R}^{d}_{-}}(\mathbb{R}^{d},w_{\gamma};X)\to H^{s,p}_{0}(\mathbb{R}^{d}% ,w_{\gamma};X),\qquad R(g,h):=g+h," class="ltx_Math" display="block" id="S6.Ex2.m1.12"><semantics id="S6.Ex2.m1.12a"><mrow id="S6.Ex2.m1.12.12.1" xref="S6.Ex2.m1.12.12.1.1.cmml"><mrow id="S6.Ex2.m1.12.12.1.1" xref="S6.Ex2.m1.12.12.1.1.cmml"><mi id="S6.Ex2.m1.12.12.1.1.4" 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xref="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.4.3.3.cmml">+</mo><mi id="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.4.3.2.3" xref="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.4.3.2.3.cmml">d</mi></msubsup><mrow id="S6.Ex2.m1.2.2.2.4" xref="S6.Ex2.m1.2.2.2.3.cmml"><mi id="S6.Ex2.m1.1.1.1.1" xref="S6.Ex2.m1.1.1.1.1.cmml">s</mi><mo id="S6.Ex2.m1.2.2.2.4.1" xref="S6.Ex2.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex2.m1.2.2.2.2" xref="S6.Ex2.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.3" xref="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.3.cmml">⁢</mo><mrow id="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.2.2" xref="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.2.3.cmml"><mo id="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.2.2.3" stretchy="false" xref="S6.Ex2.m1.12.12.1.1.1.1.1.2.2.2.2.3.cmml">(</mo><msup id="S6.Ex2.m1.12.12.1.1.1.1.1.1.1.1.1.1.1" xref="S6.Ex2.m1.12.12.1.1.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex2.m1.12.12.1.1.1.1.1.1.1.1.1.1.1.2" xref="S6.Ex2.m1.12.12.1.1.1.1.1.1.1.1.1.1.1.2.cmml">ℝ</mi><mi 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xref="S6.Ex2.m1.12.12.1.1.2.2.2.3"><plus id="S6.Ex2.m1.12.12.1.1.2.2.2.3.1.cmml" xref="S6.Ex2.m1.12.12.1.1.2.2.2.3.1"></plus><ci id="S6.Ex2.m1.12.12.1.1.2.2.2.3.2.cmml" xref="S6.Ex2.m1.12.12.1.1.2.2.2.3.2">𝑔</ci><ci id="S6.Ex2.m1.12.12.1.1.2.2.2.3.3.cmml" xref="S6.Ex2.m1.12.12.1.1.2.2.2.3.3">ℎ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex2.m1.12c">R:H^{s,p}_{\mathbb{R}^{d}_{+}}(\mathbb{R}^{d},w_{\gamma};X)\times H^{s,p}_{% \mathbb{R}^{d}_{-}}(\mathbb{R}^{d},w_{\gamma};X)\to H^{s,p}_{0}(\mathbb{R}^{d}% ,w_{\gamma};X),\qquad R(g,h):=g+h,</annotation><annotation encoding="application/x-llamapun" id="S6.Ex2.m1.12d">italic_R : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) × italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_R ( italic_g , italic_h ) := italic_g + italic_h ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.3.p3.3">is well defined and continuous. Furthermore, by Propositions <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem6" title="Proposition 4.6. ‣ 4.2.1. Bessel potential spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.6</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.Thmtheorem7" title="Proposition 4.7. ‣ 4.2.1. Bessel potential spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.7</span></a> the mapping</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex3"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S:H^{s,p}_{0}(\mathbb{R}^{d},w_{\gamma};X)\to H^{s,p}_{\mathbb{R}^{d}_{+}}(% \mathbb{R}^{d},w_{\gamma};X)\times H^{s,p}_{\mathbb{R}^{d}_{-}}(\mathbb{R}^{d}% ,w_{\gamma};X),\qquad Sf:=(\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}f,% \operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{-}}f)," class="ltx_Math" display="block" id="S6.Ex3.m1.10"><semantics id="S6.Ex3.m1.10a"><mrow id="S6.Ex3.m1.10.10.1" xref="S6.Ex3.m1.10.10.1.1.cmml"><mrow id="S6.Ex3.m1.10.10.1.1" xref="S6.Ex3.m1.10.10.1.1.cmml"><mi id="S6.Ex3.m1.10.10.1.1.4" xref="S6.Ex3.m1.10.10.1.1.4.cmml">S</mi><mo id="S6.Ex3.m1.10.10.1.1.3" lspace="0.278em" rspace="0.278em" xref="S6.Ex3.m1.10.10.1.1.3.cmml">:</mo><mrow id="S6.Ex3.m1.10.10.1.1.2.2" xref="S6.Ex3.m1.10.10.1.1.2.3.cmml"><mrow id="S6.Ex3.m1.10.10.1.1.1.1.1" xref="S6.Ex3.m1.10.10.1.1.1.1.1.cmml"><mrow id="S6.Ex3.m1.10.10.1.1.1.1.1.2" xref="S6.Ex3.m1.10.10.1.1.1.1.1.2.cmml"><msubsup id="S6.Ex3.m1.10.10.1.1.1.1.1.2.4" xref="S6.Ex3.m1.10.10.1.1.1.1.1.2.4.cmml"><mi id="S6.Ex3.m1.10.10.1.1.1.1.1.2.4.2.2" xref="S6.Ex3.m1.10.10.1.1.1.1.1.2.4.2.2.cmml">H</mi><mn id="S6.Ex3.m1.10.10.1.1.1.1.1.2.4.3" xref="S6.Ex3.m1.10.10.1.1.1.1.1.2.4.3.cmml">0</mn><mrow id="S6.Ex3.m1.2.2.2.4" xref="S6.Ex3.m1.2.2.2.3.cmml"><mi id="S6.Ex3.m1.1.1.1.1" xref="S6.Ex3.m1.1.1.1.1.cmml">s</mi><mo id="S6.Ex3.m1.2.2.2.4.1" xref="S6.Ex3.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex3.m1.2.2.2.2" xref="S6.Ex3.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S6.Ex3.m1.10.10.1.1.1.1.1.2.3" xref="S6.Ex3.m1.10.10.1.1.1.1.1.2.3.cmml">⁢</mo><mrow id="S6.Ex3.m1.10.10.1.1.1.1.1.2.2.2" 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xref="S6.Ex3.m1.10.10.1.1.2.2.2.2.2.2.2">𝑓</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex3.m1.10c">S:H^{s,p}_{0}(\mathbb{R}^{d},w_{\gamma};X)\to H^{s,p}_{\mathbb{R}^{d}_{+}}(% \mathbb{R}^{d},w_{\gamma};X)\times H^{s,p}_{\mathbb{R}^{d}_{-}}(\mathbb{R}^{d}% ,w_{\gamma};X),\qquad Sf:=(\operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{+}}f,% \operatorname{\mathbf{1}}_{\mathbb{R}^{d}_{-}}f),</annotation><annotation encoding="application/x-llamapun" id="S6.Ex3.m1.10d">italic_S : italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) × italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_S italic_f := ( bold_1 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f , bold_1 start_POSTSUBSCRIPT blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT - end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.3.p3.1">is well defined and continuous. Since <math alttext="R^{-1}=S" class="ltx_Math" display="inline" id="S6.SS1.3.p3.1.m1.1"><semantics id="S6.SS1.3.p3.1.m1.1a"><mrow id="S6.SS1.3.p3.1.m1.1.1" xref="S6.SS1.3.p3.1.m1.1.1.cmml"><msup id="S6.SS1.3.p3.1.m1.1.1.2" xref="S6.SS1.3.p3.1.m1.1.1.2.cmml"><mi id="S6.SS1.3.p3.1.m1.1.1.2.2" xref="S6.SS1.3.p3.1.m1.1.1.2.2.cmml">R</mi><mrow id="S6.SS1.3.p3.1.m1.1.1.2.3" xref="S6.SS1.3.p3.1.m1.1.1.2.3.cmml"><mo id="S6.SS1.3.p3.1.m1.1.1.2.3a" xref="S6.SS1.3.p3.1.m1.1.1.2.3.cmml">−</mo><mn id="S6.SS1.3.p3.1.m1.1.1.2.3.2" xref="S6.SS1.3.p3.1.m1.1.1.2.3.2.cmml">1</mn></mrow></msup><mo id="S6.SS1.3.p3.1.m1.1.1.1" xref="S6.SS1.3.p3.1.m1.1.1.1.cmml">=</mo><mi id="S6.SS1.3.p3.1.m1.1.1.3" xref="S6.SS1.3.p3.1.m1.1.1.3.cmml">S</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.3.p3.1.m1.1b"><apply id="S6.SS1.3.p3.1.m1.1.1.cmml" xref="S6.SS1.3.p3.1.m1.1.1"><eq id="S6.SS1.3.p3.1.m1.1.1.1.cmml" xref="S6.SS1.3.p3.1.m1.1.1.1"></eq><apply id="S6.SS1.3.p3.1.m1.1.1.2.cmml" xref="S6.SS1.3.p3.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.3.p3.1.m1.1.1.2.1.cmml" xref="S6.SS1.3.p3.1.m1.1.1.2">superscript</csymbol><ci id="S6.SS1.3.p3.1.m1.1.1.2.2.cmml" xref="S6.SS1.3.p3.1.m1.1.1.2.2">𝑅</ci><apply id="S6.SS1.3.p3.1.m1.1.1.2.3.cmml" xref="S6.SS1.3.p3.1.m1.1.1.2.3"><minus id="S6.SS1.3.p3.1.m1.1.1.2.3.1.cmml" xref="S6.SS1.3.p3.1.m1.1.1.2.3"></minus><cn id="S6.SS1.3.p3.1.m1.1.1.2.3.2.cmml" type="integer" xref="S6.SS1.3.p3.1.m1.1.1.2.3.2">1</cn></apply></apply><ci id="S6.SS1.3.p3.1.m1.1.1.3.cmml" xref="S6.SS1.3.p3.1.m1.1.1.3">𝑆</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.3.p3.1.m1.1c">R^{-1}=S</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.3.p3.1.m1.1d">italic_R start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = italic_S</annotation></semantics></math> the result now follows from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.7]</cite>.</p> </div> <div class="ltx_para" id="S6.SS1.4.p4"> <p class="ltx_p" id="S6.SS1.4.p4.6"><span class="ltx_text ltx_font_italic" id="S6.SS1.4.p4.1.1">Step 2: the case <math alttext="\mathcal{O}=\mathbb{R}^{d}_{+}" class="ltx_Math" display="inline" id="S6.SS1.4.p4.1.1.m1.1"><semantics id="S6.SS1.4.p4.1.1.m1.1a"><mrow id="S6.SS1.4.p4.1.1.m1.1.1" xref="S6.SS1.4.p4.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS1.4.p4.1.1.m1.1.1.2" xref="S6.SS1.4.p4.1.1.m1.1.1.2.cmml">𝒪</mi><mo id="S6.SS1.4.p4.1.1.m1.1.1.1" xref="S6.SS1.4.p4.1.1.m1.1.1.1.cmml">=</mo><msubsup id="S6.SS1.4.p4.1.1.m1.1.1.3" xref="S6.SS1.4.p4.1.1.m1.1.1.3.cmml"><mi id="S6.SS1.4.p4.1.1.m1.1.1.3.2.2" xref="S6.SS1.4.p4.1.1.m1.1.1.3.2.2.cmml">ℝ</mi><mo id="S6.SS1.4.p4.1.1.m1.1.1.3.3" xref="S6.SS1.4.p4.1.1.m1.1.1.3.3.cmml">+</mo><mi id="S6.SS1.4.p4.1.1.m1.1.1.3.2.3" xref="S6.SS1.4.p4.1.1.m1.1.1.3.2.3.cmml">d</mi></msubsup></mrow><annotation-xml encoding="MathML-Content" 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id="S6.SS1.4.p4.1.1.m1.1c">\mathcal{O}=\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.4.p4.1.1.m1.1d">caligraphic_O = blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math>. </span>Let <math alttext="m\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS1.4.p4.2.m1.1"><semantics id="S6.SS1.4.p4.2.m1.1a"><mrow id="S6.SS1.4.p4.2.m1.1.1" xref="S6.SS1.4.p4.2.m1.1.1.cmml"><mi id="S6.SS1.4.p4.2.m1.1.1.2" xref="S6.SS1.4.p4.2.m1.1.1.2.cmml">m</mi><mo id="S6.SS1.4.p4.2.m1.1.1.1" xref="S6.SS1.4.p4.2.m1.1.1.1.cmml">∈</mo><msub id="S6.SS1.4.p4.2.m1.1.1.3" xref="S6.SS1.4.p4.2.m1.1.1.3.cmml"><mi id="S6.SS1.4.p4.2.m1.1.1.3.2" xref="S6.SS1.4.p4.2.m1.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.4.p4.2.m1.1.1.3.3" xref="S6.SS1.4.p4.2.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.4.p4.2.m1.1b"><apply id="S6.SS1.4.p4.2.m1.1.1.cmml" xref="S6.SS1.4.p4.2.m1.1.1"><in id="S6.SS1.4.p4.2.m1.1.1.1.cmml" xref="S6.SS1.4.p4.2.m1.1.1.1"></in><ci id="S6.SS1.4.p4.2.m1.1.1.2.cmml" xref="S6.SS1.4.p4.2.m1.1.1.2">𝑚</ci><apply id="S6.SS1.4.p4.2.m1.1.1.3.cmml" xref="S6.SS1.4.p4.2.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.4.p4.2.m1.1.1.3.1.cmml" xref="S6.SS1.4.p4.2.m1.1.1.3">subscript</csymbol><ci id="S6.SS1.4.p4.2.m1.1.1.3.2.cmml" xref="S6.SS1.4.p4.2.m1.1.1.3.2">ℕ</ci><cn id="S6.SS1.4.p4.2.m1.1.1.3.3.cmml" type="integer" xref="S6.SS1.4.p4.2.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.4.p4.2.m1.1c">m\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.4.p4.2.m1.1d">italic_m ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be the smallest integer such that <math alttext="m\geq\max\{|s_{0}|,|s_{1}|\}" class="ltx_Math" display="inline" id="S6.SS1.4.p4.3.m2.3"><semantics id="S6.SS1.4.p4.3.m2.3a"><mrow id="S6.SS1.4.p4.3.m2.3.3" xref="S6.SS1.4.p4.3.m2.3.3.cmml"><mi id="S6.SS1.4.p4.3.m2.3.3.4" xref="S6.SS1.4.p4.3.m2.3.3.4.cmml">m</mi><mo id="S6.SS1.4.p4.3.m2.3.3.3" xref="S6.SS1.4.p4.3.m2.3.3.3.cmml">≥</mo><mrow id="S6.SS1.4.p4.3.m2.3.3.2.2" xref="S6.SS1.4.p4.3.m2.3.3.2.3.cmml"><mi id="S6.SS1.4.p4.3.m2.1.1" xref="S6.SS1.4.p4.3.m2.1.1.cmml">max</mi><mo id="S6.SS1.4.p4.3.m2.3.3.2.2a" xref="S6.SS1.4.p4.3.m2.3.3.2.3.cmml">⁡</mo><mrow id="S6.SS1.4.p4.3.m2.3.3.2.2.2" xref="S6.SS1.4.p4.3.m2.3.3.2.3.cmml"><mo id="S6.SS1.4.p4.3.m2.3.3.2.2.2.3" stretchy="false" xref="S6.SS1.4.p4.3.m2.3.3.2.3.cmml">{</mo><mrow id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.2.cmml"><mo id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.2.1.cmml">|</mo><msub id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.cmml"><mi id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.2" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.2.cmml">s</mi><mn id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.3" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.3.cmml">0</mn></msub><mo id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.2.1.cmml">|</mo></mrow><mo id="S6.SS1.4.p4.3.m2.3.3.2.2.2.4" xref="S6.SS1.4.p4.3.m2.3.3.2.3.cmml">,</mo><mrow id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.2.cmml"><mo id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.2" stretchy="false" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.2.1.cmml">|</mo><msub id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.cmml"><mi id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.2" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.2.cmml">s</mi><mn id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.3" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.3.cmml">1</mn></msub><mo id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.3" stretchy="false" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.2.1.cmml">|</mo></mrow><mo id="S6.SS1.4.p4.3.m2.3.3.2.2.2.5" stretchy="false" xref="S6.SS1.4.p4.3.m2.3.3.2.3.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.4.p4.3.m2.3b"><apply id="S6.SS1.4.p4.3.m2.3.3.cmml" xref="S6.SS1.4.p4.3.m2.3.3"><geq id="S6.SS1.4.p4.3.m2.3.3.3.cmml" xref="S6.SS1.4.p4.3.m2.3.3.3"></geq><ci id="S6.SS1.4.p4.3.m2.3.3.4.cmml" xref="S6.SS1.4.p4.3.m2.3.3.4">𝑚</ci><apply id="S6.SS1.4.p4.3.m2.3.3.2.3.cmml" xref="S6.SS1.4.p4.3.m2.3.3.2.2"><max id="S6.SS1.4.p4.3.m2.1.1.cmml" xref="S6.SS1.4.p4.3.m2.1.1"></max><apply id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.2.cmml" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1"><abs id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.2.1.cmml" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.2"></abs><apply id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.cmml" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.1.cmml" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.2.cmml" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.2">𝑠</ci><cn id="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.4.p4.3.m2.2.2.1.1.1.1.1.1.3">0</cn></apply></apply><apply id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.2.cmml" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1"><abs id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.2.1.cmml" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.2"></abs><apply id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.cmml" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1"><csymbol cd="ambiguous" id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.1.cmml" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1">subscript</csymbol><ci id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.2.cmml" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.2">𝑠</ci><cn id="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.3.cmml" type="integer" xref="S6.SS1.4.p4.3.m2.3.3.2.2.2.2.1.1.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.4.p4.3.m2.3c">m\geq\max\{|s_{0}|,|s_{1}|\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.4.p4.3.m2.3d">italic_m ≥ roman_max { | italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | , | italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | }</annotation></semantics></math> and let <math alttext="s>-1+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS1.4.p4.4.m3.1"><semantics id="S6.SS1.4.p4.4.m3.1a"><mrow id="S6.SS1.4.p4.4.m3.1.1" xref="S6.SS1.4.p4.4.m3.1.1.cmml"><mi id="S6.SS1.4.p4.4.m3.1.1.2" xref="S6.SS1.4.p4.4.m3.1.1.2.cmml">s</mi><mo id="S6.SS1.4.p4.4.m3.1.1.1" xref="S6.SS1.4.p4.4.m3.1.1.1.cmml">></mo><mrow id="S6.SS1.4.p4.4.m3.1.1.3" xref="S6.SS1.4.p4.4.m3.1.1.3.cmml"><mrow id="S6.SS1.4.p4.4.m3.1.1.3.2" xref="S6.SS1.4.p4.4.m3.1.1.3.2.cmml"><mo id="S6.SS1.4.p4.4.m3.1.1.3.2a" xref="S6.SS1.4.p4.4.m3.1.1.3.2.cmml">−</mo><mn id="S6.SS1.4.p4.4.m3.1.1.3.2.2" xref="S6.SS1.4.p4.4.m3.1.1.3.2.2.cmml">1</mn></mrow><mo id="S6.SS1.4.p4.4.m3.1.1.3.1" xref="S6.SS1.4.p4.4.m3.1.1.3.1.cmml">+</mo><mfrac id="S6.SS1.4.p4.4.m3.1.1.3.3" xref="S6.SS1.4.p4.4.m3.1.1.3.3.cmml"><mrow id="S6.SS1.4.p4.4.m3.1.1.3.3.2" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2.cmml"><mi id="S6.SS1.4.p4.4.m3.1.1.3.3.2.2" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS1.4.p4.4.m3.1.1.3.3.2.1" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS1.4.p4.4.m3.1.1.3.3.2.3" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS1.4.p4.4.m3.1.1.3.3.3" xref="S6.SS1.4.p4.4.m3.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.4.p4.4.m3.1b"><apply id="S6.SS1.4.p4.4.m3.1.1.cmml" xref="S6.SS1.4.p4.4.m3.1.1"><gt id="S6.SS1.4.p4.4.m3.1.1.1.cmml" xref="S6.SS1.4.p4.4.m3.1.1.1"></gt><ci id="S6.SS1.4.p4.4.m3.1.1.2.cmml" xref="S6.SS1.4.p4.4.m3.1.1.2">𝑠</ci><apply id="S6.SS1.4.p4.4.m3.1.1.3.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3"><plus id="S6.SS1.4.p4.4.m3.1.1.3.1.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.1"></plus><apply id="S6.SS1.4.p4.4.m3.1.1.3.2.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.2"><minus id="S6.SS1.4.p4.4.m3.1.1.3.2.1.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.2"></minus><cn id="S6.SS1.4.p4.4.m3.1.1.3.2.2.cmml" type="integer" xref="S6.SS1.4.p4.4.m3.1.1.3.2.2">1</cn></apply><apply id="S6.SS1.4.p4.4.m3.1.1.3.3.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.3"><divide id="S6.SS1.4.p4.4.m3.1.1.3.3.1.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.3"></divide><apply id="S6.SS1.4.p4.4.m3.1.1.3.3.2.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2"><plus id="S6.SS1.4.p4.4.m3.1.1.3.3.2.1.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2.1"></plus><ci id="S6.SS1.4.p4.4.m3.1.1.3.3.2.2.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS1.4.p4.4.m3.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS1.4.p4.4.m3.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS1.4.p4.4.m3.1.1.3.3.3.cmml" xref="S6.SS1.4.p4.4.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.4.p4.4.m3.1c">s>-1+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.4.p4.4.m3.1d">italic_s > - 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> such that <math alttext="|s|\leq m" class="ltx_Math" display="inline" id="S6.SS1.4.p4.5.m4.1"><semantics id="S6.SS1.4.p4.5.m4.1a"><mrow id="S6.SS1.4.p4.5.m4.1.2" xref="S6.SS1.4.p4.5.m4.1.2.cmml"><mrow id="S6.SS1.4.p4.5.m4.1.2.2.2" xref="S6.SS1.4.p4.5.m4.1.2.2.1.cmml"><mo id="S6.SS1.4.p4.5.m4.1.2.2.2.1" stretchy="false" xref="S6.SS1.4.p4.5.m4.1.2.2.1.1.cmml">|</mo><mi id="S6.SS1.4.p4.5.m4.1.1" xref="S6.SS1.4.p4.5.m4.1.1.cmml">s</mi><mo id="S6.SS1.4.p4.5.m4.1.2.2.2.2" stretchy="false" xref="S6.SS1.4.p4.5.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="S6.SS1.4.p4.5.m4.1.2.1" xref="S6.SS1.4.p4.5.m4.1.2.1.cmml">≤</mo><mi id="S6.SS1.4.p4.5.m4.1.2.3" xref="S6.SS1.4.p4.5.m4.1.2.3.cmml">m</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.4.p4.5.m4.1b"><apply id="S6.SS1.4.p4.5.m4.1.2.cmml" xref="S6.SS1.4.p4.5.m4.1.2"><leq id="S6.SS1.4.p4.5.m4.1.2.1.cmml" xref="S6.SS1.4.p4.5.m4.1.2.1"></leq><apply id="S6.SS1.4.p4.5.m4.1.2.2.1.cmml" xref="S6.SS1.4.p4.5.m4.1.2.2.2"><abs id="S6.SS1.4.p4.5.m4.1.2.2.1.1.cmml" xref="S6.SS1.4.p4.5.m4.1.2.2.2.1"></abs><ci id="S6.SS1.4.p4.5.m4.1.1.cmml" xref="S6.SS1.4.p4.5.m4.1.1">𝑠</ci></apply><ci id="S6.SS1.4.p4.5.m4.1.2.3.cmml" xref="S6.SS1.4.p4.5.m4.1.2.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.4.p4.5.m4.1c">|s|\leq m</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.4.p4.5.m4.1d">| italic_s | ≤ italic_m</annotation></semantics></math> and <math alttext="s\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS1.4.p4.6.m5.1"><semantics id="S6.SS1.4.p4.6.m5.1a"><mrow id="S6.SS1.4.p4.6.m5.1.1" xref="S6.SS1.4.p4.6.m5.1.1.cmml"><mi id="S6.SS1.4.p4.6.m5.1.1.2" xref="S6.SS1.4.p4.6.m5.1.1.2.cmml">s</mi><mo id="S6.SS1.4.p4.6.m5.1.1.1" xref="S6.SS1.4.p4.6.m5.1.1.1.cmml">∉</mo><mrow id="S6.SS1.4.p4.6.m5.1.1.3" xref="S6.SS1.4.p4.6.m5.1.1.3.cmml"><msub id="S6.SS1.4.p4.6.m5.1.1.3.2" xref="S6.SS1.4.p4.6.m5.1.1.3.2.cmml"><mi id="S6.SS1.4.p4.6.m5.1.1.3.2.2" xref="S6.SS1.4.p4.6.m5.1.1.3.2.2.cmml">ℕ</mi><mn id="S6.SS1.4.p4.6.m5.1.1.3.2.3" xref="S6.SS1.4.p4.6.m5.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS1.4.p4.6.m5.1.1.3.1" xref="S6.SS1.4.p4.6.m5.1.1.3.1.cmml">+</mo><mfrac id="S6.SS1.4.p4.6.m5.1.1.3.3" xref="S6.SS1.4.p4.6.m5.1.1.3.3.cmml"><mrow id="S6.SS1.4.p4.6.m5.1.1.3.3.2" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2.cmml"><mi id="S6.SS1.4.p4.6.m5.1.1.3.3.2.2" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS1.4.p4.6.m5.1.1.3.3.2.1" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS1.4.p4.6.m5.1.1.3.3.2.3" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS1.4.p4.6.m5.1.1.3.3.3" xref="S6.SS1.4.p4.6.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.4.p4.6.m5.1b"><apply id="S6.SS1.4.p4.6.m5.1.1.cmml" xref="S6.SS1.4.p4.6.m5.1.1"><notin id="S6.SS1.4.p4.6.m5.1.1.1.cmml" xref="S6.SS1.4.p4.6.m5.1.1.1"></notin><ci id="S6.SS1.4.p4.6.m5.1.1.2.cmml" xref="S6.SS1.4.p4.6.m5.1.1.2">𝑠</ci><apply id="S6.SS1.4.p4.6.m5.1.1.3.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3"><plus id="S6.SS1.4.p4.6.m5.1.1.3.1.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.1"></plus><apply id="S6.SS1.4.p4.6.m5.1.1.3.2.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS1.4.p4.6.m5.1.1.3.2.1.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.2">subscript</csymbol><ci id="S6.SS1.4.p4.6.m5.1.1.3.2.2.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.2.2">ℕ</ci><cn id="S6.SS1.4.p4.6.m5.1.1.3.2.3.cmml" type="integer" xref="S6.SS1.4.p4.6.m5.1.1.3.2.3">0</cn></apply><apply id="S6.SS1.4.p4.6.m5.1.1.3.3.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.3"><divide id="S6.SS1.4.p4.6.m5.1.1.3.3.1.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.3"></divide><apply id="S6.SS1.4.p4.6.m5.1.1.3.3.2.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2"><plus id="S6.SS1.4.p4.6.m5.1.1.3.3.2.1.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2.1"></plus><ci id="S6.SS1.4.p4.6.m5.1.1.3.3.2.2.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS1.4.p4.6.m5.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS1.4.p4.6.m5.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS1.4.p4.6.m5.1.1.3.3.3.cmml" xref="S6.SS1.4.p4.6.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.4.p4.6.m5.1c">s\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.4.p4.6.m5.1d">italic_s ∉ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Moreover, let</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{E}_{+}^{m}\in\mathcal{L}(H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X),H^{% s,p}(\mathbb{R}^{d},w_{\gamma};X))" class="ltx_Math" display="block" id="S6.Ex4.m1.8"><semantics id="S6.Ex4.m1.8a"><mrow id="S6.Ex4.m1.8.8" xref="S6.Ex4.m1.8.8.cmml"><msubsup id="S6.Ex4.m1.8.8.4" xref="S6.Ex4.m1.8.8.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex4.m1.8.8.4.2.2" xref="S6.Ex4.m1.8.8.4.2.2.cmml">ℰ</mi><mo id="S6.Ex4.m1.8.8.4.2.3" xref="S6.Ex4.m1.8.8.4.2.3.cmml">+</mo><mi id="S6.Ex4.m1.8.8.4.3" xref="S6.Ex4.m1.8.8.4.3.cmml">m</mi></msubsup><mo id="S6.Ex4.m1.8.8.3" xref="S6.Ex4.m1.8.8.3.cmml">∈</mo><mrow id="S6.Ex4.m1.8.8.2" xref="S6.Ex4.m1.8.8.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex4.m1.8.8.2.4" xref="S6.Ex4.m1.8.8.2.4.cmml">ℒ</mi><mo id="S6.Ex4.m1.8.8.2.3" xref="S6.Ex4.m1.8.8.2.3.cmml">⁢</mo><mrow id="S6.Ex4.m1.8.8.2.2.2" xref="S6.Ex4.m1.8.8.2.2.3.cmml"><mo id="S6.Ex4.m1.8.8.2.2.2.3" stretchy="false" xref="S6.Ex4.m1.8.8.2.2.3.cmml">(</mo><mrow id="S6.Ex4.m1.7.7.1.1.1.1" xref="S6.Ex4.m1.7.7.1.1.1.1.cmml"><msup id="S6.Ex4.m1.7.7.1.1.1.1.4" xref="S6.Ex4.m1.7.7.1.1.1.1.4.cmml"><mi id="S6.Ex4.m1.7.7.1.1.1.1.4.2" xref="S6.Ex4.m1.7.7.1.1.1.1.4.2.cmml">H</mi><mrow id="S6.Ex4.m1.2.2.2.4" xref="S6.Ex4.m1.2.2.2.3.cmml"><mi id="S6.Ex4.m1.1.1.1.1" xref="S6.Ex4.m1.1.1.1.1.cmml">s</mi><mo id="S6.Ex4.m1.2.2.2.4.1" xref="S6.Ex4.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex4.m1.2.2.2.2" xref="S6.Ex4.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S6.Ex4.m1.7.7.1.1.1.1.3" xref="S6.Ex4.m1.7.7.1.1.1.1.3.cmml">⁢</mo><mrow id="S6.Ex4.m1.7.7.1.1.1.1.2.2" xref="S6.Ex4.m1.7.7.1.1.1.1.2.3.cmml"><mo id="S6.Ex4.m1.7.7.1.1.1.1.2.2.3" stretchy="false" xref="S6.Ex4.m1.7.7.1.1.1.1.2.3.cmml">(</mo><msubsup id="S6.Ex4.m1.7.7.1.1.1.1.1.1.1" xref="S6.Ex4.m1.7.7.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex4.m1.7.7.1.1.1.1.1.1.1.2.2" xref="S6.Ex4.m1.7.7.1.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex4.m1.7.7.1.1.1.1.1.1.1.3" xref="S6.Ex4.m1.7.7.1.1.1.1.1.1.1.3.cmml">+</mo><mi id="S6.Ex4.m1.7.7.1.1.1.1.1.1.1.2.3" xref="S6.Ex4.m1.7.7.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex4.m1.7.7.1.1.1.1.2.2.4" xref="S6.Ex4.m1.7.7.1.1.1.1.2.3.cmml">,</mo><msub id="S6.Ex4.m1.7.7.1.1.1.1.2.2.2" xref="S6.Ex4.m1.7.7.1.1.1.1.2.2.2.cmml"><mi id="S6.Ex4.m1.7.7.1.1.1.1.2.2.2.2" xref="S6.Ex4.m1.7.7.1.1.1.1.2.2.2.2.cmml">w</mi><mi id="S6.Ex4.m1.7.7.1.1.1.1.2.2.2.3" xref="S6.Ex4.m1.7.7.1.1.1.1.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex4.m1.7.7.1.1.1.1.2.2.5" xref="S6.Ex4.m1.7.7.1.1.1.1.2.3.cmml">;</mo><mi id="S6.Ex4.m1.5.5" xref="S6.Ex4.m1.5.5.cmml">X</mi><mo id="S6.Ex4.m1.7.7.1.1.1.1.2.2.6" stretchy="false" xref="S6.Ex4.m1.7.7.1.1.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex4.m1.8.8.2.2.2.4" xref="S6.Ex4.m1.8.8.2.2.3.cmml">,</mo><mrow id="S6.Ex4.m1.8.8.2.2.2.2" xref="S6.Ex4.m1.8.8.2.2.2.2.cmml"><msup id="S6.Ex4.m1.8.8.2.2.2.2.4" xref="S6.Ex4.m1.8.8.2.2.2.2.4.cmml"><mi id="S6.Ex4.m1.8.8.2.2.2.2.4.2" xref="S6.Ex4.m1.8.8.2.2.2.2.4.2.cmml">H</mi><mrow id="S6.Ex4.m1.4.4.2.4" xref="S6.Ex4.m1.4.4.2.3.cmml"><mi id="S6.Ex4.m1.3.3.1.1" xref="S6.Ex4.m1.3.3.1.1.cmml">s</mi><mo id="S6.Ex4.m1.4.4.2.4.1" xref="S6.Ex4.m1.4.4.2.3.cmml">,</mo><mi id="S6.Ex4.m1.4.4.2.2" xref="S6.Ex4.m1.4.4.2.2.cmml">p</mi></mrow></msup><mo id="S6.Ex4.m1.8.8.2.2.2.2.3" xref="S6.Ex4.m1.8.8.2.2.2.2.3.cmml">⁢</mo><mrow id="S6.Ex4.m1.8.8.2.2.2.2.2.2" xref="S6.Ex4.m1.8.8.2.2.2.2.2.3.cmml"><mo id="S6.Ex4.m1.8.8.2.2.2.2.2.2.3" stretchy="false" xref="S6.Ex4.m1.8.8.2.2.2.2.2.3.cmml">(</mo><msup id="S6.Ex4.m1.8.8.2.2.2.2.1.1.1" xref="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.cmml"><mi id="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.2" xref="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.2.cmml">ℝ</mi><mi id="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.3" xref="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.3.cmml">d</mi></msup><mo id="S6.Ex4.m1.8.8.2.2.2.2.2.2.4" xref="S6.Ex4.m1.8.8.2.2.2.2.2.3.cmml">,</mo><msub id="S6.Ex4.m1.8.8.2.2.2.2.2.2.2" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.cmml"><mi id="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.2" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.2.cmml">w</mi><mi id="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.3" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex4.m1.8.8.2.2.2.2.2.2.5" xref="S6.Ex4.m1.8.8.2.2.2.2.2.3.cmml">;</mo><mi id="S6.Ex4.m1.6.6" xref="S6.Ex4.m1.6.6.cmml">X</mi><mo id="S6.Ex4.m1.8.8.2.2.2.2.2.2.6" stretchy="false" xref="S6.Ex4.m1.8.8.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex4.m1.8.8.2.2.2.5" stretchy="false" xref="S6.Ex4.m1.8.8.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex4.m1.8b"><apply id="S6.Ex4.m1.8.8.cmml" xref="S6.Ex4.m1.8.8"><in id="S6.Ex4.m1.8.8.3.cmml" xref="S6.Ex4.m1.8.8.3"></in><apply id="S6.Ex4.m1.8.8.4.cmml" xref="S6.Ex4.m1.8.8.4"><csymbol cd="ambiguous" 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xref="S6.Ex4.m1.8.8.2.2.2.2.4.2">𝐻</ci><list id="S6.Ex4.m1.4.4.2.3.cmml" xref="S6.Ex4.m1.4.4.2.4"><ci id="S6.Ex4.m1.3.3.1.1.cmml" xref="S6.Ex4.m1.3.3.1.1">𝑠</ci><ci id="S6.Ex4.m1.4.4.2.2.cmml" xref="S6.Ex4.m1.4.4.2.2">𝑝</ci></list></apply><vector id="S6.Ex4.m1.8.8.2.2.2.2.2.3.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2"><apply id="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.1.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.1.1.1">superscript</csymbol><ci id="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.2.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.2">ℝ</ci><ci id="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.3.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.1.1.1.3">𝑑</ci></apply><apply id="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.1.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2.2">subscript</csymbol><ci id="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.2.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.2">𝑤</ci><ci id="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.3.cmml" xref="S6.Ex4.m1.8.8.2.2.2.2.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex4.m1.6.6.cmml" xref="S6.Ex4.m1.6.6">𝑋</ci></vector></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex4.m1.8c">\mathcal{E}_{+}^{m}\in\mathcal{L}(H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X),H^{% s,p}(\mathbb{R}^{d},w_{\gamma};X))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex4.m1.8d">caligraphic_E start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT ∈ caligraphic_L ( italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.4.p4.8">be the reflection operator from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.6]</cite>. Then by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.6]</cite> the mapping</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="S:H_{0}^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\to H^{s,p}_{0}(\mathbb{R}^{d},w% _{\gamma};X),\qquad Sf:=\mathcal{E}_{+}^{m}f," class="ltx_Math" display="block" id="S6.Ex5.m1.7"><semantics id="S6.Ex5.m1.7a"><mrow id="S6.Ex5.m1.7.7.1" xref="S6.Ex5.m1.7.7.1.1.cmml"><mrow id="S6.Ex5.m1.7.7.1.1" xref="S6.Ex5.m1.7.7.1.1.cmml"><mi id="S6.Ex5.m1.7.7.1.1.4" xref="S6.Ex5.m1.7.7.1.1.4.cmml">S</mi><mo id="S6.Ex5.m1.7.7.1.1.3" lspace="0.278em" rspace="0.278em" xref="S6.Ex5.m1.7.7.1.1.3.cmml">:</mo><mrow id="S6.Ex5.m1.7.7.1.1.2.2" xref="S6.Ex5.m1.7.7.1.1.2.3.cmml"><mrow 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id="S6.Ex5.m1.7d">italic_S : italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_S italic_f := caligraphic_E start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m end_POSTSUPERSCRIPT italic_f ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.4.p4.7">is well defined and continuous. 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xref="S6.SS1.4.p4.7.m1.10.10.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.4.p4.7.m1.10.10.4.4.2.2.2.1.cmml" xref="S6.SS1.4.p4.7.m1.10.10.4.4.2.2.2">subscript</csymbol><ci id="S6.SS1.4.p4.7.m1.10.10.4.4.2.2.2.2.cmml" xref="S6.SS1.4.p4.7.m1.10.10.4.4.2.2.2.2">𝑤</ci><ci id="S6.SS1.4.p4.7.m1.10.10.4.4.2.2.2.3.cmml" xref="S6.SS1.4.p4.7.m1.10.10.4.4.2.2.2.3">𝛾</ci></apply><ci id="S6.SS1.4.p4.7.m1.6.6.cmml" xref="S6.SS1.4.p4.7.m1.6.6">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.4.p4.7.m1.10c">R:H_{0}^{s,p}(\mathbb{R}^{d},w_{\gamma};X)\to H^{s,p}_{0}(\mathbb{R}^{d}_{+},w% _{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.4.p4.7.m1.10d">italic_R : italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) → italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> be the restriction operator. The result now follows from Step 1 and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Lemma 5.3]</cite>. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS1.p2"> <p class="ltx_p" id="S6.SS1.p2.2">We continue with an interpolation result for the weighted Sobolev spaces <math alttext="W_{0}^{k,p}" class="ltx_Math" display="inline" id="S6.SS1.p2.1.m1.2"><semantics id="S6.SS1.p2.1.m1.2a"><msubsup id="S6.SS1.p2.1.m1.2.3" xref="S6.SS1.p2.1.m1.2.3.cmml"><mi id="S6.SS1.p2.1.m1.2.3.2.2" xref="S6.SS1.p2.1.m1.2.3.2.2.cmml">W</mi><mn id="S6.SS1.p2.1.m1.2.3.2.3" xref="S6.SS1.p2.1.m1.2.3.2.3.cmml">0</mn><mrow id="S6.SS1.p2.1.m1.2.2.2.4" xref="S6.SS1.p2.1.m1.2.2.2.3.cmml"><mi id="S6.SS1.p2.1.m1.1.1.1.1" xref="S6.SS1.p2.1.m1.1.1.1.1.cmml">k</mi><mo id="S6.SS1.p2.1.m1.2.2.2.4.1" xref="S6.SS1.p2.1.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS1.p2.1.m1.2.2.2.2" xref="S6.SS1.p2.1.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.1.m1.2b"><apply id="S6.SS1.p2.1.m1.2.3.cmml" xref="S6.SS1.p2.1.m1.2.3"><csymbol cd="ambiguous" id="S6.SS1.p2.1.m1.2.3.1.cmml" xref="S6.SS1.p2.1.m1.2.3">superscript</csymbol><apply id="S6.SS1.p2.1.m1.2.3.2.cmml" xref="S6.SS1.p2.1.m1.2.3"><csymbol cd="ambiguous" id="S6.SS1.p2.1.m1.2.3.2.1.cmml" xref="S6.SS1.p2.1.m1.2.3">subscript</csymbol><ci id="S6.SS1.p2.1.m1.2.3.2.2.cmml" xref="S6.SS1.p2.1.m1.2.3.2.2">𝑊</ci><cn id="S6.SS1.p2.1.m1.2.3.2.3.cmml" type="integer" xref="S6.SS1.p2.1.m1.2.3.2.3">0</cn></apply><list id="S6.SS1.p2.1.m1.2.2.2.3.cmml" xref="S6.SS1.p2.1.m1.2.2.2.4"><ci id="S6.SS1.p2.1.m1.1.1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1.1.1">𝑘</ci><ci id="S6.SS1.p2.1.m1.2.2.2.2.cmml" xref="S6.SS1.p2.1.m1.2.2.2.2">𝑝</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.1.m1.2c">W_{0}^{k,p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.1.m1.2d">italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>, see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S4.SS2.SSS2" title="4.2.2. Sobolev spaces ‣ 4.2. Density results ‣ 4. Trace spaces of Bessel potential and Sobolev spaces ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">4.2.2</span></a>. As a consequence, we obtain interpolation of the spaces <math alttext="W^{k,p}" class="ltx_Math" display="inline" id="S6.SS1.p2.2.m2.2"><semantics id="S6.SS1.p2.2.m2.2a"><msup id="S6.SS1.p2.2.m2.2.3" xref="S6.SS1.p2.2.m2.2.3.cmml"><mi id="S6.SS1.p2.2.m2.2.3.2" xref="S6.SS1.p2.2.m2.2.3.2.cmml">W</mi><mrow id="S6.SS1.p2.2.m2.2.2.2.4" xref="S6.SS1.p2.2.m2.2.2.2.3.cmml"><mi id="S6.SS1.p2.2.m2.1.1.1.1" xref="S6.SS1.p2.2.m2.1.1.1.1.cmml">k</mi><mo id="S6.SS1.p2.2.m2.2.2.2.4.1" xref="S6.SS1.p2.2.m2.2.2.2.3.cmml">,</mo><mi id="S6.SS1.p2.2.m2.2.2.2.2" xref="S6.SS1.p2.2.m2.2.2.2.2.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.2.m2.2b"><apply id="S6.SS1.p2.2.m2.2.3.cmml" xref="S6.SS1.p2.2.m2.2.3"><csymbol cd="ambiguous" id="S6.SS1.p2.2.m2.2.3.1.cmml" xref="S6.SS1.p2.2.m2.2.3">superscript</csymbol><ci id="S6.SS1.p2.2.m2.2.3.2.cmml" xref="S6.SS1.p2.2.m2.2.3.2">𝑊</ci><list id="S6.SS1.p2.2.m2.2.2.2.3.cmml" xref="S6.SS1.p2.2.m2.2.2.2.4"><ci id="S6.SS1.p2.2.m2.1.1.1.1.cmml" xref="S6.SS1.p2.2.m2.1.1.1.1">𝑘</ci><ci id="S6.SS1.p2.2.m2.2.2.2.2.cmml" xref="S6.SS1.p2.2.m2.2.2.2.2">𝑝</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.2.m2.2c">W^{k,p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p2.2.m2.2d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> as well under suitable restrictions for the weight.</p> </div> <div class="ltx_para" id="S6.SS1.p3"> <p class="ltx_p" id="S6.SS1.p3.1">For <math alttext="\kappa\in\mathbb{R}" class="ltx_Math" display="inline" id="S6.SS1.p3.1.m1.1"><semantics id="S6.SS1.p3.1.m1.1a"><mrow id="S6.SS1.p3.1.m1.1.1" xref="S6.SS1.p3.1.m1.1.1.cmml"><mi id="S6.SS1.p3.1.m1.1.1.2" xref="S6.SS1.p3.1.m1.1.1.2.cmml">κ</mi><mo id="S6.SS1.p3.1.m1.1.1.1" xref="S6.SS1.p3.1.m1.1.1.1.cmml">∈</mo><mi id="S6.SS1.p3.1.m1.1.1.3" xref="S6.SS1.p3.1.m1.1.1.3.cmml">ℝ</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.1.m1.1b"><apply id="S6.SS1.p3.1.m1.1.1.cmml" xref="S6.SS1.p3.1.m1.1.1"><in id="S6.SS1.p3.1.m1.1.1.1.cmml" xref="S6.SS1.p3.1.m1.1.1.1"></in><ci id="S6.SS1.p3.1.m1.1.1.2.cmml" xref="S6.SS1.p3.1.m1.1.1.2">𝜅</ci><ci id="S6.SS1.p3.1.m1.1.1.3.cmml" xref="S6.SS1.p3.1.m1.1.1.3">ℝ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.1.m1.1c">\kappa\in\mathbb{R}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.1.m1.1d">italic_κ ∈ blackboard_R</annotation></semantics></math> we define the pointwise multiplication operator</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="M^{\kappa}:C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X)\to C_{\mathrm{c}}^{% \infty}(\mathbb{R}^{d}_{+};X)\quad\text{ by }\quad 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id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.2.2.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.2.2.2">𝑀</ci><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.2.2.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.2.2.3">𝜅</ci></apply><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.2.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.2.3">𝑢</ci><ci id="S6.Ex6.m1.3.3.cmml" xref="S6.Ex6.m1.3.3">𝑥</ci></apply><apply id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3"><times id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.1.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.1"></times><apply id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.1.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2">subscript</csymbol><apply id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2"><csymbol cd="ambiguous" id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.2.1.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2">superscript</csymbol><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.2.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.2.2">𝑥</ci><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.2.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.2.3">𝜅</ci></apply><cn id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.3.cmml" type="integer" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.2.3">1</cn></apply><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.1.1.3.3">𝑢</ci><ci id="S6.Ex6.m1.4.4.cmml" xref="S6.Ex6.m1.4.4">𝑥</ci></apply></apply><apply id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2"><in id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.1.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.1"></in><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.2">𝑥</ci><apply id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.1.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3">subscript</csymbol><apply id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.2.1.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3">superscript</csymbol><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.2.2.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.2.2">ℝ</ci><ci id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.2.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.2.3">𝑑</ci></apply><plus id="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.3.cmml" xref="S6.Ex6.m1.6.6.1.1.2.2.2.2.2.3.3"></plus></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex6.m1.6c">M^{\kappa}:C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X)\to C_{\mathrm{c}}^{% \infty}(\mathbb{R}^{d}_{+};X)\quad\text{ by }\quad M^{\kappa}u(x)=x^{\kappa}_{% 1}u(x),\qquad x\in\mathbb{R}^{d}_{+},</annotation><annotation encoding="application/x-llamapun" id="S6.Ex6.m1.6d">italic_M start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT : italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) → italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) by italic_M start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT italic_u ( italic_x ) = italic_x start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_u ( italic_x ) , italic_x ∈ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.p3.5">which extends to an operator on <math alttext="\mathcal{D}^{\prime}(\mathbb{R}^{d}_{+};X)" class="ltx_Math" display="inline" id="S6.SS1.p3.2.m1.2"><semantics id="S6.SS1.p3.2.m1.2a"><mrow id="S6.SS1.p3.2.m1.2.2" xref="S6.SS1.p3.2.m1.2.2.cmml"><msup id="S6.SS1.p3.2.m1.2.2.3" xref="S6.SS1.p3.2.m1.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS1.p3.2.m1.2.2.3.2" xref="S6.SS1.p3.2.m1.2.2.3.2.cmml">𝒟</mi><mo id="S6.SS1.p3.2.m1.2.2.3.3" xref="S6.SS1.p3.2.m1.2.2.3.3.cmml">′</mo></msup><mo id="S6.SS1.p3.2.m1.2.2.2" xref="S6.SS1.p3.2.m1.2.2.2.cmml">⁢</mo><mrow id="S6.SS1.p3.2.m1.2.2.1.1" xref="S6.SS1.p3.2.m1.2.2.1.2.cmml"><mo id="S6.SS1.p3.2.m1.2.2.1.1.2" stretchy="false" xref="S6.SS1.p3.2.m1.2.2.1.2.cmml">(</mo><msubsup id="S6.SS1.p3.2.m1.2.2.1.1.1" xref="S6.SS1.p3.2.m1.2.2.1.1.1.cmml"><mi id="S6.SS1.p3.2.m1.2.2.1.1.1.2.2" xref="S6.SS1.p3.2.m1.2.2.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.SS1.p3.2.m1.2.2.1.1.1.3" xref="S6.SS1.p3.2.m1.2.2.1.1.1.3.cmml">+</mo><mi id="S6.SS1.p3.2.m1.2.2.1.1.1.2.3" xref="S6.SS1.p3.2.m1.2.2.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.SS1.p3.2.m1.2.2.1.1.3" xref="S6.SS1.p3.2.m1.2.2.1.2.cmml">;</mo><mi id="S6.SS1.p3.2.m1.1.1" xref="S6.SS1.p3.2.m1.1.1.cmml">X</mi><mo id="S6.SS1.p3.2.m1.2.2.1.1.4" stretchy="false" xref="S6.SS1.p3.2.m1.2.2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.2.m1.2b"><apply id="S6.SS1.p3.2.m1.2.2.cmml" xref="S6.SS1.p3.2.m1.2.2"><times id="S6.SS1.p3.2.m1.2.2.2.cmml" xref="S6.SS1.p3.2.m1.2.2.2"></times><apply id="S6.SS1.p3.2.m1.2.2.3.cmml" xref="S6.SS1.p3.2.m1.2.2.3"><csymbol cd="ambiguous" id="S6.SS1.p3.2.m1.2.2.3.1.cmml" xref="S6.SS1.p3.2.m1.2.2.3">superscript</csymbol><ci id="S6.SS1.p3.2.m1.2.2.3.2.cmml" xref="S6.SS1.p3.2.m1.2.2.3.2">𝒟</ci><ci id="S6.SS1.p3.2.m1.2.2.3.3.cmml" xref="S6.SS1.p3.2.m1.2.2.3.3">′</ci></apply><list id="S6.SS1.p3.2.m1.2.2.1.2.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1"><apply id="S6.SS1.p3.2.m1.2.2.1.1.1.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.2.m1.2.2.1.1.1.1.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1.1">subscript</csymbol><apply id="S6.SS1.p3.2.m1.2.2.1.1.1.2.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.2.m1.2.2.1.1.1.2.1.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1.1">superscript</csymbol><ci id="S6.SS1.p3.2.m1.2.2.1.1.1.2.2.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1.1.2.2">ℝ</ci><ci id="S6.SS1.p3.2.m1.2.2.1.1.1.2.3.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1.1.2.3">𝑑</ci></apply><plus id="S6.SS1.p3.2.m1.2.2.1.1.1.3.cmml" xref="S6.SS1.p3.2.m1.2.2.1.1.1.3"></plus></apply><ci id="S6.SS1.p3.2.m1.1.1.cmml" xref="S6.SS1.p3.2.m1.1.1">𝑋</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.2.m1.2c">\mathcal{D}^{\prime}(\mathbb{R}^{d}_{+};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.2.m1.2d">caligraphic_D start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> by duality and <math alttext="M^{-\kappa}" class="ltx_Math" display="inline" id="S6.SS1.p3.3.m2.1"><semantics id="S6.SS1.p3.3.m2.1a"><msup id="S6.SS1.p3.3.m2.1.1" xref="S6.SS1.p3.3.m2.1.1.cmml"><mi id="S6.SS1.p3.3.m2.1.1.2" xref="S6.SS1.p3.3.m2.1.1.2.cmml">M</mi><mrow id="S6.SS1.p3.3.m2.1.1.3" xref="S6.SS1.p3.3.m2.1.1.3.cmml"><mo id="S6.SS1.p3.3.m2.1.1.3a" xref="S6.SS1.p3.3.m2.1.1.3.cmml">−</mo><mi id="S6.SS1.p3.3.m2.1.1.3.2" xref="S6.SS1.p3.3.m2.1.1.3.2.cmml">κ</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.3.m2.1b"><apply id="S6.SS1.p3.3.m2.1.1.cmml" xref="S6.SS1.p3.3.m2.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.3.m2.1.1.1.cmml" xref="S6.SS1.p3.3.m2.1.1">superscript</csymbol><ci id="S6.SS1.p3.3.m2.1.1.2.cmml" xref="S6.SS1.p3.3.m2.1.1.2">𝑀</ci><apply id="S6.SS1.p3.3.m2.1.1.3.cmml" xref="S6.SS1.p3.3.m2.1.1.3"><minus id="S6.SS1.p3.3.m2.1.1.3.1.cmml" xref="S6.SS1.p3.3.m2.1.1.3"></minus><ci id="S6.SS1.p3.3.m2.1.1.3.2.cmml" xref="S6.SS1.p3.3.m2.1.1.3.2">𝜅</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.3.m2.1c">M^{-\kappa}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.3.m2.1d">italic_M start_POSTSUPERSCRIPT - italic_κ end_POSTSUPERSCRIPT</annotation></semantics></math> acts as inverse for <math alttext="M^{\kappa}" class="ltx_Math" display="inline" id="S6.SS1.p3.4.m3.1"><semantics id="S6.SS1.p3.4.m3.1a"><msup id="S6.SS1.p3.4.m3.1.1" xref="S6.SS1.p3.4.m3.1.1.cmml"><mi id="S6.SS1.p3.4.m3.1.1.2" xref="S6.SS1.p3.4.m3.1.1.2.cmml">M</mi><mi id="S6.SS1.p3.4.m3.1.1.3" xref="S6.SS1.p3.4.m3.1.1.3.cmml">κ</mi></msup><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.4.m3.1b"><apply id="S6.SS1.p3.4.m3.1.1.cmml" xref="S6.SS1.p3.4.m3.1.1"><csymbol cd="ambiguous" id="S6.SS1.p3.4.m3.1.1.1.cmml" xref="S6.SS1.p3.4.m3.1.1">superscript</csymbol><ci id="S6.SS1.p3.4.m3.1.1.2.cmml" xref="S6.SS1.p3.4.m3.1.1.2">𝑀</ci><ci id="S6.SS1.p3.4.m3.1.1.3.cmml" xref="S6.SS1.p3.4.m3.1.1.3">𝜅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.4.m3.1c">M^{\kappa}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.4.m3.1d">italic_M start_POSTSUPERSCRIPT italic_κ end_POSTSUPERSCRIPT</annotation></semantics></math>. Moreover, we write <math alttext="M:=M^{1}" class="ltx_Math" display="inline" id="S6.SS1.p3.5.m4.1"><semantics id="S6.SS1.p3.5.m4.1a"><mrow id="S6.SS1.p3.5.m4.1.1" xref="S6.SS1.p3.5.m4.1.1.cmml"><mi id="S6.SS1.p3.5.m4.1.1.2" xref="S6.SS1.p3.5.m4.1.1.2.cmml">M</mi><mo id="S6.SS1.p3.5.m4.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.SS1.p3.5.m4.1.1.1.cmml">:=</mo><msup id="S6.SS1.p3.5.m4.1.1.3" xref="S6.SS1.p3.5.m4.1.1.3.cmml"><mi id="S6.SS1.p3.5.m4.1.1.3.2" xref="S6.SS1.p3.5.m4.1.1.3.2.cmml">M</mi><mn id="S6.SS1.p3.5.m4.1.1.3.3" xref="S6.SS1.p3.5.m4.1.1.3.3.cmml">1</mn></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p3.5.m4.1b"><apply id="S6.SS1.p3.5.m4.1.1.cmml" xref="S6.SS1.p3.5.m4.1.1"><csymbol cd="latexml" id="S6.SS1.p3.5.m4.1.1.1.cmml" xref="S6.SS1.p3.5.m4.1.1.1">assign</csymbol><ci id="S6.SS1.p3.5.m4.1.1.2.cmml" xref="S6.SS1.p3.5.m4.1.1.2">𝑀</ci><apply id="S6.SS1.p3.5.m4.1.1.3.cmml" xref="S6.SS1.p3.5.m4.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.p3.5.m4.1.1.3.1.cmml" xref="S6.SS1.p3.5.m4.1.1.3">superscript</csymbol><ci id="S6.SS1.p3.5.m4.1.1.3.2.cmml" xref="S6.SS1.p3.5.m4.1.1.3.2">𝑀</ci><cn id="S6.SS1.p3.5.m4.1.1.3.3.cmml" type="integer" xref="S6.SS1.p3.5.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p3.5.m4.1c">M:=M^{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p3.5.m4.1d">italic_M := italic_M start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem2"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem2.1.1.1">Proposition 6.2</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem2.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem2.p1"> <p class="ltx_p" id="S6.Thmtheorem2.p1.6"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem2.p1.6.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.1.1.m1.2"><semantics id="S6.Thmtheorem2.p1.1.1.m1.2a"><mrow id="S6.Thmtheorem2.p1.1.1.m1.2.3" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.cmml"><mi id="S6.Thmtheorem2.p1.1.1.m1.2.3.2" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S6.Thmtheorem2.p1.1.1.m1.2.3.1" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S6.Thmtheorem2.p1.1.1.m1.2.3.3.2" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml"><mo id="S6.Thmtheorem2.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S6.Thmtheorem2.p1.1.1.m1.1.1" xref="S6.Thmtheorem2.p1.1.1.m1.1.1.cmml">1</mn><mo id="S6.Thmtheorem2.p1.1.1.m1.2.3.3.2.2" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem2.p1.1.1.m1.2.2" mathvariant="normal" xref="S6.Thmtheorem2.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S6.Thmtheorem2.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.1.1.m1.2b"><apply id="S6.Thmtheorem2.p1.1.1.m1.2.3.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.2.3"><in id="S6.Thmtheorem2.p1.1.1.m1.2.3.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.1"></in><ci id="S6.Thmtheorem2.p1.1.1.m1.2.3.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S6.Thmtheorem2.p1.1.1.m1.2.3.3.1.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.2.3.3.2"><cn id="S6.Thmtheorem2.p1.1.1.m1.1.1.cmml" type="integer" xref="S6.Thmtheorem2.p1.1.1.m1.1.1">1</cn><infinity id="S6.Thmtheorem2.p1.1.1.m1.2.2.cmml" xref="S6.Thmtheorem2.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="k_{0}\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.2.2.m2.1"><semantics id="S6.Thmtheorem2.p1.2.2.m2.1a"><mrow id="S6.Thmtheorem2.p1.2.2.m2.1.1" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.cmml"><msub id="S6.Thmtheorem2.p1.2.2.m2.1.1.2" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2.cmml"><mi id="S6.Thmtheorem2.p1.2.2.m2.1.1.2.2" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2.2.cmml">k</mi><mn id="S6.Thmtheorem2.p1.2.2.m2.1.1.2.3" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2.3.cmml">0</mn></msub><mo id="S6.Thmtheorem2.p1.2.2.m2.1.1.1" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.1.cmml">∈</mo><msub id="S6.Thmtheorem2.p1.2.2.m2.1.1.3" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.3.cmml"><mi id="S6.Thmtheorem2.p1.2.2.m2.1.1.3.2" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S6.Thmtheorem2.p1.2.2.m2.1.1.3.3" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.2.2.m2.1b"><apply id="S6.Thmtheorem2.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1"><in id="S6.Thmtheorem2.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.1"></in><apply id="S6.Thmtheorem2.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.2.2.m2.1.1.2.1.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem2.p1.2.2.m2.1.1.2.2.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem2.p1.2.2.m2.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.2.3">0</cn></apply><apply id="S6.Thmtheorem2.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.2.2.m2.1.1.3.1.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem2.p1.2.2.m2.1.1.3.2.cmml" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.3.2">ℕ</ci><cn id="S6.Thmtheorem2.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.2.2.m2.1c">k_{0}\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.2.2.m2.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="k_{1}\in\mathbb{N}_{1}\setminus\{1\}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.3.3.m3.1"><semantics id="S6.Thmtheorem2.p1.3.3.m3.1a"><mrow id="S6.Thmtheorem2.p1.3.3.m3.1.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.cmml"><msub id="S6.Thmtheorem2.p1.3.3.m3.1.2.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2.cmml"><mi id="S6.Thmtheorem2.p1.3.3.m3.1.2.2.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2.2.cmml">k</mi><mn id="S6.Thmtheorem2.p1.3.3.m3.1.2.2.3" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.1" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.1.cmml">∈</mo><mrow id="S6.Thmtheorem2.p1.3.3.m3.1.2.3" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.cmml"><msub id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.cmml"><mi id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.2.cmml">ℕ</mi><mn id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.3" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.1" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.1.cmml">∖</mo><mrow id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.2" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.1.cmml"><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.1.cmml">{</mo><mn id="S6.Thmtheorem2.p1.3.3.m3.1.1" xref="S6.Thmtheorem2.p1.3.3.m3.1.1.cmml">1</mn><mo id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.2.2" stretchy="false" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.3.3.m3.1b"><apply id="S6.Thmtheorem2.p1.3.3.m3.1.2.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2"><in id="S6.Thmtheorem2.p1.3.3.m3.1.2.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.1"></in><apply id="S6.Thmtheorem2.p1.3.3.m3.1.2.2.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.3.3.m3.1.2.2.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem2.p1.3.3.m3.1.2.2.2.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2.2">𝑘</ci><cn id="S6.Thmtheorem2.p1.3.3.m3.1.2.2.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.2.3">1</cn></apply><apply id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3"><setdiff id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.1"></setdiff><apply id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2">subscript</csymbol><ci id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.2.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.2">ℕ</ci><cn id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.2.3">1</cn></apply><set id="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.1.cmml" xref="S6.Thmtheorem2.p1.3.3.m3.1.2.3.3.2"><cn id="S6.Thmtheorem2.p1.3.3.m3.1.1.cmml" type="integer" xref="S6.Thmtheorem2.p1.3.3.m3.1.1">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.3.3.m3.1c">k_{1}\in\mathbb{N}_{1}\setminus\{1\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.3.3.m3.1d">italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∖ { 1 }</annotation></semantics></math>, <math alttext="\ell\in\{1,\dots,k_{1}-1\}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.4.4.m4.3"><semantics id="S6.Thmtheorem2.p1.4.4.m4.3a"><mrow id="S6.Thmtheorem2.p1.4.4.m4.3.3" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.cmml"><mi id="S6.Thmtheorem2.p1.4.4.m4.3.3.3" mathvariant="normal" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.3.cmml">ℓ</mi><mo id="S6.Thmtheorem2.p1.4.4.m4.3.3.2" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.2.cmml">∈</mo><mrow id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml"><mo id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.2" stretchy="false" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml">{</mo><mn id="S6.Thmtheorem2.p1.4.4.m4.1.1" xref="S6.Thmtheorem2.p1.4.4.m4.1.1.cmml">1</mn><mo id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.3" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml">,</mo><mi id="S6.Thmtheorem2.p1.4.4.m4.2.2" mathvariant="normal" xref="S6.Thmtheorem2.p1.4.4.m4.2.2.cmml">…</mi><mo id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.4" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml">,</mo><mrow id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.cmml"><msub id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.cmml"><mi id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.2" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.2.cmml">k</mi><mn id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.3" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.1" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.3" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.5" stretchy="false" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.4.4.m4.3b"><apply id="S6.Thmtheorem2.p1.4.4.m4.3.3.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3"><in id="S6.Thmtheorem2.p1.4.4.m4.3.3.2.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.2"></in><ci id="S6.Thmtheorem2.p1.4.4.m4.3.3.3.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.3">ℓ</ci><set id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.2.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1"><cn id="S6.Thmtheorem2.p1.4.4.m4.1.1.cmml" type="integer" xref="S6.Thmtheorem2.p1.4.4.m4.1.1">1</cn><ci id="S6.Thmtheorem2.p1.4.4.m4.2.2.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.2.2">…</ci><apply id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1"><minus id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.1"></minus><apply id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.1.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.2.cmml" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.2.3">1</cn></apply><cn id="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem2.p1.4.4.m4.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.4.4.m4.3c">\ell\in\{1,\dots,k_{1}-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.4.4.m4.3d">roman_ℓ ∈ { 1 , … , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.5.5.m5.1"><semantics id="S6.Thmtheorem2.p1.5.5.m5.1a"><mi id="S6.Thmtheorem2.p1.5.5.m5.1.1" xref="S6.Thmtheorem2.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.5.5.m5.1b"><ci id="S6.Thmtheorem2.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem2.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S6.Thmtheorem2.p1.6.6.m6.1"><semantics id="S6.Thmtheorem2.p1.6.6.m6.1a"><mi id="S6.Thmtheorem2.p1.6.6.m6.1.1" xref="S6.Thmtheorem2.p1.6.6.m6.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem2.p1.6.6.m6.1b"><ci id="S6.Thmtheorem2.p1.6.6.m6.1.1.cmml" xref="S6.Thmtheorem2.p1.6.6.m6.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem2.p1.6.6.m6.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem2.p1.6.6.m6.1d">roman_UMD</annotation></semantics></math> Banach space.</span></p> <ol class="ltx_enumerate" id="S6.I1"> <li class="ltx_item" id="S6.I1.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S6.I1.i1.p1"> <p class="ltx_p" id="S6.I1.i1.p1.1"><span class="ltx_text ltx_font_italic" id="S6.I1.i1.p1.1.1">If </span><math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S6.I1.i1.p1.1.m1.4"><semantics id="S6.I1.i1.p1.1.m1.4a"><mrow id="S6.I1.i1.p1.1.m1.4.4" xref="S6.I1.i1.p1.1.m1.4.4.cmml"><mi id="S6.I1.i1.p1.1.m1.4.4.5" xref="S6.I1.i1.p1.1.m1.4.4.5.cmml">γ</mi><mo id="S6.I1.i1.p1.1.m1.4.4.4" xref="S6.I1.i1.p1.1.m1.4.4.4.cmml">∈</mo><mrow id="S6.I1.i1.p1.1.m1.4.4.3" xref="S6.I1.i1.p1.1.m1.4.4.3.cmml"><mrow id="S6.I1.i1.p1.1.m1.2.2.1.1.1" xref="S6.I1.i1.p1.1.m1.2.2.1.1.2.cmml"><mo id="S6.I1.i1.p1.1.m1.2.2.1.1.1.2" stretchy="false" xref="S6.I1.i1.p1.1.m1.2.2.1.1.2.cmml">(</mo><mrow id="S6.I1.i1.p1.1.m1.2.2.1.1.1.1" xref="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.cmml"><mo id="S6.I1.i1.p1.1.m1.2.2.1.1.1.1a" xref="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.cmml">−</mo><mn id="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.2" xref="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.I1.i1.p1.1.m1.2.2.1.1.1.3" xref="S6.I1.i1.p1.1.m1.2.2.1.1.2.cmml">,</mo><mi id="S6.I1.i1.p1.1.m1.1.1" mathvariant="normal" xref="S6.I1.i1.p1.1.m1.1.1.cmml">∞</mi><mo id="S6.I1.i1.p1.1.m1.2.2.1.1.1.4" stretchy="false" xref="S6.I1.i1.p1.1.m1.2.2.1.1.2.cmml">)</mo></mrow><mo id="S6.I1.i1.p1.1.m1.4.4.3.4" xref="S6.I1.i1.p1.1.m1.4.4.3.4.cmml">∖</mo><mrow id="S6.I1.i1.p1.1.m1.4.4.3.3.2" xref="S6.I1.i1.p1.1.m1.4.4.3.3.3.cmml"><mo id="S6.I1.i1.p1.1.m1.4.4.3.3.2.3" stretchy="false" xref="S6.I1.i1.p1.1.m1.4.4.3.3.3.1.cmml">{</mo><mrow id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.cmml"><mrow id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.cmml"><mi id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.2" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.1" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.3" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.1" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.1.cmml">−</mo><mn id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.3" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S6.I1.i1.p1.1.m1.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.I1.i1.p1.1.m1.4.4.3.3.3.1.cmml">:</mo><mrow id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.cmml"><mi id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.2" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.2.cmml">j</mi><mo id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.1" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.cmml"><mi id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.2" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.3" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.I1.i1.p1.1.m1.4.4.3.3.2.5" stretchy="false" xref="S6.I1.i1.p1.1.m1.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i1.p1.1.m1.4b"><apply id="S6.I1.i1.p1.1.m1.4.4.cmml" xref="S6.I1.i1.p1.1.m1.4.4"><in id="S6.I1.i1.p1.1.m1.4.4.4.cmml" xref="S6.I1.i1.p1.1.m1.4.4.4"></in><ci id="S6.I1.i1.p1.1.m1.4.4.5.cmml" xref="S6.I1.i1.p1.1.m1.4.4.5">𝛾</ci><apply id="S6.I1.i1.p1.1.m1.4.4.3.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3"><setdiff id="S6.I1.i1.p1.1.m1.4.4.3.4.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.4"></setdiff><interval closure="open" id="S6.I1.i1.p1.1.m1.2.2.1.1.2.cmml" xref="S6.I1.i1.p1.1.m1.2.2.1.1.1"><apply id="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.2.2.1.1.1.1"><minus id="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.2.2.1.1.1.1"></minus><cn id="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.2.cmml" type="integer" xref="S6.I1.i1.p1.1.m1.2.2.1.1.1.1.2">1</cn></apply><infinity id="S6.I1.i1.p1.1.m1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.1.1"></infinity></interval><apply id="S6.I1.i1.p1.1.m1.4.4.3.3.3.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2"><csymbol cd="latexml" id="S6.I1.i1.p1.1.m1.4.4.3.3.3.1.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.3">conditional-set</csymbol><apply id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.cmml" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1"><minus id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.1.cmml" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.1"></minus><apply id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.cmml" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2"><times id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.1.cmml" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.1"></times><ci id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.2.cmml" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.3.cmml" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.3.cmml" type="integer" xref="S6.I1.i1.p1.1.m1.3.3.2.2.1.1.3">1</cn></apply><apply id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2"><in id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.1.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.1"></in><ci id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.2.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.2">𝑗</ci><apply id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.1.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3">subscript</csymbol><ci id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.2.cmml" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S6.I1.i1.p1.1.m1.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i1.p1.1.m1.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i1.p1.1.m1.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I1.i1.p1.1.2">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X),W^{k_{0}+k_{1},p}_{0}(% \mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}=W^{k_{0}+\ell,p}_% {0}(\mathbb{R}^{d}_{+},w_{\gamma};X)." class="ltx_Math" display="block" id="S6.Ex7.m1.10"><semantics id="S6.Ex7.m1.10a"><mrow id="S6.Ex7.m1.10.10.1" xref="S6.Ex7.m1.10.10.1.1.cmml"><mrow id="S6.Ex7.m1.10.10.1.1" xref="S6.Ex7.m1.10.10.1.1.cmml"><msub id="S6.Ex7.m1.10.10.1.1.2" xref="S6.Ex7.m1.10.10.1.1.2.cmml"><mrow id="S6.Ex7.m1.10.10.1.1.2.2.2" xref="S6.Ex7.m1.10.10.1.1.2.2.3.cmml"><mo id="S6.Ex7.m1.10.10.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex7.m1.10.10.1.1.2.2.3.cmml">[</mo><mrow id="S6.Ex7.m1.10.10.1.1.1.1.1.1" xref="S6.Ex7.m1.10.10.1.1.1.1.1.1.cmml"><msubsup id="S6.Ex7.m1.10.10.1.1.1.1.1.1.4" xref="S6.Ex7.m1.10.10.1.1.1.1.1.1.4.cmml"><mi id="S6.Ex7.m1.10.10.1.1.1.1.1.1.4.2.2" 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id="S6.Ex7.m1.10.10.1.1.4.2.2.2.3.cmml" xref="S6.Ex7.m1.10.10.1.1.4.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex7.m1.9.9.cmml" xref="S6.Ex7.m1.9.9">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex7.m1.10c">\big{[}W^{k_{0},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X),W^{k_{0}+k_{1},p}_{0}(% \mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}=W^{k_{0}+\ell,p}_% {0}(\mathbb{R}^{d}_{+},w_{\gamma};X).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex7.m1.10d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S6.I1.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S6.I1.i2.p1"> <p class="ltx_p" id="S6.I1.i2.p1.2"><span class="ltx_text ltx_font_italic" id="S6.I1.i2.p1.2.1">If </span><math alttext="\gamma>(k_{0}+\ell)p-1" class="ltx_Math" display="inline" id="S6.I1.i2.p1.1.m1.1"><semantics id="S6.I1.i2.p1.1.m1.1a"><mrow id="S6.I1.i2.p1.1.m1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.cmml"><mi id="S6.I1.i2.p1.1.m1.1.1.3" xref="S6.I1.i2.p1.1.m1.1.1.3.cmml">γ</mi><mo id="S6.I1.i2.p1.1.m1.1.1.2" xref="S6.I1.i2.p1.1.m1.1.1.2.cmml">></mo><mrow id="S6.I1.i2.p1.1.m1.1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.1.cmml"><mrow id="S6.I1.i2.p1.1.m1.1.1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.1.1.cmml"><mrow id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml"><mo id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.2" stretchy="false" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml"><msub id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.2" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mn id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.3" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.1" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.3" mathvariant="normal" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.3.cmml">ℓ</mi></mrow><mo id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.3" stretchy="false" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.I1.i2.p1.1.m1.1.1.1.1.2" xref="S6.I1.i2.p1.1.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S6.I1.i2.p1.1.m1.1.1.1.1.3" xref="S6.I1.i2.p1.1.m1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S6.I1.i2.p1.1.m1.1.1.1.2" xref="S6.I1.i2.p1.1.m1.1.1.1.2.cmml">−</mo><mn id="S6.I1.i2.p1.1.m1.1.1.1.3" xref="S6.I1.i2.p1.1.m1.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.1.m1.1b"><apply id="S6.I1.i2.p1.1.m1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1"><gt id="S6.I1.i2.p1.1.m1.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.2"></gt><ci id="S6.I1.i2.p1.1.m1.1.1.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.3">𝛾</ci><apply id="S6.I1.i2.p1.1.m1.1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1"><minus id="S6.I1.i2.p1.1.m1.1.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.2"></minus><apply id="S6.I1.i2.p1.1.m1.1.1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1"><times id="S6.I1.i2.p1.1.m1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.2"></times><apply id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1"><plus id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.1"></plus><apply id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.2">𝑘</ci><cn id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.2.3">0</cn></apply><ci id="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.1.1.1.3">ℓ</ci></apply><ci id="S6.I1.i2.p1.1.m1.1.1.1.1.3.cmml" xref="S6.I1.i2.p1.1.m1.1.1.1.1.3">𝑝</ci></apply><cn id="S6.I1.i2.p1.1.m1.1.1.1.3.cmml" type="integer" xref="S6.I1.i2.p1.1.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.1.m1.1c">\gamma>(k_{0}+\ell)p-1</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.1.m1.1d">italic_γ > ( italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ ) italic_p - 1</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I1.i2.p1.2.2"> and </span><math alttext="\gamma\notin\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S6.I1.i2.p1.2.m2.2"><semantics id="S6.I1.i2.p1.2.m2.2a"><mrow id="S6.I1.i2.p1.2.m2.2.2" xref="S6.I1.i2.p1.2.m2.2.2.cmml"><mi id="S6.I1.i2.p1.2.m2.2.2.4" xref="S6.I1.i2.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S6.I1.i2.p1.2.m2.2.2.3" xref="S6.I1.i2.p1.2.m2.2.2.3.cmml">∉</mo><mrow id="S6.I1.i2.p1.2.m2.2.2.2.2" xref="S6.I1.i2.p1.2.m2.2.2.2.3.cmml"><mo id="S6.I1.i2.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S6.I1.i2.p1.2.m2.2.2.2.3.1.cmml">{</mo><mrow id="S6.I1.i2.p1.2.m2.1.1.1.1.1" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.cmml"><mrow id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.cmml"><mi id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.2" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.2.cmml">j</mi><mo id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.1" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.3" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.3.cmml">p</mi></mrow><mo id="S6.I1.i2.p1.2.m2.1.1.1.1.1.1" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml">−</mo><mn id="S6.I1.i2.p1.2.m2.1.1.1.1.1.3" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.I1.i2.p1.2.m2.2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.I1.i2.p1.2.m2.2.2.2.3.1.cmml">:</mo><mrow id="S6.I1.i2.p1.2.m2.2.2.2.2.2" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.cmml"><mi id="S6.I1.i2.p1.2.m2.2.2.2.2.2.2" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.2.cmml">j</mi><mo id="S6.I1.i2.p1.2.m2.2.2.2.2.2.1" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.1.cmml">∈</mo><msub id="S6.I1.i2.p1.2.m2.2.2.2.2.2.3" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.cmml"><mi id="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.2" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.2.cmml">ℕ</mi><mn id="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.3" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.I1.i2.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S6.I1.i2.p1.2.m2.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I1.i2.p1.2.m2.2b"><apply id="S6.I1.i2.p1.2.m2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.2.2"><notin id="S6.I1.i2.p1.2.m2.2.2.3.cmml" xref="S6.I1.i2.p1.2.m2.2.2.3"></notin><ci id="S6.I1.i2.p1.2.m2.2.2.4.cmml" xref="S6.I1.i2.p1.2.m2.2.2.4">𝛾</ci><apply id="S6.I1.i2.p1.2.m2.2.2.2.3.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2"><csymbol cd="latexml" id="S6.I1.i2.p1.2.m2.2.2.2.3.1.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2.3">conditional-set</csymbol><apply id="S6.I1.i2.p1.2.m2.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1"><minus id="S6.I1.i2.p1.2.m2.1.1.1.1.1.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.1"></minus><apply id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.cmml" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2"><times id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.1.cmml" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.1"></times><ci id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.2.cmml" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.2">𝑗</ci><ci id="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.3.cmml" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.2.3">𝑝</ci></apply><cn id="S6.I1.i2.p1.2.m2.1.1.1.1.1.3.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.1.1.1.1.1.3">1</cn></apply><apply id="S6.I1.i2.p1.2.m2.2.2.2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2"><in id="S6.I1.i2.p1.2.m2.2.2.2.2.2.1.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.1"></in><ci id="S6.I1.i2.p1.2.m2.2.2.2.2.2.2.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.2">𝑗</ci><apply id="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.3"><csymbol cd="ambiguous" id="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.1.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.3">subscript</csymbol><ci id="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.2.cmml" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.2">ℕ</ci><cn id="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.3.cmml" type="integer" xref="S6.I1.i2.p1.2.m2.2.2.2.2.2.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I1.i2.p1.2.m2.2c">\gamma\notin\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.I1.i2.p1.2.m2.2d">italic_γ ∉ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math><span class="ltx_text ltx_font_italic" id="S6.I1.i2.p1.2.3">, then</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W^{k_{0}+k_{1},p}(\mathbb{% R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}=W^{k_{0}+\ell,p}(\mathbb{% R}^{d}_{+},w_{\gamma};X)." class="ltx_Math" display="block" id="S6.Ex8.m1.10"><semantics id="S6.Ex8.m1.10a"><mrow id="S6.Ex8.m1.10.10.1" xref="S6.Ex8.m1.10.10.1.1.cmml"><mrow id="S6.Ex8.m1.10.10.1.1" xref="S6.Ex8.m1.10.10.1.1.cmml"><msub id="S6.Ex8.m1.10.10.1.1.2" xref="S6.Ex8.m1.10.10.1.1.2.cmml"><mrow id="S6.Ex8.m1.10.10.1.1.2.2.2" xref="S6.Ex8.m1.10.10.1.1.2.2.3.cmml"><mo id="S6.Ex8.m1.10.10.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex8.m1.10.10.1.1.2.2.3.cmml">[</mo><mrow id="S6.Ex8.m1.10.10.1.1.1.1.1.1" xref="S6.Ex8.m1.10.10.1.1.1.1.1.1.cmml"><msup id="S6.Ex8.m1.10.10.1.1.1.1.1.1.4" xref="S6.Ex8.m1.10.10.1.1.1.1.1.1.4.cmml"><mi 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start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> </ol> </div> </div> <div class="ltx_proof" id="S6.SS1.5"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS1.5.p1"> <p class="ltx_p" id="S6.SS1.5.p1.5">We first consider <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.I1.i1" title="item i ‣ Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> for <math alttext="k_{0}=0" class="ltx_Math" display="inline" id="S6.SS1.5.p1.1.m1.1"><semantics id="S6.SS1.5.p1.1.m1.1a"><mrow id="S6.SS1.5.p1.1.m1.1.1" xref="S6.SS1.5.p1.1.m1.1.1.cmml"><msub id="S6.SS1.5.p1.1.m1.1.1.2" xref="S6.SS1.5.p1.1.m1.1.1.2.cmml"><mi id="S6.SS1.5.p1.1.m1.1.1.2.2" xref="S6.SS1.5.p1.1.m1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.5.p1.1.m1.1.1.2.3" xref="S6.SS1.5.p1.1.m1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.5.p1.1.m1.1.1.1" xref="S6.SS1.5.p1.1.m1.1.1.1.cmml">=</mo><mn id="S6.SS1.5.p1.1.m1.1.1.3" xref="S6.SS1.5.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.1.m1.1b"><apply id="S6.SS1.5.p1.1.m1.1.1.cmml" xref="S6.SS1.5.p1.1.m1.1.1"><eq id="S6.SS1.5.p1.1.m1.1.1.1.cmml" xref="S6.SS1.5.p1.1.m1.1.1.1"></eq><apply id="S6.SS1.5.p1.1.m1.1.1.2.cmml" xref="S6.SS1.5.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.5.p1.1.m1.1.1.2.1.cmml" xref="S6.SS1.5.p1.1.m1.1.1.2">subscript</csymbol><ci id="S6.SS1.5.p1.1.m1.1.1.2.2.cmml" xref="S6.SS1.5.p1.1.m1.1.1.2.2">𝑘</ci><cn id="S6.SS1.5.p1.1.m1.1.1.2.3.cmml" type="integer" xref="S6.SS1.5.p1.1.m1.1.1.2.3">0</cn></apply><cn id="S6.SS1.5.p1.1.m1.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.1.m1.1c">k_{0}=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.1.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math>. If <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S6.SS1.5.p1.2.m2.2"><semantics id="S6.SS1.5.p1.2.m2.2a"><mrow id="S6.SS1.5.p1.2.m2.2.2" xref="S6.SS1.5.p1.2.m2.2.2.cmml"><mi id="S6.SS1.5.p1.2.m2.2.2.4" xref="S6.SS1.5.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S6.SS1.5.p1.2.m2.2.2.3" xref="S6.SS1.5.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S6.SS1.5.p1.2.m2.2.2.2.2" xref="S6.SS1.5.p1.2.m2.2.2.2.3.cmml"><mo id="S6.SS1.5.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S6.SS1.5.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S6.SS1.5.p1.2.m2.1.1.1.1.1" xref="S6.SS1.5.p1.2.m2.1.1.1.1.1.cmml"><mo id="S6.SS1.5.p1.2.m2.1.1.1.1.1a" xref="S6.SS1.5.p1.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S6.SS1.5.p1.2.m2.1.1.1.1.1.2" xref="S6.SS1.5.p1.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.2.m2.2.2.2.2.4" xref="S6.SS1.5.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="S6.SS1.5.p1.2.m2.2.2.2.2.2" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2.cmml"><mi id="S6.SS1.5.p1.2.m2.2.2.2.2.2.2" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2.2.cmml">p</mi><mo id="S6.SS1.5.p1.2.m2.2.2.2.2.2.1" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2.1.cmml">−</mo><mn id="S6.SS1.5.p1.2.m2.2.2.2.2.2.3" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S6.SS1.5.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.2.m2.2b"><apply id="S6.SS1.5.p1.2.m2.2.2.cmml" xref="S6.SS1.5.p1.2.m2.2.2"><in id="S6.SS1.5.p1.2.m2.2.2.3.cmml" xref="S6.SS1.5.p1.2.m2.2.2.3"></in><ci id="S6.SS1.5.p1.2.m2.2.2.4.cmml" xref="S6.SS1.5.p1.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S6.SS1.5.p1.2.m2.2.2.2.3.cmml" xref="S6.SS1.5.p1.2.m2.2.2.2.2"><apply id="S6.SS1.5.p1.2.m2.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.2.m2.1.1.1.1.1"><minus id="S6.SS1.5.p1.2.m2.1.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.2.m2.1.1.1.1.1"></minus><cn id="S6.SS1.5.p1.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S6.SS1.5.p1.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S6.SS1.5.p1.2.m2.2.2.2.2.2.cmml" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2"><minus id="S6.SS1.5.p1.2.m2.2.2.2.2.2.1.cmml" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2.1"></minus><ci id="S6.SS1.5.p1.2.m2.2.2.2.2.2.2.cmml" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2.2">𝑝</ci><cn id="S6.SS1.5.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.5.p1.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.2.m2.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.2.m2.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>, then the statement follows from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>, Proposition 3.15]</cite>. For <math alttext="\gamma\in(jp-1,(j+1)p-1)" class="ltx_Math" display="inline" id="S6.SS1.5.p1.3.m3.2"><semantics id="S6.SS1.5.p1.3.m3.2a"><mrow id="S6.SS1.5.p1.3.m3.2.2" xref="S6.SS1.5.p1.3.m3.2.2.cmml"><mi id="S6.SS1.5.p1.3.m3.2.2.4" xref="S6.SS1.5.p1.3.m3.2.2.4.cmml">γ</mi><mo id="S6.SS1.5.p1.3.m3.2.2.3" xref="S6.SS1.5.p1.3.m3.2.2.3.cmml">∈</mo><mrow id="S6.SS1.5.p1.3.m3.2.2.2.2" xref="S6.SS1.5.p1.3.m3.2.2.2.3.cmml"><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.3" stretchy="false" xref="S6.SS1.5.p1.3.m3.2.2.2.3.cmml">(</mo><mrow id="S6.SS1.5.p1.3.m3.1.1.1.1.1" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.cmml"><mrow id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.cmml"><mi id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.2" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.2.cmml">j</mi><mo id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.1" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.3" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.3.cmml">p</mi></mrow><mo id="S6.SS1.5.p1.3.m3.1.1.1.1.1.1" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.1.cmml">−</mo><mn id="S6.SS1.5.p1.3.m3.1.1.1.1.1.3" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.4" xref="S6.SS1.5.p1.3.m3.2.2.2.3.cmml">,</mo><mrow id="S6.SS1.5.p1.3.m3.2.2.2.2.2" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.cmml"><mrow id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.cmml"><mrow id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.cmml"><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.2" stretchy="false" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.cmml"><mi id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.2" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.1" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.1.cmml">+</mo><mn id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.3" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.3" stretchy="false" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.2" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.2.cmml">⁢</mo><mi id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.3" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.3.cmml">p</mi></mrow><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.2.2" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.2.cmml">−</mo><mn id="S6.SS1.5.p1.3.m3.2.2.2.2.2.3" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.3.m3.2.2.2.2.5" stretchy="false" xref="S6.SS1.5.p1.3.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.3.m3.2b"><apply id="S6.SS1.5.p1.3.m3.2.2.cmml" xref="S6.SS1.5.p1.3.m3.2.2"><in id="S6.SS1.5.p1.3.m3.2.2.3.cmml" xref="S6.SS1.5.p1.3.m3.2.2.3"></in><ci id="S6.SS1.5.p1.3.m3.2.2.4.cmml" xref="S6.SS1.5.p1.3.m3.2.2.4">𝛾</ci><interval closure="open" id="S6.SS1.5.p1.3.m3.2.2.2.3.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2"><apply id="S6.SS1.5.p1.3.m3.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1"><minus id="S6.SS1.5.p1.3.m3.1.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.1"></minus><apply id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.cmml" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2"><times id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.1.cmml" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.1"></times><ci id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.2.cmml" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.2">𝑗</ci><ci id="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.3.cmml" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.2.3">𝑝</ci></apply><cn id="S6.SS1.5.p1.3.m3.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.3.m3.1.1.1.1.1.3">1</cn></apply><apply id="S6.SS1.5.p1.3.m3.2.2.2.2.2.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2"><minus id="S6.SS1.5.p1.3.m3.2.2.2.2.2.2.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.2"></minus><apply id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1"><times id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.2.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.2"></times><apply id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1"><plus id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.1"></plus><ci id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.2.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.2">𝑗</ci><cn id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.1.1.1.3">1</cn></apply><ci id="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.3.cmml" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.1.3">𝑝</ci></apply><cn id="S6.SS1.5.p1.3.m3.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.5.p1.3.m3.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.3.m3.2c">\gamma\in(jp-1,(j+1)p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.3.m3.2d">italic_γ ∈ ( italic_j italic_p - 1 , ( italic_j + 1 ) italic_p - 1 )</annotation></semantics></math> with <math alttext="j\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S6.SS1.5.p1.4.m4.1"><semantics id="S6.SS1.5.p1.4.m4.1a"><mrow id="S6.SS1.5.p1.4.m4.1.1" xref="S6.SS1.5.p1.4.m4.1.1.cmml"><mi id="S6.SS1.5.p1.4.m4.1.1.2" xref="S6.SS1.5.p1.4.m4.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.4.m4.1.1.1" xref="S6.SS1.5.p1.4.m4.1.1.1.cmml">∈</mo><msub id="S6.SS1.5.p1.4.m4.1.1.3" xref="S6.SS1.5.p1.4.m4.1.1.3.cmml"><mi id="S6.SS1.5.p1.4.m4.1.1.3.2" xref="S6.SS1.5.p1.4.m4.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.5.p1.4.m4.1.1.3.3" xref="S6.SS1.5.p1.4.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.4.m4.1b"><apply id="S6.SS1.5.p1.4.m4.1.1.cmml" xref="S6.SS1.5.p1.4.m4.1.1"><in id="S6.SS1.5.p1.4.m4.1.1.1.cmml" xref="S6.SS1.5.p1.4.m4.1.1.1"></in><ci id="S6.SS1.5.p1.4.m4.1.1.2.cmml" xref="S6.SS1.5.p1.4.m4.1.1.2">𝑗</ci><apply id="S6.SS1.5.p1.4.m4.1.1.3.cmml" xref="S6.SS1.5.p1.4.m4.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.5.p1.4.m4.1.1.3.1.cmml" xref="S6.SS1.5.p1.4.m4.1.1.3">subscript</csymbol><ci id="S6.SS1.5.p1.4.m4.1.1.3.2.cmml" xref="S6.SS1.5.p1.4.m4.1.1.3.2">ℕ</ci><cn id="S6.SS1.5.p1.4.m4.1.1.3.3.cmml" type="integer" xref="S6.SS1.5.p1.4.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.4.m4.1c">j\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.4.m4.1d">italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we obtain, applying <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Lemma 3.6]</cite>, properties of the complex interpolation method and the case <math alttext="j=0" class="ltx_Math" display="inline" id="S6.SS1.5.p1.5.m5.1"><semantics id="S6.SS1.5.p1.5.m5.1a"><mrow id="S6.SS1.5.p1.5.m5.1.1" xref="S6.SS1.5.p1.5.m5.1.1.cmml"><mi id="S6.SS1.5.p1.5.m5.1.1.2" xref="S6.SS1.5.p1.5.m5.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.5.m5.1.1.1" xref="S6.SS1.5.p1.5.m5.1.1.1.cmml">=</mo><mn id="S6.SS1.5.p1.5.m5.1.1.3" xref="S6.SS1.5.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.5.m5.1b"><apply id="S6.SS1.5.p1.5.m5.1.1.cmml" xref="S6.SS1.5.p1.5.m5.1.1"><eq id="S6.SS1.5.p1.5.m5.1.1.1.cmml" xref="S6.SS1.5.p1.5.m5.1.1.1"></eq><ci id="S6.SS1.5.p1.5.m5.1.1.2.cmml" xref="S6.SS1.5.p1.5.m5.1.1.2">𝑗</ci><cn id="S6.SS1.5.p1.5.m5.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.5.m5.1c">j=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.5.m5.1d">italic_j = 0</annotation></semantics></math>, that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx28"> <tbody id="S6.Ex9"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\big{[}L^{p}(\mathbb{R}^{d}_{+}" class="ltx_math_unparsed" display="inline" id="S6.Ex9.m1.1"><semantics id="S6.Ex9.m1.1a"><mrow id="S6.Ex9.m1.1b"><mo id="S6.Ex9.m1.1.1" maxsize="120%" minsize="120%">[</mo><msup id="S6.Ex9.m1.1.2"><mi id="S6.Ex9.m1.1.2.2">L</mi><mi id="S6.Ex9.m1.1.2.3">p</mi></msup><mrow id="S6.Ex9.m1.1.3"><mo id="S6.Ex9.m1.1.3.1" stretchy="false">(</mo><msubsup id="S6.Ex9.m1.1.3.2"><mi id="S6.Ex9.m1.1.3.2.2.2">ℝ</mi><mo id="S6.Ex9.m1.1.3.2.3">+</mo><mi id="S6.Ex9.m1.1.3.2.2.3">d</mi></msubsup></mrow></mrow><annotation encoding="application/x-tex" id="S6.Ex9.m1.1c">\displaystyle\big{[}L^{p}(\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex9.m1.1d">[ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle,w_{\gamma};X),W^{k_{1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X)% \big{]}_{\frac{\ell}{k_{1}}}" class="ltx_math_unparsed" display="inline" id="S6.Ex9.m2.2"><semantics id="S6.Ex9.m2.2a"><mrow id="S6.Ex9.m2.2b"><mrow id="S6.Ex9.m2.2.3"><mo id="S6.Ex9.m2.2.3.1">,</mo><msub id="S6.Ex9.m2.2.3.2"><mi id="S6.Ex9.m2.2.3.2.2">w</mi><mi id="S6.Ex9.m2.2.3.2.3">γ</mi></msub><mo id="S6.Ex9.m2.2.3.3">;</mo><mi id="S6.Ex9.m2.2.3.4">X</mi><mo id="S6.Ex9.m2.2.3.5" stretchy="false">)</mo></mrow><mo id="S6.Ex9.m2.2.4">,</mo><msubsup id="S6.Ex9.m2.2.5"><mi id="S6.Ex9.m2.2.5.2.2">W</mi><mn id="S6.Ex9.m2.2.5.3">0</mn><mrow id="S6.Ex9.m2.2.2.2.2"><msub id="S6.Ex9.m2.2.2.2.2.1"><mi id="S6.Ex9.m2.2.2.2.2.1.2">k</mi><mn id="S6.Ex9.m2.2.2.2.2.1.3">1</mn></msub><mo id="S6.Ex9.m2.2.2.2.2.2">,</mo><mi id="S6.Ex9.m2.1.1.1.1">p</mi></mrow></msubsup><mrow id="S6.Ex9.m2.2.6"><mo id="S6.Ex9.m2.2.6.1" stretchy="false">(</mo><msubsup id="S6.Ex9.m2.2.6.2"><mi id="S6.Ex9.m2.2.6.2.2.2">ℝ</mi><mo id="S6.Ex9.m2.2.6.2.3">+</mo><mi id="S6.Ex9.m2.2.6.2.2.3">d</mi></msubsup><mo id="S6.Ex9.m2.2.6.3">,</mo><msub id="S6.Ex9.m2.2.6.4"><mi id="S6.Ex9.m2.2.6.4.2">w</mi><mi id="S6.Ex9.m2.2.6.4.3">γ</mi></msub><mo id="S6.Ex9.m2.2.6.5">;</mo><mi id="S6.Ex9.m2.2.6.6">X</mi><mo id="S6.Ex9.m2.2.6.7" stretchy="false">)</mo></mrow><mo id="S6.Ex9.m2.2.7" maxsize="120%" minsize="120%">]</mo><msub id="S6.Ex9.m2.2.8"><mi id="S6.Ex9.m2.2.8a"></mi><mfrac id="S6.Ex9.m2.2.8.1"><mi id="S6.Ex9.m2.2.8.1.2" mathvariant="normal">ℓ</mi><msub id="S6.Ex9.m2.2.8.1.3"><mi id="S6.Ex9.m2.2.8.1.3.2">k</mi><mn id="S6.Ex9.m2.2.8.1.3.3">1</mn></msub></mfrac></msub></mrow><annotation encoding="application/x-tex" id="S6.Ex9.m2.2c">\displaystyle,w_{\gamma};X),W^{k_{1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X)% \big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex9.m2.2d">, italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex10"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{[}M^{-j}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X),M^{-j}W^{% k_{1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X)\big{]}_{\frac{\ell}{k_{1}}}" class="ltx_Math" display="inline" id="S6.Ex10.m1.6"><semantics id="S6.Ex10.m1.6a"><mrow id="S6.Ex10.m1.6.6" xref="S6.Ex10.m1.6.6.cmml"><mi id="S6.Ex10.m1.6.6.4" xref="S6.Ex10.m1.6.6.4.cmml"></mi><mo id="S6.Ex10.m1.6.6.3" xref="S6.Ex10.m1.6.6.3.cmml">=</mo><msub id="S6.Ex10.m1.6.6.2" xref="S6.Ex10.m1.6.6.2.cmml"><mrow id="S6.Ex10.m1.6.6.2.2.2" xref="S6.Ex10.m1.6.6.2.2.3.cmml"><mo id="S6.Ex10.m1.6.6.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex10.m1.6.6.2.2.3.cmml">[</mo><mrow id="S6.Ex10.m1.5.5.1.1.1.1" xref="S6.Ex10.m1.5.5.1.1.1.1.cmml"><msup id="S6.Ex10.m1.5.5.1.1.1.1.4" xref="S6.Ex10.m1.5.5.1.1.1.1.4.cmml"><mi id="S6.Ex10.m1.5.5.1.1.1.1.4.2" xref="S6.Ex10.m1.5.5.1.1.1.1.4.2.cmml">M</mi><mrow id="S6.Ex10.m1.5.5.1.1.1.1.4.3" xref="S6.Ex10.m1.5.5.1.1.1.1.4.3.cmml"><mo id="S6.Ex10.m1.5.5.1.1.1.1.4.3a" xref="S6.Ex10.m1.5.5.1.1.1.1.4.3.cmml">−</mo><mi id="S6.Ex10.m1.5.5.1.1.1.1.4.3.2" xref="S6.Ex10.m1.5.5.1.1.1.1.4.3.2.cmml">j</mi></mrow></msup><mo id="S6.Ex10.m1.5.5.1.1.1.1.3" xref="S6.Ex10.m1.5.5.1.1.1.1.3.cmml">⁢</mo><msup id="S6.Ex10.m1.5.5.1.1.1.1.5" xref="S6.Ex10.m1.5.5.1.1.1.1.5.cmml"><mi id="S6.Ex10.m1.5.5.1.1.1.1.5.2" xref="S6.Ex10.m1.5.5.1.1.1.1.5.2.cmml">L</mi><mi id="S6.Ex10.m1.5.5.1.1.1.1.5.3" xref="S6.Ex10.m1.5.5.1.1.1.1.5.3.cmml">p</mi></msup><mo id="S6.Ex10.m1.5.5.1.1.1.1.3a" xref="S6.Ex10.m1.5.5.1.1.1.1.3.cmml">⁢</mo><mrow id="S6.Ex10.m1.5.5.1.1.1.1.2.2" xref="S6.Ex10.m1.5.5.1.1.1.1.2.3.cmml"><mo id="S6.Ex10.m1.5.5.1.1.1.1.2.2.3" stretchy="false" xref="S6.Ex10.m1.5.5.1.1.1.1.2.3.cmml">(</mo><msubsup id="S6.Ex10.m1.5.5.1.1.1.1.1.1.1" xref="S6.Ex10.m1.5.5.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex10.m1.5.5.1.1.1.1.1.1.1.2.2" xref="S6.Ex10.m1.5.5.1.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex10.m1.5.5.1.1.1.1.1.1.1.3" xref="S6.Ex10.m1.5.5.1.1.1.1.1.1.1.3.cmml">+</mo><mi id="S6.Ex10.m1.5.5.1.1.1.1.1.1.1.2.3" xref="S6.Ex10.m1.5.5.1.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex10.m1.5.5.1.1.1.1.2.2.4" xref="S6.Ex10.m1.5.5.1.1.1.1.2.3.cmml">,</mo><msub id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.cmml"><mi id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.2" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.2.cmml">w</mi><mrow id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.cmml"><mi id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.2" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.2.cmml">γ</mi><mo id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.1" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.1.cmml">−</mo><mrow id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3.cmml"><mi id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3.2" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3.2.cmml">j</mi><mo id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3.1" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3.1.cmml">⁢</mo><mi id="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3.3" xref="S6.Ex10.m1.5.5.1.1.1.1.2.2.2.3.3.3.cmml">p</mi></mrow></mrow></msub><mo id="S6.Ex10.m1.5.5.1.1.1.1.2.2.5" xref="S6.Ex10.m1.5.5.1.1.1.1.2.3.cmml">;</mo><mi id="S6.Ex10.m1.3.3" xref="S6.Ex10.m1.3.3.cmml">X</mi><mo id="S6.Ex10.m1.5.5.1.1.1.1.2.2.6" stretchy="false" 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id="S6.Ex10.m1.6.6.2.4.1.cmml" xref="S6.Ex10.m1.6.6.2.4"></divide><ci id="S6.Ex10.m1.6.6.2.4.2.cmml" xref="S6.Ex10.m1.6.6.2.4.2">ℓ</ci><apply id="S6.Ex10.m1.6.6.2.4.3.cmml" xref="S6.Ex10.m1.6.6.2.4.3"><csymbol cd="ambiguous" id="S6.Ex10.m1.6.6.2.4.3.1.cmml" xref="S6.Ex10.m1.6.6.2.4.3">subscript</csymbol><ci id="S6.Ex10.m1.6.6.2.4.3.2.cmml" xref="S6.Ex10.m1.6.6.2.4.3.2">𝑘</ci><cn id="S6.Ex10.m1.6.6.2.4.3.3.cmml" type="integer" xref="S6.Ex10.m1.6.6.2.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex10.m1.6c">\displaystyle=\big{[}M^{-j}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X),M^{-j}W^{% k_{1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X)\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex10.m1.6d">= [ italic_M start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) , italic_M start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex11"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left 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xref="S6.Ex11.m1.6.6.2.2.4.3">subscript</csymbol><ci id="S6.Ex11.m1.6.6.2.2.4.3.2.cmml" xref="S6.Ex11.m1.6.6.2.2.4.3.2">𝑘</ci><cn id="S6.Ex11.m1.6.6.2.2.4.3.3.cmml" type="integer" xref="S6.Ex11.m1.6.6.2.2.4.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex11.m1.6c">\displaystyle=M^{-j}\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X),W^{k_{1},% p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X)\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex11.m1.6d">= italic_M start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT [ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT 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xref="S6.Ex12.m1.7.7.1.1.4.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex12.m1.6.6.cmml" xref="S6.Ex12.m1.6.6">𝑋</ci></vector></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex12.m1.7c">\displaystyle=M^{-j}W_{0}^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X)=W_{0}^{% \ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex12.m1.7d">= italic_M start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) = italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.5.p1.7">The case <math alttext="k_{0}\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S6.SS1.5.p1.6.m1.1"><semantics id="S6.SS1.5.p1.6.m1.1a"><mrow id="S6.SS1.5.p1.6.m1.1.1" xref="S6.SS1.5.p1.6.m1.1.1.cmml"><msub id="S6.SS1.5.p1.6.m1.1.1.2" xref="S6.SS1.5.p1.6.m1.1.1.2.cmml"><mi id="S6.SS1.5.p1.6.m1.1.1.2.2" xref="S6.SS1.5.p1.6.m1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.5.p1.6.m1.1.1.2.3" xref="S6.SS1.5.p1.6.m1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.5.p1.6.m1.1.1.1" xref="S6.SS1.5.p1.6.m1.1.1.1.cmml">∈</mo><msub id="S6.SS1.5.p1.6.m1.1.1.3" xref="S6.SS1.5.p1.6.m1.1.1.3.cmml"><mi id="S6.SS1.5.p1.6.m1.1.1.3.2" xref="S6.SS1.5.p1.6.m1.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.5.p1.6.m1.1.1.3.3" xref="S6.SS1.5.p1.6.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.6.m1.1b"><apply id="S6.SS1.5.p1.6.m1.1.1.cmml" xref="S6.SS1.5.p1.6.m1.1.1"><in id="S6.SS1.5.p1.6.m1.1.1.1.cmml" xref="S6.SS1.5.p1.6.m1.1.1.1"></in><apply id="S6.SS1.5.p1.6.m1.1.1.2.cmml" xref="S6.SS1.5.p1.6.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.5.p1.6.m1.1.1.2.1.cmml" xref="S6.SS1.5.p1.6.m1.1.1.2">subscript</csymbol><ci id="S6.SS1.5.p1.6.m1.1.1.2.2.cmml" xref="S6.SS1.5.p1.6.m1.1.1.2.2">𝑘</ci><cn id="S6.SS1.5.p1.6.m1.1.1.2.3.cmml" type="integer" xref="S6.SS1.5.p1.6.m1.1.1.2.3">0</cn></apply><apply id="S6.SS1.5.p1.6.m1.1.1.3.cmml" xref="S6.SS1.5.p1.6.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.5.p1.6.m1.1.1.3.1.cmml" xref="S6.SS1.5.p1.6.m1.1.1.3">subscript</csymbol><ci id="S6.SS1.5.p1.6.m1.1.1.3.2.cmml" xref="S6.SS1.5.p1.6.m1.1.1.3.2">ℕ</ci><cn id="S6.SS1.5.p1.6.m1.1.1.3.3.cmml" type="integer" xref="S6.SS1.5.p1.6.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.6.m1.1c">k_{0}\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.6.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> follows from the case <math alttext="k_{0}=0" class="ltx_Math" display="inline" id="S6.SS1.5.p1.7.m2.1"><semantics id="S6.SS1.5.p1.7.m2.1a"><mrow id="S6.SS1.5.p1.7.m2.1.1" xref="S6.SS1.5.p1.7.m2.1.1.cmml"><msub id="S6.SS1.5.p1.7.m2.1.1.2" xref="S6.SS1.5.p1.7.m2.1.1.2.cmml"><mi id="S6.SS1.5.p1.7.m2.1.1.2.2" xref="S6.SS1.5.p1.7.m2.1.1.2.2.cmml">k</mi><mn id="S6.SS1.5.p1.7.m2.1.1.2.3" xref="S6.SS1.5.p1.7.m2.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.5.p1.7.m2.1.1.1" xref="S6.SS1.5.p1.7.m2.1.1.1.cmml">=</mo><mn id="S6.SS1.5.p1.7.m2.1.1.3" xref="S6.SS1.5.p1.7.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.7.m2.1b"><apply id="S6.SS1.5.p1.7.m2.1.1.cmml" xref="S6.SS1.5.p1.7.m2.1.1"><eq id="S6.SS1.5.p1.7.m2.1.1.1.cmml" xref="S6.SS1.5.p1.7.m2.1.1.1"></eq><apply id="S6.SS1.5.p1.7.m2.1.1.2.cmml" xref="S6.SS1.5.p1.7.m2.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.5.p1.7.m2.1.1.2.1.cmml" xref="S6.SS1.5.p1.7.m2.1.1.2">subscript</csymbol><ci id="S6.SS1.5.p1.7.m2.1.1.2.2.cmml" xref="S6.SS1.5.p1.7.m2.1.1.2.2">𝑘</ci><cn id="S6.SS1.5.p1.7.m2.1.1.2.3.cmml" type="integer" xref="S6.SS1.5.p1.7.m2.1.1.2.3">0</cn></apply><cn id="S6.SS1.5.p1.7.m2.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.7.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.7.m2.1c">k_{0}=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.7.m2.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math> and reiteration for the complex interpolation method (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib9" title="">9</a>, Theorem 4.6.1]</cite>):</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx29"> <tbody id="S6.Ex13"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\big{[}W^{k_{0},p}_{0}(\mathbb{R}^{d}_{+}" class="ltx_math_unparsed" display="inline" id="S6.Ex13.m1.2"><semantics id="S6.Ex13.m1.2a"><mrow id="S6.Ex13.m1.2b"><mo id="S6.Ex13.m1.2.3" maxsize="120%" minsize="120%">[</mo><msubsup id="S6.Ex13.m1.2.4"><mi id="S6.Ex13.m1.2.4.2.2">W</mi><mn id="S6.Ex13.m1.2.4.3">0</mn><mrow id="S6.Ex13.m1.2.2.2.2"><msub id="S6.Ex13.m1.2.2.2.2.1"><mi id="S6.Ex13.m1.2.2.2.2.1.2">k</mi><mn id="S6.Ex13.m1.2.2.2.2.1.3">0</mn></msub><mo id="S6.Ex13.m1.2.2.2.2.2">,</mo><mi id="S6.Ex13.m1.1.1.1.1">p</mi></mrow></msubsup><mrow id="S6.Ex13.m1.2.5"><mo id="S6.Ex13.m1.2.5.1" stretchy="false">(</mo><msubsup id="S6.Ex13.m1.2.5.2"><mi id="S6.Ex13.m1.2.5.2.2.2">ℝ</mi><mo id="S6.Ex13.m1.2.5.2.3">+</mo><mi id="S6.Ex13.m1.2.5.2.2.3">d</mi></msubsup></mrow></mrow><annotation encoding="application/x-tex" id="S6.Ex13.m1.2c">\displaystyle\big{[}W^{k_{0},p}_{0}(\mathbb{R}^{d}_{+}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex13.m1.2d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle,w_{\gamma};X),W^{k_{0}+k_{1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma% };X)\big{]}_{\frac{\ell}{k_{1}}}" class="ltx_math_unparsed" display="inline" id="S6.Ex13.m2.2"><semantics id="S6.Ex13.m2.2a"><mrow id="S6.Ex13.m2.2b"><mrow id="S6.Ex13.m2.2.3"><mo id="S6.Ex13.m2.2.3.1">,</mo><msub id="S6.Ex13.m2.2.3.2"><mi id="S6.Ex13.m2.2.3.2.2">w</mi><mi id="S6.Ex13.m2.2.3.2.3">γ</mi></msub><mo id="S6.Ex13.m2.2.3.3">;</mo><mi id="S6.Ex13.m2.2.3.4">X</mi><mo id="S6.Ex13.m2.2.3.5" stretchy="false">)</mo></mrow><mo id="S6.Ex13.m2.2.4">,</mo><msubsup id="S6.Ex13.m2.2.5"><mi id="S6.Ex13.m2.2.5.2.2">W</mi><mn id="S6.Ex13.m2.2.5.3">0</mn><mrow id="S6.Ex13.m2.2.2.2.2"><mrow id="S6.Ex13.m2.2.2.2.2.1"><msub id="S6.Ex13.m2.2.2.2.2.1.2"><mi id="S6.Ex13.m2.2.2.2.2.1.2.2">k</mi><mn id="S6.Ex13.m2.2.2.2.2.1.2.3">0</mn></msub><mo id="S6.Ex13.m2.2.2.2.2.1.1">+</mo><msub id="S6.Ex13.m2.2.2.2.2.1.3"><mi id="S6.Ex13.m2.2.2.2.2.1.3.2">k</mi><mn id="S6.Ex13.m2.2.2.2.2.1.3.3">1</mn></msub></mrow><mo id="S6.Ex13.m2.2.2.2.2.2">,</mo><mi id="S6.Ex13.m2.1.1.1.1">p</mi></mrow></msubsup><mrow id="S6.Ex13.m2.2.6"><mo id="S6.Ex13.m2.2.6.1" stretchy="false">(</mo><msubsup id="S6.Ex13.m2.2.6.2"><mi id="S6.Ex13.m2.2.6.2.2.2">ℝ</mi><mo id="S6.Ex13.m2.2.6.2.3">+</mo><mi id="S6.Ex13.m2.2.6.2.2.3">d</mi></msubsup><mo id="S6.Ex13.m2.2.6.3">,</mo><msub id="S6.Ex13.m2.2.6.4"><mi id="S6.Ex13.m2.2.6.4.2">w</mi><mi id="S6.Ex13.m2.2.6.4.3">γ</mi></msub><mo id="S6.Ex13.m2.2.6.5">;</mo><mi id="S6.Ex13.m2.2.6.6">X</mi><mo id="S6.Ex13.m2.2.6.7" stretchy="false">)</mo></mrow><mo id="S6.Ex13.m2.2.7" maxsize="120%" minsize="120%">]</mo><msub id="S6.Ex13.m2.2.8"><mi id="S6.Ex13.m2.2.8a"></mi><mfrac id="S6.Ex13.m2.2.8.1"><mi id="S6.Ex13.m2.2.8.1.2" mathvariant="normal">ℓ</mi><msub id="S6.Ex13.m2.2.8.1.3"><mi id="S6.Ex13.m2.2.8.1.3.2">k</mi><mn id="S6.Ex13.m2.2.8.1.3.3">1</mn></msub></mfrac></msub></mrow><annotation encoding="application/x-tex" id="S6.Ex13.m2.2c">\displaystyle,w_{\gamma};X),W^{k_{0}+k_{1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma% };X)\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex13.m2.2d">, italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S6.Ex14.m1.1"><semantics id="S6.Ex14.m1.1a"><mo id="S6.Ex14.m1.1.1" xref="S6.Ex14.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S6.Ex14.m1.1b"><eq id="S6.Ex14.m1.1.1.cmml" xref="S6.Ex14.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex14.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S6.Ex14.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;\big{[}[L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{0}^{k_{0}+k_{% 1}}(\mathbb{R}^{d}_{+},w_{\gamma};X)]_{\frac{k_{0}}{k_{0}+k_{1}}},W^{k_{0}+k_{% 1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}" class="ltx_Math" display="inline" id="S6.Ex14.m2.7"><semantics id="S6.Ex14.m2.7a"><msub id="S6.Ex14.m2.7.7" xref="S6.Ex14.m2.7.7.cmml"><mrow id="S6.Ex14.m2.7.7.2.2" xref="S6.Ex14.m2.7.7.2.3.cmml"><mo id="S6.Ex14.m2.7.7.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex14.m2.7.7.2.3.cmml">[</mo><msub id="S6.Ex14.m2.6.6.1.1.1" xref="S6.Ex14.m2.6.6.1.1.1.cmml"><mrow id="S6.Ex14.m2.6.6.1.1.1.2.2" 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xref="S6.Ex14.m2.7.7.2.2.2.1.1.1.2.2">ℝ</ci><ci id="S6.Ex14.m2.7.7.2.2.2.1.1.1.2.3.cmml" xref="S6.Ex14.m2.7.7.2.2.2.1.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex14.m2.7.7.2.2.2.1.1.1.3.cmml" xref="S6.Ex14.m2.7.7.2.2.2.1.1.1.3"></plus></apply><apply id="S6.Ex14.m2.7.7.2.2.2.2.2.2.cmml" xref="S6.Ex14.m2.7.7.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex14.m2.7.7.2.2.2.2.2.2.1.cmml" xref="S6.Ex14.m2.7.7.2.2.2.2.2.2">subscript</csymbol><ci id="S6.Ex14.m2.7.7.2.2.2.2.2.2.2.cmml" xref="S6.Ex14.m2.7.7.2.2.2.2.2.2.2">𝑤</ci><ci id="S6.Ex14.m2.7.7.2.2.2.2.2.2.3.cmml" xref="S6.Ex14.m2.7.7.2.2.2.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex14.m2.5.5.cmml" xref="S6.Ex14.m2.5.5">𝑋</ci></vector></apply></interval><apply id="S6.Ex14.m2.7.7.4.cmml" xref="S6.Ex14.m2.7.7.4"><divide id="S6.Ex14.m2.7.7.4.1.cmml" xref="S6.Ex14.m2.7.7.4"></divide><ci id="S6.Ex14.m2.7.7.4.2.cmml" xref="S6.Ex14.m2.7.7.4.2">ℓ</ci><apply id="S6.Ex14.m2.7.7.4.3.cmml" xref="S6.Ex14.m2.7.7.4.3"><csymbol cd="ambiguous" id="S6.Ex14.m2.7.7.4.3.1.cmml" xref="S6.Ex14.m2.7.7.4.3">subscript</csymbol><ci id="S6.Ex14.m2.7.7.4.3.2.cmml" xref="S6.Ex14.m2.7.7.4.3.2">𝑘</ci><cn id="S6.Ex14.m2.7.7.4.3.3.cmml" type="integer" xref="S6.Ex14.m2.7.7.4.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex14.m2.7c">\displaystyle\;\big{[}[L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{0}^{k_{0}+k_{% 1}}(\mathbb{R}^{d}_{+},w_{\gamma};X)]_{\frac{k_{0}}{k_{0}+k_{1}}},W^{k_{0}+k_{% 1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex14.m2.7d">[ [ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex15"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S6.Ex15.m1.1"><semantics id="S6.Ex15.m1.1a"><mo id="S6.Ex15.m1.1.1" xref="S6.Ex15.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S6.Ex15.m1.1b"><eq id="S6.Ex15.m1.1.1.cmml" xref="S6.Ex15.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex15.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S6.Ex15.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{0}^{k_{0}+k_{1% },p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{k_{0}+\ell}{k_{0}+k_{1}}}=% W^{k_{0}+\ell,p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X)." class="ltx_Math" display="inline" id="S6.Ex15.m2.8"><semantics id="S6.Ex15.m2.8a"><mrow id="S6.Ex15.m2.8.8.1" xref="S6.Ex15.m2.8.8.1.1.cmml"><mrow id="S6.Ex15.m2.8.8.1.1" xref="S6.Ex15.m2.8.8.1.1.cmml"><msub id="S6.Ex15.m2.8.8.1.1.2" xref="S6.Ex15.m2.8.8.1.1.2.cmml"><mrow id="S6.Ex15.m2.8.8.1.1.2.2.2" xref="S6.Ex15.m2.8.8.1.1.2.2.3.cmml"><mo id="S6.Ex15.m2.8.8.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex15.m2.8.8.1.1.2.2.3.cmml">[</mo><mrow id="S6.Ex15.m2.8.8.1.1.1.1.1.1" xref="S6.Ex15.m2.8.8.1.1.1.1.1.1.cmml"><msup id="S6.Ex15.m2.8.8.1.1.1.1.1.1.4" xref="S6.Ex15.m2.8.8.1.1.1.1.1.1.4.cmml"><mi id="S6.Ex15.m2.8.8.1.1.1.1.1.1.4.2" xref="S6.Ex15.m2.8.8.1.1.1.1.1.1.4.2.cmml">L</mi><mi id="S6.Ex15.m2.8.8.1.1.1.1.1.1.4.3" xref="S6.Ex15.m2.8.8.1.1.1.1.1.1.4.3.cmml">p</mi></msup><mo id="S6.Ex15.m2.8.8.1.1.1.1.1.1.3" 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xref="S6.Ex15.m2.8.8.1.1.3.1.1.1">superscript</csymbol><ci id="S6.Ex15.m2.8.8.1.1.3.1.1.1.2.2.cmml" xref="S6.Ex15.m2.8.8.1.1.3.1.1.1.2.2">ℝ</ci><ci id="S6.Ex15.m2.8.8.1.1.3.1.1.1.2.3.cmml" xref="S6.Ex15.m2.8.8.1.1.3.1.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex15.m2.8.8.1.1.3.1.1.1.3.cmml" xref="S6.Ex15.m2.8.8.1.1.3.1.1.1.3"></plus></apply><apply id="S6.Ex15.m2.8.8.1.1.4.2.2.2.cmml" xref="S6.Ex15.m2.8.8.1.1.4.2.2.2"><csymbol cd="ambiguous" id="S6.Ex15.m2.8.8.1.1.4.2.2.2.1.cmml" xref="S6.Ex15.m2.8.8.1.1.4.2.2.2">subscript</csymbol><ci id="S6.Ex15.m2.8.8.1.1.4.2.2.2.2.cmml" xref="S6.Ex15.m2.8.8.1.1.4.2.2.2.2">𝑤</ci><ci id="S6.Ex15.m2.8.8.1.1.4.2.2.2.3.cmml" xref="S6.Ex15.m2.8.8.1.1.4.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex15.m2.7.7.cmml" xref="S6.Ex15.m2.7.7">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex15.m2.8c">\displaystyle\;\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{0}^{k_{0}+k_{1% },p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{k_{0}+\ell}{k_{0}+k_{1}}}=% W^{k_{0}+\ell,p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma};X).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex15.m2.8d">[ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.5.p1.8">For <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.I1.i2" title="item ii ‣ Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> with <math alttext="k_{0}=0" class="ltx_Math" display="inline" id="S6.SS1.5.p1.8.m1.1"><semantics id="S6.SS1.5.p1.8.m1.1a"><mrow id="S6.SS1.5.p1.8.m1.1.1" xref="S6.SS1.5.p1.8.m1.1.1.cmml"><msub id="S6.SS1.5.p1.8.m1.1.1.2" xref="S6.SS1.5.p1.8.m1.1.1.2.cmml"><mi id="S6.SS1.5.p1.8.m1.1.1.2.2" xref="S6.SS1.5.p1.8.m1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.5.p1.8.m1.1.1.2.3" xref="S6.SS1.5.p1.8.m1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.5.p1.8.m1.1.1.1" xref="S6.SS1.5.p1.8.m1.1.1.1.cmml">=</mo><mn id="S6.SS1.5.p1.8.m1.1.1.3" xref="S6.SS1.5.p1.8.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.8.m1.1b"><apply id="S6.SS1.5.p1.8.m1.1.1.cmml" xref="S6.SS1.5.p1.8.m1.1.1"><eq id="S6.SS1.5.p1.8.m1.1.1.1.cmml" xref="S6.SS1.5.p1.8.m1.1.1.1"></eq><apply id="S6.SS1.5.p1.8.m1.1.1.2.cmml" xref="S6.SS1.5.p1.8.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.5.p1.8.m1.1.1.2.1.cmml" xref="S6.SS1.5.p1.8.m1.1.1.2">subscript</csymbol><ci id="S6.SS1.5.p1.8.m1.1.1.2.2.cmml" xref="S6.SS1.5.p1.8.m1.1.1.2.2">𝑘</ci><cn id="S6.SS1.5.p1.8.m1.1.1.2.3.cmml" type="integer" xref="S6.SS1.5.p1.8.m1.1.1.2.3">0</cn></apply><cn id="S6.SS1.5.p1.8.m1.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.8.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.8.m1.1c">k_{0}=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.8.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math>, it follows from <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.I1.i1" title="item i ‣ Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx30"> <tbody id="S6.Ex16"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W^{\ell,p}(\mathbb{R}_{+}^{d},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.Ex16.m1.5"><semantics id="S6.Ex16.m1.5a"><mrow id="S6.Ex16.m1.5.5" xref="S6.Ex16.m1.5.5.cmml"><msup id="S6.Ex16.m1.5.5.4" xref="S6.Ex16.m1.5.5.4.cmml"><mi id="S6.Ex16.m1.5.5.4.2" xref="S6.Ex16.m1.5.5.4.2.cmml">W</mi><mrow id="S6.Ex16.m1.2.2.2.4" xref="S6.Ex16.m1.2.2.2.3.cmml"><mi id="S6.Ex16.m1.1.1.1.1" mathvariant="normal" xref="S6.Ex16.m1.1.1.1.1.cmml">ℓ</mi><mo id="S6.Ex16.m1.2.2.2.4.1" xref="S6.Ex16.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex16.m1.2.2.2.2" xref="S6.Ex16.m1.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S6.Ex16.m1.5.5.3" xref="S6.Ex16.m1.5.5.3.cmml">⁢</mo><mrow id="S6.Ex16.m1.5.5.2.2" xref="S6.Ex16.m1.5.5.2.3.cmml"><mo id="S6.Ex16.m1.5.5.2.2.3" stretchy="false" xref="S6.Ex16.m1.5.5.2.3.cmml">(</mo><msubsup id="S6.Ex16.m1.4.4.1.1.1" xref="S6.Ex16.m1.4.4.1.1.1.cmml"><mi id="S6.Ex16.m1.4.4.1.1.1.2.2" xref="S6.Ex16.m1.4.4.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex16.m1.4.4.1.1.1.2.3" xref="S6.Ex16.m1.4.4.1.1.1.2.3.cmml">+</mo><mi id="S6.Ex16.m1.4.4.1.1.1.3" xref="S6.Ex16.m1.4.4.1.1.1.3.cmml">d</mi></msubsup><mo id="S6.Ex16.m1.5.5.2.2.4" xref="S6.Ex16.m1.5.5.2.3.cmml">,</mo><msub id="S6.Ex16.m1.5.5.2.2.2" xref="S6.Ex16.m1.5.5.2.2.2.cmml"><mi id="S6.Ex16.m1.5.5.2.2.2.2" xref="S6.Ex16.m1.5.5.2.2.2.2.cmml">w</mi><mi id="S6.Ex16.m1.5.5.2.2.2.3" xref="S6.Ex16.m1.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex16.m1.5.5.2.2.5" xref="S6.Ex16.m1.5.5.2.3.cmml">;</mo><mi id="S6.Ex16.m1.3.3" xref="S6.Ex16.m1.3.3.cmml">X</mi><mo id="S6.Ex16.m1.5.5.2.2.6" stretchy="false" xref="S6.Ex16.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex16.m1.5b"><apply id="S6.Ex16.m1.5.5.cmml" xref="S6.Ex16.m1.5.5"><times id="S6.Ex16.m1.5.5.3.cmml" xref="S6.Ex16.m1.5.5.3"></times><apply id="S6.Ex16.m1.5.5.4.cmml" xref="S6.Ex16.m1.5.5.4"><csymbol cd="ambiguous" id="S6.Ex16.m1.5.5.4.1.cmml" xref="S6.Ex16.m1.5.5.4">superscript</csymbol><ci id="S6.Ex16.m1.5.5.4.2.cmml" xref="S6.Ex16.m1.5.5.4.2">𝑊</ci><list id="S6.Ex16.m1.2.2.2.3.cmml" xref="S6.Ex16.m1.2.2.2.4"><ci id="S6.Ex16.m1.1.1.1.1.cmml" xref="S6.Ex16.m1.1.1.1.1">ℓ</ci><ci id="S6.Ex16.m1.2.2.2.2.cmml" xref="S6.Ex16.m1.2.2.2.2">𝑝</ci></list></apply><vector id="S6.Ex16.m1.5.5.2.3.cmml" xref="S6.Ex16.m1.5.5.2.2"><apply id="S6.Ex16.m1.4.4.1.1.1.cmml" xref="S6.Ex16.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.Ex16.m1.4.4.1.1.1.1.cmml" xref="S6.Ex16.m1.4.4.1.1.1">superscript</csymbol><apply id="S6.Ex16.m1.4.4.1.1.1.2.cmml" xref="S6.Ex16.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.Ex16.m1.4.4.1.1.1.2.1.cmml" xref="S6.Ex16.m1.4.4.1.1.1">subscript</csymbol><ci id="S6.Ex16.m1.4.4.1.1.1.2.2.cmml" xref="S6.Ex16.m1.4.4.1.1.1.2.2">ℝ</ci><plus id="S6.Ex16.m1.4.4.1.1.1.2.3.cmml" xref="S6.Ex16.m1.4.4.1.1.1.2.3"></plus></apply><ci id="S6.Ex16.m1.4.4.1.1.1.3.cmml" xref="S6.Ex16.m1.4.4.1.1.1.3">𝑑</ci></apply><apply id="S6.Ex16.m1.5.5.2.2.2.cmml" xref="S6.Ex16.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.Ex16.m1.5.5.2.2.2.1.cmml" xref="S6.Ex16.m1.5.5.2.2.2">subscript</csymbol><ci id="S6.Ex16.m1.5.5.2.2.2.2.cmml" xref="S6.Ex16.m1.5.5.2.2.2.2">𝑤</ci><ci id="S6.Ex16.m1.5.5.2.2.2.3.cmml" xref="S6.Ex16.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex16.m1.3.3.cmml" xref="S6.Ex16.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex16.m1.5c">\displaystyle W^{\ell,p}(\mathbb{R}_{+}^{d},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex16.m1.5d">italic_W start_POSTSUPERSCRIPT roman_ℓ , 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type="integer" xref="S6.Ex16.m2.11.11.4.4.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex16.m2.11c">\displaystyle=W^{\ell,p}_{0}(\mathbb{R}_{+}^{d},w_{\gamma};X)=\big{[}L^{p}(% \mathbb{R}^{d}_{+},w_{\gamma};X),W^{k_{1},p}_{0}(\mathbb{R}^{d}_{+},w_{\gamma}% ;X)\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex16.m2.11d">= italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex17"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\hookrightarrow\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W^{k% _{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}." class="ltx_Math" display="inline" id="S6.Ex17.m1.5"><semantics 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xref="S6.Ex17.m1.5.5.1.1.2.4"><divide id="S6.Ex17.m1.5.5.1.1.2.4.1.cmml" xref="S6.Ex17.m1.5.5.1.1.2.4"></divide><ci id="S6.Ex17.m1.5.5.1.1.2.4.2.cmml" xref="S6.Ex17.m1.5.5.1.1.2.4.2">ℓ</ci><apply id="S6.Ex17.m1.5.5.1.1.2.4.3.cmml" xref="S6.Ex17.m1.5.5.1.1.2.4.3"><csymbol cd="ambiguous" id="S6.Ex17.m1.5.5.1.1.2.4.3.1.cmml" xref="S6.Ex17.m1.5.5.1.1.2.4.3">subscript</csymbol><ci id="S6.Ex17.m1.5.5.1.1.2.4.3.2.cmml" xref="S6.Ex17.m1.5.5.1.1.2.4.3.2">𝑘</ci><cn id="S6.Ex17.m1.5.5.1.1.2.4.3.3.cmml" type="integer" xref="S6.Ex17.m1.5.5.1.1.2.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex17.m1.5c">\displaystyle\hookrightarrow\big{[}L^{p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W^{k% _{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex17.m1.5d">↪ [ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.5.p1.16">To prove the other embedding, it suffices to show that</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex18"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|u\|_{W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}\leq C\|u\|_{[L^{p}(\mathbb% {R}^{d}_{+},w_{\gamma};X),W^{k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)]_{\frac% {\ell}{k_{1}}}},\qquad u\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X)," class="ltx_Math" display="block" id="S6.Ex18.m1.15"><semantics id="S6.Ex18.m1.15a"><mrow id="S6.Ex18.m1.15.15.1"><mrow id="S6.Ex18.m1.15.15.1.1.2" xref="S6.Ex18.m1.15.15.1.1.3.cmml"><mrow id="S6.Ex18.m1.15.15.1.1.1.1" xref="S6.Ex18.m1.15.15.1.1.1.1.cmml"><msub id="S6.Ex18.m1.15.15.1.1.1.1.2" xref="S6.Ex18.m1.15.15.1.1.1.1.2.cmml"><mrow id="S6.Ex18.m1.15.15.1.1.1.1.2.2.2" xref="S6.Ex18.m1.15.15.1.1.1.1.2.2.1.cmml"><mo id="S6.Ex18.m1.15.15.1.1.1.1.2.2.2.1" stretchy="false" xref="S6.Ex18.m1.15.15.1.1.1.1.2.2.1.1.cmml">‖</mo><mi id="S6.Ex18.m1.12.12" xref="S6.Ex18.m1.12.12.cmml">u</mi><mo id="S6.Ex18.m1.15.15.1.1.1.1.2.2.2.2" stretchy="false" xref="S6.Ex18.m1.15.15.1.1.1.1.2.2.1.1.cmml">‖</mo></mrow><mrow id="S6.Ex18.m1.5.5.5" 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xref="S6.Ex18.m1.15.15.1.1.2.2.1.1.1.1.2.2">ℝ</ci><ci id="S6.Ex18.m1.15.15.1.1.2.2.1.1.1.1.2.3.cmml" xref="S6.Ex18.m1.15.15.1.1.2.2.1.1.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex18.m1.15.15.1.1.2.2.1.1.1.1.3.cmml" xref="S6.Ex18.m1.15.15.1.1.2.2.1.1.1.1.3"></plus></apply><ci id="S6.Ex18.m1.14.14.cmml" xref="S6.Ex18.m1.14.14">𝑋</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex18.m1.15c">\|u\|_{W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}\leq C\|u\|_{[L^{p}(\mathbb% {R}^{d}_{+},w_{\gamma};X),W^{k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)]_{\frac% {\ell}{k_{1}}}},\qquad u\in C_{\mathrm{c}}^{\infty}(\mathbb{R}^{d}_{+};X),</annotation><annotation encoding="application/x-llamapun" id="S6.Ex18.m1.15d">∥ italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_u ∥ start_POSTSUBSCRIPT [ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_u ∈ italic_C start_POSTSUBSCRIPT roman_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.5.p1.13">by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>, Lemma 3.9]</cite>. Again, let <math alttext="j\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS1.5.p1.9.m1.1"><semantics id="S6.SS1.5.p1.9.m1.1a"><mrow id="S6.SS1.5.p1.9.m1.1.1" xref="S6.SS1.5.p1.9.m1.1.1.cmml"><mi id="S6.SS1.5.p1.9.m1.1.1.2" xref="S6.SS1.5.p1.9.m1.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.9.m1.1.1.1" xref="S6.SS1.5.p1.9.m1.1.1.1.cmml">∈</mo><msub id="S6.SS1.5.p1.9.m1.1.1.3" xref="S6.SS1.5.p1.9.m1.1.1.3.cmml"><mi id="S6.SS1.5.p1.9.m1.1.1.3.2" xref="S6.SS1.5.p1.9.m1.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.5.p1.9.m1.1.1.3.3" xref="S6.SS1.5.p1.9.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.9.m1.1b"><apply id="S6.SS1.5.p1.9.m1.1.1.cmml" xref="S6.SS1.5.p1.9.m1.1.1"><in id="S6.SS1.5.p1.9.m1.1.1.1.cmml" xref="S6.SS1.5.p1.9.m1.1.1.1"></in><ci id="S6.SS1.5.p1.9.m1.1.1.2.cmml" xref="S6.SS1.5.p1.9.m1.1.1.2">𝑗</ci><apply id="S6.SS1.5.p1.9.m1.1.1.3.cmml" xref="S6.SS1.5.p1.9.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.5.p1.9.m1.1.1.3.1.cmml" xref="S6.SS1.5.p1.9.m1.1.1.3">subscript</csymbol><ci id="S6.SS1.5.p1.9.m1.1.1.3.2.cmml" xref="S6.SS1.5.p1.9.m1.1.1.3.2">ℕ</ci><cn id="S6.SS1.5.p1.9.m1.1.1.3.3.cmml" type="integer" xref="S6.SS1.5.p1.9.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.9.m1.1c">j\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.9.m1.1d">italic_j ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be such that <math alttext="\gamma\in(jp-1,(j+1)p-1)" class="ltx_Math" display="inline" id="S6.SS1.5.p1.10.m2.2"><semantics id="S6.SS1.5.p1.10.m2.2a"><mrow id="S6.SS1.5.p1.10.m2.2.2" xref="S6.SS1.5.p1.10.m2.2.2.cmml"><mi id="S6.SS1.5.p1.10.m2.2.2.4" xref="S6.SS1.5.p1.10.m2.2.2.4.cmml">γ</mi><mo id="S6.SS1.5.p1.10.m2.2.2.3" xref="S6.SS1.5.p1.10.m2.2.2.3.cmml">∈</mo><mrow id="S6.SS1.5.p1.10.m2.2.2.2.2" xref="S6.SS1.5.p1.10.m2.2.2.2.3.cmml"><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.3" stretchy="false" xref="S6.SS1.5.p1.10.m2.2.2.2.3.cmml">(</mo><mrow id="S6.SS1.5.p1.10.m2.1.1.1.1.1" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.cmml"><mrow id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.cmml"><mi id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.2" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.2.cmml">j</mi><mo id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.1" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.3" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.3.cmml">p</mi></mrow><mo id="S6.SS1.5.p1.10.m2.1.1.1.1.1.1" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.1.cmml">−</mo><mn id="S6.SS1.5.p1.10.m2.1.1.1.1.1.3" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.4" xref="S6.SS1.5.p1.10.m2.2.2.2.3.cmml">,</mo><mrow id="S6.SS1.5.p1.10.m2.2.2.2.2.2" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.cmml"><mrow id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.cmml"><mrow id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.cmml"><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.2" stretchy="false" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.cmml"><mi id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.2" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.1" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.1.cmml">+</mo><mn id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.3" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.3" stretchy="false" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.2" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.2.cmml">⁢</mo><mi id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.3" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.3.cmml">p</mi></mrow><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.2.2" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.2.cmml">−</mo><mn id="S6.SS1.5.p1.10.m2.2.2.2.2.2.3" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS1.5.p1.10.m2.2.2.2.2.5" stretchy="false" xref="S6.SS1.5.p1.10.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.10.m2.2b"><apply id="S6.SS1.5.p1.10.m2.2.2.cmml" xref="S6.SS1.5.p1.10.m2.2.2"><in id="S6.SS1.5.p1.10.m2.2.2.3.cmml" xref="S6.SS1.5.p1.10.m2.2.2.3"></in><ci id="S6.SS1.5.p1.10.m2.2.2.4.cmml" xref="S6.SS1.5.p1.10.m2.2.2.4">𝛾</ci><interval closure="open" id="S6.SS1.5.p1.10.m2.2.2.2.3.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2"><apply id="S6.SS1.5.p1.10.m2.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1"><minus id="S6.SS1.5.p1.10.m2.1.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.1"></minus><apply id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.cmml" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2"><times id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.1.cmml" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.1"></times><ci id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.2.cmml" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.2">𝑗</ci><ci id="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.3.cmml" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.2.3">𝑝</ci></apply><cn id="S6.SS1.5.p1.10.m2.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.10.m2.1.1.1.1.1.3">1</cn></apply><apply id="S6.SS1.5.p1.10.m2.2.2.2.2.2.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2"><minus id="S6.SS1.5.p1.10.m2.2.2.2.2.2.2.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.2"></minus><apply id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1"><times id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.2.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.2"></times><apply id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1"><plus id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.1.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.1"></plus><ci id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.2.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.2">𝑗</ci><cn id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.1.1.1.3">1</cn></apply><ci id="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.3.cmml" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.1.3">𝑝</ci></apply><cn id="S6.SS1.5.p1.10.m2.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.5.p1.10.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.10.m2.2c">\gamma\in(jp-1,(j+1)p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.10.m2.2d">italic_γ ∈ ( italic_j italic_p - 1 , ( italic_j + 1 ) italic_p - 1 )</annotation></semantics></math>. The case <math alttext="j=0" class="ltx_Math" display="inline" id="S6.SS1.5.p1.11.m3.1"><semantics id="S6.SS1.5.p1.11.m3.1a"><mrow id="S6.SS1.5.p1.11.m3.1.1" xref="S6.SS1.5.p1.11.m3.1.1.cmml"><mi id="S6.SS1.5.p1.11.m3.1.1.2" xref="S6.SS1.5.p1.11.m3.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.11.m3.1.1.1" xref="S6.SS1.5.p1.11.m3.1.1.1.cmml">=</mo><mn id="S6.SS1.5.p1.11.m3.1.1.3" xref="S6.SS1.5.p1.11.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.11.m3.1b"><apply id="S6.SS1.5.p1.11.m3.1.1.cmml" xref="S6.SS1.5.p1.11.m3.1.1"><eq id="S6.SS1.5.p1.11.m3.1.1.1.cmml" xref="S6.SS1.5.p1.11.m3.1.1.1"></eq><ci id="S6.SS1.5.p1.11.m3.1.1.2.cmml" xref="S6.SS1.5.p1.11.m3.1.1.2">𝑗</ci><cn id="S6.SS1.5.p1.11.m3.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.11.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.11.m3.1c">j=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.11.m3.1d">italic_j = 0</annotation></semantics></math> follows from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.5 & 5.6]</cite>. For <math alttext="j\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S6.SS1.5.p1.12.m4.1"><semantics id="S6.SS1.5.p1.12.m4.1a"><mrow id="S6.SS1.5.p1.12.m4.1.1" xref="S6.SS1.5.p1.12.m4.1.1.cmml"><mi id="S6.SS1.5.p1.12.m4.1.1.2" xref="S6.SS1.5.p1.12.m4.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.12.m4.1.1.1" xref="S6.SS1.5.p1.12.m4.1.1.1.cmml">∈</mo><msub id="S6.SS1.5.p1.12.m4.1.1.3" xref="S6.SS1.5.p1.12.m4.1.1.3.cmml"><mi id="S6.SS1.5.p1.12.m4.1.1.3.2" xref="S6.SS1.5.p1.12.m4.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.5.p1.12.m4.1.1.3.3" xref="S6.SS1.5.p1.12.m4.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.12.m4.1b"><apply id="S6.SS1.5.p1.12.m4.1.1.cmml" xref="S6.SS1.5.p1.12.m4.1.1"><in id="S6.SS1.5.p1.12.m4.1.1.1.cmml" xref="S6.SS1.5.p1.12.m4.1.1.1"></in><ci id="S6.SS1.5.p1.12.m4.1.1.2.cmml" xref="S6.SS1.5.p1.12.m4.1.1.2">𝑗</ci><apply id="S6.SS1.5.p1.12.m4.1.1.3.cmml" xref="S6.SS1.5.p1.12.m4.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.5.p1.12.m4.1.1.3.1.cmml" xref="S6.SS1.5.p1.12.m4.1.1.3">subscript</csymbol><ci id="S6.SS1.5.p1.12.m4.1.1.3.2.cmml" xref="S6.SS1.5.p1.12.m4.1.1.3.2">ℕ</ci><cn id="S6.SS1.5.p1.12.m4.1.1.3.3.cmml" type="integer" xref="S6.SS1.5.p1.12.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.12.m4.1c">j\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.12.m4.1d">italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we obtain by applying <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Lemma 3.6]</cite>, the case <math alttext="j=0" class="ltx_Math" display="inline" id="S6.SS1.5.p1.13.m5.1"><semantics id="S6.SS1.5.p1.13.m5.1a"><mrow id="S6.SS1.5.p1.13.m5.1.1" xref="S6.SS1.5.p1.13.m5.1.1.cmml"><mi id="S6.SS1.5.p1.13.m5.1.1.2" xref="S6.SS1.5.p1.13.m5.1.1.2.cmml">j</mi><mo id="S6.SS1.5.p1.13.m5.1.1.1" xref="S6.SS1.5.p1.13.m5.1.1.1.cmml">=</mo><mn id="S6.SS1.5.p1.13.m5.1.1.3" xref="S6.SS1.5.p1.13.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.13.m5.1b"><apply id="S6.SS1.5.p1.13.m5.1.1.cmml" xref="S6.SS1.5.p1.13.m5.1.1"><eq id="S6.SS1.5.p1.13.m5.1.1.1.cmml" xref="S6.SS1.5.p1.13.m5.1.1.1"></eq><ci id="S6.SS1.5.p1.13.m5.1.1.2.cmml" xref="S6.SS1.5.p1.13.m5.1.1.2">𝑗</ci><cn id="S6.SS1.5.p1.13.m5.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.13.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.13.m5.1c">j=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.13.m5.1d">italic_j = 0</annotation></semantics></math> and properties of the complex interpolation method, that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx31"> <tbody id="S6.Ex19"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|u\|_{W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}" class="ltx_Math" display="inline" id="S6.Ex19.m1.6"><semantics id="S6.Ex19.m1.6a"><msub id="S6.Ex19.m1.6.7" xref="S6.Ex19.m1.6.7.cmml"><mrow id="S6.Ex19.m1.6.7.2.2" xref="S6.Ex19.m1.6.7.2.1.cmml"><mo id="S6.Ex19.m1.6.7.2.2.1" stretchy="false" xref="S6.Ex19.m1.6.7.2.1.1.cmml">‖</mo><mi id="S6.Ex19.m1.6.6" xref="S6.Ex19.m1.6.6.cmml">u</mi><mo id="S6.Ex19.m1.6.7.2.2.2" stretchy="false" xref="S6.Ex19.m1.6.7.2.1.1.cmml">‖</mo></mrow><mrow id="S6.Ex19.m1.5.5.5" xref="S6.Ex19.m1.5.5.5.cmml"><msup id="S6.Ex19.m1.5.5.5.7" xref="S6.Ex19.m1.5.5.5.7.cmml"><mi id="S6.Ex19.m1.5.5.5.7.2" xref="S6.Ex19.m1.5.5.5.7.2.cmml">W</mi><mrow id="S6.Ex19.m1.2.2.2.2.2.4" xref="S6.Ex19.m1.2.2.2.2.2.3.cmml"><mi id="S6.Ex19.m1.1.1.1.1.1.1" mathvariant="normal" xref="S6.Ex19.m1.1.1.1.1.1.1.cmml">ℓ</mi><mo id="S6.Ex19.m1.2.2.2.2.2.4.1" xref="S6.Ex19.m1.2.2.2.2.2.3.cmml">,</mo><mi id="S6.Ex19.m1.2.2.2.2.2.2" xref="S6.Ex19.m1.2.2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S6.Ex19.m1.5.5.5.6" xref="S6.Ex19.m1.5.5.5.6.cmml">⁢</mo><mrow id="S6.Ex19.m1.5.5.5.5.2" xref="S6.Ex19.m1.5.5.5.5.3.cmml"><mo id="S6.Ex19.m1.5.5.5.5.2.3" stretchy="false" xref="S6.Ex19.m1.5.5.5.5.3.cmml">(</mo><msubsup id="S6.Ex19.m1.4.4.4.4.1.1" xref="S6.Ex19.m1.4.4.4.4.1.1.cmml"><mi id="S6.Ex19.m1.4.4.4.4.1.1.2.2" xref="S6.Ex19.m1.4.4.4.4.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex19.m1.4.4.4.4.1.1.3" xref="S6.Ex19.m1.4.4.4.4.1.1.3.cmml">+</mo><mi id="S6.Ex19.m1.4.4.4.4.1.1.2.3" xref="S6.Ex19.m1.4.4.4.4.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex19.m1.5.5.5.5.2.4" xref="S6.Ex19.m1.5.5.5.5.3.cmml">,</mo><msub id="S6.Ex19.m1.5.5.5.5.2.2" xref="S6.Ex19.m1.5.5.5.5.2.2.cmml"><mi id="S6.Ex19.m1.5.5.5.5.2.2.2" xref="S6.Ex19.m1.5.5.5.5.2.2.2.cmml">w</mi><mi id="S6.Ex19.m1.5.5.5.5.2.2.3" xref="S6.Ex19.m1.5.5.5.5.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex19.m1.5.5.5.5.2.5" xref="S6.Ex19.m1.5.5.5.5.3.cmml">;</mo><mi id="S6.Ex19.m1.3.3.3.3" xref="S6.Ex19.m1.3.3.3.3.cmml">X</mi><mo id="S6.Ex19.m1.5.5.5.5.2.6" stretchy="false" xref="S6.Ex19.m1.5.5.5.5.3.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.Ex19.m1.6b"><apply id="S6.Ex19.m1.6.7.cmml" xref="S6.Ex19.m1.6.7"><csymbol cd="ambiguous" id="S6.Ex19.m1.6.7.1.cmml" xref="S6.Ex19.m1.6.7">subscript</csymbol><apply id="S6.Ex19.m1.6.7.2.1.cmml" xref="S6.Ex19.m1.6.7.2.2"><csymbol cd="latexml" id="S6.Ex19.m1.6.7.2.1.1.cmml" xref="S6.Ex19.m1.6.7.2.2.1">norm</csymbol><ci id="S6.Ex19.m1.6.6.cmml" xref="S6.Ex19.m1.6.6">𝑢</ci></apply><apply id="S6.Ex19.m1.5.5.5.cmml" xref="S6.Ex19.m1.5.5.5"><times id="S6.Ex19.m1.5.5.5.6.cmml" xref="S6.Ex19.m1.5.5.5.6"></times><apply id="S6.Ex19.m1.5.5.5.7.cmml" xref="S6.Ex19.m1.5.5.5.7"><csymbol cd="ambiguous" id="S6.Ex19.m1.5.5.5.7.1.cmml" 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xref="S6.Ex19.m1.4.4.4.4.1.1.3"></plus></apply><apply id="S6.Ex19.m1.5.5.5.5.2.2.cmml" xref="S6.Ex19.m1.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S6.Ex19.m1.5.5.5.5.2.2.1.cmml" xref="S6.Ex19.m1.5.5.5.5.2.2">subscript</csymbol><ci id="S6.Ex19.m1.5.5.5.5.2.2.2.cmml" xref="S6.Ex19.m1.5.5.5.5.2.2.2">𝑤</ci><ci id="S6.Ex19.m1.5.5.5.5.2.2.3.cmml" xref="S6.Ex19.m1.5.5.5.5.2.2.3">𝛾</ci></apply><ci id="S6.Ex19.m1.3.3.3.3.cmml" xref="S6.Ex19.m1.3.3.3.3">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex19.m1.6c">\displaystyle\|u\|_{W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex19.m1.6d">∥ italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) 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id="S6.Ex19.m2.6c">\displaystyle=\|M^{-j}M^{j}u\|_{W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex19.m2.6d">= ∥ italic_M start_POSTSUPERSCRIPT - italic_j end_POSTSUPERSCRIPT italic_M start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex20"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\leq 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id="S6.Ex20.m1.4.4.4.4.1.1.2.2.cmml" xref="S6.Ex20.m1.4.4.4.4.1.1.2.2">ℝ</ci><ci id="S6.Ex20.m1.4.4.4.4.1.1.2.3.cmml" xref="S6.Ex20.m1.4.4.4.4.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex20.m1.4.4.4.4.1.1.3.cmml" xref="S6.Ex20.m1.4.4.4.4.1.1.3"></plus></apply><apply id="S6.Ex20.m1.5.5.5.5.2.2.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2"><csymbol cd="ambiguous" id="S6.Ex20.m1.5.5.5.5.2.2.1.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2">subscript</csymbol><ci id="S6.Ex20.m1.5.5.5.5.2.2.2.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.2">𝑤</ci><apply id="S6.Ex20.m1.5.5.5.5.2.2.3.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.3"><minus id="S6.Ex20.m1.5.5.5.5.2.2.3.1.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.3.1"></minus><ci id="S6.Ex20.m1.5.5.5.5.2.2.3.2.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.3.2">𝛾</ci><apply id="S6.Ex20.m1.5.5.5.5.2.2.3.3.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.3.3"><times id="S6.Ex20.m1.5.5.5.5.2.2.3.3.1.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.3.3.1"></times><ci id="S6.Ex20.m1.5.5.5.5.2.2.3.3.2.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.3.3.2">𝑗</ci><ci id="S6.Ex20.m1.5.5.5.5.2.2.3.3.3.cmml" xref="S6.Ex20.m1.5.5.5.5.2.2.3.3.3">𝑝</ci></apply></apply></apply><ci id="S6.Ex20.m1.3.3.3.3.cmml" xref="S6.Ex20.m1.3.3.3.3">𝑋</ci></vector></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex20.m1.6c">\displaystyle\leq C\|M^{j}u\|_{W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma-jp};X)}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex20.m1.6d">≤ italic_C ∥ italic_M start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody 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encoding="application/x-llamapun" id="S6.Ex21.m1.7d">≤ italic_C ∥ italic_M start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT [ italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody 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id="S6.SS1.5.p1.14.m1.1.1" xref="S6.SS1.5.p1.14.m1.1.1.cmml"><msub id="S6.SS1.5.p1.14.m1.1.1.2" xref="S6.SS1.5.p1.14.m1.1.1.2.cmml"><mi id="S6.SS1.5.p1.14.m1.1.1.2.2" xref="S6.SS1.5.p1.14.m1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.5.p1.14.m1.1.1.2.3" xref="S6.SS1.5.p1.14.m1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.5.p1.14.m1.1.1.1" xref="S6.SS1.5.p1.14.m1.1.1.1.cmml">∈</mo><msub id="S6.SS1.5.p1.14.m1.1.1.3" xref="S6.SS1.5.p1.14.m1.1.1.3.cmml"><mi id="S6.SS1.5.p1.14.m1.1.1.3.2" xref="S6.SS1.5.p1.14.m1.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.5.p1.14.m1.1.1.3.3" xref="S6.SS1.5.p1.14.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.14.m1.1b"><apply id="S6.SS1.5.p1.14.m1.1.1.cmml" xref="S6.SS1.5.p1.14.m1.1.1"><in id="S6.SS1.5.p1.14.m1.1.1.1.cmml" xref="S6.SS1.5.p1.14.m1.1.1.1"></in><apply id="S6.SS1.5.p1.14.m1.1.1.2.cmml" xref="S6.SS1.5.p1.14.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.5.p1.14.m1.1.1.2.1.cmml" xref="S6.SS1.5.p1.14.m1.1.1.2">subscript</csymbol><ci id="S6.SS1.5.p1.14.m1.1.1.2.2.cmml" xref="S6.SS1.5.p1.14.m1.1.1.2.2">𝑘</ci><cn id="S6.SS1.5.p1.14.m1.1.1.2.3.cmml" type="integer" xref="S6.SS1.5.p1.14.m1.1.1.2.3">0</cn></apply><apply id="S6.SS1.5.p1.14.m1.1.1.3.cmml" xref="S6.SS1.5.p1.14.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.5.p1.14.m1.1.1.3.1.cmml" xref="S6.SS1.5.p1.14.m1.1.1.3">subscript</csymbol><ci id="S6.SS1.5.p1.14.m1.1.1.3.2.cmml" xref="S6.SS1.5.p1.14.m1.1.1.3.2">ℕ</ci><cn id="S6.SS1.5.p1.14.m1.1.1.3.3.cmml" type="integer" xref="S6.SS1.5.p1.14.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.14.m1.1c">k_{0}\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.14.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> again follows from the case <math alttext="k_{0}=0" class="ltx_Math" display="inline" id="S6.SS1.5.p1.15.m2.1"><semantics id="S6.SS1.5.p1.15.m2.1a"><mrow id="S6.SS1.5.p1.15.m2.1.1" xref="S6.SS1.5.p1.15.m2.1.1.cmml"><msub id="S6.SS1.5.p1.15.m2.1.1.2" xref="S6.SS1.5.p1.15.m2.1.1.2.cmml"><mi id="S6.SS1.5.p1.15.m2.1.1.2.2" xref="S6.SS1.5.p1.15.m2.1.1.2.2.cmml">k</mi><mn id="S6.SS1.5.p1.15.m2.1.1.2.3" xref="S6.SS1.5.p1.15.m2.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.5.p1.15.m2.1.1.1" xref="S6.SS1.5.p1.15.m2.1.1.1.cmml">=</mo><mn id="S6.SS1.5.p1.15.m2.1.1.3" xref="S6.SS1.5.p1.15.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.5.p1.15.m2.1b"><apply id="S6.SS1.5.p1.15.m2.1.1.cmml" xref="S6.SS1.5.p1.15.m2.1.1"><eq id="S6.SS1.5.p1.15.m2.1.1.1.cmml" xref="S6.SS1.5.p1.15.m2.1.1.1"></eq><apply id="S6.SS1.5.p1.15.m2.1.1.2.cmml" xref="S6.SS1.5.p1.15.m2.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.5.p1.15.m2.1.1.2.1.cmml" xref="S6.SS1.5.p1.15.m2.1.1.2">subscript</csymbol><ci id="S6.SS1.5.p1.15.m2.1.1.2.2.cmml" xref="S6.SS1.5.p1.15.m2.1.1.2.2">𝑘</ci><cn id="S6.SS1.5.p1.15.m2.1.1.2.3.cmml" type="integer" xref="S6.SS1.5.p1.15.m2.1.1.2.3">0</cn></apply><cn id="S6.SS1.5.p1.15.m2.1.1.3.cmml" type="integer" xref="S6.SS1.5.p1.15.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.5.p1.15.m2.1c">k_{0}=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.5.p1.15.m2.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = 0</annotation></semantics></math> and reiteration. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS1.p4"> <p class="ltx_p" id="S6.SS1.p4.1">For weighted Sobolev spaces without boundary conditions and with <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S6.SS1.p4.1.m1.4"><semantics id="S6.SS1.p4.1.m1.4a"><mrow id="S6.SS1.p4.1.m1.4.4" xref="S6.SS1.p4.1.m1.4.4.cmml"><mi id="S6.SS1.p4.1.m1.4.4.5" xref="S6.SS1.p4.1.m1.4.4.5.cmml">γ</mi><mo id="S6.SS1.p4.1.m1.4.4.4" xref="S6.SS1.p4.1.m1.4.4.4.cmml">∈</mo><mrow id="S6.SS1.p4.1.m1.4.4.3" xref="S6.SS1.p4.1.m1.4.4.3.cmml"><mrow id="S6.SS1.p4.1.m1.2.2.1.1.1" xref="S6.SS1.p4.1.m1.2.2.1.1.2.cmml"><mo id="S6.SS1.p4.1.m1.2.2.1.1.1.2" stretchy="false" xref="S6.SS1.p4.1.m1.2.2.1.1.2.cmml">(</mo><mrow id="S6.SS1.p4.1.m1.2.2.1.1.1.1" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.cmml"><mo id="S6.SS1.p4.1.m1.2.2.1.1.1.1a" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.cmml">−</mo><mn id="S6.SS1.p4.1.m1.2.2.1.1.1.1.2" xref="S6.SS1.p4.1.m1.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.SS1.p4.1.m1.2.2.1.1.1.3" xref="S6.SS1.p4.1.m1.2.2.1.1.2.cmml">,</mo><mi id="S6.SS1.p4.1.m1.1.1" mathvariant="normal" xref="S6.SS1.p4.1.m1.1.1.cmml">∞</mi><mo id="S6.SS1.p4.1.m1.2.2.1.1.1.4" stretchy="false" xref="S6.SS1.p4.1.m1.2.2.1.1.2.cmml">)</mo></mrow><mo id="S6.SS1.p4.1.m1.4.4.3.4" xref="S6.SS1.p4.1.m1.4.4.3.4.cmml">∖</mo><mrow id="S6.SS1.p4.1.m1.4.4.3.3.2" xref="S6.SS1.p4.1.m1.4.4.3.3.3.cmml"><mo id="S6.SS1.p4.1.m1.4.4.3.3.2.3" stretchy="false" xref="S6.SS1.p4.1.m1.4.4.3.3.3.1.cmml">{</mo><mrow id="S6.SS1.p4.1.m1.3.3.2.2.1.1" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.cmml"><mrow id="S6.SS1.p4.1.m1.3.3.2.2.1.1.2" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.cmml"><mi id="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.2" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.1" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.3" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S6.SS1.p4.1.m1.3.3.2.2.1.1.1" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.1.cmml">−</mo><mn id="S6.SS1.p4.1.m1.3.3.2.2.1.1.3" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.p4.1.m1.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS1.p4.1.m1.4.4.3.3.3.1.cmml">:</mo><mrow id="S6.SS1.p4.1.m1.4.4.3.3.2.2" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.cmml"><mi id="S6.SS1.p4.1.m1.4.4.3.3.2.2.2" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.2.cmml">j</mi><mo id="S6.SS1.p4.1.m1.4.4.3.3.2.2.1" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S6.SS1.p4.1.m1.4.4.3.3.2.2.3" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.cmml"><mi id="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.2" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.3" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS1.p4.1.m1.4.4.3.3.2.5" stretchy="false" xref="S6.SS1.p4.1.m1.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.1.m1.4b"><apply id="S6.SS1.p4.1.m1.4.4.cmml" xref="S6.SS1.p4.1.m1.4.4"><in id="S6.SS1.p4.1.m1.4.4.4.cmml" xref="S6.SS1.p4.1.m1.4.4.4"></in><ci id="S6.SS1.p4.1.m1.4.4.5.cmml" xref="S6.SS1.p4.1.m1.4.4.5">𝛾</ci><apply id="S6.SS1.p4.1.m1.4.4.3.cmml" xref="S6.SS1.p4.1.m1.4.4.3"><setdiff id="S6.SS1.p4.1.m1.4.4.3.4.cmml" xref="S6.SS1.p4.1.m1.4.4.3.4"></setdiff><interval closure="open" id="S6.SS1.p4.1.m1.2.2.1.1.2.cmml" xref="S6.SS1.p4.1.m1.2.2.1.1.1"><apply id="S6.SS1.p4.1.m1.2.2.1.1.1.1.cmml" 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xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S6.SS1.p4.1.m1.3.3.2.2.1.1.3.cmml" type="integer" xref="S6.SS1.p4.1.m1.3.3.2.2.1.1.3">1</cn></apply><apply id="S6.SS1.p4.1.m1.4.4.3.3.2.2.cmml" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2"><in id="S6.SS1.p4.1.m1.4.4.3.3.2.2.1.cmml" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.1"></in><ci id="S6.SS1.p4.1.m1.4.4.3.3.2.2.2.cmml" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.2">𝑗</ci><apply id="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.cmml" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.1.cmml" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.3">subscript</csymbol><ci id="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.2.cmml" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S6.SS1.p4.1.m1.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.1.m1.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.p4.1.m1.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, we can now prove one of the embeddings for the complex interpolation space. The converse embedding will be proved in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem6" title="Proposition 6.6. ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.6</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem3"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem3.1.1.1">Proposition 6.3</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem3.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem3.p1"> <p class="ltx_p" id="S6.Thmtheorem3.p1.7"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem3.p1.7.7">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.1.1.m1.2"><semantics id="S6.Thmtheorem3.p1.1.1.m1.2a"><mrow id="S6.Thmtheorem3.p1.1.1.m1.2.3" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.cmml"><mi id="S6.Thmtheorem3.p1.1.1.m1.2.3.2" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S6.Thmtheorem3.p1.1.1.m1.2.3.1" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S6.Thmtheorem3.p1.1.1.m1.2.3.3.2" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml"><mo id="S6.Thmtheorem3.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S6.Thmtheorem3.p1.1.1.m1.1.1" xref="S6.Thmtheorem3.p1.1.1.m1.1.1.cmml">1</mn><mo id="S6.Thmtheorem3.p1.1.1.m1.2.3.3.2.2" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem3.p1.1.1.m1.2.2" mathvariant="normal" xref="S6.Thmtheorem3.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S6.Thmtheorem3.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.1.1.m1.2b"><apply id="S6.Thmtheorem3.p1.1.1.m1.2.3.cmml" xref="S6.Thmtheorem3.p1.1.1.m1.2.3"><in id="S6.Thmtheorem3.p1.1.1.m1.2.3.1.cmml" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.1"></in><ci id="S6.Thmtheorem3.p1.1.1.m1.2.3.2.cmml" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S6.Thmtheorem3.p1.1.1.m1.2.3.3.1.cmml" xref="S6.Thmtheorem3.p1.1.1.m1.2.3.3.2"><cn id="S6.Thmtheorem3.p1.1.1.m1.1.1.cmml" type="integer" xref="S6.Thmtheorem3.p1.1.1.m1.1.1">1</cn><infinity id="S6.Thmtheorem3.p1.1.1.m1.2.2.cmml" xref="S6.Thmtheorem3.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="k_{0}\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.2.2.m2.1"><semantics id="S6.Thmtheorem3.p1.2.2.m2.1a"><mrow id="S6.Thmtheorem3.p1.2.2.m2.1.1" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.cmml"><msub id="S6.Thmtheorem3.p1.2.2.m2.1.1.2" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.2.cmml"><mi id="S6.Thmtheorem3.p1.2.2.m2.1.1.2.2" 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id="S6.Thmtheorem3.p1.2.2.m2.1.1.2.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem3.p1.2.2.m2.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.2.3">0</cn></apply><apply id="S6.Thmtheorem3.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem3.p1.2.2.m2.1.1.3.1.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem3.p1.2.2.m2.1.1.3.2.cmml" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.3.2">ℕ</ci><cn id="S6.Thmtheorem3.p1.2.2.m2.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem3.p1.2.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.2.2.m2.1c">k_{0}\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.2.2.m2.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math 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xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2"><csymbol cd="latexml" id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.3.1.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.3">conditional-set</csymbol><apply id="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1"><minus id="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.1.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.1"></minus><apply id="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2"><times id="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2.1.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2.1"></times><ci id="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2.2.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2.3.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.3.cmml" type="integer" xref="S6.Thmtheorem3.p1.4.4.m4.3.3.2.2.1.1.3">1</cn></apply><apply id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2"><in id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.1.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.1"></in><ci id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.2.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.2">𝑗</ci><apply id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3.1.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3">subscript</csymbol><ci id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3.2.cmml" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S6.Thmtheorem3.p1.4.4.m4.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.4.4.m4.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.4.4.m4.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, <math alttext="\ell\in\{1,\dots,k_{1}-1\}" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.5.5.m5.3"><semantics id="S6.Thmtheorem3.p1.5.5.m5.3a"><mrow id="S6.Thmtheorem3.p1.5.5.m5.3.3" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.cmml"><mi id="S6.Thmtheorem3.p1.5.5.m5.3.3.3" mathvariant="normal" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.3.cmml">ℓ</mi><mo id="S6.Thmtheorem3.p1.5.5.m5.3.3.2" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.2.cmml">∈</mo><mrow id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.2.cmml"><mo id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.2" stretchy="false" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.2.cmml">{</mo><mn id="S6.Thmtheorem3.p1.5.5.m5.1.1" xref="S6.Thmtheorem3.p1.5.5.m5.1.1.cmml">1</mn><mo id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.3" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.2.cmml">,</mo><mi id="S6.Thmtheorem3.p1.5.5.m5.2.2" mathvariant="normal" xref="S6.Thmtheorem3.p1.5.5.m5.2.2.cmml">…</mi><mo id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.4" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.2.cmml">,</mo><mrow id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.cmml"><msub id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.cmml"><mi id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.2" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.2.cmml">k</mi><mn id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.3" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.1" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.3" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.5" stretchy="false" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.5.5.m5.3b"><apply id="S6.Thmtheorem3.p1.5.5.m5.3.3.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3"><in id="S6.Thmtheorem3.p1.5.5.m5.3.3.2.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.2"></in><ci id="S6.Thmtheorem3.p1.5.5.m5.3.3.3.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.3">ℓ</ci><set id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.2.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1"><cn id="S6.Thmtheorem3.p1.5.5.m5.1.1.cmml" type="integer" xref="S6.Thmtheorem3.p1.5.5.m5.1.1">1</cn><ci id="S6.Thmtheorem3.p1.5.5.m5.2.2.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.2.2">…</ci><apply id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1"><minus id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.1"></minus><apply id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.1.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.2.cmml" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.2.3">1</cn></apply><cn id="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem3.p1.5.5.m5.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.5.5.m5.3c">\ell\in\{1,\dots,k_{1}-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.5.5.m5.3d">roman_ℓ ∈ { 1 , … , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.6.6.m6.1"><semantics id="S6.Thmtheorem3.p1.6.6.m6.1a"><mi id="S6.Thmtheorem3.p1.6.6.m6.1.1" xref="S6.Thmtheorem3.p1.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.6.6.m6.1b"><ci id="S6.Thmtheorem3.p1.6.6.m6.1.1.cmml" xref="S6.Thmtheorem3.p1.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.6.6.m6.1d">italic_X</annotation></semantics></math> be a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S6.Thmtheorem3.p1.7.7.m7.1"><semantics id="S6.Thmtheorem3.p1.7.7.m7.1a"><mi id="S6.Thmtheorem3.p1.7.7.m7.1.1" xref="S6.Thmtheorem3.p1.7.7.m7.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem3.p1.7.7.m7.1b"><ci id="S6.Thmtheorem3.p1.7.7.m7.1.1.cmml" xref="S6.Thmtheorem3.p1.7.7.m7.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem3.p1.7.7.m7.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem3.p1.7.7.m7.1d">roman_UMD</annotation></semantics></math> Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.E1"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W^{k_{0}+k_{1},p}(\mathbb{% R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}\hookrightarrow W^{k_{0}+% \ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)." class="ltx_Math" display="block" id="S6.E1.m1.10"><semantics id="S6.E1.m1.10a"><mrow id="S6.E1.m1.10.10.1" xref="S6.E1.m1.10.10.1.1.cmml"><mrow id="S6.E1.m1.10.10.1.1" xref="S6.E1.m1.10.10.1.1.cmml"><msub id="S6.E1.m1.10.10.1.1.2" xref="S6.E1.m1.10.10.1.1.2.cmml"><mrow id="S6.E1.m1.10.10.1.1.2.2.2" xref="S6.E1.m1.10.10.1.1.2.2.3.cmml"><mo id="S6.E1.m1.10.10.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.E1.m1.10.10.1.1.2.2.3.cmml">[</mo><mrow id="S6.E1.m1.10.10.1.1.1.1.1.1" xref="S6.E1.m1.10.10.1.1.1.1.1.1.cmml"><msup 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\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X).</annotation><annotation encoding="application/x-llamapun" id="S6.E1.m1.10d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.1)</span></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S6.SS1.10"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS1.6.p1"> <p class="ltx_p" id="S6.SS1.6.p1.6">For notational convenience we write <math alttext="W^{k,p}(w_{\gamma}):=W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS1.6.p1.1.m1.8"><semantics id="S6.SS1.6.p1.1.m1.8a"><mrow id="S6.SS1.6.p1.1.m1.8.8" xref="S6.SS1.6.p1.1.m1.8.8.cmml"><mrow id="S6.SS1.6.p1.1.m1.6.6.1" 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id="S6.SS1.6.p1.1.m1.8.8.3.2.2.6" stretchy="false" xref="S6.SS1.6.p1.1.m1.8.8.3.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.6.p1.1.m1.8b"><apply id="S6.SS1.6.p1.1.m1.8.8.cmml" xref="S6.SS1.6.p1.1.m1.8.8"><csymbol cd="latexml" id="S6.SS1.6.p1.1.m1.8.8.4.cmml" xref="S6.SS1.6.p1.1.m1.8.8.4">assign</csymbol><apply id="S6.SS1.6.p1.1.m1.6.6.1.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1"><times id="S6.SS1.6.p1.1.m1.6.6.1.2.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.2"></times><apply id="S6.SS1.6.p1.1.m1.6.6.1.3.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.3"><csymbol cd="ambiguous" id="S6.SS1.6.p1.1.m1.6.6.1.3.1.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.3">superscript</csymbol><ci id="S6.SS1.6.p1.1.m1.6.6.1.3.2.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.3.2">𝑊</ci><list id="S6.SS1.6.p1.1.m1.2.2.2.3.cmml" xref="S6.SS1.6.p1.1.m1.2.2.2.4"><ci id="S6.SS1.6.p1.1.m1.1.1.1.1.cmml" xref="S6.SS1.6.p1.1.m1.1.1.1.1">𝑘</ci><ci id="S6.SS1.6.p1.1.m1.2.2.2.2.cmml" xref="S6.SS1.6.p1.1.m1.2.2.2.2">𝑝</ci></list></apply><apply id="S6.SS1.6.p1.1.m1.6.6.1.1.1.1.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.6.p1.1.m1.6.6.1.1.1.1.1.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.1.1">subscript</csymbol><ci id="S6.SS1.6.p1.1.m1.6.6.1.1.1.1.2.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.1.1.1.2">𝑤</ci><ci id="S6.SS1.6.p1.1.m1.6.6.1.1.1.1.3.cmml" xref="S6.SS1.6.p1.1.m1.6.6.1.1.1.1.3">𝛾</ci></apply></apply><apply id="S6.SS1.6.p1.1.m1.8.8.3.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3"><times id="S6.SS1.6.p1.1.m1.8.8.3.3.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.3"></times><apply id="S6.SS1.6.p1.1.m1.8.8.3.4.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.4"><csymbol cd="ambiguous" id="S6.SS1.6.p1.1.m1.8.8.3.4.1.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.4">superscript</csymbol><ci id="S6.SS1.6.p1.1.m1.8.8.3.4.2.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.4.2">𝑊</ci><list id="S6.SS1.6.p1.1.m1.4.4.2.3.cmml" xref="S6.SS1.6.p1.1.m1.4.4.2.4"><ci id="S6.SS1.6.p1.1.m1.3.3.1.1.cmml" xref="S6.SS1.6.p1.1.m1.3.3.1.1">𝑘</ci><ci id="S6.SS1.6.p1.1.m1.4.4.2.2.cmml" xref="S6.SS1.6.p1.1.m1.4.4.2.2">𝑝</ci></list></apply><vector id="S6.SS1.6.p1.1.m1.8.8.3.2.3.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.2.2"><apply id="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.cmml" xref="S6.SS1.6.p1.1.m1.7.7.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.1.cmml" xref="S6.SS1.6.p1.1.m1.7.7.2.1.1.1">subscript</csymbol><apply id="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.2.cmml" xref="S6.SS1.6.p1.1.m1.7.7.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.2.1.cmml" xref="S6.SS1.6.p1.1.m1.7.7.2.1.1.1">superscript</csymbol><ci id="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.2.2.cmml" xref="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.2.2">ℝ</ci><ci id="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.2.3.cmml" xref="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.2.3">𝑑</ci></apply><plus id="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.3.cmml" xref="S6.SS1.6.p1.1.m1.7.7.2.1.1.1.3"></plus></apply><apply id="S6.SS1.6.p1.1.m1.8.8.3.2.2.2.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.6.p1.1.m1.8.8.3.2.2.2.1.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.2.2.2">subscript</csymbol><ci id="S6.SS1.6.p1.1.m1.8.8.3.2.2.2.2.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.2.2.2.2">𝑤</ci><ci id="S6.SS1.6.p1.1.m1.8.8.3.2.2.2.3.cmml" xref="S6.SS1.6.p1.1.m1.8.8.3.2.2.2.3">𝛾</ci></apply><ci id="S6.SS1.6.p1.1.m1.5.5.cmml" xref="S6.SS1.6.p1.1.m1.5.5">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.6.p1.1.m1.8c">W^{k,p}(w_{\gamma}):=W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.6.p1.1.m1.8d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) := italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS1.6.p1.2.m2.1"><semantics id="S6.SS1.6.p1.2.m2.1a"><mrow id="S6.SS1.6.p1.2.m2.1.1" xref="S6.SS1.6.p1.2.m2.1.1.cmml"><mi id="S6.SS1.6.p1.2.m2.1.1.2" xref="S6.SS1.6.p1.2.m2.1.1.2.cmml">k</mi><mo id="S6.SS1.6.p1.2.m2.1.1.1" xref="S6.SS1.6.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S6.SS1.6.p1.2.m2.1.1.3" xref="S6.SS1.6.p1.2.m2.1.1.3.cmml"><mi id="S6.SS1.6.p1.2.m2.1.1.3.2" xref="S6.SS1.6.p1.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.6.p1.2.m2.1.1.3.3" xref="S6.SS1.6.p1.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.6.p1.2.m2.1b"><apply id="S6.SS1.6.p1.2.m2.1.1.cmml" xref="S6.SS1.6.p1.2.m2.1.1"><in id="S6.SS1.6.p1.2.m2.1.1.1.cmml" xref="S6.SS1.6.p1.2.m2.1.1.1"></in><ci id="S6.SS1.6.p1.2.m2.1.1.2.cmml" xref="S6.SS1.6.p1.2.m2.1.1.2">𝑘</ci><apply id="S6.SS1.6.p1.2.m2.1.1.3.cmml" xref="S6.SS1.6.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.6.p1.2.m2.1.1.3.1.cmml" xref="S6.SS1.6.p1.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS1.6.p1.2.m2.1.1.3.2.cmml" xref="S6.SS1.6.p1.2.m2.1.1.3.2">ℕ</ci><cn id="S6.SS1.6.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS1.6.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.6.p1.2.m2.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.6.p1.2.m2.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>. Let <math alttext="j\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS1.6.p1.3.m3.1"><semantics id="S6.SS1.6.p1.3.m3.1a"><mrow id="S6.SS1.6.p1.3.m3.1.1" xref="S6.SS1.6.p1.3.m3.1.1.cmml"><mi id="S6.SS1.6.p1.3.m3.1.1.2" xref="S6.SS1.6.p1.3.m3.1.1.2.cmml">j</mi><mo id="S6.SS1.6.p1.3.m3.1.1.1" xref="S6.SS1.6.p1.3.m3.1.1.1.cmml">∈</mo><msub id="S6.SS1.6.p1.3.m3.1.1.3" xref="S6.SS1.6.p1.3.m3.1.1.3.cmml"><mi id="S6.SS1.6.p1.3.m3.1.1.3.2" xref="S6.SS1.6.p1.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.6.p1.3.m3.1.1.3.3" xref="S6.SS1.6.p1.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.6.p1.3.m3.1b"><apply id="S6.SS1.6.p1.3.m3.1.1.cmml" xref="S6.SS1.6.p1.3.m3.1.1"><in id="S6.SS1.6.p1.3.m3.1.1.1.cmml" xref="S6.SS1.6.p1.3.m3.1.1.1"></in><ci id="S6.SS1.6.p1.3.m3.1.1.2.cmml" xref="S6.SS1.6.p1.3.m3.1.1.2">𝑗</ci><apply id="S6.SS1.6.p1.3.m3.1.1.3.cmml" xref="S6.SS1.6.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.6.p1.3.m3.1.1.3.1.cmml" xref="S6.SS1.6.p1.3.m3.1.1.3">subscript</csymbol><ci id="S6.SS1.6.p1.3.m3.1.1.3.2.cmml" xref="S6.SS1.6.p1.3.m3.1.1.3.2">ℕ</ci><cn id="S6.SS1.6.p1.3.m3.1.1.3.3.cmml" type="integer" xref="S6.SS1.6.p1.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.6.p1.3.m3.1c">j\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.6.p1.3.m3.1d">italic_j ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be such that <math alttext="\gamma\in(jp-1,(j+1)p-1)" class="ltx_Math" display="inline" id="S6.SS1.6.p1.4.m4.2"><semantics id="S6.SS1.6.p1.4.m4.2a"><mrow id="S6.SS1.6.p1.4.m4.2.2" xref="S6.SS1.6.p1.4.m4.2.2.cmml"><mi id="S6.SS1.6.p1.4.m4.2.2.4" xref="S6.SS1.6.p1.4.m4.2.2.4.cmml">γ</mi><mo id="S6.SS1.6.p1.4.m4.2.2.3" xref="S6.SS1.6.p1.4.m4.2.2.3.cmml">∈</mo><mrow id="S6.SS1.6.p1.4.m4.2.2.2.2" xref="S6.SS1.6.p1.4.m4.2.2.2.3.cmml"><mo id="S6.SS1.6.p1.4.m4.2.2.2.2.3" stretchy="false" xref="S6.SS1.6.p1.4.m4.2.2.2.3.cmml">(</mo><mrow id="S6.SS1.6.p1.4.m4.1.1.1.1.1" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.cmml"><mrow id="S6.SS1.6.p1.4.m4.1.1.1.1.1.2" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.cmml"><mi id="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.2" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.2.cmml">j</mi><mo id="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.1" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.3" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.3.cmml">p</mi></mrow><mo id="S6.SS1.6.p1.4.m4.1.1.1.1.1.1" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.1.cmml">−</mo><mn id="S6.SS1.6.p1.4.m4.1.1.1.1.1.3" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.6.p1.4.m4.2.2.2.2.4" xref="S6.SS1.6.p1.4.m4.2.2.2.3.cmml">,</mo><mrow id="S6.SS1.6.p1.4.m4.2.2.2.2.2" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.cmml"><mrow id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.cmml"><mrow id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.cmml"><mo id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.2" stretchy="false" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.cmml"><mi id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.2" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.2.cmml">j</mi><mo id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.1" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.1.cmml">+</mo><mn id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.3" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.3" stretchy="false" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.2" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.2.cmml">⁢</mo><mi id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.3" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.3.cmml">p</mi></mrow><mo id="S6.SS1.6.p1.4.m4.2.2.2.2.2.2" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.2.cmml">−</mo><mn id="S6.SS1.6.p1.4.m4.2.2.2.2.2.3" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.3.cmml">1</mn></mrow><mo 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xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.2.3">𝑝</ci></apply><cn id="S6.SS1.6.p1.4.m4.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.6.p1.4.m4.1.1.1.1.1.3">1</cn></apply><apply id="S6.SS1.6.p1.4.m4.2.2.2.2.2.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2"><minus id="S6.SS1.6.p1.4.m4.2.2.2.2.2.2.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.2"></minus><apply id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1"><times id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.2.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.2"></times><apply id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1"><plus id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.1.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.1"></plus><ci id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.2.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.2">𝑗</ci><cn id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.1.1.1.3">1</cn></apply><ci id="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.3.cmml" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.1.3">𝑝</ci></apply><cn id="S6.SS1.6.p1.4.m4.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.6.p1.4.m4.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.6.p1.4.m4.2c">\gamma\in(jp-1,(j+1)p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.6.p1.4.m4.2d">italic_γ ∈ ( italic_j italic_p - 1 , ( italic_j + 1 ) italic_p - 1 )</annotation></semantics></math>. Note that for <math alttext="j=0" class="ltx_Math" display="inline" id="S6.SS1.6.p1.5.m5.1"><semantics id="S6.SS1.6.p1.5.m5.1a"><mrow id="S6.SS1.6.p1.5.m5.1.1" xref="S6.SS1.6.p1.5.m5.1.1.cmml"><mi id="S6.SS1.6.p1.5.m5.1.1.2" xref="S6.SS1.6.p1.5.m5.1.1.2.cmml">j</mi><mo id="S6.SS1.6.p1.5.m5.1.1.1" xref="S6.SS1.6.p1.5.m5.1.1.1.cmml">=</mo><mn id="S6.SS1.6.p1.5.m5.1.1.3" xref="S6.SS1.6.p1.5.m5.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.6.p1.5.m5.1b"><apply id="S6.SS1.6.p1.5.m5.1.1.cmml" xref="S6.SS1.6.p1.5.m5.1.1"><eq id="S6.SS1.6.p1.5.m5.1.1.1.cmml" xref="S6.SS1.6.p1.5.m5.1.1.1"></eq><ci id="S6.SS1.6.p1.5.m5.1.1.2.cmml" xref="S6.SS1.6.p1.5.m5.1.1.2">𝑗</ci><cn id="S6.SS1.6.p1.5.m5.1.1.3.cmml" type="integer" xref="S6.SS1.6.p1.5.m5.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.6.p1.5.m5.1c">j=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.6.p1.5.m5.1d">italic_j = 0</annotation></semantics></math> the result follows from <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.5 & 5.6]</cite>, so we may assume that <math alttext="j\geq 1" class="ltx_Math" display="inline" id="S6.SS1.6.p1.6.m6.1"><semantics id="S6.SS1.6.p1.6.m6.1a"><mrow id="S6.SS1.6.p1.6.m6.1.1" xref="S6.SS1.6.p1.6.m6.1.1.cmml"><mi id="S6.SS1.6.p1.6.m6.1.1.2" xref="S6.SS1.6.p1.6.m6.1.1.2.cmml">j</mi><mo id="S6.SS1.6.p1.6.m6.1.1.1" xref="S6.SS1.6.p1.6.m6.1.1.1.cmml">≥</mo><mn id="S6.SS1.6.p1.6.m6.1.1.3" xref="S6.SS1.6.p1.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.6.p1.6.m6.1b"><apply id="S6.SS1.6.p1.6.m6.1.1.cmml" xref="S6.SS1.6.p1.6.m6.1.1"><geq id="S6.SS1.6.p1.6.m6.1.1.1.cmml" xref="S6.SS1.6.p1.6.m6.1.1.1"></geq><ci id="S6.SS1.6.p1.6.m6.1.1.2.cmml" xref="S6.SS1.6.p1.6.m6.1.1.2">𝑗</ci><cn id="S6.SS1.6.p1.6.m6.1.1.3.cmml" type="integer" xref="S6.SS1.6.p1.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.6.p1.6.m6.1c">j\geq 1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.6.p1.6.m6.1d">italic_j ≥ 1</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS1.7.p2"> <p class="ltx_p" id="S6.SS1.7.p2.4"><span class="ltx_text ltx_font_italic" id="S6.SS1.7.p2.1.1">Step 1: the case <math alttext="j\in\{1,\dots,k_{0}\}" class="ltx_Math" display="inline" id="S6.SS1.7.p2.1.1.m1.3"><semantics id="S6.SS1.7.p2.1.1.m1.3a"><mrow id="S6.SS1.7.p2.1.1.m1.3.3" xref="S6.SS1.7.p2.1.1.m1.3.3.cmml"><mi id="S6.SS1.7.p2.1.1.m1.3.3.3" xref="S6.SS1.7.p2.1.1.m1.3.3.3.cmml">j</mi><mo id="S6.SS1.7.p2.1.1.m1.3.3.2" xref="S6.SS1.7.p2.1.1.m1.3.3.2.cmml">∈</mo><mrow id="S6.SS1.7.p2.1.1.m1.3.3.1.1" xref="S6.SS1.7.p2.1.1.m1.3.3.1.2.cmml"><mo id="S6.SS1.7.p2.1.1.m1.3.3.1.1.2" stretchy="false" xref="S6.SS1.7.p2.1.1.m1.3.3.1.2.cmml">{</mo><mn id="S6.SS1.7.p2.1.1.m1.1.1" xref="S6.SS1.7.p2.1.1.m1.1.1.cmml">1</mn><mo id="S6.SS1.7.p2.1.1.m1.3.3.1.1.3" xref="S6.SS1.7.p2.1.1.m1.3.3.1.2.cmml">,</mo><mi id="S6.SS1.7.p2.1.1.m1.2.2" mathvariant="normal" xref="S6.SS1.7.p2.1.1.m1.2.2.cmml">…</mi><mo id="S6.SS1.7.p2.1.1.m1.3.3.1.1.4" xref="S6.SS1.7.p2.1.1.m1.3.3.1.2.cmml">,</mo><msub id="S6.SS1.7.p2.1.1.m1.3.3.1.1.1" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.cmml"><mi id="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.2" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.2.cmml">k</mi><mn id="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.3" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.3.cmml">0</mn></msub><mo id="S6.SS1.7.p2.1.1.m1.3.3.1.1.5" stretchy="false" xref="S6.SS1.7.p2.1.1.m1.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.1.1.m1.3b"><apply id="S6.SS1.7.p2.1.1.m1.3.3.cmml" xref="S6.SS1.7.p2.1.1.m1.3.3"><in id="S6.SS1.7.p2.1.1.m1.3.3.2.cmml" xref="S6.SS1.7.p2.1.1.m1.3.3.2"></in><ci id="S6.SS1.7.p2.1.1.m1.3.3.3.cmml" xref="S6.SS1.7.p2.1.1.m1.3.3.3">𝑗</ci><set id="S6.SS1.7.p2.1.1.m1.3.3.1.2.cmml" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1"><cn id="S6.SS1.7.p2.1.1.m1.1.1.cmml" type="integer" xref="S6.SS1.7.p2.1.1.m1.1.1">1</cn><ci id="S6.SS1.7.p2.1.1.m1.2.2.cmml" xref="S6.SS1.7.p2.1.1.m1.2.2">…</ci><apply id="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.cmml" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.1.cmml" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1.1">subscript</csymbol><ci id="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.2.cmml" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.2">𝑘</ci><cn id="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.3.cmml" type="integer" xref="S6.SS1.7.p2.1.1.m1.3.3.1.1.1.3">0</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.1.1.m1.3c">j\in\{1,\dots,k_{0}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.1.1.m1.3d">italic_j ∈ { 1 , … , italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT }</annotation></semantics></math>.</span> We prove the embedding (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E1" title="In Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.1</span></a>) for <math alttext="j\in\{1,\dots,k_{0}\}" class="ltx_Math" display="inline" id="S6.SS1.7.p2.2.m1.3"><semantics id="S6.SS1.7.p2.2.m1.3a"><mrow id="S6.SS1.7.p2.2.m1.3.3" xref="S6.SS1.7.p2.2.m1.3.3.cmml"><mi id="S6.SS1.7.p2.2.m1.3.3.3" xref="S6.SS1.7.p2.2.m1.3.3.3.cmml">j</mi><mo id="S6.SS1.7.p2.2.m1.3.3.2" xref="S6.SS1.7.p2.2.m1.3.3.2.cmml">∈</mo><mrow id="S6.SS1.7.p2.2.m1.3.3.1.1" xref="S6.SS1.7.p2.2.m1.3.3.1.2.cmml"><mo id="S6.SS1.7.p2.2.m1.3.3.1.1.2" stretchy="false" xref="S6.SS1.7.p2.2.m1.3.3.1.2.cmml">{</mo><mn id="S6.SS1.7.p2.2.m1.1.1" xref="S6.SS1.7.p2.2.m1.1.1.cmml">1</mn><mo id="S6.SS1.7.p2.2.m1.3.3.1.1.3" xref="S6.SS1.7.p2.2.m1.3.3.1.2.cmml">,</mo><mi id="S6.SS1.7.p2.2.m1.2.2" mathvariant="normal" xref="S6.SS1.7.p2.2.m1.2.2.cmml">…</mi><mo id="S6.SS1.7.p2.2.m1.3.3.1.1.4" xref="S6.SS1.7.p2.2.m1.3.3.1.2.cmml">,</mo><msub id="S6.SS1.7.p2.2.m1.3.3.1.1.1" xref="S6.SS1.7.p2.2.m1.3.3.1.1.1.cmml"><mi id="S6.SS1.7.p2.2.m1.3.3.1.1.1.2" xref="S6.SS1.7.p2.2.m1.3.3.1.1.1.2.cmml">k</mi><mn id="S6.SS1.7.p2.2.m1.3.3.1.1.1.3" xref="S6.SS1.7.p2.2.m1.3.3.1.1.1.3.cmml">0</mn></msub><mo id="S6.SS1.7.p2.2.m1.3.3.1.1.5" stretchy="false" xref="S6.SS1.7.p2.2.m1.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.2.m1.3b"><apply id="S6.SS1.7.p2.2.m1.3.3.cmml" xref="S6.SS1.7.p2.2.m1.3.3"><in id="S6.SS1.7.p2.2.m1.3.3.2.cmml" xref="S6.SS1.7.p2.2.m1.3.3.2"></in><ci id="S6.SS1.7.p2.2.m1.3.3.3.cmml" xref="S6.SS1.7.p2.2.m1.3.3.3">𝑗</ci><set id="S6.SS1.7.p2.2.m1.3.3.1.2.cmml" xref="S6.SS1.7.p2.2.m1.3.3.1.1"><cn id="S6.SS1.7.p2.2.m1.1.1.cmml" type="integer" xref="S6.SS1.7.p2.2.m1.1.1">1</cn><ci id="S6.SS1.7.p2.2.m1.2.2.cmml" xref="S6.SS1.7.p2.2.m1.2.2">…</ci><apply id="S6.SS1.7.p2.2.m1.3.3.1.1.1.cmml" xref="S6.SS1.7.p2.2.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.7.p2.2.m1.3.3.1.1.1.1.cmml" xref="S6.SS1.7.p2.2.m1.3.3.1.1.1">subscript</csymbol><ci id="S6.SS1.7.p2.2.m1.3.3.1.1.1.2.cmml" xref="S6.SS1.7.p2.2.m1.3.3.1.1.1.2">𝑘</ci><cn id="S6.SS1.7.p2.2.m1.3.3.1.1.1.3.cmml" type="integer" xref="S6.SS1.7.p2.2.m1.3.3.1.1.1.3">0</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.2.m1.3c">j\in\{1,\dots,k_{0}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.2.m1.3d">italic_j ∈ { 1 , … , italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT }</annotation></semantics></math>. Note that <math alttext="k_{0}-j\geq 0" class="ltx_Math" display="inline" id="S6.SS1.7.p2.3.m2.1"><semantics id="S6.SS1.7.p2.3.m2.1a"><mrow id="S6.SS1.7.p2.3.m2.1.1" xref="S6.SS1.7.p2.3.m2.1.1.cmml"><mrow id="S6.SS1.7.p2.3.m2.1.1.2" xref="S6.SS1.7.p2.3.m2.1.1.2.cmml"><msub id="S6.SS1.7.p2.3.m2.1.1.2.2" xref="S6.SS1.7.p2.3.m2.1.1.2.2.cmml"><mi id="S6.SS1.7.p2.3.m2.1.1.2.2.2" xref="S6.SS1.7.p2.3.m2.1.1.2.2.2.cmml">k</mi><mn id="S6.SS1.7.p2.3.m2.1.1.2.2.3" xref="S6.SS1.7.p2.3.m2.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.SS1.7.p2.3.m2.1.1.2.1" xref="S6.SS1.7.p2.3.m2.1.1.2.1.cmml">−</mo><mi id="S6.SS1.7.p2.3.m2.1.1.2.3" xref="S6.SS1.7.p2.3.m2.1.1.2.3.cmml">j</mi></mrow><mo id="S6.SS1.7.p2.3.m2.1.1.1" xref="S6.SS1.7.p2.3.m2.1.1.1.cmml">≥</mo><mn id="S6.SS1.7.p2.3.m2.1.1.3" xref="S6.SS1.7.p2.3.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.3.m2.1b"><apply id="S6.SS1.7.p2.3.m2.1.1.cmml" xref="S6.SS1.7.p2.3.m2.1.1"><geq id="S6.SS1.7.p2.3.m2.1.1.1.cmml" xref="S6.SS1.7.p2.3.m2.1.1.1"></geq><apply id="S6.SS1.7.p2.3.m2.1.1.2.cmml" xref="S6.SS1.7.p2.3.m2.1.1.2"><minus id="S6.SS1.7.p2.3.m2.1.1.2.1.cmml" xref="S6.SS1.7.p2.3.m2.1.1.2.1"></minus><apply id="S6.SS1.7.p2.3.m2.1.1.2.2.cmml" xref="S6.SS1.7.p2.3.m2.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS1.7.p2.3.m2.1.1.2.2.1.cmml" xref="S6.SS1.7.p2.3.m2.1.1.2.2">subscript</csymbol><ci id="S6.SS1.7.p2.3.m2.1.1.2.2.2.cmml" xref="S6.SS1.7.p2.3.m2.1.1.2.2.2">𝑘</ci><cn id="S6.SS1.7.p2.3.m2.1.1.2.2.3.cmml" type="integer" xref="S6.SS1.7.p2.3.m2.1.1.2.2.3">0</cn></apply><ci id="S6.SS1.7.p2.3.m2.1.1.2.3.cmml" xref="S6.SS1.7.p2.3.m2.1.1.2.3">𝑗</ci></apply><cn id="S6.SS1.7.p2.3.m2.1.1.3.cmml" type="integer" xref="S6.SS1.7.p2.3.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.3.m2.1c">k_{0}-j\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.3.m2.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_j ≥ 0</annotation></semantics></math> and <math alttext="\gamma-jp\in(-1,p-1)" class="ltx_Math" display="inline" id="S6.SS1.7.p2.4.m3.2"><semantics id="S6.SS1.7.p2.4.m3.2a"><mrow id="S6.SS1.7.p2.4.m3.2.2" xref="S6.SS1.7.p2.4.m3.2.2.cmml"><mrow id="S6.SS1.7.p2.4.m3.2.2.4" xref="S6.SS1.7.p2.4.m3.2.2.4.cmml"><mi id="S6.SS1.7.p2.4.m3.2.2.4.2" xref="S6.SS1.7.p2.4.m3.2.2.4.2.cmml">γ</mi><mo id="S6.SS1.7.p2.4.m3.2.2.4.1" xref="S6.SS1.7.p2.4.m3.2.2.4.1.cmml">−</mo><mrow id="S6.SS1.7.p2.4.m3.2.2.4.3" xref="S6.SS1.7.p2.4.m3.2.2.4.3.cmml"><mi id="S6.SS1.7.p2.4.m3.2.2.4.3.2" xref="S6.SS1.7.p2.4.m3.2.2.4.3.2.cmml">j</mi><mo id="S6.SS1.7.p2.4.m3.2.2.4.3.1" xref="S6.SS1.7.p2.4.m3.2.2.4.3.1.cmml">⁢</mo><mi id="S6.SS1.7.p2.4.m3.2.2.4.3.3" xref="S6.SS1.7.p2.4.m3.2.2.4.3.3.cmml">p</mi></mrow></mrow><mo id="S6.SS1.7.p2.4.m3.2.2.3" xref="S6.SS1.7.p2.4.m3.2.2.3.cmml">∈</mo><mrow id="S6.SS1.7.p2.4.m3.2.2.2.2" xref="S6.SS1.7.p2.4.m3.2.2.2.3.cmml"><mo id="S6.SS1.7.p2.4.m3.2.2.2.2.3" stretchy="false" xref="S6.SS1.7.p2.4.m3.2.2.2.3.cmml">(</mo><mrow id="S6.SS1.7.p2.4.m3.1.1.1.1.1" xref="S6.SS1.7.p2.4.m3.1.1.1.1.1.cmml"><mo id="S6.SS1.7.p2.4.m3.1.1.1.1.1a" xref="S6.SS1.7.p2.4.m3.1.1.1.1.1.cmml">−</mo><mn id="S6.SS1.7.p2.4.m3.1.1.1.1.1.2" xref="S6.SS1.7.p2.4.m3.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.SS1.7.p2.4.m3.2.2.2.2.4" xref="S6.SS1.7.p2.4.m3.2.2.2.3.cmml">,</mo><mrow id="S6.SS1.7.p2.4.m3.2.2.2.2.2" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2.cmml"><mi id="S6.SS1.7.p2.4.m3.2.2.2.2.2.2" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2.2.cmml">p</mi><mo id="S6.SS1.7.p2.4.m3.2.2.2.2.2.1" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2.1.cmml">−</mo><mn id="S6.SS1.7.p2.4.m3.2.2.2.2.2.3" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS1.7.p2.4.m3.2.2.2.2.5" stretchy="false" xref="S6.SS1.7.p2.4.m3.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.4.m3.2b"><apply id="S6.SS1.7.p2.4.m3.2.2.cmml" xref="S6.SS1.7.p2.4.m3.2.2"><in id="S6.SS1.7.p2.4.m3.2.2.3.cmml" xref="S6.SS1.7.p2.4.m3.2.2.3"></in><apply id="S6.SS1.7.p2.4.m3.2.2.4.cmml" xref="S6.SS1.7.p2.4.m3.2.2.4"><minus id="S6.SS1.7.p2.4.m3.2.2.4.1.cmml" xref="S6.SS1.7.p2.4.m3.2.2.4.1"></minus><ci id="S6.SS1.7.p2.4.m3.2.2.4.2.cmml" xref="S6.SS1.7.p2.4.m3.2.2.4.2">𝛾</ci><apply id="S6.SS1.7.p2.4.m3.2.2.4.3.cmml" xref="S6.SS1.7.p2.4.m3.2.2.4.3"><times id="S6.SS1.7.p2.4.m3.2.2.4.3.1.cmml" xref="S6.SS1.7.p2.4.m3.2.2.4.3.1"></times><ci id="S6.SS1.7.p2.4.m3.2.2.4.3.2.cmml" xref="S6.SS1.7.p2.4.m3.2.2.4.3.2">𝑗</ci><ci id="S6.SS1.7.p2.4.m3.2.2.4.3.3.cmml" xref="S6.SS1.7.p2.4.m3.2.2.4.3.3">𝑝</ci></apply></apply><interval closure="open" id="S6.SS1.7.p2.4.m3.2.2.2.3.cmml" xref="S6.SS1.7.p2.4.m3.2.2.2.2"><apply id="S6.SS1.7.p2.4.m3.1.1.1.1.1.cmml" xref="S6.SS1.7.p2.4.m3.1.1.1.1.1"><minus id="S6.SS1.7.p2.4.m3.1.1.1.1.1.1.cmml" xref="S6.SS1.7.p2.4.m3.1.1.1.1.1"></minus><cn id="S6.SS1.7.p2.4.m3.1.1.1.1.1.2.cmml" type="integer" xref="S6.SS1.7.p2.4.m3.1.1.1.1.1.2">1</cn></apply><apply id="S6.SS1.7.p2.4.m3.2.2.2.2.2.cmml" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2"><minus id="S6.SS1.7.p2.4.m3.2.2.2.2.2.1.cmml" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2.1"></minus><ci id="S6.SS1.7.p2.4.m3.2.2.2.2.2.2.cmml" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2.2">𝑝</ci><cn id="S6.SS1.7.p2.4.m3.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.7.p2.4.m3.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.4.m3.2c">\gamma-jp\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.4.m3.2d">italic_γ - italic_j italic_p ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math>, so <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib50" title="">50</a>, Proposition 5.5 & 5.6]</cite> imply that</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex23"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0}-j,p}(w_{\gamma-jp}),W^{k_{0}+k_{1}-j,p}(w_{\gamma-jp})\big{]}_% {\frac{\ell}{k_{1}}}\hookrightarrow W^{k_{0}+\ell-j}(w_{\gamma-jp})." class="ltx_Math" display="block" id="S6.Ex23.m1.5"><semantics id="S6.Ex23.m1.5a"><mrow id="S6.Ex23.m1.5.5.1" xref="S6.Ex23.m1.5.5.1.1.cmml"><mrow id="S6.Ex23.m1.5.5.1.1" xref="S6.Ex23.m1.5.5.1.1.cmml"><msub id="S6.Ex23.m1.5.5.1.1.2" xref="S6.Ex23.m1.5.5.1.1.2.cmml"><mrow id="S6.Ex23.m1.5.5.1.1.2.2.2" xref="S6.Ex23.m1.5.5.1.1.2.2.3.cmml"><mo id="S6.Ex23.m1.5.5.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex23.m1.5.5.1.1.2.2.3.cmml">[</mo><mrow id="S6.Ex23.m1.5.5.1.1.1.1.1.1" xref="S6.Ex23.m1.5.5.1.1.1.1.1.1.cmml"><msup id="S6.Ex23.m1.5.5.1.1.1.1.1.1.3" xref="S6.Ex23.m1.5.5.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex23.m1.5.5.1.1.1.1.1.1.3.2" xref="S6.Ex23.m1.5.5.1.1.1.1.1.1.3.2.cmml">W</mi><mrow id="S6.Ex23.m1.2.2.2.2" xref="S6.Ex23.m1.2.2.2.3.cmml"><mrow id="S6.Ex23.m1.2.2.2.2.1" 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id="S6.Ex23.m1.5.5.1.1.3.1.1.1.3.3.2.cmml" xref="S6.Ex23.m1.5.5.1.1.3.1.1.1.3.3.2">𝑗</ci><ci id="S6.Ex23.m1.5.5.1.1.3.1.1.1.3.3.3.cmml" xref="S6.Ex23.m1.5.5.1.1.3.1.1.1.3.3.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex23.m1.5c">\big{[}W^{k_{0}-j,p}(w_{\gamma-jp}),W^{k_{0}+k_{1}-j,p}(w_{\gamma-jp})\big{]}_% {\frac{\ell}{k_{1}}}\hookrightarrow W^{k_{0}+\ell-j}(w_{\gamma-jp}).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex23.m1.5d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_j , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ - italic_j end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ - italic_j italic_p end_POSTSUBSCRIPT ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.7.p2.5">We prove by induction on <math alttext="j_{1}\geq 0" class="ltx_Math" display="inline" id="S6.SS1.7.p2.5.m1.1"><semantics id="S6.SS1.7.p2.5.m1.1a"><mrow id="S6.SS1.7.p2.5.m1.1.1" xref="S6.SS1.7.p2.5.m1.1.1.cmml"><msub id="S6.SS1.7.p2.5.m1.1.1.2" xref="S6.SS1.7.p2.5.m1.1.1.2.cmml"><mi id="S6.SS1.7.p2.5.m1.1.1.2.2" xref="S6.SS1.7.p2.5.m1.1.1.2.2.cmml">j</mi><mn id="S6.SS1.7.p2.5.m1.1.1.2.3" xref="S6.SS1.7.p2.5.m1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS1.7.p2.5.m1.1.1.1" xref="S6.SS1.7.p2.5.m1.1.1.1.cmml">≥</mo><mn id="S6.SS1.7.p2.5.m1.1.1.3" xref="S6.SS1.7.p2.5.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.5.m1.1b"><apply id="S6.SS1.7.p2.5.m1.1.1.cmml" xref="S6.SS1.7.p2.5.m1.1.1"><geq id="S6.SS1.7.p2.5.m1.1.1.1.cmml" xref="S6.SS1.7.p2.5.m1.1.1.1"></geq><apply id="S6.SS1.7.p2.5.m1.1.1.2.cmml" xref="S6.SS1.7.p2.5.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.7.p2.5.m1.1.1.2.1.cmml" xref="S6.SS1.7.p2.5.m1.1.1.2">subscript</csymbol><ci id="S6.SS1.7.p2.5.m1.1.1.2.2.cmml" xref="S6.SS1.7.p2.5.m1.1.1.2.2">𝑗</ci><cn id="S6.SS1.7.p2.5.m1.1.1.2.3.cmml" type="integer" xref="S6.SS1.7.p2.5.m1.1.1.2.3">1</cn></apply><cn id="S6.SS1.7.p2.5.m1.1.1.3.cmml" type="integer" xref="S6.SS1.7.p2.5.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.5.m1.1c">j_{1}\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.5.m1.1d">italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math> that</p> <table class="ltx_equation ltx_eqn_table" id="S6.E2"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0}+j_{1}-j,p}(w_{\gamma+(j_{1}-j)p}),W^{k_{0}+k_{1}+j_{1}-j,p}(w_% {\gamma+(j_{1}-j)p})\big{]}_{\frac{\ell}{k_{1}}}\hookrightarrow W^{k_{0}+\ell+% j_{1}-j}(w_{\gamma+(j_{1}-j)p})," class="ltx_Math" display="block" id="S6.E2.m1.8"><semantics id="S6.E2.m1.8a"><mrow id="S6.E2.m1.8.8.1" xref="S6.E2.m1.8.8.1.1.cmml"><mrow id="S6.E2.m1.8.8.1.1" xref="S6.E2.m1.8.8.1.1.cmml"><msub id="S6.E2.m1.8.8.1.1.2" xref="S6.E2.m1.8.8.1.1.2.cmml"><mrow id="S6.E2.m1.8.8.1.1.2.2.2" xref="S6.E2.m1.8.8.1.1.2.2.3.cmml"><mo id="S6.E2.m1.8.8.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.E2.m1.8.8.1.1.2.2.3.cmml">[</mo><mrow id="S6.E2.m1.8.8.1.1.1.1.1.1" xref="S6.E2.m1.8.8.1.1.1.1.1.1.cmml"><msup id="S6.E2.m1.8.8.1.1.1.1.1.1.3" xref="S6.E2.m1.8.8.1.1.1.1.1.1.3.cmml"><mi id="S6.E2.m1.8.8.1.1.1.1.1.1.3.2" xref="S6.E2.m1.8.8.1.1.1.1.1.1.3.2.cmml">W</mi><mrow id="S6.E2.m1.2.2.2.2" xref="S6.E2.m1.2.2.2.3.cmml"><mrow id="S6.E2.m1.2.2.2.2.1" xref="S6.E2.m1.2.2.2.2.1.cmml"><mrow id="S6.E2.m1.2.2.2.2.1.2" xref="S6.E2.m1.2.2.2.2.1.2.cmml"><msub id="S6.E2.m1.2.2.2.2.1.2.2" xref="S6.E2.m1.2.2.2.2.1.2.2.cmml"><mi id="S6.E2.m1.2.2.2.2.1.2.2.2" xref="S6.E2.m1.2.2.2.2.1.2.2.2.cmml">k</mi><mn id="S6.E2.m1.2.2.2.2.1.2.2.3" xref="S6.E2.m1.2.2.2.2.1.2.2.3.cmml">0</mn></msub><mo id="S6.E2.m1.2.2.2.2.1.2.1" xref="S6.E2.m1.2.2.2.2.1.2.1.cmml">+</mo><msub id="S6.E2.m1.2.2.2.2.1.2.3" xref="S6.E2.m1.2.2.2.2.1.2.3.cmml"><mi id="S6.E2.m1.2.2.2.2.1.2.3.2" xref="S6.E2.m1.2.2.2.2.1.2.3.2.cmml">j</mi><mn id="S6.E2.m1.2.2.2.2.1.2.3.3" xref="S6.E2.m1.2.2.2.2.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.E2.m1.2.2.2.2.1.1" xref="S6.E2.m1.2.2.2.2.1.1.cmml">−</mo><mi id="S6.E2.m1.2.2.2.2.1.3" xref="S6.E2.m1.2.2.2.2.1.3.cmml">j</mi></mrow><mo id="S6.E2.m1.2.2.2.2.2" xref="S6.E2.m1.2.2.2.3.cmml">,</mo><mi id="S6.E2.m1.1.1.1.1" xref="S6.E2.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.E2.m1.8.8.1.1.1.1.1.1.2" xref="S6.E2.m1.8.8.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.E2.m1.8.8.1.1.1.1.1.1.1.1" xref="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.2" stretchy="false" xref="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.1" xref="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.1.cmml"><mi id="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.1.2" xref="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.1.2.cmml">w</mi><mrow id="S6.E2.m1.3.3.1" xref="S6.E2.m1.3.3.1.cmml"><mi id="S6.E2.m1.3.3.1.3" xref="S6.E2.m1.3.3.1.3.cmml">γ</mi><mo id="S6.E2.m1.3.3.1.2" xref="S6.E2.m1.3.3.1.2.cmml">+</mo><mrow id="S6.E2.m1.3.3.1.1" xref="S6.E2.m1.3.3.1.1.cmml"><mrow id="S6.E2.m1.3.3.1.1.1.1" xref="S6.E2.m1.3.3.1.1.1.1.1.cmml"><mo id="S6.E2.m1.3.3.1.1.1.1.2" stretchy="false" xref="S6.E2.m1.3.3.1.1.1.1.1.cmml">(</mo><mrow id="S6.E2.m1.3.3.1.1.1.1.1" xref="S6.E2.m1.3.3.1.1.1.1.1.cmml"><msub id="S6.E2.m1.3.3.1.1.1.1.1.2" xref="S6.E2.m1.3.3.1.1.1.1.1.2.cmml"><mi id="S6.E2.m1.3.3.1.1.1.1.1.2.2" xref="S6.E2.m1.3.3.1.1.1.1.1.2.2.cmml">j</mi><mn id="S6.E2.m1.3.3.1.1.1.1.1.2.3" xref="S6.E2.m1.3.3.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.E2.m1.3.3.1.1.1.1.1.1" xref="S6.E2.m1.3.3.1.1.1.1.1.1.cmml">−</mo><mi id="S6.E2.m1.3.3.1.1.1.1.1.3" xref="S6.E2.m1.3.3.1.1.1.1.1.3.cmml">j</mi></mrow><mo id="S6.E2.m1.3.3.1.1.1.1.3" stretchy="false" xref="S6.E2.m1.3.3.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.E2.m1.3.3.1.1.2" xref="S6.E2.m1.3.3.1.1.2.cmml">⁢</mo><mi id="S6.E2.m1.3.3.1.1.3" xref="S6.E2.m1.3.3.1.1.3.cmml">p</mi></mrow></mrow></msub><mo id="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.3" stretchy="false" xref="S6.E2.m1.8.8.1.1.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.E2.m1.8.8.1.1.2.2.2.4" xref="S6.E2.m1.8.8.1.1.2.2.3.cmml">,</mo><mrow id="S6.E2.m1.8.8.1.1.2.2.2.2" xref="S6.E2.m1.8.8.1.1.2.2.2.2.cmml"><msup id="S6.E2.m1.8.8.1.1.2.2.2.2.3" xref="S6.E2.m1.8.8.1.1.2.2.2.2.3.cmml"><mi id="S6.E2.m1.8.8.1.1.2.2.2.2.3.2" xref="S6.E2.m1.8.8.1.1.2.2.2.2.3.2.cmml">W</mi><mrow id="S6.E2.m1.5.5.2.2" xref="S6.E2.m1.5.5.2.3.cmml"><mrow id="S6.E2.m1.5.5.2.2.1" xref="S6.E2.m1.5.5.2.2.1.cmml"><mrow id="S6.E2.m1.5.5.2.2.1.2" xref="S6.E2.m1.5.5.2.2.1.2.cmml"><msub id="S6.E2.m1.5.5.2.2.1.2.2" xref="S6.E2.m1.5.5.2.2.1.2.2.cmml"><mi id="S6.E2.m1.5.5.2.2.1.2.2.2" xref="S6.E2.m1.5.5.2.2.1.2.2.2.cmml">k</mi><mn id="S6.E2.m1.5.5.2.2.1.2.2.3" xref="S6.E2.m1.5.5.2.2.1.2.2.3.cmml">0</mn></msub><mo id="S6.E2.m1.5.5.2.2.1.2.1" xref="S6.E2.m1.5.5.2.2.1.2.1.cmml">+</mo><msub id="S6.E2.m1.5.5.2.2.1.2.3" xref="S6.E2.m1.5.5.2.2.1.2.3.cmml"><mi id="S6.E2.m1.5.5.2.2.1.2.3.2" xref="S6.E2.m1.5.5.2.2.1.2.3.2.cmml">k</mi><mn id="S6.E2.m1.5.5.2.2.1.2.3.3" xref="S6.E2.m1.5.5.2.2.1.2.3.3.cmml">1</mn></msub><mo id="S6.E2.m1.5.5.2.2.1.2.1a" xref="S6.E2.m1.5.5.2.2.1.2.1.cmml">+</mo><msub id="S6.E2.m1.5.5.2.2.1.2.4" xref="S6.E2.m1.5.5.2.2.1.2.4.cmml"><mi id="S6.E2.m1.5.5.2.2.1.2.4.2" xref="S6.E2.m1.5.5.2.2.1.2.4.2.cmml">j</mi><mn id="S6.E2.m1.5.5.2.2.1.2.4.3" xref="S6.E2.m1.5.5.2.2.1.2.4.3.cmml">1</mn></msub></mrow><mo id="S6.E2.m1.5.5.2.2.1.1" xref="S6.E2.m1.5.5.2.2.1.1.cmml">−</mo><mi id="S6.E2.m1.5.5.2.2.1.3" xref="S6.E2.m1.5.5.2.2.1.3.cmml">j</mi></mrow><mo id="S6.E2.m1.5.5.2.2.2" xref="S6.E2.m1.5.5.2.3.cmml">,</mo><mi id="S6.E2.m1.4.4.1.1" xref="S6.E2.m1.4.4.1.1.cmml">p</mi></mrow></msup><mo id="S6.E2.m1.8.8.1.1.2.2.2.2.2" xref="S6.E2.m1.8.8.1.1.2.2.2.2.2.cmml">⁢</mo><mrow id="S6.E2.m1.8.8.1.1.2.2.2.2.1.1" xref="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.1.cmml"><mo id="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.2" stretchy="false" xref="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.1.cmml">(</mo><msub id="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.1" xref="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.1.cmml"><mi id="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.1.2" xref="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.1.2.cmml">w</mi><mrow id="S6.E2.m1.6.6.1" xref="S6.E2.m1.6.6.1.cmml"><mi id="S6.E2.m1.6.6.1.3" xref="S6.E2.m1.6.6.1.3.cmml">γ</mi><mo id="S6.E2.m1.6.6.1.2" xref="S6.E2.m1.6.6.1.2.cmml">+</mo><mrow id="S6.E2.m1.6.6.1.1" xref="S6.E2.m1.6.6.1.1.cmml"><mrow id="S6.E2.m1.6.6.1.1.1.1" xref="S6.E2.m1.6.6.1.1.1.1.1.cmml"><mo id="S6.E2.m1.6.6.1.1.1.1.2" stretchy="false" xref="S6.E2.m1.6.6.1.1.1.1.1.cmml">(</mo><mrow id="S6.E2.m1.6.6.1.1.1.1.1" xref="S6.E2.m1.6.6.1.1.1.1.1.cmml"><msub id="S6.E2.m1.6.6.1.1.1.1.1.2" xref="S6.E2.m1.6.6.1.1.1.1.1.2.cmml"><mi id="S6.E2.m1.6.6.1.1.1.1.1.2.2" xref="S6.E2.m1.6.6.1.1.1.1.1.2.2.cmml">j</mi><mn id="S6.E2.m1.6.6.1.1.1.1.1.2.3" xref="S6.E2.m1.6.6.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.E2.m1.6.6.1.1.1.1.1.1" xref="S6.E2.m1.6.6.1.1.1.1.1.1.cmml">−</mo><mi id="S6.E2.m1.6.6.1.1.1.1.1.3" xref="S6.E2.m1.6.6.1.1.1.1.1.3.cmml">j</mi></mrow><mo id="S6.E2.m1.6.6.1.1.1.1.3" stretchy="false" xref="S6.E2.m1.6.6.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.E2.m1.6.6.1.1.2" xref="S6.E2.m1.6.6.1.1.2.cmml">⁢</mo><mi id="S6.E2.m1.6.6.1.1.3" xref="S6.E2.m1.6.6.1.1.3.cmml">p</mi></mrow></mrow></msub><mo id="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.3" stretchy="false" xref="S6.E2.m1.8.8.1.1.2.2.2.2.1.1.1.cmml">)</mo></mrow></mrow><mo id="S6.E2.m1.8.8.1.1.2.2.2.5" maxsize="120%" minsize="120%" xref="S6.E2.m1.8.8.1.1.2.2.3.cmml">]</mo></mrow><mfrac id="S6.E2.m1.8.8.1.1.2.4" xref="S6.E2.m1.8.8.1.1.2.4.cmml"><mi id="S6.E2.m1.8.8.1.1.2.4.2" mathvariant="normal" xref="S6.E2.m1.8.8.1.1.2.4.2.cmml">ℓ</mi><msub id="S6.E2.m1.8.8.1.1.2.4.3" xref="S6.E2.m1.8.8.1.1.2.4.3.cmml"><mi id="S6.E2.m1.8.8.1.1.2.4.3.2" xref="S6.E2.m1.8.8.1.1.2.4.3.2.cmml">k</mi><mn id="S6.E2.m1.8.8.1.1.2.4.3.3" xref="S6.E2.m1.8.8.1.1.2.4.3.3.cmml">1</mn></msub></mfrac></msub><mo id="S6.E2.m1.8.8.1.1.4" stretchy="false" xref="S6.E2.m1.8.8.1.1.4.cmml">↪</mo><mrow id="S6.E2.m1.8.8.1.1.3" xref="S6.E2.m1.8.8.1.1.3.cmml"><msup id="S6.E2.m1.8.8.1.1.3.3" xref="S6.E2.m1.8.8.1.1.3.3.cmml"><mi id="S6.E2.m1.8.8.1.1.3.3.2" 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cd="ambiguous" id="S6.E2.m1.8.8.1.1.3.1.1.1.1.cmml" xref="S6.E2.m1.8.8.1.1.3.1.1">subscript</csymbol><ci id="S6.E2.m1.8.8.1.1.3.1.1.1.2.cmml" xref="S6.E2.m1.8.8.1.1.3.1.1.1.2">𝑤</ci><apply id="S6.E2.m1.7.7.1.cmml" xref="S6.E2.m1.7.7.1"><plus id="S6.E2.m1.7.7.1.2.cmml" xref="S6.E2.m1.7.7.1.2"></plus><ci id="S6.E2.m1.7.7.1.3.cmml" xref="S6.E2.m1.7.7.1.3">𝛾</ci><apply id="S6.E2.m1.7.7.1.1.cmml" xref="S6.E2.m1.7.7.1.1"><times id="S6.E2.m1.7.7.1.1.2.cmml" xref="S6.E2.m1.7.7.1.1.2"></times><apply id="S6.E2.m1.7.7.1.1.1.1.1.cmml" xref="S6.E2.m1.7.7.1.1.1.1"><minus id="S6.E2.m1.7.7.1.1.1.1.1.1.cmml" xref="S6.E2.m1.7.7.1.1.1.1.1.1"></minus><apply id="S6.E2.m1.7.7.1.1.1.1.1.2.cmml" xref="S6.E2.m1.7.7.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.E2.m1.7.7.1.1.1.1.1.2.1.cmml" xref="S6.E2.m1.7.7.1.1.1.1.1.2">subscript</csymbol><ci id="S6.E2.m1.7.7.1.1.1.1.1.2.2.cmml" xref="S6.E2.m1.7.7.1.1.1.1.1.2.2">𝑗</ci><cn id="S6.E2.m1.7.7.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.E2.m1.7.7.1.1.1.1.1.2.3">1</cn></apply><ci id="S6.E2.m1.7.7.1.1.1.1.1.3.cmml" xref="S6.E2.m1.7.7.1.1.1.1.1.3">𝑗</ci></apply><ci id="S6.E2.m1.7.7.1.1.3.cmml" xref="S6.E2.m1.7.7.1.1.3">𝑝</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E2.m1.8c">\big{[}W^{k_{0}+j_{1}-j,p}(w_{\gamma+(j_{1}-j)p}),W^{k_{0}+k_{1}+j_{1}-j,p}(w_% {\gamma+(j_{1}-j)p})\big{]}_{\frac{\ell}{k_{1}}}\hookrightarrow W^{k_{0}+\ell+% j_{1}-j}(w_{\gamma+(j_{1}-j)p}),</annotation><annotation encoding="application/x-llamapun" id="S6.E2.m1.8d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ + ( italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j ) italic_p end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ + ( italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j ) italic_p end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ + italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ + ( italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j ) italic_p end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.2)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.7.p2.8">so that (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E1" title="In Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.1</span></a>) for <math alttext="\gamma<(k_{0}+1)p-1" class="ltx_Math" display="inline" id="S6.SS1.7.p2.6.m1.1"><semantics id="S6.SS1.7.p2.6.m1.1a"><mrow id="S6.SS1.7.p2.6.m1.1.1" xref="S6.SS1.7.p2.6.m1.1.1.cmml"><mi id="S6.SS1.7.p2.6.m1.1.1.3" xref="S6.SS1.7.p2.6.m1.1.1.3.cmml">γ</mi><mo id="S6.SS1.7.p2.6.m1.1.1.2" xref="S6.SS1.7.p2.6.m1.1.1.2.cmml"><</mo><mrow id="S6.SS1.7.p2.6.m1.1.1.1" xref="S6.SS1.7.p2.6.m1.1.1.1.cmml"><mrow id="S6.SS1.7.p2.6.m1.1.1.1.1" xref="S6.SS1.7.p2.6.m1.1.1.1.1.cmml"><mrow id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.cmml"><mo id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.cmml"><msub id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.cmml"><mi id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.2" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.3" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.1" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.1.cmml">+</mo><mn id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.3" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS1.7.p2.6.m1.1.1.1.1.2" xref="S6.SS1.7.p2.6.m1.1.1.1.1.2.cmml">⁢</mo><mi id="S6.SS1.7.p2.6.m1.1.1.1.1.3" xref="S6.SS1.7.p2.6.m1.1.1.1.1.3.cmml">p</mi></mrow><mo id="S6.SS1.7.p2.6.m1.1.1.1.2" xref="S6.SS1.7.p2.6.m1.1.1.1.2.cmml">−</mo><mn id="S6.SS1.7.p2.6.m1.1.1.1.3" xref="S6.SS1.7.p2.6.m1.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.6.m1.1b"><apply id="S6.SS1.7.p2.6.m1.1.1.cmml" xref="S6.SS1.7.p2.6.m1.1.1"><lt id="S6.SS1.7.p2.6.m1.1.1.2.cmml" xref="S6.SS1.7.p2.6.m1.1.1.2"></lt><ci id="S6.SS1.7.p2.6.m1.1.1.3.cmml" xref="S6.SS1.7.p2.6.m1.1.1.3">𝛾</ci><apply id="S6.SS1.7.p2.6.m1.1.1.1.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1"><minus id="S6.SS1.7.p2.6.m1.1.1.1.2.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.2"></minus><apply id="S6.SS1.7.p2.6.m1.1.1.1.1.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1"><times id="S6.SS1.7.p2.6.m1.1.1.1.1.2.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1.2"></times><apply id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1"><plus id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.1"></plus><apply id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.2.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.2">𝑘</ci><cn id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.2.3">0</cn></apply><cn id="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.3.cmml" type="integer" xref="S6.SS1.7.p2.6.m1.1.1.1.1.1.1.1.3">1</cn></apply><ci id="S6.SS1.7.p2.6.m1.1.1.1.1.3.cmml" xref="S6.SS1.7.p2.6.m1.1.1.1.1.3">𝑝</ci></apply><cn id="S6.SS1.7.p2.6.m1.1.1.1.3.cmml" type="integer" xref="S6.SS1.7.p2.6.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.6.m1.1c">\gamma<(k_{0}+1)p-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.6.m1.1d">italic_γ < ( italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 ) italic_p - 1</annotation></semantics></math> follows from (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E2" title="In Proof. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a>) with <math alttext="j_{1}=j" class="ltx_Math" display="inline" id="S6.SS1.7.p2.7.m2.1"><semantics id="S6.SS1.7.p2.7.m2.1a"><mrow id="S6.SS1.7.p2.7.m2.1.1" xref="S6.SS1.7.p2.7.m2.1.1.cmml"><msub id="S6.SS1.7.p2.7.m2.1.1.2" xref="S6.SS1.7.p2.7.m2.1.1.2.cmml"><mi id="S6.SS1.7.p2.7.m2.1.1.2.2" xref="S6.SS1.7.p2.7.m2.1.1.2.2.cmml">j</mi><mn id="S6.SS1.7.p2.7.m2.1.1.2.3" xref="S6.SS1.7.p2.7.m2.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS1.7.p2.7.m2.1.1.1" xref="S6.SS1.7.p2.7.m2.1.1.1.cmml">=</mo><mi id="S6.SS1.7.p2.7.m2.1.1.3" xref="S6.SS1.7.p2.7.m2.1.1.3.cmml">j</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.7.m2.1b"><apply id="S6.SS1.7.p2.7.m2.1.1.cmml" xref="S6.SS1.7.p2.7.m2.1.1"><eq id="S6.SS1.7.p2.7.m2.1.1.1.cmml" xref="S6.SS1.7.p2.7.m2.1.1.1"></eq><apply id="S6.SS1.7.p2.7.m2.1.1.2.cmml" xref="S6.SS1.7.p2.7.m2.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.7.p2.7.m2.1.1.2.1.cmml" xref="S6.SS1.7.p2.7.m2.1.1.2">subscript</csymbol><ci id="S6.SS1.7.p2.7.m2.1.1.2.2.cmml" xref="S6.SS1.7.p2.7.m2.1.1.2.2">𝑗</ci><cn id="S6.SS1.7.p2.7.m2.1.1.2.3.cmml" type="integer" xref="S6.SS1.7.p2.7.m2.1.1.2.3">1</cn></apply><ci id="S6.SS1.7.p2.7.m2.1.1.3.cmml" xref="S6.SS1.7.p2.7.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.7.m2.1c">j_{1}=j</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.7.m2.1d">italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_j</annotation></semantics></math>. Assume that (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E2" title="In Proof. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a>) holds for some <math alttext="j_{1}\geq 0" class="ltx_Math" display="inline" id="S6.SS1.7.p2.8.m3.1"><semantics id="S6.SS1.7.p2.8.m3.1a"><mrow id="S6.SS1.7.p2.8.m3.1.1" xref="S6.SS1.7.p2.8.m3.1.1.cmml"><msub id="S6.SS1.7.p2.8.m3.1.1.2" xref="S6.SS1.7.p2.8.m3.1.1.2.cmml"><mi id="S6.SS1.7.p2.8.m3.1.1.2.2" xref="S6.SS1.7.p2.8.m3.1.1.2.2.cmml">j</mi><mn id="S6.SS1.7.p2.8.m3.1.1.2.3" xref="S6.SS1.7.p2.8.m3.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS1.7.p2.8.m3.1.1.1" xref="S6.SS1.7.p2.8.m3.1.1.1.cmml">≥</mo><mn id="S6.SS1.7.p2.8.m3.1.1.3" xref="S6.SS1.7.p2.8.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.8.m3.1b"><apply id="S6.SS1.7.p2.8.m3.1.1.cmml" xref="S6.SS1.7.p2.8.m3.1.1"><geq id="S6.SS1.7.p2.8.m3.1.1.1.cmml" xref="S6.SS1.7.p2.8.m3.1.1.1"></geq><apply id="S6.SS1.7.p2.8.m3.1.1.2.cmml" xref="S6.SS1.7.p2.8.m3.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.7.p2.8.m3.1.1.2.1.cmml" xref="S6.SS1.7.p2.8.m3.1.1.2">subscript</csymbol><ci id="S6.SS1.7.p2.8.m3.1.1.2.2.cmml" xref="S6.SS1.7.p2.8.m3.1.1.2.2">𝑗</ci><cn id="S6.SS1.7.p2.8.m3.1.1.2.3.cmml" type="integer" xref="S6.SS1.7.p2.8.m3.1.1.2.3">1</cn></apply><cn id="S6.SS1.7.p2.8.m3.1.1.3.cmml" type="integer" xref="S6.SS1.7.p2.8.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.8.m3.1c">j_{1}\geq 0</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.8.m3.1d">italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ 0</annotation></semantics></math>. Let</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex24"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{k}:=k_{0}+j_{1}-j\quad\text{ and }\quad\widetilde{\gamma}:=\gamma+(% j_{1}-j)p." class="ltx_Math" display="block" id="S6.Ex24.m1.2"><semantics id="S6.Ex24.m1.2a"><mrow id="S6.Ex24.m1.2.2.1"><mrow id="S6.Ex24.m1.2.2.1.1.2" xref="S6.Ex24.m1.2.2.1.1.3.cmml"><mrow id="S6.Ex24.m1.2.2.1.1.1.1" xref="S6.Ex24.m1.2.2.1.1.1.1.cmml"><mover accent="true" id="S6.Ex24.m1.2.2.1.1.1.1.3" xref="S6.Ex24.m1.2.2.1.1.1.1.3.cmml"><mi id="S6.Ex24.m1.2.2.1.1.1.1.3.2" xref="S6.Ex24.m1.2.2.1.1.1.1.3.2.cmml">k</mi><mo id="S6.Ex24.m1.2.2.1.1.1.1.3.1" xref="S6.Ex24.m1.2.2.1.1.1.1.3.1.cmml">~</mo></mover><mo id="S6.Ex24.m1.2.2.1.1.1.1.2" lspace="0.278em" rspace="0.278em" xref="S6.Ex24.m1.2.2.1.1.1.1.2.cmml">:=</mo><mrow id="S6.Ex24.m1.2.2.1.1.1.1.1.1" xref="S6.Ex24.m1.2.2.1.1.1.1.1.2.cmml"><mrow id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.cmml"><mrow id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.cmml"><msub id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.cmml"><mi id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.2" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.2.cmml">k</mi><mn id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.3" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.1" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.1.cmml">+</mo><msub id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.cmml"><mi id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.2" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.2.cmml">j</mi><mn id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.3" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.1" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.3" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.3.cmml">j</mi></mrow><mspace id="S6.Ex24.m1.2.2.1.1.1.1.1.1.2" width="1em" xref="S6.Ex24.m1.2.2.1.1.1.1.1.2.cmml"></mspace><mtext id="S6.Ex24.m1.1.1" xref="S6.Ex24.m1.1.1a.cmml"> and </mtext></mrow></mrow><mspace id="S6.Ex24.m1.2.2.1.1.2.3" width="1em" xref="S6.Ex24.m1.2.2.1.1.3a.cmml"></mspace><mrow id="S6.Ex24.m1.2.2.1.1.2.2" xref="S6.Ex24.m1.2.2.1.1.2.2.cmml"><mover accent="true" id="S6.Ex24.m1.2.2.1.1.2.2.3" xref="S6.Ex24.m1.2.2.1.1.2.2.3.cmml"><mi id="S6.Ex24.m1.2.2.1.1.2.2.3.2" xref="S6.Ex24.m1.2.2.1.1.2.2.3.2.cmml">γ</mi><mo id="S6.Ex24.m1.2.2.1.1.2.2.3.1" xref="S6.Ex24.m1.2.2.1.1.2.2.3.1.cmml">~</mo></mover><mo id="S6.Ex24.m1.2.2.1.1.2.2.2" lspace="0.278em" rspace="0.278em" xref="S6.Ex24.m1.2.2.1.1.2.2.2.cmml">:=</mo><mrow id="S6.Ex24.m1.2.2.1.1.2.2.1" xref="S6.Ex24.m1.2.2.1.1.2.2.1.cmml"><mi id="S6.Ex24.m1.2.2.1.1.2.2.1.3" xref="S6.Ex24.m1.2.2.1.1.2.2.1.3.cmml">γ</mi><mo id="S6.Ex24.m1.2.2.1.1.2.2.1.2" xref="S6.Ex24.m1.2.2.1.1.2.2.1.2.cmml">+</mo><mrow id="S6.Ex24.m1.2.2.1.1.2.2.1.1" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.cmml"><mrow id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.cmml"><mo id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.2" stretchy="false" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.cmml"><msub id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.cmml"><mi id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.2" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.2.cmml">j</mi><mn id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.3" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.1" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml">−</mo><mi id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.3" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.3.cmml">j</mi></mrow><mo id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.3" stretchy="false" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex24.m1.2.2.1.1.2.2.1.1.2" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.2.cmml">⁢</mo><mi id="S6.Ex24.m1.2.2.1.1.2.2.1.1.3" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.3.cmml">p</mi></mrow></mrow></mrow></mrow><mo id="S6.Ex24.m1.2.2.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex24.m1.2b"><apply id="S6.Ex24.m1.2.2.1.1.3.cmml" xref="S6.Ex24.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="S6.Ex24.m1.2.2.1.1.3a.cmml" xref="S6.Ex24.m1.2.2.1.1.2.3">formulae-sequence</csymbol><apply id="S6.Ex24.m1.2.2.1.1.1.1.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1"><csymbol cd="latexml" id="S6.Ex24.m1.2.2.1.1.1.1.2.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.2">assign</csymbol><apply id="S6.Ex24.m1.2.2.1.1.1.1.3.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.3"><ci id="S6.Ex24.m1.2.2.1.1.1.1.3.1.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.3.1">~</ci><ci id="S6.Ex24.m1.2.2.1.1.1.1.3.2.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.3.2">𝑘</ci></apply><list id="S6.Ex24.m1.2.2.1.1.1.1.1.2.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1"><apply id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1"><minus id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.1"></minus><apply id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2"><plus id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.1.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.1"></plus><apply id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.1.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.2.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.2">𝑘</ci><cn id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.3.cmml" type="integer" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.2.3">0</cn></apply><apply id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.1.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3">subscript</csymbol><ci id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.2.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.2">𝑗</ci><cn id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.3.cmml" type="integer" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.3.cmml" xref="S6.Ex24.m1.2.2.1.1.1.1.1.1.1.3">𝑗</ci></apply><ci id="S6.Ex24.m1.1.1a.cmml" xref="S6.Ex24.m1.1.1"><mtext id="S6.Ex24.m1.1.1.cmml" xref="S6.Ex24.m1.1.1"> and </mtext></ci></list></apply><apply id="S6.Ex24.m1.2.2.1.1.2.2.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2"><csymbol cd="latexml" id="S6.Ex24.m1.2.2.1.1.2.2.2.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.2">assign</csymbol><apply id="S6.Ex24.m1.2.2.1.1.2.2.3.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.3"><ci id="S6.Ex24.m1.2.2.1.1.2.2.3.1.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.3.1">~</ci><ci id="S6.Ex24.m1.2.2.1.1.2.2.3.2.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.3.2">𝛾</ci></apply><apply id="S6.Ex24.m1.2.2.1.1.2.2.1.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1"><plus id="S6.Ex24.m1.2.2.1.1.2.2.1.2.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.2"></plus><ci id="S6.Ex24.m1.2.2.1.1.2.2.1.3.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.3">𝛾</ci><apply id="S6.Ex24.m1.2.2.1.1.2.2.1.1.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1"><times id="S6.Ex24.m1.2.2.1.1.2.2.1.1.2.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.2"></times><apply id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1"><minus id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.1.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.1"></minus><apply id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.1.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2">subscript</csymbol><ci id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.2.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.2">𝑗</ci><cn id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.2.3">1</cn></apply><ci id="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.3.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.1.1.1.3">𝑗</ci></apply><ci id="S6.Ex24.m1.2.2.1.1.2.2.1.1.3.cmml" xref="S6.Ex24.m1.2.2.1.1.2.2.1.1.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex24.m1.2c">\widetilde{k}:=k_{0}+j_{1}-j\quad\text{ and }\quad\widetilde{\gamma}:=\gamma+(% j_{1}-j)p.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex24.m1.2d">over~ start_ARG italic_k end_ARG := italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j and over~ start_ARG italic_γ end_ARG := italic_γ + ( italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j ) italic_p .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.7.p2.9">Let <math alttext="u\in W^{\widetilde{k}+k_{1}+1}(w_{\widetilde{\gamma}+p})" class="ltx_Math" display="inline" id="S6.SS1.7.p2.9.m1.1"><semantics id="S6.SS1.7.p2.9.m1.1a"><mrow id="S6.SS1.7.p2.9.m1.1.1" xref="S6.SS1.7.p2.9.m1.1.1.cmml"><mi id="S6.SS1.7.p2.9.m1.1.1.3" xref="S6.SS1.7.p2.9.m1.1.1.3.cmml">u</mi><mo id="S6.SS1.7.p2.9.m1.1.1.2" xref="S6.SS1.7.p2.9.m1.1.1.2.cmml">∈</mo><mrow id="S6.SS1.7.p2.9.m1.1.1.1" xref="S6.SS1.7.p2.9.m1.1.1.1.cmml"><msup id="S6.SS1.7.p2.9.m1.1.1.1.3" xref="S6.SS1.7.p2.9.m1.1.1.1.3.cmml"><mi id="S6.SS1.7.p2.9.m1.1.1.1.3.2" xref="S6.SS1.7.p2.9.m1.1.1.1.3.2.cmml">W</mi><mrow id="S6.SS1.7.p2.9.m1.1.1.1.3.3" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.cmml"><mover accent="true" id="S6.SS1.7.p2.9.m1.1.1.1.3.3.2" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.cmml"><mi id="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.2" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.2.cmml">k</mi><mo id="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.1" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.1.cmml">~</mo></mover><mo id="S6.SS1.7.p2.9.m1.1.1.1.3.3.1" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.1.cmml">+</mo><msub id="S6.SS1.7.p2.9.m1.1.1.1.3.3.3" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.cmml"><mi id="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.2" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.2.cmml">k</mi><mn id="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.3" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.3.cmml">1</mn></msub><mo id="S6.SS1.7.p2.9.m1.1.1.1.3.3.1a" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.1.cmml">+</mo><mn id="S6.SS1.7.p2.9.m1.1.1.1.3.3.4" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.4.cmml">1</mn></mrow></msup><mo id="S6.SS1.7.p2.9.m1.1.1.1.2" xref="S6.SS1.7.p2.9.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.SS1.7.p2.9.m1.1.1.1.1.1" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.cmml"><mo id="S6.SS1.7.p2.9.m1.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.cmml"><mi id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.2" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.2.cmml">w</mi><mrow id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.cmml"><mover accent="true" id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.2" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.2.cmml">γ</mi><mo id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.1" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.1.cmml">~</mo></mover><mo id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.1" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.3" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.3.cmml">p</mi></mrow></msub><mo id="S6.SS1.7.p2.9.m1.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.9.m1.1b"><apply id="S6.SS1.7.p2.9.m1.1.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1"><in id="S6.SS1.7.p2.9.m1.1.1.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.2"></in><ci id="S6.SS1.7.p2.9.m1.1.1.3.cmml" xref="S6.SS1.7.p2.9.m1.1.1.3">𝑢</ci><apply id="S6.SS1.7.p2.9.m1.1.1.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1"><times id="S6.SS1.7.p2.9.m1.1.1.1.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.2"></times><apply id="S6.SS1.7.p2.9.m1.1.1.1.3.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.7.p2.9.m1.1.1.1.3.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3">superscript</csymbol><ci id="S6.SS1.7.p2.9.m1.1.1.1.3.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.2">𝑊</ci><apply id="S6.SS1.7.p2.9.m1.1.1.1.3.3.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3"><plus id="S6.SS1.7.p2.9.m1.1.1.1.3.3.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.1"></plus><apply id="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.2"><ci id="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.1">~</ci><ci id="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.2.2">𝑘</ci></apply><apply id="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.3">subscript</csymbol><ci id="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.2">𝑘</ci><cn id="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.3.cmml" type="integer" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.3.3">1</cn></apply><cn id="S6.SS1.7.p2.9.m1.1.1.1.3.3.4.cmml" type="integer" xref="S6.SS1.7.p2.9.m1.1.1.1.3.3.4">1</cn></apply></apply><apply id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1">subscript</csymbol><ci id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.2">𝑤</ci><apply id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3"><plus id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.1"></plus><apply id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2"><ci id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.1.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.1">~</ci><ci id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.2.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.2.2">𝛾</ci></apply><ci id="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.3.cmml" xref="S6.SS1.7.p2.9.m1.1.1.1.1.1.1.3.3">𝑝</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.9.m1.1c">u\in W^{\widetilde{k}+k_{1}+1}(w_{\widetilde{\gamma}+p})</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.9.m1.1d">italic_u ∈ italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT )</annotation></semantics></math>, then</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx32"> <tbody id="S6.Ex25"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\|u\|_{W^{\widetilde{k}+\ell+1}(w_{\widetilde{\gamma}+p})}=\|u\|_% {W^{\widetilde{k}+\ell}(w_{\widetilde{\gamma}})}+\sum_{|\alpha|=1}\sum_{|\beta% |=\widetilde{k}+\ell}\|M\partial^{\alpha+\beta}u\|_{L^{p}(w_{\widetilde{\gamma% }})}." class="ltx_Math" display="inline" id="S6.Ex25.m1.8"><semantics id="S6.Ex25.m1.8a"><mrow id="S6.Ex25.m1.8.8.1" xref="S6.Ex25.m1.8.8.1.1.cmml"><mrow id="S6.Ex25.m1.8.8.1.1" xref="S6.Ex25.m1.8.8.1.1.cmml"><msub id="S6.Ex25.m1.8.8.1.1.3" xref="S6.Ex25.m1.8.8.1.1.3.cmml"><mrow id="S6.Ex25.m1.8.8.1.1.3.2.2" xref="S6.Ex25.m1.8.8.1.1.3.2.1.cmml"><mo id="S6.Ex25.m1.8.8.1.1.3.2.2.1" stretchy="false" xref="S6.Ex25.m1.8.8.1.1.3.2.1.1.cmml">‖</mo><mi id="S6.Ex25.m1.6.6" xref="S6.Ex25.m1.6.6.cmml">u</mi><mo id="S6.Ex25.m1.8.8.1.1.3.2.2.2" stretchy="false" xref="S6.Ex25.m1.8.8.1.1.3.2.1.1.cmml">‖</mo></mrow><mrow id="S6.Ex25.m1.1.1.1" xref="S6.Ex25.m1.1.1.1.cmml"><msup id="S6.Ex25.m1.1.1.1.3" xref="S6.Ex25.m1.1.1.1.3.cmml"><mi id="S6.Ex25.m1.1.1.1.3.2" xref="S6.Ex25.m1.1.1.1.3.2.cmml">W</mi><mrow id="S6.Ex25.m1.1.1.1.3.3" xref="S6.Ex25.m1.1.1.1.3.3.cmml"><mover accent="true" id="S6.Ex25.m1.1.1.1.3.3.2" xref="S6.Ex25.m1.1.1.1.3.3.2.cmml"><mi id="S6.Ex25.m1.1.1.1.3.3.2.2" xref="S6.Ex25.m1.1.1.1.3.3.2.2.cmml">k</mi><mo id="S6.Ex25.m1.1.1.1.3.3.2.1" xref="S6.Ex25.m1.1.1.1.3.3.2.1.cmml">~</mo></mover><mo id="S6.Ex25.m1.1.1.1.3.3.1" xref="S6.Ex25.m1.1.1.1.3.3.1.cmml">+</mo><mi id="S6.Ex25.m1.1.1.1.3.3.3" mathvariant="normal" xref="S6.Ex25.m1.1.1.1.3.3.3.cmml">ℓ</mi><mo id="S6.Ex25.m1.1.1.1.3.3.1a" xref="S6.Ex25.m1.1.1.1.3.3.1.cmml">+</mo><mn id="S6.Ex25.m1.1.1.1.3.3.4" xref="S6.Ex25.m1.1.1.1.3.3.4.cmml">1</mn></mrow></msup><mo id="S6.Ex25.m1.1.1.1.2" xref="S6.Ex25.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex25.m1.1.1.1.1.1" xref="S6.Ex25.m1.1.1.1.1.1.1.cmml"><mo id="S6.Ex25.m1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex25.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex25.m1.1.1.1.1.1.1" xref="S6.Ex25.m1.1.1.1.1.1.1.cmml"><mi id="S6.Ex25.m1.1.1.1.1.1.1.2" xref="S6.Ex25.m1.1.1.1.1.1.1.2.cmml">w</mi><mrow id="S6.Ex25.m1.1.1.1.1.1.1.3" xref="S6.Ex25.m1.1.1.1.1.1.1.3.cmml"><mover accent="true" id="S6.Ex25.m1.1.1.1.1.1.1.3.2" xref="S6.Ex25.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S6.Ex25.m1.1.1.1.1.1.1.3.2.2" xref="S6.Ex25.m1.1.1.1.1.1.1.3.2.2.cmml">γ</mi><mo id="S6.Ex25.m1.1.1.1.1.1.1.3.2.1" xref="S6.Ex25.m1.1.1.1.1.1.1.3.2.1.cmml">~</mo></mover><mo id="S6.Ex25.m1.1.1.1.1.1.1.3.1" xref="S6.Ex25.m1.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="S6.Ex25.m1.1.1.1.1.1.1.3.3" xref="S6.Ex25.m1.1.1.1.1.1.1.3.3.cmml">p</mi></mrow></msub><mo id="S6.Ex25.m1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex25.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Ex25.m1.8.8.1.1.2" xref="S6.Ex25.m1.8.8.1.1.2.cmml">=</mo><mrow id="S6.Ex25.m1.8.8.1.1.1" xref="S6.Ex25.m1.8.8.1.1.1.cmml"><msub id="S6.Ex25.m1.8.8.1.1.1.3" xref="S6.Ex25.m1.8.8.1.1.1.3.cmml"><mrow id="S6.Ex25.m1.8.8.1.1.1.3.2.2" 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{W^{\widetilde{k}+\ell}(w_{\widetilde{\gamma}})}+\sum_{|\alpha|=1}\sum_{|\beta% |=\widetilde{k}+\ell}\|M\partial^{\alpha+\beta}u\|_{L^{p}(w_{\widetilde{\gamma% }})}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex25.m1.8d">∥ italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + roman_ℓ + 1 end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT = ∥ italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + roman_ℓ end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT | italic_α | = 1 end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT | italic_β | = over~ start_ARG italic_k end_ARG + roman_ℓ end_POSTSUBSCRIPT ∥ italic_M ∂ start_POSTSUPERSCRIPT italic_α + italic_β end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.7.p2.11">Using <math alttext="\partial_{1}^{n}M=M\partial_{1}^{n}+n\partial_{1}^{n-1}" class="ltx_Math" display="inline" id="S6.SS1.7.p2.10.m1.1"><semantics id="S6.SS1.7.p2.10.m1.1a"><mrow id="S6.SS1.7.p2.10.m1.1.1" xref="S6.SS1.7.p2.10.m1.1.1.cmml"><mrow id="S6.SS1.7.p2.10.m1.1.1.2" xref="S6.SS1.7.p2.10.m1.1.1.2.cmml"><msubsup id="S6.SS1.7.p2.10.m1.1.1.2.1" xref="S6.SS1.7.p2.10.m1.1.1.2.1.cmml"><mo id="S6.SS1.7.p2.10.m1.1.1.2.1.2.2" xref="S6.SS1.7.p2.10.m1.1.1.2.1.2.2.cmml">∂</mo><mn id="S6.SS1.7.p2.10.m1.1.1.2.1.2.3" xref="S6.SS1.7.p2.10.m1.1.1.2.1.2.3.cmml">1</mn><mi id="S6.SS1.7.p2.10.m1.1.1.2.1.3" xref="S6.SS1.7.p2.10.m1.1.1.2.1.3.cmml">n</mi></msubsup><mi id="S6.SS1.7.p2.10.m1.1.1.2.2" xref="S6.SS1.7.p2.10.m1.1.1.2.2.cmml">M</mi></mrow><mo id="S6.SS1.7.p2.10.m1.1.1.1" xref="S6.SS1.7.p2.10.m1.1.1.1.cmml">=</mo><mrow id="S6.SS1.7.p2.10.m1.1.1.3" xref="S6.SS1.7.p2.10.m1.1.1.3.cmml"><mrow id="S6.SS1.7.p2.10.m1.1.1.3.2" xref="S6.SS1.7.p2.10.m1.1.1.3.2.cmml"><mi id="S6.SS1.7.p2.10.m1.1.1.3.2.2" xref="S6.SS1.7.p2.10.m1.1.1.3.2.2.cmml">M</mi><mo id="S6.SS1.7.p2.10.m1.1.1.3.2.1" lspace="0em" xref="S6.SS1.7.p2.10.m1.1.1.3.2.1.cmml">⁢</mo><msubsup id="S6.SS1.7.p2.10.m1.1.1.3.2.3" xref="S6.SS1.7.p2.10.m1.1.1.3.2.3.cmml"><mo id="S6.SS1.7.p2.10.m1.1.1.3.2.3.2.2" rspace="0em" xref="S6.SS1.7.p2.10.m1.1.1.3.2.3.2.2.cmml">∂</mo><mn id="S6.SS1.7.p2.10.m1.1.1.3.2.3.2.3" xref="S6.SS1.7.p2.10.m1.1.1.3.2.3.2.3.cmml">1</mn><mi id="S6.SS1.7.p2.10.m1.1.1.3.2.3.3" xref="S6.SS1.7.p2.10.m1.1.1.3.2.3.3.cmml">n</mi></msubsup></mrow><mo id="S6.SS1.7.p2.10.m1.1.1.3.1" xref="S6.SS1.7.p2.10.m1.1.1.3.1.cmml">+</mo><mrow id="S6.SS1.7.p2.10.m1.1.1.3.3" xref="S6.SS1.7.p2.10.m1.1.1.3.3.cmml"><mi id="S6.SS1.7.p2.10.m1.1.1.3.3.2" xref="S6.SS1.7.p2.10.m1.1.1.3.3.2.cmml">n</mi><mo id="S6.SS1.7.p2.10.m1.1.1.3.3.1" lspace="0em" xref="S6.SS1.7.p2.10.m1.1.1.3.3.1.cmml">⁢</mo><msubsup id="S6.SS1.7.p2.10.m1.1.1.3.3.3" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.cmml"><mo id="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.2" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.2.cmml">∂</mo><mn id="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.3" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.3.cmml">1</mn><mrow id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.cmml"><mi id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.2" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.2.cmml">n</mi><mo id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.1" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.1.cmml">−</mo><mn id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.3" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.3.cmml">1</mn></mrow></msubsup></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.10.m1.1b"><apply id="S6.SS1.7.p2.10.m1.1.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1"><eq id="S6.SS1.7.p2.10.m1.1.1.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.1"></eq><apply id="S6.SS1.7.p2.10.m1.1.1.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2"><apply id="S6.SS1.7.p2.10.m1.1.1.2.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2.1"><csymbol cd="ambiguous" id="S6.SS1.7.p2.10.m1.1.1.2.1.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2.1">superscript</csymbol><apply id="S6.SS1.7.p2.10.m1.1.1.2.1.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2.1"><csymbol cd="ambiguous" id="S6.SS1.7.p2.10.m1.1.1.2.1.2.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2.1">subscript</csymbol><partialdiff id="S6.SS1.7.p2.10.m1.1.1.2.1.2.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2.1.2.2"></partialdiff><cn id="S6.SS1.7.p2.10.m1.1.1.2.1.2.3.cmml" type="integer" xref="S6.SS1.7.p2.10.m1.1.1.2.1.2.3">1</cn></apply><ci id="S6.SS1.7.p2.10.m1.1.1.2.1.3.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2.1.3">𝑛</ci></apply><ci id="S6.SS1.7.p2.10.m1.1.1.2.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.2.2">𝑀</ci></apply><apply id="S6.SS1.7.p2.10.m1.1.1.3.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3"><plus 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xref="S6.SS1.7.p2.10.m1.1.1.3.2.3.3">𝑛</ci></apply></apply><apply id="S6.SS1.7.p2.10.m1.1.1.3.3.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3"><times id="S6.SS1.7.p2.10.m1.1.1.3.3.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.1"></times><ci id="S6.SS1.7.p2.10.m1.1.1.3.3.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.2">𝑛</ci><apply id="S6.SS1.7.p2.10.m1.1.1.3.3.3.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S6.SS1.7.p2.10.m1.1.1.3.3.3.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3">superscript</csymbol><apply id="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3"><csymbol cd="ambiguous" id="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3">subscript</csymbol><partialdiff id="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.2"></partialdiff><cn id="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.3.cmml" type="integer" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.2.3">1</cn></apply><apply id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3"><minus id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.1.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.1"></minus><ci id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.2.cmml" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.2">𝑛</ci><cn id="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.3.cmml" type="integer" xref="S6.SS1.7.p2.10.m1.1.1.3.3.3.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.10.m1.1c">\partial_{1}^{n}M=M\partial_{1}^{n}+n\partial_{1}^{n-1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.10.m1.1d">∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT italic_M = italic_M ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT + italic_n ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n - 1 end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="n\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S6.SS1.7.p2.11.m2.1"><semantics id="S6.SS1.7.p2.11.m2.1a"><mrow id="S6.SS1.7.p2.11.m2.1.1" xref="S6.SS1.7.p2.11.m2.1.1.cmml"><mi id="S6.SS1.7.p2.11.m2.1.1.2" xref="S6.SS1.7.p2.11.m2.1.1.2.cmml">n</mi><mo id="S6.SS1.7.p2.11.m2.1.1.1" xref="S6.SS1.7.p2.11.m2.1.1.1.cmml">∈</mo><msub id="S6.SS1.7.p2.11.m2.1.1.3" xref="S6.SS1.7.p2.11.m2.1.1.3.cmml"><mi id="S6.SS1.7.p2.11.m2.1.1.3.2" xref="S6.SS1.7.p2.11.m2.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS1.7.p2.11.m2.1.1.3.3" xref="S6.SS1.7.p2.11.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.11.m2.1b"><apply id="S6.SS1.7.p2.11.m2.1.1.cmml" xref="S6.SS1.7.p2.11.m2.1.1"><in id="S6.SS1.7.p2.11.m2.1.1.1.cmml" xref="S6.SS1.7.p2.11.m2.1.1.1"></in><ci id="S6.SS1.7.p2.11.m2.1.1.2.cmml" xref="S6.SS1.7.p2.11.m2.1.1.2">𝑛</ci><apply id="S6.SS1.7.p2.11.m2.1.1.3.cmml" xref="S6.SS1.7.p2.11.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.7.p2.11.m2.1.1.3.1.cmml" xref="S6.SS1.7.p2.11.m2.1.1.3">subscript</csymbol><ci id="S6.SS1.7.p2.11.m2.1.1.3.2.cmml" xref="S6.SS1.7.p2.11.m2.1.1.3.2">ℕ</ci><cn id="S6.SS1.7.p2.11.m2.1.1.3.3.cmml" type="integer" xref="S6.SS1.7.p2.11.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.11.m2.1c">n\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.11.m2.1d">italic_n ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and the induction hypothesis (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E2" title="In Proof. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a>), we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx33"> <tbody id="S6.Ex26"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\sum_{|\alpha|=1}\sum_{|\beta|=\widetilde{k}+\ell}" class="ltx_Math" display="inline" id="S6.Ex26.m1.2"><semantics id="S6.Ex26.m1.2a"><mrow id="S6.Ex26.m1.2.3" xref="S6.Ex26.m1.2.3.cmml"><mstyle displaystyle="true" id="S6.Ex26.m1.2.3.1" xref="S6.Ex26.m1.2.3.1.cmml"><munder id="S6.Ex26.m1.2.3.1a" xref="S6.Ex26.m1.2.3.1.cmml"><mo id="S6.Ex26.m1.2.3.1.2" movablelimits="false" xref="S6.Ex26.m1.2.3.1.2.cmml">∑</mo><mrow id="S6.Ex26.m1.1.1.1" xref="S6.Ex26.m1.1.1.1.cmml"><mrow id="S6.Ex26.m1.1.1.1.3.2" xref="S6.Ex26.m1.1.1.1.3.1.cmml"><mo id="S6.Ex26.m1.1.1.1.3.2.1" stretchy="false" xref="S6.Ex26.m1.1.1.1.3.1.1.cmml">|</mo><mi id="S6.Ex26.m1.1.1.1.1" xref="S6.Ex26.m1.1.1.1.1.cmml">α</mi><mo id="S6.Ex26.m1.1.1.1.3.2.2" stretchy="false" xref="S6.Ex26.m1.1.1.1.3.1.1.cmml">|</mo></mrow><mo id="S6.Ex26.m1.1.1.1.2" xref="S6.Ex26.m1.1.1.1.2.cmml">=</mo><mn id="S6.Ex26.m1.1.1.1.4" xref="S6.Ex26.m1.1.1.1.4.cmml">1</mn></mrow></munder></mstyle><mstyle displaystyle="true" id="S6.Ex26.m1.2.3.2" xref="S6.Ex26.m1.2.3.2.cmml"><munder id="S6.Ex26.m1.2.3.2a" xref="S6.Ex26.m1.2.3.2.cmml"><mo id="S6.Ex26.m1.2.3.2.2" movablelimits="false" xref="S6.Ex26.m1.2.3.2.2.cmml">∑</mo><mrow id="S6.Ex26.m1.2.2.1" xref="S6.Ex26.m1.2.2.1.cmml"><mrow id="S6.Ex26.m1.2.2.1.3.2" xref="S6.Ex26.m1.2.2.1.3.1.cmml"><mo id="S6.Ex26.m1.2.2.1.3.2.1" stretchy="false" xref="S6.Ex26.m1.2.2.1.3.1.1.cmml">|</mo><mi id="S6.Ex26.m1.2.2.1.1" xref="S6.Ex26.m1.2.2.1.1.cmml">β</mi><mo id="S6.Ex26.m1.2.2.1.3.2.2" stretchy="false" xref="S6.Ex26.m1.2.2.1.3.1.1.cmml">|</mo></mrow><mo id="S6.Ex26.m1.2.2.1.2" xref="S6.Ex26.m1.2.2.1.2.cmml">=</mo><mrow id="S6.Ex26.m1.2.2.1.4" xref="S6.Ex26.m1.2.2.1.4.cmml"><mover accent="true" id="S6.Ex26.m1.2.2.1.4.2" xref="S6.Ex26.m1.2.2.1.4.2.cmml"><mi id="S6.Ex26.m1.2.2.1.4.2.2" xref="S6.Ex26.m1.2.2.1.4.2.2.cmml">k</mi><mo id="S6.Ex26.m1.2.2.1.4.2.1" xref="S6.Ex26.m1.2.2.1.4.2.1.cmml">~</mo></mover><mo id="S6.Ex26.m1.2.2.1.4.1" xref="S6.Ex26.m1.2.2.1.4.1.cmml">+</mo><mi id="S6.Ex26.m1.2.2.1.4.3" mathvariant="normal" xref="S6.Ex26.m1.2.2.1.4.3.cmml">ℓ</mi></mrow></mrow></munder></mstyle></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex26.m1.2b"><apply id="S6.Ex26.m1.2.3.cmml" xref="S6.Ex26.m1.2.3"><apply id="S6.Ex26.m1.2.3.1.cmml" xref="S6.Ex26.m1.2.3.1"><csymbol cd="ambiguous" id="S6.Ex26.m1.2.3.1.1.cmml" xref="S6.Ex26.m1.2.3.1">subscript</csymbol><sum id="S6.Ex26.m1.2.3.1.2.cmml" xref="S6.Ex26.m1.2.3.1.2"></sum><apply id="S6.Ex26.m1.1.1.1.cmml" 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xref="S6.Ex26.m1.2.2.1.4"><plus id="S6.Ex26.m1.2.2.1.4.1.cmml" xref="S6.Ex26.m1.2.2.1.4.1"></plus><apply id="S6.Ex26.m1.2.2.1.4.2.cmml" xref="S6.Ex26.m1.2.2.1.4.2"><ci id="S6.Ex26.m1.2.2.1.4.2.1.cmml" xref="S6.Ex26.m1.2.2.1.4.2.1">~</ci><ci id="S6.Ex26.m1.2.2.1.4.2.2.cmml" xref="S6.Ex26.m1.2.2.1.4.2.2">𝑘</ci></apply><ci id="S6.Ex26.m1.2.2.1.4.3.cmml" xref="S6.Ex26.m1.2.2.1.4.3">ℓ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex26.m1.2c">\displaystyle\sum_{|\alpha|=1}\sum_{|\beta|=\widetilde{k}+\ell}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex26.m1.2d">∑ start_POSTSUBSCRIPT | italic_α | = 1 end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT | italic_β | = over~ start_ARG italic_k end_ARG + roman_ℓ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\|M\partial^{\alpha+\beta}u\|_{L^{p}(w_{\widetilde{\gamma}})}" class="ltx_Math" display="inline" id="S6.Ex26.m2.2"><semantics id="S6.Ex26.m2.2a"><msub id="S6.Ex26.m2.2.2" xref="S6.Ex26.m2.2.2.cmml"><mrow id="S6.Ex26.m2.2.2.1.1" xref="S6.Ex26.m2.2.2.1.2.cmml"><mo id="S6.Ex26.m2.2.2.1.1.2" stretchy="false" xref="S6.Ex26.m2.2.2.1.2.1.cmml">‖</mo><mrow id="S6.Ex26.m2.2.2.1.1.1" xref="S6.Ex26.m2.2.2.1.1.1.cmml"><mi id="S6.Ex26.m2.2.2.1.1.1.2" xref="S6.Ex26.m2.2.2.1.1.1.2.cmml">M</mi><mo id="S6.Ex26.m2.2.2.1.1.1.1" lspace="0em" xref="S6.Ex26.m2.2.2.1.1.1.1.cmml">⁢</mo><mrow id="S6.Ex26.m2.2.2.1.1.1.3" xref="S6.Ex26.m2.2.2.1.1.1.3.cmml"><msup id="S6.Ex26.m2.2.2.1.1.1.3.1" xref="S6.Ex26.m2.2.2.1.1.1.3.1.cmml"><mo id="S6.Ex26.m2.2.2.1.1.1.3.1.2" rspace="0em" xref="S6.Ex26.m2.2.2.1.1.1.3.1.2.cmml">∂</mo><mrow id="S6.Ex26.m2.2.2.1.1.1.3.1.3" xref="S6.Ex26.m2.2.2.1.1.1.3.1.3.cmml"><mi id="S6.Ex26.m2.2.2.1.1.1.3.1.3.2" xref="S6.Ex26.m2.2.2.1.1.1.3.1.3.2.cmml">α</mi><mo id="S6.Ex26.m2.2.2.1.1.1.3.1.3.1" xref="S6.Ex26.m2.2.2.1.1.1.3.1.3.1.cmml">+</mo><mi id="S6.Ex26.m2.2.2.1.1.1.3.1.3.3" xref="S6.Ex26.m2.2.2.1.1.1.3.1.3.3.cmml">β</mi></mrow></msup><mi id="S6.Ex26.m2.2.2.1.1.1.3.2" xref="S6.Ex26.m2.2.2.1.1.1.3.2.cmml">u</mi></mrow></mrow><mo id="S6.Ex26.m2.2.2.1.1.3" stretchy="false" xref="S6.Ex26.m2.2.2.1.2.1.cmml">‖</mo></mrow><mrow id="S6.Ex26.m2.1.1.1" xref="S6.Ex26.m2.1.1.1.cmml"><msup id="S6.Ex26.m2.1.1.1.3" xref="S6.Ex26.m2.1.1.1.3.cmml"><mi id="S6.Ex26.m2.1.1.1.3.2" xref="S6.Ex26.m2.1.1.1.3.2.cmml">L</mi><mi id="S6.Ex26.m2.1.1.1.3.3" xref="S6.Ex26.m2.1.1.1.3.3.cmml">p</mi></msup><mo id="S6.Ex26.m2.1.1.1.2" xref="S6.Ex26.m2.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex26.m2.1.1.1.1.1" xref="S6.Ex26.m2.1.1.1.1.1.1.cmml"><mo id="S6.Ex26.m2.1.1.1.1.1.2" stretchy="false" xref="S6.Ex26.m2.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex26.m2.1.1.1.1.1.1" xref="S6.Ex26.m2.1.1.1.1.1.1.cmml"><mi id="S6.Ex26.m2.1.1.1.1.1.1.2" xref="S6.Ex26.m2.1.1.1.1.1.1.2.cmml">w</mi><mover accent="true" id="S6.Ex26.m2.1.1.1.1.1.1.3" xref="S6.Ex26.m2.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex26.m2.1.1.1.1.1.1.3.2" xref="S6.Ex26.m2.1.1.1.1.1.1.3.2.cmml">γ</mi><mo id="S6.Ex26.m2.1.1.1.1.1.1.3.1" xref="S6.Ex26.m2.1.1.1.1.1.1.3.1.cmml">~</mo></mover></msub><mo id="S6.Ex26.m2.1.1.1.1.1.3" stretchy="false" xref="S6.Ex26.m2.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.Ex26.m2.2b"><apply id="S6.Ex26.m2.2.2.cmml" xref="S6.Ex26.m2.2.2"><csymbol cd="ambiguous" id="S6.Ex26.m2.2.2.2.cmml" xref="S6.Ex26.m2.2.2">subscript</csymbol><apply id="S6.Ex26.m2.2.2.1.2.cmml" xref="S6.Ex26.m2.2.2.1.1"><csymbol cd="latexml" id="S6.Ex26.m2.2.2.1.2.1.cmml" xref="S6.Ex26.m2.2.2.1.1.2">norm</csymbol><apply id="S6.Ex26.m2.2.2.1.1.1.cmml" xref="S6.Ex26.m2.2.2.1.1.1"><times id="S6.Ex26.m2.2.2.1.1.1.1.cmml" xref="S6.Ex26.m2.2.2.1.1.1.1"></times><ci id="S6.Ex26.m2.2.2.1.1.1.2.cmml" xref="S6.Ex26.m2.2.2.1.1.1.2">𝑀</ci><apply id="S6.Ex26.m2.2.2.1.1.1.3.cmml" xref="S6.Ex26.m2.2.2.1.1.1.3"><apply id="S6.Ex26.m2.2.2.1.1.1.3.1.cmml" xref="S6.Ex26.m2.2.2.1.1.1.3.1"><csymbol cd="ambiguous" 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xref="S6.Ex26.m2.1.1.1.3.2">𝐿</ci><ci id="S6.Ex26.m2.1.1.1.3.3.cmml" xref="S6.Ex26.m2.1.1.1.3.3">𝑝</ci></apply><apply id="S6.Ex26.m2.1.1.1.1.1.1.cmml" xref="S6.Ex26.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex26.m2.1.1.1.1.1.1.1.cmml" xref="S6.Ex26.m2.1.1.1.1.1">subscript</csymbol><ci id="S6.Ex26.m2.1.1.1.1.1.1.2.cmml" xref="S6.Ex26.m2.1.1.1.1.1.1.2">𝑤</ci><apply id="S6.Ex26.m2.1.1.1.1.1.1.3.cmml" xref="S6.Ex26.m2.1.1.1.1.1.1.3"><ci id="S6.Ex26.m2.1.1.1.1.1.1.3.1.cmml" xref="S6.Ex26.m2.1.1.1.1.1.1.3.1">~</ci><ci id="S6.Ex26.m2.1.1.1.1.1.1.3.2.cmml" xref="S6.Ex26.m2.1.1.1.1.1.1.3.2">𝛾</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex26.m2.2c">\displaystyle\|M\partial^{\alpha+\beta}u\|_{L^{p}(w_{\widetilde{\gamma}})}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex26.m2.2d">∥ italic_M ∂ start_POSTSUPERSCRIPT italic_α + italic_β end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex27"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S6.Ex27.m1.1"><semantics id="S6.Ex27.m1.1a"><mo id="S6.Ex27.m1.1.1" xref="S6.Ex27.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S6.Ex27.m1.1b"><leq id="S6.Ex27.m1.1.1.cmml" xref="S6.Ex27.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex27.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S6.Ex27.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;\sum_{|\alpha|=1}\|M\partial^{\alpha}u\|_{W^{\widetilde{k}+\ell% ,p}(w_{\widetilde{\gamma}})}+\sum_{\begin{subarray}{c}|\beta|=\widetilde{k}+% \ell\\ \beta_{1}>0\end{subarray}}\|\partial^{\beta}M\partial^{\alpha}u-\beta_{1}% \partial_{1}^{\beta_{1}-1}\partial^{\widetilde{\beta}+\alpha}u\|_{L^{p}(w_{% \widetilde{\gamma}})}" class="ltx_Math" display="inline" id="S6.Ex27.m2.8"><semantics id="S6.Ex27.m2.8a"><mrow id="S6.Ex27.m2.8.8" xref="S6.Ex27.m2.8.8.cmml"><mrow id="S6.Ex27.m2.7.7.1" xref="S6.Ex27.m2.7.7.1.cmml"><mstyle displaystyle="true" id="S6.Ex27.m2.7.7.1.2" xref="S6.Ex27.m2.7.7.1.2.cmml"><munder id="S6.Ex27.m2.7.7.1.2a" xref="S6.Ex27.m2.7.7.1.2.cmml"><mo id="S6.Ex27.m2.7.7.1.2.2" movablelimits="false" xref="S6.Ex27.m2.7.7.1.2.2.cmml">∑</mo><mrow id="S6.Ex27.m2.2.2.1" xref="S6.Ex27.m2.2.2.1.cmml"><mrow id="S6.Ex27.m2.2.2.1.3.2" xref="S6.Ex27.m2.2.2.1.3.1.cmml"><mo id="S6.Ex27.m2.2.2.1.3.2.1" stretchy="false" xref="S6.Ex27.m2.2.2.1.3.1.1.cmml">|</mo><mi 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xref="S6.Ex27.m2.4.4.2.2.2.2.1.1.cmml">+</mo><mi id="S6.Ex27.m2.4.4.2.2.2.2.1.3" mathvariant="normal" xref="S6.Ex27.m2.4.4.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.Ex27.m2.4.4.2.2.2.2.2" xref="S6.Ex27.m2.4.4.2.2.2.3.cmml">,</mo><mi id="S6.Ex27.m2.3.3.1.1.1.1" xref="S6.Ex27.m2.3.3.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.Ex27.m2.5.5.3.4" xref="S6.Ex27.m2.5.5.3.4.cmml">⁢</mo><mrow id="S6.Ex27.m2.5.5.3.3.1" xref="S6.Ex27.m2.5.5.3.3.1.1.cmml"><mo id="S6.Ex27.m2.5.5.3.3.1.2" stretchy="false" xref="S6.Ex27.m2.5.5.3.3.1.1.cmml">(</mo><msub id="S6.Ex27.m2.5.5.3.3.1.1" xref="S6.Ex27.m2.5.5.3.3.1.1.cmml"><mi id="S6.Ex27.m2.5.5.3.3.1.1.2" xref="S6.Ex27.m2.5.5.3.3.1.1.2.cmml">w</mi><mover accent="true" id="S6.Ex27.m2.5.5.3.3.1.1.3" xref="S6.Ex27.m2.5.5.3.3.1.1.3.cmml"><mi id="S6.Ex27.m2.5.5.3.3.1.1.3.2" xref="S6.Ex27.m2.5.5.3.3.1.1.3.2.cmml">γ</mi><mo id="S6.Ex27.m2.5.5.3.3.1.1.3.1" xref="S6.Ex27.m2.5.5.3.3.1.1.3.1.cmml">~</mo></mover></msub><mo id="S6.Ex27.m2.5.5.3.3.1.3" stretchy="false" 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xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.2.3.cmml">1</mn><mrow id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.cmml"><msub id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.2" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.2.cmml"><mi id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.2.2" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.2.2.cmml">β</mi><mn id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.2.3" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.2.3.cmml">1</mn></msub><mo id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.1" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.1.cmml">−</mo><mn id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.3" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.1.3.3.cmml">1</mn></mrow></msubsup><mrow id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.2" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.2.cmml"><msup id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.2.1" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.2.1.cmml"><mo id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.2.1.2" lspace="0em" rspace="0em" xref="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.2.1.2.cmml">∂</mo><mrow id="S6.Ex27.m2.8.8.2.1.1.1.1.3.3.2.1.3" 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xref="S6.Ex27.m2.6.6.1.1.1">subscript</csymbol><ci id="S6.Ex27.m2.6.6.1.1.1.1.2.cmml" xref="S6.Ex27.m2.6.6.1.1.1.1.2">𝑤</ci><apply id="S6.Ex27.m2.6.6.1.1.1.1.3.cmml" xref="S6.Ex27.m2.6.6.1.1.1.1.3"><ci id="S6.Ex27.m2.6.6.1.1.1.1.3.1.cmml" xref="S6.Ex27.m2.6.6.1.1.1.1.3.1">~</ci><ci id="S6.Ex27.m2.6.6.1.1.1.1.3.2.cmml" xref="S6.Ex27.m2.6.6.1.1.1.1.3.2">𝛾</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex27.m2.8c">\displaystyle\;\sum_{|\alpha|=1}\|M\partial^{\alpha}u\|_{W^{\widetilde{k}+\ell% ,p}(w_{\widetilde{\gamma}})}+\sum_{\begin{subarray}{c}|\beta|=\widetilde{k}+% \ell\\ \beta_{1}>0\end{subarray}}\|\partial^{\beta}M\partial^{\alpha}u-\beta_{1}% \partial_{1}^{\beta_{1}-1}\partial^{\widetilde{\beta}+\alpha}u\|_{L^{p}(w_{% \widetilde{\gamma}})}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex27.m2.8d">∑ start_POSTSUBSCRIPT | italic_α | = 1 end_POSTSUBSCRIPT ∥ italic_M ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT + ∑ start_POSTSUBSCRIPT start_ARG start_ROW start_CELL | italic_β | = over~ start_ARG italic_k end_ARG + roman_ℓ end_CELL end_ROW start_ROW start_CELL italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > 0 end_CELL end_ROW end_ARG end_POSTSUBSCRIPT ∥ ∂ start_POSTSUPERSCRIPT italic_β end_POSTSUPERSCRIPT italic_M ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_u - italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∂ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_β start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT ∂ start_POSTSUPERSCRIPT over~ start_ARG italic_β end_ARG + italic_α end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex28"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S6.Ex28.m1.1"><semantics id="S6.Ex28.m1.1a"><mo id="S6.Ex28.m1.1.1" xref="S6.Ex28.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S6.Ex28.m1.1b"><leq id="S6.Ex28.m1.1.1.cmml" xref="S6.Ex28.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex28.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S6.Ex28.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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xref="S6.Ex28.m2.7.7.3.3.1.1.3.2">𝛾</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex28.m2.9c">\displaystyle\;C\Big{(}\sum_{|\alpha|=1}\|M\partial^{\alpha}u\|_{W^{\widetilde% {k}+\ell,p}(w_{\widetilde{\gamma}})}+\|u\|_{W^{\widetilde{k}+\ell,p}(w_{% \widetilde{\gamma}})}\Big{)}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex28.m2.9d">italic_C ( ∑ start_POSTSUBSCRIPT | italic_α | = 1 end_POSTSUBSCRIPT ∥ italic_M ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT + ∥ italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex29"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S6.Ex29.m1.1"><semantics id="S6.Ex29.m1.1a"><mo id="S6.Ex29.m1.1.1" xref="S6.Ex29.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S6.Ex29.m1.1b"><leq id="S6.Ex29.m1.1.1.cmml" xref="S6.Ex29.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex29.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S6.Ex29.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;C\Big{(}\sum_{|\alpha|=1}\|M\partial^{\alpha}u\|_{[W^{% 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id="S6.Ex29.m2.13.13.6.8.3.1.cmml" xref="S6.Ex29.m2.13.13.6.8.3">subscript</csymbol><ci id="S6.Ex29.m2.13.13.6.8.3.2.cmml" xref="S6.Ex29.m2.13.13.6.8.3.2">𝑘</ci><cn id="S6.Ex29.m2.13.13.6.8.3.3.cmml" type="integer" xref="S6.Ex29.m2.13.13.6.8.3.3">1</cn></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex29.m2.15c">\displaystyle\;C\Big{(}\sum_{|\alpha|=1}\|M\partial^{\alpha}u\|_{[W^{% \widetilde{k},p}(w_{\widetilde{\gamma}}),W^{\widetilde{k}+k_{1},p}(w_{% \widetilde{\gamma}})]_{\frac{\ell}{k_{1}}}}+\|u\|_{[W^{\widetilde{k},p}(w_{% \widetilde{\gamma}}),W^{\widetilde{k}+k_{1},p}(w_{\widetilde{\gamma}})]_{\frac% {\ell}{k_{1}}}}\Big{)}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex29.m2.15d">italic_C ( ∑ start_POSTSUBSCRIPT | italic_α | = 1 end_POSTSUBSCRIPT ∥ italic_M ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT italic_u ∥ start_POSTSUBSCRIPT [ italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT + ∥ italic_u ∥ start_POSTSUBSCRIPT [ italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex30"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\leq" class="ltx_Math" display="inline" id="S6.Ex30.m1.1"><semantics id="S6.Ex30.m1.1a"><mo id="S6.Ex30.m1.1.1" xref="S6.Ex30.m1.1.1.cmml">≤</mo><annotation-xml encoding="MathML-Content" id="S6.Ex30.m1.1b"><leq id="S6.Ex30.m1.1.1.cmml" xref="S6.Ex30.m1.1.1"></leq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex30.m1.1c">\displaystyle\leq</annotation><annotation encoding="application/x-llamapun" id="S6.Ex30.m1.1d">≤</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;C\|u\|_{[W^{\widetilde{k}+1,p}(w_{\widetilde{\gamma}+p}),W^{% \widetilde{k}+k_{1}+1,p}(w_{\widetilde{\gamma}+p})]_{\frac{\ell}{k_{1}}}}\Big{% )}," class="ltx_math_unparsed" display="inline" id="S6.Ex30.m2.7"><semantics id="S6.Ex30.m2.7a"><mrow id="S6.Ex30.m2.7b"><mi id="S6.Ex30.m2.7.8">C</mi><mo id="S6.Ex30.m2.7.9" lspace="0em" rspace="0.167em">∥</mo><mi id="S6.Ex30.m2.7.7">u</mi><msub id="S6.Ex30.m2.7.10"><mo id="S6.Ex30.m2.7.10.2" lspace="0em" rspace="0.167em">∥</mo><msub id="S6.Ex30.m2.6.6.6"><mrow id="S6.Ex30.m2.6.6.6.6.2"><mo id="S6.Ex30.m2.6.6.6.6.2.3" stretchy="false">[</mo><mrow id="S6.Ex30.m2.5.5.5.5.1.1"><msup id="S6.Ex30.m2.5.5.5.5.1.1.3"><mi id="S6.Ex30.m2.5.5.5.5.1.1.3.2">W</mi><mrow id="S6.Ex30.m2.2.2.2.2.2.2"><mrow id="S6.Ex30.m2.2.2.2.2.2.2.1"><mover accent="true" id="S6.Ex30.m2.2.2.2.2.2.2.1.2"><mi id="S6.Ex30.m2.2.2.2.2.2.2.1.2.2">k</mi><mo id="S6.Ex30.m2.2.2.2.2.2.2.1.2.1">~</mo></mover><mo id="S6.Ex30.m2.2.2.2.2.2.2.1.1">+</mo><mn id="S6.Ex30.m2.2.2.2.2.2.2.1.3">1</mn></mrow><mo id="S6.Ex30.m2.2.2.2.2.2.2.2">,</mo><mi id="S6.Ex30.m2.1.1.1.1.1.1">p</mi></mrow></msup><mo id="S6.Ex30.m2.5.5.5.5.1.1.2">⁢</mo><mrow id="S6.Ex30.m2.5.5.5.5.1.1.1.1"><mo id="S6.Ex30.m2.5.5.5.5.1.1.1.1.2" stretchy="false">(</mo><msub id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1"><mi id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1.2">w</mi><mrow id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1.3"><mover accent="true" id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1.3.2"><mi id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1.3.2.2">γ</mi><mo id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1.3.2.1">~</mo></mover><mo id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1.3.1">+</mo><mi id="S6.Ex30.m2.5.5.5.5.1.1.1.1.1.3.3">p</mi></mrow></msub><mo id="S6.Ex30.m2.5.5.5.5.1.1.1.1.3" stretchy="false">)</mo></mrow></mrow><mo id="S6.Ex30.m2.6.6.6.6.2.4">,</mo><mrow id="S6.Ex30.m2.6.6.6.6.2.2"><msup id="S6.Ex30.m2.6.6.6.6.2.2.3"><mi id="S6.Ex30.m2.6.6.6.6.2.2.3.2">W</mi><mrow id="S6.Ex30.m2.4.4.4.4.2.2"><mrow id="S6.Ex30.m2.4.4.4.4.2.2.1"><mover accent="true" 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id="S6.Ex30.m2.6.6.6.6.2.2.1.1.1.3.3">p</mi></mrow></msub><mo id="S6.Ex30.m2.6.6.6.6.2.2.1.1.3" stretchy="false">)</mo></mrow></mrow><mo id="S6.Ex30.m2.6.6.6.6.2.5" stretchy="false">]</mo></mrow><mfrac id="S6.Ex30.m2.6.6.6.8"><mi id="S6.Ex30.m2.6.6.6.8.2" mathvariant="normal">ℓ</mi><msub id="S6.Ex30.m2.6.6.6.8.3"><mi id="S6.Ex30.m2.6.6.6.8.3.2">k</mi><mn id="S6.Ex30.m2.6.6.6.8.3.3">1</mn></msub></mfrac></msub></msub><mo id="S6.Ex30.m2.7.11" maxsize="160%" minsize="160%">)</mo><mo id="S6.Ex30.m2.7.12">,</mo></mrow><annotation encoding="application/x-tex" id="S6.Ex30.m2.7c">\displaystyle\;C\|u\|_{[W^{\widetilde{k}+1,p}(w_{\widetilde{\gamma}+p}),W^{% \widetilde{k}+k_{1}+1,p}(w_{\widetilde{\gamma}+p})]_{\frac{\ell}{k_{1}}}}\Big{% )},</annotation><annotation encoding="application/x-llamapun" id="S6.Ex30.m2.7d">italic_C ∥ italic_u ∥ start_POSTSUBSCRIPT [ italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + 1 , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.7.p2.12">where the last estimate follows from the fact that for <math alttext="|\alpha|=1" class="ltx_Math" display="inline" id="S6.SS1.7.p2.12.m1.1"><semantics id="S6.SS1.7.p2.12.m1.1a"><mrow id="S6.SS1.7.p2.12.m1.1.2" xref="S6.SS1.7.p2.12.m1.1.2.cmml"><mrow id="S6.SS1.7.p2.12.m1.1.2.2.2" xref="S6.SS1.7.p2.12.m1.1.2.2.1.cmml"><mo id="S6.SS1.7.p2.12.m1.1.2.2.2.1" stretchy="false" xref="S6.SS1.7.p2.12.m1.1.2.2.1.1.cmml">|</mo><mi id="S6.SS1.7.p2.12.m1.1.1" xref="S6.SS1.7.p2.12.m1.1.1.cmml">α</mi><mo id="S6.SS1.7.p2.12.m1.1.2.2.2.2" stretchy="false" xref="S6.SS1.7.p2.12.m1.1.2.2.1.1.cmml">|</mo></mrow><mo id="S6.SS1.7.p2.12.m1.1.2.1" xref="S6.SS1.7.p2.12.m1.1.2.1.cmml">=</mo><mn id="S6.SS1.7.p2.12.m1.1.2.3" xref="S6.SS1.7.p2.12.m1.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.12.m1.1b"><apply id="S6.SS1.7.p2.12.m1.1.2.cmml" xref="S6.SS1.7.p2.12.m1.1.2"><eq id="S6.SS1.7.p2.12.m1.1.2.1.cmml" xref="S6.SS1.7.p2.12.m1.1.2.1"></eq><apply id="S6.SS1.7.p2.12.m1.1.2.2.1.cmml" xref="S6.SS1.7.p2.12.m1.1.2.2.2"><abs id="S6.SS1.7.p2.12.m1.1.2.2.1.1.cmml" xref="S6.SS1.7.p2.12.m1.1.2.2.2.1"></abs><ci id="S6.SS1.7.p2.12.m1.1.1.cmml" xref="S6.SS1.7.p2.12.m1.1.1">𝛼</ci></apply><cn id="S6.SS1.7.p2.12.m1.1.2.3.cmml" type="integer" xref="S6.SS1.7.p2.12.m1.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.12.m1.1c">|\alpha|=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.12.m1.1d">| italic_α | = 1</annotation></semantics></math> the operators</p> <table class="ltx_equationgroup ltx_eqn_table" id="S6.E3"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S6.E3X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle M\partial^{\alpha},\operatorname{id}" class="ltx_Math" display="inline" id="S6.E3X.2.1.1.m1.2"><semantics id="S6.E3X.2.1.1.m1.2a"><mrow id="S6.E3X.2.1.1.m1.2.2.1" xref="S6.E3X.2.1.1.m1.2.2.2.cmml"><mrow id="S6.E3X.2.1.1.m1.2.2.1.1" xref="S6.E3X.2.1.1.m1.2.2.1.1.cmml"><mi id="S6.E3X.2.1.1.m1.2.2.1.1.2" xref="S6.E3X.2.1.1.m1.2.2.1.1.2.cmml">M</mi><mo id="S6.E3X.2.1.1.m1.2.2.1.1.1" lspace="0em" xref="S6.E3X.2.1.1.m1.2.2.1.1.1.cmml">⁢</mo><msup id="S6.E3X.2.1.1.m1.2.2.1.1.3" xref="S6.E3X.2.1.1.m1.2.2.1.1.3.cmml"><mo id="S6.E3X.2.1.1.m1.2.2.1.1.3.2" rspace="0em" xref="S6.E3X.2.1.1.m1.2.2.1.1.3.2.cmml">∂</mo><mi id="S6.E3X.2.1.1.m1.2.2.1.1.3.3" xref="S6.E3X.2.1.1.m1.2.2.1.1.3.3.cmml">α</mi></msup></mrow><mo id="S6.E3X.2.1.1.m1.2.2.1.2" xref="S6.E3X.2.1.1.m1.2.2.2.cmml">,</mo><mi id="S6.E3X.2.1.1.m1.1.1" xref="S6.E3X.2.1.1.m1.1.1.cmml">id</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.E3X.2.1.1.m1.2b"><list id="S6.E3X.2.1.1.m1.2.2.2.cmml" xref="S6.E3X.2.1.1.m1.2.2.1"><apply id="S6.E3X.2.1.1.m1.2.2.1.1.cmml" xref="S6.E3X.2.1.1.m1.2.2.1.1"><times id="S6.E3X.2.1.1.m1.2.2.1.1.1.cmml" xref="S6.E3X.2.1.1.m1.2.2.1.1.1"></times><ci id="S6.E3X.2.1.1.m1.2.2.1.1.2.cmml" xref="S6.E3X.2.1.1.m1.2.2.1.1.2">𝑀</ci><apply id="S6.E3X.2.1.1.m1.2.2.1.1.3.cmml" xref="S6.E3X.2.1.1.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.E3X.2.1.1.m1.2.2.1.1.3.1.cmml" xref="S6.E3X.2.1.1.m1.2.2.1.1.3">superscript</csymbol><partialdiff id="S6.E3X.2.1.1.m1.2.2.1.1.3.2.cmml" xref="S6.E3X.2.1.1.m1.2.2.1.1.3.2"></partialdiff><ci id="S6.E3X.2.1.1.m1.2.2.1.1.3.3.cmml" xref="S6.E3X.2.1.1.m1.2.2.1.1.3.3">𝛼</ci></apply></apply><ci id="S6.E3X.2.1.1.m1.1.1.cmml" xref="S6.E3X.2.1.1.m1.1.1">id</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S6.E3X.2.1.1.m1.2c">\displaystyle M\partial^{\alpha},\operatorname{id}</annotation><annotation encoding="application/x-llamapun" id="S6.E3X.2.1.1.m1.2d">italic_M ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT , roman_id</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle:W^{\widetilde{k}+1,p}(w_{\widetilde{\gamma}+p})\to W^{\widetilde% {k},p}(w_{\widetilde{\gamma}})\quad\text{ and }" class="ltx_Math" display="inline" id="S6.E3X.3.2.2.m1.7"><semantics id="S6.E3X.3.2.2.m1.7a"><mrow id="S6.E3X.3.2.2.m1.7.7" xref="S6.E3X.3.2.2.m1.7.7.cmml"><mi id="S6.E3X.3.2.2.m1.7.7.4" xref="S6.E3X.3.2.2.m1.7.7.4.cmml"></mi><mo id="S6.E3X.3.2.2.m1.7.7.3" lspace="0.278em" rspace="0.278em" 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start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + 1 , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) → italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT ) and</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(6.3)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S6.E3Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle M\partial^{\alpha},\operatorname{id}" class="ltx_Math" display="inline" id="S6.E3Xa.2.1.1.m1.2"><semantics id="S6.E3Xa.2.1.1.m1.2a"><mrow id="S6.E3Xa.2.1.1.m1.2.2.1" xref="S6.E3Xa.2.1.1.m1.2.2.2.cmml"><mrow id="S6.E3Xa.2.1.1.m1.2.2.1.1" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.cmml"><mi id="S6.E3Xa.2.1.1.m1.2.2.1.1.2" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.2.cmml">M</mi><mo id="S6.E3Xa.2.1.1.m1.2.2.1.1.1" lspace="0em" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.1.cmml">⁢</mo><msup id="S6.E3Xa.2.1.1.m1.2.2.1.1.3" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.3.cmml"><mo id="S6.E3Xa.2.1.1.m1.2.2.1.1.3.2" rspace="0em" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.3.2.cmml">∂</mo><mi id="S6.E3Xa.2.1.1.m1.2.2.1.1.3.3" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.3.3.cmml">α</mi></msup></mrow><mo id="S6.E3Xa.2.1.1.m1.2.2.1.2" xref="S6.E3Xa.2.1.1.m1.2.2.2.cmml">,</mo><mi id="S6.E3Xa.2.1.1.m1.1.1" xref="S6.E3Xa.2.1.1.m1.1.1.cmml">id</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.E3Xa.2.1.1.m1.2b"><list id="S6.E3Xa.2.1.1.m1.2.2.2.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1"><apply id="S6.E3Xa.2.1.1.m1.2.2.1.1.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1.1"><times id="S6.E3Xa.2.1.1.m1.2.2.1.1.1.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.1"></times><ci id="S6.E3Xa.2.1.1.m1.2.2.1.1.2.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.2">𝑀</ci><apply id="S6.E3Xa.2.1.1.m1.2.2.1.1.3.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.E3Xa.2.1.1.m1.2.2.1.1.3.1.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.3">superscript</csymbol><partialdiff id="S6.E3Xa.2.1.1.m1.2.2.1.1.3.2.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.3.2"></partialdiff><ci id="S6.E3Xa.2.1.1.m1.2.2.1.1.3.3.cmml" xref="S6.E3Xa.2.1.1.m1.2.2.1.1.3.3">𝛼</ci></apply></apply><ci id="S6.E3Xa.2.1.1.m1.1.1.cmml" xref="S6.E3Xa.2.1.1.m1.1.1">id</ci></list></annotation-xml><annotation encoding="application/x-tex" id="S6.E3Xa.2.1.1.m1.2c">\displaystyle M\partial^{\alpha},\operatorname{id}</annotation><annotation encoding="application/x-llamapun" id="S6.E3Xa.2.1.1.m1.2d">italic_M ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT , roman_id</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.1.cmml" xref="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1">subscript</csymbol><ci id="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.2.cmml" xref="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.2">𝑤</ci><apply id="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.3.cmml" xref="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.3"><ci id="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.3.1.cmml" xref="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.3.1">~</ci><ci id="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.3.2.cmml" xref="S6.E3Xa.3.2.2.m1.6.6.2.2.1.1.1.3.2">𝛾</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E3Xa.3.2.2.m1.6c">\displaystyle:W^{\widetilde{k}+k_{1}+1,p}(w_{\widetilde{\gamma}+p})\to W^{% \widetilde{k}+k_{1},p}(w_{\widetilde{\gamma}})</annotation><annotation encoding="application/x-llamapun" id="S6.E3Xa.3.2.2.m1.6d">: italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) → italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG end_POSTSUBSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S6.SS1.7.p2.16">are bounded. Indeed, from Hardy’s inequality (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Corollary 3.3]</cite> using <math alttext="\widetilde{\gamma}>j_{1}p-1\geq-1" class="ltx_Math" display="inline" id="S6.SS1.7.p2.13.m1.1"><semantics id="S6.SS1.7.p2.13.m1.1a"><mrow id="S6.SS1.7.p2.13.m1.1.1" xref="S6.SS1.7.p2.13.m1.1.1.cmml"><mover accent="true" id="S6.SS1.7.p2.13.m1.1.1.2" xref="S6.SS1.7.p2.13.m1.1.1.2.cmml"><mi id="S6.SS1.7.p2.13.m1.1.1.2.2" xref="S6.SS1.7.p2.13.m1.1.1.2.2.cmml">γ</mi><mo id="S6.SS1.7.p2.13.m1.1.1.2.1" xref="S6.SS1.7.p2.13.m1.1.1.2.1.cmml">~</mo></mover><mo id="S6.SS1.7.p2.13.m1.1.1.3" xref="S6.SS1.7.p2.13.m1.1.1.3.cmml">></mo><mrow id="S6.SS1.7.p2.13.m1.1.1.4" xref="S6.SS1.7.p2.13.m1.1.1.4.cmml"><mrow id="S6.SS1.7.p2.13.m1.1.1.4.2" xref="S6.SS1.7.p2.13.m1.1.1.4.2.cmml"><msub id="S6.SS1.7.p2.13.m1.1.1.4.2.2" xref="S6.SS1.7.p2.13.m1.1.1.4.2.2.cmml"><mi id="S6.SS1.7.p2.13.m1.1.1.4.2.2.2" xref="S6.SS1.7.p2.13.m1.1.1.4.2.2.2.cmml">j</mi><mn id="S6.SS1.7.p2.13.m1.1.1.4.2.2.3" xref="S6.SS1.7.p2.13.m1.1.1.4.2.2.3.cmml">1</mn></msub><mo id="S6.SS1.7.p2.13.m1.1.1.4.2.1" xref="S6.SS1.7.p2.13.m1.1.1.4.2.1.cmml">⁢</mo><mi id="S6.SS1.7.p2.13.m1.1.1.4.2.3" xref="S6.SS1.7.p2.13.m1.1.1.4.2.3.cmml">p</mi></mrow><mo id="S6.SS1.7.p2.13.m1.1.1.4.1" xref="S6.SS1.7.p2.13.m1.1.1.4.1.cmml">−</mo><mn id="S6.SS1.7.p2.13.m1.1.1.4.3" xref="S6.SS1.7.p2.13.m1.1.1.4.3.cmml">1</mn></mrow><mo id="S6.SS1.7.p2.13.m1.1.1.5" xref="S6.SS1.7.p2.13.m1.1.1.5.cmml">≥</mo><mrow id="S6.SS1.7.p2.13.m1.1.1.6" xref="S6.SS1.7.p2.13.m1.1.1.6.cmml"><mo id="S6.SS1.7.p2.13.m1.1.1.6a" xref="S6.SS1.7.p2.13.m1.1.1.6.cmml">−</mo><mn id="S6.SS1.7.p2.13.m1.1.1.6.2" xref="S6.SS1.7.p2.13.m1.1.1.6.2.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.13.m1.1b"><apply id="S6.SS1.7.p2.13.m1.1.1.cmml" xref="S6.SS1.7.p2.13.m1.1.1"><and id="S6.SS1.7.p2.13.m1.1.1a.cmml" xref="S6.SS1.7.p2.13.m1.1.1"></and><apply id="S6.SS1.7.p2.13.m1.1.1b.cmml" xref="S6.SS1.7.p2.13.m1.1.1"><gt id="S6.SS1.7.p2.13.m1.1.1.3.cmml" xref="S6.SS1.7.p2.13.m1.1.1.3"></gt><apply id="S6.SS1.7.p2.13.m1.1.1.2.cmml" xref="S6.SS1.7.p2.13.m1.1.1.2"><ci id="S6.SS1.7.p2.13.m1.1.1.2.1.cmml" xref="S6.SS1.7.p2.13.m1.1.1.2.1">~</ci><ci id="S6.SS1.7.p2.13.m1.1.1.2.2.cmml" xref="S6.SS1.7.p2.13.m1.1.1.2.2">𝛾</ci></apply><apply id="S6.SS1.7.p2.13.m1.1.1.4.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4"><minus id="S6.SS1.7.p2.13.m1.1.1.4.1.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4.1"></minus><apply id="S6.SS1.7.p2.13.m1.1.1.4.2.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4.2"><times id="S6.SS1.7.p2.13.m1.1.1.4.2.1.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4.2.1"></times><apply id="S6.SS1.7.p2.13.m1.1.1.4.2.2.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4.2.2"><csymbol cd="ambiguous" id="S6.SS1.7.p2.13.m1.1.1.4.2.2.1.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4.2.2">subscript</csymbol><ci id="S6.SS1.7.p2.13.m1.1.1.4.2.2.2.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4.2.2.2">𝑗</ci><cn id="S6.SS1.7.p2.13.m1.1.1.4.2.2.3.cmml" type="integer" xref="S6.SS1.7.p2.13.m1.1.1.4.2.2.3">1</cn></apply><ci id="S6.SS1.7.p2.13.m1.1.1.4.2.3.cmml" xref="S6.SS1.7.p2.13.m1.1.1.4.2.3">𝑝</ci></apply><cn id="S6.SS1.7.p2.13.m1.1.1.4.3.cmml" type="integer" xref="S6.SS1.7.p2.13.m1.1.1.4.3">1</cn></apply></apply><apply id="S6.SS1.7.p2.13.m1.1.1c.cmml" xref="S6.SS1.7.p2.13.m1.1.1"><geq id="S6.SS1.7.p2.13.m1.1.1.5.cmml" xref="S6.SS1.7.p2.13.m1.1.1.5"></geq><share href="https://arxiv.org/html/2503.14636v1#S6.SS1.7.p2.13.m1.1.1.4.cmml" id="S6.SS1.7.p2.13.m1.1.1d.cmml" xref="S6.SS1.7.p2.13.m1.1.1"></share><apply id="S6.SS1.7.p2.13.m1.1.1.6.cmml" xref="S6.SS1.7.p2.13.m1.1.1.6"><minus id="S6.SS1.7.p2.13.m1.1.1.6.1.cmml" xref="S6.SS1.7.p2.13.m1.1.1.6"></minus><cn id="S6.SS1.7.p2.13.m1.1.1.6.2.cmml" type="integer" xref="S6.SS1.7.p2.13.m1.1.1.6.2">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.13.m1.1c">\widetilde{\gamma}>j_{1}p-1\geq-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.13.m1.1d">over~ start_ARG italic_γ end_ARG > italic_j start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_p - 1 ≥ - 1</annotation></semantics></math>) it follows that the operator <math alttext="\operatorname{id}" class="ltx_Math" display="inline" id="S6.SS1.7.p2.14.m2.1"><semantics id="S6.SS1.7.p2.14.m2.1a"><mi id="S6.SS1.7.p2.14.m2.1.1" xref="S6.SS1.7.p2.14.m2.1.1.cmml">id</mi><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.14.m2.1b"><ci id="S6.SS1.7.p2.14.m2.1.1.cmml" xref="S6.SS1.7.p2.14.m2.1.1">id</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.14.m2.1c">\operatorname{id}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.14.m2.1d">roman_id</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E3" title="In Proof. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a>) is bounded. Furthermore, by <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib49" title="">49</a>, Lemma 3.6]</cite> it follows that the operator <math alttext="M\partial^{\alpha}" class="ltx_Math" display="inline" id="S6.SS1.7.p2.15.m3.1"><semantics id="S6.SS1.7.p2.15.m3.1a"><mrow id="S6.SS1.7.p2.15.m3.1.1" xref="S6.SS1.7.p2.15.m3.1.1.cmml"><mi id="S6.SS1.7.p2.15.m3.1.1.2" xref="S6.SS1.7.p2.15.m3.1.1.2.cmml">M</mi><mo id="S6.SS1.7.p2.15.m3.1.1.1" lspace="0em" xref="S6.SS1.7.p2.15.m3.1.1.1.cmml">⁢</mo><msup id="S6.SS1.7.p2.15.m3.1.1.3" xref="S6.SS1.7.p2.15.m3.1.1.3.cmml"><mo id="S6.SS1.7.p2.15.m3.1.1.3.2" xref="S6.SS1.7.p2.15.m3.1.1.3.2.cmml">∂</mo><mi id="S6.SS1.7.p2.15.m3.1.1.3.3" xref="S6.SS1.7.p2.15.m3.1.1.3.3.cmml">α</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.15.m3.1b"><apply id="S6.SS1.7.p2.15.m3.1.1.cmml" xref="S6.SS1.7.p2.15.m3.1.1"><times id="S6.SS1.7.p2.15.m3.1.1.1.cmml" xref="S6.SS1.7.p2.15.m3.1.1.1"></times><ci id="S6.SS1.7.p2.15.m3.1.1.2.cmml" xref="S6.SS1.7.p2.15.m3.1.1.2">𝑀</ci><apply id="S6.SS1.7.p2.15.m3.1.1.3.cmml" xref="S6.SS1.7.p2.15.m3.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.7.p2.15.m3.1.1.3.1.cmml" xref="S6.SS1.7.p2.15.m3.1.1.3">superscript</csymbol><partialdiff id="S6.SS1.7.p2.15.m3.1.1.3.2.cmml" xref="S6.SS1.7.p2.15.m3.1.1.3.2"></partialdiff><ci id="S6.SS1.7.p2.15.m3.1.1.3.3.cmml" xref="S6.SS1.7.p2.15.m3.1.1.3.3">𝛼</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.15.m3.1c">M\partial^{\alpha}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.15.m3.1d">italic_M ∂ start_POSTSUPERSCRIPT italic_α end_POSTSUPERSCRIPT</annotation></semantics></math> with <math alttext="|\alpha|=1" class="ltx_Math" display="inline" id="S6.SS1.7.p2.16.m4.1"><semantics id="S6.SS1.7.p2.16.m4.1a"><mrow id="S6.SS1.7.p2.16.m4.1.2" xref="S6.SS1.7.p2.16.m4.1.2.cmml"><mrow id="S6.SS1.7.p2.16.m4.1.2.2.2" xref="S6.SS1.7.p2.16.m4.1.2.2.1.cmml"><mo id="S6.SS1.7.p2.16.m4.1.2.2.2.1" stretchy="false" xref="S6.SS1.7.p2.16.m4.1.2.2.1.1.cmml">|</mo><mi id="S6.SS1.7.p2.16.m4.1.1" xref="S6.SS1.7.p2.16.m4.1.1.cmml">α</mi><mo id="S6.SS1.7.p2.16.m4.1.2.2.2.2" stretchy="false" xref="S6.SS1.7.p2.16.m4.1.2.2.1.1.cmml">|</mo></mrow><mo id="S6.SS1.7.p2.16.m4.1.2.1" xref="S6.SS1.7.p2.16.m4.1.2.1.cmml">=</mo><mn id="S6.SS1.7.p2.16.m4.1.2.3" xref="S6.SS1.7.p2.16.m4.1.2.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.7.p2.16.m4.1b"><apply id="S6.SS1.7.p2.16.m4.1.2.cmml" xref="S6.SS1.7.p2.16.m4.1.2"><eq id="S6.SS1.7.p2.16.m4.1.2.1.cmml" xref="S6.SS1.7.p2.16.m4.1.2.1"></eq><apply id="S6.SS1.7.p2.16.m4.1.2.2.1.cmml" xref="S6.SS1.7.p2.16.m4.1.2.2.2"><abs id="S6.SS1.7.p2.16.m4.1.2.2.1.1.cmml" xref="S6.SS1.7.p2.16.m4.1.2.2.2.1"></abs><ci id="S6.SS1.7.p2.16.m4.1.1.cmml" xref="S6.SS1.7.p2.16.m4.1.1">𝛼</ci></apply><cn id="S6.SS1.7.p2.16.m4.1.2.3.cmml" type="integer" xref="S6.SS1.7.p2.16.m4.1.2.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.7.p2.16.m4.1c">|\alpha|=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.7.p2.16.m4.1d">| italic_α | = 1</annotation></semantics></math> in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E3" title="In Proof. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a>) is bounded.</p> </div> <div class="ltx_para" id="S6.SS1.8.p3"> <p class="ltx_p" id="S6.SS1.8.p3.2">Thus, we have proved</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex31"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\|u\|_{W^{\widetilde{k}+\ell+1}(w_{\widetilde{\gamma}+p})}\leq C\|u\|_{[W^{% \widetilde{k}+1,p}(w_{\widetilde{\gamma}+p}),W^{\widetilde{k}+k_{1}+1,p}(w_{% \widetilde{\gamma}+p})]_{\frac{\ell}{k_{1}}}},\quad u\in W^{\widetilde{k}+k_{1% }+1}(w_{\widetilde{\gamma}+p})," class="ltx_Math" display="block" id="S6.Ex31.m1.10"><semantics id="S6.Ex31.m1.10a"><mrow id="S6.Ex31.m1.10.10.1"><mrow id="S6.Ex31.m1.10.10.1.1.2" xref="S6.Ex31.m1.10.10.1.1.3.cmml"><mrow id="S6.Ex31.m1.10.10.1.1.1.1" xref="S6.Ex31.m1.10.10.1.1.1.1.cmml"><msub id="S6.Ex31.m1.10.10.1.1.1.1.2" xref="S6.Ex31.m1.10.10.1.1.1.1.2.cmml"><mrow id="S6.Ex31.m1.10.10.1.1.1.1.2.2.2" xref="S6.Ex31.m1.10.10.1.1.1.1.2.2.1.cmml"><mo id="S6.Ex31.m1.10.10.1.1.1.1.2.2.2.1" stretchy="false" xref="S6.Ex31.m1.10.10.1.1.1.1.2.2.1.1.cmml">‖</mo><mi id="S6.Ex31.m1.8.8" xref="S6.Ex31.m1.8.8.cmml">u</mi><mo id="S6.Ex31.m1.10.10.1.1.1.1.2.2.2.2" stretchy="false" xref="S6.Ex31.m1.10.10.1.1.1.1.2.2.1.1.cmml">‖</mo></mrow><mrow id="S6.Ex31.m1.1.1.1" xref="S6.Ex31.m1.1.1.1.cmml"><msup id="S6.Ex31.m1.1.1.1.3" xref="S6.Ex31.m1.1.1.1.3.cmml"><mi id="S6.Ex31.m1.1.1.1.3.2" xref="S6.Ex31.m1.1.1.1.3.2.cmml">W</mi><mrow id="S6.Ex31.m1.1.1.1.3.3" xref="S6.Ex31.m1.1.1.1.3.3.cmml"><mover accent="true" id="S6.Ex31.m1.1.1.1.3.3.2" xref="S6.Ex31.m1.1.1.1.3.3.2.cmml"><mi id="S6.Ex31.m1.1.1.1.3.3.2.2" xref="S6.Ex31.m1.1.1.1.3.3.2.2.cmml">k</mi><mo id="S6.Ex31.m1.1.1.1.3.3.2.1" xref="S6.Ex31.m1.1.1.1.3.3.2.1.cmml">~</mo></mover><mo id="S6.Ex31.m1.1.1.1.3.3.1" xref="S6.Ex31.m1.1.1.1.3.3.1.cmml">+</mo><mi id="S6.Ex31.m1.1.1.1.3.3.3" mathvariant="normal" xref="S6.Ex31.m1.1.1.1.3.3.3.cmml">ℓ</mi><mo id="S6.Ex31.m1.1.1.1.3.3.1a" xref="S6.Ex31.m1.1.1.1.3.3.1.cmml">+</mo><mn id="S6.Ex31.m1.1.1.1.3.3.4" xref="S6.Ex31.m1.1.1.1.3.3.4.cmml">1</mn></mrow></msup><mo id="S6.Ex31.m1.1.1.1.2" xref="S6.Ex31.m1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex31.m1.1.1.1.1.1" xref="S6.Ex31.m1.1.1.1.1.1.1.cmml"><mo id="S6.Ex31.m1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex31.m1.1.1.1.1.1.1.cmml">(</mo><msub id="S6.Ex31.m1.1.1.1.1.1.1" xref="S6.Ex31.m1.1.1.1.1.1.1.cmml"><mi id="S6.Ex31.m1.1.1.1.1.1.1.2" xref="S6.Ex31.m1.1.1.1.1.1.1.2.cmml">w</mi><mrow id="S6.Ex31.m1.1.1.1.1.1.1.3" xref="S6.Ex31.m1.1.1.1.1.1.1.3.cmml"><mover accent="true" id="S6.Ex31.m1.1.1.1.1.1.1.3.2" xref="S6.Ex31.m1.1.1.1.1.1.1.3.2.cmml"><mi id="S6.Ex31.m1.1.1.1.1.1.1.3.2.2" xref="S6.Ex31.m1.1.1.1.1.1.1.3.2.2.cmml">γ</mi><mo id="S6.Ex31.m1.1.1.1.1.1.1.3.2.1" xref="S6.Ex31.m1.1.1.1.1.1.1.3.2.1.cmml">~</mo></mover><mo id="S6.Ex31.m1.1.1.1.1.1.1.3.1" xref="S6.Ex31.m1.1.1.1.1.1.1.3.1.cmml">+</mo><mi id="S6.Ex31.m1.1.1.1.1.1.1.3.3" xref="S6.Ex31.m1.1.1.1.1.1.1.3.3.cmml">p</mi></mrow></msub><mo id="S6.Ex31.m1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex31.m1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></msub><mo id="S6.Ex31.m1.10.10.1.1.1.1.1" xref="S6.Ex31.m1.10.10.1.1.1.1.1.cmml">≤</mo><mrow id="S6.Ex31.m1.10.10.1.1.1.1.3" xref="S6.Ex31.m1.10.10.1.1.1.1.3.cmml"><mi id="S6.Ex31.m1.10.10.1.1.1.1.3.2" xref="S6.Ex31.m1.10.10.1.1.1.1.3.2.cmml">C</mi><mo id="S6.Ex31.m1.10.10.1.1.1.1.3.1" xref="S6.Ex31.m1.10.10.1.1.1.1.3.1.cmml">⁢</mo><msub id="S6.Ex31.m1.10.10.1.1.1.1.3.3" xref="S6.Ex31.m1.10.10.1.1.1.1.3.3.cmml"><mrow id="S6.Ex31.m1.10.10.1.1.1.1.3.3.2.2" xref="S6.Ex31.m1.10.10.1.1.1.1.3.3.2.1.cmml"><mo id="S6.Ex31.m1.10.10.1.1.1.1.3.3.2.2.1" stretchy="false" xref="S6.Ex31.m1.10.10.1.1.1.1.3.3.2.1.1.cmml">‖</mo><mi id="S6.Ex31.m1.9.9" 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encoding="application/x-tex" id="S6.Ex31.m1.10c">\|u\|_{W^{\widetilde{k}+\ell+1}(w_{\widetilde{\gamma}+p})}\leq C\|u\|_{[W^{% \widetilde{k}+1,p}(w_{\widetilde{\gamma}+p}),W^{\widetilde{k}+k_{1}+1,p}(w_{% \widetilde{\gamma}+p})]_{\frac{\ell}{k_{1}}}},\quad u\in W^{\widetilde{k}+k_{1% }+1}(w_{\widetilde{\gamma}+p}),</annotation><annotation encoding="application/x-llamapun" id="S6.Ex31.m1.10d">∥ italic_u ∥ start_POSTSUBSCRIPT italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + roman_ℓ + 1 end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) end_POSTSUBSCRIPT ≤ italic_C ∥ italic_u ∥ start_POSTSUBSCRIPT [ italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + 1 , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_u ∈ italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.8.p3.1">and the induction is completed by noting that <math alttext="W^{\widetilde{k}+k_{1}+1}(w_{\widetilde{\gamma}+p})" class="ltx_Math" display="inline" id="S6.SS1.8.p3.1.m1.1"><semantics id="S6.SS1.8.p3.1.m1.1a"><mrow id="S6.SS1.8.p3.1.m1.1.1" xref="S6.SS1.8.p3.1.m1.1.1.cmml"><msup id="S6.SS1.8.p3.1.m1.1.1.3" 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xref="S6.SS1.8.p3.1.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.8.p3.1.m1.1b"><apply id="S6.SS1.8.p3.1.m1.1.1.cmml" xref="S6.SS1.8.p3.1.m1.1.1"><times id="S6.SS1.8.p3.1.m1.1.1.2.cmml" xref="S6.SS1.8.p3.1.m1.1.1.2"></times><apply id="S6.SS1.8.p3.1.m1.1.1.3.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS1.8.p3.1.m1.1.1.3.1.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3">superscript</csymbol><ci id="S6.SS1.8.p3.1.m1.1.1.3.2.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3.2">𝑊</ci><apply id="S6.SS1.8.p3.1.m1.1.1.3.3.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3.3"><plus id="S6.SS1.8.p3.1.m1.1.1.3.3.1.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3.3.1"></plus><apply id="S6.SS1.8.p3.1.m1.1.1.3.3.2.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3.3.2"><ci id="S6.SS1.8.p3.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3.3.2.1">~</ci><ci id="S6.SS1.8.p3.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS1.8.p3.1.m1.1.1.3.3.2.2">𝑘</ci></apply><apply id="S6.SS1.8.p3.1.m1.1.1.3.3.3.cmml" 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id="S6.SS1.8.p3.1.m1.1.1.1.1.1.3.2.1.cmml" xref="S6.SS1.8.p3.1.m1.1.1.1.1.1.3.2.1">~</ci><ci id="S6.SS1.8.p3.1.m1.1.1.1.1.1.3.2.2.cmml" xref="S6.SS1.8.p3.1.m1.1.1.1.1.1.3.2.2">𝛾</ci></apply><ci id="S6.SS1.8.p3.1.m1.1.1.1.1.1.3.3.cmml" xref="S6.SS1.8.p3.1.m1.1.1.1.1.1.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.8.p3.1.m1.1c">W^{\widetilde{k}+k_{1}+1}(w_{\widetilde{\gamma}+p})</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.8.p3.1.m1.1d">italic_W start_POSTSUPERSCRIPT over~ start_ARG italic_k end_ARG + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + 1 end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT over~ start_ARG italic_γ end_ARG + italic_p end_POSTSUBSCRIPT )</annotation></semantics></math> is dense in the interpolation space (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib67" title="">67</a>, Theorem 1.9.3]</cite>).</p> </div> <div class="ltx_para" id="S6.SS1.9.p4"> <p class="ltx_p" id="S6.SS1.9.p4.6"><span class="ltx_text ltx_font_italic" id="S6.SS1.9.p4.1.1">Step 2: the case <math alttext="j\in\{k_{0}+1,\dots,k_{0}+\ell-1\}" class="ltx_Math" display="inline" id="S6.SS1.9.p4.1.1.m1.3"><semantics id="S6.SS1.9.p4.1.1.m1.3a"><mrow id="S6.SS1.9.p4.1.1.m1.3.3" xref="S6.SS1.9.p4.1.1.m1.3.3.cmml"><mi id="S6.SS1.9.p4.1.1.m1.3.3.4" xref="S6.SS1.9.p4.1.1.m1.3.3.4.cmml">j</mi><mo id="S6.SS1.9.p4.1.1.m1.3.3.3" xref="S6.SS1.9.p4.1.1.m1.3.3.3.cmml">∈</mo><mrow id="S6.SS1.9.p4.1.1.m1.3.3.2.2" xref="S6.SS1.9.p4.1.1.m1.3.3.2.3.cmml"><mo id="S6.SS1.9.p4.1.1.m1.3.3.2.2.3" stretchy="false" xref="S6.SS1.9.p4.1.1.m1.3.3.2.3.cmml">{</mo><mrow id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.cmml"><msub id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.cmml"><mi id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.2" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.3" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.1" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.1.cmml">+</mo><mn id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.3" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.9.p4.1.1.m1.3.3.2.2.4" xref="S6.SS1.9.p4.1.1.m1.3.3.2.3.cmml">,</mo><mi id="S6.SS1.9.p4.1.1.m1.1.1" mathvariant="normal" xref="S6.SS1.9.p4.1.1.m1.1.1.cmml">…</mi><mo id="S6.SS1.9.p4.1.1.m1.3.3.2.2.5" xref="S6.SS1.9.p4.1.1.m1.3.3.2.3.cmml">,</mo><mrow id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.cmml"><mrow id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.cmml"><msub id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.cmml"><mi id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.2" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.2.cmml">k</mi><mn id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.3" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.3.cmml">0</mn></msub><mo id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.1" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.1.cmml">+</mo><mi id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.3" mathvariant="normal" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.3.cmml">ℓ</mi></mrow><mo id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.1" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.1.cmml">−</mo><mn id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.3" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS1.9.p4.1.1.m1.3.3.2.2.6" stretchy="false" xref="S6.SS1.9.p4.1.1.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.9.p4.1.1.m1.3b"><apply id="S6.SS1.9.p4.1.1.m1.3.3.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3"><in id="S6.SS1.9.p4.1.1.m1.3.3.3.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.3"></in><ci id="S6.SS1.9.p4.1.1.m1.3.3.4.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.4">𝑗</ci><set id="S6.SS1.9.p4.1.1.m1.3.3.2.3.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2"><apply id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.cmml" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1"><plus id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.1.cmml" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.1"></plus><apply id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.cmml" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.1.cmml" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2">subscript</csymbol><ci id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.2.cmml" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.2">𝑘</ci><cn id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.2.3">0</cn></apply><cn id="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.3.cmml" type="integer" xref="S6.SS1.9.p4.1.1.m1.2.2.1.1.1.3">1</cn></apply><ci id="S6.SS1.9.p4.1.1.m1.1.1.cmml" xref="S6.SS1.9.p4.1.1.m1.1.1">…</ci><apply id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2"><minus id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.1.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.1"></minus><apply id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2"><plus id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.1.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.1"></plus><apply id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.1.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2">subscript</csymbol><ci id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.2.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.2">𝑘</ci><cn id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.2.3">0</cn></apply><ci id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.3.cmml" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.2.3">ℓ</ci></apply><cn id="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.3.cmml" type="integer" xref="S6.SS1.9.p4.1.1.m1.3.3.2.2.2.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.1.1.m1.3c">j\in\{k_{0}+1,\dots,k_{0}+\ell-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.1.1.m1.3d">italic_j ∈ { italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 , … , italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ - 1 }</annotation></semantics></math>.</span> Assume that <math alttext="\ell\in\{2,\dots,k_{1}-1\}" class="ltx_Math" display="inline" id="S6.SS1.9.p4.2.m1.3"><semantics id="S6.SS1.9.p4.2.m1.3a"><mrow id="S6.SS1.9.p4.2.m1.3.3" xref="S6.SS1.9.p4.2.m1.3.3.cmml"><mi id="S6.SS1.9.p4.2.m1.3.3.3" mathvariant="normal" xref="S6.SS1.9.p4.2.m1.3.3.3.cmml">ℓ</mi><mo id="S6.SS1.9.p4.2.m1.3.3.2" xref="S6.SS1.9.p4.2.m1.3.3.2.cmml">∈</mo><mrow id="S6.SS1.9.p4.2.m1.3.3.1.1" xref="S6.SS1.9.p4.2.m1.3.3.1.2.cmml"><mo id="S6.SS1.9.p4.2.m1.3.3.1.1.2" stretchy="false" xref="S6.SS1.9.p4.2.m1.3.3.1.2.cmml">{</mo><mn id="S6.SS1.9.p4.2.m1.1.1" xref="S6.SS1.9.p4.2.m1.1.1.cmml">2</mn><mo id="S6.SS1.9.p4.2.m1.3.3.1.1.3" xref="S6.SS1.9.p4.2.m1.3.3.1.2.cmml">,</mo><mi id="S6.SS1.9.p4.2.m1.2.2" mathvariant="normal" xref="S6.SS1.9.p4.2.m1.2.2.cmml">…</mi><mo id="S6.SS1.9.p4.2.m1.3.3.1.1.4" xref="S6.SS1.9.p4.2.m1.3.3.1.2.cmml">,</mo><mrow id="S6.SS1.9.p4.2.m1.3.3.1.1.1" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.cmml"><msub id="S6.SS1.9.p4.2.m1.3.3.1.1.1.2" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.cmml"><mi id="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.2" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.3" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS1.9.p4.2.m1.3.3.1.1.1.1" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.1.cmml">−</mo><mn id="S6.SS1.9.p4.2.m1.3.3.1.1.1.3" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.9.p4.2.m1.3.3.1.1.5" stretchy="false" xref="S6.SS1.9.p4.2.m1.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.9.p4.2.m1.3b"><apply id="S6.SS1.9.p4.2.m1.3.3.cmml" xref="S6.SS1.9.p4.2.m1.3.3"><in id="S6.SS1.9.p4.2.m1.3.3.2.cmml" xref="S6.SS1.9.p4.2.m1.3.3.2"></in><ci id="S6.SS1.9.p4.2.m1.3.3.3.cmml" xref="S6.SS1.9.p4.2.m1.3.3.3">ℓ</ci><set id="S6.SS1.9.p4.2.m1.3.3.1.2.cmml" xref="S6.SS1.9.p4.2.m1.3.3.1.1"><cn id="S6.SS1.9.p4.2.m1.1.1.cmml" type="integer" xref="S6.SS1.9.p4.2.m1.1.1">2</cn><ci id="S6.SS1.9.p4.2.m1.2.2.cmml" xref="S6.SS1.9.p4.2.m1.2.2">…</ci><apply id="S6.SS1.9.p4.2.m1.3.3.1.1.1.cmml" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1"><minus id="S6.SS1.9.p4.2.m1.3.3.1.1.1.1.cmml" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.1"></minus><apply id="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.cmml" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.1.cmml" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.2">subscript</csymbol><ci id="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.2.cmml" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.2">𝑘</ci><cn id="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.2.3">1</cn></apply><cn id="S6.SS1.9.p4.2.m1.3.3.1.1.1.3.cmml" type="integer" xref="S6.SS1.9.p4.2.m1.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.2.m1.3c">\ell\in\{2,\dots,k_{1}-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.2.m1.3d">roman_ℓ ∈ { 2 , … , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 }</annotation></semantics></math> (the case <math alttext="\ell=1" class="ltx_Math" display="inline" id="S6.SS1.9.p4.3.m2.1"><semantics id="S6.SS1.9.p4.3.m2.1a"><mrow id="S6.SS1.9.p4.3.m2.1.1" xref="S6.SS1.9.p4.3.m2.1.1.cmml"><mi id="S6.SS1.9.p4.3.m2.1.1.2" mathvariant="normal" xref="S6.SS1.9.p4.3.m2.1.1.2.cmml">ℓ</mi><mo id="S6.SS1.9.p4.3.m2.1.1.1" xref="S6.SS1.9.p4.3.m2.1.1.1.cmml">=</mo><mn id="S6.SS1.9.p4.3.m2.1.1.3" xref="S6.SS1.9.p4.3.m2.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.9.p4.3.m2.1b"><apply id="S6.SS1.9.p4.3.m2.1.1.cmml" xref="S6.SS1.9.p4.3.m2.1.1"><eq id="S6.SS1.9.p4.3.m2.1.1.1.cmml" xref="S6.SS1.9.p4.3.m2.1.1.1"></eq><ci id="S6.SS1.9.p4.3.m2.1.1.2.cmml" xref="S6.SS1.9.p4.3.m2.1.1.2">ℓ</ci><cn id="S6.SS1.9.p4.3.m2.1.1.3.cmml" type="integer" xref="S6.SS1.9.p4.3.m2.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.3.m2.1c">\ell=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.3.m2.1d">roman_ℓ = 1</annotation></semantics></math> is covered by Step 1 and 3) and <math alttext="j\in\{k_{0}+1,\dots,k_{0}+\ell-1\}" class="ltx_Math" display="inline" id="S6.SS1.9.p4.4.m3.3"><semantics id="S6.SS1.9.p4.4.m3.3a"><mrow id="S6.SS1.9.p4.4.m3.3.3" xref="S6.SS1.9.p4.4.m3.3.3.cmml"><mi id="S6.SS1.9.p4.4.m3.3.3.4" xref="S6.SS1.9.p4.4.m3.3.3.4.cmml">j</mi><mo id="S6.SS1.9.p4.4.m3.3.3.3" xref="S6.SS1.9.p4.4.m3.3.3.3.cmml">∈</mo><mrow id="S6.SS1.9.p4.4.m3.3.3.2.2" xref="S6.SS1.9.p4.4.m3.3.3.2.3.cmml"><mo id="S6.SS1.9.p4.4.m3.3.3.2.2.3" stretchy="false" xref="S6.SS1.9.p4.4.m3.3.3.2.3.cmml">{</mo><mrow id="S6.SS1.9.p4.4.m3.2.2.1.1.1" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.cmml"><msub id="S6.SS1.9.p4.4.m3.2.2.1.1.1.2" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.cmml"><mi id="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.2" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.3" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.9.p4.4.m3.2.2.1.1.1.1" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.1.cmml">+</mo><mn id="S6.SS1.9.p4.4.m3.2.2.1.1.1.3" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.3.cmml">1</mn></mrow><mo id="S6.SS1.9.p4.4.m3.3.3.2.2.4" xref="S6.SS1.9.p4.4.m3.3.3.2.3.cmml">,</mo><mi id="S6.SS1.9.p4.4.m3.1.1" mathvariant="normal" xref="S6.SS1.9.p4.4.m3.1.1.cmml">…</mi><mo id="S6.SS1.9.p4.4.m3.3.3.2.2.5" xref="S6.SS1.9.p4.4.m3.3.3.2.3.cmml">,</mo><mrow id="S6.SS1.9.p4.4.m3.3.3.2.2.2" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.cmml"><mrow id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.cmml"><msub id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.cmml"><mi id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.2" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.2.cmml">k</mi><mn id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.3" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.3.cmml">0</mn></msub><mo id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.1" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.1.cmml">+</mo><mi id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.3" mathvariant="normal" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.3.cmml">ℓ</mi></mrow><mo id="S6.SS1.9.p4.4.m3.3.3.2.2.2.1" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.1.cmml">−</mo><mn id="S6.SS1.9.p4.4.m3.3.3.2.2.2.3" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS1.9.p4.4.m3.3.3.2.2.6" stretchy="false" xref="S6.SS1.9.p4.4.m3.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.9.p4.4.m3.3b"><apply id="S6.SS1.9.p4.4.m3.3.3.cmml" xref="S6.SS1.9.p4.4.m3.3.3"><in id="S6.SS1.9.p4.4.m3.3.3.3.cmml" xref="S6.SS1.9.p4.4.m3.3.3.3"></in><ci id="S6.SS1.9.p4.4.m3.3.3.4.cmml" xref="S6.SS1.9.p4.4.m3.3.3.4">𝑗</ci><set id="S6.SS1.9.p4.4.m3.3.3.2.3.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2"><apply id="S6.SS1.9.p4.4.m3.2.2.1.1.1.cmml" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1"><plus id="S6.SS1.9.p4.4.m3.2.2.1.1.1.1.cmml" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.1"></plus><apply id="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.cmml" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.1.cmml" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.2">subscript</csymbol><ci id="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.2.cmml" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.2">𝑘</ci><cn id="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.2.3">0</cn></apply><cn id="S6.SS1.9.p4.4.m3.2.2.1.1.1.3.cmml" type="integer" xref="S6.SS1.9.p4.4.m3.2.2.1.1.1.3">1</cn></apply><ci id="S6.SS1.9.p4.4.m3.1.1.cmml" xref="S6.SS1.9.p4.4.m3.1.1">…</ci><apply id="S6.SS1.9.p4.4.m3.3.3.2.2.2.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2"><minus id="S6.SS1.9.p4.4.m3.3.3.2.2.2.1.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.1"></minus><apply id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2"><plus id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.1.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.1"></plus><apply id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.1.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2">subscript</csymbol><ci id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.2.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.2">𝑘</ci><cn id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.2.3">0</cn></apply><ci id="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.3.cmml" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.2.3">ℓ</ci></apply><cn id="S6.SS1.9.p4.4.m3.3.3.2.2.2.3.cmml" type="integer" xref="S6.SS1.9.p4.4.m3.3.3.2.2.2.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.4.m3.3c">j\in\{k_{0}+1,\dots,k_{0}+\ell-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.4.m3.3d">italic_j ∈ { italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + 1 , … , italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ - 1 }</annotation></semantics></math>. We apply Step 1, Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem2" title="Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.I1.i2" title="item ii ‣ Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> and Wolff interpolation to prove (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E1" title="In Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.1</span></a>). Since <math alttext="\gamma>jp-1" class="ltx_Math" display="inline" id="S6.SS1.9.p4.5.m4.1"><semantics id="S6.SS1.9.p4.5.m4.1a"><mrow id="S6.SS1.9.p4.5.m4.1.1" xref="S6.SS1.9.p4.5.m4.1.1.cmml"><mi id="S6.SS1.9.p4.5.m4.1.1.2" xref="S6.SS1.9.p4.5.m4.1.1.2.cmml">γ</mi><mo id="S6.SS1.9.p4.5.m4.1.1.1" xref="S6.SS1.9.p4.5.m4.1.1.1.cmml">></mo><mrow id="S6.SS1.9.p4.5.m4.1.1.3" xref="S6.SS1.9.p4.5.m4.1.1.3.cmml"><mrow id="S6.SS1.9.p4.5.m4.1.1.3.2" xref="S6.SS1.9.p4.5.m4.1.1.3.2.cmml"><mi id="S6.SS1.9.p4.5.m4.1.1.3.2.2" xref="S6.SS1.9.p4.5.m4.1.1.3.2.2.cmml">j</mi><mo id="S6.SS1.9.p4.5.m4.1.1.3.2.1" xref="S6.SS1.9.p4.5.m4.1.1.3.2.1.cmml">⁢</mo><mi id="S6.SS1.9.p4.5.m4.1.1.3.2.3" xref="S6.SS1.9.p4.5.m4.1.1.3.2.3.cmml">p</mi></mrow><mo id="S6.SS1.9.p4.5.m4.1.1.3.1" xref="S6.SS1.9.p4.5.m4.1.1.3.1.cmml">−</mo><mn id="S6.SS1.9.p4.5.m4.1.1.3.3" xref="S6.SS1.9.p4.5.m4.1.1.3.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.9.p4.5.m4.1b"><apply id="S6.SS1.9.p4.5.m4.1.1.cmml" xref="S6.SS1.9.p4.5.m4.1.1"><gt id="S6.SS1.9.p4.5.m4.1.1.1.cmml" xref="S6.SS1.9.p4.5.m4.1.1.1"></gt><ci id="S6.SS1.9.p4.5.m4.1.1.2.cmml" xref="S6.SS1.9.p4.5.m4.1.1.2">𝛾</ci><apply id="S6.SS1.9.p4.5.m4.1.1.3.cmml" xref="S6.SS1.9.p4.5.m4.1.1.3"><minus id="S6.SS1.9.p4.5.m4.1.1.3.1.cmml" xref="S6.SS1.9.p4.5.m4.1.1.3.1"></minus><apply id="S6.SS1.9.p4.5.m4.1.1.3.2.cmml" xref="S6.SS1.9.p4.5.m4.1.1.3.2"><times id="S6.SS1.9.p4.5.m4.1.1.3.2.1.cmml" xref="S6.SS1.9.p4.5.m4.1.1.3.2.1"></times><ci id="S6.SS1.9.p4.5.m4.1.1.3.2.2.cmml" xref="S6.SS1.9.p4.5.m4.1.1.3.2.2">𝑗</ci><ci id="S6.SS1.9.p4.5.m4.1.1.3.2.3.cmml" xref="S6.SS1.9.p4.5.m4.1.1.3.2.3">𝑝</ci></apply><cn id="S6.SS1.9.p4.5.m4.1.1.3.3.cmml" type="integer" xref="S6.SS1.9.p4.5.m4.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.5.m4.1c">\gamma>jp-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.5.m4.1d">italic_γ > italic_j italic_p - 1</annotation></semantics></math> and <math alttext="1\leq j-k_{0}\leq\ell-1" class="ltx_Math" display="inline" id="S6.SS1.9.p4.6.m5.1"><semantics id="S6.SS1.9.p4.6.m5.1a"><mrow id="S6.SS1.9.p4.6.m5.1.1" xref="S6.SS1.9.p4.6.m5.1.1.cmml"><mn id="S6.SS1.9.p4.6.m5.1.1.2" xref="S6.SS1.9.p4.6.m5.1.1.2.cmml">1</mn><mo id="S6.SS1.9.p4.6.m5.1.1.3" xref="S6.SS1.9.p4.6.m5.1.1.3.cmml">≤</mo><mrow id="S6.SS1.9.p4.6.m5.1.1.4" xref="S6.SS1.9.p4.6.m5.1.1.4.cmml"><mi id="S6.SS1.9.p4.6.m5.1.1.4.2" xref="S6.SS1.9.p4.6.m5.1.1.4.2.cmml">j</mi><mo id="S6.SS1.9.p4.6.m5.1.1.4.1" xref="S6.SS1.9.p4.6.m5.1.1.4.1.cmml">−</mo><msub id="S6.SS1.9.p4.6.m5.1.1.4.3" xref="S6.SS1.9.p4.6.m5.1.1.4.3.cmml"><mi id="S6.SS1.9.p4.6.m5.1.1.4.3.2" xref="S6.SS1.9.p4.6.m5.1.1.4.3.2.cmml">k</mi><mn id="S6.SS1.9.p4.6.m5.1.1.4.3.3" xref="S6.SS1.9.p4.6.m5.1.1.4.3.3.cmml">0</mn></msub></mrow><mo id="S6.SS1.9.p4.6.m5.1.1.5" xref="S6.SS1.9.p4.6.m5.1.1.5.cmml">≤</mo><mrow id="S6.SS1.9.p4.6.m5.1.1.6" xref="S6.SS1.9.p4.6.m5.1.1.6.cmml"><mi id="S6.SS1.9.p4.6.m5.1.1.6.2" mathvariant="normal" xref="S6.SS1.9.p4.6.m5.1.1.6.2.cmml">ℓ</mi><mo id="S6.SS1.9.p4.6.m5.1.1.6.1" xref="S6.SS1.9.p4.6.m5.1.1.6.1.cmml">−</mo><mn id="S6.SS1.9.p4.6.m5.1.1.6.3" xref="S6.SS1.9.p4.6.m5.1.1.6.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.9.p4.6.m5.1b"><apply id="S6.SS1.9.p4.6.m5.1.1.cmml" xref="S6.SS1.9.p4.6.m5.1.1"><and id="S6.SS1.9.p4.6.m5.1.1a.cmml" xref="S6.SS1.9.p4.6.m5.1.1"></and><apply id="S6.SS1.9.p4.6.m5.1.1b.cmml" xref="S6.SS1.9.p4.6.m5.1.1"><leq id="S6.SS1.9.p4.6.m5.1.1.3.cmml" xref="S6.SS1.9.p4.6.m5.1.1.3"></leq><cn id="S6.SS1.9.p4.6.m5.1.1.2.cmml" type="integer" xref="S6.SS1.9.p4.6.m5.1.1.2">1</cn><apply id="S6.SS1.9.p4.6.m5.1.1.4.cmml" xref="S6.SS1.9.p4.6.m5.1.1.4"><minus id="S6.SS1.9.p4.6.m5.1.1.4.1.cmml" xref="S6.SS1.9.p4.6.m5.1.1.4.1"></minus><ci id="S6.SS1.9.p4.6.m5.1.1.4.2.cmml" xref="S6.SS1.9.p4.6.m5.1.1.4.2">𝑗</ci><apply id="S6.SS1.9.p4.6.m5.1.1.4.3.cmml" xref="S6.SS1.9.p4.6.m5.1.1.4.3"><csymbol cd="ambiguous" id="S6.SS1.9.p4.6.m5.1.1.4.3.1.cmml" xref="S6.SS1.9.p4.6.m5.1.1.4.3">subscript</csymbol><ci id="S6.SS1.9.p4.6.m5.1.1.4.3.2.cmml" xref="S6.SS1.9.p4.6.m5.1.1.4.3.2">𝑘</ci><cn id="S6.SS1.9.p4.6.m5.1.1.4.3.3.cmml" type="integer" xref="S6.SS1.9.p4.6.m5.1.1.4.3.3">0</cn></apply></apply></apply><apply id="S6.SS1.9.p4.6.m5.1.1c.cmml" xref="S6.SS1.9.p4.6.m5.1.1"><leq id="S6.SS1.9.p4.6.m5.1.1.5.cmml" xref="S6.SS1.9.p4.6.m5.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.SS1.9.p4.6.m5.1.1.4.cmml" id="S6.SS1.9.p4.6.m5.1.1d.cmml" xref="S6.SS1.9.p4.6.m5.1.1"></share><apply id="S6.SS1.9.p4.6.m5.1.1.6.cmml" xref="S6.SS1.9.p4.6.m5.1.1.6"><minus id="S6.SS1.9.p4.6.m5.1.1.6.1.cmml" xref="S6.SS1.9.p4.6.m5.1.1.6.1"></minus><ci id="S6.SS1.9.p4.6.m5.1.1.6.2.cmml" xref="S6.SS1.9.p4.6.m5.1.1.6.2">ℓ</ci><cn id="S6.SS1.9.p4.6.m5.1.1.6.3.cmml" type="integer" xref="S6.SS1.9.p4.6.m5.1.1.6.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.6.m5.1c">1\leq j-k_{0}\leq\ell-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.6.m5.1d">1 ≤ italic_j - italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ≤ roman_ℓ - 1</annotation></semantics></math>, Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem2" title="Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.I1.i2" title="item ii ‣ Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. 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id="S6.Ex32.m1.7.7.cmml" type="integer" xref="S6.Ex32.m1.7.7">0</cn><cn id="S6.Ex32.m1.8.8.cmml" type="integer" xref="S6.Ex32.m1.8.8">1</cn></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex32.m1.9c">\big{[}W^{k_{0},p}(w_{\gamma}),W^{k_{0}+\ell,p}(w_{\gamma})\big{]}_{\lambda}=W% ^{j,p}(w_{\gamma}),\quad\text{ with }\lambda:=\frac{j-k_{0}}{\ell}\in[0,1],</annotation><annotation encoding="application/x-llamapun" id="S6.Ex32.m1.9d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_j , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , with italic_λ := divide start_ARG italic_j - italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_ARG start_ARG roman_ℓ end_ARG ∈ [ 0 , 1 ] ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.9.p4.7">and since <math alttext="\gamma<(j+1)p-1" class="ltx_Math" display="inline" id="S6.SS1.9.p4.7.m1.1"><semantics id="S6.SS1.9.p4.7.m1.1a"><mrow id="S6.SS1.9.p4.7.m1.1.1" xref="S6.SS1.9.p4.7.m1.1.1.cmml"><mi id="S6.SS1.9.p4.7.m1.1.1.3" xref="S6.SS1.9.p4.7.m1.1.1.3.cmml">γ</mi><mo id="S6.SS1.9.p4.7.m1.1.1.2" xref="S6.SS1.9.p4.7.m1.1.1.2.cmml"><</mo><mrow id="S6.SS1.9.p4.7.m1.1.1.1" xref="S6.SS1.9.p4.7.m1.1.1.1.cmml"><mrow id="S6.SS1.9.p4.7.m1.1.1.1.1" xref="S6.SS1.9.p4.7.m1.1.1.1.1.cmml"><mrow id="S6.SS1.9.p4.7.m1.1.1.1.1.1.1" xref="S6.SS1.9.p4.7.m1.1.1.1.1.1.1.1.cmml"><mo id="S6.SS1.9.p4.7.m1.1.1.1.1.1.1.2" stretchy="false" 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id="S6.SS1.9.p4.7.m1.1.1.1.3.cmml" type="integer" xref="S6.SS1.9.p4.7.m1.1.1.1.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.7.m1.1c">\gamma<(j+1)p-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.7.m1.1d">italic_γ < ( italic_j + 1 ) italic_p - 1</annotation></semantics></math>, Step 1 implies</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex33"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{j,p}(w_{\gamma}),W^{k_{0}+k_{1},p}(w_{\gamma})\big{]}_{\mu}% \hookrightarrow W^{k_{0}+\ell,p}(w_{\gamma}),\quad\text{ with }\mu:=\frac{k_{0% }+\ell-j}{k_{0}+k_{1}-j}\in[0,1]." class="ltx_Math" display="block" id="S6.Ex33.m1.9"><semantics id="S6.Ex33.m1.9a"><mrow id="S6.Ex33.m1.9.9.1"><mrow id="S6.Ex33.m1.9.9.1.1.2" xref="S6.Ex33.m1.9.9.1.1.3.cmml"><mrow 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id="S6.Ex33.m1.7.7.cmml" type="integer" xref="S6.Ex33.m1.7.7">0</cn><cn id="S6.Ex33.m1.8.8.cmml" type="integer" xref="S6.Ex33.m1.8.8">1</cn></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex33.m1.9c">\big{[}W^{j,p}(w_{\gamma}),W^{k_{0}+k_{1},p}(w_{\gamma})\big{]}_{\mu}% \hookrightarrow W^{k_{0}+\ell,p}(w_{\gamma}),\quad\text{ with }\mu:=\frac{k_{0% }+\ell-j}{k_{0}+k_{1}-j}\in[0,1].</annotation><annotation encoding="application/x-llamapun" id="S6.Ex33.m1.9d">[ italic_W start_POSTSUPERSCRIPT italic_j , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT italic_μ end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , with italic_μ := divide start_ARG italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ - italic_j end_ARG start_ARG italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j end_ARG ∈ [ 0 , 1 ] .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.9.p4.9">Wolff interpolation <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib33" title="">33</a>, Corollary 3]</cite> gives that</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex34"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) ] start_POSTSUBSCRIPT italic_η end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ) , with italic_η := divide start_ARG italic_μ end_ARG start_ARG 1 + italic_λ ( italic_μ - 1 ) end_ARG .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS1.9.p4.8">A straightforward calculation shows that <math alttext="\eta=\frac{\ell}{k_{1}}" class="ltx_Math" display="inline" id="S6.SS1.9.p4.8.m1.1"><semantics id="S6.SS1.9.p4.8.m1.1a"><mrow id="S6.SS1.9.p4.8.m1.1.1" xref="S6.SS1.9.p4.8.m1.1.1.cmml"><mi id="S6.SS1.9.p4.8.m1.1.1.2" xref="S6.SS1.9.p4.8.m1.1.1.2.cmml">η</mi><mo id="S6.SS1.9.p4.8.m1.1.1.1" xref="S6.SS1.9.p4.8.m1.1.1.1.cmml">=</mo><mfrac 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xref="S6.SS1.9.p4.8.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS1.9.p4.8.m1.1.1.3.3.1.cmml" xref="S6.SS1.9.p4.8.m1.1.1.3.3">subscript</csymbol><ci id="S6.SS1.9.p4.8.m1.1.1.3.3.2.cmml" xref="S6.SS1.9.p4.8.m1.1.1.3.3.2">𝑘</ci><cn id="S6.SS1.9.p4.8.m1.1.1.3.3.3.cmml" type="integer" xref="S6.SS1.9.p4.8.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.9.p4.8.m1.1c">\eta=\frac{\ell}{k_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.9.p4.8.m1.1d">italic_η = divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS1.10.p5"> <p class="ltx_p" id="S6.SS1.10.p5.3"><span class="ltx_text ltx_font_italic" id="S6.SS1.10.p5.1.1">Step 3: the case <math alttext="j\geq k_{0}+\ell" class="ltx_Math" display="inline" id="S6.SS1.10.p5.1.1.m1.1"><semantics id="S6.SS1.10.p5.1.1.m1.1a"><mrow 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id="S6.SS1.10.p5.1.1.m1.1.1.2.cmml" xref="S6.SS1.10.p5.1.1.m1.1.1.2">𝑗</ci><apply id="S6.SS1.10.p5.1.1.m1.1.1.3.cmml" xref="S6.SS1.10.p5.1.1.m1.1.1.3"><plus id="S6.SS1.10.p5.1.1.m1.1.1.3.1.cmml" xref="S6.SS1.10.p5.1.1.m1.1.1.3.1"></plus><apply id="S6.SS1.10.p5.1.1.m1.1.1.3.2.cmml" xref="S6.SS1.10.p5.1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS1.10.p5.1.1.m1.1.1.3.2.1.cmml" xref="S6.SS1.10.p5.1.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS1.10.p5.1.1.m1.1.1.3.2.2.cmml" xref="S6.SS1.10.p5.1.1.m1.1.1.3.2.2">𝑘</ci><cn id="S6.SS1.10.p5.1.1.m1.1.1.3.2.3.cmml" type="integer" xref="S6.SS1.10.p5.1.1.m1.1.1.3.2.3">0</cn></apply><ci id="S6.SS1.10.p5.1.1.m1.1.1.3.3.cmml" xref="S6.SS1.10.p5.1.1.m1.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.10.p5.1.1.m1.1c">j\geq k_{0}+\ell</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.10.p5.1.1.m1.1d">italic_j ≥ italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ</annotation></semantics></math>.</span> If <math alttext="j\geq k_{0}+\ell" class="ltx_Math" display="inline" id="S6.SS1.10.p5.2.m1.1"><semantics id="S6.SS1.10.p5.2.m1.1a"><mrow id="S6.SS1.10.p5.2.m1.1.1" xref="S6.SS1.10.p5.2.m1.1.1.cmml"><mi id="S6.SS1.10.p5.2.m1.1.1.2" xref="S6.SS1.10.p5.2.m1.1.1.2.cmml">j</mi><mo id="S6.SS1.10.p5.2.m1.1.1.1" xref="S6.SS1.10.p5.2.m1.1.1.1.cmml">≥</mo><mrow id="S6.SS1.10.p5.2.m1.1.1.3" xref="S6.SS1.10.p5.2.m1.1.1.3.cmml"><msub id="S6.SS1.10.p5.2.m1.1.1.3.2" xref="S6.SS1.10.p5.2.m1.1.1.3.2.cmml"><mi id="S6.SS1.10.p5.2.m1.1.1.3.2.2" xref="S6.SS1.10.p5.2.m1.1.1.3.2.2.cmml">k</mi><mn id="S6.SS1.10.p5.2.m1.1.1.3.2.3" xref="S6.SS1.10.p5.2.m1.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS1.10.p5.2.m1.1.1.3.1" xref="S6.SS1.10.p5.2.m1.1.1.3.1.cmml">+</mo><mi id="S6.SS1.10.p5.2.m1.1.1.3.3" mathvariant="normal" xref="S6.SS1.10.p5.2.m1.1.1.3.3.cmml">ℓ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.10.p5.2.m1.1b"><apply id="S6.SS1.10.p5.2.m1.1.1.cmml" xref="S6.SS1.10.p5.2.m1.1.1"><geq id="S6.SS1.10.p5.2.m1.1.1.1.cmml" xref="S6.SS1.10.p5.2.m1.1.1.1"></geq><ci id="S6.SS1.10.p5.2.m1.1.1.2.cmml" xref="S6.SS1.10.p5.2.m1.1.1.2">𝑗</ci><apply id="S6.SS1.10.p5.2.m1.1.1.3.cmml" xref="S6.SS1.10.p5.2.m1.1.1.3"><plus id="S6.SS1.10.p5.2.m1.1.1.3.1.cmml" xref="S6.SS1.10.p5.2.m1.1.1.3.1"></plus><apply id="S6.SS1.10.p5.2.m1.1.1.3.2.cmml" xref="S6.SS1.10.p5.2.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS1.10.p5.2.m1.1.1.3.2.1.cmml" xref="S6.SS1.10.p5.2.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS1.10.p5.2.m1.1.1.3.2.2.cmml" xref="S6.SS1.10.p5.2.m1.1.1.3.2.2">𝑘</ci><cn id="S6.SS1.10.p5.2.m1.1.1.3.2.3.cmml" type="integer" xref="S6.SS1.10.p5.2.m1.1.1.3.2.3">0</cn></apply><ci id="S6.SS1.10.p5.2.m1.1.1.3.3.cmml" xref="S6.SS1.10.p5.2.m1.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.10.p5.2.m1.1c">j\geq k_{0}+\ell</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.10.p5.2.m1.1d">italic_j ≥ italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ</annotation></semantics></math>, then <math alttext="\gamma>jp-1\geq(k_{0}+\ell)p-1" class="ltx_Math" display="inline" id="S6.SS1.10.p5.3.m2.1"><semantics id="S6.SS1.10.p5.3.m2.1a"><mrow id="S6.SS1.10.p5.3.m2.1.1" xref="S6.SS1.10.p5.3.m2.1.1.cmml"><mi id="S6.SS1.10.p5.3.m2.1.1.3" xref="S6.SS1.10.p5.3.m2.1.1.3.cmml">γ</mi><mo id="S6.SS1.10.p5.3.m2.1.1.4" xref="S6.SS1.10.p5.3.m2.1.1.4.cmml">></mo><mrow id="S6.SS1.10.p5.3.m2.1.1.5" xref="S6.SS1.10.p5.3.m2.1.1.5.cmml"><mrow id="S6.SS1.10.p5.3.m2.1.1.5.2" xref="S6.SS1.10.p5.3.m2.1.1.5.2.cmml"><mi id="S6.SS1.10.p5.3.m2.1.1.5.2.2" xref="S6.SS1.10.p5.3.m2.1.1.5.2.2.cmml">j</mi><mo id="S6.SS1.10.p5.3.m2.1.1.5.2.1" xref="S6.SS1.10.p5.3.m2.1.1.5.2.1.cmml">⁢</mo><mi id="S6.SS1.10.p5.3.m2.1.1.5.2.3" xref="S6.SS1.10.p5.3.m2.1.1.5.2.3.cmml">p</mi></mrow><mo id="S6.SS1.10.p5.3.m2.1.1.5.1" xref="S6.SS1.10.p5.3.m2.1.1.5.1.cmml">−</mo><mn id="S6.SS1.10.p5.3.m2.1.1.5.3" xref="S6.SS1.10.p5.3.m2.1.1.5.3.cmml">1</mn></mrow><mo id="S6.SS1.10.p5.3.m2.1.1.6" xref="S6.SS1.10.p5.3.m2.1.1.6.cmml">≥</mo><mrow id="S6.SS1.10.p5.3.m2.1.1.1" xref="S6.SS1.10.p5.3.m2.1.1.1.cmml"><mrow id="S6.SS1.10.p5.3.m2.1.1.1.1" xref="S6.SS1.10.p5.3.m2.1.1.1.1.cmml"><mrow id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.cmml"><mo id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.2" stretchy="false" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.cmml"><msub id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.cmml"><mi id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.2" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.2.cmml">k</mi><mn id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.3" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.1" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.1.cmml">+</mo><mi id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.3" mathvariant="normal" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.3" stretchy="false" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS1.10.p5.3.m2.1.1.1.1.2" xref="S6.SS1.10.p5.3.m2.1.1.1.1.2.cmml">⁢</mo><mi id="S6.SS1.10.p5.3.m2.1.1.1.1.3" xref="S6.SS1.10.p5.3.m2.1.1.1.1.3.cmml">p</mi></mrow><mo id="S6.SS1.10.p5.3.m2.1.1.1.2" xref="S6.SS1.10.p5.3.m2.1.1.1.2.cmml">−</mo><mn id="S6.SS1.10.p5.3.m2.1.1.1.3" xref="S6.SS1.10.p5.3.m2.1.1.1.3.cmml">1</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.10.p5.3.m2.1b"><apply id="S6.SS1.10.p5.3.m2.1.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1"><and id="S6.SS1.10.p5.3.m2.1.1a.cmml" xref="S6.SS1.10.p5.3.m2.1.1"></and><apply id="S6.SS1.10.p5.3.m2.1.1b.cmml" xref="S6.SS1.10.p5.3.m2.1.1"><gt id="S6.SS1.10.p5.3.m2.1.1.4.cmml" xref="S6.SS1.10.p5.3.m2.1.1.4"></gt><ci id="S6.SS1.10.p5.3.m2.1.1.3.cmml" xref="S6.SS1.10.p5.3.m2.1.1.3">𝛾</ci><apply id="S6.SS1.10.p5.3.m2.1.1.5.cmml" xref="S6.SS1.10.p5.3.m2.1.1.5"><minus id="S6.SS1.10.p5.3.m2.1.1.5.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1.5.1"></minus><apply id="S6.SS1.10.p5.3.m2.1.1.5.2.cmml" xref="S6.SS1.10.p5.3.m2.1.1.5.2"><times id="S6.SS1.10.p5.3.m2.1.1.5.2.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1.5.2.1"></times><ci id="S6.SS1.10.p5.3.m2.1.1.5.2.2.cmml" xref="S6.SS1.10.p5.3.m2.1.1.5.2.2">𝑗</ci><ci id="S6.SS1.10.p5.3.m2.1.1.5.2.3.cmml" xref="S6.SS1.10.p5.3.m2.1.1.5.2.3">𝑝</ci></apply><cn id="S6.SS1.10.p5.3.m2.1.1.5.3.cmml" type="integer" xref="S6.SS1.10.p5.3.m2.1.1.5.3">1</cn></apply></apply><apply id="S6.SS1.10.p5.3.m2.1.1c.cmml" xref="S6.SS1.10.p5.3.m2.1.1"><geq id="S6.SS1.10.p5.3.m2.1.1.6.cmml" xref="S6.SS1.10.p5.3.m2.1.1.6"></geq><share href="https://arxiv.org/html/2503.14636v1#S6.SS1.10.p5.3.m2.1.1.5.cmml" id="S6.SS1.10.p5.3.m2.1.1d.cmml" xref="S6.SS1.10.p5.3.m2.1.1"></share><apply id="S6.SS1.10.p5.3.m2.1.1.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1"><minus id="S6.SS1.10.p5.3.m2.1.1.1.2.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.2"></minus><apply id="S6.SS1.10.p5.3.m2.1.1.1.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1"><times id="S6.SS1.10.p5.3.m2.1.1.1.1.2.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.2"></times><apply id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1"><plus id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.1"></plus><apply id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.1.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.2.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.2">𝑘</ci><cn id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.2.3">0</cn></apply><ci id="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.3.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.1.1.1.3">ℓ</ci></apply><ci id="S6.SS1.10.p5.3.m2.1.1.1.1.3.cmml" xref="S6.SS1.10.p5.3.m2.1.1.1.1.3">𝑝</ci></apply><cn id="S6.SS1.10.p5.3.m2.1.1.1.3.cmml" type="integer" xref="S6.SS1.10.p5.3.m2.1.1.1.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.10.p5.3.m2.1c">\gamma>jp-1\geq(k_{0}+\ell)p-1</annotation><annotation encoding="application/x-llamapun" id="S6.SS1.10.p5.3.m2.1d">italic_γ > italic_j italic_p - 1 ≥ ( italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ ) italic_p - 1</annotation></semantics></math> and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E1" title="In Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.1</span></a>) follows from Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem2" title="Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.I1.i2" title="item ii ‣ Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a>. This finishes the proof. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S6.SS2"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.2. </span>Main results about complex interpolation of weighted spaces</h3> <div class="ltx_para" id="S6.SS2.p1"> <p class="ltx_p" id="S6.SS2.p1.2">We start with the definition of weighted spaces with vanishing boundary conditions on the half-space. Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.SS2.p1.1.m1.2"><semantics id="S6.SS2.p1.1.m1.2a"><mrow id="S6.SS2.p1.1.m1.2.3" xref="S6.SS2.p1.1.m1.2.3.cmml"><mi id="S6.SS2.p1.1.m1.2.3.2" xref="S6.SS2.p1.1.m1.2.3.2.cmml">p</mi><mo id="S6.SS2.p1.1.m1.2.3.1" xref="S6.SS2.p1.1.m1.2.3.1.cmml">∈</mo><mrow id="S6.SS2.p1.1.m1.2.3.3.2" xref="S6.SS2.p1.1.m1.2.3.3.1.cmml"><mo id="S6.SS2.p1.1.m1.2.3.3.2.1" stretchy="false" xref="S6.SS2.p1.1.m1.2.3.3.1.cmml">(</mo><mn id="S6.SS2.p1.1.m1.1.1" xref="S6.SS2.p1.1.m1.1.1.cmml">1</mn><mo id="S6.SS2.p1.1.m1.2.3.3.2.2" xref="S6.SS2.p1.1.m1.2.3.3.1.cmml">,</mo><mi id="S6.SS2.p1.1.m1.2.2" mathvariant="normal" xref="S6.SS2.p1.1.m1.2.2.cmml">∞</mi><mo id="S6.SS2.p1.1.m1.2.3.3.2.3" stretchy="false" xref="S6.SS2.p1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.1.m1.2b"><apply id="S6.SS2.p1.1.m1.2.3.cmml" xref="S6.SS2.p1.1.m1.2.3"><in id="S6.SS2.p1.1.m1.2.3.1.cmml" xref="S6.SS2.p1.1.m1.2.3.1"></in><ci id="S6.SS2.p1.1.m1.2.3.2.cmml" xref="S6.SS2.p1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S6.SS2.p1.1.m1.2.3.3.1.cmml" xref="S6.SS2.p1.1.m1.2.3.3.2"><cn id="S6.SS2.p1.1.m1.1.1.cmml" type="integer" xref="S6.SS2.p1.1.m1.1.1">1</cn><infinity id="S6.SS2.p1.1.m1.2.2.cmml" xref="S6.SS2.p1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.SS2.p1.2.m2.1"><semantics id="S6.SS2.p1.2.m2.1a"><mi id="S6.SS2.p1.2.m2.1.1" xref="S6.SS2.p1.2.m2.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.2.m2.1b"><ci id="S6.SS2.p1.2.m2.1.1.cmml" xref="S6.SS2.p1.2.m2.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.2.m2.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.2.m2.1d">italic_X</annotation></semantics></math> be a Banach space. Assume that the conditions <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.I4.i1" title="item i ‣ Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a>-<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.I4.i2" title="item ii ‣ Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">ii</span></a> of Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem4" title="Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.4</span></a> hold.</p> <ol class="ltx_enumerate" id="S6.I2"> <li class="ltx_item" id="S6.I2.i1" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(i)</span> <div class="ltx_para" id="S6.I2.i1.p1"> <p class="ltx_p" id="S6.I2.i1.p1.5">If <math alttext="s>0" class="ltx_Math" display="inline" id="S6.I2.i1.p1.1.m1.1"><semantics id="S6.I2.i1.p1.1.m1.1a"><mrow id="S6.I2.i1.p1.1.m1.1.1" xref="S6.I2.i1.p1.1.m1.1.1.cmml"><mi id="S6.I2.i1.p1.1.m1.1.1.2" xref="S6.I2.i1.p1.1.m1.1.1.2.cmml">s</mi><mo id="S6.I2.i1.p1.1.m1.1.1.1" xref="S6.I2.i1.p1.1.m1.1.1.1.cmml">></mo><mn id="S6.I2.i1.p1.1.m1.1.1.3" xref="S6.I2.i1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.1.m1.1b"><apply id="S6.I2.i1.p1.1.m1.1.1.cmml" xref="S6.I2.i1.p1.1.m1.1.1"><gt id="S6.I2.i1.p1.1.m1.1.1.1.cmml" xref="S6.I2.i1.p1.1.m1.1.1.1"></gt><ci id="S6.I2.i1.p1.1.m1.1.1.2.cmml" xref="S6.I2.i1.p1.1.m1.1.1.2">𝑠</ci><cn id="S6.I2.i1.p1.1.m1.1.1.3.cmml" type="integer" xref="S6.I2.i1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.1.m1.1c">s>0</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.1.m1.1d">italic_s > 0</annotation></semantics></math>, <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S6.I2.i1.p1.2.m2.2"><semantics id="S6.I2.i1.p1.2.m2.2a"><mrow id="S6.I2.i1.p1.2.m2.2.2" xref="S6.I2.i1.p1.2.m2.2.2.cmml"><mi id="S6.I2.i1.p1.2.m2.2.2.4" xref="S6.I2.i1.p1.2.m2.2.2.4.cmml">γ</mi><mo id="S6.I2.i1.p1.2.m2.2.2.3" xref="S6.I2.i1.p1.2.m2.2.2.3.cmml">∈</mo><mrow id="S6.I2.i1.p1.2.m2.2.2.2.2" xref="S6.I2.i1.p1.2.m2.2.2.2.3.cmml"><mo id="S6.I2.i1.p1.2.m2.2.2.2.2.3" stretchy="false" xref="S6.I2.i1.p1.2.m2.2.2.2.3.cmml">(</mo><mrow id="S6.I2.i1.p1.2.m2.1.1.1.1.1" xref="S6.I2.i1.p1.2.m2.1.1.1.1.1.cmml"><mo id="S6.I2.i1.p1.2.m2.1.1.1.1.1a" xref="S6.I2.i1.p1.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S6.I2.i1.p1.2.m2.1.1.1.1.1.2" xref="S6.I2.i1.p1.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.I2.i1.p1.2.m2.2.2.2.2.4" xref="S6.I2.i1.p1.2.m2.2.2.2.3.cmml">,</mo><mrow id="S6.I2.i1.p1.2.m2.2.2.2.2.2" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2.cmml"><mi id="S6.I2.i1.p1.2.m2.2.2.2.2.2.2" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2.2.cmml">p</mi><mo id="S6.I2.i1.p1.2.m2.2.2.2.2.2.1" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2.1.cmml">−</mo><mn id="S6.I2.i1.p1.2.m2.2.2.2.2.2.3" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.I2.i1.p1.2.m2.2.2.2.2.5" stretchy="false" xref="S6.I2.i1.p1.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.2.m2.2b"><apply id="S6.I2.i1.p1.2.m2.2.2.cmml" xref="S6.I2.i1.p1.2.m2.2.2"><in id="S6.I2.i1.p1.2.m2.2.2.3.cmml" xref="S6.I2.i1.p1.2.m2.2.2.3"></in><ci id="S6.I2.i1.p1.2.m2.2.2.4.cmml" xref="S6.I2.i1.p1.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S6.I2.i1.p1.2.m2.2.2.2.3.cmml" xref="S6.I2.i1.p1.2.m2.2.2.2.2"><apply id="S6.I2.i1.p1.2.m2.1.1.1.1.1.cmml" xref="S6.I2.i1.p1.2.m2.1.1.1.1.1"><minus id="S6.I2.i1.p1.2.m2.1.1.1.1.1.1.cmml" xref="S6.I2.i1.p1.2.m2.1.1.1.1.1"></minus><cn id="S6.I2.i1.p1.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S6.I2.i1.p1.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S6.I2.i1.p1.2.m2.2.2.2.2.2.cmml" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2"><minus id="S6.I2.i1.p1.2.m2.2.2.2.2.2.1.cmml" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2.1"></minus><ci id="S6.I2.i1.p1.2.m2.2.2.2.2.2.2.cmml" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2.2">𝑝</ci><cn id="S6.I2.i1.p1.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S6.I2.i1.p1.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.2.m2.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.2.m2.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S6.I2.i1.p1.3.m3.1"><semantics id="S6.I2.i1.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S6.I2.i1.p1.3.m3.1.1" xref="S6.I2.i1.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.3.m3.1b"><ci id="S6.I2.i1.p1.3.m3.1.1.cmml" xref="S6.I2.i1.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.3.m3.1d">caligraphic_B</annotation></semantics></math> is of type <math alttext="(p,s,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S6.I2.i1.p1.4.m4.5"><semantics id="S6.I2.i1.p1.4.m4.5a"><mrow id="S6.I2.i1.p1.4.m4.5.6.2" xref="S6.I2.i1.p1.4.m4.5.6.1.cmml"><mo id="S6.I2.i1.p1.4.m4.5.6.2.1" stretchy="false" xref="S6.I2.i1.p1.4.m4.5.6.1.cmml">(</mo><mi id="S6.I2.i1.p1.4.m4.1.1" xref="S6.I2.i1.p1.4.m4.1.1.cmml">p</mi><mo id="S6.I2.i1.p1.4.m4.5.6.2.2" xref="S6.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S6.I2.i1.p1.4.m4.2.2" xref="S6.I2.i1.p1.4.m4.2.2.cmml">s</mi><mo id="S6.I2.i1.p1.4.m4.5.6.2.3" xref="S6.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mi id="S6.I2.i1.p1.4.m4.3.3" xref="S6.I2.i1.p1.4.m4.3.3.cmml">γ</mi><mo id="S6.I2.i1.p1.4.m4.5.6.2.4" xref="S6.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S6.I2.i1.p1.4.m4.4.4" xref="S6.I2.i1.p1.4.m4.4.4.cmml"><mi id="S6.I2.i1.p1.4.m4.4.4.2" xref="S6.I2.i1.p1.4.m4.4.4.2.cmml">m</mi><mo id="S6.I2.i1.p1.4.m4.4.4.1" xref="S6.I2.i1.p1.4.m4.4.4.1.cmml">¯</mo></mover><mo id="S6.I2.i1.p1.4.m4.5.6.2.5" xref="S6.I2.i1.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S6.I2.i1.p1.4.m4.5.5" xref="S6.I2.i1.p1.4.m4.5.5.cmml"><mi id="S6.I2.i1.p1.4.m4.5.5.2" xref="S6.I2.i1.p1.4.m4.5.5.2.cmml">Y</mi><mo id="S6.I2.i1.p1.4.m4.5.5.1" xref="S6.I2.i1.p1.4.m4.5.5.1.cmml">¯</mo></mover><mo id="S6.I2.i1.p1.4.m4.5.6.2.6" stretchy="false" xref="S6.I2.i1.p1.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.4.m4.5b"><vector id="S6.I2.i1.p1.4.m4.5.6.1.cmml" xref="S6.I2.i1.p1.4.m4.5.6.2"><ci id="S6.I2.i1.p1.4.m4.1.1.cmml" xref="S6.I2.i1.p1.4.m4.1.1">𝑝</ci><ci id="S6.I2.i1.p1.4.m4.2.2.cmml" xref="S6.I2.i1.p1.4.m4.2.2">𝑠</ci><ci id="S6.I2.i1.p1.4.m4.3.3.cmml" xref="S6.I2.i1.p1.4.m4.3.3">𝛾</ci><apply id="S6.I2.i1.p1.4.m4.4.4.cmml" xref="S6.I2.i1.p1.4.m4.4.4"><ci id="S6.I2.i1.p1.4.m4.4.4.1.cmml" xref="S6.I2.i1.p1.4.m4.4.4.1">¯</ci><ci id="S6.I2.i1.p1.4.m4.4.4.2.cmml" xref="S6.I2.i1.p1.4.m4.4.4.2">𝑚</ci></apply><apply id="S6.I2.i1.p1.4.m4.5.5.cmml" xref="S6.I2.i1.p1.4.m4.5.5"><ci id="S6.I2.i1.p1.4.m4.5.5.1.cmml" xref="S6.I2.i1.p1.4.m4.5.5.1">¯</ci><ci id="S6.I2.i1.p1.4.m4.5.5.2.cmml" xref="S6.I2.i1.p1.4.m4.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.4.m4.5c">(p,s,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.4.m4.5d">( italic_p , italic_s , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math>, then we define for <math alttext="t\leq s" class="ltx_Math" display="inline" id="S6.I2.i1.p1.5.m5.1"><semantics id="S6.I2.i1.p1.5.m5.1a"><mrow id="S6.I2.i1.p1.5.m5.1.1" xref="S6.I2.i1.p1.5.m5.1.1.cmml"><mi id="S6.I2.i1.p1.5.m5.1.1.2" xref="S6.I2.i1.p1.5.m5.1.1.2.cmml">t</mi><mo id="S6.I2.i1.p1.5.m5.1.1.1" xref="S6.I2.i1.p1.5.m5.1.1.1.cmml">≤</mo><mi id="S6.I2.i1.p1.5.m5.1.1.3" xref="S6.I2.i1.p1.5.m5.1.1.3.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.5.m5.1b"><apply id="S6.I2.i1.p1.5.m5.1.1.cmml" xref="S6.I2.i1.p1.5.m5.1.1"><leq id="S6.I2.i1.p1.5.m5.1.1.1.cmml" xref="S6.I2.i1.p1.5.m5.1.1.1"></leq><ci id="S6.I2.i1.p1.5.m5.1.1.2.cmml" xref="S6.I2.i1.p1.5.m5.1.1.2">𝑡</ci><ci id="S6.I2.i1.p1.5.m5.1.1.3.cmml" xref="S6.I2.i1.p1.5.m5.1.1.3">𝑠</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.5.m5.1c">t\leq s</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i1.p1.5.m5.1d">italic_t ≤ italic_s</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx34"> <tbody id="S6.Ex35"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H^{t,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\Big{\{}f% \in H^{t,p}(\mathbb{R}^{d}_{+},w_{\gamma};X):\mathcal{B}^{m_{i}}f=0\text{ if }% m_{i}+\frac{\gamma+1}{p}<t\Big{\}}." class="ltx_Math" display="inline" id="S6.Ex35.m1.7"><semantics id="S6.Ex35.m1.7a"><mrow id="S6.Ex35.m1.7.7.1" xref="S6.Ex35.m1.7.7.1.1.cmml"><mrow id="S6.Ex35.m1.7.7.1.1" xref="S6.Ex35.m1.7.7.1.1.cmml"><mrow id="S6.Ex35.m1.7.7.1.1.2" xref="S6.Ex35.m1.7.7.1.1.2.cmml"><msubsup id="S6.Ex35.m1.7.7.1.1.2.4" xref="S6.Ex35.m1.7.7.1.1.2.4.cmml"><mi id="S6.Ex35.m1.7.7.1.1.2.4.2.2" xref="S6.Ex35.m1.7.7.1.1.2.4.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex35.m1.7.7.1.1.2.4.3" xref="S6.Ex35.m1.7.7.1.1.2.4.3.cmml">ℬ</mi><mrow id="S6.Ex35.m1.2.2.2.4" xref="S6.Ex35.m1.2.2.2.3.cmml"><mi id="S6.Ex35.m1.1.1.1.1" xref="S6.Ex35.m1.1.1.1.1.cmml">t</mi><mo id="S6.Ex35.m1.2.2.2.4.1" xref="S6.Ex35.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex35.m1.2.2.2.2" xref="S6.Ex35.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S6.Ex35.m1.7.7.1.1.2.3" xref="S6.Ex35.m1.7.7.1.1.2.3.cmml">⁢</mo><mrow id="S6.Ex35.m1.7.7.1.1.2.2.2" xref="S6.Ex35.m1.7.7.1.1.2.2.3.cmml"><mo id="S6.Ex35.m1.7.7.1.1.2.2.2.3" stretchy="false" xref="S6.Ex35.m1.7.7.1.1.2.2.3.cmml">(</mo><msubsup id="S6.Ex35.m1.7.7.1.1.1.1.1.1" xref="S6.Ex35.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S6.Ex35.m1.7.7.1.1.1.1.1.1.2.2" xref="S6.Ex35.m1.7.7.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex35.m1.7.7.1.1.1.1.1.1.3" xref="S6.Ex35.m1.7.7.1.1.1.1.1.1.3.cmml">+</mo><mi id="S6.Ex35.m1.7.7.1.1.1.1.1.1.2.3" xref="S6.Ex35.m1.7.7.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex35.m1.7.7.1.1.2.2.2.4" xref="S6.Ex35.m1.7.7.1.1.2.2.3.cmml">,</mo><msub id="S6.Ex35.m1.7.7.1.1.2.2.2.2" xref="S6.Ex35.m1.7.7.1.1.2.2.2.2.cmml"><mi id="S6.Ex35.m1.7.7.1.1.2.2.2.2.2" xref="S6.Ex35.m1.7.7.1.1.2.2.2.2.2.cmml">w</mi><mi id="S6.Ex35.m1.7.7.1.1.2.2.2.2.3" xref="S6.Ex35.m1.7.7.1.1.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex35.m1.7.7.1.1.2.2.2.5" xref="S6.Ex35.m1.7.7.1.1.2.2.3.cmml">;</mo><mi id="S6.Ex35.m1.5.5" xref="S6.Ex35.m1.5.5.cmml">X</mi><mo id="S6.Ex35.m1.7.7.1.1.2.2.2.6" stretchy="false" xref="S6.Ex35.m1.7.7.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex35.m1.7.7.1.1.5" xref="S6.Ex35.m1.7.7.1.1.5.cmml">=</mo><mrow id="S6.Ex35.m1.7.7.1.1.4.2" xref="S6.Ex35.m1.7.7.1.1.4.3.cmml"><mo id="S6.Ex35.m1.7.7.1.1.4.2.3" maxsize="160%" minsize="160%" 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xref="S6.Ex35.m1.6.6.cmml">X</mi><mo id="S6.Ex35.m1.7.7.1.1.3.1.1.2.2.2.6" rspace="0.278em" stretchy="false" xref="S6.Ex35.m1.7.7.1.1.3.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Ex35.m1.7.7.1.1.4.2.4" rspace="0.278em" xref="S6.Ex35.m1.7.7.1.1.4.3.1.cmml">:</mo><mrow id="S6.Ex35.m1.7.7.1.1.4.2.2" xref="S6.Ex35.m1.7.7.1.1.4.2.2.cmml"><mrow id="S6.Ex35.m1.7.7.1.1.4.2.2.2" xref="S6.Ex35.m1.7.7.1.1.4.2.2.2.cmml"><msup id="S6.Ex35.m1.7.7.1.1.4.2.2.2.2" xref="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.2" xref="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.2.cmml">ℬ</mi><msub id="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.3" xref="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.3.cmml"><mi id="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.3.2" xref="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.3.2.cmml">m</mi><mi id="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.3.3" xref="S6.Ex35.m1.7.7.1.1.4.2.2.2.2.3.3.cmml">i</mi></msub></msup><mo id="S6.Ex35.m1.7.7.1.1.4.2.2.2.1" xref="S6.Ex35.m1.7.7.1.1.4.2.2.2.1.cmml">⁢</mo><mi 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id="S6.Ex35.m1.7.7.1.1.4.2.2c.cmml" xref="S6.Ex35.m1.7.7.1.1.4.2.2"><lt id="S6.Ex35.m1.7.7.1.1.4.2.2.5.cmml" xref="S6.Ex35.m1.7.7.1.1.4.2.2.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S6.Ex35.m1.7.7.1.1.4.2.2.4.cmml" id="S6.Ex35.m1.7.7.1.1.4.2.2d.cmml" xref="S6.Ex35.m1.7.7.1.1.4.2.2"></share><ci id="S6.Ex35.m1.7.7.1.1.4.2.2.6.cmml" xref="S6.Ex35.m1.7.7.1.1.4.2.2.6">𝑡</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex35.m1.7c">\displaystyle H^{t,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\Big{\{}f% \in H^{t,p}(\mathbb{R}^{d}_{+},w_{\gamma};X):\mathcal{B}^{m_{i}}f=0\text{ if }% m_{i}+\frac{\gamma+1}{p}<t\Big{\}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex35.m1.7d">italic_H start_POSTSUPERSCRIPT italic_t , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = { italic_f ∈ italic_H start_POSTSUPERSCRIPT italic_t , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) : caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f = 0 if italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_t } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> <li class="ltx_item" id="S6.I2.i2" style="list-style-type:none;"> <span class="ltx_tag ltx_tag_item">(ii)</span> <div class="ltx_para" id="S6.I2.i2.p1"> <p class="ltx_p" id="S6.I2.i2.p1.5">If <math alttext="k\in\mathbb{N}_{1}" class="ltx_Math" display="inline" id="S6.I2.i2.p1.1.m1.1"><semantics id="S6.I2.i2.p1.1.m1.1a"><mrow id="S6.I2.i2.p1.1.m1.1.1" xref="S6.I2.i2.p1.1.m1.1.1.cmml"><mi id="S6.I2.i2.p1.1.m1.1.1.2" xref="S6.I2.i2.p1.1.m1.1.1.2.cmml">k</mi><mo id="S6.I2.i2.p1.1.m1.1.1.1" xref="S6.I2.i2.p1.1.m1.1.1.1.cmml">∈</mo><msub id="S6.I2.i2.p1.1.m1.1.1.3" xref="S6.I2.i2.p1.1.m1.1.1.3.cmml"><mi id="S6.I2.i2.p1.1.m1.1.1.3.2" xref="S6.I2.i2.p1.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S6.I2.i2.p1.1.m1.1.1.3.3" xref="S6.I2.i2.p1.1.m1.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.1.m1.1b"><apply id="S6.I2.i2.p1.1.m1.1.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1"><in id="S6.I2.i2.p1.1.m1.1.1.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1.1"></in><ci id="S6.I2.i2.p1.1.m1.1.1.2.cmml" xref="S6.I2.i2.p1.1.m1.1.1.2">𝑘</ci><apply id="S6.I2.i2.p1.1.m1.1.1.3.cmml" xref="S6.I2.i2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.I2.i2.p1.1.m1.1.1.3.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1.3">subscript</csymbol><ci id="S6.I2.i2.p1.1.m1.1.1.3.2.cmml" xref="S6.I2.i2.p1.1.m1.1.1.3.2">ℕ</ci><cn id="S6.I2.i2.p1.1.m1.1.1.3.3.cmml" type="integer" xref="S6.I2.i2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.1.m1.1c">k\in\mathbb{N}_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.1.m1.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S6.I2.i2.p1.2.m2.4"><semantics id="S6.I2.i2.p1.2.m2.4a"><mrow id="S6.I2.i2.p1.2.m2.4.4" xref="S6.I2.i2.p1.2.m2.4.4.cmml"><mi id="S6.I2.i2.p1.2.m2.4.4.5" xref="S6.I2.i2.p1.2.m2.4.4.5.cmml">γ</mi><mo id="S6.I2.i2.p1.2.m2.4.4.4" xref="S6.I2.i2.p1.2.m2.4.4.4.cmml">∈</mo><mrow id="S6.I2.i2.p1.2.m2.4.4.3" xref="S6.I2.i2.p1.2.m2.4.4.3.cmml"><mrow id="S6.I2.i2.p1.2.m2.2.2.1.1.1" xref="S6.I2.i2.p1.2.m2.2.2.1.1.2.cmml"><mo id="S6.I2.i2.p1.2.m2.2.2.1.1.1.2" stretchy="false" xref="S6.I2.i2.p1.2.m2.2.2.1.1.2.cmml">(</mo><mrow id="S6.I2.i2.p1.2.m2.2.2.1.1.1.1" xref="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.cmml"><mo id="S6.I2.i2.p1.2.m2.2.2.1.1.1.1a" xref="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.cmml">−</mo><mn id="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.2" xref="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.I2.i2.p1.2.m2.2.2.1.1.1.3" xref="S6.I2.i2.p1.2.m2.2.2.1.1.2.cmml">,</mo><mi id="S6.I2.i2.p1.2.m2.1.1" mathvariant="normal" xref="S6.I2.i2.p1.2.m2.1.1.cmml">∞</mi><mo id="S6.I2.i2.p1.2.m2.2.2.1.1.1.4" stretchy="false" xref="S6.I2.i2.p1.2.m2.2.2.1.1.2.cmml">)</mo></mrow><mo id="S6.I2.i2.p1.2.m2.4.4.3.4" xref="S6.I2.i2.p1.2.m2.4.4.3.4.cmml">∖</mo><mrow id="S6.I2.i2.p1.2.m2.4.4.3.3.2" xref="S6.I2.i2.p1.2.m2.4.4.3.3.3.cmml"><mo id="S6.I2.i2.p1.2.m2.4.4.3.3.2.3" stretchy="false" xref="S6.I2.i2.p1.2.m2.4.4.3.3.3.1.cmml">{</mo><mrow id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.cmml"><mrow id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.cmml"><mi id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.2" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.1" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.3" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.1" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.1.cmml">−</mo><mn id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.3" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S6.I2.i2.p1.2.m2.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.I2.i2.p1.2.m2.4.4.3.3.3.1.cmml">:</mo><mrow id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.cmml"><mi id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.2" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.2.cmml">j</mi><mo id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.1" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.cmml"><mi id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.2" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.3" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.I2.i2.p1.2.m2.4.4.3.3.2.5" stretchy="false" xref="S6.I2.i2.p1.2.m2.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.2.m2.4b"><apply id="S6.I2.i2.p1.2.m2.4.4.cmml" xref="S6.I2.i2.p1.2.m2.4.4"><in id="S6.I2.i2.p1.2.m2.4.4.4.cmml" xref="S6.I2.i2.p1.2.m2.4.4.4"></in><ci id="S6.I2.i2.p1.2.m2.4.4.5.cmml" xref="S6.I2.i2.p1.2.m2.4.4.5">𝛾</ci><apply id="S6.I2.i2.p1.2.m2.4.4.3.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3"><setdiff id="S6.I2.i2.p1.2.m2.4.4.3.4.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.4"></setdiff><interval closure="open" id="S6.I2.i2.p1.2.m2.2.2.1.1.2.cmml" xref="S6.I2.i2.p1.2.m2.2.2.1.1.1"><apply id="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.cmml" xref="S6.I2.i2.p1.2.m2.2.2.1.1.1.1"><minus id="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S6.I2.i2.p1.2.m2.2.2.1.1.1.1"></minus><cn id="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.2.cmml" type="integer" xref="S6.I2.i2.p1.2.m2.2.2.1.1.1.1.2">1</cn></apply><infinity id="S6.I2.i2.p1.2.m2.1.1.cmml" xref="S6.I2.i2.p1.2.m2.1.1"></infinity></interval><apply id="S6.I2.i2.p1.2.m2.4.4.3.3.3.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2"><csymbol cd="latexml" id="S6.I2.i2.p1.2.m2.4.4.3.3.3.1.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.3">conditional-set</csymbol><apply id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.cmml" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1"><minus id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.1.cmml" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.1"></minus><apply id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.cmml" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2"><times id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.1.cmml" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.1"></times><ci id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.2.cmml" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.3.cmml" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.3.cmml" type="integer" xref="S6.I2.i2.p1.2.m2.3.3.2.2.1.1.3">1</cn></apply><apply id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2"><in id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.1.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.1"></in><ci id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.2.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.2">𝑗</ci><apply id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.1.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3">subscript</csymbol><ci id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.2.cmml" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S6.I2.i2.p1.2.m2.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.2.m2.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.2.m2.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> and <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S6.I2.i2.p1.3.m3.1"><semantics id="S6.I2.i2.p1.3.m3.1a"><mi class="ltx_font_mathcaligraphic" id="S6.I2.i2.p1.3.m3.1.1" xref="S6.I2.i2.p1.3.m3.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.3.m3.1b"><ci id="S6.I2.i2.p1.3.m3.1.1.cmml" xref="S6.I2.i2.p1.3.m3.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.3.m3.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.3.m3.1d">caligraphic_B</annotation></semantics></math> is of type <math alttext="(p,k,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S6.I2.i2.p1.4.m4.5"><semantics id="S6.I2.i2.p1.4.m4.5a"><mrow id="S6.I2.i2.p1.4.m4.5.6.2" xref="S6.I2.i2.p1.4.m4.5.6.1.cmml"><mo id="S6.I2.i2.p1.4.m4.5.6.2.1" stretchy="false" xref="S6.I2.i2.p1.4.m4.5.6.1.cmml">(</mo><mi id="S6.I2.i2.p1.4.m4.1.1" xref="S6.I2.i2.p1.4.m4.1.1.cmml">p</mi><mo id="S6.I2.i2.p1.4.m4.5.6.2.2" xref="S6.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S6.I2.i2.p1.4.m4.2.2" xref="S6.I2.i2.p1.4.m4.2.2.cmml">k</mi><mo id="S6.I2.i2.p1.4.m4.5.6.2.3" xref="S6.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mi id="S6.I2.i2.p1.4.m4.3.3" xref="S6.I2.i2.p1.4.m4.3.3.cmml">γ</mi><mo id="S6.I2.i2.p1.4.m4.5.6.2.4" xref="S6.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S6.I2.i2.p1.4.m4.4.4" xref="S6.I2.i2.p1.4.m4.4.4.cmml"><mi id="S6.I2.i2.p1.4.m4.4.4.2" xref="S6.I2.i2.p1.4.m4.4.4.2.cmml">m</mi><mo id="S6.I2.i2.p1.4.m4.4.4.1" xref="S6.I2.i2.p1.4.m4.4.4.1.cmml">¯</mo></mover><mo id="S6.I2.i2.p1.4.m4.5.6.2.5" xref="S6.I2.i2.p1.4.m4.5.6.1.cmml">,</mo><mover accent="true" id="S6.I2.i2.p1.4.m4.5.5" xref="S6.I2.i2.p1.4.m4.5.5.cmml"><mi id="S6.I2.i2.p1.4.m4.5.5.2" xref="S6.I2.i2.p1.4.m4.5.5.2.cmml">Y</mi><mo id="S6.I2.i2.p1.4.m4.5.5.1" xref="S6.I2.i2.p1.4.m4.5.5.1.cmml">¯</mo></mover><mo id="S6.I2.i2.p1.4.m4.5.6.2.6" stretchy="false" xref="S6.I2.i2.p1.4.m4.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.4.m4.5b"><vector id="S6.I2.i2.p1.4.m4.5.6.1.cmml" xref="S6.I2.i2.p1.4.m4.5.6.2"><ci id="S6.I2.i2.p1.4.m4.1.1.cmml" xref="S6.I2.i2.p1.4.m4.1.1">𝑝</ci><ci id="S6.I2.i2.p1.4.m4.2.2.cmml" xref="S6.I2.i2.p1.4.m4.2.2">𝑘</ci><ci id="S6.I2.i2.p1.4.m4.3.3.cmml" xref="S6.I2.i2.p1.4.m4.3.3">𝛾</ci><apply id="S6.I2.i2.p1.4.m4.4.4.cmml" xref="S6.I2.i2.p1.4.m4.4.4"><ci id="S6.I2.i2.p1.4.m4.4.4.1.cmml" xref="S6.I2.i2.p1.4.m4.4.4.1">¯</ci><ci id="S6.I2.i2.p1.4.m4.4.4.2.cmml" xref="S6.I2.i2.p1.4.m4.4.4.2">𝑚</ci></apply><apply id="S6.I2.i2.p1.4.m4.5.5.cmml" xref="S6.I2.i2.p1.4.m4.5.5"><ci id="S6.I2.i2.p1.4.m4.5.5.1.cmml" xref="S6.I2.i2.p1.4.m4.5.5.1">¯</ci><ci id="S6.I2.i2.p1.4.m4.5.5.2.cmml" xref="S6.I2.i2.p1.4.m4.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.4.m4.5c">(p,k,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.4.m4.5d">( italic_p , italic_k , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math>, then we define for <math alttext="\ell\leq k" class="ltx_Math" display="inline" id="S6.I2.i2.p1.5.m5.1"><semantics id="S6.I2.i2.p1.5.m5.1a"><mrow id="S6.I2.i2.p1.5.m5.1.1" xref="S6.I2.i2.p1.5.m5.1.1.cmml"><mi id="S6.I2.i2.p1.5.m5.1.1.2" mathvariant="normal" xref="S6.I2.i2.p1.5.m5.1.1.2.cmml">ℓ</mi><mo id="S6.I2.i2.p1.5.m5.1.1.1" xref="S6.I2.i2.p1.5.m5.1.1.1.cmml">≤</mo><mi id="S6.I2.i2.p1.5.m5.1.1.3" xref="S6.I2.i2.p1.5.m5.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.5.m5.1b"><apply id="S6.I2.i2.p1.5.m5.1.1.cmml" xref="S6.I2.i2.p1.5.m5.1.1"><leq id="S6.I2.i2.p1.5.m5.1.1.1.cmml" xref="S6.I2.i2.p1.5.m5.1.1.1"></leq><ci id="S6.I2.i2.p1.5.m5.1.1.2.cmml" xref="S6.I2.i2.p1.5.m5.1.1.2">ℓ</ci><ci id="S6.I2.i2.p1.5.m5.1.1.3.cmml" xref="S6.I2.i2.p1.5.m5.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.5.m5.1c">\ell\leq k</annotation><annotation encoding="application/x-llamapun" id="S6.I2.i2.p1.5.m5.1d">roman_ℓ ≤ italic_k</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx35"> <tbody id="S6.Ex36"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W^{\ell,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\Big{% \{}f\in W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X):\mathcal{B}^{m_{i}}f=0% \text{ if }m_{i}+\frac{\gamma+1}{p}<\ell\Big{\}}." class="ltx_Math" display="inline" id="S6.Ex36.m1.7"><semantics id="S6.Ex36.m1.7a"><mrow id="S6.Ex36.m1.7.7.1" xref="S6.Ex36.m1.7.7.1.1.cmml"><mrow id="S6.Ex36.m1.7.7.1.1" xref="S6.Ex36.m1.7.7.1.1.cmml"><mrow id="S6.Ex36.m1.7.7.1.1.2" xref="S6.Ex36.m1.7.7.1.1.2.cmml"><msubsup id="S6.Ex36.m1.7.7.1.1.2.4" xref="S6.Ex36.m1.7.7.1.1.2.4.cmml"><mi id="S6.Ex36.m1.7.7.1.1.2.4.2.2" xref="S6.Ex36.m1.7.7.1.1.2.4.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex36.m1.7.7.1.1.2.4.3" xref="S6.Ex36.m1.7.7.1.1.2.4.3.cmml">ℬ</mi><mrow id="S6.Ex36.m1.2.2.2.4" xref="S6.Ex36.m1.2.2.2.3.cmml"><mi id="S6.Ex36.m1.1.1.1.1" mathvariant="normal" xref="S6.Ex36.m1.1.1.1.1.cmml">ℓ</mi><mo id="S6.Ex36.m1.2.2.2.4.1" xref="S6.Ex36.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex36.m1.2.2.2.2" xref="S6.Ex36.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S6.Ex36.m1.7.7.1.1.2.3" xref="S6.Ex36.m1.7.7.1.1.2.3.cmml">⁢</mo><mrow id="S6.Ex36.m1.7.7.1.1.2.2.2" xref="S6.Ex36.m1.7.7.1.1.2.2.3.cmml"><mo id="S6.Ex36.m1.7.7.1.1.2.2.2.3" stretchy="false" xref="S6.Ex36.m1.7.7.1.1.2.2.3.cmml">(</mo><msubsup id="S6.Ex36.m1.7.7.1.1.1.1.1.1" xref="S6.Ex36.m1.7.7.1.1.1.1.1.1.cmml"><mi id="S6.Ex36.m1.7.7.1.1.1.1.1.1.2.2" xref="S6.Ex36.m1.7.7.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex36.m1.7.7.1.1.1.1.1.1.3" xref="S6.Ex36.m1.7.7.1.1.1.1.1.1.3.cmml">+</mo><mi id="S6.Ex36.m1.7.7.1.1.1.1.1.1.2.3" xref="S6.Ex36.m1.7.7.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex36.m1.7.7.1.1.2.2.2.4" xref="S6.Ex36.m1.7.7.1.1.2.2.3.cmml">,</mo><msub id="S6.Ex36.m1.7.7.1.1.2.2.2.2" xref="S6.Ex36.m1.7.7.1.1.2.2.2.2.cmml"><mi id="S6.Ex36.m1.7.7.1.1.2.2.2.2.2" xref="S6.Ex36.m1.7.7.1.1.2.2.2.2.2.cmml">w</mi><mi id="S6.Ex36.m1.7.7.1.1.2.2.2.2.3" xref="S6.Ex36.m1.7.7.1.1.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex36.m1.7.7.1.1.2.2.2.5" xref="S6.Ex36.m1.7.7.1.1.2.2.3.cmml">;</mo><mi id="S6.Ex36.m1.5.5" xref="S6.Ex36.m1.5.5.cmml">X</mi><mo id="S6.Ex36.m1.7.7.1.1.2.2.2.6" stretchy="false" xref="S6.Ex36.m1.7.7.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex36.m1.7.7.1.1.5" xref="S6.Ex36.m1.7.7.1.1.5.cmml">=</mo><mrow id="S6.Ex36.m1.7.7.1.1.4.2" xref="S6.Ex36.m1.7.7.1.1.4.3.cmml"><mo id="S6.Ex36.m1.7.7.1.1.4.2.3" maxsize="160%" minsize="160%" xref="S6.Ex36.m1.7.7.1.1.4.3.1.cmml">{</mo><mrow id="S6.Ex36.m1.7.7.1.1.3.1.1" xref="S6.Ex36.m1.7.7.1.1.3.1.1.cmml"><mi id="S6.Ex36.m1.7.7.1.1.3.1.1.4" xref="S6.Ex36.m1.7.7.1.1.3.1.1.4.cmml">f</mi><mo id="S6.Ex36.m1.7.7.1.1.3.1.1.3" xref="S6.Ex36.m1.7.7.1.1.3.1.1.3.cmml">∈</mo><mrow id="S6.Ex36.m1.7.7.1.1.3.1.1.2" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.cmml"><msup id="S6.Ex36.m1.7.7.1.1.3.1.1.2.4" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.4.cmml"><mi id="S6.Ex36.m1.7.7.1.1.3.1.1.2.4.2" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.4.2.cmml">W</mi><mrow id="S6.Ex36.m1.4.4.2.4" xref="S6.Ex36.m1.4.4.2.3.cmml"><mi id="S6.Ex36.m1.3.3.1.1" mathvariant="normal" xref="S6.Ex36.m1.3.3.1.1.cmml">ℓ</mi><mo id="S6.Ex36.m1.4.4.2.4.1" xref="S6.Ex36.m1.4.4.2.3.cmml">,</mo><mi id="S6.Ex36.m1.4.4.2.2" xref="S6.Ex36.m1.4.4.2.2.cmml">p</mi></mrow></msup><mo id="S6.Ex36.m1.7.7.1.1.3.1.1.2.3" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.3.cmml">⁢</mo><mrow id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.3.cmml"><mo id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.3" stretchy="false" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.3.cmml">(</mo><msubsup id="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1" xref="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1.cmml"><mi id="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1.2.2" xref="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1.3" xref="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1.3.cmml">+</mo><mi id="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1.2.3" xref="S6.Ex36.m1.7.7.1.1.3.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.4" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.3.cmml">,</mo><msub id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.2" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.2.cmml"><mi id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.2.2" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.2.2.cmml">w</mi><mi id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.2.3" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.5" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.3.cmml">;</mo><mi id="S6.Ex36.m1.6.6" xref="S6.Ex36.m1.6.6.cmml">X</mi><mo id="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.2.6" rspace="0.278em" stretchy="false" xref="S6.Ex36.m1.7.7.1.1.3.1.1.2.2.3.cmml">)</mo></mrow></mrow></mrow><mo id="S6.Ex36.m1.7.7.1.1.4.2.4" rspace="0.278em" xref="S6.Ex36.m1.7.7.1.1.4.3.1.cmml">:</mo><mrow id="S6.Ex36.m1.7.7.1.1.4.2.2" xref="S6.Ex36.m1.7.7.1.1.4.2.2.cmml"><mrow id="S6.Ex36.m1.7.7.1.1.4.2.2.2" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.cmml"><msup id="S6.Ex36.m1.7.7.1.1.4.2.2.2.2" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.2" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.2.cmml">ℬ</mi><msub id="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.3" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.3.cmml"><mi id="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.3.2" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.3.2.cmml">m</mi><mi id="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.3.3" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.2.3.3.cmml">i</mi></msub></msup><mo id="S6.Ex36.m1.7.7.1.1.4.2.2.2.1" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.1.cmml">⁢</mo><mi id="S6.Ex36.m1.7.7.1.1.4.2.2.2.3" xref="S6.Ex36.m1.7.7.1.1.4.2.2.2.3.cmml">f</mi></mrow><mo id="S6.Ex36.m1.7.7.1.1.4.2.2.3" xref="S6.Ex36.m1.7.7.1.1.4.2.2.3.cmml">=</mo><mrow id="S6.Ex36.m1.7.7.1.1.4.2.2.4" xref="S6.Ex36.m1.7.7.1.1.4.2.2.4.cmml"><mrow id="S6.Ex36.m1.7.7.1.1.4.2.2.4.2" xref="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.cmml"><mn id="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.2" xref="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.2.cmml">0</mn><mo id="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.1" xref="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.1.cmml">⁢</mo><mtext id="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.3" xref="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.3a.cmml"> if </mtext><mo id="S6.Ex36.m1.7.7.1.1.4.2.2.4.2.1a" 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\{}f\in W^{\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X):\mathcal{B}^{m_{i}}f=0% \text{ if }m_{i}+\frac{\gamma+1}{p}<\ell\Big{\}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex36.m1.7d">italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = { italic_f ∈ italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) : caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f = 0 if italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < roman_ℓ } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </li> </ol> <p class="ltx_p" id="S6.SS2.p1.5">All the traces in the above definition are well defined by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem6" title="Theorem 5.6. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.6</span></a>. We note that if <math alttext="t,\ell<m_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS2.p1.3.m1.2"><semantics id="S6.SS2.p1.3.m1.2a"><mrow id="S6.SS2.p1.3.m1.2.3" xref="S6.SS2.p1.3.m1.2.3.cmml"><mrow id="S6.SS2.p1.3.m1.2.3.2.2" xref="S6.SS2.p1.3.m1.2.3.2.1.cmml"><mi id="S6.SS2.p1.3.m1.1.1" xref="S6.SS2.p1.3.m1.1.1.cmml">t</mi><mo id="S6.SS2.p1.3.m1.2.3.2.2.1" xref="S6.SS2.p1.3.m1.2.3.2.1.cmml">,</mo><mi id="S6.SS2.p1.3.m1.2.2" mathvariant="normal" xref="S6.SS2.p1.3.m1.2.2.cmml">ℓ</mi></mrow><mo id="S6.SS2.p1.3.m1.2.3.1" xref="S6.SS2.p1.3.m1.2.3.1.cmml"><</mo><mrow id="S6.SS2.p1.3.m1.2.3.3" xref="S6.SS2.p1.3.m1.2.3.3.cmml"><msub id="S6.SS2.p1.3.m1.2.3.3.2" xref="S6.SS2.p1.3.m1.2.3.3.2.cmml"><mi id="S6.SS2.p1.3.m1.2.3.3.2.2" xref="S6.SS2.p1.3.m1.2.3.3.2.2.cmml">m</mi><mn id="S6.SS2.p1.3.m1.2.3.3.2.3" xref="S6.SS2.p1.3.m1.2.3.3.2.3.cmml">0</mn></msub><mo id="S6.SS2.p1.3.m1.2.3.3.1" xref="S6.SS2.p1.3.m1.2.3.3.1.cmml">+</mo><mfrac id="S6.SS2.p1.3.m1.2.3.3.3" xref="S6.SS2.p1.3.m1.2.3.3.3.cmml"><mrow id="S6.SS2.p1.3.m1.2.3.3.3.2" xref="S6.SS2.p1.3.m1.2.3.3.3.2.cmml"><mi id="S6.SS2.p1.3.m1.2.3.3.3.2.2" xref="S6.SS2.p1.3.m1.2.3.3.3.2.2.cmml">γ</mi><mo id="S6.SS2.p1.3.m1.2.3.3.3.2.1" xref="S6.SS2.p1.3.m1.2.3.3.3.2.1.cmml">+</mo><mn id="S6.SS2.p1.3.m1.2.3.3.3.2.3" xref="S6.SS2.p1.3.m1.2.3.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS2.p1.3.m1.2.3.3.3.3" xref="S6.SS2.p1.3.m1.2.3.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.3.m1.2b"><apply id="S6.SS2.p1.3.m1.2.3.cmml" xref="S6.SS2.p1.3.m1.2.3"><lt id="S6.SS2.p1.3.m1.2.3.1.cmml" xref="S6.SS2.p1.3.m1.2.3.1"></lt><list id="S6.SS2.p1.3.m1.2.3.2.1.cmml" xref="S6.SS2.p1.3.m1.2.3.2.2"><ci id="S6.SS2.p1.3.m1.1.1.cmml" xref="S6.SS2.p1.3.m1.1.1">𝑡</ci><ci id="S6.SS2.p1.3.m1.2.2.cmml" xref="S6.SS2.p1.3.m1.2.2">ℓ</ci></list><apply id="S6.SS2.p1.3.m1.2.3.3.cmml" xref="S6.SS2.p1.3.m1.2.3.3"><plus id="S6.SS2.p1.3.m1.2.3.3.1.cmml" xref="S6.SS2.p1.3.m1.2.3.3.1"></plus><apply id="S6.SS2.p1.3.m1.2.3.3.2.cmml" xref="S6.SS2.p1.3.m1.2.3.3.2"><csymbol cd="ambiguous" id="S6.SS2.p1.3.m1.2.3.3.2.1.cmml" xref="S6.SS2.p1.3.m1.2.3.3.2">subscript</csymbol><ci id="S6.SS2.p1.3.m1.2.3.3.2.2.cmml" xref="S6.SS2.p1.3.m1.2.3.3.2.2">𝑚</ci><cn id="S6.SS2.p1.3.m1.2.3.3.2.3.cmml" type="integer" xref="S6.SS2.p1.3.m1.2.3.3.2.3">0</cn></apply><apply id="S6.SS2.p1.3.m1.2.3.3.3.cmml" xref="S6.SS2.p1.3.m1.2.3.3.3"><divide id="S6.SS2.p1.3.m1.2.3.3.3.1.cmml" xref="S6.SS2.p1.3.m1.2.3.3.3"></divide><apply id="S6.SS2.p1.3.m1.2.3.3.3.2.cmml" xref="S6.SS2.p1.3.m1.2.3.3.3.2"><plus id="S6.SS2.p1.3.m1.2.3.3.3.2.1.cmml" xref="S6.SS2.p1.3.m1.2.3.3.3.2.1"></plus><ci id="S6.SS2.p1.3.m1.2.3.3.3.2.2.cmml" xref="S6.SS2.p1.3.m1.2.3.3.3.2.2">𝛾</ci><cn id="S6.SS2.p1.3.m1.2.3.3.3.2.3.cmml" type="integer" xref="S6.SS2.p1.3.m1.2.3.3.3.2.3">1</cn></apply><ci id="S6.SS2.p1.3.m1.2.3.3.3.3.cmml" xref="S6.SS2.p1.3.m1.2.3.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.3.m1.2c">t,\ell<m_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.3.m1.2d">italic_t , roman_ℓ < italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>, then the condition <math alttext="\mathcal{B}^{m_{i}}f=0" class="ltx_Math" display="inline" id="S6.SS2.p1.4.m2.1"><semantics id="S6.SS2.p1.4.m2.1a"><mrow id="S6.SS2.p1.4.m2.1.1" xref="S6.SS2.p1.4.m2.1.1.cmml"><mrow id="S6.SS2.p1.4.m2.1.1.2" xref="S6.SS2.p1.4.m2.1.1.2.cmml"><msup id="S6.SS2.p1.4.m2.1.1.2.2" xref="S6.SS2.p1.4.m2.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS2.p1.4.m2.1.1.2.2.2" xref="S6.SS2.p1.4.m2.1.1.2.2.2.cmml">ℬ</mi><msub id="S6.SS2.p1.4.m2.1.1.2.2.3" xref="S6.SS2.p1.4.m2.1.1.2.2.3.cmml"><mi id="S6.SS2.p1.4.m2.1.1.2.2.3.2" xref="S6.SS2.p1.4.m2.1.1.2.2.3.2.cmml">m</mi><mi id="S6.SS2.p1.4.m2.1.1.2.2.3.3" xref="S6.SS2.p1.4.m2.1.1.2.2.3.3.cmml">i</mi></msub></msup><mo id="S6.SS2.p1.4.m2.1.1.2.1" xref="S6.SS2.p1.4.m2.1.1.2.1.cmml">⁢</mo><mi id="S6.SS2.p1.4.m2.1.1.2.3" xref="S6.SS2.p1.4.m2.1.1.2.3.cmml">f</mi></mrow><mo id="S6.SS2.p1.4.m2.1.1.1" xref="S6.SS2.p1.4.m2.1.1.1.cmml">=</mo><mn id="S6.SS2.p1.4.m2.1.1.3" xref="S6.SS2.p1.4.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.4.m2.1b"><apply id="S6.SS2.p1.4.m2.1.1.cmml" xref="S6.SS2.p1.4.m2.1.1"><eq id="S6.SS2.p1.4.m2.1.1.1.cmml" xref="S6.SS2.p1.4.m2.1.1.1"></eq><apply id="S6.SS2.p1.4.m2.1.1.2.cmml" xref="S6.SS2.p1.4.m2.1.1.2"><times id="S6.SS2.p1.4.m2.1.1.2.1.cmml" xref="S6.SS2.p1.4.m2.1.1.2.1"></times><apply id="S6.SS2.p1.4.m2.1.1.2.2.cmml" xref="S6.SS2.p1.4.m2.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.p1.4.m2.1.1.2.2.1.cmml" xref="S6.SS2.p1.4.m2.1.1.2.2">superscript</csymbol><ci id="S6.SS2.p1.4.m2.1.1.2.2.2.cmml" xref="S6.SS2.p1.4.m2.1.1.2.2.2">ℬ</ci><apply id="S6.SS2.p1.4.m2.1.1.2.2.3.cmml" xref="S6.SS2.p1.4.m2.1.1.2.2.3"><csymbol cd="ambiguous" id="S6.SS2.p1.4.m2.1.1.2.2.3.1.cmml" xref="S6.SS2.p1.4.m2.1.1.2.2.3">subscript</csymbol><ci id="S6.SS2.p1.4.m2.1.1.2.2.3.2.cmml" xref="S6.SS2.p1.4.m2.1.1.2.2.3.2">𝑚</ci><ci id="S6.SS2.p1.4.m2.1.1.2.2.3.3.cmml" xref="S6.SS2.p1.4.m2.1.1.2.2.3.3">𝑖</ci></apply></apply><ci id="S6.SS2.p1.4.m2.1.1.2.3.cmml" xref="S6.SS2.p1.4.m2.1.1.2.3">𝑓</ci></apply><cn id="S6.SS2.p1.4.m2.1.1.3.cmml" type="integer" xref="S6.SS2.p1.4.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.4.m2.1c">\mathcal{B}^{m_{i}}f=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.4.m2.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f = 0</annotation></semantics></math> is empty for all <math alttext="i\in\{0,\dots,n\}" class="ltx_Math" display="inline" id="S6.SS2.p1.5.m3.3"><semantics id="S6.SS2.p1.5.m3.3a"><mrow id="S6.SS2.p1.5.m3.3.4" xref="S6.SS2.p1.5.m3.3.4.cmml"><mi id="S6.SS2.p1.5.m3.3.4.2" xref="S6.SS2.p1.5.m3.3.4.2.cmml">i</mi><mo id="S6.SS2.p1.5.m3.3.4.1" xref="S6.SS2.p1.5.m3.3.4.1.cmml">∈</mo><mrow id="S6.SS2.p1.5.m3.3.4.3.2" xref="S6.SS2.p1.5.m3.3.4.3.1.cmml"><mo id="S6.SS2.p1.5.m3.3.4.3.2.1" stretchy="false" xref="S6.SS2.p1.5.m3.3.4.3.1.cmml">{</mo><mn id="S6.SS2.p1.5.m3.1.1" xref="S6.SS2.p1.5.m3.1.1.cmml">0</mn><mo id="S6.SS2.p1.5.m3.3.4.3.2.2" xref="S6.SS2.p1.5.m3.3.4.3.1.cmml">,</mo><mi id="S6.SS2.p1.5.m3.2.2" mathvariant="normal" xref="S6.SS2.p1.5.m3.2.2.cmml">…</mi><mo id="S6.SS2.p1.5.m3.3.4.3.2.3" xref="S6.SS2.p1.5.m3.3.4.3.1.cmml">,</mo><mi id="S6.SS2.p1.5.m3.3.3" xref="S6.SS2.p1.5.m3.3.3.cmml">n</mi><mo id="S6.SS2.p1.5.m3.3.4.3.2.4" stretchy="false" xref="S6.SS2.p1.5.m3.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p1.5.m3.3b"><apply id="S6.SS2.p1.5.m3.3.4.cmml" xref="S6.SS2.p1.5.m3.3.4"><in id="S6.SS2.p1.5.m3.3.4.1.cmml" xref="S6.SS2.p1.5.m3.3.4.1"></in><ci id="S6.SS2.p1.5.m3.3.4.2.cmml" xref="S6.SS2.p1.5.m3.3.4.2">𝑖</ci><set id="S6.SS2.p1.5.m3.3.4.3.1.cmml" xref="S6.SS2.p1.5.m3.3.4.3.2"><cn id="S6.SS2.p1.5.m3.1.1.cmml" type="integer" xref="S6.SS2.p1.5.m3.1.1">0</cn><ci id="S6.SS2.p1.5.m3.2.2.cmml" xref="S6.SS2.p1.5.m3.2.2">…</ci><ci id="S6.SS2.p1.5.m3.3.3.cmml" xref="S6.SS2.p1.5.m3.3.3">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p1.5.m3.3c">i\in\{0,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p1.5.m3.3d">italic_i ∈ { 0 , … , italic_n }</annotation></semantics></math>, so that the spaces with and without boundary conditions coincide. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S6.SS2.p2"> <p class="ltx_p" id="S6.SS2.p2.1">The following two theorems characterise the complex interpolation spaces for weighted spaces with boundary conditions on the half-space. The proofs of these theorems are given in Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.SS3" title="6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S6.Thmtheorem4"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem4.1.1.1">Theorem 6.4</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem4.2.2"> </span>(Complex interpolation of weighted Bessel potential spaces)<span class="ltx_text ltx_font_bold" id="S6.Thmtheorem4.3.3">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem4.p1"> <p class="ltx_p" id="S6.Thmtheorem4.p1.11"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem4.p1.11.11">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.1.1.m1.2"><semantics id="S6.Thmtheorem4.p1.1.1.m1.2a"><mrow id="S6.Thmtheorem4.p1.1.1.m1.2.3" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.cmml"><mi id="S6.Thmtheorem4.p1.1.1.m1.2.3.2" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S6.Thmtheorem4.p1.1.1.m1.2.3.1" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S6.Thmtheorem4.p1.1.1.m1.2.3.3.2" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml"><mo id="S6.Thmtheorem4.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S6.Thmtheorem4.p1.1.1.m1.1.1" xref="S6.Thmtheorem4.p1.1.1.m1.1.1.cmml">1</mn><mo id="S6.Thmtheorem4.p1.1.1.m1.2.3.3.2.2" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem4.p1.1.1.m1.2.2" mathvariant="normal" xref="S6.Thmtheorem4.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S6.Thmtheorem4.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.1.1.m1.2b"><apply id="S6.Thmtheorem4.p1.1.1.m1.2.3.cmml" xref="S6.Thmtheorem4.p1.1.1.m1.2.3"><in id="S6.Thmtheorem4.p1.1.1.m1.2.3.1.cmml" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.1"></in><ci id="S6.Thmtheorem4.p1.1.1.m1.2.3.2.cmml" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S6.Thmtheorem4.p1.1.1.m1.2.3.3.1.cmml" xref="S6.Thmtheorem4.p1.1.1.m1.2.3.3.2"><cn id="S6.Thmtheorem4.p1.1.1.m1.1.1.cmml" type="integer" xref="S6.Thmtheorem4.p1.1.1.m1.1.1">1</cn><infinity id="S6.Thmtheorem4.p1.1.1.m1.2.2.cmml" xref="S6.Thmtheorem4.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.2.2.m2.2"><semantics id="S6.Thmtheorem4.p1.2.2.m2.2a"><mrow id="S6.Thmtheorem4.p1.2.2.m2.2.2" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.cmml"><mi id="S6.Thmtheorem4.p1.2.2.m2.2.2.4" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.4.cmml">γ</mi><mo id="S6.Thmtheorem4.p1.2.2.m2.2.2.3" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.3.cmml">∈</mo><mrow id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.3.cmml"><mo id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.3" stretchy="false" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.3.cmml">(</mo><mrow id="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1" xref="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1a" xref="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.2" xref="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.4" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.3.cmml">,</mo><mrow id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.cmml"><mi id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.2" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.2.cmml">p</mi><mo id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.1" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.1.cmml">−</mo><mn id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.3" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.5" stretchy="false" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.2.2.m2.2b"><apply id="S6.Thmtheorem4.p1.2.2.m2.2.2.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.2.2"><in id="S6.Thmtheorem4.p1.2.2.m2.2.2.3.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.3"></in><ci id="S6.Thmtheorem4.p1.2.2.m2.2.2.4.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.4">𝛾</ci><interval closure="open" id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.3.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2"><apply id="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1"><minus id="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1"></minus><cn id="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem4.p1.2.2.m2.1.1.1.1.1.2">1</cn></apply><apply id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2"><minus id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.1.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.1"></minus><ci id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.2.cmml" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.2">𝑝</ci><cn id="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.3.cmml" type="integer" xref="S6.Thmtheorem4.p1.2.2.m2.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.2.2.m2.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.2.2.m2.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.3.3.m3.1"><semantics id="S6.Thmtheorem4.p1.3.3.m3.1a"><mi id="S6.Thmtheorem4.p1.3.3.m3.1.1" xref="S6.Thmtheorem4.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.3.3.m3.1b"><ci id="S6.Thmtheorem4.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem4.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.3.3.m3.1d">italic_X</annotation></semantics></math> be a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.4.4.m4.1"><semantics id="S6.Thmtheorem4.p1.4.4.m4.1a"><mi id="S6.Thmtheorem4.p1.4.4.m4.1.1" xref="S6.Thmtheorem4.p1.4.4.m4.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.4.4.m4.1b"><ci id="S6.Thmtheorem4.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem4.p1.4.4.m4.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.4.4.m4.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.4.4.m4.1d">roman_UMD</annotation></semantics></math> Banach space. Let <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.5.5.m5.1"><semantics id="S6.Thmtheorem4.p1.5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem4.p1.5.5.m5.1.1" xref="S6.Thmtheorem4.p1.5.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.5.5.m5.1b"><ci id="S6.Thmtheorem4.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem4.p1.5.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.5.5.m5.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.5.5.m5.1d">caligraphic_B</annotation></semantics></math> be a normal boundary operator of type <math alttext="(p,s,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.6.6.m6.5"><semantics id="S6.Thmtheorem4.p1.6.6.m6.5a"><mrow id="S6.Thmtheorem4.p1.6.6.m6.5.6.2" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml"><mo id="S6.Thmtheorem4.p1.6.6.m6.5.6.2.1" stretchy="false" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml">(</mo><mi id="S6.Thmtheorem4.p1.6.6.m6.1.1" xref="S6.Thmtheorem4.p1.6.6.m6.1.1.cmml">p</mi><mo id="S6.Thmtheorem4.p1.6.6.m6.5.6.2.2" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml">,</mo><mi id="S6.Thmtheorem4.p1.6.6.m6.2.2" xref="S6.Thmtheorem4.p1.6.6.m6.2.2.cmml">s</mi><mo id="S6.Thmtheorem4.p1.6.6.m6.5.6.2.3" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml">,</mo><mi id="S6.Thmtheorem4.p1.6.6.m6.3.3" xref="S6.Thmtheorem4.p1.6.6.m6.3.3.cmml">γ</mi><mo id="S6.Thmtheorem4.p1.6.6.m6.5.6.2.4" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml">,</mo><mover accent="true" id="S6.Thmtheorem4.p1.6.6.m6.4.4" xref="S6.Thmtheorem4.p1.6.6.m6.4.4.cmml"><mi id="S6.Thmtheorem4.p1.6.6.m6.4.4.2" xref="S6.Thmtheorem4.p1.6.6.m6.4.4.2.cmml">m</mi><mo id="S6.Thmtheorem4.p1.6.6.m6.4.4.1" xref="S6.Thmtheorem4.p1.6.6.m6.4.4.1.cmml">¯</mo></mover><mo id="S6.Thmtheorem4.p1.6.6.m6.5.6.2.5" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml">,</mo><mover accent="true" id="S6.Thmtheorem4.p1.6.6.m6.5.5" xref="S6.Thmtheorem4.p1.6.6.m6.5.5.cmml"><mi id="S6.Thmtheorem4.p1.6.6.m6.5.5.2" xref="S6.Thmtheorem4.p1.6.6.m6.5.5.2.cmml">Y</mi><mo id="S6.Thmtheorem4.p1.6.6.m6.5.5.1" xref="S6.Thmtheorem4.p1.6.6.m6.5.5.1.cmml">¯</mo></mover><mo id="S6.Thmtheorem4.p1.6.6.m6.5.6.2.6" stretchy="false" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.6.6.m6.5b"><vector id="S6.Thmtheorem4.p1.6.6.m6.5.6.1.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.5.6.2"><ci id="S6.Thmtheorem4.p1.6.6.m6.1.1.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.1.1">𝑝</ci><ci id="S6.Thmtheorem4.p1.6.6.m6.2.2.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.2.2">𝑠</ci><ci id="S6.Thmtheorem4.p1.6.6.m6.3.3.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.3.3">𝛾</ci><apply id="S6.Thmtheorem4.p1.6.6.m6.4.4.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.4.4"><ci id="S6.Thmtheorem4.p1.6.6.m6.4.4.1.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.4.4.1">¯</ci><ci id="S6.Thmtheorem4.p1.6.6.m6.4.4.2.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.4.4.2">𝑚</ci></apply><apply id="S6.Thmtheorem4.p1.6.6.m6.5.5.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.5.5"><ci id="S6.Thmtheorem4.p1.6.6.m6.5.5.1.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.5.5.1">¯</ci><ci id="S6.Thmtheorem4.p1.6.6.m6.5.5.2.cmml" xref="S6.Thmtheorem4.p1.6.6.m6.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.6.6.m6.5c">(p,s,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.6.6.m6.5d">( italic_p , italic_s , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math> as in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem4" title="Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.4</span></a>. Assume that <math alttext="\theta\in(0,1)" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.7.7.m7.2"><semantics id="S6.Thmtheorem4.p1.7.7.m7.2a"><mrow id="S6.Thmtheorem4.p1.7.7.m7.2.3" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.cmml"><mi id="S6.Thmtheorem4.p1.7.7.m7.2.3.2" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.2.cmml">θ</mi><mo id="S6.Thmtheorem4.p1.7.7.m7.2.3.1" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.1.cmml">∈</mo><mrow id="S6.Thmtheorem4.p1.7.7.m7.2.3.3.2" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.3.1.cmml"><mo id="S6.Thmtheorem4.p1.7.7.m7.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.3.1.cmml">(</mo><mn id="S6.Thmtheorem4.p1.7.7.m7.1.1" xref="S6.Thmtheorem4.p1.7.7.m7.1.1.cmml">0</mn><mo id="S6.Thmtheorem4.p1.7.7.m7.2.3.3.2.2" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.3.1.cmml">,</mo><mn id="S6.Thmtheorem4.p1.7.7.m7.2.2" xref="S6.Thmtheorem4.p1.7.7.m7.2.2.cmml">1</mn><mo id="S6.Thmtheorem4.p1.7.7.m7.2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.7.7.m7.2b"><apply id="S6.Thmtheorem4.p1.7.7.m7.2.3.cmml" xref="S6.Thmtheorem4.p1.7.7.m7.2.3"><in id="S6.Thmtheorem4.p1.7.7.m7.2.3.1.cmml" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.1"></in><ci id="S6.Thmtheorem4.p1.7.7.m7.2.3.2.cmml" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.2">𝜃</ci><interval closure="open" id="S6.Thmtheorem4.p1.7.7.m7.2.3.3.1.cmml" xref="S6.Thmtheorem4.p1.7.7.m7.2.3.3.2"><cn id="S6.Thmtheorem4.p1.7.7.m7.1.1.cmml" type="integer" xref="S6.Thmtheorem4.p1.7.7.m7.1.1">0</cn><cn id="S6.Thmtheorem4.p1.7.7.m7.2.2.cmml" type="integer" xref="S6.Thmtheorem4.p1.7.7.m7.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.7.7.m7.2c">\theta\in(0,1)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.7.7.m7.2d">italic_θ ∈ ( 0 , 1 )</annotation></semantics></math>, <math alttext="s_{0}<s_{\theta}<s_{1}\leq s" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.8.8.m8.1"><semantics id="S6.Thmtheorem4.p1.8.8.m8.1a"><mrow id="S6.Thmtheorem4.p1.8.8.m8.1.1" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.cmml"><msub id="S6.Thmtheorem4.p1.8.8.m8.1.1.2" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.2.cmml"><mi id="S6.Thmtheorem4.p1.8.8.m8.1.1.2.2" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.2.2.cmml">s</mi><mn id="S6.Thmtheorem4.p1.8.8.m8.1.1.2.3" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.2.3.cmml">0</mn></msub><mo id="S6.Thmtheorem4.p1.8.8.m8.1.1.3" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.3.cmml"><</mo><msub id="S6.Thmtheorem4.p1.8.8.m8.1.1.4" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.4.cmml"><mi id="S6.Thmtheorem4.p1.8.8.m8.1.1.4.2" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.4.2.cmml">s</mi><mi id="S6.Thmtheorem4.p1.8.8.m8.1.1.4.3" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.4.3.cmml">θ</mi></msub><mo id="S6.Thmtheorem4.p1.8.8.m8.1.1.5" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.5.cmml"><</mo><msub id="S6.Thmtheorem4.p1.8.8.m8.1.1.6" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.6.cmml"><mi id="S6.Thmtheorem4.p1.8.8.m8.1.1.6.2" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.6.2.cmml">s</mi><mn id="S6.Thmtheorem4.p1.8.8.m8.1.1.6.3" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.6.3.cmml">1</mn></msub><mo id="S6.Thmtheorem4.p1.8.8.m8.1.1.7" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.7.cmml">≤</mo><mi id="S6.Thmtheorem4.p1.8.8.m8.1.1.8" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.8.cmml">s</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.8.8.m8.1b"><apply id="S6.Thmtheorem4.p1.8.8.m8.1.1.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1"><and id="S6.Thmtheorem4.p1.8.8.m8.1.1a.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1"></and><apply id="S6.Thmtheorem4.p1.8.8.m8.1.1b.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1"><lt id="S6.Thmtheorem4.p1.8.8.m8.1.1.3.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.3"></lt><apply id="S6.Thmtheorem4.p1.8.8.m8.1.1.2.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.8.8.m8.1.1.2.1.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem4.p1.8.8.m8.1.1.2.2.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.2.2">𝑠</ci><cn id="S6.Thmtheorem4.p1.8.8.m8.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.2.3">0</cn></apply><apply id="S6.Thmtheorem4.p1.8.8.m8.1.1.4.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.4"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.8.8.m8.1.1.4.1.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.4">subscript</csymbol><ci id="S6.Thmtheorem4.p1.8.8.m8.1.1.4.2.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.4.2">𝑠</ci><ci id="S6.Thmtheorem4.p1.8.8.m8.1.1.4.3.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.4.3">𝜃</ci></apply></apply><apply id="S6.Thmtheorem4.p1.8.8.m8.1.1c.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1"><lt id="S6.Thmtheorem4.p1.8.8.m8.1.1.5.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.5"></lt><share href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4.p1.8.8.m8.1.1.4.cmml" id="S6.Thmtheorem4.p1.8.8.m8.1.1d.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1"></share><apply id="S6.Thmtheorem4.p1.8.8.m8.1.1.6.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.6"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.8.8.m8.1.1.6.1.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.6">subscript</csymbol><ci id="S6.Thmtheorem4.p1.8.8.m8.1.1.6.2.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.6.2">𝑠</ci><cn id="S6.Thmtheorem4.p1.8.8.m8.1.1.6.3.cmml" type="integer" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.6.3">1</cn></apply></apply><apply id="S6.Thmtheorem4.p1.8.8.m8.1.1e.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1"><leq id="S6.Thmtheorem4.p1.8.8.m8.1.1.7.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.7"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4.p1.8.8.m8.1.1.6.cmml" id="S6.Thmtheorem4.p1.8.8.m8.1.1f.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1"></share><ci id="S6.Thmtheorem4.p1.8.8.m8.1.1.8.cmml" xref="S6.Thmtheorem4.p1.8.8.m8.1.1.8">𝑠</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.8.8.m8.1c">s_{0}<s_{\theta}<s_{1}\leq s</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.8.8.m8.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT < italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_s</annotation></semantics></math> with <math alttext="s_{\theta}=(1-\theta)s_{0}+\theta s_{1}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.9.9.m9.1"><semantics id="S6.Thmtheorem4.p1.9.9.m9.1a"><mrow id="S6.Thmtheorem4.p1.9.9.m9.1.1" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.cmml"><msub id="S6.Thmtheorem4.p1.9.9.m9.1.1.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.3.cmml"><mi id="S6.Thmtheorem4.p1.9.9.m9.1.1.3.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.3.2.cmml">s</mi><mi id="S6.Thmtheorem4.p1.9.9.m9.1.1.3.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.3.3.cmml">θ</mi></msub><mo id="S6.Thmtheorem4.p1.9.9.m9.1.1.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.2.cmml">=</mo><mrow id="S6.Thmtheorem4.p1.9.9.m9.1.1.1" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.cmml"><mrow id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.cmml"><mrow id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.cmml"><mo id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.2" stretchy="false" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.cmml"><mn id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.1" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.1.cmml">−</mo><mi id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.3.cmml">θ</mi></mrow><mo id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.3" stretchy="false" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.2.cmml">⁢</mo><msub id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.cmml"><mi id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.2.cmml">s</mi><mn id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.2.cmml">+</mo><mrow id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.cmml"><mi id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.2.cmml">θ</mi><mo id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.1" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.1.cmml">⁢</mo><msub id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.cmml"><mi id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.2" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.2.cmml">s</mi><mn id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.3" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.9.9.m9.1b"><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1"><eq id="S6.Thmtheorem4.p1.9.9.m9.1.1.2.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.2"></eq><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.3.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.9.9.m9.1.1.3.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem4.p1.9.9.m9.1.1.3.2.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.3.2">𝑠</ci><ci id="S6.Thmtheorem4.p1.9.9.m9.1.1.3.3.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.3.3">𝜃</ci></apply><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1"><plus id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.2.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.2"></plus><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1"><times id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.2.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.2"></times><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1"><minus id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.1"></minus><cn id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.2">1</cn><ci id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.3.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.1.1.1.3">𝜃</ci></apply><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.2.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.2">𝑠</ci><cn id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.1.3.3">0</cn></apply></apply><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3"><times id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.1"></times><ci id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.2.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.2">𝜃</ci><apply id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.1.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3">subscript</csymbol><ci id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.2.cmml" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.2">𝑠</ci><cn id="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.3.cmml" type="integer" xref="S6.Thmtheorem4.p1.9.9.m9.1.1.1.3.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.9.9.m9.1c">s_{\theta}=(1-\theta)s_{0}+\theta s_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.9.9.m9.1d">italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = ( 1 - italic_θ ) italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_θ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> satisfy <math alttext="s_{0},s_{\theta},s_{1}\notin\{m_{i}+\frac{\gamma+1}{p}:0\leq i\leq n\}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.10.10.m10.5"><semantics id="S6.Thmtheorem4.p1.10.10.m10.5a"><mrow id="S6.Thmtheorem4.p1.10.10.m10.5.5" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.cmml"><mrow id="S6.Thmtheorem4.p1.10.10.m10.3.3.3.3" xref="S6.Thmtheorem4.p1.10.10.m10.3.3.3.4.cmml"><msub id="S6.Thmtheorem4.p1.10.10.m10.1.1.1.1.1" xref="S6.Thmtheorem4.p1.10.10.m10.1.1.1.1.1.cmml"><mi id="S6.Thmtheorem4.p1.10.10.m10.1.1.1.1.1.2" 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id="S6.Thmtheorem4.p1.10.10.m10.4.4.4.1.1.3.3.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.4.4.4.1.1.3.3">𝑝</ci></apply></apply><apply id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2"><and id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2a.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2"></and><apply id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2b.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2"><leq id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.3.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.3"></leq><cn id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.2.cmml" type="integer" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.2">0</cn><ci id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.4.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.4">𝑖</ci></apply><apply id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2c.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2"><leq id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.5.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.4.cmml" id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2d.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2"></share><ci id="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.6.cmml" xref="S6.Thmtheorem4.p1.10.10.m10.5.5.5.2.2.6">𝑛</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.10.10.m10.5c">s_{0},s_{\theta},s_{1}\notin\{m_{i}+\frac{\gamma+1}{p}:0\leq i\leq n\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.10.10.m10.5d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∉ { italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG : 0 ≤ italic_i ≤ italic_n }</annotation></semantics></math> and <math alttext="s_{0}>-1+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.Thmtheorem4.p1.11.11.m11.1"><semantics id="S6.Thmtheorem4.p1.11.11.m11.1a"><mrow id="S6.Thmtheorem4.p1.11.11.m11.1.1" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.cmml"><msub id="S6.Thmtheorem4.p1.11.11.m11.1.1.2" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.2.cmml"><mi id="S6.Thmtheorem4.p1.11.11.m11.1.1.2.2" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.2.2.cmml">s</mi><mn id="S6.Thmtheorem4.p1.11.11.m11.1.1.2.3" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.2.3.cmml">0</mn></msub><mo id="S6.Thmtheorem4.p1.11.11.m11.1.1.1" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.1.cmml">></mo><mrow id="S6.Thmtheorem4.p1.11.11.m11.1.1.3" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.cmml"><mrow id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.2" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.2.cmml"><mo id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.2a" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.2.cmml">−</mo><mn id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.2.2" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.2.2.cmml">1</mn></mrow><mo id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.1" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.1.cmml">+</mo><mfrac id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.cmml"><mrow id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.cmml"><mi id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.2" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.1" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.1.cmml">+</mo><mn id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.3" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.3" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem4.p1.11.11.m11.1b"><apply id="S6.Thmtheorem4.p1.11.11.m11.1.1.cmml" xref="S6.Thmtheorem4.p1.11.11.m11.1.1"><gt id="S6.Thmtheorem4.p1.11.11.m11.1.1.1.cmml" 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xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.2.2">1</cn></apply><apply id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.cmml" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3"><divide id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.1.cmml" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3"></divide><apply id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.cmml" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2"><plus id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.1.cmml" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.1"></plus><ci id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.2.cmml" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.2">𝛾</ci><cn id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.3.cmml" type="integer" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.2.3">1</cn></apply><ci id="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.3.cmml" xref="S6.Thmtheorem4.p1.11.11.m11.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem4.p1.11.11.m11.1c">s_{0}>-1+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem4.p1.11.11.m11.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > - 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Then</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx36"> <tbody id="S6.Ex37"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle H_{\mathcal{B}}^{s_{\theta},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.Ex37.m1.5"><semantics id="S6.Ex37.m1.5a"><mrow id="S6.Ex37.m1.5.5" xref="S6.Ex37.m1.5.5.cmml"><msubsup id="S6.Ex37.m1.5.5.4" xref="S6.Ex37.m1.5.5.4.cmml"><mi id="S6.Ex37.m1.5.5.4.2.2" xref="S6.Ex37.m1.5.5.4.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex37.m1.5.5.4.2.3" xref="S6.Ex37.m1.5.5.4.2.3.cmml">ℬ</mi><mrow id="S6.Ex37.m1.2.2.2.2" xref="S6.Ex37.m1.2.2.2.3.cmml"><msub id="S6.Ex37.m1.2.2.2.2.1" xref="S6.Ex37.m1.2.2.2.2.1.cmml"><mi id="S6.Ex37.m1.2.2.2.2.1.2" xref="S6.Ex37.m1.2.2.2.2.1.2.cmml">s</mi><mi id="S6.Ex37.m1.2.2.2.2.1.3" xref="S6.Ex37.m1.2.2.2.2.1.3.cmml">θ</mi></msub><mo id="S6.Ex37.m1.2.2.2.2.2" xref="S6.Ex37.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex37.m1.1.1.1.1" xref="S6.Ex37.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex37.m1.5.5.3" xref="S6.Ex37.m1.5.5.3.cmml">⁢</mo><mrow id="S6.Ex37.m1.5.5.2.2" xref="S6.Ex37.m1.5.5.2.3.cmml"><mo id="S6.Ex37.m1.5.5.2.2.3" stretchy="false" xref="S6.Ex37.m1.5.5.2.3.cmml">(</mo><msubsup id="S6.Ex37.m1.4.4.1.1.1" xref="S6.Ex37.m1.4.4.1.1.1.cmml"><mi id="S6.Ex37.m1.4.4.1.1.1.2.2" xref="S6.Ex37.m1.4.4.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex37.m1.4.4.1.1.1.3" xref="S6.Ex37.m1.4.4.1.1.1.3.cmml">+</mo><mi id="S6.Ex37.m1.4.4.1.1.1.2.3" xref="S6.Ex37.m1.4.4.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex37.m1.5.5.2.2.4" xref="S6.Ex37.m1.5.5.2.3.cmml">,</mo><msub id="S6.Ex37.m1.5.5.2.2.2" xref="S6.Ex37.m1.5.5.2.2.2.cmml"><mi id="S6.Ex37.m1.5.5.2.2.2.2" xref="S6.Ex37.m1.5.5.2.2.2.2.cmml">w</mi><mi id="S6.Ex37.m1.5.5.2.2.2.3" xref="S6.Ex37.m1.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex37.m1.5.5.2.2.5" xref="S6.Ex37.m1.5.5.2.3.cmml">;</mo><mi id="S6.Ex37.m1.3.3" xref="S6.Ex37.m1.3.3.cmml">X</mi><mo id="S6.Ex37.m1.5.5.2.2.6" stretchy="false" xref="S6.Ex37.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex37.m1.5b"><apply id="S6.Ex37.m1.5.5.cmml" xref="S6.Ex37.m1.5.5"><times id="S6.Ex37.m1.5.5.3.cmml" xref="S6.Ex37.m1.5.5.3"></times><apply id="S6.Ex37.m1.5.5.4.cmml" xref="S6.Ex37.m1.5.5.4"><csymbol cd="ambiguous" id="S6.Ex37.m1.5.5.4.1.cmml" xref="S6.Ex37.m1.5.5.4">superscript</csymbol><apply id="S6.Ex37.m1.5.5.4.2.cmml" xref="S6.Ex37.m1.5.5.4"><csymbol cd="ambiguous" id="S6.Ex37.m1.5.5.4.2.1.cmml" xref="S6.Ex37.m1.5.5.4">subscript</csymbol><ci id="S6.Ex37.m1.5.5.4.2.2.cmml" xref="S6.Ex37.m1.5.5.4.2.2">𝐻</ci><ci id="S6.Ex37.m1.5.5.4.2.3.cmml" xref="S6.Ex37.m1.5.5.4.2.3">ℬ</ci></apply><list id="S6.Ex37.m1.2.2.2.3.cmml" xref="S6.Ex37.m1.2.2.2.2"><apply id="S6.Ex37.m1.2.2.2.2.1.cmml" xref="S6.Ex37.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.Ex37.m1.2.2.2.2.1.1.cmml" xref="S6.Ex37.m1.2.2.2.2.1">subscript</csymbol><ci id="S6.Ex37.m1.2.2.2.2.1.2.cmml" xref="S6.Ex37.m1.2.2.2.2.1.2">𝑠</ci><ci id="S6.Ex37.m1.2.2.2.2.1.3.cmml" xref="S6.Ex37.m1.2.2.2.2.1.3">𝜃</ci></apply><ci id="S6.Ex37.m1.1.1.1.1.cmml" xref="S6.Ex37.m1.1.1.1.1">𝑝</ci></list></apply><vector id="S6.Ex37.m1.5.5.2.3.cmml" xref="S6.Ex37.m1.5.5.2.2"><apply id="S6.Ex37.m1.4.4.1.1.1.cmml" xref="S6.Ex37.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.Ex37.m1.4.4.1.1.1.1.cmml" xref="S6.Ex37.m1.4.4.1.1.1">subscript</csymbol><apply id="S6.Ex37.m1.4.4.1.1.1.2.cmml" xref="S6.Ex37.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.Ex37.m1.4.4.1.1.1.2.1.cmml" xref="S6.Ex37.m1.4.4.1.1.1">superscript</csymbol><ci id="S6.Ex37.m1.4.4.1.1.1.2.2.cmml" xref="S6.Ex37.m1.4.4.1.1.1.2.2">ℝ</ci><ci id="S6.Ex37.m1.4.4.1.1.1.2.3.cmml" xref="S6.Ex37.m1.4.4.1.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex37.m1.4.4.1.1.1.3.cmml" xref="S6.Ex37.m1.4.4.1.1.1.3"></plus></apply><apply id="S6.Ex37.m1.5.5.2.2.2.cmml" xref="S6.Ex37.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.Ex37.m1.5.5.2.2.2.1.cmml" xref="S6.Ex37.m1.5.5.2.2.2">subscript</csymbol><ci id="S6.Ex37.m1.5.5.2.2.2.2.cmml" xref="S6.Ex37.m1.5.5.2.2.2.2">𝑤</ci><ci id="S6.Ex37.m1.5.5.2.2.2.3.cmml" xref="S6.Ex37.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex37.m1.3.3.cmml" xref="S6.Ex37.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex37.m1.5c">\displaystyle H_{\mathcal{B}}^{s_{\theta},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex37.m1.5d">italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{[}H^{s_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X),H_{\mathcal{% B}}^{s_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\theta}" class="ltx_Math" display="inline" id="S6.Ex37.m2.8"><semantics id="S6.Ex37.m2.8a"><mrow id="S6.Ex37.m2.8.8" xref="S6.Ex37.m2.8.8.cmml"><mi id="S6.Ex37.m2.8.8.4" xref="S6.Ex37.m2.8.8.4.cmml"></mi><mo id="S6.Ex37.m2.8.8.3" xref="S6.Ex37.m2.8.8.3.cmml">=</mo><msub id="S6.Ex37.m2.8.8.2" xref="S6.Ex37.m2.8.8.2.cmml"><mrow id="S6.Ex37.m2.8.8.2.2.2" xref="S6.Ex37.m2.8.8.2.2.3.cmml"><mo id="S6.Ex37.m2.8.8.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex37.m2.8.8.2.2.3.cmml">[</mo><mrow id="S6.Ex37.m2.7.7.1.1.1.1" xref="S6.Ex37.m2.7.7.1.1.1.1.cmml"><msup id="S6.Ex37.m2.7.7.1.1.1.1.4" xref="S6.Ex37.m2.7.7.1.1.1.1.4.cmml"><mi id="S6.Ex37.m2.7.7.1.1.1.1.4.2" xref="S6.Ex37.m2.7.7.1.1.1.1.4.2.cmml">H</mi><mrow id="S6.Ex37.m2.2.2.2.2" 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id="S6.Ex37.m2.8.8.2.2.2.2.2.2.2.3.cmml" xref="S6.Ex37.m2.8.8.2.2.2.2.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex37.m2.6.6.cmml" xref="S6.Ex37.m2.6.6">𝑋</ci></vector></apply></interval><ci id="S6.Ex37.m2.8.8.2.4.cmml" xref="S6.Ex37.m2.8.8.2.4">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex37.m2.8c">\displaystyle=\big{[}H^{s_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X),H_{\mathcal{% B}}^{s_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex37.m2.8d">= [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex38"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{[}H^{s_{0},p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X% ),H_{\mathcal{B}}^{s_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\theta}." class="ltx_Math" display="inline" id="S6.Ex38.m1.7"><semantics id="S6.Ex38.m1.7a"><mrow id="S6.Ex38.m1.7.7.1" xref="S6.Ex38.m1.7.7.1.1.cmml"><mrow id="S6.Ex38.m1.7.7.1.1" xref="S6.Ex38.m1.7.7.1.1.cmml"><mi id="S6.Ex38.m1.7.7.1.1.4" 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end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_theorem ltx_theorem_theorem" id="S6.Thmtheorem5"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem5.1.1.1">Theorem 6.5</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem5.2.2"> </span>(Complex interpolation of weighted Sobolev spaces)<span class="ltx_text ltx_font_bold" id="S6.Thmtheorem5.3.3">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem5.p1"> <p class="ltx_p" id="S6.Thmtheorem5.p1.10"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem5.p1.10.10">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.1.1.m1.2"><semantics 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id="S6.Thmtheorem5.p1.1.1.m1.2.3.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.2.3"><in id="S6.Thmtheorem5.p1.1.1.m1.2.3.1.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.2.3.1"></in><ci id="S6.Thmtheorem5.p1.1.1.m1.2.3.2.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S6.Thmtheorem5.p1.1.1.m1.2.3.3.1.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.2.3.3.2"><cn id="S6.Thmtheorem5.p1.1.1.m1.1.1.cmml" type="integer" xref="S6.Thmtheorem5.p1.1.1.m1.1.1">1</cn><infinity id="S6.Thmtheorem5.p1.1.1.m1.2.2.cmml" xref="S6.Thmtheorem5.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.2.2.m2.4"><semantics id="S6.Thmtheorem5.p1.2.2.m2.4a"><mrow id="S6.Thmtheorem5.p1.2.2.m2.4.4" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.cmml"><mi id="S6.Thmtheorem5.p1.2.2.m2.4.4.5" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.5.cmml">γ</mi><mo id="S6.Thmtheorem5.p1.2.2.m2.4.4.4" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.4.cmml">∈</mo><mrow id="S6.Thmtheorem5.p1.2.2.m2.4.4.3" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.cmml"><mrow id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.2.cmml"><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.2" stretchy="false" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.2.cmml">(</mo><mrow id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.cmml"><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1a" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.2" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.3" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.2.cmml">,</mo><mi id="S6.Thmtheorem5.p1.2.2.m2.1.1" mathvariant="normal" xref="S6.Thmtheorem5.p1.2.2.m2.1.1.cmml">∞</mi><mo id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.4" stretchy="false" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.2.cmml">)</mo></mrow><mo id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.4" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.4.cmml">∖</mo><mrow id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.3.cmml"><mo id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.3" stretchy="false" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.3.1.cmml">{</mo><mrow id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.cmml"><mrow id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.cmml"><mi id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.2" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.1" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.3" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.1" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.3" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.3.1.cmml">:</mo><mrow id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.cmml"><mi id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.2" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.2.cmml">j</mi><mo id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.1" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.cmml"><mi id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.2" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.3" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.5" stretchy="false" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.2.2.m2.4b"><apply id="S6.Thmtheorem5.p1.2.2.m2.4.4.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4"><in id="S6.Thmtheorem5.p1.2.2.m2.4.4.4.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.4"></in><ci id="S6.Thmtheorem5.p1.2.2.m2.4.4.5.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.5">𝛾</ci><apply id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3"><setdiff id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.4.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.4"></setdiff><interval closure="open" id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1"><apply id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1"><minus id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1"></minus><cn id="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem5.p1.2.2.m2.2.2.1.1.1.1.2">1</cn></apply><infinity id="S6.Thmtheorem5.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.1.1"></infinity></interval><apply id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2"><csymbol cd="latexml" id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.3.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.3">conditional-set</csymbol><apply id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1"><minus id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.1"></minus><apply id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2"><times id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.1"></times><ci id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.2.2.m2.3.3.2.2.1.1.3">1</cn></apply><apply id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2"><in id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.1"></in><ci id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.2">𝑗</ci><apply id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.1.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3">subscript</csymbol><ci id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.2.cmml" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.2.2.m2.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.2.2.m2.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.2.2.m2.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.3.3.m3.1"><semantics id="S6.Thmtheorem5.p1.3.3.m3.1a"><mi id="S6.Thmtheorem5.p1.3.3.m3.1.1" xref="S6.Thmtheorem5.p1.3.3.m3.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.3.3.m3.1b"><ci id="S6.Thmtheorem5.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem5.p1.3.3.m3.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.3.3.m3.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.3.3.m3.1d">italic_X</annotation></semantics></math> be a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.4.4.m4.1"><semantics id="S6.Thmtheorem5.p1.4.4.m4.1a"><mi id="S6.Thmtheorem5.p1.4.4.m4.1.1" xref="S6.Thmtheorem5.p1.4.4.m4.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.4.4.m4.1b"><ci id="S6.Thmtheorem5.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem5.p1.4.4.m4.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.4.4.m4.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.4.4.m4.1d">roman_UMD</annotation></semantics></math> Banach space. Let <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.5.5.m5.1"><semantics id="S6.Thmtheorem5.p1.5.5.m5.1a"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem5.p1.5.5.m5.1.1" xref="S6.Thmtheorem5.p1.5.5.m5.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.5.5.m5.1b"><ci id="S6.Thmtheorem5.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem5.p1.5.5.m5.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.5.5.m5.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.5.5.m5.1d">caligraphic_B</annotation></semantics></math> be a normal boundary operator of type <math alttext="(p,k,\gamma,\overline{m},\overline{Y})" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.6.6.m6.5"><semantics id="S6.Thmtheorem5.p1.6.6.m6.5a"><mrow id="S6.Thmtheorem5.p1.6.6.m6.5.6.2" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml"><mo id="S6.Thmtheorem5.p1.6.6.m6.5.6.2.1" stretchy="false" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml">(</mo><mi id="S6.Thmtheorem5.p1.6.6.m6.1.1" xref="S6.Thmtheorem5.p1.6.6.m6.1.1.cmml">p</mi><mo id="S6.Thmtheorem5.p1.6.6.m6.5.6.2.2" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml">,</mo><mi id="S6.Thmtheorem5.p1.6.6.m6.2.2" xref="S6.Thmtheorem5.p1.6.6.m6.2.2.cmml">k</mi><mo id="S6.Thmtheorem5.p1.6.6.m6.5.6.2.3" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml">,</mo><mi id="S6.Thmtheorem5.p1.6.6.m6.3.3" xref="S6.Thmtheorem5.p1.6.6.m6.3.3.cmml">γ</mi><mo id="S6.Thmtheorem5.p1.6.6.m6.5.6.2.4" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml">,</mo><mover accent="true" id="S6.Thmtheorem5.p1.6.6.m6.4.4" xref="S6.Thmtheorem5.p1.6.6.m6.4.4.cmml"><mi id="S6.Thmtheorem5.p1.6.6.m6.4.4.2" xref="S6.Thmtheorem5.p1.6.6.m6.4.4.2.cmml">m</mi><mo id="S6.Thmtheorem5.p1.6.6.m6.4.4.1" xref="S6.Thmtheorem5.p1.6.6.m6.4.4.1.cmml">¯</mo></mover><mo id="S6.Thmtheorem5.p1.6.6.m6.5.6.2.5" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml">,</mo><mover accent="true" id="S6.Thmtheorem5.p1.6.6.m6.5.5" xref="S6.Thmtheorem5.p1.6.6.m6.5.5.cmml"><mi id="S6.Thmtheorem5.p1.6.6.m6.5.5.2" xref="S6.Thmtheorem5.p1.6.6.m6.5.5.2.cmml">Y</mi><mo id="S6.Thmtheorem5.p1.6.6.m6.5.5.1" xref="S6.Thmtheorem5.p1.6.6.m6.5.5.1.cmml">¯</mo></mover><mo id="S6.Thmtheorem5.p1.6.6.m6.5.6.2.6" stretchy="false" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.6.6.m6.5b"><vector id="S6.Thmtheorem5.p1.6.6.m6.5.6.1.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.5.6.2"><ci id="S6.Thmtheorem5.p1.6.6.m6.1.1.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.1.1">𝑝</ci><ci id="S6.Thmtheorem5.p1.6.6.m6.2.2.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.2.2">𝑘</ci><ci id="S6.Thmtheorem5.p1.6.6.m6.3.3.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.3.3">𝛾</ci><apply id="S6.Thmtheorem5.p1.6.6.m6.4.4.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.4.4"><ci id="S6.Thmtheorem5.p1.6.6.m6.4.4.1.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.4.4.1">¯</ci><ci id="S6.Thmtheorem5.p1.6.6.m6.4.4.2.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.4.4.2">𝑚</ci></apply><apply id="S6.Thmtheorem5.p1.6.6.m6.5.5.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.5.5"><ci id="S6.Thmtheorem5.p1.6.6.m6.5.5.1.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.5.5.1">¯</ci><ci id="S6.Thmtheorem5.p1.6.6.m6.5.5.2.cmml" xref="S6.Thmtheorem5.p1.6.6.m6.5.5.2">𝑌</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.6.6.m6.5c">(p,k,\gamma,\overline{m},\overline{Y})</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.6.6.m6.5d">( italic_p , italic_k , italic_γ , over¯ start_ARG italic_m end_ARG , over¯ start_ARG italic_Y end_ARG )</annotation></semantics></math> as in Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem4" title="Definition 5.4 (System of normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.4</span></a>. Assume that <math alttext="k_{0}\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.7.7.m7.1"><semantics id="S6.Thmtheorem5.p1.7.7.m7.1a"><mrow id="S6.Thmtheorem5.p1.7.7.m7.1.1" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.cmml"><msub id="S6.Thmtheorem5.p1.7.7.m7.1.1.2" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.2.cmml"><mi id="S6.Thmtheorem5.p1.7.7.m7.1.1.2.2" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.2.2.cmml">k</mi><mn id="S6.Thmtheorem5.p1.7.7.m7.1.1.2.3" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.2.3.cmml">0</mn></msub><mo id="S6.Thmtheorem5.p1.7.7.m7.1.1.1" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.1.cmml">∈</mo><msub id="S6.Thmtheorem5.p1.7.7.m7.1.1.3" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.3.cmml"><mi id="S6.Thmtheorem5.p1.7.7.m7.1.1.3.2" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.3.2.cmml">ℕ</mi><mn id="S6.Thmtheorem5.p1.7.7.m7.1.1.3.3" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.7.7.m7.1b"><apply id="S6.Thmtheorem5.p1.7.7.m7.1.1.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1"><in id="S6.Thmtheorem5.p1.7.7.m7.1.1.1.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.1"></in><apply id="S6.Thmtheorem5.p1.7.7.m7.1.1.2.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.7.7.m7.1.1.2.1.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem5.p1.7.7.m7.1.1.2.2.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem5.p1.7.7.m7.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.2.3">0</cn></apply><apply id="S6.Thmtheorem5.p1.7.7.m7.1.1.3.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.7.7.m7.1.1.3.1.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem5.p1.7.7.m7.1.1.3.2.cmml" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.3.2">ℕ</ci><cn id="S6.Thmtheorem5.p1.7.7.m7.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.7.7.m7.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.7.7.m7.1c">k_{0}\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.7.7.m7.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and <math alttext="k_{1}\in\mathbb{N}_{1}\setminus\{1\}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.8.8.m8.1"><semantics id="S6.Thmtheorem5.p1.8.8.m8.1a"><mrow id="S6.Thmtheorem5.p1.8.8.m8.1.2" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.cmml"><msub id="S6.Thmtheorem5.p1.8.8.m8.1.2.2" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.2.cmml"><mi id="S6.Thmtheorem5.p1.8.8.m8.1.2.2.2" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.2.2.cmml">k</mi><mn id="S6.Thmtheorem5.p1.8.8.m8.1.2.2.3" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem5.p1.8.8.m8.1.2.1" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.1.cmml">∈</mo><mrow id="S6.Thmtheorem5.p1.8.8.m8.1.2.3" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.cmml"><msub id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.cmml"><mi id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.2" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.2.cmml">ℕ</mi><mn id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.3" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.1" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.1.cmml">∖</mo><mrow id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.2" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.1.cmml"><mo id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.1.cmml">{</mo><mn id="S6.Thmtheorem5.p1.8.8.m8.1.1" xref="S6.Thmtheorem5.p1.8.8.m8.1.1.cmml">1</mn><mo id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.2.2" stretchy="false" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.8.8.m8.1b"><apply id="S6.Thmtheorem5.p1.8.8.m8.1.2.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2"><in id="S6.Thmtheorem5.p1.8.8.m8.1.2.1.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.1"></in><apply id="S6.Thmtheorem5.p1.8.8.m8.1.2.2.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.8.8.m8.1.2.2.1.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem5.p1.8.8.m8.1.2.2.2.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.2.2">𝑘</ci><cn id="S6.Thmtheorem5.p1.8.8.m8.1.2.2.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.2.3">1</cn></apply><apply id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3"><setdiff id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.1.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.1"></setdiff><apply id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.1.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2">subscript</csymbol><ci id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.2.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.2">ℕ</ci><cn id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.2.3">1</cn></apply><set id="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.1.cmml" xref="S6.Thmtheorem5.p1.8.8.m8.1.2.3.3.2"><cn id="S6.Thmtheorem5.p1.8.8.m8.1.1.cmml" type="integer" xref="S6.Thmtheorem5.p1.8.8.m8.1.1">1</cn></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.8.8.m8.1c">k_{1}\in\mathbb{N}_{1}\setminus\{1\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.8.8.m8.1d">italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∖ { 1 }</annotation></semantics></math> are such that <math alttext="k_{0}+k_{1}\leq k" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.9.9.m9.1"><semantics id="S6.Thmtheorem5.p1.9.9.m9.1a"><mrow id="S6.Thmtheorem5.p1.9.9.m9.1.1" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.cmml"><mrow id="S6.Thmtheorem5.p1.9.9.m9.1.1.2" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.cmml"><msub id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.cmml"><mi id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.2" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.2.cmml">k</mi><mn id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.3" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.1" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.1.cmml">+</mo><msub id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.cmml"><mi id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.2" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.2.cmml">k</mi><mn id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.3" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.Thmtheorem5.p1.9.9.m9.1.1.1" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.1.cmml">≤</mo><mi id="S6.Thmtheorem5.p1.9.9.m9.1.1.3" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.9.9.m9.1b"><apply id="S6.Thmtheorem5.p1.9.9.m9.1.1.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1"><leq id="S6.Thmtheorem5.p1.9.9.m9.1.1.1.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.1"></leq><apply id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2"><plus id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.1.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.1"></plus><apply id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.1.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2">subscript</csymbol><ci id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.2.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.2">𝑘</ci><cn id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.2.3">0</cn></apply><apply id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.1.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3">subscript</csymbol><ci id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.2.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.2">𝑘</ci><cn id="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.2.3.3">1</cn></apply></apply><ci id="S6.Thmtheorem5.p1.9.9.m9.1.1.3.cmml" xref="S6.Thmtheorem5.p1.9.9.m9.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.9.9.m9.1c">k_{0}+k_{1}\leq k</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.9.9.m9.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_k</annotation></semantics></math>. Then for <math alttext="\ell\in\{1,\dots,k_{1}-1\}" class="ltx_Math" display="inline" id="S6.Thmtheorem5.p1.10.10.m10.3"><semantics id="S6.Thmtheorem5.p1.10.10.m10.3a"><mrow id="S6.Thmtheorem5.p1.10.10.m10.3.3" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.cmml"><mi id="S6.Thmtheorem5.p1.10.10.m10.3.3.3" mathvariant="normal" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.3.cmml">ℓ</mi><mo id="S6.Thmtheorem5.p1.10.10.m10.3.3.2" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.2.cmml">∈</mo><mrow id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.2.cmml"><mo id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.2" stretchy="false" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.2.cmml">{</mo><mn id="S6.Thmtheorem5.p1.10.10.m10.1.1" xref="S6.Thmtheorem5.p1.10.10.m10.1.1.cmml">1</mn><mo id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.3" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.2.cmml">,</mo><mi id="S6.Thmtheorem5.p1.10.10.m10.2.2" mathvariant="normal" xref="S6.Thmtheorem5.p1.10.10.m10.2.2.cmml">…</mi><mo id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.4" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.2.cmml">,</mo><mrow id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.cmml"><msub id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.cmml"><mi id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.2" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.2.cmml">k</mi><mn id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.3" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.1" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.3" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.5" stretchy="false" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem5.p1.10.10.m10.3b"><apply id="S6.Thmtheorem5.p1.10.10.m10.3.3.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3"><in id="S6.Thmtheorem5.p1.10.10.m10.3.3.2.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.2"></in><ci id="S6.Thmtheorem5.p1.10.10.m10.3.3.3.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.3">ℓ</ci><set id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.2.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1"><cn id="S6.Thmtheorem5.p1.10.10.m10.1.1.cmml" type="integer" xref="S6.Thmtheorem5.p1.10.10.m10.1.1">1</cn><ci id="S6.Thmtheorem5.p1.10.10.m10.2.2.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.2.2">…</ci><apply id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1"><minus id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.1"></minus><apply id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.1.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.2.cmml" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.2.3">1</cn></apply><cn id="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem5.p1.10.10.m10.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem5.p1.10.10.m10.3c">\ell\in\{1,\dots,k_{1}-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem5.p1.10.10.m10.3d">roman_ℓ ∈ { 1 , … , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 }</annotation></semantics></math> we have</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx37"> <tbody id="S6.Ex39"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W_{\mathcal{B}}^{k_{0}+\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.Ex39.m1.5"><semantics id="S6.Ex39.m1.5a"><mrow id="S6.Ex39.m1.5.5" xref="S6.Ex39.m1.5.5.cmml"><msubsup id="S6.Ex39.m1.5.5.4" xref="S6.Ex39.m1.5.5.4.cmml"><mi id="S6.Ex39.m1.5.5.4.2.2" xref="S6.Ex39.m1.5.5.4.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex39.m1.5.5.4.2.3" xref="S6.Ex39.m1.5.5.4.2.3.cmml">ℬ</mi><mrow id="S6.Ex39.m1.2.2.2.2" xref="S6.Ex39.m1.2.2.2.3.cmml"><mrow id="S6.Ex39.m1.2.2.2.2.1" xref="S6.Ex39.m1.2.2.2.2.1.cmml"><msub id="S6.Ex39.m1.2.2.2.2.1.2" xref="S6.Ex39.m1.2.2.2.2.1.2.cmml"><mi id="S6.Ex39.m1.2.2.2.2.1.2.2" xref="S6.Ex39.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.Ex39.m1.2.2.2.2.1.2.3" xref="S6.Ex39.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.Ex39.m1.2.2.2.2.1.1" xref="S6.Ex39.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.Ex39.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.Ex39.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.Ex39.m1.2.2.2.2.2" xref="S6.Ex39.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex39.m1.1.1.1.1" xref="S6.Ex39.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex39.m1.5.5.3" xref="S6.Ex39.m1.5.5.3.cmml">⁢</mo><mrow id="S6.Ex39.m1.5.5.2.2" xref="S6.Ex39.m1.5.5.2.3.cmml"><mo id="S6.Ex39.m1.5.5.2.2.3" stretchy="false" xref="S6.Ex39.m1.5.5.2.3.cmml">(</mo><msubsup id="S6.Ex39.m1.4.4.1.1.1" xref="S6.Ex39.m1.4.4.1.1.1.cmml"><mi id="S6.Ex39.m1.4.4.1.1.1.2.2" xref="S6.Ex39.m1.4.4.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex39.m1.4.4.1.1.1.3" xref="S6.Ex39.m1.4.4.1.1.1.3.cmml">+</mo><mi id="S6.Ex39.m1.4.4.1.1.1.2.3" xref="S6.Ex39.m1.4.4.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex39.m1.5.5.2.2.4" xref="S6.Ex39.m1.5.5.2.3.cmml">,</mo><msub id="S6.Ex39.m1.5.5.2.2.2" xref="S6.Ex39.m1.5.5.2.2.2.cmml"><mi id="S6.Ex39.m1.5.5.2.2.2.2" xref="S6.Ex39.m1.5.5.2.2.2.2.cmml">w</mi><mi id="S6.Ex39.m1.5.5.2.2.2.3" xref="S6.Ex39.m1.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex39.m1.5.5.2.2.5" xref="S6.Ex39.m1.5.5.2.3.cmml">;</mo><mi id="S6.Ex39.m1.3.3" xref="S6.Ex39.m1.3.3.cmml">X</mi><mo id="S6.Ex39.m1.5.5.2.2.6" stretchy="false" xref="S6.Ex39.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex39.m1.5b"><apply id="S6.Ex39.m1.5.5.cmml" xref="S6.Ex39.m1.5.5"><times id="S6.Ex39.m1.5.5.3.cmml" xref="S6.Ex39.m1.5.5.3"></times><apply id="S6.Ex39.m1.5.5.4.cmml" xref="S6.Ex39.m1.5.5.4"><csymbol cd="ambiguous" id="S6.Ex39.m1.5.5.4.1.cmml" xref="S6.Ex39.m1.5.5.4">superscript</csymbol><apply id="S6.Ex39.m1.5.5.4.2.cmml" xref="S6.Ex39.m1.5.5.4"><csymbol cd="ambiguous" id="S6.Ex39.m1.5.5.4.2.1.cmml" xref="S6.Ex39.m1.5.5.4">subscript</csymbol><ci id="S6.Ex39.m1.5.5.4.2.2.cmml" xref="S6.Ex39.m1.5.5.4.2.2">𝑊</ci><ci id="S6.Ex39.m1.5.5.4.2.3.cmml" xref="S6.Ex39.m1.5.5.4.2.3">ℬ</ci></apply><list id="S6.Ex39.m1.2.2.2.3.cmml" xref="S6.Ex39.m1.2.2.2.2"><apply id="S6.Ex39.m1.2.2.2.2.1.cmml" xref="S6.Ex39.m1.2.2.2.2.1"><plus id="S6.Ex39.m1.2.2.2.2.1.1.cmml" xref="S6.Ex39.m1.2.2.2.2.1.1"></plus><apply id="S6.Ex39.m1.2.2.2.2.1.2.cmml" xref="S6.Ex39.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.Ex39.m1.2.2.2.2.1.2.1.cmml" xref="S6.Ex39.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S6.Ex39.m1.2.2.2.2.1.2.2.cmml" xref="S6.Ex39.m1.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.Ex39.m1.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.Ex39.m1.2.2.2.2.1.2.3">0</cn></apply><ci id="S6.Ex39.m1.2.2.2.2.1.3.cmml" xref="S6.Ex39.m1.2.2.2.2.1.3">ℓ</ci></apply><ci id="S6.Ex39.m1.1.1.1.1.cmml" xref="S6.Ex39.m1.1.1.1.1">𝑝</ci></list></apply><vector id="S6.Ex39.m1.5.5.2.3.cmml" xref="S6.Ex39.m1.5.5.2.2"><apply id="S6.Ex39.m1.4.4.1.1.1.cmml" xref="S6.Ex39.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.Ex39.m1.4.4.1.1.1.1.cmml" xref="S6.Ex39.m1.4.4.1.1.1">subscript</csymbol><apply id="S6.Ex39.m1.4.4.1.1.1.2.cmml" xref="S6.Ex39.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.Ex39.m1.4.4.1.1.1.2.1.cmml" xref="S6.Ex39.m1.4.4.1.1.1">superscript</csymbol><ci id="S6.Ex39.m1.4.4.1.1.1.2.2.cmml" xref="S6.Ex39.m1.4.4.1.1.1.2.2">ℝ</ci><ci id="S6.Ex39.m1.4.4.1.1.1.2.3.cmml" xref="S6.Ex39.m1.4.4.1.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex39.m1.4.4.1.1.1.3.cmml" xref="S6.Ex39.m1.4.4.1.1.1.3"></plus></apply><apply id="S6.Ex39.m1.5.5.2.2.2.cmml" xref="S6.Ex39.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.Ex39.m1.5.5.2.2.2.1.cmml" xref="S6.Ex39.m1.5.5.2.2.2">subscript</csymbol><ci id="S6.Ex39.m1.5.5.2.2.2.2.cmml" xref="S6.Ex39.m1.5.5.2.2.2.2">𝑤</ci><ci id="S6.Ex39.m1.5.5.2.2.2.3.cmml" xref="S6.Ex39.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex39.m1.3.3.cmml" xref="S6.Ex39.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex39.m1.5c">\displaystyle W_{\mathcal{B}}^{k_{0}+\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex39.m1.5d">italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{[}W^{k_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{\mathcal{% B}}^{k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}" class="ltx_Math" display="inline" id="S6.Ex39.m2.8"><semantics id="S6.Ex39.m2.8a"><mrow id="S6.Ex39.m2.8.8" xref="S6.Ex39.m2.8.8.cmml"><mi id="S6.Ex39.m2.8.8.4" xref="S6.Ex39.m2.8.8.4.cmml"></mi><mo id="S6.Ex39.m2.8.8.3" xref="S6.Ex39.m2.8.8.3.cmml">=</mo><msub id="S6.Ex39.m2.8.8.2" xref="S6.Ex39.m2.8.8.2.cmml"><mrow id="S6.Ex39.m2.8.8.2.2.2" xref="S6.Ex39.m2.8.8.2.2.3.cmml"><mo id="S6.Ex39.m2.8.8.2.2.2.3" 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cd="ambiguous" id="S6.Ex39.m2.8.8.2.4.3.1.cmml" xref="S6.Ex39.m2.8.8.2.4.3">subscript</csymbol><ci id="S6.Ex39.m2.8.8.2.4.3.2.cmml" xref="S6.Ex39.m2.8.8.2.4.3.2">𝑘</ci><cn id="S6.Ex39.m2.8.8.2.4.3.3.cmml" type="integer" xref="S6.Ex39.m2.8.8.2.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex39.m2.8c">\displaystyle=\big{[}W^{k_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{\mathcal{% B}}^{k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex39.m2.8d">= [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex40"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{[}W^{k_{0},p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X% ),W_{\mathcal{B}}^{k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{% \frac{\ell}{k_{1}}}." class="ltx_Math" display="inline" 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encoding="application/x-tex" id="S6.Ex40.m1.7c">\displaystyle=\big{[}W^{k_{0},p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X% ),W_{\mathcal{B}}^{k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{% \frac{\ell}{k_{1}}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex40.m1.7d">= [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S6.SS2.p3"> <p class="ltx_p" id="S6.SS2.p3.1">As a consequence of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a> we can characterise the complex interpolation space of weighted Sobolev spaces without boundary conditions. One of the embeddings, which is needed in the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>, was already proved in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem3" title="Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem6"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem6.1.1.1">Proposition 6.6</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem6.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem6.p1"> <p class="ltx_p" id="S6.Thmtheorem6.p1.7"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem6.p1.7.7">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.1.1.m1.2"><semantics id="S6.Thmtheorem6.p1.1.1.m1.2a"><mrow id="S6.Thmtheorem6.p1.1.1.m1.2.3" xref="S6.Thmtheorem6.p1.1.1.m1.2.3.cmml"><mi id="S6.Thmtheorem6.p1.1.1.m1.2.3.2" xref="S6.Thmtheorem6.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S6.Thmtheorem6.p1.1.1.m1.2.3.1" 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id="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2"><in id="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.1.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.1"></in><ci id="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.2.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.2">𝑗</ci><apply id="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3.1.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3">subscript</csymbol><ci id="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3.2.cmml" xref="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.4.4.m4.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.4.4.m4.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.4.4.m4.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math>, <math alttext="\ell\in\{1,\dots,k_{1}-1\}" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.5.5.m5.3"><semantics id="S6.Thmtheorem6.p1.5.5.m5.3a"><mrow id="S6.Thmtheorem6.p1.5.5.m5.3.3" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.cmml"><mi id="S6.Thmtheorem6.p1.5.5.m5.3.3.3" mathvariant="normal" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.3.cmml">ℓ</mi><mo id="S6.Thmtheorem6.p1.5.5.m5.3.3.2" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.2.cmml">∈</mo><mrow id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.2.cmml"><mo id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.2" stretchy="false" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.2.cmml">{</mo><mn id="S6.Thmtheorem6.p1.5.5.m5.1.1" xref="S6.Thmtheorem6.p1.5.5.m5.1.1.cmml">1</mn><mo id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.3" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.2.cmml">,</mo><mi id="S6.Thmtheorem6.p1.5.5.m5.2.2" mathvariant="normal" xref="S6.Thmtheorem6.p1.5.5.m5.2.2.cmml">…</mi><mo id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.4" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.2.cmml">,</mo><mrow id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.cmml"><msub id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.cmml"><mi id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.2" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.2.cmml">k</mi><mn id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.3" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.1" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.3" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.5" stretchy="false" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.5.5.m5.3b"><apply id="S6.Thmtheorem6.p1.5.5.m5.3.3.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3"><in id="S6.Thmtheorem6.p1.5.5.m5.3.3.2.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.2"></in><ci id="S6.Thmtheorem6.p1.5.5.m5.3.3.3.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.3">ℓ</ci><set id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.2.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1"><cn id="S6.Thmtheorem6.p1.5.5.m5.1.1.cmml" type="integer" xref="S6.Thmtheorem6.p1.5.5.m5.1.1">1</cn><ci id="S6.Thmtheorem6.p1.5.5.m5.2.2.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.2.2">…</ci><apply id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1"><minus id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.1"></minus><apply id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.1.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.2.cmml" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.2.3">1</cn></apply><cn id="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem6.p1.5.5.m5.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.5.5.m5.3c">\ell\in\{1,\dots,k_{1}-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.5.5.m5.3d">roman_ℓ ∈ { 1 , … , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.6.6.m6.1"><semantics id="S6.Thmtheorem6.p1.6.6.m6.1a"><mi id="S6.Thmtheorem6.p1.6.6.m6.1.1" xref="S6.Thmtheorem6.p1.6.6.m6.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.6.6.m6.1b"><ci id="S6.Thmtheorem6.p1.6.6.m6.1.1.cmml" xref="S6.Thmtheorem6.p1.6.6.m6.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.6.6.m6.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.6.6.m6.1d">italic_X</annotation></semantics></math> be a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S6.Thmtheorem6.p1.7.7.m7.1"><semantics id="S6.Thmtheorem6.p1.7.7.m7.1a"><mi id="S6.Thmtheorem6.p1.7.7.m7.1.1" xref="S6.Thmtheorem6.p1.7.7.m7.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem6.p1.7.7.m7.1b"><ci id="S6.Thmtheorem6.p1.7.7.m7.1.1.cmml" xref="S6.Thmtheorem6.p1.7.7.m7.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem6.p1.7.7.m7.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem6.p1.7.7.m7.1d">roman_UMD</annotation></semantics></math> Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex41"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="W^{k_{0}+\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big{[}W^{k_{0},p}(\mathbb{R% }^{d}_{+},w_{\gamma};X),W^{k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big% {]}_{\frac{\ell}{k_{1}}}." class="ltx_Math" display="block" id="S6.Ex41.m1.10"><semantics id="S6.Ex41.m1.10a"><mrow id="S6.Ex41.m1.10.10.1" xref="S6.Ex41.m1.10.10.1.1.cmml"><mrow id="S6.Ex41.m1.10.10.1.1" xref="S6.Ex41.m1.10.10.1.1.cmml"><mrow id="S6.Ex41.m1.10.10.1.1.2" xref="S6.Ex41.m1.10.10.1.1.2.cmml"><msup id="S6.Ex41.m1.10.10.1.1.2.4" xref="S6.Ex41.m1.10.10.1.1.2.4.cmml"><mi id="S6.Ex41.m1.10.10.1.1.2.4.2" xref="S6.Ex41.m1.10.10.1.1.2.4.2.cmml">W</mi><mrow id="S6.Ex41.m1.2.2.2.2" xref="S6.Ex41.m1.2.2.2.3.cmml"><mrow id="S6.Ex41.m1.2.2.2.2.1" xref="S6.Ex41.m1.2.2.2.2.1.cmml"><msub id="S6.Ex41.m1.2.2.2.2.1.2" xref="S6.Ex41.m1.2.2.2.2.1.2.cmml"><mi id="S6.Ex41.m1.2.2.2.2.1.2.2" xref="S6.Ex41.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.Ex41.m1.2.2.2.2.1.2.3" xref="S6.Ex41.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.Ex41.m1.2.2.2.2.1.1" xref="S6.Ex41.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.Ex41.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.Ex41.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.Ex41.m1.2.2.2.2.2" xref="S6.Ex41.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex41.m1.1.1.1.1" xref="S6.Ex41.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.Ex41.m1.10.10.1.1.2.3" xref="S6.Ex41.m1.10.10.1.1.2.3.cmml">⁢</mo><mrow id="S6.Ex41.m1.10.10.1.1.2.2.2" xref="S6.Ex41.m1.10.10.1.1.2.2.3.cmml"><mo id="S6.Ex41.m1.10.10.1.1.2.2.2.3" stretchy="false" xref="S6.Ex41.m1.10.10.1.1.2.2.3.cmml">(</mo><msubsup id="S6.Ex41.m1.10.10.1.1.1.1.1.1" xref="S6.Ex41.m1.10.10.1.1.1.1.1.1.cmml"><mi id="S6.Ex41.m1.10.10.1.1.1.1.1.1.2.2" xref="S6.Ex41.m1.10.10.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo 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+ end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S6.SS2.1"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.1.p1"> <p class="ltx_p" id="S6.SS2.1.p1.3">Note that the embedding “<math alttext="\hookleftarrow" class="ltx_Math" display="inline" id="S6.SS2.1.p1.1.m1.1"><semantics id="S6.SS2.1.p1.1.m1.1a"><mo id="S6.SS2.1.p1.1.m1.1.1" stretchy="false" xref="S6.SS2.1.p1.1.m1.1.1.cmml">↩</mo><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.1.m1.1b"><ci id="S6.SS2.1.p1.1.m1.1.1.cmml" xref="S6.SS2.1.p1.1.m1.1.1">↩</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.1.m1.1c">\hookleftarrow</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.1.m1.1d">↩</annotation></semantics></math>” has been proved in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem3" title="Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a>. For the other embedding, let <math alttext="j\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS2.1.p1.2.m2.1"><semantics id="S6.SS2.1.p1.2.m2.1a"><mrow id="S6.SS2.1.p1.2.m2.1.1" xref="S6.SS2.1.p1.2.m2.1.1.cmml"><mi id="S6.SS2.1.p1.2.m2.1.1.2" xref="S6.SS2.1.p1.2.m2.1.1.2.cmml">j</mi><mo id="S6.SS2.1.p1.2.m2.1.1.1" xref="S6.SS2.1.p1.2.m2.1.1.1.cmml">∈</mo><msub id="S6.SS2.1.p1.2.m2.1.1.3" xref="S6.SS2.1.p1.2.m2.1.1.3.cmml"><mi id="S6.SS2.1.p1.2.m2.1.1.3.2" xref="S6.SS2.1.p1.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS2.1.p1.2.m2.1.1.3.3" xref="S6.SS2.1.p1.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.2.m2.1b"><apply id="S6.SS2.1.p1.2.m2.1.1.cmml" xref="S6.SS2.1.p1.2.m2.1.1"><in id="S6.SS2.1.p1.2.m2.1.1.1.cmml" xref="S6.SS2.1.p1.2.m2.1.1.1"></in><ci id="S6.SS2.1.p1.2.m2.1.1.2.cmml" xref="S6.SS2.1.p1.2.m2.1.1.2">𝑗</ci><apply id="S6.SS2.1.p1.2.m2.1.1.3.cmml" xref="S6.SS2.1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.1.p1.2.m2.1.1.3.1.cmml" xref="S6.SS2.1.p1.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS2.1.p1.2.m2.1.1.3.2.cmml" xref="S6.SS2.1.p1.2.m2.1.1.3.2">ℕ</ci><cn id="S6.SS2.1.p1.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS2.1.p1.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.2.m2.1c">j\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.2.m2.1d">italic_j ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> be the smallest integer such that <math alttext="k_{0}+\ell<j+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS2.1.p1.3.m3.1"><semantics id="S6.SS2.1.p1.3.m3.1a"><mrow id="S6.SS2.1.p1.3.m3.1.1" xref="S6.SS2.1.p1.3.m3.1.1.cmml"><mrow id="S6.SS2.1.p1.3.m3.1.1.2" xref="S6.SS2.1.p1.3.m3.1.1.2.cmml"><msub id="S6.SS2.1.p1.3.m3.1.1.2.2" xref="S6.SS2.1.p1.3.m3.1.1.2.2.cmml"><mi id="S6.SS2.1.p1.3.m3.1.1.2.2.2" xref="S6.SS2.1.p1.3.m3.1.1.2.2.2.cmml">k</mi><mn id="S6.SS2.1.p1.3.m3.1.1.2.2.3" xref="S6.SS2.1.p1.3.m3.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.SS2.1.p1.3.m3.1.1.2.1" xref="S6.SS2.1.p1.3.m3.1.1.2.1.cmml">+</mo><mi id="S6.SS2.1.p1.3.m3.1.1.2.3" mathvariant="normal" xref="S6.SS2.1.p1.3.m3.1.1.2.3.cmml">ℓ</mi></mrow><mo id="S6.SS2.1.p1.3.m3.1.1.1" xref="S6.SS2.1.p1.3.m3.1.1.1.cmml"><</mo><mrow id="S6.SS2.1.p1.3.m3.1.1.3" xref="S6.SS2.1.p1.3.m3.1.1.3.cmml"><mi id="S6.SS2.1.p1.3.m3.1.1.3.2" xref="S6.SS2.1.p1.3.m3.1.1.3.2.cmml">j</mi><mo id="S6.SS2.1.p1.3.m3.1.1.3.1" xref="S6.SS2.1.p1.3.m3.1.1.3.1.cmml">+</mo><mfrac id="S6.SS2.1.p1.3.m3.1.1.3.3" xref="S6.SS2.1.p1.3.m3.1.1.3.3.cmml"><mrow id="S6.SS2.1.p1.3.m3.1.1.3.3.2" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2.cmml"><mi id="S6.SS2.1.p1.3.m3.1.1.3.3.2.2" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS2.1.p1.3.m3.1.1.3.3.2.1" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS2.1.p1.3.m3.1.1.3.3.2.3" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS2.1.p1.3.m3.1.1.3.3.3" xref="S6.SS2.1.p1.3.m3.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.1.p1.3.m3.1b"><apply id="S6.SS2.1.p1.3.m3.1.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1"><lt id="S6.SS2.1.p1.3.m3.1.1.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1.1"></lt><apply id="S6.SS2.1.p1.3.m3.1.1.2.cmml" xref="S6.SS2.1.p1.3.m3.1.1.2"><plus id="S6.SS2.1.p1.3.m3.1.1.2.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1.2.1"></plus><apply id="S6.SS2.1.p1.3.m3.1.1.2.2.cmml" xref="S6.SS2.1.p1.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.1.p1.3.m3.1.1.2.2.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1.2.2">subscript</csymbol><ci id="S6.SS2.1.p1.3.m3.1.1.2.2.2.cmml" xref="S6.SS2.1.p1.3.m3.1.1.2.2.2">𝑘</ci><cn id="S6.SS2.1.p1.3.m3.1.1.2.2.3.cmml" type="integer" xref="S6.SS2.1.p1.3.m3.1.1.2.2.3">0</cn></apply><ci id="S6.SS2.1.p1.3.m3.1.1.2.3.cmml" xref="S6.SS2.1.p1.3.m3.1.1.2.3">ℓ</ci></apply><apply id="S6.SS2.1.p1.3.m3.1.1.3.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3"><plus id="S6.SS2.1.p1.3.m3.1.1.3.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.1"></plus><ci id="S6.SS2.1.p1.3.m3.1.1.3.2.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.2">𝑗</ci><apply id="S6.SS2.1.p1.3.m3.1.1.3.3.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.3"><divide id="S6.SS2.1.p1.3.m3.1.1.3.3.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.3"></divide><apply id="S6.SS2.1.p1.3.m3.1.1.3.3.2.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2"><plus id="S6.SS2.1.p1.3.m3.1.1.3.3.2.1.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2.1"></plus><ci id="S6.SS2.1.p1.3.m3.1.1.3.3.2.2.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS2.1.p1.3.m3.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS2.1.p1.3.m3.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS2.1.p1.3.m3.1.1.3.3.3.cmml" xref="S6.SS2.1.p1.3.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.1.p1.3.m3.1c">k_{0}+\ell<j+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.1.p1.3.m3.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ < italic_j + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Then by Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a> we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx38"> <tbody id="S6.Ex42"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W^{k_{0}+\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.Ex42.m1.5"><semantics id="S6.Ex42.m1.5a"><mrow id="S6.Ex42.m1.5.5" xref="S6.Ex42.m1.5.5.cmml"><msup id="S6.Ex42.m1.5.5.4" xref="S6.Ex42.m1.5.5.4.cmml"><mi id="S6.Ex42.m1.5.5.4.2" xref="S6.Ex42.m1.5.5.4.2.cmml">W</mi><mrow id="S6.Ex42.m1.2.2.2.2" xref="S6.Ex42.m1.2.2.2.3.cmml"><mrow id="S6.Ex42.m1.2.2.2.2.1" xref="S6.Ex42.m1.2.2.2.2.1.cmml"><msub id="S6.Ex42.m1.2.2.2.2.1.2" xref="S6.Ex42.m1.2.2.2.2.1.2.cmml"><mi id="S6.Ex42.m1.2.2.2.2.1.2.2" xref="S6.Ex42.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.Ex42.m1.2.2.2.2.1.2.3" xref="S6.Ex42.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.Ex42.m1.2.2.2.2.1.1" xref="S6.Ex42.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.Ex42.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.Ex42.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.Ex42.m1.2.2.2.2.2" xref="S6.Ex42.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex42.m1.1.1.1.1" xref="S6.Ex42.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.Ex42.m1.5.5.3" xref="S6.Ex42.m1.5.5.3.cmml">⁢</mo><mrow id="S6.Ex42.m1.5.5.2.2" xref="S6.Ex42.m1.5.5.2.3.cmml"><mo id="S6.Ex42.m1.5.5.2.2.3" stretchy="false" xref="S6.Ex42.m1.5.5.2.3.cmml">(</mo><msubsup id="S6.Ex42.m1.4.4.1.1.1" xref="S6.Ex42.m1.4.4.1.1.1.cmml"><mi id="S6.Ex42.m1.4.4.1.1.1.2.2" xref="S6.Ex42.m1.4.4.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.Ex42.m1.4.4.1.1.1.3" xref="S6.Ex42.m1.4.4.1.1.1.3.cmml">+</mo><mi id="S6.Ex42.m1.4.4.1.1.1.2.3" xref="S6.Ex42.m1.4.4.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.Ex42.m1.5.5.2.2.4" xref="S6.Ex42.m1.5.5.2.3.cmml">,</mo><msub id="S6.Ex42.m1.5.5.2.2.2" xref="S6.Ex42.m1.5.5.2.2.2.cmml"><mi id="S6.Ex42.m1.5.5.2.2.2.2" xref="S6.Ex42.m1.5.5.2.2.2.2.cmml">w</mi><mi id="S6.Ex42.m1.5.5.2.2.2.3" xref="S6.Ex42.m1.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex42.m1.5.5.2.2.5" xref="S6.Ex42.m1.5.5.2.3.cmml">;</mo><mi id="S6.Ex42.m1.3.3" xref="S6.Ex42.m1.3.3.cmml">X</mi><mo id="S6.Ex42.m1.5.5.2.2.6" stretchy="false" xref="S6.Ex42.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex42.m1.5b"><apply id="S6.Ex42.m1.5.5.cmml" xref="S6.Ex42.m1.5.5"><times id="S6.Ex42.m1.5.5.3.cmml" xref="S6.Ex42.m1.5.5.3"></times><apply id="S6.Ex42.m1.5.5.4.cmml" xref="S6.Ex42.m1.5.5.4"><csymbol cd="ambiguous" id="S6.Ex42.m1.5.5.4.1.cmml" xref="S6.Ex42.m1.5.5.4">superscript</csymbol><ci id="S6.Ex42.m1.5.5.4.2.cmml" xref="S6.Ex42.m1.5.5.4.2">𝑊</ci><list id="S6.Ex42.m1.2.2.2.3.cmml" xref="S6.Ex42.m1.2.2.2.2"><apply id="S6.Ex42.m1.2.2.2.2.1.cmml" 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xref="S6.Ex42.m1.4.4.1.1.1">superscript</csymbol><ci id="S6.Ex42.m1.4.4.1.1.1.2.2.cmml" xref="S6.Ex42.m1.4.4.1.1.1.2.2">ℝ</ci><ci id="S6.Ex42.m1.4.4.1.1.1.2.3.cmml" xref="S6.Ex42.m1.4.4.1.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex42.m1.4.4.1.1.1.3.cmml" xref="S6.Ex42.m1.4.4.1.1.1.3"></plus></apply><apply id="S6.Ex42.m1.5.5.2.2.2.cmml" xref="S6.Ex42.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.Ex42.m1.5.5.2.2.2.1.cmml" xref="S6.Ex42.m1.5.5.2.2.2">subscript</csymbol><ci id="S6.Ex42.m1.5.5.2.2.2.2.cmml" xref="S6.Ex42.m1.5.5.2.2.2.2">𝑤</ci><ci id="S6.Ex42.m1.5.5.2.2.2.3.cmml" xref="S6.Ex42.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex42.m1.3.3.cmml" xref="S6.Ex42.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex42.m1.5c">\displaystyle W^{k_{0}+\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex42.m1.5d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=W^{k_{0}+\ell,p}_{\operatorname{Tr}_{j}}(\mathbb{R}^{d}_{+},w_{% \gamma};X)" class="ltx_Math" display="inline" id="S6.Ex42.m2.5"><semantics id="S6.Ex42.m2.5a"><mrow id="S6.Ex42.m2.5.5" xref="S6.Ex42.m2.5.5.cmml"><mi id="S6.Ex42.m2.5.5.4" xref="S6.Ex42.m2.5.5.4.cmml"></mi><mo id="S6.Ex42.m2.5.5.3" xref="S6.Ex42.m2.5.5.3.cmml">=</mo><mrow id="S6.Ex42.m2.5.5.2" xref="S6.Ex42.m2.5.5.2.cmml"><msubsup id="S6.Ex42.m2.5.5.2.4" xref="S6.Ex42.m2.5.5.2.4.cmml"><mi id="S6.Ex42.m2.5.5.2.4.2.2" xref="S6.Ex42.m2.5.5.2.4.2.2.cmml">W</mi><msub id="S6.Ex42.m2.5.5.2.4.3" xref="S6.Ex42.m2.5.5.2.4.3.cmml"><mi id="S6.Ex42.m2.5.5.2.4.3.2" xref="S6.Ex42.m2.5.5.2.4.3.2.cmml">Tr</mi><mi 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xref="S6.Ex43.m1.8.8.2.4.3">subscript</csymbol><ci id="S6.Ex43.m1.8.8.2.4.3.2.cmml" xref="S6.Ex43.m1.8.8.2.4.3.2">𝑘</ci><cn id="S6.Ex43.m1.8.8.2.4.3.3.cmml" type="integer" xref="S6.Ex43.m1.8.8.2.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex43.m1.8c">\displaystyle=\big{[}W^{k_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W^{k_{0}+k_{% 1},p}_{\operatorname{Tr}_{j}}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{% \ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex43.m1.8d">= [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex44"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\hookrightarrow\big{[}W^{k_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X% ),W^{k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}% }}." class="ltx_Math" display="inline" 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id="S6.Ex44.m1.7.7.1.1.2.4.3.2.cmml" xref="S6.Ex44.m1.7.7.1.1.2.4.3.2">𝑘</ci><cn id="S6.Ex44.m1.7.7.1.1.2.4.3.3.cmml" type="integer" xref="S6.Ex44.m1.7.7.1.1.2.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex44.m1.7c">\displaystyle\hookrightarrow\big{[}W^{k_{0},p}(\mathbb{R}^{d}_{+},w_{\gamma};X% ),W^{k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}% }}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex44.m1.7d">↪ [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS2.1.p1.4">This completes the proof. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS2.p4"> <p class="ltx_p" id="S6.SS2.p4.4">By standard localisation techniques, the complex interpolation results above can be extended to smooth bounded domains. Using localisation the definition of the traces on the half-space and the function space <math alttext="W^{\ell,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS2.p4.1.m1.5"><semantics id="S6.SS2.p4.1.m1.5a"><mrow id="S6.SS2.p4.1.m1.5.5" xref="S6.SS2.p4.1.m1.5.5.cmml"><msubsup id="S6.SS2.p4.1.m1.5.5.4" xref="S6.SS2.p4.1.m1.5.5.4.cmml"><mi id="S6.SS2.p4.1.m1.5.5.4.2.2" xref="S6.SS2.p4.1.m1.5.5.4.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS2.p4.1.m1.5.5.4.3" xref="S6.SS2.p4.1.m1.5.5.4.3.cmml">ℬ</mi><mrow id="S6.SS2.p4.1.m1.2.2.2.4" xref="S6.SS2.p4.1.m1.2.2.2.3.cmml"><mi id="S6.SS2.p4.1.m1.1.1.1.1" mathvariant="normal" xref="S6.SS2.p4.1.m1.1.1.1.1.cmml">ℓ</mi><mo id="S6.SS2.p4.1.m1.2.2.2.4.1" xref="S6.SS2.p4.1.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS2.p4.1.m1.2.2.2.2" xref="S6.SS2.p4.1.m1.2.2.2.2.cmml">p</mi></mrow></msubsup><mo id="S6.SS2.p4.1.m1.5.5.3" xref="S6.SS2.p4.1.m1.5.5.3.cmml">⁢</mo><mrow id="S6.SS2.p4.1.m1.5.5.2.2" xref="S6.SS2.p4.1.m1.5.5.2.3.cmml"><mo id="S6.SS2.p4.1.m1.5.5.2.2.3" stretchy="false" xref="S6.SS2.p4.1.m1.5.5.2.3.cmml">(</mo><msubsup id="S6.SS2.p4.1.m1.4.4.1.1.1" xref="S6.SS2.p4.1.m1.4.4.1.1.1.cmml"><mi id="S6.SS2.p4.1.m1.4.4.1.1.1.2.2" xref="S6.SS2.p4.1.m1.4.4.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.SS2.p4.1.m1.4.4.1.1.1.3" xref="S6.SS2.p4.1.m1.4.4.1.1.1.3.cmml">+</mo><mi id="S6.SS2.p4.1.m1.4.4.1.1.1.2.3" xref="S6.SS2.p4.1.m1.4.4.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.SS2.p4.1.m1.5.5.2.2.4" xref="S6.SS2.p4.1.m1.5.5.2.3.cmml">,</mo><msub id="S6.SS2.p4.1.m1.5.5.2.2.2" xref="S6.SS2.p4.1.m1.5.5.2.2.2.cmml"><mi id="S6.SS2.p4.1.m1.5.5.2.2.2.2" xref="S6.SS2.p4.1.m1.5.5.2.2.2.2.cmml">w</mi><mi id="S6.SS2.p4.1.m1.5.5.2.2.2.3" xref="S6.SS2.p4.1.m1.5.5.2.2.2.3.cmml">γ</mi></msub><mo id="S6.SS2.p4.1.m1.5.5.2.2.5" xref="S6.SS2.p4.1.m1.5.5.2.3.cmml">;</mo><mi id="S6.SS2.p4.1.m1.3.3" xref="S6.SS2.p4.1.m1.3.3.cmml">X</mi><mo id="S6.SS2.p4.1.m1.5.5.2.2.6" stretchy="false" xref="S6.SS2.p4.1.m1.5.5.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.1.m1.5b"><apply id="S6.SS2.p4.1.m1.5.5.cmml" xref="S6.SS2.p4.1.m1.5.5"><times id="S6.SS2.p4.1.m1.5.5.3.cmml" xref="S6.SS2.p4.1.m1.5.5.3"></times><apply id="S6.SS2.p4.1.m1.5.5.4.cmml" xref="S6.SS2.p4.1.m1.5.5.4"><csymbol cd="ambiguous" id="S6.SS2.p4.1.m1.5.5.4.1.cmml" xref="S6.SS2.p4.1.m1.5.5.4">subscript</csymbol><apply id="S6.SS2.p4.1.m1.5.5.4.2.cmml" xref="S6.SS2.p4.1.m1.5.5.4"><csymbol cd="ambiguous" id="S6.SS2.p4.1.m1.5.5.4.2.1.cmml" xref="S6.SS2.p4.1.m1.5.5.4">superscript</csymbol><ci id="S6.SS2.p4.1.m1.5.5.4.2.2.cmml" xref="S6.SS2.p4.1.m1.5.5.4.2.2">𝑊</ci><list id="S6.SS2.p4.1.m1.2.2.2.3.cmml" xref="S6.SS2.p4.1.m1.2.2.2.4"><ci id="S6.SS2.p4.1.m1.1.1.1.1.cmml" xref="S6.SS2.p4.1.m1.1.1.1.1">ℓ</ci><ci id="S6.SS2.p4.1.m1.2.2.2.2.cmml" xref="S6.SS2.p4.1.m1.2.2.2.2">𝑝</ci></list></apply><ci id="S6.SS2.p4.1.m1.5.5.4.3.cmml" xref="S6.SS2.p4.1.m1.5.5.4.3">ℬ</ci></apply><vector id="S6.SS2.p4.1.m1.5.5.2.3.cmml" xref="S6.SS2.p4.1.m1.5.5.2.2"><apply id="S6.SS2.p4.1.m1.4.4.1.1.1.cmml" xref="S6.SS2.p4.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p4.1.m1.4.4.1.1.1.1.cmml" xref="S6.SS2.p4.1.m1.4.4.1.1.1">subscript</csymbol><apply id="S6.SS2.p4.1.m1.4.4.1.1.1.2.cmml" xref="S6.SS2.p4.1.m1.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p4.1.m1.4.4.1.1.1.2.1.cmml" xref="S6.SS2.p4.1.m1.4.4.1.1.1">superscript</csymbol><ci id="S6.SS2.p4.1.m1.4.4.1.1.1.2.2.cmml" xref="S6.SS2.p4.1.m1.4.4.1.1.1.2.2">ℝ</ci><ci id="S6.SS2.p4.1.m1.4.4.1.1.1.2.3.cmml" xref="S6.SS2.p4.1.m1.4.4.1.1.1.2.3">𝑑</ci></apply><plus id="S6.SS2.p4.1.m1.4.4.1.1.1.3.cmml" xref="S6.SS2.p4.1.m1.4.4.1.1.1.3"></plus></apply><apply id="S6.SS2.p4.1.m1.5.5.2.2.2.cmml" xref="S6.SS2.p4.1.m1.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.p4.1.m1.5.5.2.2.2.1.cmml" xref="S6.SS2.p4.1.m1.5.5.2.2.2">subscript</csymbol><ci id="S6.SS2.p4.1.m1.5.5.2.2.2.2.cmml" xref="S6.SS2.p4.1.m1.5.5.2.2.2.2">𝑤</ci><ci id="S6.SS2.p4.1.m1.5.5.2.2.2.3.cmml" xref="S6.SS2.p4.1.m1.5.5.2.2.2.3">𝛾</ci></apply><ci id="S6.SS2.p4.1.m1.3.3.cmml" xref="S6.SS2.p4.1.m1.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.1.m1.5c">W^{\ell,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.1.m1.5d">italic_W start_POSTSUPERSCRIPT roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> can be extended to smooth bounded domains <math alttext="\mathcal{O}\subseteq\mathbb{R}^{d}" class="ltx_Math" display="inline" id="S6.SS2.p4.2.m2.1"><semantics id="S6.SS2.p4.2.m2.1a"><mrow id="S6.SS2.p4.2.m2.1.1" xref="S6.SS2.p4.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS2.p4.2.m2.1.1.2" xref="S6.SS2.p4.2.m2.1.1.2.cmml">𝒪</mi><mo id="S6.SS2.p4.2.m2.1.1.1" xref="S6.SS2.p4.2.m2.1.1.1.cmml">⊆</mo><msup id="S6.SS2.p4.2.m2.1.1.3" xref="S6.SS2.p4.2.m2.1.1.3.cmml"><mi id="S6.SS2.p4.2.m2.1.1.3.2" xref="S6.SS2.p4.2.m2.1.1.3.2.cmml">ℝ</mi><mi id="S6.SS2.p4.2.m2.1.1.3.3" xref="S6.SS2.p4.2.m2.1.1.3.3.cmml">d</mi></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.2.m2.1b"><apply id="S6.SS2.p4.2.m2.1.1.cmml" xref="S6.SS2.p4.2.m2.1.1"><subset id="S6.SS2.p4.2.m2.1.1.1.cmml" xref="S6.SS2.p4.2.m2.1.1.1"></subset><ci id="S6.SS2.p4.2.m2.1.1.2.cmml" xref="S6.SS2.p4.2.m2.1.1.2">𝒪</ci><apply id="S6.SS2.p4.2.m2.1.1.3.cmml" xref="S6.SS2.p4.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS2.p4.2.m2.1.1.3.1.cmml" xref="S6.SS2.p4.2.m2.1.1.3">superscript</csymbol><ci id="S6.SS2.p4.2.m2.1.1.3.2.cmml" xref="S6.SS2.p4.2.m2.1.1.3.2">ℝ</ci><ci id="S6.SS2.p4.2.m2.1.1.3.3.cmml" xref="S6.SS2.p4.2.m2.1.1.3.3">𝑑</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.2.m2.1c">\mathcal{O}\subseteq\mathbb{R}^{d}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.2.m2.1d">caligraphic_O ⊆ blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT</annotation></semantics></math>, we refer to <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>, Section 5.2]</cite> for details. Recall that <math alttext="w_{\gamma}^{\partial\mathcal{O}}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}" class="ltx_math_unparsed" display="inline" id="S6.SS2.p4.3.m3.3"><semantics id="S6.SS2.p4.3.m3.3a"><mrow id="S6.SS2.p4.3.m3.3b"><msubsup id="S6.SS2.p4.3.m3.3.4"><mi id="S6.SS2.p4.3.m3.3.4.2.2">w</mi><mi id="S6.SS2.p4.3.m3.3.4.2.3">γ</mi><mrow id="S6.SS2.p4.3.m3.3.4.3"><mo id="S6.SS2.p4.3.m3.3.4.3.1" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S6.SS2.p4.3.m3.3.4.3.2">𝒪</mi></mrow></msubsup><mrow id="S6.SS2.p4.3.m3.3.5"><mo id="S6.SS2.p4.3.m3.3.5.1" stretchy="false">(</mo><mi id="S6.SS2.p4.3.m3.1.1">x</mi><mo id="S6.SS2.p4.3.m3.3.5.2" rspace="0.278em" stretchy="false">)</mo></mrow><mo id="S6.SS2.p4.3.m3.3.6" rspace="0.278em">:=</mo><mi id="S6.SS2.p4.3.m3.2.2">dist</mi><msup id="S6.SS2.p4.3.m3.3.7"><mrow id="S6.SS2.p4.3.m3.3.7.2"><mo id="S6.SS2.p4.3.m3.3.7.2.1" stretchy="false">(</mo><mi id="S6.SS2.p4.3.m3.3.3">x</mi><mo id="S6.SS2.p4.3.m3.3.7.2.2">,</mo><mo id="S6.SS2.p4.3.m3.3.7.2.3" lspace="0em" rspace="0em">∂</mo><mi class="ltx_font_mathcaligraphic" id="S6.SS2.p4.3.m3.3.7.2.4">𝒪</mi><mo id="S6.SS2.p4.3.m3.3.7.2.5" stretchy="false">)</mo></mrow><mi id="S6.SS2.p4.3.m3.3.7.3">γ</mi></msup></mrow><annotation encoding="application/x-tex" id="S6.SS2.p4.3.m3.3c">w_{\gamma}^{\partial\mathcal{O}}(x):=\operatorname{dist}(x,\partial\mathcal{O}% )^{\gamma}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.3.m3.3d">italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT ( italic_x ) := roman_dist ( italic_x , ∂ caligraphic_O ) start_POSTSUPERSCRIPT italic_γ end_POSTSUPERSCRIPT</annotation></semantics></math> for <math alttext="x\in\mathcal{O}" class="ltx_Math" display="inline" id="S6.SS2.p4.4.m4.1"><semantics id="S6.SS2.p4.4.m4.1a"><mrow id="S6.SS2.p4.4.m4.1.1" xref="S6.SS2.p4.4.m4.1.1.cmml"><mi id="S6.SS2.p4.4.m4.1.1.2" xref="S6.SS2.p4.4.m4.1.1.2.cmml">x</mi><mo id="S6.SS2.p4.4.m4.1.1.1" xref="S6.SS2.p4.4.m4.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="S6.SS2.p4.4.m4.1.1.3" xref="S6.SS2.p4.4.m4.1.1.3.cmml">𝒪</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p4.4.m4.1b"><apply id="S6.SS2.p4.4.m4.1.1.cmml" xref="S6.SS2.p4.4.m4.1.1"><in id="S6.SS2.p4.4.m4.1.1.1.cmml" xref="S6.SS2.p4.4.m4.1.1.1"></in><ci id="S6.SS2.p4.4.m4.1.1.2.cmml" xref="S6.SS2.p4.4.m4.1.1.2">𝑥</ci><ci id="S6.SS2.p4.4.m4.1.1.3.cmml" xref="S6.SS2.p4.4.m4.1.1.3">𝒪</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p4.4.m4.1c">x\in\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p4.4.m4.1d">italic_x ∈ caligraphic_O</annotation></semantics></math>. From Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a> and Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem6" title="Proposition 6.6. ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.6</span></a> we obtain the following result.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem7"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem7.1.1.1">Proposition 6.7</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem7.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem7.p1"> <p class="ltx_p" id="S6.Thmtheorem7.p1.3"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem7.p1.3.3">Assume that the conditions of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a> hold and let <math alttext="\mathcal{O}" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.1.1.m1.1"><semantics id="S6.Thmtheorem7.p1.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S6.Thmtheorem7.p1.1.1.m1.1.1" xref="S6.Thmtheorem7.p1.1.1.m1.1.1.cmml">𝒪</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.1.1.m1.1b"><ci id="S6.Thmtheorem7.p1.1.1.m1.1.1.cmml" xref="S6.Thmtheorem7.p1.1.1.m1.1.1">𝒪</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.1.1.m1.1c">\mathcal{O}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.1.1.m1.1d">caligraphic_O</annotation></semantics></math> be a bounded <math alttext="C^{\infty}" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.2.2.m2.1"><semantics id="S6.Thmtheorem7.p1.2.2.m2.1a"><msup id="S6.Thmtheorem7.p1.2.2.m2.1.1" xref="S6.Thmtheorem7.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmtheorem7.p1.2.2.m2.1.1.2" xref="S6.Thmtheorem7.p1.2.2.m2.1.1.2.cmml">C</mi><mi id="S6.Thmtheorem7.p1.2.2.m2.1.1.3" mathvariant="normal" xref="S6.Thmtheorem7.p1.2.2.m2.1.1.3.cmml">∞</mi></msup><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.2.2.m2.1b"><apply id="S6.Thmtheorem7.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem7.p1.2.2.m2.1.1"><csymbol cd="ambiguous" id="S6.Thmtheorem7.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem7.p1.2.2.m2.1.1">superscript</csymbol><ci id="S6.Thmtheorem7.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem7.p1.2.2.m2.1.1.2">𝐶</ci><infinity id="S6.Thmtheorem7.p1.2.2.m2.1.1.3.cmml" xref="S6.Thmtheorem7.p1.2.2.m2.1.1.3"></infinity></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.2.2.m2.1c">C^{\infty}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.2.2.m2.1d">italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT</annotation></semantics></math>-domain. Then for <math alttext="\ell\in\{1,\dots,k_{1}-1\}" class="ltx_Math" display="inline" id="S6.Thmtheorem7.p1.3.3.m3.3"><semantics id="S6.Thmtheorem7.p1.3.3.m3.3a"><mrow id="S6.Thmtheorem7.p1.3.3.m3.3.3" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.cmml"><mi id="S6.Thmtheorem7.p1.3.3.m3.3.3.3" mathvariant="normal" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.3.cmml">ℓ</mi><mo id="S6.Thmtheorem7.p1.3.3.m3.3.3.2" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.2.cmml">∈</mo><mrow id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.2.cmml"><mo id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.2" stretchy="false" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.2.cmml">{</mo><mn id="S6.Thmtheorem7.p1.3.3.m3.1.1" xref="S6.Thmtheorem7.p1.3.3.m3.1.1.cmml">1</mn><mo id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.3" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.2.cmml">,</mo><mi id="S6.Thmtheorem7.p1.3.3.m3.2.2" mathvariant="normal" xref="S6.Thmtheorem7.p1.3.3.m3.2.2.cmml">…</mi><mo id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.4" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.2.cmml">,</mo><mrow id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.cmml"><msub id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.cmml"><mi id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.2" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.2.cmml">k</mi><mn id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.3" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.1" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.3" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.5" stretchy="false" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem7.p1.3.3.m3.3b"><apply id="S6.Thmtheorem7.p1.3.3.m3.3.3.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3"><in id="S6.Thmtheorem7.p1.3.3.m3.3.3.2.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.2"></in><ci id="S6.Thmtheorem7.p1.3.3.m3.3.3.3.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.3">ℓ</ci><set id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.2.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1"><cn id="S6.Thmtheorem7.p1.3.3.m3.1.1.cmml" type="integer" xref="S6.Thmtheorem7.p1.3.3.m3.1.1">1</cn><ci id="S6.Thmtheorem7.p1.3.3.m3.2.2.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.2.2">…</ci><apply id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1"><minus id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.1.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.1"></minus><apply id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2"><csymbol cd="ambiguous" id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.1.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2">subscript</csymbol><ci id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.2.cmml" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.2">𝑘</ci><cn id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.3.cmml" type="integer" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.2.3">1</cn></apply><cn id="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.3.cmml" type="integer" xref="S6.Thmtheorem7.p1.3.3.m3.3.3.1.1.1.3">1</cn></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem7.p1.3.3.m3.3c">\ell\in\{1,\dots,k_{1}-1\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem7.p1.3.3.m3.3d">roman_ℓ ∈ { 1 , … , italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - 1 }</annotation></semantics></math> we have</span></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx39"> <tbody id="S6.Ex45"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W_{\mathcal{B}}^{k_{0}+\ell,p}(\mathcal{O},w^{\partial\mathcal{O% }}_{\gamma};X)" class="ltx_Math" display="inline" id="S6.Ex45.m1.5"><semantics id="S6.Ex45.m1.5a"><mrow id="S6.Ex45.m1.5.5" xref="S6.Ex45.m1.5.5.cmml"><msubsup id="S6.Ex45.m1.5.5.3" xref="S6.Ex45.m1.5.5.3.cmml"><mi id="S6.Ex45.m1.5.5.3.2.2" xref="S6.Ex45.m1.5.5.3.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex45.m1.5.5.3.2.3" xref="S6.Ex45.m1.5.5.3.2.3.cmml">ℬ</mi><mrow id="S6.Ex45.m1.2.2.2.2" xref="S6.Ex45.m1.2.2.2.3.cmml"><mrow id="S6.Ex45.m1.2.2.2.2.1" xref="S6.Ex45.m1.2.2.2.2.1.cmml"><msub id="S6.Ex45.m1.2.2.2.2.1.2" xref="S6.Ex45.m1.2.2.2.2.1.2.cmml"><mi id="S6.Ex45.m1.2.2.2.2.1.2.2" xref="S6.Ex45.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.Ex45.m1.2.2.2.2.1.2.3" xref="S6.Ex45.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.Ex45.m1.2.2.2.2.1.1" xref="S6.Ex45.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.Ex45.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.Ex45.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.Ex45.m1.2.2.2.2.2" xref="S6.Ex45.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex45.m1.1.1.1.1" xref="S6.Ex45.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex45.m1.5.5.2" xref="S6.Ex45.m1.5.5.2.cmml">⁢</mo><mrow id="S6.Ex45.m1.5.5.1.1" xref="S6.Ex45.m1.5.5.1.2.cmml"><mo id="S6.Ex45.m1.5.5.1.1.2" stretchy="false" xref="S6.Ex45.m1.5.5.1.2.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex45.m1.3.3" xref="S6.Ex45.m1.3.3.cmml">𝒪</mi><mo id="S6.Ex45.m1.5.5.1.1.3" xref="S6.Ex45.m1.5.5.1.2.cmml">,</mo><msubsup id="S6.Ex45.m1.5.5.1.1.1" xref="S6.Ex45.m1.5.5.1.1.1.cmml"><mi id="S6.Ex45.m1.5.5.1.1.1.2.2" xref="S6.Ex45.m1.5.5.1.1.1.2.2.cmml">w</mi><mi id="S6.Ex45.m1.5.5.1.1.1.3" xref="S6.Ex45.m1.5.5.1.1.1.3.cmml">γ</mi><mrow id="S6.Ex45.m1.5.5.1.1.1.2.3" xref="S6.Ex45.m1.5.5.1.1.1.2.3.cmml"><mo id="S6.Ex45.m1.5.5.1.1.1.2.3.1" rspace="0em" xref="S6.Ex45.m1.5.5.1.1.1.2.3.1.cmml">∂</mo><mi class="ltx_font_mathcaligraphic" id="S6.Ex45.m1.5.5.1.1.1.2.3.2" xref="S6.Ex45.m1.5.5.1.1.1.2.3.2.cmml">𝒪</mi></mrow></msubsup><mo id="S6.Ex45.m1.5.5.1.1.4" xref="S6.Ex45.m1.5.5.1.2.cmml">;</mo><mi id="S6.Ex45.m1.4.4" xref="S6.Ex45.m1.4.4.cmml">X</mi><mo id="S6.Ex45.m1.5.5.1.1.5" stretchy="false" xref="S6.Ex45.m1.5.5.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex45.m1.5b"><apply id="S6.Ex45.m1.5.5.cmml" xref="S6.Ex45.m1.5.5"><times id="S6.Ex45.m1.5.5.2.cmml" xref="S6.Ex45.m1.5.5.2"></times><apply id="S6.Ex45.m1.5.5.3.cmml" xref="S6.Ex45.m1.5.5.3"><csymbol cd="ambiguous" id="S6.Ex45.m1.5.5.3.1.cmml" xref="S6.Ex45.m1.5.5.3">superscript</csymbol><apply id="S6.Ex45.m1.5.5.3.2.cmml" xref="S6.Ex45.m1.5.5.3"><csymbol cd="ambiguous" id="S6.Ex45.m1.5.5.3.2.1.cmml" xref="S6.Ex45.m1.5.5.3">subscript</csymbol><ci id="S6.Ex45.m1.5.5.3.2.2.cmml" xref="S6.Ex45.m1.5.5.3.2.2">𝑊</ci><ci id="S6.Ex45.m1.5.5.3.2.3.cmml" xref="S6.Ex45.m1.5.5.3.2.3">ℬ</ci></apply><list id="S6.Ex45.m1.2.2.2.3.cmml" xref="S6.Ex45.m1.2.2.2.2"><apply id="S6.Ex45.m1.2.2.2.2.1.cmml" xref="S6.Ex45.m1.2.2.2.2.1"><plus id="S6.Ex45.m1.2.2.2.2.1.1.cmml" xref="S6.Ex45.m1.2.2.2.2.1.1"></plus><apply id="S6.Ex45.m1.2.2.2.2.1.2.cmml" xref="S6.Ex45.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.Ex45.m1.2.2.2.2.1.2.1.cmml" xref="S6.Ex45.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S6.Ex45.m1.2.2.2.2.1.2.2.cmml" xref="S6.Ex45.m1.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.Ex45.m1.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.Ex45.m1.2.2.2.2.1.2.3">0</cn></apply><ci id="S6.Ex45.m1.2.2.2.2.1.3.cmml" xref="S6.Ex45.m1.2.2.2.2.1.3">ℓ</ci></apply><ci id="S6.Ex45.m1.1.1.1.1.cmml" xref="S6.Ex45.m1.1.1.1.1">𝑝</ci></list></apply><vector id="S6.Ex45.m1.5.5.1.2.cmml" xref="S6.Ex45.m1.5.5.1.1"><ci id="S6.Ex45.m1.3.3.cmml" xref="S6.Ex45.m1.3.3">𝒪</ci><apply id="S6.Ex45.m1.5.5.1.1.1.cmml" xref="S6.Ex45.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S6.Ex45.m1.5.5.1.1.1.1.cmml" xref="S6.Ex45.m1.5.5.1.1.1">subscript</csymbol><apply id="S6.Ex45.m1.5.5.1.1.1.2.cmml" xref="S6.Ex45.m1.5.5.1.1.1"><csymbol cd="ambiguous" id="S6.Ex45.m1.5.5.1.1.1.2.1.cmml" xref="S6.Ex45.m1.5.5.1.1.1">superscript</csymbol><ci id="S6.Ex45.m1.5.5.1.1.1.2.2.cmml" xref="S6.Ex45.m1.5.5.1.1.1.2.2">𝑤</ci><apply id="S6.Ex45.m1.5.5.1.1.1.2.3.cmml" xref="S6.Ex45.m1.5.5.1.1.1.2.3"><partialdiff id="S6.Ex45.m1.5.5.1.1.1.2.3.1.cmml" xref="S6.Ex45.m1.5.5.1.1.1.2.3.1"></partialdiff><ci id="S6.Ex45.m1.5.5.1.1.1.2.3.2.cmml" xref="S6.Ex45.m1.5.5.1.1.1.2.3.2">𝒪</ci></apply></apply><ci id="S6.Ex45.m1.5.5.1.1.1.3.cmml" xref="S6.Ex45.m1.5.5.1.1.1.3">𝛾</ci></apply><ci id="S6.Ex45.m1.4.4.cmml" xref="S6.Ex45.m1.4.4">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex45.m1.5c">\displaystyle W_{\mathcal{B}}^{k_{0}+\ell,p}(\mathcal{O},w^{\partial\mathcal{O% }}_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex45.m1.5d">italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O 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id="S6.Ex45.m2.10.10.2.4.3.1.cmml" xref="S6.Ex45.m2.10.10.2.4.3">subscript</csymbol><ci id="S6.Ex45.m2.10.10.2.4.3.2.cmml" xref="S6.Ex45.m2.10.10.2.4.3.2">𝑘</ci><cn id="S6.Ex45.m2.10.10.2.4.3.3.cmml" type="integer" xref="S6.Ex45.m2.10.10.2.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex45.m2.10c">\displaystyle=\big{[}W^{k_{0},p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};% X),W_{\mathcal{B}}^{k_{0}+k_{1},p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma% };X)\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex45.m2.10d">= [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex46"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{[}W^{k_{0},p}_{\mathcal{B}}(\mathcal{O},w^{\partial\mathcal% {O}}_{\gamma};X),W_{\mathcal{B}}^{k_{0}+k_{1},p}(\mathcal{O},w^{\partial% \mathcal{O}}_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}," class="ltx_Math" display="inline" 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id="S6.Ex46.m1.9c">\displaystyle=\big{[}W^{k_{0},p}_{\mathcal{B}}(\mathcal{O},w^{\partial\mathcal% {O}}_{\gamma};X),W_{\mathcal{B}}^{k_{0}+k_{1},p}(\mathcal{O},w^{\partial% \mathcal{O}}_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}},</annotation><annotation encoding="application/x-llamapun" id="S6.Ex46.m1.9d">= [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.Thmtheorem7.p1.4"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem7.p1.4.1">and</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex47"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="W^{k_{0}+\ell,p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X)=\big{[}W^{k_{% 0},p}(\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X),W^{k_{0}+k_{1},p}(% \mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}." class="ltx_Math" display="block" id="S6.Ex47.m1.13"><semantics id="S6.Ex47.m1.13a"><mrow id="S6.Ex47.m1.13.13.1" xref="S6.Ex47.m1.13.13.1.1.cmml"><mrow id="S6.Ex47.m1.13.13.1.1" 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\mathcal{O},w^{\partial\mathcal{O}}_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex47.m1.13d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( caligraphic_O , italic_w start_POSTSUPERSCRIPT ∂ caligraphic_O end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_para" id="S6.SS2.p5"> <p class="ltx_p" id="S6.SS2.p5.1">For simplicity, we only deal with smooth domains to avoid problems with the space-dependent coefficients in the boundary operators, although this assumption can be weakened. Similar interpolation results can be obtained for Bessel potential spaces on domains with <math alttext="\gamma\in(-1,p-1)" class="ltx_Math" display="inline" id="S6.SS2.p5.1.m1.2"><semantics id="S6.SS2.p5.1.m1.2a"><mrow id="S6.SS2.p5.1.m1.2.2" xref="S6.SS2.p5.1.m1.2.2.cmml"><mi id="S6.SS2.p5.1.m1.2.2.4" xref="S6.SS2.p5.1.m1.2.2.4.cmml">γ</mi><mo id="S6.SS2.p5.1.m1.2.2.3" xref="S6.SS2.p5.1.m1.2.2.3.cmml">∈</mo><mrow id="S6.SS2.p5.1.m1.2.2.2.2" xref="S6.SS2.p5.1.m1.2.2.2.3.cmml"><mo id="S6.SS2.p5.1.m1.2.2.2.2.3" stretchy="false" xref="S6.SS2.p5.1.m1.2.2.2.3.cmml">(</mo><mrow id="S6.SS2.p5.1.m1.1.1.1.1.1" xref="S6.SS2.p5.1.m1.1.1.1.1.1.cmml"><mo id="S6.SS2.p5.1.m1.1.1.1.1.1a" xref="S6.SS2.p5.1.m1.1.1.1.1.1.cmml">−</mo><mn id="S6.SS2.p5.1.m1.1.1.1.1.1.2" xref="S6.SS2.p5.1.m1.1.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.SS2.p5.1.m1.2.2.2.2.4" xref="S6.SS2.p5.1.m1.2.2.2.3.cmml">,</mo><mrow id="S6.SS2.p5.1.m1.2.2.2.2.2" xref="S6.SS2.p5.1.m1.2.2.2.2.2.cmml"><mi id="S6.SS2.p5.1.m1.2.2.2.2.2.2" xref="S6.SS2.p5.1.m1.2.2.2.2.2.2.cmml">p</mi><mo id="S6.SS2.p5.1.m1.2.2.2.2.2.1" xref="S6.SS2.p5.1.m1.2.2.2.2.2.1.cmml">−</mo><mn id="S6.SS2.p5.1.m1.2.2.2.2.2.3" xref="S6.SS2.p5.1.m1.2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS2.p5.1.m1.2.2.2.2.5" stretchy="false" xref="S6.SS2.p5.1.m1.2.2.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.p5.1.m1.2b"><apply id="S6.SS2.p5.1.m1.2.2.cmml" xref="S6.SS2.p5.1.m1.2.2"><in id="S6.SS2.p5.1.m1.2.2.3.cmml" xref="S6.SS2.p5.1.m1.2.2.3"></in><ci id="S6.SS2.p5.1.m1.2.2.4.cmml" xref="S6.SS2.p5.1.m1.2.2.4">𝛾</ci><interval closure="open" id="S6.SS2.p5.1.m1.2.2.2.3.cmml" xref="S6.SS2.p5.1.m1.2.2.2.2"><apply id="S6.SS2.p5.1.m1.1.1.1.1.1.cmml" xref="S6.SS2.p5.1.m1.1.1.1.1.1"><minus id="S6.SS2.p5.1.m1.1.1.1.1.1.1.cmml" xref="S6.SS2.p5.1.m1.1.1.1.1.1"></minus><cn id="S6.SS2.p5.1.m1.1.1.1.1.1.2.cmml" type="integer" xref="S6.SS2.p5.1.m1.1.1.1.1.1.2">1</cn></apply><apply id="S6.SS2.p5.1.m1.2.2.2.2.2.cmml" xref="S6.SS2.p5.1.m1.2.2.2.2.2"><minus id="S6.SS2.p5.1.m1.2.2.2.2.2.1.cmml" xref="S6.SS2.p5.1.m1.2.2.2.2.2.1"></minus><ci id="S6.SS2.p5.1.m1.2.2.2.2.2.2.cmml" xref="S6.SS2.p5.1.m1.2.2.2.2.2.2">𝑝</ci><cn id="S6.SS2.p5.1.m1.2.2.2.2.2.3.cmml" type="integer" xref="S6.SS2.p5.1.m1.2.2.2.2.2.3">1</cn></apply></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p5.1.m1.2c">\gamma\in(-1,p-1)</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.p5.1.m1.2d">italic_γ ∈ ( - 1 , italic_p - 1 )</annotation></semantics></math> using Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4" title="Theorem 6.4 (Complex interpolation of weighted Bessel potential spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a>. <br class="ltx_break"/></p> </div> <div class="ltx_para" id="S6.SS2.p6"> <p class="ltx_p" id="S6.SS2.p6.1">We close this section with an application of the complex interpolation result in Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem6" title="Proposition 6.6. ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.6</span></a> to derive Fubini’s property (mixed derivative theorem) for weighted Sobolev spaces on the half-space. It extends the range of weights obtained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>, Corollary 3.19]</cite>. The Fubini property for unweighted spaces is studied in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib69" title="">69</a>, Section I.4]</cite> and some results for weighted spaces are contained in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib47" title="">47</a>, Section 5]</cite>.</p> </div> <div class="ltx_theorem ltx_theorem_proposition" id="S6.Thmtheorem8"> <h6 class="ltx_title ltx_runin ltx_title_theorem"> <span class="ltx_tag ltx_tag_theorem"><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem8.1.1.1">Proposition 6.8</span></span><span class="ltx_text ltx_font_bold" id="S6.Thmtheorem8.2.2">.</span> </h6> <div class="ltx_para" id="S6.Thmtheorem8.p1"> <p class="ltx_p" id="S6.Thmtheorem8.p1.6"><span class="ltx_text ltx_font_italic" id="S6.Thmtheorem8.p1.6.6">Let <math alttext="p\in(1,\infty)" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.1.1.m1.2"><semantics id="S6.Thmtheorem8.p1.1.1.m1.2a"><mrow id="S6.Thmtheorem8.p1.1.1.m1.2.3" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.cmml"><mi id="S6.Thmtheorem8.p1.1.1.m1.2.3.2" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.2.cmml">p</mi><mo id="S6.Thmtheorem8.p1.1.1.m1.2.3.1" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.1.cmml">∈</mo><mrow id="S6.Thmtheorem8.p1.1.1.m1.2.3.3.2" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml"><mo id="S6.Thmtheorem8.p1.1.1.m1.2.3.3.2.1" stretchy="false" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">(</mo><mn id="S6.Thmtheorem8.p1.1.1.m1.1.1" xref="S6.Thmtheorem8.p1.1.1.m1.1.1.cmml">1</mn><mo id="S6.Thmtheorem8.p1.1.1.m1.2.3.3.2.2" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">,</mo><mi id="S6.Thmtheorem8.p1.1.1.m1.2.2" mathvariant="normal" xref="S6.Thmtheorem8.p1.1.1.m1.2.2.cmml">∞</mi><mo id="S6.Thmtheorem8.p1.1.1.m1.2.3.3.2.3" stretchy="false" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.1.1.m1.2b"><apply id="S6.Thmtheorem8.p1.1.1.m1.2.3.cmml" xref="S6.Thmtheorem8.p1.1.1.m1.2.3"><in id="S6.Thmtheorem8.p1.1.1.m1.2.3.1.cmml" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.1"></in><ci id="S6.Thmtheorem8.p1.1.1.m1.2.3.2.cmml" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.2">𝑝</ci><interval closure="open" id="S6.Thmtheorem8.p1.1.1.m1.2.3.3.1.cmml" xref="S6.Thmtheorem8.p1.1.1.m1.2.3.3.2"><cn id="S6.Thmtheorem8.p1.1.1.m1.1.1.cmml" type="integer" xref="S6.Thmtheorem8.p1.1.1.m1.1.1">1</cn><infinity id="S6.Thmtheorem8.p1.1.1.m1.2.2.cmml" xref="S6.Thmtheorem8.p1.1.1.m1.2.2"></infinity></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.1.1.m1.2c">p\in(1,\infty)</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.1.1.m1.2d">italic_p ∈ ( 1 , ∞ )</annotation></semantics></math>, <math alttext="d\geq 2" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.2.2.m2.1"><semantics id="S6.Thmtheorem8.p1.2.2.m2.1a"><mrow id="S6.Thmtheorem8.p1.2.2.m2.1.1" xref="S6.Thmtheorem8.p1.2.2.m2.1.1.cmml"><mi id="S6.Thmtheorem8.p1.2.2.m2.1.1.2" xref="S6.Thmtheorem8.p1.2.2.m2.1.1.2.cmml">d</mi><mo id="S6.Thmtheorem8.p1.2.2.m2.1.1.1" xref="S6.Thmtheorem8.p1.2.2.m2.1.1.1.cmml">≥</mo><mn id="S6.Thmtheorem8.p1.2.2.m2.1.1.3" xref="S6.Thmtheorem8.p1.2.2.m2.1.1.3.cmml">2</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.2.2.m2.1b"><apply id="S6.Thmtheorem8.p1.2.2.m2.1.1.cmml" xref="S6.Thmtheorem8.p1.2.2.m2.1.1"><geq id="S6.Thmtheorem8.p1.2.2.m2.1.1.1.cmml" xref="S6.Thmtheorem8.p1.2.2.m2.1.1.1"></geq><ci id="S6.Thmtheorem8.p1.2.2.m2.1.1.2.cmml" xref="S6.Thmtheorem8.p1.2.2.m2.1.1.2">𝑑</ci><cn id="S6.Thmtheorem8.p1.2.2.m2.1.1.3.cmml" type="integer" xref="S6.Thmtheorem8.p1.2.2.m2.1.1.3">2</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.2.2.m2.1c">d\geq 2</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.2.2.m2.1d">italic_d ≥ 2</annotation></semantics></math>, <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.3.3.m3.1"><semantics id="S6.Thmtheorem8.p1.3.3.m3.1a"><mrow id="S6.Thmtheorem8.p1.3.3.m3.1.1" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.cmml"><mi id="S6.Thmtheorem8.p1.3.3.m3.1.1.2" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.2.cmml">k</mi><mo id="S6.Thmtheorem8.p1.3.3.m3.1.1.1" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.1.cmml">∈</mo><msub id="S6.Thmtheorem8.p1.3.3.m3.1.1.3" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.3.cmml"><mi id="S6.Thmtheorem8.p1.3.3.m3.1.1.3.2" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.3.2.cmml">ℕ</mi><mn id="S6.Thmtheorem8.p1.3.3.m3.1.1.3.3" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.3.3.m3.1b"><apply id="S6.Thmtheorem8.p1.3.3.m3.1.1.cmml" xref="S6.Thmtheorem8.p1.3.3.m3.1.1"><in id="S6.Thmtheorem8.p1.3.3.m3.1.1.1.cmml" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.1"></in><ci id="S6.Thmtheorem8.p1.3.3.m3.1.1.2.cmml" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.2">𝑘</ci><apply id="S6.Thmtheorem8.p1.3.3.m3.1.1.3.cmml" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="S6.Thmtheorem8.p1.3.3.m3.1.1.3.1.cmml" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.3">subscript</csymbol><ci id="S6.Thmtheorem8.p1.3.3.m3.1.1.3.2.cmml" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.3.2">ℕ</ci><cn id="S6.Thmtheorem8.p1.3.3.m3.1.1.3.3.cmml" type="integer" xref="S6.Thmtheorem8.p1.3.3.m3.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.3.3.m3.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.3.3.m3.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, <math alttext="\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.4.4.m4.4"><semantics id="S6.Thmtheorem8.p1.4.4.m4.4a"><mrow id="S6.Thmtheorem8.p1.4.4.m4.4.4" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.cmml"><mi id="S6.Thmtheorem8.p1.4.4.m4.4.4.5" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.5.cmml">γ</mi><mo id="S6.Thmtheorem8.p1.4.4.m4.4.4.4" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.4.cmml">∈</mo><mrow id="S6.Thmtheorem8.p1.4.4.m4.4.4.3" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.cmml"><mrow id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.2.cmml"><mo id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.2" stretchy="false" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.2.cmml">(</mo><mrow id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.cmml"><mo id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1a" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.2" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.2.cmml">1</mn></mrow><mo id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.3" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.2.cmml">,</mo><mi id="S6.Thmtheorem8.p1.4.4.m4.1.1" mathvariant="normal" xref="S6.Thmtheorem8.p1.4.4.m4.1.1.cmml">∞</mi><mo id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.4" stretchy="false" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.2.cmml">)</mo></mrow><mo id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.4" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.4.cmml">∖</mo><mrow id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.3.cmml"><mo id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.3" stretchy="false" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.3.1.cmml">{</mo><mrow id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.cmml"><mrow id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.cmml"><mi id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.2" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.2.cmml">j</mi><mo id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.1" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.1.cmml">⁢</mo><mi id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.3" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.3.cmml">p</mi></mrow><mo id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.1" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.1.cmml">−</mo><mn id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.3" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.3.cmml">1</mn></mrow><mo id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.4" lspace="0.278em" rspace="0.278em" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.3.1.cmml">:</mo><mrow id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.cmml"><mi id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.2" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.2.cmml">j</mi><mo id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.1" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.1.cmml">∈</mo><msub id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.cmml"><mi id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.2" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.2.cmml">ℕ</mi><mn id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.3" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.5" stretchy="false" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.3.1.cmml">}</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.4.4.m4.4b"><apply id="S6.Thmtheorem8.p1.4.4.m4.4.4.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4"><in id="S6.Thmtheorem8.p1.4.4.m4.4.4.4.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.4"></in><ci id="S6.Thmtheorem8.p1.4.4.m4.4.4.5.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.5">𝛾</ci><apply id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3"><setdiff id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.4.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.4"></setdiff><interval closure="open" id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.2.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1"><apply id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1"><minus id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1"></minus><cn id="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.2.cmml" type="integer" xref="S6.Thmtheorem8.p1.4.4.m4.2.2.1.1.1.1.2">1</cn></apply><infinity id="S6.Thmtheorem8.p1.4.4.m4.1.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.1.1"></infinity></interval><apply id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.3.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2"><csymbol cd="latexml" id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.3.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.3">conditional-set</csymbol><apply id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1"><minus id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.1"></minus><apply id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2"><times id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.1"></times><ci id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.2.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.2">𝑗</ci><ci id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.3.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.2.3">𝑝</ci></apply><cn id="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.3.cmml" type="integer" xref="S6.Thmtheorem8.p1.4.4.m4.3.3.2.2.1.1.3">1</cn></apply><apply id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2"><in id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.1"></in><ci id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.2.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.2">𝑗</ci><apply id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3"><csymbol cd="ambiguous" id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.1.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3">subscript</csymbol><ci id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.2.cmml" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.2">ℕ</ci><cn id="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.3.cmml" type="integer" xref="S6.Thmtheorem8.p1.4.4.m4.4.4.3.3.2.2.3.3">1</cn></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.4.4.m4.4c">\gamma\in(-1,\infty)\setminus\{jp-1:j\in\mathbb{N}_{1}\}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.4.4.m4.4d">italic_γ ∈ ( - 1 , ∞ ) ∖ { italic_j italic_p - 1 : italic_j ∈ blackboard_N start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT }</annotation></semantics></math> and let <math alttext="X" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.5.5.m5.1"><semantics id="S6.Thmtheorem8.p1.5.5.m5.1a"><mi id="S6.Thmtheorem8.p1.5.5.m5.1.1" xref="S6.Thmtheorem8.p1.5.5.m5.1.1.cmml">X</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.5.5.m5.1b"><ci id="S6.Thmtheorem8.p1.5.5.m5.1.1.cmml" xref="S6.Thmtheorem8.p1.5.5.m5.1.1">𝑋</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.5.5.m5.1c">X</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.5.5.m5.1d">italic_X</annotation></semantics></math> be a <math alttext="\operatorname{UMD}" class="ltx_Math" display="inline" id="S6.Thmtheorem8.p1.6.6.m6.1"><semantics id="S6.Thmtheorem8.p1.6.6.m6.1a"><mi id="S6.Thmtheorem8.p1.6.6.m6.1.1" xref="S6.Thmtheorem8.p1.6.6.m6.1.1.cmml">UMD</mi><annotation-xml encoding="MathML-Content" id="S6.Thmtheorem8.p1.6.6.m6.1b"><ci id="S6.Thmtheorem8.p1.6.6.m6.1.1.cmml" xref="S6.Thmtheorem8.p1.6.6.m6.1.1">UMD</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.Thmtheorem8.p1.6.6.m6.1c">\operatorname{UMD}</annotation><annotation encoding="application/x-llamapun" id="S6.Thmtheorem8.p1.6.6.m6.1d">roman_UMD</annotation></semantics></math> Banach space. Then</span></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex48"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="L^{p}(\mathbb{R}^{d-1};W^{k,p}(\mathbb{R}_{+},w_{\gamma};X))\cap W^{k,p}(% \mathbb{R}^{d-1};L^{p}(\mathbb{R}_{+},w_{\gamma};X))=W^{k,p}(\mathbb{R}^{d}_{+% },w_{\gamma};X)." class="ltx_Math" display="block" id="S6.Ex48.m1.10"><semantics id="S6.Ex48.m1.10a"><mrow id="S6.Ex48.m1.10.10.1" xref="S6.Ex48.m1.10.10.1.1.cmml"><mrow id="S6.Ex48.m1.10.10.1.1" xref="S6.Ex48.m1.10.10.1.1.cmml"><mrow id="S6.Ex48.m1.10.10.1.1.4" xref="S6.Ex48.m1.10.10.1.1.4.cmml"><mrow id="S6.Ex48.m1.10.10.1.1.2.2" xref="S6.Ex48.m1.10.10.1.1.2.2.cmml"><msup id="S6.Ex48.m1.10.10.1.1.2.2.4" xref="S6.Ex48.m1.10.10.1.1.2.2.4.cmml"><mi id="S6.Ex48.m1.10.10.1.1.2.2.4.2" xref="S6.Ex48.m1.10.10.1.1.2.2.4.2.cmml">L</mi><mi id="S6.Ex48.m1.10.10.1.1.2.2.4.3" xref="S6.Ex48.m1.10.10.1.1.2.2.4.3.cmml">p</mi></msup><mo id="S6.Ex48.m1.10.10.1.1.2.2.3" xref="S6.Ex48.m1.10.10.1.1.2.2.3.cmml">⁢</mo><mrow id="S6.Ex48.m1.10.10.1.1.2.2.2.2" xref="S6.Ex48.m1.10.10.1.1.2.2.2.3.cmml"><mo id="S6.Ex48.m1.10.10.1.1.2.2.2.2.3" stretchy="false" xref="S6.Ex48.m1.10.10.1.1.2.2.2.3.cmml">(</mo><msup id="S6.Ex48.m1.10.10.1.1.1.1.1.1.1" xref="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.cmml"><mi id="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.2" xref="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.2.cmml">ℝ</mi><mrow id="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3" xref="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3.2" xref="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3.2.cmml">d</mi><mo id="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3.1" xref="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3.1.cmml">−</mo><mn id="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3.3" xref="S6.Ex48.m1.10.10.1.1.1.1.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S6.Ex48.m1.10.10.1.1.2.2.2.2.4" xref="S6.Ex48.m1.10.10.1.1.2.2.2.3.cmml">;</mo><mrow 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xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.1.1.1.2.cmml">ℝ</mi><mo id="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.1.1.1.3" xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.1.1.1.3.cmml">+</mo></msub><mo id="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.4" xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.3.cmml">,</mo><msub id="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.2" xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.2.cmml"><mi id="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.2.2" xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.2.2.cmml">w</mi><mi id="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.2.3" xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.5" xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.3.cmml">;</mo><mi id="S6.Ex48.m1.7.7" xref="S6.Ex48.m1.7.7.cmml">X</mi><mo id="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.2.6" stretchy="false" xref="S6.Ex48.m1.10.10.1.1.2.2.2.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex48.m1.10.10.1.1.2.2.2.2.5" stretchy="false" xref="S6.Ex48.m1.10.10.1.1.2.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.Ex48.m1.10.10.1.1.4.5" xref="S6.Ex48.m1.10.10.1.1.4.5.cmml">∩</mo><mrow id="S6.Ex48.m1.10.10.1.1.4.4" xref="S6.Ex48.m1.10.10.1.1.4.4.cmml"><msup id="S6.Ex48.m1.10.10.1.1.4.4.4" xref="S6.Ex48.m1.10.10.1.1.4.4.4.cmml"><mi id="S6.Ex48.m1.10.10.1.1.4.4.4.2" xref="S6.Ex48.m1.10.10.1.1.4.4.4.2.cmml">W</mi><mrow id="S6.Ex48.m1.4.4.2.4" xref="S6.Ex48.m1.4.4.2.3.cmml"><mi id="S6.Ex48.m1.3.3.1.1" xref="S6.Ex48.m1.3.3.1.1.cmml">k</mi><mo id="S6.Ex48.m1.4.4.2.4.1" xref="S6.Ex48.m1.4.4.2.3.cmml">,</mo><mi id="S6.Ex48.m1.4.4.2.2" xref="S6.Ex48.m1.4.4.2.2.cmml">p</mi></mrow></msup><mo id="S6.Ex48.m1.10.10.1.1.4.4.3" xref="S6.Ex48.m1.10.10.1.1.4.4.3.cmml">⁢</mo><mrow id="S6.Ex48.m1.10.10.1.1.4.4.2.2" xref="S6.Ex48.m1.10.10.1.1.4.4.2.3.cmml"><mo id="S6.Ex48.m1.10.10.1.1.4.4.2.2.3" stretchy="false" xref="S6.Ex48.m1.10.10.1.1.4.4.2.3.cmml">(</mo><msup id="S6.Ex48.m1.10.10.1.1.3.3.1.1.1" xref="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.cmml"><mi id="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.2" xref="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.2.cmml">ℝ</mi><mrow id="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3" xref="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3.cmml"><mi id="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3.2" xref="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3.2.cmml">d</mi><mo id="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3.1" xref="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3.1.cmml">−</mo><mn id="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3.3" xref="S6.Ex48.m1.10.10.1.1.3.3.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S6.Ex48.m1.10.10.1.1.4.4.2.2.4" xref="S6.Ex48.m1.10.10.1.1.4.4.2.3.cmml">;</mo><mrow id="S6.Ex48.m1.10.10.1.1.4.4.2.2.2" xref="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.cmml"><msup id="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.4" xref="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.4.cmml"><mi id="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.4.2" xref="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.4.2.cmml">L</mi><mi id="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.4.3" xref="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.4.3.cmml">p</mi></msup><mo id="S6.Ex48.m1.10.10.1.1.4.4.2.2.2.3" 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italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) = italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> </div> <div class="ltx_proof" id="S6.SS2.2"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof.</h6> <div class="ltx_para" id="S6.SS2.2.p1"> <p class="ltx_p" id="S6.SS2.2.p1.3">The embedding “<math alttext="\hookleftarrow" class="ltx_Math" display="inline" id="S6.SS2.2.p1.1.m1.1"><semantics id="S6.SS2.2.p1.1.m1.1a"><mo id="S6.SS2.2.p1.1.m1.1.1" stretchy="false" xref="S6.SS2.2.p1.1.m1.1.1.cmml">↩</mo><annotation-xml encoding="MathML-Content" id="S6.SS2.2.p1.1.m1.1b"><ci id="S6.SS2.2.p1.1.m1.1.1.cmml" xref="S6.SS2.2.p1.1.m1.1.1">↩</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.2.p1.1.m1.1c">\hookleftarrow</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.2.p1.1.m1.1d">↩</annotation></semantics></math>” is clear. For the other embedding, let <math alttext="k_{1},k_{2}\in\{0,\dots,k\}" class="ltx_Math" display="inline" id="S6.SS2.2.p1.2.m2.5"><semantics id="S6.SS2.2.p1.2.m2.5a"><mrow id="S6.SS2.2.p1.2.m2.5.5" xref="S6.SS2.2.p1.2.m2.5.5.cmml"><mrow id="S6.SS2.2.p1.2.m2.5.5.2.2" xref="S6.SS2.2.p1.2.m2.5.5.2.3.cmml"><msub id="S6.SS2.2.p1.2.m2.4.4.1.1.1" xref="S6.SS2.2.p1.2.m2.4.4.1.1.1.cmml"><mi id="S6.SS2.2.p1.2.m2.4.4.1.1.1.2" xref="S6.SS2.2.p1.2.m2.4.4.1.1.1.2.cmml">k</mi><mn id="S6.SS2.2.p1.2.m2.4.4.1.1.1.3" xref="S6.SS2.2.p1.2.m2.4.4.1.1.1.3.cmml">1</mn></msub><mo id="S6.SS2.2.p1.2.m2.5.5.2.2.3" xref="S6.SS2.2.p1.2.m2.5.5.2.3.cmml">,</mo><msub id="S6.SS2.2.p1.2.m2.5.5.2.2.2" xref="S6.SS2.2.p1.2.m2.5.5.2.2.2.cmml"><mi id="S6.SS2.2.p1.2.m2.5.5.2.2.2.2" xref="S6.SS2.2.p1.2.m2.5.5.2.2.2.2.cmml">k</mi><mn id="S6.SS2.2.p1.2.m2.5.5.2.2.2.3" xref="S6.SS2.2.p1.2.m2.5.5.2.2.2.3.cmml">2</mn></msub></mrow><mo id="S6.SS2.2.p1.2.m2.5.5.3" xref="S6.SS2.2.p1.2.m2.5.5.3.cmml">∈</mo><mrow id="S6.SS2.2.p1.2.m2.5.5.4.2" xref="S6.SS2.2.p1.2.m2.5.5.4.1.cmml"><mo id="S6.SS2.2.p1.2.m2.5.5.4.2.1" stretchy="false" xref="S6.SS2.2.p1.2.m2.5.5.4.1.cmml">{</mo><mn id="S6.SS2.2.p1.2.m2.1.1" xref="S6.SS2.2.p1.2.m2.1.1.cmml">0</mn><mo id="S6.SS2.2.p1.2.m2.5.5.4.2.2" xref="S6.SS2.2.p1.2.m2.5.5.4.1.cmml">,</mo><mi id="S6.SS2.2.p1.2.m2.2.2" mathvariant="normal" xref="S6.SS2.2.p1.2.m2.2.2.cmml">…</mi><mo id="S6.SS2.2.p1.2.m2.5.5.4.2.3" xref="S6.SS2.2.p1.2.m2.5.5.4.1.cmml">,</mo><mi id="S6.SS2.2.p1.2.m2.3.3" xref="S6.SS2.2.p1.2.m2.3.3.cmml">k</mi><mo id="S6.SS2.2.p1.2.m2.5.5.4.2.4" stretchy="false" xref="S6.SS2.2.p1.2.m2.5.5.4.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.2.p1.2.m2.5b"><apply id="S6.SS2.2.p1.2.m2.5.5.cmml" xref="S6.SS2.2.p1.2.m2.5.5"><in id="S6.SS2.2.p1.2.m2.5.5.3.cmml" xref="S6.SS2.2.p1.2.m2.5.5.3"></in><list id="S6.SS2.2.p1.2.m2.5.5.2.3.cmml" xref="S6.SS2.2.p1.2.m2.5.5.2.2"><apply id="S6.SS2.2.p1.2.m2.4.4.1.1.1.cmml" xref="S6.SS2.2.p1.2.m2.4.4.1.1.1"><csymbol cd="ambiguous" id="S6.SS2.2.p1.2.m2.4.4.1.1.1.1.cmml" xref="S6.SS2.2.p1.2.m2.4.4.1.1.1">subscript</csymbol><ci id="S6.SS2.2.p1.2.m2.4.4.1.1.1.2.cmml" xref="S6.SS2.2.p1.2.m2.4.4.1.1.1.2">𝑘</ci><cn id="S6.SS2.2.p1.2.m2.4.4.1.1.1.3.cmml" type="integer" xref="S6.SS2.2.p1.2.m2.4.4.1.1.1.3">1</cn></apply><apply id="S6.SS2.2.p1.2.m2.5.5.2.2.2.cmml" xref="S6.SS2.2.p1.2.m2.5.5.2.2.2"><csymbol cd="ambiguous" id="S6.SS2.2.p1.2.m2.5.5.2.2.2.1.cmml" xref="S6.SS2.2.p1.2.m2.5.5.2.2.2">subscript</csymbol><ci id="S6.SS2.2.p1.2.m2.5.5.2.2.2.2.cmml" xref="S6.SS2.2.p1.2.m2.5.5.2.2.2.2">𝑘</ci><cn id="S6.SS2.2.p1.2.m2.5.5.2.2.2.3.cmml" type="integer" xref="S6.SS2.2.p1.2.m2.5.5.2.2.2.3">2</cn></apply></list><set id="S6.SS2.2.p1.2.m2.5.5.4.1.cmml" xref="S6.SS2.2.p1.2.m2.5.5.4.2"><cn id="S6.SS2.2.p1.2.m2.1.1.cmml" type="integer" xref="S6.SS2.2.p1.2.m2.1.1">0</cn><ci id="S6.SS2.2.p1.2.m2.2.2.cmml" xref="S6.SS2.2.p1.2.m2.2.2">…</ci><ci id="S6.SS2.2.p1.2.m2.3.3.cmml" xref="S6.SS2.2.p1.2.m2.3.3">𝑘</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.2.p1.2.m2.5c">k_{1},k_{2}\in\{0,\dots,k\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.2.p1.2.m2.5d">italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ∈ { 0 , … , italic_k }</annotation></semantics></math> such that <math alttext="k_{1}+k_{2}=k" class="ltx_Math" display="inline" id="S6.SS2.2.p1.3.m3.1"><semantics id="S6.SS2.2.p1.3.m3.1a"><mrow id="S6.SS2.2.p1.3.m3.1.1" xref="S6.SS2.2.p1.3.m3.1.1.cmml"><mrow id="S6.SS2.2.p1.3.m3.1.1.2" xref="S6.SS2.2.p1.3.m3.1.1.2.cmml"><msub id="S6.SS2.2.p1.3.m3.1.1.2.2" xref="S6.SS2.2.p1.3.m3.1.1.2.2.cmml"><mi id="S6.SS2.2.p1.3.m3.1.1.2.2.2" xref="S6.SS2.2.p1.3.m3.1.1.2.2.2.cmml">k</mi><mn id="S6.SS2.2.p1.3.m3.1.1.2.2.3" xref="S6.SS2.2.p1.3.m3.1.1.2.2.3.cmml">1</mn></msub><mo id="S6.SS2.2.p1.3.m3.1.1.2.1" xref="S6.SS2.2.p1.3.m3.1.1.2.1.cmml">+</mo><msub id="S6.SS2.2.p1.3.m3.1.1.2.3" xref="S6.SS2.2.p1.3.m3.1.1.2.3.cmml"><mi id="S6.SS2.2.p1.3.m3.1.1.2.3.2" xref="S6.SS2.2.p1.3.m3.1.1.2.3.2.cmml">k</mi><mn id="S6.SS2.2.p1.3.m3.1.1.2.3.3" xref="S6.SS2.2.p1.3.m3.1.1.2.3.3.cmml">2</mn></msub></mrow><mo id="S6.SS2.2.p1.3.m3.1.1.1" xref="S6.SS2.2.p1.3.m3.1.1.1.cmml">=</mo><mi id="S6.SS2.2.p1.3.m3.1.1.3" xref="S6.SS2.2.p1.3.m3.1.1.3.cmml">k</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.2.p1.3.m3.1b"><apply id="S6.SS2.2.p1.3.m3.1.1.cmml" xref="S6.SS2.2.p1.3.m3.1.1"><eq id="S6.SS2.2.p1.3.m3.1.1.1.cmml" xref="S6.SS2.2.p1.3.m3.1.1.1"></eq><apply id="S6.SS2.2.p1.3.m3.1.1.2.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2"><plus id="S6.SS2.2.p1.3.m3.1.1.2.1.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2.1"></plus><apply id="S6.SS2.2.p1.3.m3.1.1.2.2.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS2.2.p1.3.m3.1.1.2.2.1.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2.2">subscript</csymbol><ci id="S6.SS2.2.p1.3.m3.1.1.2.2.2.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2.2.2">𝑘</ci><cn id="S6.SS2.2.p1.3.m3.1.1.2.2.3.cmml" type="integer" xref="S6.SS2.2.p1.3.m3.1.1.2.2.3">1</cn></apply><apply id="S6.SS2.2.p1.3.m3.1.1.2.3.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS2.2.p1.3.m3.1.1.2.3.1.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2.3">subscript</csymbol><ci id="S6.SS2.2.p1.3.m3.1.1.2.3.2.cmml" xref="S6.SS2.2.p1.3.m3.1.1.2.3.2">𝑘</ci><cn id="S6.SS2.2.p1.3.m3.1.1.2.3.3.cmml" type="integer" xref="S6.SS2.2.p1.3.m3.1.1.2.3.3">2</cn></apply></apply><ci id="S6.SS2.2.p1.3.m3.1.1.3.cmml" xref="S6.SS2.2.p1.3.m3.1.1.3">𝑘</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.2.p1.3.m3.1c">k_{1}+k_{2}=k</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.2.p1.3.m3.1d">italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_k</annotation></semantics></math>. By <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib51" title="">51</a>, Theorem 3.18]</cite> and Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem6" title="Proposition 6.6. ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.6</span></a>, we have</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx40"> <tbody id="S6.Ex49"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle L^{p}(\mathbb{R}^{d-1};W^{k,p}" class="ltx_math_unparsed" display="inline" id="S6.Ex49.m1.2"><semantics id="S6.Ex49.m1.2a"><mrow id="S6.Ex49.m1.2b"><msup id="S6.Ex49.m1.2.3"><mi id="S6.Ex49.m1.2.3.2">L</mi><mi id="S6.Ex49.m1.2.3.3">p</mi></msup><mrow id="S6.Ex49.m1.2.4"><mo id="S6.Ex49.m1.2.4.1" stretchy="false">(</mo><msup id="S6.Ex49.m1.2.4.2"><mi id="S6.Ex49.m1.2.4.2.2">ℝ</mi><mrow id="S6.Ex49.m1.2.4.2.3"><mi id="S6.Ex49.m1.2.4.2.3.2">d</mi><mo id="S6.Ex49.m1.2.4.2.3.1">−</mo><mn id="S6.Ex49.m1.2.4.2.3.3">1</mn></mrow></msup><mo id="S6.Ex49.m1.2.4.3">;</mo><msup id="S6.Ex49.m1.2.4.4"><mi id="S6.Ex49.m1.2.4.4.2">W</mi><mrow id="S6.Ex49.m1.2.2.2.4"><mi id="S6.Ex49.m1.1.1.1.1">k</mi><mo id="S6.Ex49.m1.2.2.2.4.1">,</mo><mi id="S6.Ex49.m1.2.2.2.2">p</mi></mrow></msup></mrow></mrow><annotation encoding="application/x-tex" id="S6.Ex49.m1.2c">\displaystyle L^{p}(\mathbb{R}^{d-1};W^{k,p}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex49.m1.2d">italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle(\mathbb{R}_{+},w_{\gamma};X))\cap W^{k,p}(\mathbb{R}^{d-1};L^{p}% (\mathbb{R}_{+},w_{\gamma};X))" class="ltx_math_unparsed" display="inline" id="S6.Ex49.m2.3"><semantics id="S6.Ex49.m2.3a"><mrow id="S6.Ex49.m2.3b"><mrow id="S6.Ex49.m2.3.4"><mo id="S6.Ex49.m2.3.4.1" stretchy="false">(</mo><msub id="S6.Ex49.m2.3.4.2"><mi id="S6.Ex49.m2.3.4.2.2">ℝ</mi><mo id="S6.Ex49.m2.3.4.2.3">+</mo></msub><mo id="S6.Ex49.m2.3.4.3">,</mo><msub id="S6.Ex49.m2.3.4.4"><mi id="S6.Ex49.m2.3.4.4.2">w</mi><mi id="S6.Ex49.m2.3.4.4.3">γ</mi></msub><mo id="S6.Ex49.m2.3.4.5">;</mo><mi id="S6.Ex49.m2.3.3">X</mi><mo id="S6.Ex49.m2.3.4.6" stretchy="false">)</mo></mrow><mo id="S6.Ex49.m2.3.5" stretchy="false">)</mo><mo id="S6.Ex49.m2.3.6">∩</mo><mi id="S6.Ex49.m2.3.7">W</mi><msup id="S6.Ex49.m2.2.2"><mi id="S6.Ex49.m2.2.2a"></mi><mrow id="S6.Ex49.m2.2.2.2.4"><mi id="S6.Ex49.m2.1.1.1.1">k</mi><mo id="S6.Ex49.m2.2.2.2.4.1">,</mo><mi id="S6.Ex49.m2.2.2.2.2">p</mi></mrow></msup><mo id="S6.Ex49.m2.3.8" stretchy="false">(</mo><msup id="S6.Ex49.m2.3.9"><mi id="S6.Ex49.m2.3.9.2">ℝ</mi><mrow id="S6.Ex49.m2.3.9.3"><mi id="S6.Ex49.m2.3.9.3.2">d</mi><mo id="S6.Ex49.m2.3.9.3.1">−</mo><mn id="S6.Ex49.m2.3.9.3.3">1</mn></mrow></msup><mo id="S6.Ex49.m2.3.10">;</mo><msup id="S6.Ex49.m2.3.11"><mi id="S6.Ex49.m2.3.11.2">L</mi><mi id="S6.Ex49.m2.3.11.3">p</mi></msup><mrow id="S6.Ex49.m2.3.12"><mo id="S6.Ex49.m2.3.12.1" stretchy="false">(</mo><msub id="S6.Ex49.m2.3.12.2"><mi id="S6.Ex49.m2.3.12.2.2">ℝ</mi><mo id="S6.Ex49.m2.3.12.2.3">+</mo></msub><mo id="S6.Ex49.m2.3.12.3">,</mo><msub id="S6.Ex49.m2.3.12.4"><mi id="S6.Ex49.m2.3.12.4.2">w</mi><mi id="S6.Ex49.m2.3.12.4.3">γ</mi></msub><mo id="S6.Ex49.m2.3.12.5">;</mo><mi id="S6.Ex49.m2.3.12.6">X</mi><mo id="S6.Ex49.m2.3.12.7" stretchy="false">)</mo></mrow><mo id="S6.Ex49.m2.3.13" stretchy="false">)</mo></mrow><annotation encoding="application/x-tex" id="S6.Ex49.m2.3c">\displaystyle(\mathbb{R}_{+},w_{\gamma};X))\cap W^{k,p}(\mathbb{R}^{d-1};L^{p}% (\mathbb{R}_{+},w_{\gamma};X))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex49.m2.3d">( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) ∩ italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex50"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S6.Ex50.m1.1"><semantics id="S6.Ex50.m1.1a"><mo id="S6.Ex50.m1.1.1" xref="S6.Ex50.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S6.Ex50.m1.1b"><eq id="S6.Ex50.m1.1.1.cmml" xref="S6.Ex50.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex50.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S6.Ex50.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle\;H^{0,p}(\mathbb{R}^{d-1};W^{k,p}(\mathbb{R}_{+},w_{\gamma};X))% \cap H^{k,p}(\mathbb{R}^{d-1};L^{p}(\mathbb{R}_{+},w_{\gamma};X))" class="ltx_Math" display="inline" id="S6.Ex50.m2.12"><semantics id="S6.Ex50.m2.12a"><mrow id="S6.Ex50.m2.12.12" xref="S6.Ex50.m2.12.12.cmml"><mrow id="S6.Ex50.m2.10.10.2" xref="S6.Ex50.m2.10.10.2.cmml"><msup id="S6.Ex50.m2.10.10.2.4" xref="S6.Ex50.m2.10.10.2.4.cmml"><mi id="S6.Ex50.m2.10.10.2.4.2" xref="S6.Ex50.m2.10.10.2.4.2.cmml">H</mi><mrow id="S6.Ex50.m2.2.2.2.4" xref="S6.Ex50.m2.2.2.2.3.cmml"><mn id="S6.Ex50.m2.1.1.1.1" xref="S6.Ex50.m2.1.1.1.1.cmml">0</mn><mo id="S6.Ex50.m2.2.2.2.4.1" xref="S6.Ex50.m2.2.2.2.3.cmml">,</mo><mi id="S6.Ex50.m2.2.2.2.2" xref="S6.Ex50.m2.2.2.2.2.cmml">p</mi></mrow></msup><mo id="S6.Ex50.m2.10.10.2.3" 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xref="S6.Ex50.m2.12.12.4.2.2.2.1.1.1.3"></plus></apply><apply id="S6.Ex50.m2.12.12.4.2.2.2.2.2.2.cmml" xref="S6.Ex50.m2.12.12.4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex50.m2.12.12.4.2.2.2.2.2.2.1.cmml" xref="S6.Ex50.m2.12.12.4.2.2.2.2.2.2">subscript</csymbol><ci id="S6.Ex50.m2.12.12.4.2.2.2.2.2.2.2.cmml" xref="S6.Ex50.m2.12.12.4.2.2.2.2.2.2.2">𝑤</ci><ci id="S6.Ex50.m2.12.12.4.2.2.2.2.2.2.3.cmml" xref="S6.Ex50.m2.12.12.4.2.2.2.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex50.m2.8.8.cmml" xref="S6.Ex50.m2.8.8">𝑋</ci></vector></apply></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex50.m2.12c">\displaystyle\;H^{0,p}(\mathbb{R}^{d-1};W^{k,p}(\mathbb{R}_{+},w_{\gamma};X))% \cap H^{k,p}(\mathbb{R}^{d-1};L^{p}(\mathbb{R}_{+},w_{\gamma};X))</annotation><annotation encoding="application/x-llamapun" id="S6.Ex50.m2.12d">italic_H start_POSTSUPERSCRIPT 0 , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) ∩ italic_H start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex51"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\hookrightarrow" class="ltx_Math" display="inline" id="S6.Ex51.m1.1"><semantics id="S6.Ex51.m1.1a"><mo id="S6.Ex51.m1.1.1" stretchy="false" 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id="S6.Ex51.m2.10c">\displaystyle\;\big{[}H^{0,p}(\mathbb{R}^{d-1};W^{k,p}(\mathbb{R}_{+},w_{% \gamma};X)),H^{k,p}(\mathbb{R}^{d-1};L^{p}(\mathbb{R}_{+},w_{\gamma};X))\big{]% }_{\frac{k_{1}}{k}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex51.m2.10d">[ italic_H start_POSTSUPERSCRIPT 0 , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) , italic_H start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_L start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) ] start_POSTSUBSCRIPT divide start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex52"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle=" class="ltx_Math" display="inline" id="S6.Ex52.m1.1"><semantics id="S6.Ex52.m1.1a"><mo id="S6.Ex52.m1.1.1" xref="S6.Ex52.m1.1.1.cmml">=</mo><annotation-xml encoding="MathML-Content" id="S6.Ex52.m1.1b"><eq id="S6.Ex52.m1.1.1.cmml" xref="S6.Ex52.m1.1.1"></eq></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex52.m1.1c">\displaystyle=</annotation><annotation encoding="application/x-llamapun" id="S6.Ex52.m1.1d">=</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math 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id="S6.Ex53.m2.6.6.1.1.2.2.2.1.1.1.1.cmml" xref="S6.Ex53.m2.6.6.1.1.2.2.2.1.1.1">subscript</csymbol><ci id="S6.Ex53.m2.6.6.1.1.2.2.2.1.1.1.2.cmml" xref="S6.Ex53.m2.6.6.1.1.2.2.2.1.1.1.2">ℝ</ci><plus id="S6.Ex53.m2.6.6.1.1.2.2.2.1.1.1.3.cmml" xref="S6.Ex53.m2.6.6.1.1.2.2.2.1.1.1.3"></plus></apply><apply id="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2.cmml" xref="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2.1.cmml" xref="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2">subscript</csymbol><ci id="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2.2.cmml" xref="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2.2">𝑤</ci><ci id="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2.3.cmml" xref="S6.Ex53.m2.6.6.1.1.2.2.2.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex53.m2.5.5.cmml" xref="S6.Ex53.m2.5.5">𝑋</ci></vector></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex53.m2.6c">\displaystyle\;W^{k_{1},p}(\mathbb{R}^{d-1};W^{k_{2},p}(\mathbb{R}_{+},w_{% \gamma};X)),</annotation><annotation encoding="application/x-llamapun" id="S6.Ex53.m2.6d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS2.2.p1.4">where it was used that <math alttext="1-\frac{k_{1}}{k}=\frac{k_{2}}{k}" class="ltx_Math" display="inline" id="S6.SS2.2.p1.4.m1.1"><semantics id="S6.SS2.2.p1.4.m1.1a"><mrow id="S6.SS2.2.p1.4.m1.1.1" xref="S6.SS2.2.p1.4.m1.1.1.cmml"><mrow id="S6.SS2.2.p1.4.m1.1.1.2" xref="S6.SS2.2.p1.4.m1.1.1.2.cmml"><mn id="S6.SS2.2.p1.4.m1.1.1.2.2" xref="S6.SS2.2.p1.4.m1.1.1.2.2.cmml">1</mn><mo id="S6.SS2.2.p1.4.m1.1.1.2.1" xref="S6.SS2.2.p1.4.m1.1.1.2.1.cmml">−</mo><mfrac id="S6.SS2.2.p1.4.m1.1.1.2.3" xref="S6.SS2.2.p1.4.m1.1.1.2.3.cmml"><msub id="S6.SS2.2.p1.4.m1.1.1.2.3.2" xref="S6.SS2.2.p1.4.m1.1.1.2.3.2.cmml"><mi id="S6.SS2.2.p1.4.m1.1.1.2.3.2.2" xref="S6.SS2.2.p1.4.m1.1.1.2.3.2.2.cmml">k</mi><mn id="S6.SS2.2.p1.4.m1.1.1.2.3.2.3" xref="S6.SS2.2.p1.4.m1.1.1.2.3.2.3.cmml">1</mn></msub><mi id="S6.SS2.2.p1.4.m1.1.1.2.3.3" xref="S6.SS2.2.p1.4.m1.1.1.2.3.3.cmml">k</mi></mfrac></mrow><mo id="S6.SS2.2.p1.4.m1.1.1.1" xref="S6.SS2.2.p1.4.m1.1.1.1.cmml">=</mo><mfrac id="S6.SS2.2.p1.4.m1.1.1.3" xref="S6.SS2.2.p1.4.m1.1.1.3.cmml"><msub id="S6.SS2.2.p1.4.m1.1.1.3.2" xref="S6.SS2.2.p1.4.m1.1.1.3.2.cmml"><mi id="S6.SS2.2.p1.4.m1.1.1.3.2.2" xref="S6.SS2.2.p1.4.m1.1.1.3.2.2.cmml">k</mi><mn id="S6.SS2.2.p1.4.m1.1.1.3.2.3" xref="S6.SS2.2.p1.4.m1.1.1.3.2.3.cmml">2</mn></msub><mi id="S6.SS2.2.p1.4.m1.1.1.3.3" xref="S6.SS2.2.p1.4.m1.1.1.3.3.cmml">k</mi></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.SS2.2.p1.4.m1.1b"><apply id="S6.SS2.2.p1.4.m1.1.1.cmml" xref="S6.SS2.2.p1.4.m1.1.1"><eq id="S6.SS2.2.p1.4.m1.1.1.1.cmml" xref="S6.SS2.2.p1.4.m1.1.1.1"></eq><apply id="S6.SS2.2.p1.4.m1.1.1.2.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2"><minus id="S6.SS2.2.p1.4.m1.1.1.2.1.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2.1"></minus><cn id="S6.SS2.2.p1.4.m1.1.1.2.2.cmml" type="integer" xref="S6.SS2.2.p1.4.m1.1.1.2.2">1</cn><apply id="S6.SS2.2.p1.4.m1.1.1.2.3.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2.3"><divide id="S6.SS2.2.p1.4.m1.1.1.2.3.1.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2.3"></divide><apply id="S6.SS2.2.p1.4.m1.1.1.2.3.2.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2.3.2"><csymbol cd="ambiguous" id="S6.SS2.2.p1.4.m1.1.1.2.3.2.1.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2.3.2">subscript</csymbol><ci id="S6.SS2.2.p1.4.m1.1.1.2.3.2.2.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2.3.2.2">𝑘</ci><cn id="S6.SS2.2.p1.4.m1.1.1.2.3.2.3.cmml" type="integer" xref="S6.SS2.2.p1.4.m1.1.1.2.3.2.3">1</cn></apply><ci id="S6.SS2.2.p1.4.m1.1.1.2.3.3.cmml" xref="S6.SS2.2.p1.4.m1.1.1.2.3.3">𝑘</ci></apply></apply><apply id="S6.SS2.2.p1.4.m1.1.1.3.cmml" xref="S6.SS2.2.p1.4.m1.1.1.3"><divide id="S6.SS2.2.p1.4.m1.1.1.3.1.cmml" xref="S6.SS2.2.p1.4.m1.1.1.3"></divide><apply id="S6.SS2.2.p1.4.m1.1.1.3.2.cmml" xref="S6.SS2.2.p1.4.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS2.2.p1.4.m1.1.1.3.2.1.cmml" xref="S6.SS2.2.p1.4.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS2.2.p1.4.m1.1.1.3.2.2.cmml" xref="S6.SS2.2.p1.4.m1.1.1.3.2.2">𝑘</ci><cn id="S6.SS2.2.p1.4.m1.1.1.3.2.3.cmml" type="integer" xref="S6.SS2.2.p1.4.m1.1.1.3.2.3">2</cn></apply><ci id="S6.SS2.2.p1.4.m1.1.1.3.3.cmml" xref="S6.SS2.2.p1.4.m1.1.1.3.3">𝑘</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.2.p1.4.m1.1c">1-\frac{k_{1}}{k}=\frac{k_{2}}{k}</annotation><annotation encoding="application/x-llamapun" id="S6.SS2.2.p1.4.m1.1d">1 - divide start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG = divide start_ARG italic_k start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_ARG start_ARG italic_k end_ARG</annotation></semantics></math>. This proves the result. ∎</p> </div> </div> </section> <section class="ltx_subsection" id="S6.SS3"> <h3 class="ltx_title ltx_title_subsection"> <span class="ltx_tag ltx_tag_subsection">6.3. </span>The proofs of Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4" title="Theorem 6.4 (Complex interpolation of weighted Bessel potential spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a> </h3> <div class="ltx_para" id="S6.SS3.p1"> <p class="ltx_p" id="S6.SS3.p1.1">Using the trace theorem for <math alttext="\mathcal{B}" class="ltx_Math" display="inline" id="S6.SS3.p1.1.m1.1"><semantics id="S6.SS3.p1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.p1.1.m1.1.1" xref="S6.SS3.p1.1.m1.1.1.cmml">ℬ</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.p1.1.m1.1b"><ci id="S6.SS3.p1.1.m1.1.1.cmml" xref="S6.SS3.p1.1.m1.1.1">ℬ</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.p1.1.m1.1c">\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.p1.1.m1.1d">caligraphic_B</annotation></semantics></math> from Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5" title="5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5</span></a>, we will prove Theorems <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4" title="Theorem 6.4 (Complex interpolation of weighted Bessel potential spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a> and <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>. The proofs follow the arguments in <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib7" title="">7</a>, Theorems VIII.2.4.3, VIII.2.4.4 & VIII.2.4.8]</cite> for unweighted Bessel potential spaces. We will provide the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a> in full detail and afterwards indicate how the arguments should be adapted to prove Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4" title="Theorem 6.4 (Complex interpolation of weighted Bessel potential spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a>.</p> </div> <div class="ltx_proof" id="S6.SS3.10"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>.</h6> <div class="ltx_para" id="S6.SS3.1.p1"> <p class="ltx_p" id="S6.SS3.1.p1.1">First, we prove in Steps 1 and 2 that for <math alttext="\theta:=\frac{\ell}{k_{1}}" class="ltx_Math" display="inline" id="S6.SS3.1.p1.1.m1.1"><semantics id="S6.SS3.1.p1.1.m1.1a"><mrow id="S6.SS3.1.p1.1.m1.1.1" xref="S6.SS3.1.p1.1.m1.1.1.cmml"><mi id="S6.SS3.1.p1.1.m1.1.1.2" xref="S6.SS3.1.p1.1.m1.1.1.2.cmml">θ</mi><mo id="S6.SS3.1.p1.1.m1.1.1.1" lspace="0.278em" rspace="0.278em" xref="S6.SS3.1.p1.1.m1.1.1.1.cmml">:=</mo><mfrac id="S6.SS3.1.p1.1.m1.1.1.3" xref="S6.SS3.1.p1.1.m1.1.1.3.cmml"><mi id="S6.SS3.1.p1.1.m1.1.1.3.2" mathvariant="normal" xref="S6.SS3.1.p1.1.m1.1.1.3.2.cmml">ℓ</mi><msub id="S6.SS3.1.p1.1.m1.1.1.3.3" xref="S6.SS3.1.p1.1.m1.1.1.3.3.cmml"><mi id="S6.SS3.1.p1.1.m1.1.1.3.3.2" xref="S6.SS3.1.p1.1.m1.1.1.3.3.2.cmml">k</mi><mn id="S6.SS3.1.p1.1.m1.1.1.3.3.3" xref="S6.SS3.1.p1.1.m1.1.1.3.3.3.cmml">1</mn></msub></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.1.p1.1.m1.1b"><apply id="S6.SS3.1.p1.1.m1.1.1.cmml" xref="S6.SS3.1.p1.1.m1.1.1"><csymbol cd="latexml" id="S6.SS3.1.p1.1.m1.1.1.1.cmml" xref="S6.SS3.1.p1.1.m1.1.1.1">assign</csymbol><ci id="S6.SS3.1.p1.1.m1.1.1.2.cmml" xref="S6.SS3.1.p1.1.m1.1.1.2">𝜃</ci><apply id="S6.SS3.1.p1.1.m1.1.1.3.cmml" xref="S6.SS3.1.p1.1.m1.1.1.3"><divide id="S6.SS3.1.p1.1.m1.1.1.3.1.cmml" xref="S6.SS3.1.p1.1.m1.1.1.3"></divide><ci id="S6.SS3.1.p1.1.m1.1.1.3.2.cmml" xref="S6.SS3.1.p1.1.m1.1.1.3.2">ℓ</ci><apply id="S6.SS3.1.p1.1.m1.1.1.3.3.cmml" xref="S6.SS3.1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS3.1.p1.1.m1.1.1.3.3.1.cmml" xref="S6.SS3.1.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S6.SS3.1.p1.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.1.p1.1.m1.1.1.3.3.2">𝑘</ci><cn id="S6.SS3.1.p1.1.m1.1.1.3.3.3.cmml" type="integer" xref="S6.SS3.1.p1.1.m1.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.1.p1.1.m1.1c">\theta:=\frac{\ell}{k_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.1.p1.1.m1.1d">italic_θ := divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math> we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.E4"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="W_{\mathcal{B}}^{k_{0}+\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big{[}W^{k_{0% },p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{\mathcal{B}}^{k_{0}+k_{1},p}(\mathbb{% R}^{d}_{+},w_{\gamma};X)\big{]}_{\theta}." class="ltx_Math" display="block" id="S6.E4.m1.10"><semantics id="S6.E4.m1.10a"><mrow id="S6.E4.m1.10.10.1" xref="S6.E4.m1.10.10.1.1.cmml"><mrow id="S6.E4.m1.10.10.1.1" xref="S6.E4.m1.10.10.1.1.cmml"><mrow id="S6.E4.m1.10.10.1.1.2" xref="S6.E4.m1.10.10.1.1.2.cmml"><msubsup id="S6.E4.m1.10.10.1.1.2.4" xref="S6.E4.m1.10.10.1.1.2.4.cmml"><mi id="S6.E4.m1.10.10.1.1.2.4.2.2" xref="S6.E4.m1.10.10.1.1.2.4.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.E4.m1.10.10.1.1.2.4.2.3" xref="S6.E4.m1.10.10.1.1.2.4.2.3.cmml">ℬ</mi><mrow id="S6.E4.m1.2.2.2.2" xref="S6.E4.m1.2.2.2.3.cmml"><mrow id="S6.E4.m1.2.2.2.2.1" xref="S6.E4.m1.2.2.2.2.1.cmml"><msub id="S6.E4.m1.2.2.2.2.1.2" xref="S6.E4.m1.2.2.2.2.1.2.cmml"><mi id="S6.E4.m1.2.2.2.2.1.2.2" xref="S6.E4.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.E4.m1.2.2.2.2.1.2.3" xref="S6.E4.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.E4.m1.2.2.2.2.1.1" xref="S6.E4.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.E4.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.E4.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.E4.m1.2.2.2.2.2" xref="S6.E4.m1.2.2.2.3.cmml">,</mo><mi id="S6.E4.m1.1.1.1.1" xref="S6.E4.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.E4.m1.10.10.1.1.2.3" xref="S6.E4.m1.10.10.1.1.2.3.cmml">⁢</mo><mrow id="S6.E4.m1.10.10.1.1.2.2.2" xref="S6.E4.m1.10.10.1.1.2.2.3.cmml"><mo id="S6.E4.m1.10.10.1.1.2.2.2.3" stretchy="false" xref="S6.E4.m1.10.10.1.1.2.2.3.cmml">(</mo><msubsup id="S6.E4.m1.10.10.1.1.1.1.1.1" xref="S6.E4.m1.10.10.1.1.1.1.1.1.cmml"><mi id="S6.E4.m1.10.10.1.1.1.1.1.1.2.2" xref="S6.E4.m1.10.10.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.E4.m1.10.10.1.1.1.1.1.1.3" xref="S6.E4.m1.10.10.1.1.1.1.1.1.3.cmml">+</mo><mi id="S6.E4.m1.10.10.1.1.1.1.1.1.2.3" xref="S6.E4.m1.10.10.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.E4.m1.10.10.1.1.2.2.2.4" xref="S6.E4.m1.10.10.1.1.2.2.3.cmml">,</mo><msub id="S6.E4.m1.10.10.1.1.2.2.2.2" xref="S6.E4.m1.10.10.1.1.2.2.2.2.cmml"><mi id="S6.E4.m1.10.10.1.1.2.2.2.2.2" xref="S6.E4.m1.10.10.1.1.2.2.2.2.2.cmml">w</mi><mi id="S6.E4.m1.10.10.1.1.2.2.2.2.3" xref="S6.E4.m1.10.10.1.1.2.2.2.2.3.cmml">γ</mi></msub><mo id="S6.E4.m1.10.10.1.1.2.2.2.5" xref="S6.E4.m1.10.10.1.1.2.2.3.cmml">;</mo><mi id="S6.E4.m1.7.7" xref="S6.E4.m1.7.7.cmml">X</mi><mo id="S6.E4.m1.10.10.1.1.2.2.2.6" stretchy="false" xref="S6.E4.m1.10.10.1.1.2.2.3.cmml">)</mo></mrow></mrow><mo id="S6.E4.m1.10.10.1.1.5" xref="S6.E4.m1.10.10.1.1.5.cmml">=</mo><msub id="S6.E4.m1.10.10.1.1.4" xref="S6.E4.m1.10.10.1.1.4.cmml"><mrow id="S6.E4.m1.10.10.1.1.4.2.2" xref="S6.E4.m1.10.10.1.1.4.2.3.cmml"><mo id="S6.E4.m1.10.10.1.1.4.2.2.3" maxsize="120%" minsize="120%" xref="S6.E4.m1.10.10.1.1.4.2.3.cmml">[</mo><mrow id="S6.E4.m1.10.10.1.1.3.1.1.1" xref="S6.E4.m1.10.10.1.1.3.1.1.1.cmml"><msup id="S6.E4.m1.10.10.1.1.3.1.1.1.4" xref="S6.E4.m1.10.10.1.1.3.1.1.1.4.cmml"><mi id="S6.E4.m1.10.10.1.1.3.1.1.1.4.2" xref="S6.E4.m1.10.10.1.1.3.1.1.1.4.2.cmml">W</mi><mrow id="S6.E4.m1.4.4.2.2" xref="S6.E4.m1.4.4.2.3.cmml"><msub id="S6.E4.m1.4.4.2.2.1" xref="S6.E4.m1.4.4.2.2.1.cmml"><mi id="S6.E4.m1.4.4.2.2.1.2" xref="S6.E4.m1.4.4.2.2.1.2.cmml">k</mi><mn id="S6.E4.m1.4.4.2.2.1.3" xref="S6.E4.m1.4.4.2.2.1.3.cmml">0</mn></msub><mo id="S6.E4.m1.4.4.2.2.2" xref="S6.E4.m1.4.4.2.3.cmml">,</mo><mi id="S6.E4.m1.3.3.1.1" xref="S6.E4.m1.3.3.1.1.cmml">p</mi></mrow></msup><mo id="S6.E4.m1.10.10.1.1.3.1.1.1.3" xref="S6.E4.m1.10.10.1.1.3.1.1.1.3.cmml">⁢</mo><mrow id="S6.E4.m1.10.10.1.1.3.1.1.1.2.2" xref="S6.E4.m1.10.10.1.1.3.1.1.1.2.3.cmml"><mo id="S6.E4.m1.10.10.1.1.3.1.1.1.2.2.3" stretchy="false" xref="S6.E4.m1.10.10.1.1.3.1.1.1.2.3.cmml">(</mo><msubsup id="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1" xref="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1.cmml"><mi id="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1.2.2" xref="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1.2.2.cmml">ℝ</mi><mo id="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1.3" xref="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1.3.cmml">+</mo><mi id="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1.2.3" xref="S6.E4.m1.10.10.1.1.3.1.1.1.1.1.1.2.3.cmml">d</mi></msubsup><mo id="S6.E4.m1.10.10.1.1.3.1.1.1.2.2.4" xref="S6.E4.m1.10.10.1.1.3.1.1.1.2.3.cmml">,</mo><msub 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encoding="application/x-tex" id="S6.E4.m1.10c">W_{\mathcal{B}}^{k_{0}+\ell,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big{[}W^{k_{0% },p}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{\mathcal{B}}^{k_{0}+k_{1},p}(\mathbb{% R}^{d}_{+},w_{\gamma};X)\big{]}_{\theta}.</annotation><annotation encoding="application/x-llamapun" id="S6.E4.m1.10d">italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.4)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.1.p1.3">For notational convenience we write <math alttext="W^{k,p}:=W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS3.1.p1.2.m1.7"><semantics id="S6.SS3.1.p1.2.m1.7a"><mrow id="S6.SS3.1.p1.2.m1.7.7" xref="S6.SS3.1.p1.2.m1.7.7.cmml"><msup 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xref="S6.SS3.1.p1.2.m1.7.7.2.2.2.2.3">𝛾</ci></apply><ci id="S6.SS3.1.p1.2.m1.5.5.cmml" xref="S6.SS3.1.p1.2.m1.5.5">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.1.p1.2.m1.7c">W^{k,p}:=W^{k,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.1.p1.2.m1.7d">italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT := italic_W start_POSTSUPERSCRIPT italic_k , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math> for <math alttext="k\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS3.1.p1.3.m2.1"><semantics id="S6.SS3.1.p1.3.m2.1a"><mrow id="S6.SS3.1.p1.3.m2.1.1" xref="S6.SS3.1.p1.3.m2.1.1.cmml"><mi id="S6.SS3.1.p1.3.m2.1.1.2" xref="S6.SS3.1.p1.3.m2.1.1.2.cmml">k</mi><mo id="S6.SS3.1.p1.3.m2.1.1.1" xref="S6.SS3.1.p1.3.m2.1.1.1.cmml">∈</mo><msub id="S6.SS3.1.p1.3.m2.1.1.3" xref="S6.SS3.1.p1.3.m2.1.1.3.cmml"><mi id="S6.SS3.1.p1.3.m2.1.1.3.2" xref="S6.SS3.1.p1.3.m2.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS3.1.p1.3.m2.1.1.3.3" xref="S6.SS3.1.p1.3.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.1.p1.3.m2.1b"><apply id="S6.SS3.1.p1.3.m2.1.1.cmml" xref="S6.SS3.1.p1.3.m2.1.1"><in id="S6.SS3.1.p1.3.m2.1.1.1.cmml" xref="S6.SS3.1.p1.3.m2.1.1.1"></in><ci id="S6.SS3.1.p1.3.m2.1.1.2.cmml" xref="S6.SS3.1.p1.3.m2.1.1.2">𝑘</ci><apply id="S6.SS3.1.p1.3.m2.1.1.3.cmml" xref="S6.SS3.1.p1.3.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.1.p1.3.m2.1.1.3.1.cmml" xref="S6.SS3.1.p1.3.m2.1.1.3">subscript</csymbol><ci id="S6.SS3.1.p1.3.m2.1.1.3.2.cmml" xref="S6.SS3.1.p1.3.m2.1.1.3.2">ℕ</ci><cn id="S6.SS3.1.p1.3.m2.1.1.3.3.cmml" type="integer" xref="S6.SS3.1.p1.3.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.1.p1.3.m2.1c">k\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.1.p1.3.m2.1d">italic_k ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS3.2.p2"> <p class="ltx_p" id="S6.SS3.2.p2.1"><span class="ltx_text ltx_font_italic" id="S6.SS3.2.p2.1.1">Step 1: the embedding “<math alttext="\hookleftarrow" class="ltx_Math" display="inline" id="S6.SS3.2.p2.1.1.m1.1"><semantics id="S6.SS3.2.p2.1.1.m1.1a"><mo id="S6.SS3.2.p2.1.1.m1.1.1" stretchy="false" xref="S6.SS3.2.p2.1.1.m1.1.1.cmml">↩</mo><annotation-xml encoding="MathML-Content" id="S6.SS3.2.p2.1.1.m1.1b"><ci id="S6.SS3.2.p2.1.1.m1.1.1.cmml" xref="S6.SS3.2.p2.1.1.m1.1.1">↩</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.2.p2.1.1.m1.1c">\hookleftarrow</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.2.p2.1.1.m1.1d">↩</annotation></semantics></math>”. </span>Note that by Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem3" title="Proposition 6.3. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.3</span></a> it holds that</p> <table class="ltx_equation ltx_eqn_table" id="S6.E5"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}\big{]}_{\theta}% \hookrightarrow\big{[}W^{k_{0},p},W^{k_{0}+k_{1},p}\big{]}_{\theta}% \hookrightarrow W^{k_{0}+\ell,p}." class="ltx_Math" display="block" id="S6.E5.m1.11"><semantics id="S6.E5.m1.11a"><mrow id="S6.E5.m1.11.11.1" xref="S6.E5.m1.11.11.1.1.cmml"><mrow id="S6.E5.m1.11.11.1.1" xref="S6.E5.m1.11.11.1.1.cmml"><msub id="S6.E5.m1.11.11.1.1.2" xref="S6.E5.m1.11.11.1.1.2.cmml"><mrow id="S6.E5.m1.11.11.1.1.2.2.2" xref="S6.E5.m1.11.11.1.1.2.2.3.cmml"><mo id="S6.E5.m1.11.11.1.1.2.2.2.3" maxsize="120%" minsize="120%" 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id="S6.E5.m1.8.8.2.2.1.3.cmml" xref="S6.E5.m1.8.8.2.2.1.3"><csymbol cd="ambiguous" id="S6.E5.m1.8.8.2.2.1.3.1.cmml" xref="S6.E5.m1.8.8.2.2.1.3">subscript</csymbol><ci id="S6.E5.m1.8.8.2.2.1.3.2.cmml" xref="S6.E5.m1.8.8.2.2.1.3.2">𝑘</ci><cn id="S6.E5.m1.8.8.2.2.1.3.3.cmml" type="integer" xref="S6.E5.m1.8.8.2.2.1.3.3">1</cn></apply></apply><ci id="S6.E5.m1.7.7.1.1.cmml" xref="S6.E5.m1.7.7.1.1">𝑝</ci></list></apply></interval><ci id="S6.E5.m1.11.11.1.1.4.4.cmml" xref="S6.E5.m1.11.11.1.1.4.4">𝜃</ci></apply></apply><apply id="S6.E5.m1.11.11.1.1c.cmml" xref="S6.E5.m1.11.11.1"><ci id="S6.E5.m1.11.11.1.1.7.cmml" xref="S6.E5.m1.11.11.1.1.7">↪</ci><share href="https://arxiv.org/html/2503.14636v1#S6.E5.m1.11.11.1.1.4.cmml" id="S6.E5.m1.11.11.1.1d.cmml" xref="S6.E5.m1.11.11.1"></share><apply id="S6.E5.m1.11.11.1.1.8.cmml" xref="S6.E5.m1.11.11.1.1.8"><csymbol cd="ambiguous" id="S6.E5.m1.11.11.1.1.8.1.cmml" xref="S6.E5.m1.11.11.1.1.8">superscript</csymbol><ci id="S6.E5.m1.11.11.1.1.8.2.cmml" xref="S6.E5.m1.11.11.1.1.8.2">𝑊</ci><list id="S6.E5.m1.10.10.2.3.cmml" xref="S6.E5.m1.10.10.2.2"><apply id="S6.E5.m1.10.10.2.2.1.cmml" xref="S6.E5.m1.10.10.2.2.1"><plus id="S6.E5.m1.10.10.2.2.1.1.cmml" xref="S6.E5.m1.10.10.2.2.1.1"></plus><apply id="S6.E5.m1.10.10.2.2.1.2.cmml" xref="S6.E5.m1.10.10.2.2.1.2"><csymbol cd="ambiguous" id="S6.E5.m1.10.10.2.2.1.2.1.cmml" xref="S6.E5.m1.10.10.2.2.1.2">subscript</csymbol><ci id="S6.E5.m1.10.10.2.2.1.2.2.cmml" xref="S6.E5.m1.10.10.2.2.1.2.2">𝑘</ci><cn id="S6.E5.m1.10.10.2.2.1.2.3.cmml" type="integer" xref="S6.E5.m1.10.10.2.2.1.2.3">0</cn></apply><ci id="S6.E5.m1.10.10.2.2.1.3.cmml" xref="S6.E5.m1.10.10.2.2.1.3">ℓ</ci></apply><ci id="S6.E5.m1.9.9.1.1.cmml" xref="S6.E5.m1.9.9.1.1">𝑝</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E5.m1.11c">\big{[}W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}\big{]}_{\theta}% \hookrightarrow\big{[}W^{k_{0},p},W^{k_{0}+k_{1},p}\big{]}_{\theta}% \hookrightarrow W^{k_{0}+\ell,p}.</annotation><annotation encoding="application/x-llamapun" id="S6.E5.m1.11d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.5)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.2.p2.3">Now suppose that <math alttext="u\in[W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}]_{\theta}" class="ltx_Math" display="inline" id="S6.SS3.2.p2.2.m1.6"><semantics id="S6.SS3.2.p2.2.m1.6a"><mrow id="S6.SS3.2.p2.2.m1.6.6" xref="S6.SS3.2.p2.2.m1.6.6.cmml"><mi id="S6.SS3.2.p2.2.m1.6.6.4" xref="S6.SS3.2.p2.2.m1.6.6.4.cmml">u</mi><mo id="S6.SS3.2.p2.2.m1.6.6.3" xref="S6.SS3.2.p2.2.m1.6.6.3.cmml">∈</mo><msub id="S6.SS3.2.p2.2.m1.6.6.2" xref="S6.SS3.2.p2.2.m1.6.6.2.cmml"><mrow id="S6.SS3.2.p2.2.m1.6.6.2.2.2" xref="S6.SS3.2.p2.2.m1.6.6.2.2.3.cmml"><mo id="S6.SS3.2.p2.2.m1.6.6.2.2.2.3" stretchy="false" xref="S6.SS3.2.p2.2.m1.6.6.2.2.3.cmml">[</mo><msup id="S6.SS3.2.p2.2.m1.5.5.1.1.1.1" xref="S6.SS3.2.p2.2.m1.5.5.1.1.1.1.cmml"><mi id="S6.SS3.2.p2.2.m1.5.5.1.1.1.1.2" xref="S6.SS3.2.p2.2.m1.5.5.1.1.1.1.2.cmml">W</mi><mrow id="S6.SS3.2.p2.2.m1.2.2.2.2" xref="S6.SS3.2.p2.2.m1.2.2.2.3.cmml"><msub id="S6.SS3.2.p2.2.m1.2.2.2.2.1" xref="S6.SS3.2.p2.2.m1.2.2.2.2.1.cmml"><mi id="S6.SS3.2.p2.2.m1.2.2.2.2.1.2" xref="S6.SS3.2.p2.2.m1.2.2.2.2.1.2.cmml">k</mi><mn id="S6.SS3.2.p2.2.m1.2.2.2.2.1.3" xref="S6.SS3.2.p2.2.m1.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.SS3.2.p2.2.m1.2.2.2.2.2" xref="S6.SS3.2.p2.2.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS3.2.p2.2.m1.1.1.1.1" xref="S6.SS3.2.p2.2.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.SS3.2.p2.2.m1.6.6.2.2.2.4" xref="S6.SS3.2.p2.2.m1.6.6.2.2.3.cmml">,</mo><msubsup id="S6.SS3.2.p2.2.m1.6.6.2.2.2.2" xref="S6.SS3.2.p2.2.m1.6.6.2.2.2.2.cmml"><mi id="S6.SS3.2.p2.2.m1.6.6.2.2.2.2.2.2" xref="S6.SS3.2.p2.2.m1.6.6.2.2.2.2.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.2.p2.2.m1.6.6.2.2.2.2.2.3" xref="S6.SS3.2.p2.2.m1.6.6.2.2.2.2.2.3.cmml">ℬ</mi><mrow id="S6.SS3.2.p2.2.m1.4.4.2.2" xref="S6.SS3.2.p2.2.m1.4.4.2.3.cmml"><mrow id="S6.SS3.2.p2.2.m1.4.4.2.2.1" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.cmml"><msub id="S6.SS3.2.p2.2.m1.4.4.2.2.1.2" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.2.cmml"><mi id="S6.SS3.2.p2.2.m1.4.4.2.2.1.2.2" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.2.p2.2.m1.4.4.2.2.1.2.3" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.2.p2.2.m1.4.4.2.2.1.1" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.1.cmml">+</mo><msub id="S6.SS3.2.p2.2.m1.4.4.2.2.1.3" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.cmml"><mi id="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.2" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.2.cmml">k</mi><mn id="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.3" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.2.p2.2.m1.4.4.2.2.2" xref="S6.SS3.2.p2.2.m1.4.4.2.3.cmml">,</mo><mi id="S6.SS3.2.p2.2.m1.3.3.1.1" xref="S6.SS3.2.p2.2.m1.3.3.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.SS3.2.p2.2.m1.6.6.2.2.2.5" stretchy="false" xref="S6.SS3.2.p2.2.m1.6.6.2.2.3.cmml">]</mo></mrow><mi id="S6.SS3.2.p2.2.m1.6.6.2.4" xref="S6.SS3.2.p2.2.m1.6.6.2.4.cmml">θ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.2.p2.2.m1.6b"><apply id="S6.SS3.2.p2.2.m1.6.6.cmml" xref="S6.SS3.2.p2.2.m1.6.6"><in id="S6.SS3.2.p2.2.m1.6.6.3.cmml" xref="S6.SS3.2.p2.2.m1.6.6.3"></in><ci id="S6.SS3.2.p2.2.m1.6.6.4.cmml" xref="S6.SS3.2.p2.2.m1.6.6.4">𝑢</ci><apply id="S6.SS3.2.p2.2.m1.6.6.2.cmml" xref="S6.SS3.2.p2.2.m1.6.6.2"><csymbol cd="ambiguous" id="S6.SS3.2.p2.2.m1.6.6.2.3.cmml" xref="S6.SS3.2.p2.2.m1.6.6.2">subscript</csymbol><interval closure="closed" id="S6.SS3.2.p2.2.m1.6.6.2.2.3.cmml" xref="S6.SS3.2.p2.2.m1.6.6.2.2.2"><apply id="S6.SS3.2.p2.2.m1.5.5.1.1.1.1.cmml" xref="S6.SS3.2.p2.2.m1.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.2.p2.2.m1.5.5.1.1.1.1.1.cmml" xref="S6.SS3.2.p2.2.m1.5.5.1.1.1.1">superscript</csymbol><ci id="S6.SS3.2.p2.2.m1.5.5.1.1.1.1.2.cmml" xref="S6.SS3.2.p2.2.m1.5.5.1.1.1.1.2">𝑊</ci><list 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xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.3">subscript</csymbol><ci id="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.2.cmml" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.2">𝑘</ci><cn id="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.3.cmml" type="integer" xref="S6.SS3.2.p2.2.m1.4.4.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.2.p2.2.m1.3.3.1.1.cmml" xref="S6.SS3.2.p2.2.m1.3.3.1.1">𝑝</ci></list></apply></interval><ci id="S6.SS3.2.p2.2.m1.6.6.2.4.cmml" xref="S6.SS3.2.p2.2.m1.6.6.2.4">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.2.p2.2.m1.6c">u\in[W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}]_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.2.p2.2.m1.6d">italic_u ∈ [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math>. We prove that <math alttext="u\in W^{k_{0}+\ell,p}_{\mathcal{B}}" class="ltx_Math" display="inline" id="S6.SS3.2.p2.3.m2.2"><semantics id="S6.SS3.2.p2.3.m2.2a"><mrow id="S6.SS3.2.p2.3.m2.2.3" xref="S6.SS3.2.p2.3.m2.2.3.cmml"><mi id="S6.SS3.2.p2.3.m2.2.3.2" xref="S6.SS3.2.p2.3.m2.2.3.2.cmml">u</mi><mo id="S6.SS3.2.p2.3.m2.2.3.1" xref="S6.SS3.2.p2.3.m2.2.3.1.cmml">∈</mo><msubsup id="S6.SS3.2.p2.3.m2.2.3.3" xref="S6.SS3.2.p2.3.m2.2.3.3.cmml"><mi id="S6.SS3.2.p2.3.m2.2.3.3.2.2" xref="S6.SS3.2.p2.3.m2.2.3.3.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.2.p2.3.m2.2.3.3.3" xref="S6.SS3.2.p2.3.m2.2.3.3.3.cmml">ℬ</mi><mrow id="S6.SS3.2.p2.3.m2.2.2.2.2" xref="S6.SS3.2.p2.3.m2.2.2.2.3.cmml"><mrow id="S6.SS3.2.p2.3.m2.2.2.2.2.1" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.cmml"><msub id="S6.SS3.2.p2.3.m2.2.2.2.2.1.2" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.cmml"><mi id="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.2" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.3" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.2.p2.3.m2.2.2.2.2.1.1" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.1.cmml">+</mo><mi id="S6.SS3.2.p2.3.m2.2.2.2.2.1.3" mathvariant="normal" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.2.p2.3.m2.2.2.2.2.2" xref="S6.SS3.2.p2.3.m2.2.2.2.3.cmml">,</mo><mi id="S6.SS3.2.p2.3.m2.1.1.1.1" xref="S6.SS3.2.p2.3.m2.1.1.1.1.cmml">p</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.2.p2.3.m2.2b"><apply id="S6.SS3.2.p2.3.m2.2.3.cmml" xref="S6.SS3.2.p2.3.m2.2.3"><in id="S6.SS3.2.p2.3.m2.2.3.1.cmml" xref="S6.SS3.2.p2.3.m2.2.3.1"></in><ci id="S6.SS3.2.p2.3.m2.2.3.2.cmml" xref="S6.SS3.2.p2.3.m2.2.3.2">𝑢</ci><apply id="S6.SS3.2.p2.3.m2.2.3.3.cmml" xref="S6.SS3.2.p2.3.m2.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.2.p2.3.m2.2.3.3.1.cmml" xref="S6.SS3.2.p2.3.m2.2.3.3">subscript</csymbol><apply id="S6.SS3.2.p2.3.m2.2.3.3.2.cmml" xref="S6.SS3.2.p2.3.m2.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.2.p2.3.m2.2.3.3.2.1.cmml" xref="S6.SS3.2.p2.3.m2.2.3.3">superscript</csymbol><ci id="S6.SS3.2.p2.3.m2.2.3.3.2.2.cmml" xref="S6.SS3.2.p2.3.m2.2.3.3.2.2">𝑊</ci><list id="S6.SS3.2.p2.3.m2.2.2.2.3.cmml" xref="S6.SS3.2.p2.3.m2.2.2.2.2"><apply id="S6.SS3.2.p2.3.m2.2.2.2.2.1.cmml" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1"><plus id="S6.SS3.2.p2.3.m2.2.2.2.2.1.1.cmml" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.1"></plus><apply id="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.cmml" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.1.cmml" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.2">subscript</csymbol><ci id="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.2.cmml" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.2.3">0</cn></apply><ci id="S6.SS3.2.p2.3.m2.2.2.2.2.1.3.cmml" xref="S6.SS3.2.p2.3.m2.2.2.2.2.1.3">ℓ</ci></apply><ci id="S6.SS3.2.p2.3.m2.1.1.1.1.cmml" xref="S6.SS3.2.p2.3.m2.1.1.1.1">𝑝</ci></list></apply><ci id="S6.SS3.2.p2.3.m2.2.3.3.3.cmml" xref="S6.SS3.2.p2.3.m2.2.3.3.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.2.p2.3.m2.2c">u\in W^{k_{0}+\ell,p}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.2.p2.3.m2.2d">italic_u ∈ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> in Step 1a and 1b.</p> </div> <div class="ltx_para" id="S6.SS3.3.p3"> <p class="ltx_p" id="S6.SS3.3.p3.2"><span class="ltx_text ltx_font_italic" id="S6.SS3.3.p3.2.1">Step 1a. </span>Assume that <math alttext="k_{0}+\ell<m_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.3.p3.1.m1.1"><semantics id="S6.SS3.3.p3.1.m1.1a"><mrow id="S6.SS3.3.p3.1.m1.1.1" xref="S6.SS3.3.p3.1.m1.1.1.cmml"><mrow id="S6.SS3.3.p3.1.m1.1.1.2" xref="S6.SS3.3.p3.1.m1.1.1.2.cmml"><msub id="S6.SS3.3.p3.1.m1.1.1.2.2" xref="S6.SS3.3.p3.1.m1.1.1.2.2.cmml"><mi id="S6.SS3.3.p3.1.m1.1.1.2.2.2" xref="S6.SS3.3.p3.1.m1.1.1.2.2.2.cmml">k</mi><mn id="S6.SS3.3.p3.1.m1.1.1.2.2.3" xref="S6.SS3.3.p3.1.m1.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.SS3.3.p3.1.m1.1.1.2.1" xref="S6.SS3.3.p3.1.m1.1.1.2.1.cmml">+</mo><mi id="S6.SS3.3.p3.1.m1.1.1.2.3" mathvariant="normal" xref="S6.SS3.3.p3.1.m1.1.1.2.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.3.p3.1.m1.1.1.1" xref="S6.SS3.3.p3.1.m1.1.1.1.cmml"><</mo><mrow id="S6.SS3.3.p3.1.m1.1.1.3" xref="S6.SS3.3.p3.1.m1.1.1.3.cmml"><msub id="S6.SS3.3.p3.1.m1.1.1.3.2" xref="S6.SS3.3.p3.1.m1.1.1.3.2.cmml"><mi id="S6.SS3.3.p3.1.m1.1.1.3.2.2" xref="S6.SS3.3.p3.1.m1.1.1.3.2.2.cmml">m</mi><mn id="S6.SS3.3.p3.1.m1.1.1.3.2.3" xref="S6.SS3.3.p3.1.m1.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS3.3.p3.1.m1.1.1.3.1" xref="S6.SS3.3.p3.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.3.p3.1.m1.1.1.3.3" xref="S6.SS3.3.p3.1.m1.1.1.3.3.cmml"><mrow id="S6.SS3.3.p3.1.m1.1.1.3.3.2" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS3.3.p3.1.m1.1.1.3.3.2.2" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.3.p3.1.m1.1.1.3.3.2.1" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.3.p3.1.m1.1.1.3.3.2.3" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.3.p3.1.m1.1.1.3.3.3" xref="S6.SS3.3.p3.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.3.p3.1.m1.1b"><apply id="S6.SS3.3.p3.1.m1.1.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1"><lt id="S6.SS3.3.p3.1.m1.1.1.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1.1"></lt><apply id="S6.SS3.3.p3.1.m1.1.1.2.cmml" xref="S6.SS3.3.p3.1.m1.1.1.2"><plus id="S6.SS3.3.p3.1.m1.1.1.2.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1.2.1"></plus><apply id="S6.SS3.3.p3.1.m1.1.1.2.2.cmml" xref="S6.SS3.3.p3.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS3.3.p3.1.m1.1.1.2.2.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1.2.2">subscript</csymbol><ci id="S6.SS3.3.p3.1.m1.1.1.2.2.2.cmml" xref="S6.SS3.3.p3.1.m1.1.1.2.2.2">𝑘</ci><cn id="S6.SS3.3.p3.1.m1.1.1.2.2.3.cmml" type="integer" xref="S6.SS3.3.p3.1.m1.1.1.2.2.3">0</cn></apply><ci id="S6.SS3.3.p3.1.m1.1.1.2.3.cmml" xref="S6.SS3.3.p3.1.m1.1.1.2.3">ℓ</ci></apply><apply id="S6.SS3.3.p3.1.m1.1.1.3.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3"><plus id="S6.SS3.3.p3.1.m1.1.1.3.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.1"></plus><apply id="S6.SS3.3.p3.1.m1.1.1.3.2.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.3.p3.1.m1.1.1.3.2.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS3.3.p3.1.m1.1.1.3.2.2.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.2.2">𝑚</ci><cn id="S6.SS3.3.p3.1.m1.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.3.p3.1.m1.1.1.3.2.3">0</cn></apply><apply id="S6.SS3.3.p3.1.m1.1.1.3.3.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.3"><divide id="S6.SS3.3.p3.1.m1.1.1.3.3.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.3"></divide><apply id="S6.SS3.3.p3.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2"><plus id="S6.SS3.3.p3.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS3.3.p3.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.3.p3.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.3.p3.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.3.p3.1.m1.1.1.3.3.3.cmml" xref="S6.SS3.3.p3.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.3.p3.1.m1.1c">k_{0}+\ell<m_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.3.p3.1.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ < italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Then it holds that <math alttext="W^{k_{0}+\ell,p}_{\mathcal{B}}=W^{k_{0}+\ell,p}" class="ltx_Math" display="inline" id="S6.SS3.3.p3.2.m2.4"><semantics id="S6.SS3.3.p3.2.m2.4a"><mrow id="S6.SS3.3.p3.2.m2.4.5" xref="S6.SS3.3.p3.2.m2.4.5.cmml"><msubsup id="S6.SS3.3.p3.2.m2.4.5.2" xref="S6.SS3.3.p3.2.m2.4.5.2.cmml"><mi id="S6.SS3.3.p3.2.m2.4.5.2.2.2" xref="S6.SS3.3.p3.2.m2.4.5.2.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.3.p3.2.m2.4.5.2.3" xref="S6.SS3.3.p3.2.m2.4.5.2.3.cmml">ℬ</mi><mrow id="S6.SS3.3.p3.2.m2.2.2.2.2" xref="S6.SS3.3.p3.2.m2.2.2.2.3.cmml"><mrow id="S6.SS3.3.p3.2.m2.2.2.2.2.1" xref="S6.SS3.3.p3.2.m2.2.2.2.2.1.cmml"><msub id="S6.SS3.3.p3.2.m2.2.2.2.2.1.2" xref="S6.SS3.3.p3.2.m2.2.2.2.2.1.2.cmml"><mi id="S6.SS3.3.p3.2.m2.2.2.2.2.1.2.2" xref="S6.SS3.3.p3.2.m2.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.3.p3.2.m2.2.2.2.2.1.2.3" xref="S6.SS3.3.p3.2.m2.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.3.p3.2.m2.2.2.2.2.1.1" xref="S6.SS3.3.p3.2.m2.2.2.2.2.1.1.cmml">+</mo><mi id="S6.SS3.3.p3.2.m2.2.2.2.2.1.3" mathvariant="normal" xref="S6.SS3.3.p3.2.m2.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.3.p3.2.m2.2.2.2.2.2" xref="S6.SS3.3.p3.2.m2.2.2.2.3.cmml">,</mo><mi id="S6.SS3.3.p3.2.m2.1.1.1.1" xref="S6.SS3.3.p3.2.m2.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.SS3.3.p3.2.m2.4.5.1" xref="S6.SS3.3.p3.2.m2.4.5.1.cmml">=</mo><msup id="S6.SS3.3.p3.2.m2.4.5.3" xref="S6.SS3.3.p3.2.m2.4.5.3.cmml"><mi id="S6.SS3.3.p3.2.m2.4.5.3.2" xref="S6.SS3.3.p3.2.m2.4.5.3.2.cmml">W</mi><mrow id="S6.SS3.3.p3.2.m2.4.4.2.2" xref="S6.SS3.3.p3.2.m2.4.4.2.3.cmml"><mrow id="S6.SS3.3.p3.2.m2.4.4.2.2.1" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.cmml"><msub id="S6.SS3.3.p3.2.m2.4.4.2.2.1.2" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.cmml"><mi id="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.2" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.3" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.3.p3.2.m2.4.4.2.2.1.1" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.1.cmml">+</mo><mi id="S6.SS3.3.p3.2.m2.4.4.2.2.1.3" mathvariant="normal" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.3.p3.2.m2.4.4.2.2.2" xref="S6.SS3.3.p3.2.m2.4.4.2.3.cmml">,</mo><mi id="S6.SS3.3.p3.2.m2.3.3.1.1" xref="S6.SS3.3.p3.2.m2.3.3.1.1.cmml">p</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.3.p3.2.m2.4b"><apply id="S6.SS3.3.p3.2.m2.4.5.cmml" xref="S6.SS3.3.p3.2.m2.4.5"><eq id="S6.SS3.3.p3.2.m2.4.5.1.cmml" xref="S6.SS3.3.p3.2.m2.4.5.1"></eq><apply id="S6.SS3.3.p3.2.m2.4.5.2.cmml" xref="S6.SS3.3.p3.2.m2.4.5.2"><csymbol cd="ambiguous" id="S6.SS3.3.p3.2.m2.4.5.2.1.cmml" xref="S6.SS3.3.p3.2.m2.4.5.2">subscript</csymbol><apply id="S6.SS3.3.p3.2.m2.4.5.2.2.cmml" xref="S6.SS3.3.p3.2.m2.4.5.2"><csymbol cd="ambiguous" id="S6.SS3.3.p3.2.m2.4.5.2.2.1.cmml" xref="S6.SS3.3.p3.2.m2.4.5.2">superscript</csymbol><ci id="S6.SS3.3.p3.2.m2.4.5.2.2.2.cmml" 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xref="S6.SS3.3.p3.2.m2.4.5.3"><csymbol cd="ambiguous" id="S6.SS3.3.p3.2.m2.4.5.3.1.cmml" xref="S6.SS3.3.p3.2.m2.4.5.3">superscript</csymbol><ci id="S6.SS3.3.p3.2.m2.4.5.3.2.cmml" xref="S6.SS3.3.p3.2.m2.4.5.3.2">𝑊</ci><list id="S6.SS3.3.p3.2.m2.4.4.2.3.cmml" xref="S6.SS3.3.p3.2.m2.4.4.2.2"><apply id="S6.SS3.3.p3.2.m2.4.4.2.2.1.cmml" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1"><plus id="S6.SS3.3.p3.2.m2.4.4.2.2.1.1.cmml" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.1"></plus><apply id="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.cmml" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.1.cmml" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.2">subscript</csymbol><ci id="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.2.cmml" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.2.3">0</cn></apply><ci id="S6.SS3.3.p3.2.m2.4.4.2.2.1.3.cmml" xref="S6.SS3.3.p3.2.m2.4.4.2.2.1.3">ℓ</ci></apply><ci id="S6.SS3.3.p3.2.m2.3.3.1.1.cmml" xref="S6.SS3.3.p3.2.m2.3.3.1.1">𝑝</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.3.p3.2.m2.4c">W^{k_{0}+\ell,p}_{\mathcal{B}}=W^{k_{0}+\ell,p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.3.p3.2.m2.4d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, the embedding</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex54"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}\big{]}_{\theta}% \hookrightarrow W^{k_{0}+\ell,p}_{\mathcal{B}}" class="ltx_Math" display="block" id="S6.Ex54.m1.8"><semantics id="S6.Ex54.m1.8a"><mrow id="S6.Ex54.m1.8.8" xref="S6.Ex54.m1.8.8.cmml"><msub id="S6.Ex54.m1.8.8.2" xref="S6.Ex54.m1.8.8.2.cmml"><mrow id="S6.Ex54.m1.8.8.2.2.2" xref="S6.Ex54.m1.8.8.2.2.3.cmml"><mo id="S6.Ex54.m1.8.8.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex54.m1.8.8.2.2.3.cmml">[</mo><msup id="S6.Ex54.m1.7.7.1.1.1.1" xref="S6.Ex54.m1.7.7.1.1.1.1.cmml"><mi id="S6.Ex54.m1.7.7.1.1.1.1.2" xref="S6.Ex54.m1.7.7.1.1.1.1.2.cmml">W</mi><mrow id="S6.Ex54.m1.2.2.2.2" xref="S6.Ex54.m1.2.2.2.3.cmml"><msub id="S6.Ex54.m1.2.2.2.2.1" xref="S6.Ex54.m1.2.2.2.2.1.cmml"><mi id="S6.Ex54.m1.2.2.2.2.1.2" xref="S6.Ex54.m1.2.2.2.2.1.2.cmml">k</mi><mn id="S6.Ex54.m1.2.2.2.2.1.3" xref="S6.Ex54.m1.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.Ex54.m1.2.2.2.2.2" xref="S6.Ex54.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex54.m1.1.1.1.1" xref="S6.Ex54.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.Ex54.m1.8.8.2.2.2.4" xref="S6.Ex54.m1.8.8.2.2.3.cmml">,</mo><msubsup id="S6.Ex54.m1.8.8.2.2.2.2" xref="S6.Ex54.m1.8.8.2.2.2.2.cmml"><mi id="S6.Ex54.m1.8.8.2.2.2.2.2.2" xref="S6.Ex54.m1.8.8.2.2.2.2.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex54.m1.8.8.2.2.2.2.2.3" xref="S6.Ex54.m1.8.8.2.2.2.2.2.3.cmml">ℬ</mi><mrow id="S6.Ex54.m1.4.4.2.2" xref="S6.Ex54.m1.4.4.2.3.cmml"><mrow id="S6.Ex54.m1.4.4.2.2.1" xref="S6.Ex54.m1.4.4.2.2.1.cmml"><msub id="S6.Ex54.m1.4.4.2.2.1.2" xref="S6.Ex54.m1.4.4.2.2.1.2.cmml"><mi id="S6.Ex54.m1.4.4.2.2.1.2.2" xref="S6.Ex54.m1.4.4.2.2.1.2.2.cmml">k</mi><mn id="S6.Ex54.m1.4.4.2.2.1.2.3" 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xref="S6.Ex54.m1.6.6.2.2.1"><plus id="S6.Ex54.m1.6.6.2.2.1.1.cmml" xref="S6.Ex54.m1.6.6.2.2.1.1"></plus><apply id="S6.Ex54.m1.6.6.2.2.1.2.cmml" xref="S6.Ex54.m1.6.6.2.2.1.2"><csymbol cd="ambiguous" id="S6.Ex54.m1.6.6.2.2.1.2.1.cmml" xref="S6.Ex54.m1.6.6.2.2.1.2">subscript</csymbol><ci id="S6.Ex54.m1.6.6.2.2.1.2.2.cmml" xref="S6.Ex54.m1.6.6.2.2.1.2.2">𝑘</ci><cn id="S6.Ex54.m1.6.6.2.2.1.2.3.cmml" type="integer" xref="S6.Ex54.m1.6.6.2.2.1.2.3">0</cn></apply><ci id="S6.Ex54.m1.6.6.2.2.1.3.cmml" xref="S6.Ex54.m1.6.6.2.2.1.3">ℓ</ci></apply><ci id="S6.Ex54.m1.5.5.1.1.cmml" xref="S6.Ex54.m1.5.5.1.1">𝑝</ci></list></apply><ci id="S6.Ex54.m1.8.8.4.3.cmml" xref="S6.Ex54.m1.8.8.4.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex54.m1.8c">\big{[}W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}\big{]}_{\theta}% \hookrightarrow W^{k_{0}+\ell,p}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex54.m1.8d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.3.p3.3">follows from (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E5" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>).</p> </div> <div class="ltx_para" id="S6.SS3.4.p4"> <p class="ltx_p" id="S6.SS3.4.p4.16"><span class="ltx_text ltx_font_italic" id="S6.SS3.4.p4.16.1">Step 1b. </span>Assume that <math alttext="k_{0}+\ell>m_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.1.m1.1"><semantics id="S6.SS3.4.p4.1.m1.1a"><mrow id="S6.SS3.4.p4.1.m1.1.1" xref="S6.SS3.4.p4.1.m1.1.1.cmml"><mrow id="S6.SS3.4.p4.1.m1.1.1.2" xref="S6.SS3.4.p4.1.m1.1.1.2.cmml"><msub id="S6.SS3.4.p4.1.m1.1.1.2.2" xref="S6.SS3.4.p4.1.m1.1.1.2.2.cmml"><mi id="S6.SS3.4.p4.1.m1.1.1.2.2.2" xref="S6.SS3.4.p4.1.m1.1.1.2.2.2.cmml">k</mi><mn id="S6.SS3.4.p4.1.m1.1.1.2.2.3" xref="S6.SS3.4.p4.1.m1.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.1.m1.1.1.2.1" xref="S6.SS3.4.p4.1.m1.1.1.2.1.cmml">+</mo><mi id="S6.SS3.4.p4.1.m1.1.1.2.3" mathvariant="normal" xref="S6.SS3.4.p4.1.m1.1.1.2.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.4.p4.1.m1.1.1.1" xref="S6.SS3.4.p4.1.m1.1.1.1.cmml">></mo><mrow id="S6.SS3.4.p4.1.m1.1.1.3" xref="S6.SS3.4.p4.1.m1.1.1.3.cmml"><msub id="S6.SS3.4.p4.1.m1.1.1.3.2" xref="S6.SS3.4.p4.1.m1.1.1.3.2.cmml"><mi id="S6.SS3.4.p4.1.m1.1.1.3.2.2" xref="S6.SS3.4.p4.1.m1.1.1.3.2.2.cmml">m</mi><mn id="S6.SS3.4.p4.1.m1.1.1.3.2.3" xref="S6.SS3.4.p4.1.m1.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.1.m1.1.1.3.1" xref="S6.SS3.4.p4.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.4.p4.1.m1.1.1.3.3" xref="S6.SS3.4.p4.1.m1.1.1.3.3.cmml"><mrow id="S6.SS3.4.p4.1.m1.1.1.3.3.2" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS3.4.p4.1.m1.1.1.3.3.2.2" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.4.p4.1.m1.1.1.3.3.2.1" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.4.p4.1.m1.1.1.3.3.2.3" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.4.p4.1.m1.1.1.3.3.3" xref="S6.SS3.4.p4.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.1.m1.1b"><apply id="S6.SS3.4.p4.1.m1.1.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1"><gt id="S6.SS3.4.p4.1.m1.1.1.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1.1"></gt><apply id="S6.SS3.4.p4.1.m1.1.1.2.cmml" xref="S6.SS3.4.p4.1.m1.1.1.2"><plus id="S6.SS3.4.p4.1.m1.1.1.2.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1.2.1"></plus><apply id="S6.SS3.4.p4.1.m1.1.1.2.2.cmml" xref="S6.SS3.4.p4.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.1.m1.1.1.2.2.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1.2.2">subscript</csymbol><ci id="S6.SS3.4.p4.1.m1.1.1.2.2.2.cmml" xref="S6.SS3.4.p4.1.m1.1.1.2.2.2">𝑘</ci><cn id="S6.SS3.4.p4.1.m1.1.1.2.2.3.cmml" type="integer" xref="S6.SS3.4.p4.1.m1.1.1.2.2.3">0</cn></apply><ci id="S6.SS3.4.p4.1.m1.1.1.2.3.cmml" xref="S6.SS3.4.p4.1.m1.1.1.2.3">ℓ</ci></apply><apply id="S6.SS3.4.p4.1.m1.1.1.3.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3"><plus id="S6.SS3.4.p4.1.m1.1.1.3.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.1"></plus><apply id="S6.SS3.4.p4.1.m1.1.1.3.2.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.1.m1.1.1.3.2.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS3.4.p4.1.m1.1.1.3.2.2.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.2.2">𝑚</ci><cn id="S6.SS3.4.p4.1.m1.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.4.p4.1.m1.1.1.3.2.3">0</cn></apply><apply id="S6.SS3.4.p4.1.m1.1.1.3.3.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.3"><divide id="S6.SS3.4.p4.1.m1.1.1.3.3.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.3"></divide><apply id="S6.SS3.4.p4.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2"><plus id="S6.SS3.4.p4.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS3.4.p4.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.4.p4.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.4.p4.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.4.p4.1.m1.1.1.3.3.3.cmml" xref="S6.SS3.4.p4.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.1.m1.1c">k_{0}+\ell>m_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.1.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ > italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and let <math alttext="i\leq n" class="ltx_Math" display="inline" id="S6.SS3.4.p4.2.m2.1"><semantics id="S6.SS3.4.p4.2.m2.1a"><mrow id="S6.SS3.4.p4.2.m2.1.1" xref="S6.SS3.4.p4.2.m2.1.1.cmml"><mi id="S6.SS3.4.p4.2.m2.1.1.2" xref="S6.SS3.4.p4.2.m2.1.1.2.cmml">i</mi><mo id="S6.SS3.4.p4.2.m2.1.1.1" xref="S6.SS3.4.p4.2.m2.1.1.1.cmml">≤</mo><mi id="S6.SS3.4.p4.2.m2.1.1.3" xref="S6.SS3.4.p4.2.m2.1.1.3.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.2.m2.1b"><apply id="S6.SS3.4.p4.2.m2.1.1.cmml" xref="S6.SS3.4.p4.2.m2.1.1"><leq id="S6.SS3.4.p4.2.m2.1.1.1.cmml" xref="S6.SS3.4.p4.2.m2.1.1.1"></leq><ci id="S6.SS3.4.p4.2.m2.1.1.2.cmml" xref="S6.SS3.4.p4.2.m2.1.1.2">𝑖</ci><ci id="S6.SS3.4.p4.2.m2.1.1.3.cmml" xref="S6.SS3.4.p4.2.m2.1.1.3">𝑛</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.2.m2.1c">i\leq n</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.2.m2.1d">italic_i ≤ italic_n</annotation></semantics></math> be the largest integer such that <math alttext="m_{i}+\frac{\gamma+1}{p}<k_{0}+\ell" class="ltx_Math" display="inline" id="S6.SS3.4.p4.3.m3.1"><semantics id="S6.SS3.4.p4.3.m3.1a"><mrow id="S6.SS3.4.p4.3.m3.1.1" xref="S6.SS3.4.p4.3.m3.1.1.cmml"><mrow id="S6.SS3.4.p4.3.m3.1.1.2" xref="S6.SS3.4.p4.3.m3.1.1.2.cmml"><msub id="S6.SS3.4.p4.3.m3.1.1.2.2" xref="S6.SS3.4.p4.3.m3.1.1.2.2.cmml"><mi id="S6.SS3.4.p4.3.m3.1.1.2.2.2" xref="S6.SS3.4.p4.3.m3.1.1.2.2.2.cmml">m</mi><mi id="S6.SS3.4.p4.3.m3.1.1.2.2.3" xref="S6.SS3.4.p4.3.m3.1.1.2.2.3.cmml">i</mi></msub><mo id="S6.SS3.4.p4.3.m3.1.1.2.1" xref="S6.SS3.4.p4.3.m3.1.1.2.1.cmml">+</mo><mfrac id="S6.SS3.4.p4.3.m3.1.1.2.3" xref="S6.SS3.4.p4.3.m3.1.1.2.3.cmml"><mrow id="S6.SS3.4.p4.3.m3.1.1.2.3.2" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2.cmml"><mi id="S6.SS3.4.p4.3.m3.1.1.2.3.2.2" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2.2.cmml">γ</mi><mo id="S6.SS3.4.p4.3.m3.1.1.2.3.2.1" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2.1.cmml">+</mo><mn id="S6.SS3.4.p4.3.m3.1.1.2.3.2.3" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.4.p4.3.m3.1.1.2.3.3" xref="S6.SS3.4.p4.3.m3.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS3.4.p4.3.m3.1.1.1" xref="S6.SS3.4.p4.3.m3.1.1.1.cmml"><</mo><mrow id="S6.SS3.4.p4.3.m3.1.1.3" xref="S6.SS3.4.p4.3.m3.1.1.3.cmml"><msub id="S6.SS3.4.p4.3.m3.1.1.3.2" xref="S6.SS3.4.p4.3.m3.1.1.3.2.cmml"><mi id="S6.SS3.4.p4.3.m3.1.1.3.2.2" xref="S6.SS3.4.p4.3.m3.1.1.3.2.2.cmml">k</mi><mn id="S6.SS3.4.p4.3.m3.1.1.3.2.3" xref="S6.SS3.4.p4.3.m3.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.3.m3.1.1.3.1" xref="S6.SS3.4.p4.3.m3.1.1.3.1.cmml">+</mo><mi id="S6.SS3.4.p4.3.m3.1.1.3.3" mathvariant="normal" xref="S6.SS3.4.p4.3.m3.1.1.3.3.cmml">ℓ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.3.m3.1b"><apply id="S6.SS3.4.p4.3.m3.1.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1"><lt id="S6.SS3.4.p4.3.m3.1.1.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1.1"></lt><apply id="S6.SS3.4.p4.3.m3.1.1.2.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2"><plus id="S6.SS3.4.p4.3.m3.1.1.2.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.1"></plus><apply id="S6.SS3.4.p4.3.m3.1.1.2.2.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.3.m3.1.1.2.2.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.2">subscript</csymbol><ci id="S6.SS3.4.p4.3.m3.1.1.2.2.2.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.2.2">𝑚</ci><ci id="S6.SS3.4.p4.3.m3.1.1.2.2.3.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.2.3">𝑖</ci></apply><apply id="S6.SS3.4.p4.3.m3.1.1.2.3.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.3"><divide id="S6.SS3.4.p4.3.m3.1.1.2.3.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.3"></divide><apply id="S6.SS3.4.p4.3.m3.1.1.2.3.2.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2"><plus id="S6.SS3.4.p4.3.m3.1.1.2.3.2.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2.1"></plus><ci id="S6.SS3.4.p4.3.m3.1.1.2.3.2.2.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2.2">𝛾</ci><cn id="S6.SS3.4.p4.3.m3.1.1.2.3.2.3.cmml" type="integer" xref="S6.SS3.4.p4.3.m3.1.1.2.3.2.3">1</cn></apply><ci id="S6.SS3.4.p4.3.m3.1.1.2.3.3.cmml" xref="S6.SS3.4.p4.3.m3.1.1.2.3.3">𝑝</ci></apply></apply><apply id="S6.SS3.4.p4.3.m3.1.1.3.cmml" xref="S6.SS3.4.p4.3.m3.1.1.3"><plus id="S6.SS3.4.p4.3.m3.1.1.3.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1.3.1"></plus><apply id="S6.SS3.4.p4.3.m3.1.1.3.2.cmml" xref="S6.SS3.4.p4.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.3.m3.1.1.3.2.1.cmml" xref="S6.SS3.4.p4.3.m3.1.1.3.2">subscript</csymbol><ci id="S6.SS3.4.p4.3.m3.1.1.3.2.2.cmml" xref="S6.SS3.4.p4.3.m3.1.1.3.2.2">𝑘</ci><cn id="S6.SS3.4.p4.3.m3.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.4.p4.3.m3.1.1.3.2.3">0</cn></apply><ci id="S6.SS3.4.p4.3.m3.1.1.3.3.cmml" xref="S6.SS3.4.p4.3.m3.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.3.m3.1c">m_{i}+\frac{\gamma+1}{p}<k_{0}+\ell</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.3.m3.1d">italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ</annotation></semantics></math>. By definition of the complex interpolation method (see Section <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.SS5" title="2.5. The complex interpolation method ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.5</span></a>), there exists a <math alttext="f\in\mathscr{H}(W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p})" class="ltx_Math" display="inline" id="S6.SS3.4.p4.4.m4.6"><semantics id="S6.SS3.4.p4.4.m4.6a"><mrow id="S6.SS3.4.p4.4.m4.6.6" xref="S6.SS3.4.p4.4.m4.6.6.cmml"><mi id="S6.SS3.4.p4.4.m4.6.6.4" xref="S6.SS3.4.p4.4.m4.6.6.4.cmml">f</mi><mo id="S6.SS3.4.p4.4.m4.6.6.3" xref="S6.SS3.4.p4.4.m4.6.6.3.cmml">∈</mo><mrow id="S6.SS3.4.p4.4.m4.6.6.2" xref="S6.SS3.4.p4.4.m4.6.6.2.cmml"><mi class="ltx_font_mathscript" id="S6.SS3.4.p4.4.m4.6.6.2.4" xref="S6.SS3.4.p4.4.m4.6.6.2.4.cmml">ℋ</mi><mo id="S6.SS3.4.p4.4.m4.6.6.2.3" xref="S6.SS3.4.p4.4.m4.6.6.2.3.cmml">⁢</mo><mrow id="S6.SS3.4.p4.4.m4.6.6.2.2.2" xref="S6.SS3.4.p4.4.m4.6.6.2.2.3.cmml"><mo id="S6.SS3.4.p4.4.m4.6.6.2.2.2.3" stretchy="false" xref="S6.SS3.4.p4.4.m4.6.6.2.2.3.cmml">(</mo><msup id="S6.SS3.4.p4.4.m4.5.5.1.1.1.1" xref="S6.SS3.4.p4.4.m4.5.5.1.1.1.1.cmml"><mi id="S6.SS3.4.p4.4.m4.5.5.1.1.1.1.2" xref="S6.SS3.4.p4.4.m4.5.5.1.1.1.1.2.cmml">W</mi><mrow id="S6.SS3.4.p4.4.m4.2.2.2.2" xref="S6.SS3.4.p4.4.m4.2.2.2.3.cmml"><msub id="S6.SS3.4.p4.4.m4.2.2.2.2.1" xref="S6.SS3.4.p4.4.m4.2.2.2.2.1.cmml"><mi id="S6.SS3.4.p4.4.m4.2.2.2.2.1.2" xref="S6.SS3.4.p4.4.m4.2.2.2.2.1.2.cmml">k</mi><mn id="S6.SS3.4.p4.4.m4.2.2.2.2.1.3" xref="S6.SS3.4.p4.4.m4.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.4.m4.2.2.2.2.2" xref="S6.SS3.4.p4.4.m4.2.2.2.3.cmml">,</mo><mi id="S6.SS3.4.p4.4.m4.1.1.1.1" xref="S6.SS3.4.p4.4.m4.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.SS3.4.p4.4.m4.6.6.2.2.2.4" xref="S6.SS3.4.p4.4.m4.6.6.2.2.3.cmml">,</mo><msubsup id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.cmml"><mi id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.2" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.3" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.3.cmml">ℬ</mi><mrow id="S6.SS3.4.p4.4.m4.4.4.2.2" xref="S6.SS3.4.p4.4.m4.4.4.2.3.cmml"><mrow id="S6.SS3.4.p4.4.m4.4.4.2.2.1" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.cmml"><msub id="S6.SS3.4.p4.4.m4.4.4.2.2.1.2" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.cmml"><mi id="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.2" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.3" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.4.m4.4.4.2.2.1.1" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.1.cmml">+</mo><msub id="S6.SS3.4.p4.4.m4.4.4.2.2.1.3" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.cmml"><mi id="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.2" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.2.cmml">k</mi><mn id="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.3" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.4.p4.4.m4.4.4.2.2.2" xref="S6.SS3.4.p4.4.m4.4.4.2.3.cmml">,</mo><mi id="S6.SS3.4.p4.4.m4.3.3.1.1" xref="S6.SS3.4.p4.4.m4.3.3.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.SS3.4.p4.4.m4.6.6.2.2.2.5" stretchy="false" xref="S6.SS3.4.p4.4.m4.6.6.2.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.4.m4.6b"><apply id="S6.SS3.4.p4.4.m4.6.6.cmml" xref="S6.SS3.4.p4.4.m4.6.6"><in id="S6.SS3.4.p4.4.m4.6.6.3.cmml" xref="S6.SS3.4.p4.4.m4.6.6.3"></in><ci id="S6.SS3.4.p4.4.m4.6.6.4.cmml" xref="S6.SS3.4.p4.4.m4.6.6.4">𝑓</ci><apply id="S6.SS3.4.p4.4.m4.6.6.2.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2"><times id="S6.SS3.4.p4.4.m4.6.6.2.3.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.3"></times><ci id="S6.SS3.4.p4.4.m4.6.6.2.4.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.4">ℋ</ci><interval closure="open" id="S6.SS3.4.p4.4.m4.6.6.2.2.3.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2"><apply id="S6.SS3.4.p4.4.m4.5.5.1.1.1.1.cmml" xref="S6.SS3.4.p4.4.m4.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.4.p4.4.m4.5.5.1.1.1.1.1.cmml" xref="S6.SS3.4.p4.4.m4.5.5.1.1.1.1">superscript</csymbol><ci id="S6.SS3.4.p4.4.m4.5.5.1.1.1.1.2.cmml" xref="S6.SS3.4.p4.4.m4.5.5.1.1.1.1.2">𝑊</ci><list id="S6.SS3.4.p4.4.m4.2.2.2.3.cmml" xref="S6.SS3.4.p4.4.m4.2.2.2.2"><apply id="S6.SS3.4.p4.4.m4.2.2.2.2.1.cmml" xref="S6.SS3.4.p4.4.m4.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.4.p4.4.m4.2.2.2.2.1.1.cmml" xref="S6.SS3.4.p4.4.m4.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.4.p4.4.m4.2.2.2.2.1.2.cmml" xref="S6.SS3.4.p4.4.m4.2.2.2.2.1.2">𝑘</ci><cn id="S6.SS3.4.p4.4.m4.2.2.2.2.1.3.cmml" type="integer" xref="S6.SS3.4.p4.4.m4.2.2.2.2.1.3">0</cn></apply><ci id="S6.SS3.4.p4.4.m4.1.1.1.1.cmml" xref="S6.SS3.4.p4.4.m4.1.1.1.1">𝑝</ci></list></apply><apply id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.1.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2">superscript</csymbol><apply id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.1.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2">subscript</csymbol><ci id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.2.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.2">𝑊</ci><ci id="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.3.cmml" xref="S6.SS3.4.p4.4.m4.6.6.2.2.2.2.2.3">ℬ</ci></apply><list id="S6.SS3.4.p4.4.m4.4.4.2.3.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2"><apply id="S6.SS3.4.p4.4.m4.4.4.2.2.1.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1"><plus id="S6.SS3.4.p4.4.m4.4.4.2.2.1.1.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.1"></plus><apply id="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.1.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.2">subscript</csymbol><ci id="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.2.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.2.3">0</cn></apply><apply id="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.1.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.3">subscript</csymbol><ci id="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.2.cmml" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.2">𝑘</ci><cn id="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.3.cmml" type="integer" xref="S6.SS3.4.p4.4.m4.4.4.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.4.p4.4.m4.3.3.1.1.cmml" xref="S6.SS3.4.p4.4.m4.3.3.1.1">𝑝</ci></list></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.4.m4.6c">f\in\mathscr{H}(W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.4.m4.6d">italic_f ∈ script_H ( italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT )</annotation></semantics></math> such that <math alttext="f(\theta)=u" class="ltx_Math" display="inline" id="S6.SS3.4.p4.5.m5.1"><semantics id="S6.SS3.4.p4.5.m5.1a"><mrow id="S6.SS3.4.p4.5.m5.1.2" xref="S6.SS3.4.p4.5.m5.1.2.cmml"><mrow id="S6.SS3.4.p4.5.m5.1.2.2" xref="S6.SS3.4.p4.5.m5.1.2.2.cmml"><mi id="S6.SS3.4.p4.5.m5.1.2.2.2" xref="S6.SS3.4.p4.5.m5.1.2.2.2.cmml">f</mi><mo id="S6.SS3.4.p4.5.m5.1.2.2.1" xref="S6.SS3.4.p4.5.m5.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS3.4.p4.5.m5.1.2.2.3.2" xref="S6.SS3.4.p4.5.m5.1.2.2.cmml"><mo id="S6.SS3.4.p4.5.m5.1.2.2.3.2.1" stretchy="false" xref="S6.SS3.4.p4.5.m5.1.2.2.cmml">(</mo><mi id="S6.SS3.4.p4.5.m5.1.1" xref="S6.SS3.4.p4.5.m5.1.1.cmml">θ</mi><mo id="S6.SS3.4.p4.5.m5.1.2.2.3.2.2" stretchy="false" xref="S6.SS3.4.p4.5.m5.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS3.4.p4.5.m5.1.2.1" xref="S6.SS3.4.p4.5.m5.1.2.1.cmml">=</mo><mi id="S6.SS3.4.p4.5.m5.1.2.3" xref="S6.SS3.4.p4.5.m5.1.2.3.cmml">u</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.5.m5.1b"><apply id="S6.SS3.4.p4.5.m5.1.2.cmml" xref="S6.SS3.4.p4.5.m5.1.2"><eq id="S6.SS3.4.p4.5.m5.1.2.1.cmml" xref="S6.SS3.4.p4.5.m5.1.2.1"></eq><apply id="S6.SS3.4.p4.5.m5.1.2.2.cmml" xref="S6.SS3.4.p4.5.m5.1.2.2"><times id="S6.SS3.4.p4.5.m5.1.2.2.1.cmml" xref="S6.SS3.4.p4.5.m5.1.2.2.1"></times><ci id="S6.SS3.4.p4.5.m5.1.2.2.2.cmml" xref="S6.SS3.4.p4.5.m5.1.2.2.2">𝑓</ci><ci id="S6.SS3.4.p4.5.m5.1.1.cmml" xref="S6.SS3.4.p4.5.m5.1.1">𝜃</ci></apply><ci id="S6.SS3.4.p4.5.m5.1.2.3.cmml" xref="S6.SS3.4.p4.5.m5.1.2.3">𝑢</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.5.m5.1c">f(\theta)=u</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.5.m5.1d">italic_f ( italic_θ ) = italic_u</annotation></semantics></math>. By (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S2.E10" title="In 2.5. The complex interpolation method ‣ 2. Preliminaries ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">2.10</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E5" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>) we have that the restriction <math alttext="f|_{\mathbb{S}_{[\theta,1]}}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.6.m6.3"><semantics id="S6.SS3.4.p4.6.m6.3a"><msub id="S6.SS3.4.p4.6.m6.3.4.2" xref="S6.SS3.4.p4.6.m6.3.4.1.cmml"><mrow id="S6.SS3.4.p4.6.m6.3.4.2.2" xref="S6.SS3.4.p4.6.m6.3.4.1.cmml"><mi id="S6.SS3.4.p4.6.m6.3.3" xref="S6.SS3.4.p4.6.m6.3.3.cmml">f</mi><mo id="S6.SS3.4.p4.6.m6.3.4.2.2.1" stretchy="false" xref="S6.SS3.4.p4.6.m6.3.4.1.1.cmml">|</mo></mrow><msub id="S6.SS3.4.p4.6.m6.2.2.2" xref="S6.SS3.4.p4.6.m6.2.2.2.cmml"><mi id="S6.SS3.4.p4.6.m6.2.2.2.4" xref="S6.SS3.4.p4.6.m6.2.2.2.4.cmml">𝕊</mi><mrow id="S6.SS3.4.p4.6.m6.2.2.2.2.2.4" xref="S6.SS3.4.p4.6.m6.2.2.2.2.2.3.cmml"><mo id="S6.SS3.4.p4.6.m6.2.2.2.2.2.4.1" stretchy="false" xref="S6.SS3.4.p4.6.m6.2.2.2.2.2.3.cmml">[</mo><mi id="S6.SS3.4.p4.6.m6.1.1.1.1.1.1" xref="S6.SS3.4.p4.6.m6.1.1.1.1.1.1.cmml">θ</mi><mo id="S6.SS3.4.p4.6.m6.2.2.2.2.2.4.2" xref="S6.SS3.4.p4.6.m6.2.2.2.2.2.3.cmml">,</mo><mn id="S6.SS3.4.p4.6.m6.2.2.2.2.2.2" xref="S6.SS3.4.p4.6.m6.2.2.2.2.2.2.cmml">1</mn><mo id="S6.SS3.4.p4.6.m6.2.2.2.2.2.4.3" stretchy="false" xref="S6.SS3.4.p4.6.m6.2.2.2.2.2.3.cmml">]</mo></mrow></msub></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.6.m6.3b"><apply id="S6.SS3.4.p4.6.m6.3.4.1.cmml" xref="S6.SS3.4.p4.6.m6.3.4.2"><csymbol cd="latexml" id="S6.SS3.4.p4.6.m6.3.4.1.1.cmml" xref="S6.SS3.4.p4.6.m6.3.4.2.2.1">evaluated-at</csymbol><ci id="S6.SS3.4.p4.6.m6.3.3.cmml" xref="S6.SS3.4.p4.6.m6.3.3">𝑓</ci><apply id="S6.SS3.4.p4.6.m6.2.2.2.cmml" xref="S6.SS3.4.p4.6.m6.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.6.m6.2.2.2.3.cmml" xref="S6.SS3.4.p4.6.m6.2.2.2">subscript</csymbol><ci id="S6.SS3.4.p4.6.m6.2.2.2.4.cmml" xref="S6.SS3.4.p4.6.m6.2.2.2.4">𝕊</ci><interval closure="closed" id="S6.SS3.4.p4.6.m6.2.2.2.2.2.3.cmml" xref="S6.SS3.4.p4.6.m6.2.2.2.2.2.4"><ci id="S6.SS3.4.p4.6.m6.1.1.1.1.1.1.cmml" xref="S6.SS3.4.p4.6.m6.1.1.1.1.1.1">𝜃</ci><cn id="S6.SS3.4.p4.6.m6.2.2.2.2.2.2.cmml" type="integer" xref="S6.SS3.4.p4.6.m6.2.2.2.2.2.2">1</cn></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.6.m6.3c">f|_{\mathbb{S}_{[\theta,1]}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.6.m6.3d">italic_f | start_POSTSUBSCRIPT blackboard_S start_POSTSUBSCRIPT [ italic_θ , 1 ] end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is bounded and continuous with values in <math alttext="W^{k_{0}+\ell,p}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.7.m7.2"><semantics id="S6.SS3.4.p4.7.m7.2a"><msup id="S6.SS3.4.p4.7.m7.2.3" xref="S6.SS3.4.p4.7.m7.2.3.cmml"><mi id="S6.SS3.4.p4.7.m7.2.3.2" xref="S6.SS3.4.p4.7.m7.2.3.2.cmml">W</mi><mrow id="S6.SS3.4.p4.7.m7.2.2.2.2" xref="S6.SS3.4.p4.7.m7.2.2.2.3.cmml"><mrow id="S6.SS3.4.p4.7.m7.2.2.2.2.1" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.cmml"><msub id="S6.SS3.4.p4.7.m7.2.2.2.2.1.2" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.cmml"><mi id="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.2" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.3" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.7.m7.2.2.2.2.1.1" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.1.cmml">+</mo><mi id="S6.SS3.4.p4.7.m7.2.2.2.2.1.3" mathvariant="normal" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.4.p4.7.m7.2.2.2.2.2" xref="S6.SS3.4.p4.7.m7.2.2.2.3.cmml">,</mo><mi id="S6.SS3.4.p4.7.m7.1.1.1.1" xref="S6.SS3.4.p4.7.m7.1.1.1.1.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.7.m7.2b"><apply id="S6.SS3.4.p4.7.m7.2.3.cmml" xref="S6.SS3.4.p4.7.m7.2.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.7.m7.2.3.1.cmml" xref="S6.SS3.4.p4.7.m7.2.3">superscript</csymbol><ci id="S6.SS3.4.p4.7.m7.2.3.2.cmml" xref="S6.SS3.4.p4.7.m7.2.3.2">𝑊</ci><list id="S6.SS3.4.p4.7.m7.2.2.2.3.cmml" xref="S6.SS3.4.p4.7.m7.2.2.2.2"><apply id="S6.SS3.4.p4.7.m7.2.2.2.2.1.cmml" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1"><plus id="S6.SS3.4.p4.7.m7.2.2.2.2.1.1.cmml" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.1"></plus><apply id="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.cmml" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.1.cmml" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.2">subscript</csymbol><ci id="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.2.cmml" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.2.3">0</cn></apply><ci id="S6.SS3.4.p4.7.m7.2.2.2.2.1.3.cmml" xref="S6.SS3.4.p4.7.m7.2.2.2.2.1.3">ℓ</ci></apply><ci id="S6.SS3.4.p4.7.m7.1.1.1.1.cmml" xref="S6.SS3.4.p4.7.m7.1.1.1.1">𝑝</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.7.m7.2c">W^{k_{0}+\ell,p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.7.m7.2d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> and is holomorphic on <math alttext="{\mathbb{S}}_{(\theta,1)}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.8.m8.2"><semantics id="S6.SS3.4.p4.8.m8.2a"><msub id="S6.SS3.4.p4.8.m8.2.3" xref="S6.SS3.4.p4.8.m8.2.3.cmml"><mi id="S6.SS3.4.p4.8.m8.2.3.2" xref="S6.SS3.4.p4.8.m8.2.3.2.cmml">𝕊</mi><mrow id="S6.SS3.4.p4.8.m8.2.2.2.4" xref="S6.SS3.4.p4.8.m8.2.2.2.3.cmml"><mo id="S6.SS3.4.p4.8.m8.2.2.2.4.1" stretchy="false" xref="S6.SS3.4.p4.8.m8.2.2.2.3.cmml">(</mo><mi id="S6.SS3.4.p4.8.m8.1.1.1.1" xref="S6.SS3.4.p4.8.m8.1.1.1.1.cmml">θ</mi><mo id="S6.SS3.4.p4.8.m8.2.2.2.4.2" xref="S6.SS3.4.p4.8.m8.2.2.2.3.cmml">,</mo><mn id="S6.SS3.4.p4.8.m8.2.2.2.2" xref="S6.SS3.4.p4.8.m8.2.2.2.2.cmml">1</mn><mo id="S6.SS3.4.p4.8.m8.2.2.2.4.3" stretchy="false" xref="S6.SS3.4.p4.8.m8.2.2.2.3.cmml">)</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.8.m8.2b"><apply id="S6.SS3.4.p4.8.m8.2.3.cmml" xref="S6.SS3.4.p4.8.m8.2.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.8.m8.2.3.1.cmml" xref="S6.SS3.4.p4.8.m8.2.3">subscript</csymbol><ci id="S6.SS3.4.p4.8.m8.2.3.2.cmml" xref="S6.SS3.4.p4.8.m8.2.3.2">𝕊</ci><interval closure="open" id="S6.SS3.4.p4.8.m8.2.2.2.3.cmml" xref="S6.SS3.4.p4.8.m8.2.2.2.4"><ci id="S6.SS3.4.p4.8.m8.1.1.1.1.cmml" xref="S6.SS3.4.p4.8.m8.1.1.1.1">𝜃</ci><cn id="S6.SS3.4.p4.8.m8.2.2.2.2.cmml" type="integer" xref="S6.SS3.4.p4.8.m8.2.2.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.8.m8.2c">{\mathbb{S}}_{(\theta,1)}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.8.m8.2d">blackboard_S start_POSTSUBSCRIPT ( italic_θ , 1 ) end_POSTSUBSCRIPT</annotation></semantics></math>. Therefore, if <math alttext="0\leq j\leq i" class="ltx_Math" display="inline" id="S6.SS3.4.p4.9.m9.1"><semantics id="S6.SS3.4.p4.9.m9.1a"><mrow id="S6.SS3.4.p4.9.m9.1.1" xref="S6.SS3.4.p4.9.m9.1.1.cmml"><mn id="S6.SS3.4.p4.9.m9.1.1.2" xref="S6.SS3.4.p4.9.m9.1.1.2.cmml">0</mn><mo id="S6.SS3.4.p4.9.m9.1.1.3" xref="S6.SS3.4.p4.9.m9.1.1.3.cmml">≤</mo><mi id="S6.SS3.4.p4.9.m9.1.1.4" xref="S6.SS3.4.p4.9.m9.1.1.4.cmml">j</mi><mo id="S6.SS3.4.p4.9.m9.1.1.5" xref="S6.SS3.4.p4.9.m9.1.1.5.cmml">≤</mo><mi id="S6.SS3.4.p4.9.m9.1.1.6" xref="S6.SS3.4.p4.9.m9.1.1.6.cmml">i</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.9.m9.1b"><apply id="S6.SS3.4.p4.9.m9.1.1.cmml" xref="S6.SS3.4.p4.9.m9.1.1"><and id="S6.SS3.4.p4.9.m9.1.1a.cmml" xref="S6.SS3.4.p4.9.m9.1.1"></and><apply id="S6.SS3.4.p4.9.m9.1.1b.cmml" xref="S6.SS3.4.p4.9.m9.1.1"><leq id="S6.SS3.4.p4.9.m9.1.1.3.cmml" xref="S6.SS3.4.p4.9.m9.1.1.3"></leq><cn id="S6.SS3.4.p4.9.m9.1.1.2.cmml" type="integer" xref="S6.SS3.4.p4.9.m9.1.1.2">0</cn><ci id="S6.SS3.4.p4.9.m9.1.1.4.cmml" xref="S6.SS3.4.p4.9.m9.1.1.4">𝑗</ci></apply><apply id="S6.SS3.4.p4.9.m9.1.1c.cmml" xref="S6.SS3.4.p4.9.m9.1.1"><leq id="S6.SS3.4.p4.9.m9.1.1.5.cmml" xref="S6.SS3.4.p4.9.m9.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.4.p4.9.m9.1.1.4.cmml" id="S6.SS3.4.p4.9.m9.1.1d.cmml" xref="S6.SS3.4.p4.9.m9.1.1"></share><ci id="S6.SS3.4.p4.9.m9.1.1.6.cmml" xref="S6.SS3.4.p4.9.m9.1.1.6">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.9.m9.1c">0\leq j\leq i</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.9.m9.1d">0 ≤ italic_j ≤ italic_i</annotation></semantics></math>, then <math alttext="\mathcal{B}^{m_{j}}f" class="ltx_Math" display="inline" id="S6.SS3.4.p4.10.m10.1"><semantics id="S6.SS3.4.p4.10.m10.1a"><mrow id="S6.SS3.4.p4.10.m10.1.1" xref="S6.SS3.4.p4.10.m10.1.1.cmml"><msup id="S6.SS3.4.p4.10.m10.1.1.2" xref="S6.SS3.4.p4.10.m10.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.4.p4.10.m10.1.1.2.2" xref="S6.SS3.4.p4.10.m10.1.1.2.2.cmml">ℬ</mi><msub id="S6.SS3.4.p4.10.m10.1.1.2.3" xref="S6.SS3.4.p4.10.m10.1.1.2.3.cmml"><mi id="S6.SS3.4.p4.10.m10.1.1.2.3.2" xref="S6.SS3.4.p4.10.m10.1.1.2.3.2.cmml">m</mi><mi id="S6.SS3.4.p4.10.m10.1.1.2.3.3" xref="S6.SS3.4.p4.10.m10.1.1.2.3.3.cmml">j</mi></msub></msup><mo id="S6.SS3.4.p4.10.m10.1.1.1" xref="S6.SS3.4.p4.10.m10.1.1.1.cmml">⁢</mo><mi id="S6.SS3.4.p4.10.m10.1.1.3" xref="S6.SS3.4.p4.10.m10.1.1.3.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.10.m10.1b"><apply id="S6.SS3.4.p4.10.m10.1.1.cmml" xref="S6.SS3.4.p4.10.m10.1.1"><times id="S6.SS3.4.p4.10.m10.1.1.1.cmml" xref="S6.SS3.4.p4.10.m10.1.1.1"></times><apply id="S6.SS3.4.p4.10.m10.1.1.2.cmml" xref="S6.SS3.4.p4.10.m10.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.10.m10.1.1.2.1.cmml" xref="S6.SS3.4.p4.10.m10.1.1.2">superscript</csymbol><ci id="S6.SS3.4.p4.10.m10.1.1.2.2.cmml" xref="S6.SS3.4.p4.10.m10.1.1.2.2">ℬ</ci><apply id="S6.SS3.4.p4.10.m10.1.1.2.3.cmml" xref="S6.SS3.4.p4.10.m10.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.10.m10.1.1.2.3.1.cmml" xref="S6.SS3.4.p4.10.m10.1.1.2.3">subscript</csymbol><ci id="S6.SS3.4.p4.10.m10.1.1.2.3.2.cmml" xref="S6.SS3.4.p4.10.m10.1.1.2.3.2">𝑚</ci><ci id="S6.SS3.4.p4.10.m10.1.1.2.3.3.cmml" xref="S6.SS3.4.p4.10.m10.1.1.2.3.3">𝑗</ci></apply></apply><ci id="S6.SS3.4.p4.10.m10.1.1.3.cmml" xref="S6.SS3.4.p4.10.m10.1.1.3">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.10.m10.1c">\mathcal{B}^{m_{j}}f</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.10.m10.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f</annotation></semantics></math> is bounded and continuous on <math alttext="{\mathbb{S}}_{[\theta,1]}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.11.m11.2"><semantics id="S6.SS3.4.p4.11.m11.2a"><msub id="S6.SS3.4.p4.11.m11.2.3" xref="S6.SS3.4.p4.11.m11.2.3.cmml"><mi id="S6.SS3.4.p4.11.m11.2.3.2" xref="S6.SS3.4.p4.11.m11.2.3.2.cmml">𝕊</mi><mrow id="S6.SS3.4.p4.11.m11.2.2.2.4" xref="S6.SS3.4.p4.11.m11.2.2.2.3.cmml"><mo id="S6.SS3.4.p4.11.m11.2.2.2.4.1" stretchy="false" xref="S6.SS3.4.p4.11.m11.2.2.2.3.cmml">[</mo><mi id="S6.SS3.4.p4.11.m11.1.1.1.1" xref="S6.SS3.4.p4.11.m11.1.1.1.1.cmml">θ</mi><mo id="S6.SS3.4.p4.11.m11.2.2.2.4.2" xref="S6.SS3.4.p4.11.m11.2.2.2.3.cmml">,</mo><mn id="S6.SS3.4.p4.11.m11.2.2.2.2" xref="S6.SS3.4.p4.11.m11.2.2.2.2.cmml">1</mn><mo id="S6.SS3.4.p4.11.m11.2.2.2.4.3" stretchy="false" xref="S6.SS3.4.p4.11.m11.2.2.2.3.cmml">]</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.11.m11.2b"><apply id="S6.SS3.4.p4.11.m11.2.3.cmml" xref="S6.SS3.4.p4.11.m11.2.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.11.m11.2.3.1.cmml" xref="S6.SS3.4.p4.11.m11.2.3">subscript</csymbol><ci id="S6.SS3.4.p4.11.m11.2.3.2.cmml" xref="S6.SS3.4.p4.11.m11.2.3.2">𝕊</ci><interval closure="closed" id="S6.SS3.4.p4.11.m11.2.2.2.3.cmml" xref="S6.SS3.4.p4.11.m11.2.2.2.4"><ci id="S6.SS3.4.p4.11.m11.1.1.1.1.cmml" xref="S6.SS3.4.p4.11.m11.1.1.1.1">𝜃</ci><cn id="S6.SS3.4.p4.11.m11.2.2.2.2.cmml" type="integer" xref="S6.SS3.4.p4.11.m11.2.2.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.11.m11.2c">{\mathbb{S}}_{[\theta,1]}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.11.m11.2d">blackboard_S start_POSTSUBSCRIPT [ italic_θ , 1 ] end_POSTSUBSCRIPT</annotation></semantics></math> with values in <math alttext="B_{p,p}^{k_{0}+\ell-m_{j}-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};Y_{j})" class="ltx_Math" display="inline" id="S6.SS3.4.p4.12.m12.4"><semantics id="S6.SS3.4.p4.12.m12.4a"><mrow id="S6.SS3.4.p4.12.m12.4.4" xref="S6.SS3.4.p4.12.m12.4.4.cmml"><msubsup id="S6.SS3.4.p4.12.m12.4.4.4" xref="S6.SS3.4.p4.12.m12.4.4.4.cmml"><mi id="S6.SS3.4.p4.12.m12.4.4.4.2.2" xref="S6.SS3.4.p4.12.m12.4.4.4.2.2.cmml">B</mi><mrow id="S6.SS3.4.p4.12.m12.2.2.2.4" xref="S6.SS3.4.p4.12.m12.2.2.2.3.cmml"><mi id="S6.SS3.4.p4.12.m12.1.1.1.1" xref="S6.SS3.4.p4.12.m12.1.1.1.1.cmml">p</mi><mo id="S6.SS3.4.p4.12.m12.2.2.2.4.1" xref="S6.SS3.4.p4.12.m12.2.2.2.3.cmml">,</mo><mi id="S6.SS3.4.p4.12.m12.2.2.2.2" xref="S6.SS3.4.p4.12.m12.2.2.2.2.cmml">p</mi></mrow><mrow id="S6.SS3.4.p4.12.m12.4.4.4.3" xref="S6.SS3.4.p4.12.m12.4.4.4.3.cmml"><mrow id="S6.SS3.4.p4.12.m12.4.4.4.3.2" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.cmml"><msub id="S6.SS3.4.p4.12.m12.4.4.4.3.2.2" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.cmml"><mi id="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.2" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.2.cmml">k</mi><mn id="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.3" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.12.m12.4.4.4.3.2.1" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.1.cmml">+</mo><mi id="S6.SS3.4.p4.12.m12.4.4.4.3.2.3" mathvariant="normal" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.4.p4.12.m12.4.4.4.3.1" xref="S6.SS3.4.p4.12.m12.4.4.4.3.1.cmml">−</mo><msub id="S6.SS3.4.p4.12.m12.4.4.4.3.3" xref="S6.SS3.4.p4.12.m12.4.4.4.3.3.cmml"><mi id="S6.SS3.4.p4.12.m12.4.4.4.3.3.2" xref="S6.SS3.4.p4.12.m12.4.4.4.3.3.2.cmml">m</mi><mi id="S6.SS3.4.p4.12.m12.4.4.4.3.3.3" xref="S6.SS3.4.p4.12.m12.4.4.4.3.3.3.cmml">j</mi></msub><mo id="S6.SS3.4.p4.12.m12.4.4.4.3.1a" xref="S6.SS3.4.p4.12.m12.4.4.4.3.1.cmml">−</mo><mfrac id="S6.SS3.4.p4.12.m12.4.4.4.3.4" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.cmml"><mrow id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.cmml"><mi id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.2" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.2.cmml">γ</mi><mo id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.1" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.1.cmml">+</mo><mn id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.3" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.3.cmml">1</mn></mrow><mi id="S6.SS3.4.p4.12.m12.4.4.4.3.4.3" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S6.SS3.4.p4.12.m12.4.4.3" xref="S6.SS3.4.p4.12.m12.4.4.3.cmml">⁢</mo><mrow id="S6.SS3.4.p4.12.m12.4.4.2.2" xref="S6.SS3.4.p4.12.m12.4.4.2.3.cmml"><mo id="S6.SS3.4.p4.12.m12.4.4.2.2.3" stretchy="false" xref="S6.SS3.4.p4.12.m12.4.4.2.3.cmml">(</mo><msup id="S6.SS3.4.p4.12.m12.3.3.1.1.1" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.cmml"><mi id="S6.SS3.4.p4.12.m12.3.3.1.1.1.2" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.2.cmml">ℝ</mi><mrow id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.cmml"><mi id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.2" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.2.cmml">d</mi><mo id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.1" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.1.cmml">−</mo><mn id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.3" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.3.cmml">1</mn></mrow></msup><mo id="S6.SS3.4.p4.12.m12.4.4.2.2.4" xref="S6.SS3.4.p4.12.m12.4.4.2.3.cmml">;</mo><msub id="S6.SS3.4.p4.12.m12.4.4.2.2.2" xref="S6.SS3.4.p4.12.m12.4.4.2.2.2.cmml"><mi id="S6.SS3.4.p4.12.m12.4.4.2.2.2.2" xref="S6.SS3.4.p4.12.m12.4.4.2.2.2.2.cmml">Y</mi><mi id="S6.SS3.4.p4.12.m12.4.4.2.2.2.3" xref="S6.SS3.4.p4.12.m12.4.4.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS3.4.p4.12.m12.4.4.2.2.5" stretchy="false" xref="S6.SS3.4.p4.12.m12.4.4.2.3.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.12.m12.4b"><apply id="S6.SS3.4.p4.12.m12.4.4.cmml" xref="S6.SS3.4.p4.12.m12.4.4"><times id="S6.SS3.4.p4.12.m12.4.4.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.3"></times><apply id="S6.SS3.4.p4.12.m12.4.4.4.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4"><csymbol cd="ambiguous" id="S6.SS3.4.p4.12.m12.4.4.4.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4">superscript</csymbol><apply id="S6.SS3.4.p4.12.m12.4.4.4.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4"><csymbol cd="ambiguous" id="S6.SS3.4.p4.12.m12.4.4.4.2.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4">subscript</csymbol><ci id="S6.SS3.4.p4.12.m12.4.4.4.2.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.2.2">𝐵</ci><list id="S6.SS3.4.p4.12.m12.2.2.2.3.cmml" xref="S6.SS3.4.p4.12.m12.2.2.2.4"><ci id="S6.SS3.4.p4.12.m12.1.1.1.1.cmml" xref="S6.SS3.4.p4.12.m12.1.1.1.1">𝑝</ci><ci id="S6.SS3.4.p4.12.m12.2.2.2.2.cmml" xref="S6.SS3.4.p4.12.m12.2.2.2.2">𝑝</ci></list></apply><apply id="S6.SS3.4.p4.12.m12.4.4.4.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3"><minus id="S6.SS3.4.p4.12.m12.4.4.4.3.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.1"></minus><apply id="S6.SS3.4.p4.12.m12.4.4.4.3.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2"><plus id="S6.SS3.4.p4.12.m12.4.4.4.3.2.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.1"></plus><apply id="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.2">subscript</csymbol><ci id="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.2">𝑘</ci><cn id="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.3.cmml" type="integer" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.2.3">0</cn></apply><ci id="S6.SS3.4.p4.12.m12.4.4.4.3.2.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.2.3">ℓ</ci></apply><apply id="S6.SS3.4.p4.12.m12.4.4.4.3.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.12.m12.4.4.4.3.3.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.3">subscript</csymbol><ci id="S6.SS3.4.p4.12.m12.4.4.4.3.3.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.3.2">𝑚</ci><ci id="S6.SS3.4.p4.12.m12.4.4.4.3.3.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.3.3">𝑗</ci></apply><apply id="S6.SS3.4.p4.12.m12.4.4.4.3.4.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4"><divide id="S6.SS3.4.p4.12.m12.4.4.4.3.4.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4"></divide><apply id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2"><plus id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.1"></plus><ci id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.2">𝛾</ci><cn id="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.3.cmml" type="integer" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.2.3">1</cn></apply><ci id="S6.SS3.4.p4.12.m12.4.4.4.3.4.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.4.3.4.3">𝑝</ci></apply></apply></apply><list id="S6.SS3.4.p4.12.m12.4.4.2.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.2.2"><apply id="S6.SS3.4.p4.12.m12.3.3.1.1.1.cmml" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.4.p4.12.m12.3.3.1.1.1.1.cmml" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1">superscript</csymbol><ci id="S6.SS3.4.p4.12.m12.3.3.1.1.1.2.cmml" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.2">ℝ</ci><apply id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.cmml" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3"><minus id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.1.cmml" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.1"></minus><ci id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.2.cmml" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.2">𝑑</ci><cn id="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.3.cmml" type="integer" xref="S6.SS3.4.p4.12.m12.3.3.1.1.1.3.3">1</cn></apply></apply><apply id="S6.SS3.4.p4.12.m12.4.4.2.2.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.12.m12.4.4.2.2.2.1.cmml" xref="S6.SS3.4.p4.12.m12.4.4.2.2.2">subscript</csymbol><ci id="S6.SS3.4.p4.12.m12.4.4.2.2.2.2.cmml" xref="S6.SS3.4.p4.12.m12.4.4.2.2.2.2">𝑌</ci><ci id="S6.SS3.4.p4.12.m12.4.4.2.2.2.3.cmml" xref="S6.SS3.4.p4.12.m12.4.4.2.2.2.3">𝑗</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.12.m12.4c">B_{p,p}^{k_{0}+\ell-m_{j}-\frac{\gamma+1}{p}}(\mathbb{R}^{d-1};Y_{j})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.12.m12.4d">italic_B start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ - italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_Y start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT )</annotation></semantics></math> (see Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem6" title="Theorem 5.6. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.6</span></a>) and is holomorphic on <math alttext="{\mathbb{S}}_{(\theta,1)}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.13.m13.2"><semantics id="S6.SS3.4.p4.13.m13.2a"><msub id="S6.SS3.4.p4.13.m13.2.3" xref="S6.SS3.4.p4.13.m13.2.3.cmml"><mi id="S6.SS3.4.p4.13.m13.2.3.2" xref="S6.SS3.4.p4.13.m13.2.3.2.cmml">𝕊</mi><mrow id="S6.SS3.4.p4.13.m13.2.2.2.4" xref="S6.SS3.4.p4.13.m13.2.2.2.3.cmml"><mo id="S6.SS3.4.p4.13.m13.2.2.2.4.1" stretchy="false" xref="S6.SS3.4.p4.13.m13.2.2.2.3.cmml">(</mo><mi id="S6.SS3.4.p4.13.m13.1.1.1.1" xref="S6.SS3.4.p4.13.m13.1.1.1.1.cmml">θ</mi><mo id="S6.SS3.4.p4.13.m13.2.2.2.4.2" xref="S6.SS3.4.p4.13.m13.2.2.2.3.cmml">,</mo><mn id="S6.SS3.4.p4.13.m13.2.2.2.2" xref="S6.SS3.4.p4.13.m13.2.2.2.2.cmml">1</mn><mo id="S6.SS3.4.p4.13.m13.2.2.2.4.3" stretchy="false" xref="S6.SS3.4.p4.13.m13.2.2.2.3.cmml">)</mo></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.13.m13.2b"><apply id="S6.SS3.4.p4.13.m13.2.3.cmml" xref="S6.SS3.4.p4.13.m13.2.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.13.m13.2.3.1.cmml" xref="S6.SS3.4.p4.13.m13.2.3">subscript</csymbol><ci id="S6.SS3.4.p4.13.m13.2.3.2.cmml" xref="S6.SS3.4.p4.13.m13.2.3.2">𝕊</ci><interval closure="open" id="S6.SS3.4.p4.13.m13.2.2.2.3.cmml" xref="S6.SS3.4.p4.13.m13.2.2.2.4"><ci id="S6.SS3.4.p4.13.m13.1.1.1.1.cmml" xref="S6.SS3.4.p4.13.m13.1.1.1.1">𝜃</ci><cn id="S6.SS3.4.p4.13.m13.2.2.2.2.cmml" type="integer" xref="S6.SS3.4.p4.13.m13.2.2.2.2">1</cn></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.13.m13.2c">{\mathbb{S}}_{(\theta,1)}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.13.m13.2d">blackboard_S start_POSTSUBSCRIPT ( italic_θ , 1 ) end_POSTSUBSCRIPT</annotation></semantics></math>. Moreover, for <math alttext="\operatorname{Re}(z)=1" class="ltx_Math" display="inline" id="S6.SS3.4.p4.14.m14.2"><semantics id="S6.SS3.4.p4.14.m14.2a"><mrow id="S6.SS3.4.p4.14.m14.2.3" xref="S6.SS3.4.p4.14.m14.2.3.cmml"><mrow id="S6.SS3.4.p4.14.m14.2.3.2.2" xref="S6.SS3.4.p4.14.m14.2.3.2.1.cmml"><mi id="S6.SS3.4.p4.14.m14.1.1" xref="S6.SS3.4.p4.14.m14.1.1.cmml">Re</mi><mo id="S6.SS3.4.p4.14.m14.2.3.2.2a" xref="S6.SS3.4.p4.14.m14.2.3.2.1.cmml">⁡</mo><mrow id="S6.SS3.4.p4.14.m14.2.3.2.2.1" xref="S6.SS3.4.p4.14.m14.2.3.2.1.cmml"><mo id="S6.SS3.4.p4.14.m14.2.3.2.2.1.1" stretchy="false" xref="S6.SS3.4.p4.14.m14.2.3.2.1.cmml">(</mo><mi id="S6.SS3.4.p4.14.m14.2.2" xref="S6.SS3.4.p4.14.m14.2.2.cmml">z</mi><mo id="S6.SS3.4.p4.14.m14.2.3.2.2.1.2" stretchy="false" xref="S6.SS3.4.p4.14.m14.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="S6.SS3.4.p4.14.m14.2.3.1" xref="S6.SS3.4.p4.14.m14.2.3.1.cmml">=</mo><mn id="S6.SS3.4.p4.14.m14.2.3.3" xref="S6.SS3.4.p4.14.m14.2.3.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.14.m14.2b"><apply id="S6.SS3.4.p4.14.m14.2.3.cmml" xref="S6.SS3.4.p4.14.m14.2.3"><eq id="S6.SS3.4.p4.14.m14.2.3.1.cmml" xref="S6.SS3.4.p4.14.m14.2.3.1"></eq><apply id="S6.SS3.4.p4.14.m14.2.3.2.1.cmml" xref="S6.SS3.4.p4.14.m14.2.3.2.2"><ci id="S6.SS3.4.p4.14.m14.1.1.cmml" xref="S6.SS3.4.p4.14.m14.1.1">Re</ci><ci id="S6.SS3.4.p4.14.m14.2.2.cmml" xref="S6.SS3.4.p4.14.m14.2.2">𝑧</ci></apply><cn id="S6.SS3.4.p4.14.m14.2.3.3.cmml" type="integer" xref="S6.SS3.4.p4.14.m14.2.3.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.14.m14.2c">\operatorname{Re}(z)=1</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.14.m14.2d">roman_Re ( italic_z ) = 1</annotation></semantics></math> we have <math alttext="\mathcal{B}^{m_{j}}f(z)=0" class="ltx_Math" display="inline" id="S6.SS3.4.p4.15.m15.1"><semantics id="S6.SS3.4.p4.15.m15.1a"><mrow id="S6.SS3.4.p4.15.m15.1.2" xref="S6.SS3.4.p4.15.m15.1.2.cmml"><mrow id="S6.SS3.4.p4.15.m15.1.2.2" xref="S6.SS3.4.p4.15.m15.1.2.2.cmml"><msup id="S6.SS3.4.p4.15.m15.1.2.2.2" xref="S6.SS3.4.p4.15.m15.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.4.p4.15.m15.1.2.2.2.2" xref="S6.SS3.4.p4.15.m15.1.2.2.2.2.cmml">ℬ</mi><msub id="S6.SS3.4.p4.15.m15.1.2.2.2.3" xref="S6.SS3.4.p4.15.m15.1.2.2.2.3.cmml"><mi id="S6.SS3.4.p4.15.m15.1.2.2.2.3.2" xref="S6.SS3.4.p4.15.m15.1.2.2.2.3.2.cmml">m</mi><mi id="S6.SS3.4.p4.15.m15.1.2.2.2.3.3" xref="S6.SS3.4.p4.15.m15.1.2.2.2.3.3.cmml">j</mi></msub></msup><mo id="S6.SS3.4.p4.15.m15.1.2.2.1" xref="S6.SS3.4.p4.15.m15.1.2.2.1.cmml">⁢</mo><mi id="S6.SS3.4.p4.15.m15.1.2.2.3" xref="S6.SS3.4.p4.15.m15.1.2.2.3.cmml">f</mi><mo id="S6.SS3.4.p4.15.m15.1.2.2.1a" xref="S6.SS3.4.p4.15.m15.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS3.4.p4.15.m15.1.2.2.4.2" xref="S6.SS3.4.p4.15.m15.1.2.2.cmml"><mo id="S6.SS3.4.p4.15.m15.1.2.2.4.2.1" stretchy="false" xref="S6.SS3.4.p4.15.m15.1.2.2.cmml">(</mo><mi id="S6.SS3.4.p4.15.m15.1.1" xref="S6.SS3.4.p4.15.m15.1.1.cmml">z</mi><mo id="S6.SS3.4.p4.15.m15.1.2.2.4.2.2" stretchy="false" xref="S6.SS3.4.p4.15.m15.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS3.4.p4.15.m15.1.2.1" xref="S6.SS3.4.p4.15.m15.1.2.1.cmml">=</mo><mn id="S6.SS3.4.p4.15.m15.1.2.3" xref="S6.SS3.4.p4.15.m15.1.2.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.15.m15.1b"><apply id="S6.SS3.4.p4.15.m15.1.2.cmml" xref="S6.SS3.4.p4.15.m15.1.2"><eq id="S6.SS3.4.p4.15.m15.1.2.1.cmml" xref="S6.SS3.4.p4.15.m15.1.2.1"></eq><apply id="S6.SS3.4.p4.15.m15.1.2.2.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2"><times id="S6.SS3.4.p4.15.m15.1.2.2.1.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.1"></times><apply id="S6.SS3.4.p4.15.m15.1.2.2.2.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.15.m15.1.2.2.2.1.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.2">superscript</csymbol><ci id="S6.SS3.4.p4.15.m15.1.2.2.2.2.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.2.2">ℬ</ci><apply id="S6.SS3.4.p4.15.m15.1.2.2.2.3.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.2.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.15.m15.1.2.2.2.3.1.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.2.3">subscript</csymbol><ci id="S6.SS3.4.p4.15.m15.1.2.2.2.3.2.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.2.3.2">𝑚</ci><ci id="S6.SS3.4.p4.15.m15.1.2.2.2.3.3.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.2.3.3">𝑗</ci></apply></apply><ci id="S6.SS3.4.p4.15.m15.1.2.2.3.cmml" xref="S6.SS3.4.p4.15.m15.1.2.2.3">𝑓</ci><ci id="S6.SS3.4.p4.15.m15.1.1.cmml" xref="S6.SS3.4.p4.15.m15.1.1">𝑧</ci></apply><cn id="S6.SS3.4.p4.15.m15.1.2.3.cmml" type="integer" xref="S6.SS3.4.p4.15.m15.1.2.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.15.m15.1c">\mathcal{B}^{m_{j}}f(z)=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.15.m15.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f ( italic_z ) = 0</annotation></semantics></math>. By the three lines lemma (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib9" title="">9</a>, Lemma 1.1.2]</cite>), <math alttext="f(z)" class="ltx_Math" display="inline" id="S6.SS3.4.p4.16.m16.1"><semantics id="S6.SS3.4.p4.16.m16.1a"><mrow id="S6.SS3.4.p4.16.m16.1.2" xref="S6.SS3.4.p4.16.m16.1.2.cmml"><mi id="S6.SS3.4.p4.16.m16.1.2.2" xref="S6.SS3.4.p4.16.m16.1.2.2.cmml">f</mi><mo id="S6.SS3.4.p4.16.m16.1.2.1" xref="S6.SS3.4.p4.16.m16.1.2.1.cmml">⁢</mo><mrow id="S6.SS3.4.p4.16.m16.1.2.3.2" xref="S6.SS3.4.p4.16.m16.1.2.cmml"><mo id="S6.SS3.4.p4.16.m16.1.2.3.2.1" stretchy="false" xref="S6.SS3.4.p4.16.m16.1.2.cmml">(</mo><mi id="S6.SS3.4.p4.16.m16.1.1" xref="S6.SS3.4.p4.16.m16.1.1.cmml">z</mi><mo id="S6.SS3.4.p4.16.m16.1.2.3.2.2" stretchy="false" xref="S6.SS3.4.p4.16.m16.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.16.m16.1b"><apply id="S6.SS3.4.p4.16.m16.1.2.cmml" xref="S6.SS3.4.p4.16.m16.1.2"><times id="S6.SS3.4.p4.16.m16.1.2.1.cmml" xref="S6.SS3.4.p4.16.m16.1.2.1"></times><ci id="S6.SS3.4.p4.16.m16.1.2.2.cmml" xref="S6.SS3.4.p4.16.m16.1.2.2">𝑓</ci><ci id="S6.SS3.4.p4.16.m16.1.1.cmml" xref="S6.SS3.4.p4.16.m16.1.1">𝑧</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.16.m16.1c">f(z)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.16.m16.1d">italic_f ( italic_z )</annotation></semantics></math> vanishes identically and thus</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex55"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{B}^{m_{j}}u=\mathcal{B}^{m_{j}}f(\theta)=0\qquad\text{ for }0\leq j% \leq i." class="ltx_Math" display="block" id="S6.Ex55.m1.2"><semantics id="S6.Ex55.m1.2a"><mrow id="S6.Ex55.m1.2.2.1"><mrow id="S6.Ex55.m1.2.2.1.1.2" xref="S6.Ex55.m1.2.2.1.1.3.cmml"><mrow id="S6.Ex55.m1.2.2.1.1.1.1" xref="S6.Ex55.m1.2.2.1.1.1.1.cmml"><mrow id="S6.Ex55.m1.2.2.1.1.1.1.2" xref="S6.Ex55.m1.2.2.1.1.1.1.2.cmml"><msup id="S6.Ex55.m1.2.2.1.1.1.1.2.2" xref="S6.Ex55.m1.2.2.1.1.1.1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex55.m1.2.2.1.1.1.1.2.2.2" xref="S6.Ex55.m1.2.2.1.1.1.1.2.2.2.cmml">ℬ</mi><msub id="S6.Ex55.m1.2.2.1.1.1.1.2.2.3" xref="S6.Ex55.m1.2.2.1.1.1.1.2.2.3.cmml"><mi id="S6.Ex55.m1.2.2.1.1.1.1.2.2.3.2" xref="S6.Ex55.m1.2.2.1.1.1.1.2.2.3.2.cmml">m</mi><mi id="S6.Ex55.m1.2.2.1.1.1.1.2.2.3.3" xref="S6.Ex55.m1.2.2.1.1.1.1.2.2.3.3.cmml">j</mi></msub></msup><mo id="S6.Ex55.m1.2.2.1.1.1.1.2.1" xref="S6.Ex55.m1.2.2.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.Ex55.m1.2.2.1.1.1.1.2.3" xref="S6.Ex55.m1.2.2.1.1.1.1.2.3.cmml">u</mi></mrow><mo id="S6.Ex55.m1.2.2.1.1.1.1.3" xref="S6.Ex55.m1.2.2.1.1.1.1.3.cmml">=</mo><mrow id="S6.Ex55.m1.2.2.1.1.1.1.4" xref="S6.Ex55.m1.2.2.1.1.1.1.4.cmml"><msup id="S6.Ex55.m1.2.2.1.1.1.1.4.2" xref="S6.Ex55.m1.2.2.1.1.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex55.m1.2.2.1.1.1.1.4.2.2" xref="S6.Ex55.m1.2.2.1.1.1.1.4.2.2.cmml">ℬ</mi><msub id="S6.Ex55.m1.2.2.1.1.1.1.4.2.3" xref="S6.Ex55.m1.2.2.1.1.1.1.4.2.3.cmml"><mi id="S6.Ex55.m1.2.2.1.1.1.1.4.2.3.2" xref="S6.Ex55.m1.2.2.1.1.1.1.4.2.3.2.cmml">m</mi><mi id="S6.Ex55.m1.2.2.1.1.1.1.4.2.3.3" xref="S6.Ex55.m1.2.2.1.1.1.1.4.2.3.3.cmml">j</mi></msub></msup><mo id="S6.Ex55.m1.2.2.1.1.1.1.4.1" xref="S6.Ex55.m1.2.2.1.1.1.1.4.1.cmml">⁢</mo><mi id="S6.Ex55.m1.2.2.1.1.1.1.4.3" xref="S6.Ex55.m1.2.2.1.1.1.1.4.3.cmml">f</mi><mo id="S6.Ex55.m1.2.2.1.1.1.1.4.1a" xref="S6.Ex55.m1.2.2.1.1.1.1.4.1.cmml">⁢</mo><mrow id="S6.Ex55.m1.2.2.1.1.1.1.4.4.2" xref="S6.Ex55.m1.2.2.1.1.1.1.4.cmml"><mo id="S6.Ex55.m1.2.2.1.1.1.1.4.4.2.1" stretchy="false" xref="S6.Ex55.m1.2.2.1.1.1.1.4.cmml">(</mo><mi id="S6.Ex55.m1.1.1" xref="S6.Ex55.m1.1.1.cmml">θ</mi><mo id="S6.Ex55.m1.2.2.1.1.1.1.4.4.2.2" stretchy="false" xref="S6.Ex55.m1.2.2.1.1.1.1.4.cmml">)</mo></mrow></mrow><mo id="S6.Ex55.m1.2.2.1.1.1.1.5" xref="S6.Ex55.m1.2.2.1.1.1.1.5.cmml">=</mo><mn id="S6.Ex55.m1.2.2.1.1.1.1.6" xref="S6.Ex55.m1.2.2.1.1.1.1.6.cmml">0</mn></mrow><mspace id="S6.Ex55.m1.2.2.1.1.2.3" width="2em" xref="S6.Ex55.m1.2.2.1.1.3a.cmml"></mspace><mrow id="S6.Ex55.m1.2.2.1.1.2.2" xref="S6.Ex55.m1.2.2.1.1.2.2.cmml"><mrow id="S6.Ex55.m1.2.2.1.1.2.2.2" xref="S6.Ex55.m1.2.2.1.1.2.2.2.cmml"><mtext id="S6.Ex55.m1.2.2.1.1.2.2.2.2" xref="S6.Ex55.m1.2.2.1.1.2.2.2.2a.cmml"> for </mtext><mo id="S6.Ex55.m1.2.2.1.1.2.2.2.1" xref="S6.Ex55.m1.2.2.1.1.2.2.2.1.cmml">⁢</mo><mn id="S6.Ex55.m1.2.2.1.1.2.2.2.3" xref="S6.Ex55.m1.2.2.1.1.2.2.2.3.cmml">0</mn></mrow><mo id="S6.Ex55.m1.2.2.1.1.2.2.3" xref="S6.Ex55.m1.2.2.1.1.2.2.3.cmml">≤</mo><mi id="S6.Ex55.m1.2.2.1.1.2.2.4" xref="S6.Ex55.m1.2.2.1.1.2.2.4.cmml">j</mi><mo id="S6.Ex55.m1.2.2.1.1.2.2.5" xref="S6.Ex55.m1.2.2.1.1.2.2.5.cmml">≤</mo><mi id="S6.Ex55.m1.2.2.1.1.2.2.6" xref="S6.Ex55.m1.2.2.1.1.2.2.6.cmml">i</mi></mrow></mrow><mo id="S6.Ex55.m1.2.2.1.2" lspace="0em">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex55.m1.2b"><apply id="S6.Ex55.m1.2.2.1.1.3.cmml" xref="S6.Ex55.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="S6.Ex55.m1.2.2.1.1.3a.cmml" xref="S6.Ex55.m1.2.2.1.1.2.3">formulae-sequence</csymbol><apply id="S6.Ex55.m1.2.2.1.1.1.1.cmml" xref="S6.Ex55.m1.2.2.1.1.1.1"><and id="S6.Ex55.m1.2.2.1.1.1.1a.cmml" xref="S6.Ex55.m1.2.2.1.1.1.1"></and><apply id="S6.Ex55.m1.2.2.1.1.1.1b.cmml" xref="S6.Ex55.m1.2.2.1.1.1.1"><eq id="S6.Ex55.m1.2.2.1.1.1.1.3.cmml" xref="S6.Ex55.m1.2.2.1.1.1.1.3"></eq><apply id="S6.Ex55.m1.2.2.1.1.1.1.2.cmml" xref="S6.Ex55.m1.2.2.1.1.1.1.2"><times id="S6.Ex55.m1.2.2.1.1.1.1.2.1.cmml" xref="S6.Ex55.m1.2.2.1.1.1.1.2.1"></times><apply id="S6.Ex55.m1.2.2.1.1.1.1.2.2.cmml" xref="S6.Ex55.m1.2.2.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.Ex55.m1.2.2.1.1.1.1.2.2.1.cmml" 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xref="S6.Ex55.m1.2.2.1.1.1.1.6">0</cn></apply></apply><apply id="S6.Ex55.m1.2.2.1.1.2.2.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2"><and id="S6.Ex55.m1.2.2.1.1.2.2a.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2"></and><apply id="S6.Ex55.m1.2.2.1.1.2.2b.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2"><leq id="S6.Ex55.m1.2.2.1.1.2.2.3.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.3"></leq><apply id="S6.Ex55.m1.2.2.1.1.2.2.2.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.2"><times id="S6.Ex55.m1.2.2.1.1.2.2.2.1.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.2.1"></times><ci id="S6.Ex55.m1.2.2.1.1.2.2.2.2a.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.2.2"><mtext id="S6.Ex55.m1.2.2.1.1.2.2.2.2.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.2.2"> for </mtext></ci><cn id="S6.Ex55.m1.2.2.1.1.2.2.2.3.cmml" type="integer" xref="S6.Ex55.m1.2.2.1.1.2.2.2.3">0</cn></apply><ci id="S6.Ex55.m1.2.2.1.1.2.2.4.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.4">𝑗</ci></apply><apply id="S6.Ex55.m1.2.2.1.1.2.2c.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2"><leq id="S6.Ex55.m1.2.2.1.1.2.2.5.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.Ex55.m1.2.2.1.1.2.2.4.cmml" id="S6.Ex55.m1.2.2.1.1.2.2d.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2"></share><ci id="S6.Ex55.m1.2.2.1.1.2.2.6.cmml" xref="S6.Ex55.m1.2.2.1.1.2.2.6">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex55.m1.2c">\mathcal{B}^{m_{j}}u=\mathcal{B}^{m_{j}}f(\theta)=0\qquad\text{ for }0\leq j% \leq i.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex55.m1.2d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_u = caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_f ( italic_θ ) = 0 for 0 ≤ italic_j ≤ italic_i .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.4.p4.18">This proves <math alttext="u\in W^{k_{0}+\ell,p}_{\mathcal{B}}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.17.m1.2"><semantics id="S6.SS3.4.p4.17.m1.2a"><mrow id="S6.SS3.4.p4.17.m1.2.3" xref="S6.SS3.4.p4.17.m1.2.3.cmml"><mi id="S6.SS3.4.p4.17.m1.2.3.2" xref="S6.SS3.4.p4.17.m1.2.3.2.cmml">u</mi><mo id="S6.SS3.4.p4.17.m1.2.3.1" xref="S6.SS3.4.p4.17.m1.2.3.1.cmml">∈</mo><msubsup id="S6.SS3.4.p4.17.m1.2.3.3" xref="S6.SS3.4.p4.17.m1.2.3.3.cmml"><mi id="S6.SS3.4.p4.17.m1.2.3.3.2.2" xref="S6.SS3.4.p4.17.m1.2.3.3.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.4.p4.17.m1.2.3.3.3" xref="S6.SS3.4.p4.17.m1.2.3.3.3.cmml">ℬ</mi><mrow id="S6.SS3.4.p4.17.m1.2.2.2.2" xref="S6.SS3.4.p4.17.m1.2.2.2.3.cmml"><mrow id="S6.SS3.4.p4.17.m1.2.2.2.2.1" xref="S6.SS3.4.p4.17.m1.2.2.2.2.1.cmml"><msub id="S6.SS3.4.p4.17.m1.2.2.2.2.1.2" xref="S6.SS3.4.p4.17.m1.2.2.2.2.1.2.cmml"><mi id="S6.SS3.4.p4.17.m1.2.2.2.2.1.2.2" xref="S6.SS3.4.p4.17.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.4.p4.17.m1.2.2.2.2.1.2.3" xref="S6.SS3.4.p4.17.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.17.m1.2.2.2.2.1.1" xref="S6.SS3.4.p4.17.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.SS3.4.p4.17.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.SS3.4.p4.17.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.4.p4.17.m1.2.2.2.2.2" xref="S6.SS3.4.p4.17.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS3.4.p4.17.m1.1.1.1.1" xref="S6.SS3.4.p4.17.m1.1.1.1.1.cmml">p</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.17.m1.2b"><apply id="S6.SS3.4.p4.17.m1.2.3.cmml" xref="S6.SS3.4.p4.17.m1.2.3"><in id="S6.SS3.4.p4.17.m1.2.3.1.cmml" xref="S6.SS3.4.p4.17.m1.2.3.1"></in><ci id="S6.SS3.4.p4.17.m1.2.3.2.cmml" xref="S6.SS3.4.p4.17.m1.2.3.2">𝑢</ci><apply id="S6.SS3.4.p4.17.m1.2.3.3.cmml" xref="S6.SS3.4.p4.17.m1.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.17.m1.2.3.3.1.cmml" xref="S6.SS3.4.p4.17.m1.2.3.3">subscript</csymbol><apply id="S6.SS3.4.p4.17.m1.2.3.3.2.cmml" xref="S6.SS3.4.p4.17.m1.2.3.3"><csymbol 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xref="S6.SS3.4.p4.17.m1.1.1.1.1">𝑝</ci></list></apply><ci id="S6.SS3.4.p4.17.m1.2.3.3.3.cmml" xref="S6.SS3.4.p4.17.m1.2.3.3.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.17.m1.2c">u\in W^{k_{0}+\ell,p}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.17.m1.2d">italic_u ∈ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> and thus <math alttext="[W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}]_{\theta}\hookrightarrow W^{k_{0}% +\ell,p}_{\mathcal{B}}" class="ltx_Math" display="inline" id="S6.SS3.4.p4.18.m2.8"><semantics id="S6.SS3.4.p4.18.m2.8a"><mrow id="S6.SS3.4.p4.18.m2.8.8" xref="S6.SS3.4.p4.18.m2.8.8.cmml"><msub id="S6.SS3.4.p4.18.m2.8.8.2" xref="S6.SS3.4.p4.18.m2.8.8.2.cmml"><mrow id="S6.SS3.4.p4.18.m2.8.8.2.2.2" 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xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.4.p4.18.m2.4.4.2.2.2" xref="S6.SS3.4.p4.18.m2.4.4.2.3.cmml">,</mo><mi id="S6.SS3.4.p4.18.m2.3.3.1.1" xref="S6.SS3.4.p4.18.m2.3.3.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.SS3.4.p4.18.m2.8.8.2.2.2.5" stretchy="false" xref="S6.SS3.4.p4.18.m2.8.8.2.2.3.cmml">]</mo></mrow><mi id="S6.SS3.4.p4.18.m2.8.8.2.4" xref="S6.SS3.4.p4.18.m2.8.8.2.4.cmml">θ</mi></msub><mo id="S6.SS3.4.p4.18.m2.8.8.3" stretchy="false" xref="S6.SS3.4.p4.18.m2.8.8.3.cmml">↪</mo><msubsup id="S6.SS3.4.p4.18.m2.8.8.4" xref="S6.SS3.4.p4.18.m2.8.8.4.cmml"><mi id="S6.SS3.4.p4.18.m2.8.8.4.2.2" xref="S6.SS3.4.p4.18.m2.8.8.4.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.4.p4.18.m2.8.8.4.3" xref="S6.SS3.4.p4.18.m2.8.8.4.3.cmml">ℬ</mi><mrow id="S6.SS3.4.p4.18.m2.6.6.2.2" xref="S6.SS3.4.p4.18.m2.6.6.2.3.cmml"><mrow id="S6.SS3.4.p4.18.m2.6.6.2.2.1" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.cmml"><msub id="S6.SS3.4.p4.18.m2.6.6.2.2.1.2" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.cmml"><mi id="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.2" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.3" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.4.p4.18.m2.6.6.2.2.1.1" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.1.cmml">+</mo><mi id="S6.SS3.4.p4.18.m2.6.6.2.2.1.3" mathvariant="normal" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.4.p4.18.m2.6.6.2.2.2" xref="S6.SS3.4.p4.18.m2.6.6.2.3.cmml">,</mo><mi id="S6.SS3.4.p4.18.m2.5.5.1.1" xref="S6.SS3.4.p4.18.m2.5.5.1.1.cmml">p</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.4.p4.18.m2.8b"><apply id="S6.SS3.4.p4.18.m2.8.8.cmml" xref="S6.SS3.4.p4.18.m2.8.8"><ci id="S6.SS3.4.p4.18.m2.8.8.3.cmml" xref="S6.SS3.4.p4.18.m2.8.8.3">↪</ci><apply id="S6.SS3.4.p4.18.m2.8.8.2.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.8.8.2.3.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2">subscript</csymbol><interval closure="closed" id="S6.SS3.4.p4.18.m2.8.8.2.2.3.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.2.2"><apply id="S6.SS3.4.p4.18.m2.7.7.1.1.1.1.cmml" xref="S6.SS3.4.p4.18.m2.7.7.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.7.7.1.1.1.1.1.cmml" xref="S6.SS3.4.p4.18.m2.7.7.1.1.1.1">superscript</csymbol><ci id="S6.SS3.4.p4.18.m2.7.7.1.1.1.1.2.cmml" xref="S6.SS3.4.p4.18.m2.7.7.1.1.1.1.2">𝑊</ci><list id="S6.SS3.4.p4.18.m2.2.2.2.3.cmml" xref="S6.SS3.4.p4.18.m2.2.2.2.2"><apply id="S6.SS3.4.p4.18.m2.2.2.2.2.1.cmml" xref="S6.SS3.4.p4.18.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.2.2.2.2.1.1.cmml" xref="S6.SS3.4.p4.18.m2.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.4.p4.18.m2.2.2.2.2.1.2.cmml" xref="S6.SS3.4.p4.18.m2.2.2.2.2.1.2">𝑘</ci><cn id="S6.SS3.4.p4.18.m2.2.2.2.2.1.3.cmml" type="integer" xref="S6.SS3.4.p4.18.m2.2.2.2.2.1.3">0</cn></apply><ci id="S6.SS3.4.p4.18.m2.1.1.1.1.cmml" xref="S6.SS3.4.p4.18.m2.1.1.1.1">𝑝</ci></list></apply><apply id="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.1.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.2.2.2">superscript</csymbol><apply id="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.2.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.2.1.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.2.2.2">subscript</csymbol><ci id="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.2.2.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.2.2">𝑊</ci><ci id="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.2.3.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.2.2.2.2.3">ℬ</ci></apply><list id="S6.SS3.4.p4.18.m2.4.4.2.3.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2"><apply id="S6.SS3.4.p4.18.m2.4.4.2.2.1.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1"><plus id="S6.SS3.4.p4.18.m2.4.4.2.2.1.1.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.1"></plus><apply id="S6.SS3.4.p4.18.m2.4.4.2.2.1.2.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.4.4.2.2.1.2.1.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.2">subscript</csymbol><ci id="S6.SS3.4.p4.18.m2.4.4.2.2.1.2.2.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.4.p4.18.m2.4.4.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.2.3">0</cn></apply><apply id="S6.SS3.4.p4.18.m2.4.4.2.2.1.3.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.4.4.2.2.1.3.1.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.3">subscript</csymbol><ci id="S6.SS3.4.p4.18.m2.4.4.2.2.1.3.2.cmml" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.3.2">𝑘</ci><cn id="S6.SS3.4.p4.18.m2.4.4.2.2.1.3.3.cmml" type="integer" xref="S6.SS3.4.p4.18.m2.4.4.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.4.p4.18.m2.3.3.1.1.cmml" xref="S6.SS3.4.p4.18.m2.3.3.1.1">𝑝</ci></list></apply></interval><ci id="S6.SS3.4.p4.18.m2.8.8.2.4.cmml" xref="S6.SS3.4.p4.18.m2.8.8.2.4">𝜃</ci></apply><apply id="S6.SS3.4.p4.18.m2.8.8.4.cmml" xref="S6.SS3.4.p4.18.m2.8.8.4"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.8.8.4.1.cmml" xref="S6.SS3.4.p4.18.m2.8.8.4">subscript</csymbol><apply id="S6.SS3.4.p4.18.m2.8.8.4.2.cmml" xref="S6.SS3.4.p4.18.m2.8.8.4"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.8.8.4.2.1.cmml" xref="S6.SS3.4.p4.18.m2.8.8.4">superscript</csymbol><ci id="S6.SS3.4.p4.18.m2.8.8.4.2.2.cmml" xref="S6.SS3.4.p4.18.m2.8.8.4.2.2">𝑊</ci><list id="S6.SS3.4.p4.18.m2.6.6.2.3.cmml" xref="S6.SS3.4.p4.18.m2.6.6.2.2"><apply id="S6.SS3.4.p4.18.m2.6.6.2.2.1.cmml" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1"><plus id="S6.SS3.4.p4.18.m2.6.6.2.2.1.1.cmml" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.1"></plus><apply id="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.cmml" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.1.cmml" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.2">subscript</csymbol><ci id="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.2.cmml" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.2.3">0</cn></apply><ci id="S6.SS3.4.p4.18.m2.6.6.2.2.1.3.cmml" xref="S6.SS3.4.p4.18.m2.6.6.2.2.1.3">ℓ</ci></apply><ci id="S6.SS3.4.p4.18.m2.5.5.1.1.cmml" xref="S6.SS3.4.p4.18.m2.5.5.1.1">𝑝</ci></list></apply><ci id="S6.SS3.4.p4.18.m2.8.8.4.3.cmml" xref="S6.SS3.4.p4.18.m2.8.8.4.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.4.p4.18.m2.8c">[W^{k_{0},p},W_{\mathcal{B}}^{k_{0}+k_{1},p}]_{\theta}\hookrightarrow W^{k_{0}% +\ell,p}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.4.p4.18.m2.8d">[ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS3.5.p5"> <p class="ltx_p" id="S6.SS3.5.p5.6"><span class="ltx_text ltx_font_italic" id="S6.SS3.5.p5.1.1">Step 2: the embedding “<math alttext="\hookrightarrow" class="ltx_Math" display="inline" id="S6.SS3.5.p5.1.1.m1.1"><semantics id="S6.SS3.5.p5.1.1.m1.1a"><mo id="S6.SS3.5.p5.1.1.m1.1.1" stretchy="false" xref="S6.SS3.5.p5.1.1.m1.1.1.cmml">↪</mo><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.1.1.m1.1b"><ci id="S6.SS3.5.p5.1.1.m1.1.1.cmml" xref="S6.SS3.5.p5.1.1.m1.1.1">↪</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.1.1.m1.1c">\hookrightarrow</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.1.1.m1.1d">↪</annotation></semantics></math>”.</span> Let <math alttext="0\leq i\leq n" class="ltx_Math" display="inline" id="S6.SS3.5.p5.2.m1.1"><semantics id="S6.SS3.5.p5.2.m1.1a"><mrow id="S6.SS3.5.p5.2.m1.1.1" xref="S6.SS3.5.p5.2.m1.1.1.cmml"><mn id="S6.SS3.5.p5.2.m1.1.1.2" xref="S6.SS3.5.p5.2.m1.1.1.2.cmml">0</mn><mo id="S6.SS3.5.p5.2.m1.1.1.3" xref="S6.SS3.5.p5.2.m1.1.1.3.cmml">≤</mo><mi id="S6.SS3.5.p5.2.m1.1.1.4" xref="S6.SS3.5.p5.2.m1.1.1.4.cmml">i</mi><mo id="S6.SS3.5.p5.2.m1.1.1.5" xref="S6.SS3.5.p5.2.m1.1.1.5.cmml">≤</mo><mi id="S6.SS3.5.p5.2.m1.1.1.6" xref="S6.SS3.5.p5.2.m1.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.2.m1.1b"><apply id="S6.SS3.5.p5.2.m1.1.1.cmml" xref="S6.SS3.5.p5.2.m1.1.1"><and id="S6.SS3.5.p5.2.m1.1.1a.cmml" xref="S6.SS3.5.p5.2.m1.1.1"></and><apply id="S6.SS3.5.p5.2.m1.1.1b.cmml" xref="S6.SS3.5.p5.2.m1.1.1"><leq id="S6.SS3.5.p5.2.m1.1.1.3.cmml" xref="S6.SS3.5.p5.2.m1.1.1.3"></leq><cn id="S6.SS3.5.p5.2.m1.1.1.2.cmml" type="integer" xref="S6.SS3.5.p5.2.m1.1.1.2">0</cn><ci id="S6.SS3.5.p5.2.m1.1.1.4.cmml" xref="S6.SS3.5.p5.2.m1.1.1.4">𝑖</ci></apply><apply id="S6.SS3.5.p5.2.m1.1.1c.cmml" xref="S6.SS3.5.p5.2.m1.1.1"><leq id="S6.SS3.5.p5.2.m1.1.1.5.cmml" xref="S6.SS3.5.p5.2.m1.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.5.p5.2.m1.1.1.4.cmml" id="S6.SS3.5.p5.2.m1.1.1d.cmml" xref="S6.SS3.5.p5.2.m1.1.1"></share><ci id="S6.SS3.5.p5.2.m1.1.1.6.cmml" xref="S6.SS3.5.p5.2.m1.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.2.m1.1c">0\leq i\leq n</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.2.m1.1d">0 ≤ italic_i ≤ italic_n</annotation></semantics></math> and let <math alttext="b_{i,m_{i}}" class="ltx_Math" display="inline" id="S6.SS3.5.p5.3.m2.2"><semantics id="S6.SS3.5.p5.3.m2.2a"><msub id="S6.SS3.5.p5.3.m2.2.3" xref="S6.SS3.5.p5.3.m2.2.3.cmml"><mi id="S6.SS3.5.p5.3.m2.2.3.2" xref="S6.SS3.5.p5.3.m2.2.3.2.cmml">b</mi><mrow id="S6.SS3.5.p5.3.m2.2.2.2.2" xref="S6.SS3.5.p5.3.m2.2.2.2.3.cmml"><mi id="S6.SS3.5.p5.3.m2.1.1.1.1" xref="S6.SS3.5.p5.3.m2.1.1.1.1.cmml">i</mi><mo id="S6.SS3.5.p5.3.m2.2.2.2.2.2" xref="S6.SS3.5.p5.3.m2.2.2.2.3.cmml">,</mo><msub id="S6.SS3.5.p5.3.m2.2.2.2.2.1" xref="S6.SS3.5.p5.3.m2.2.2.2.2.1.cmml"><mi id="S6.SS3.5.p5.3.m2.2.2.2.2.1.2" xref="S6.SS3.5.p5.3.m2.2.2.2.2.1.2.cmml">m</mi><mi id="S6.SS3.5.p5.3.m2.2.2.2.2.1.3" xref="S6.SS3.5.p5.3.m2.2.2.2.2.1.3.cmml">i</mi></msub></mrow></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.3.m2.2b"><apply id="S6.SS3.5.p5.3.m2.2.3.cmml" xref="S6.SS3.5.p5.3.m2.2.3"><csymbol cd="ambiguous" id="S6.SS3.5.p5.3.m2.2.3.1.cmml" xref="S6.SS3.5.p5.3.m2.2.3">subscript</csymbol><ci id="S6.SS3.5.p5.3.m2.2.3.2.cmml" xref="S6.SS3.5.p5.3.m2.2.3.2">𝑏</ci><list id="S6.SS3.5.p5.3.m2.2.2.2.3.cmml" xref="S6.SS3.5.p5.3.m2.2.2.2.2"><ci id="S6.SS3.5.p5.3.m2.1.1.1.1.cmml" xref="S6.SS3.5.p5.3.m2.1.1.1.1">𝑖</ci><apply id="S6.SS3.5.p5.3.m2.2.2.2.2.1.cmml" xref="S6.SS3.5.p5.3.m2.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.5.p5.3.m2.2.2.2.2.1.1.cmml" xref="S6.SS3.5.p5.3.m2.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.5.p5.3.m2.2.2.2.2.1.2.cmml" xref="S6.SS3.5.p5.3.m2.2.2.2.2.1.2">𝑚</ci><ci id="S6.SS3.5.p5.3.m2.2.2.2.2.1.3.cmml" xref="S6.SS3.5.p5.3.m2.2.2.2.2.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.3.m2.2c">b_{i,m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.3.m2.2d">italic_b start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> be the leading order coefficient of <math alttext="\mathcal{B}^{m_{i}}" class="ltx_Math" display="inline" id="S6.SS3.5.p5.4.m3.1"><semantics id="S6.SS3.5.p5.4.m3.1a"><msup id="S6.SS3.5.p5.4.m3.1.1" xref="S6.SS3.5.p5.4.m3.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.5.p5.4.m3.1.1.2" xref="S6.SS3.5.p5.4.m3.1.1.2.cmml">ℬ</mi><msub id="S6.SS3.5.p5.4.m3.1.1.3" xref="S6.SS3.5.p5.4.m3.1.1.3.cmml"><mi id="S6.SS3.5.p5.4.m3.1.1.3.2" xref="S6.SS3.5.p5.4.m3.1.1.3.2.cmml">m</mi><mi id="S6.SS3.5.p5.4.m3.1.1.3.3" xref="S6.SS3.5.p5.4.m3.1.1.3.3.cmml">i</mi></msub></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.4.m3.1b"><apply id="S6.SS3.5.p5.4.m3.1.1.cmml" xref="S6.SS3.5.p5.4.m3.1.1"><csymbol cd="ambiguous" id="S6.SS3.5.p5.4.m3.1.1.1.cmml" xref="S6.SS3.5.p5.4.m3.1.1">superscript</csymbol><ci id="S6.SS3.5.p5.4.m3.1.1.2.cmml" xref="S6.SS3.5.p5.4.m3.1.1.2">ℬ</ci><apply id="S6.SS3.5.p5.4.m3.1.1.3.cmml" xref="S6.SS3.5.p5.4.m3.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.5.p5.4.m3.1.1.3.1.cmml" xref="S6.SS3.5.p5.4.m3.1.1.3">subscript</csymbol><ci id="S6.SS3.5.p5.4.m3.1.1.3.2.cmml" xref="S6.SS3.5.p5.4.m3.1.1.3.2">𝑚</ci><ci id="S6.SS3.5.p5.4.m3.1.1.3.3.cmml" xref="S6.SS3.5.p5.4.m3.1.1.3.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.4.m3.1c">\mathcal{B}^{m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.4.m3.1d">caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT</annotation></semantics></math> for which <math alttext="b^{{\rm c}}_{i,m_{i}}" class="ltx_Math" display="inline" id="S6.SS3.5.p5.5.m4.2"><semantics id="S6.SS3.5.p5.5.m4.2a"><msubsup id="S6.SS3.5.p5.5.m4.2.3" xref="S6.SS3.5.p5.5.m4.2.3.cmml"><mi id="S6.SS3.5.p5.5.m4.2.3.2.2" xref="S6.SS3.5.p5.5.m4.2.3.2.2.cmml">b</mi><mrow id="S6.SS3.5.p5.5.m4.2.2.2.2" xref="S6.SS3.5.p5.5.m4.2.2.2.3.cmml"><mi id="S6.SS3.5.p5.5.m4.1.1.1.1" xref="S6.SS3.5.p5.5.m4.1.1.1.1.cmml">i</mi><mo id="S6.SS3.5.p5.5.m4.2.2.2.2.2" xref="S6.SS3.5.p5.5.m4.2.2.2.3.cmml">,</mo><msub id="S6.SS3.5.p5.5.m4.2.2.2.2.1" xref="S6.SS3.5.p5.5.m4.2.2.2.2.1.cmml"><mi id="S6.SS3.5.p5.5.m4.2.2.2.2.1.2" xref="S6.SS3.5.p5.5.m4.2.2.2.2.1.2.cmml">m</mi><mi id="S6.SS3.5.p5.5.m4.2.2.2.2.1.3" xref="S6.SS3.5.p5.5.m4.2.2.2.2.1.3.cmml">i</mi></msub></mrow><mi id="S6.SS3.5.p5.5.m4.2.3.2.3" mathvariant="normal" xref="S6.SS3.5.p5.5.m4.2.3.2.3.cmml">c</mi></msubsup><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.5.m4.2b"><apply id="S6.SS3.5.p5.5.m4.2.3.cmml" xref="S6.SS3.5.p5.5.m4.2.3"><csymbol cd="ambiguous" id="S6.SS3.5.p5.5.m4.2.3.1.cmml" xref="S6.SS3.5.p5.5.m4.2.3">subscript</csymbol><apply id="S6.SS3.5.p5.5.m4.2.3.2.cmml" xref="S6.SS3.5.p5.5.m4.2.3"><csymbol cd="ambiguous" id="S6.SS3.5.p5.5.m4.2.3.2.1.cmml" xref="S6.SS3.5.p5.5.m4.2.3">superscript</csymbol><ci id="S6.SS3.5.p5.5.m4.2.3.2.2.cmml" xref="S6.SS3.5.p5.5.m4.2.3.2.2">𝑏</ci><ci id="S6.SS3.5.p5.5.m4.2.3.2.3.cmml" xref="S6.SS3.5.p5.5.m4.2.3.2.3">c</ci></apply><list id="S6.SS3.5.p5.5.m4.2.2.2.3.cmml" xref="S6.SS3.5.p5.5.m4.2.2.2.2"><ci id="S6.SS3.5.p5.5.m4.1.1.1.1.cmml" xref="S6.SS3.5.p5.5.m4.1.1.1.1">𝑖</ci><apply id="S6.SS3.5.p5.5.m4.2.2.2.2.1.cmml" xref="S6.SS3.5.p5.5.m4.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.5.p5.5.m4.2.2.2.2.1.1.cmml" xref="S6.SS3.5.p5.5.m4.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.5.p5.5.m4.2.2.2.2.1.2.cmml" xref="S6.SS3.5.p5.5.m4.2.2.2.2.1.2">𝑚</ci><ci id="S6.SS3.5.p5.5.m4.2.2.2.2.1.3.cmml" xref="S6.SS3.5.p5.5.m4.2.2.2.2.1.3">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.5.m4.2c">b^{{\rm c}}_{i,m_{i}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.5.m4.2d">italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT</annotation></semantics></math> is the coretraction from Definition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem3" title="Definition 5.3 (Normal boundary operators). ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.3</span></a>. Define the boundary operator <math alttext="\widetilde{\mathcal{B}}^{m_{i}}:=b^{{\rm c}}_{i,m_{i}}\mathcal{B}" class="ltx_Math" display="inline" id="S6.SS3.5.p5.6.m5.2"><semantics id="S6.SS3.5.p5.6.m5.2a"><mrow id="S6.SS3.5.p5.6.m5.2.3" xref="S6.SS3.5.p5.6.m5.2.3.cmml"><msup id="S6.SS3.5.p5.6.m5.2.3.2" xref="S6.SS3.5.p5.6.m5.2.3.2.cmml"><mover accent="true" id="S6.SS3.5.p5.6.m5.2.3.2.2" xref="S6.SS3.5.p5.6.m5.2.3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.5.p5.6.m5.2.3.2.2.2" xref="S6.SS3.5.p5.6.m5.2.3.2.2.2.cmml">ℬ</mi><mo id="S6.SS3.5.p5.6.m5.2.3.2.2.1" xref="S6.SS3.5.p5.6.m5.2.3.2.2.1.cmml">~</mo></mover><msub id="S6.SS3.5.p5.6.m5.2.3.2.3" xref="S6.SS3.5.p5.6.m5.2.3.2.3.cmml"><mi id="S6.SS3.5.p5.6.m5.2.3.2.3.2" xref="S6.SS3.5.p5.6.m5.2.3.2.3.2.cmml">m</mi><mi id="S6.SS3.5.p5.6.m5.2.3.2.3.3" xref="S6.SS3.5.p5.6.m5.2.3.2.3.3.cmml">i</mi></msub></msup><mo id="S6.SS3.5.p5.6.m5.2.3.1" lspace="0.278em" rspace="0.278em" xref="S6.SS3.5.p5.6.m5.2.3.1.cmml">:=</mo><mrow id="S6.SS3.5.p5.6.m5.2.3.3" xref="S6.SS3.5.p5.6.m5.2.3.3.cmml"><msubsup id="S6.SS3.5.p5.6.m5.2.3.3.2" xref="S6.SS3.5.p5.6.m5.2.3.3.2.cmml"><mi id="S6.SS3.5.p5.6.m5.2.3.3.2.2.2" xref="S6.SS3.5.p5.6.m5.2.3.3.2.2.2.cmml">b</mi><mrow id="S6.SS3.5.p5.6.m5.2.2.2.2" xref="S6.SS3.5.p5.6.m5.2.2.2.3.cmml"><mi id="S6.SS3.5.p5.6.m5.1.1.1.1" xref="S6.SS3.5.p5.6.m5.1.1.1.1.cmml">i</mi><mo id="S6.SS3.5.p5.6.m5.2.2.2.2.2" xref="S6.SS3.5.p5.6.m5.2.2.2.3.cmml">,</mo><msub id="S6.SS3.5.p5.6.m5.2.2.2.2.1" xref="S6.SS3.5.p5.6.m5.2.2.2.2.1.cmml"><mi id="S6.SS3.5.p5.6.m5.2.2.2.2.1.2" xref="S6.SS3.5.p5.6.m5.2.2.2.2.1.2.cmml">m</mi><mi id="S6.SS3.5.p5.6.m5.2.2.2.2.1.3" xref="S6.SS3.5.p5.6.m5.2.2.2.2.1.3.cmml">i</mi></msub></mrow><mi id="S6.SS3.5.p5.6.m5.2.3.3.2.2.3" mathvariant="normal" xref="S6.SS3.5.p5.6.m5.2.3.3.2.2.3.cmml">c</mi></msubsup><mo id="S6.SS3.5.p5.6.m5.2.3.3.1" xref="S6.SS3.5.p5.6.m5.2.3.3.1.cmml">⁢</mo><mi class="ltx_font_mathcaligraphic" id="S6.SS3.5.p5.6.m5.2.3.3.3" xref="S6.SS3.5.p5.6.m5.2.3.3.3.cmml">ℬ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.6.m5.2b"><apply id="S6.SS3.5.p5.6.m5.2.3.cmml" xref="S6.SS3.5.p5.6.m5.2.3"><csymbol cd="latexml" id="S6.SS3.5.p5.6.m5.2.3.1.cmml" xref="S6.SS3.5.p5.6.m5.2.3.1">assign</csymbol><apply id="S6.SS3.5.p5.6.m5.2.3.2.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2"><csymbol cd="ambiguous" id="S6.SS3.5.p5.6.m5.2.3.2.1.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2">superscript</csymbol><apply id="S6.SS3.5.p5.6.m5.2.3.2.2.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2.2"><ci id="S6.SS3.5.p5.6.m5.2.3.2.2.1.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2.2.1">~</ci><ci id="S6.SS3.5.p5.6.m5.2.3.2.2.2.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2.2.2">ℬ</ci></apply><apply id="S6.SS3.5.p5.6.m5.2.3.2.3.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2.3"><csymbol cd="ambiguous" id="S6.SS3.5.p5.6.m5.2.3.2.3.1.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2.3">subscript</csymbol><ci id="S6.SS3.5.p5.6.m5.2.3.2.3.2.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2.3.2">𝑚</ci><ci id="S6.SS3.5.p5.6.m5.2.3.2.3.3.cmml" xref="S6.SS3.5.p5.6.m5.2.3.2.3.3">𝑖</ci></apply></apply><apply id="S6.SS3.5.p5.6.m5.2.3.3.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3"><times id="S6.SS3.5.p5.6.m5.2.3.3.1.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.1"></times><apply id="S6.SS3.5.p5.6.m5.2.3.3.2.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.2"><csymbol cd="ambiguous" id="S6.SS3.5.p5.6.m5.2.3.3.2.1.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.2">subscript</csymbol><apply id="S6.SS3.5.p5.6.m5.2.3.3.2.2.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.2"><csymbol cd="ambiguous" id="S6.SS3.5.p5.6.m5.2.3.3.2.2.1.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.2">superscript</csymbol><ci id="S6.SS3.5.p5.6.m5.2.3.3.2.2.2.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.2.2.2">𝑏</ci><ci id="S6.SS3.5.p5.6.m5.2.3.3.2.2.3.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.2.2.3">c</ci></apply><list id="S6.SS3.5.p5.6.m5.2.2.2.3.cmml" xref="S6.SS3.5.p5.6.m5.2.2.2.2"><ci id="S6.SS3.5.p5.6.m5.1.1.1.1.cmml" xref="S6.SS3.5.p5.6.m5.1.1.1.1">𝑖</ci><apply id="S6.SS3.5.p5.6.m5.2.2.2.2.1.cmml" xref="S6.SS3.5.p5.6.m5.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.5.p5.6.m5.2.2.2.2.1.1.cmml" xref="S6.SS3.5.p5.6.m5.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.5.p5.6.m5.2.2.2.2.1.2.cmml" xref="S6.SS3.5.p5.6.m5.2.2.2.2.1.2">𝑚</ci><ci id="S6.SS3.5.p5.6.m5.2.2.2.2.1.3.cmml" xref="S6.SS3.5.p5.6.m5.2.2.2.2.1.3">𝑖</ci></apply></list></apply><ci id="S6.SS3.5.p5.6.m5.2.3.3.3.cmml" xref="S6.SS3.5.p5.6.m5.2.3.3.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.6.m5.2c">\widetilde{\mathcal{B}}^{m_{i}}:=b^{{\rm c}}_{i,m_{i}}\mathcal{B}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.6.m5.2d">over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT := italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT caligraphic_B</annotation></semantics></math>. Then it is straightforward to verify that</p> <table class="ltx_equation ltx_eqn_table" id="S6.E6"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="{\rm ker}(\widetilde{\mathcal{B}}^{m_{i}})={\rm ker}(\mathcal{B}^{m_{i}})% \qquad\text{ for }0\leq i\leq n." class="ltx_Math" display="block" id="S6.E6.m1.1"><semantics id="S6.E6.m1.1a"><mrow id="S6.E6.m1.1.1.1"><mrow id="S6.E6.m1.1.1.1.1.2" xref="S6.E6.m1.1.1.1.1.3.cmml"><mrow id="S6.E6.m1.1.1.1.1.1.1" xref="S6.E6.m1.1.1.1.1.1.1.cmml"><mrow id="S6.E6.m1.1.1.1.1.1.1.1" xref="S6.E6.m1.1.1.1.1.1.1.1.cmml"><mi id="S6.E6.m1.1.1.1.1.1.1.1.3" xref="S6.E6.m1.1.1.1.1.1.1.1.3.cmml">ker</mi><mo id="S6.E6.m1.1.1.1.1.1.1.1.2" xref="S6.E6.m1.1.1.1.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.E6.m1.1.1.1.1.1.1.1.1.1" xref="S6.E6.m1.1.1.1.1.1.1.1.1.1.1.cmml"><mo id="S6.E6.m1.1.1.1.1.1.1.1.1.1.2" stretchy="false" 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id="S6.E6.m1.1.1.1.1.2.2.4.cmml" xref="S6.E6.m1.1.1.1.1.2.2.4">𝑖</ci></apply><apply id="S6.E6.m1.1.1.1.1.2.2c.cmml" xref="S6.E6.m1.1.1.1.1.2.2"><leq id="S6.E6.m1.1.1.1.1.2.2.5.cmml" xref="S6.E6.m1.1.1.1.1.2.2.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.E6.m1.1.1.1.1.2.2.4.cmml" id="S6.E6.m1.1.1.1.1.2.2d.cmml" xref="S6.E6.m1.1.1.1.1.2.2"></share><ci id="S6.E6.m1.1.1.1.1.2.2.6.cmml" xref="S6.E6.m1.1.1.1.1.2.2.6">𝑛</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E6.m1.1c">{\rm ker}(\widetilde{\mathcal{B}}^{m_{i}})={\rm ker}(\mathcal{B}^{m_{i}})% \qquad\text{ for }0\leq i\leq n.</annotation><annotation encoding="application/x-llamapun" id="S6.E6.m1.1d">roman_ker ( over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) = roman_ker ( caligraphic_B start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT ) for 0 ≤ italic_i ≤ italic_n .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.6)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.5.p5.10">Similar to (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.E1" title="In 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.1</span></a>) we write</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex56"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{\mathcal{B}}^{m_{i}}=\sum_{j=0}^{m_{i}}\widetilde{b}_{i,j}% \operatorname{Tr}_{j}\qquad\text{ where }\widetilde{b}_{i,j}:=b^{{\rm c}}_{i,m% _{i}}b_{i,j}." class="ltx_Math" display="block" id="S6.Ex56.m1.9"><semantics id="S6.Ex56.m1.9a"><mrow id="S6.Ex56.m1.9.9.1"><mrow id="S6.Ex56.m1.9.9.1.1.2" xref="S6.Ex56.m1.9.9.1.1.3.cmml"><mrow id="S6.Ex56.m1.9.9.1.1.1.1" xref="S6.Ex56.m1.9.9.1.1.1.1.cmml"><msup id="S6.Ex56.m1.9.9.1.1.1.1.2" xref="S6.Ex56.m1.9.9.1.1.1.1.2.cmml"><mover accent="true" id="S6.Ex56.m1.9.9.1.1.1.1.2.2" xref="S6.Ex56.m1.9.9.1.1.1.1.2.2.cmml"><mi 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xref="S6.Ex56.m1.8.8.2.2">𝑗</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex56.m1.9c">\widetilde{\mathcal{B}}^{m_{i}}=\sum_{j=0}^{m_{i}}\widetilde{b}_{i,j}% \operatorname{Tr}_{j}\qquad\text{ where }\widetilde{b}_{i,j}:=b^{{\rm c}}_{i,m% _{i}}b_{i,j}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex56.m1.9d">over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT over~ start_ARG italic_b end_ARG start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT where over~ start_ARG italic_b end_ARG start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT := italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.5.p5.8">We claim that <math alttext="\pi_{m_{i}}:=\widetilde{b}_{i,m_{i}}:\mathbb{R}^{d-1}\to\mathcal{L}(X)" class="ltx_Math" display="inline" id="S6.SS3.5.p5.7.m1.3"><semantics id="S6.SS3.5.p5.7.m1.3a"><mrow id="S6.SS3.5.p5.7.m1.3.4" xref="S6.SS3.5.p5.7.m1.3.4.cmml"><mrow id="S6.SS3.5.p5.7.m1.3.4.2" xref="S6.SS3.5.p5.7.m1.3.4.2.cmml"><msub id="S6.SS3.5.p5.7.m1.3.4.2.2" xref="S6.SS3.5.p5.7.m1.3.4.2.2.cmml"><mi id="S6.SS3.5.p5.7.m1.3.4.2.2.2" xref="S6.SS3.5.p5.7.m1.3.4.2.2.2.cmml">π</mi><msub id="S6.SS3.5.p5.7.m1.3.4.2.2.3" xref="S6.SS3.5.p5.7.m1.3.4.2.2.3.cmml"><mi id="S6.SS3.5.p5.7.m1.3.4.2.2.3.2" xref="S6.SS3.5.p5.7.m1.3.4.2.2.3.2.cmml">m</mi><mi 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xref="S6.SS3.5.p5.7.m1.3.4.3.3.2.cmml">ℒ</mi><mo id="S6.SS3.5.p5.7.m1.3.4.3.3.1" xref="S6.SS3.5.p5.7.m1.3.4.3.3.1.cmml">⁢</mo><mrow id="S6.SS3.5.p5.7.m1.3.4.3.3.3.2" xref="S6.SS3.5.p5.7.m1.3.4.3.3.cmml"><mo id="S6.SS3.5.p5.7.m1.3.4.3.3.3.2.1" stretchy="false" xref="S6.SS3.5.p5.7.m1.3.4.3.3.cmml">(</mo><mi id="S6.SS3.5.p5.7.m1.3.3" xref="S6.SS3.5.p5.7.m1.3.3.cmml">X</mi><mo id="S6.SS3.5.p5.7.m1.3.4.3.3.3.2.2" stretchy="false" xref="S6.SS3.5.p5.7.m1.3.4.3.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.7.m1.3b"><apply id="S6.SS3.5.p5.7.m1.3.4.cmml" xref="S6.SS3.5.p5.7.m1.3.4"><ci id="S6.SS3.5.p5.7.m1.3.4.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.1">:</ci><apply id="S6.SS3.5.p5.7.m1.3.4.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2"><csymbol cd="latexml" id="S6.SS3.5.p5.7.m1.3.4.2.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.1">assign</csymbol><apply id="S6.SS3.5.p5.7.m1.3.4.2.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.2"><csymbol cd="ambiguous" id="S6.SS3.5.p5.7.m1.3.4.2.2.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.2">subscript</csymbol><ci id="S6.SS3.5.p5.7.m1.3.4.2.2.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.2.2">𝜋</ci><apply id="S6.SS3.5.p5.7.m1.3.4.2.2.3.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.2.3"><csymbol cd="ambiguous" id="S6.SS3.5.p5.7.m1.3.4.2.2.3.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.2.3">subscript</csymbol><ci id="S6.SS3.5.p5.7.m1.3.4.2.2.3.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.2.3.2">𝑚</ci><ci id="S6.SS3.5.p5.7.m1.3.4.2.2.3.3.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.2.3.3">𝑖</ci></apply></apply><apply id="S6.SS3.5.p5.7.m1.3.4.2.3.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.3"><csymbol cd="ambiguous" id="S6.SS3.5.p5.7.m1.3.4.2.3.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.3">subscript</csymbol><apply id="S6.SS3.5.p5.7.m1.3.4.2.3.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.3.2"><ci id="S6.SS3.5.p5.7.m1.3.4.2.3.2.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.3.2.1">~</ci><ci id="S6.SS3.5.p5.7.m1.3.4.2.3.2.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.2.3.2.2">𝑏</ci></apply><list id="S6.SS3.5.p5.7.m1.2.2.2.3.cmml" xref="S6.SS3.5.p5.7.m1.2.2.2.2"><ci id="S6.SS3.5.p5.7.m1.1.1.1.1.cmml" xref="S6.SS3.5.p5.7.m1.1.1.1.1">𝑖</ci><apply id="S6.SS3.5.p5.7.m1.2.2.2.2.1.cmml" xref="S6.SS3.5.p5.7.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.5.p5.7.m1.2.2.2.2.1.1.cmml" xref="S6.SS3.5.p5.7.m1.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.5.p5.7.m1.2.2.2.2.1.2.cmml" xref="S6.SS3.5.p5.7.m1.2.2.2.2.1.2">𝑚</ci><ci id="S6.SS3.5.p5.7.m1.2.2.2.2.1.3.cmml" xref="S6.SS3.5.p5.7.m1.2.2.2.2.1.3">𝑖</ci></apply></list></apply></apply><apply id="S6.SS3.5.p5.7.m1.3.4.3.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3"><ci id="S6.SS3.5.p5.7.m1.3.4.3.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.1">→</ci><apply id="S6.SS3.5.p5.7.m1.3.4.3.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.2"><csymbol cd="ambiguous" id="S6.SS3.5.p5.7.m1.3.4.3.2.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.2">superscript</csymbol><ci id="S6.SS3.5.p5.7.m1.3.4.3.2.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.2.2">ℝ</ci><apply id="S6.SS3.5.p5.7.m1.3.4.3.2.3.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.2.3"><minus id="S6.SS3.5.p5.7.m1.3.4.3.2.3.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.2.3.1"></minus><ci id="S6.SS3.5.p5.7.m1.3.4.3.2.3.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.2.3.2">𝑑</ci><cn id="S6.SS3.5.p5.7.m1.3.4.3.2.3.3.cmml" type="integer" xref="S6.SS3.5.p5.7.m1.3.4.3.2.3.3">1</cn></apply></apply><apply id="S6.SS3.5.p5.7.m1.3.4.3.3.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.3"><times id="S6.SS3.5.p5.7.m1.3.4.3.3.1.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.3.1"></times><ci id="S6.SS3.5.p5.7.m1.3.4.3.3.2.cmml" xref="S6.SS3.5.p5.7.m1.3.4.3.3.2">ℒ</ci><ci id="S6.SS3.5.p5.7.m1.3.3.cmml" xref="S6.SS3.5.p5.7.m1.3.3">𝑋</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.7.m1.3c">\pi_{m_{i}}:=\widetilde{b}_{i,m_{i}}:\mathbb{R}^{d-1}\to\mathcal{L}(X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.7.m1.3d">italic_π start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT := over~ start_ARG italic_b end_ARG start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT : blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT → caligraphic_L ( italic_X )</annotation></semantics></math> is a projection. Indeed, for <math alttext="0\leq j\leq m_{i}" class="ltx_Math" display="inline" id="S6.SS3.5.p5.8.m2.1"><semantics id="S6.SS3.5.p5.8.m2.1a"><mrow id="S6.SS3.5.p5.8.m2.1.1" xref="S6.SS3.5.p5.8.m2.1.1.cmml"><mn id="S6.SS3.5.p5.8.m2.1.1.2" xref="S6.SS3.5.p5.8.m2.1.1.2.cmml">0</mn><mo id="S6.SS3.5.p5.8.m2.1.1.3" xref="S6.SS3.5.p5.8.m2.1.1.3.cmml">≤</mo><mi id="S6.SS3.5.p5.8.m2.1.1.4" xref="S6.SS3.5.p5.8.m2.1.1.4.cmml">j</mi><mo id="S6.SS3.5.p5.8.m2.1.1.5" xref="S6.SS3.5.p5.8.m2.1.1.5.cmml">≤</mo><msub id="S6.SS3.5.p5.8.m2.1.1.6" xref="S6.SS3.5.p5.8.m2.1.1.6.cmml"><mi id="S6.SS3.5.p5.8.m2.1.1.6.2" xref="S6.SS3.5.p5.8.m2.1.1.6.2.cmml">m</mi><mi id="S6.SS3.5.p5.8.m2.1.1.6.3" xref="S6.SS3.5.p5.8.m2.1.1.6.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.8.m2.1b"><apply id="S6.SS3.5.p5.8.m2.1.1.cmml" xref="S6.SS3.5.p5.8.m2.1.1"><and id="S6.SS3.5.p5.8.m2.1.1a.cmml" xref="S6.SS3.5.p5.8.m2.1.1"></and><apply id="S6.SS3.5.p5.8.m2.1.1b.cmml" xref="S6.SS3.5.p5.8.m2.1.1"><leq id="S6.SS3.5.p5.8.m2.1.1.3.cmml" xref="S6.SS3.5.p5.8.m2.1.1.3"></leq><cn id="S6.SS3.5.p5.8.m2.1.1.2.cmml" type="integer" xref="S6.SS3.5.p5.8.m2.1.1.2">0</cn><ci id="S6.SS3.5.p5.8.m2.1.1.4.cmml" xref="S6.SS3.5.p5.8.m2.1.1.4">𝑗</ci></apply><apply id="S6.SS3.5.p5.8.m2.1.1c.cmml" xref="S6.SS3.5.p5.8.m2.1.1"><leq id="S6.SS3.5.p5.8.m2.1.1.5.cmml" xref="S6.SS3.5.p5.8.m2.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.5.p5.8.m2.1.1.4.cmml" id="S6.SS3.5.p5.8.m2.1.1d.cmml" xref="S6.SS3.5.p5.8.m2.1.1"></share><apply id="S6.SS3.5.p5.8.m2.1.1.6.cmml" xref="S6.SS3.5.p5.8.m2.1.1.6"><csymbol cd="ambiguous" id="S6.SS3.5.p5.8.m2.1.1.6.1.cmml" xref="S6.SS3.5.p5.8.m2.1.1.6">subscript</csymbol><ci id="S6.SS3.5.p5.8.m2.1.1.6.2.cmml" xref="S6.SS3.5.p5.8.m2.1.1.6.2">𝑚</ci><ci id="S6.SS3.5.p5.8.m2.1.1.6.3.cmml" xref="S6.SS3.5.p5.8.m2.1.1.6.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.8.m2.1c">0\leq j\leq m_{i}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.8.m2.1d">0 ≤ italic_j ≤ italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.E7"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\pi_{m_{i}}\widetilde{b}_{i,j}=b^{{\rm c}}_{i,m_{i}}b_{i,m_{i}}b^{{\rm c}}_{i,% m_{i}}b_{i,j}=b^{{\rm c}}_{i,m_{i}}b_{i,j}=\widetilde{b}_{i,j}." class="ltx_Math" display="block" id="S6.E7.m1.17"><semantics id="S6.E7.m1.17a"><mrow id="S6.E7.m1.17.17.1" xref="S6.E7.m1.17.17.1.1.cmml"><mrow id="S6.E7.m1.17.17.1.1" xref="S6.E7.m1.17.17.1.1.cmml"><mrow id="S6.E7.m1.17.17.1.1.2" xref="S6.E7.m1.17.17.1.1.2.cmml"><msub id="S6.E7.m1.17.17.1.1.2.2" xref="S6.E7.m1.17.17.1.1.2.2.cmml"><mi id="S6.E7.m1.17.17.1.1.2.2.2" 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href="https://arxiv.org/html/2503.14636v1#S6.E7.m1.17.17.1.1.4.cmml" id="S6.E7.m1.17.17.1.1d.cmml" xref="S6.E7.m1.17.17.1"></share><apply id="S6.E7.m1.17.17.1.1.6.cmml" xref="S6.E7.m1.17.17.1.1.6"><times id="S6.E7.m1.17.17.1.1.6.1.cmml" xref="S6.E7.m1.17.17.1.1.6.1"></times><apply id="S6.E7.m1.17.17.1.1.6.2.cmml" xref="S6.E7.m1.17.17.1.1.6.2"><csymbol cd="ambiguous" id="S6.E7.m1.17.17.1.1.6.2.1.cmml" xref="S6.E7.m1.17.17.1.1.6.2">subscript</csymbol><apply id="S6.E7.m1.17.17.1.1.6.2.2.cmml" xref="S6.E7.m1.17.17.1.1.6.2"><csymbol cd="ambiguous" id="S6.E7.m1.17.17.1.1.6.2.2.1.cmml" xref="S6.E7.m1.17.17.1.1.6.2">superscript</csymbol><ci id="S6.E7.m1.17.17.1.1.6.2.2.2.cmml" xref="S6.E7.m1.17.17.1.1.6.2.2.2">𝑏</ci><ci id="S6.E7.m1.17.17.1.1.6.2.2.3.cmml" xref="S6.E7.m1.17.17.1.1.6.2.2.3">c</ci></apply><list id="S6.E7.m1.12.12.2.3.cmml" xref="S6.E7.m1.12.12.2.2"><ci id="S6.E7.m1.11.11.1.1.cmml" xref="S6.E7.m1.11.11.1.1">𝑖</ci><apply id="S6.E7.m1.12.12.2.2.1.cmml" 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id="S6.E7.m1.17.17.1.1f.cmml" xref="S6.E7.m1.17.17.1"></share><apply id="S6.E7.m1.17.17.1.1.8.cmml" xref="S6.E7.m1.17.17.1.1.8"><csymbol cd="ambiguous" id="S6.E7.m1.17.17.1.1.8.1.cmml" xref="S6.E7.m1.17.17.1.1.8">subscript</csymbol><apply id="S6.E7.m1.17.17.1.1.8.2.cmml" xref="S6.E7.m1.17.17.1.1.8.2"><ci id="S6.E7.m1.17.17.1.1.8.2.1.cmml" xref="S6.E7.m1.17.17.1.1.8.2.1">~</ci><ci id="S6.E7.m1.17.17.1.1.8.2.2.cmml" xref="S6.E7.m1.17.17.1.1.8.2.2">𝑏</ci></apply><list id="S6.E7.m1.16.16.2.3.cmml" xref="S6.E7.m1.16.16.2.4"><ci id="S6.E7.m1.15.15.1.1.cmml" xref="S6.E7.m1.15.15.1.1">𝑖</ci><ci id="S6.E7.m1.16.16.2.2.cmml" xref="S6.E7.m1.16.16.2.2">𝑗</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E7.m1.17c">\pi_{m_{i}}\widetilde{b}_{i,j}=b^{{\rm c}}_{i,m_{i}}b_{i,m_{i}}b^{{\rm c}}_{i,% m_{i}}b_{i,j}=b^{{\rm c}}_{i,m_{i}}b_{i,j}=\widetilde{b}_{i,j}.</annotation><annotation encoding="application/x-llamapun" id="S6.E7.m1.17d">italic_π start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT over~ start_ARG italic_b end_ARG start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = italic_b start_POSTSUPERSCRIPT roman_c end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_i , italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_b start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT = over~ start_ARG italic_b end_ARG start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.7)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.5.p5.9">Moreover, (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E7" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.7</span></a>) implies that for <math alttext="i\in\{0,\dots,n\}" class="ltx_Math" display="inline" id="S6.SS3.5.p5.9.m1.3"><semantics id="S6.SS3.5.p5.9.m1.3a"><mrow id="S6.SS3.5.p5.9.m1.3.4" xref="S6.SS3.5.p5.9.m1.3.4.cmml"><mi id="S6.SS3.5.p5.9.m1.3.4.2" xref="S6.SS3.5.p5.9.m1.3.4.2.cmml">i</mi><mo id="S6.SS3.5.p5.9.m1.3.4.1" xref="S6.SS3.5.p5.9.m1.3.4.1.cmml">∈</mo><mrow id="S6.SS3.5.p5.9.m1.3.4.3.2" xref="S6.SS3.5.p5.9.m1.3.4.3.1.cmml"><mo id="S6.SS3.5.p5.9.m1.3.4.3.2.1" stretchy="false" xref="S6.SS3.5.p5.9.m1.3.4.3.1.cmml">{</mo><mn id="S6.SS3.5.p5.9.m1.1.1" xref="S6.SS3.5.p5.9.m1.1.1.cmml">0</mn><mo id="S6.SS3.5.p5.9.m1.3.4.3.2.2" xref="S6.SS3.5.p5.9.m1.3.4.3.1.cmml">,</mo><mi id="S6.SS3.5.p5.9.m1.2.2" mathvariant="normal" xref="S6.SS3.5.p5.9.m1.2.2.cmml">…</mi><mo id="S6.SS3.5.p5.9.m1.3.4.3.2.3" xref="S6.SS3.5.p5.9.m1.3.4.3.1.cmml">,</mo><mi id="S6.SS3.5.p5.9.m1.3.3" xref="S6.SS3.5.p5.9.m1.3.3.cmml">n</mi><mo id="S6.SS3.5.p5.9.m1.3.4.3.2.4" stretchy="false" xref="S6.SS3.5.p5.9.m1.3.4.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.5.p5.9.m1.3b"><apply id="S6.SS3.5.p5.9.m1.3.4.cmml" xref="S6.SS3.5.p5.9.m1.3.4"><in id="S6.SS3.5.p5.9.m1.3.4.1.cmml" xref="S6.SS3.5.p5.9.m1.3.4.1"></in><ci id="S6.SS3.5.p5.9.m1.3.4.2.cmml" xref="S6.SS3.5.p5.9.m1.3.4.2">𝑖</ci><set id="S6.SS3.5.p5.9.m1.3.4.3.1.cmml" xref="S6.SS3.5.p5.9.m1.3.4.3.2"><cn id="S6.SS3.5.p5.9.m1.1.1.cmml" type="integer" xref="S6.SS3.5.p5.9.m1.1.1">0</cn><ci id="S6.SS3.5.p5.9.m1.2.2.cmml" xref="S6.SS3.5.p5.9.m1.2.2">…</ci><ci id="S6.SS3.5.p5.9.m1.3.3.cmml" xref="S6.SS3.5.p5.9.m1.3.3">𝑛</ci></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.5.p5.9.m1.3c">i\in\{0,\dots,n\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.5.p5.9.m1.3d">italic_i ∈ { 0 , … , italic_n }</annotation></semantics></math> we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.E8"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\pi_{m_{i}}\widetilde{\mathcal{B}}^{m_{i}}=\widetilde{\mathcal{B}}^{m_{i}}." class="ltx_Math" display="block" id="S6.E8.m1.1"><semantics id="S6.E8.m1.1a"><mrow id="S6.E8.m1.1.1.1" xref="S6.E8.m1.1.1.1.1.cmml"><mrow id="S6.E8.m1.1.1.1.1" xref="S6.E8.m1.1.1.1.1.cmml"><mrow id="S6.E8.m1.1.1.1.1.2" xref="S6.E8.m1.1.1.1.1.2.cmml"><msub id="S6.E8.m1.1.1.1.1.2.2" xref="S6.E8.m1.1.1.1.1.2.2.cmml"><mi id="S6.E8.m1.1.1.1.1.2.2.2" xref="S6.E8.m1.1.1.1.1.2.2.2.cmml">π</mi><msub id="S6.E8.m1.1.1.1.1.2.2.3" xref="S6.E8.m1.1.1.1.1.2.2.3.cmml"><mi id="S6.E8.m1.1.1.1.1.2.2.3.2" xref="S6.E8.m1.1.1.1.1.2.2.3.2.cmml">m</mi><mi id="S6.E8.m1.1.1.1.1.2.2.3.3" xref="S6.E8.m1.1.1.1.1.2.2.3.3.cmml">i</mi></msub></msub><mo 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xref="S6.E8.m1.1.1.1.1.3.2.1.cmml">~</mo></mover><msub id="S6.E8.m1.1.1.1.1.3.3" xref="S6.E8.m1.1.1.1.1.3.3.cmml"><mi id="S6.E8.m1.1.1.1.1.3.3.2" xref="S6.E8.m1.1.1.1.1.3.3.2.cmml">m</mi><mi id="S6.E8.m1.1.1.1.1.3.3.3" xref="S6.E8.m1.1.1.1.1.3.3.3.cmml">i</mi></msub></msup></mrow><mo id="S6.E8.m1.1.1.1.2" lspace="0em" xref="S6.E8.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.E8.m1.1b"><apply id="S6.E8.m1.1.1.1.1.cmml" xref="S6.E8.m1.1.1.1"><eq id="S6.E8.m1.1.1.1.1.1.cmml" xref="S6.E8.m1.1.1.1.1.1"></eq><apply id="S6.E8.m1.1.1.1.1.2.cmml" xref="S6.E8.m1.1.1.1.1.2"><times id="S6.E8.m1.1.1.1.1.2.1.cmml" xref="S6.E8.m1.1.1.1.1.2.1"></times><apply id="S6.E8.m1.1.1.1.1.2.2.cmml" xref="S6.E8.m1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S6.E8.m1.1.1.1.1.2.2.1.cmml" xref="S6.E8.m1.1.1.1.1.2.2">subscript</csymbol><ci id="S6.E8.m1.1.1.1.1.2.2.2.cmml" xref="S6.E8.m1.1.1.1.1.2.2.2">𝜋</ci><apply id="S6.E8.m1.1.1.1.1.2.2.3.cmml" xref="S6.E8.m1.1.1.1.1.2.2.3"><csymbol 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xref="S6.E8.m1.1.1.1.1.2.3.3.3">𝑖</ci></apply></apply></apply><apply id="S6.E8.m1.1.1.1.1.3.cmml" xref="S6.E8.m1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.E8.m1.1.1.1.1.3.1.cmml" xref="S6.E8.m1.1.1.1.1.3">superscript</csymbol><apply id="S6.E8.m1.1.1.1.1.3.2.cmml" xref="S6.E8.m1.1.1.1.1.3.2"><ci id="S6.E8.m1.1.1.1.1.3.2.1.cmml" xref="S6.E8.m1.1.1.1.1.3.2.1">~</ci><ci id="S6.E8.m1.1.1.1.1.3.2.2.cmml" xref="S6.E8.m1.1.1.1.1.3.2.2">ℬ</ci></apply><apply id="S6.E8.m1.1.1.1.1.3.3.cmml" xref="S6.E8.m1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S6.E8.m1.1.1.1.1.3.3.1.cmml" xref="S6.E8.m1.1.1.1.1.3.3">subscript</csymbol><ci id="S6.E8.m1.1.1.1.1.3.3.2.cmml" xref="S6.E8.m1.1.1.1.1.3.3.2">𝑚</ci><ci id="S6.E8.m1.1.1.1.1.3.3.3.cmml" xref="S6.E8.m1.1.1.1.1.3.3.3">𝑖</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E8.m1.1c">\pi_{m_{i}}\widetilde{\mathcal{B}}^{m_{i}}=\widetilde{\mathcal{B}}^{m_{i}}.</annotation><annotation encoding="application/x-llamapun" id="S6.E8.m1.1d">italic_π start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT = over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.8)</span></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S6.SS3.6.p6"> <p class="ltx_p" id="S6.SS3.6.p6.7"><span class="ltx_text ltx_font_italic" id="S6.SS3.6.p6.7.1">Step 2a. </span>Assume that <math alttext="k_{0}+k_{1}>m_{n}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.6.p6.1.m1.1"><semantics id="S6.SS3.6.p6.1.m1.1a"><mrow id="S6.SS3.6.p6.1.m1.1.1" xref="S6.SS3.6.p6.1.m1.1.1.cmml"><mrow id="S6.SS3.6.p6.1.m1.1.1.2" xref="S6.SS3.6.p6.1.m1.1.1.2.cmml"><msub id="S6.SS3.6.p6.1.m1.1.1.2.2" xref="S6.SS3.6.p6.1.m1.1.1.2.2.cmml"><mi id="S6.SS3.6.p6.1.m1.1.1.2.2.2" xref="S6.SS3.6.p6.1.m1.1.1.2.2.2.cmml">k</mi><mn id="S6.SS3.6.p6.1.m1.1.1.2.2.3" xref="S6.SS3.6.p6.1.m1.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.SS3.6.p6.1.m1.1.1.2.1" xref="S6.SS3.6.p6.1.m1.1.1.2.1.cmml">+</mo><msub id="S6.SS3.6.p6.1.m1.1.1.2.3" xref="S6.SS3.6.p6.1.m1.1.1.2.3.cmml"><mi id="S6.SS3.6.p6.1.m1.1.1.2.3.2" xref="S6.SS3.6.p6.1.m1.1.1.2.3.2.cmml">k</mi><mn id="S6.SS3.6.p6.1.m1.1.1.2.3.3" xref="S6.SS3.6.p6.1.m1.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.6.p6.1.m1.1.1.1" xref="S6.SS3.6.p6.1.m1.1.1.1.cmml">></mo><mrow id="S6.SS3.6.p6.1.m1.1.1.3" xref="S6.SS3.6.p6.1.m1.1.1.3.cmml"><msub id="S6.SS3.6.p6.1.m1.1.1.3.2" xref="S6.SS3.6.p6.1.m1.1.1.3.2.cmml"><mi id="S6.SS3.6.p6.1.m1.1.1.3.2.2" xref="S6.SS3.6.p6.1.m1.1.1.3.2.2.cmml">m</mi><mi id="S6.SS3.6.p6.1.m1.1.1.3.2.3" xref="S6.SS3.6.p6.1.m1.1.1.3.2.3.cmml">n</mi></msub><mo id="S6.SS3.6.p6.1.m1.1.1.3.1" xref="S6.SS3.6.p6.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.6.p6.1.m1.1.1.3.3" xref="S6.SS3.6.p6.1.m1.1.1.3.3.cmml"><mrow id="S6.SS3.6.p6.1.m1.1.1.3.3.2" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS3.6.p6.1.m1.1.1.3.3.2.2" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.6.p6.1.m1.1.1.3.3.2.1" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.6.p6.1.m1.1.1.3.3.2.3" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.6.p6.1.m1.1.1.3.3.3" xref="S6.SS3.6.p6.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.1.m1.1b"><apply id="S6.SS3.6.p6.1.m1.1.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1"><gt id="S6.SS3.6.p6.1.m1.1.1.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.1"></gt><apply id="S6.SS3.6.p6.1.m1.1.1.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2"><plus id="S6.SS3.6.p6.1.m1.1.1.2.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2.1"></plus><apply id="S6.SS3.6.p6.1.m1.1.1.2.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.1.m1.1.1.2.2.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2.2">subscript</csymbol><ci id="S6.SS3.6.p6.1.m1.1.1.2.2.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2.2.2">𝑘</ci><cn id="S6.SS3.6.p6.1.m1.1.1.2.2.3.cmml" type="integer" xref="S6.SS3.6.p6.1.m1.1.1.2.2.3">0</cn></apply><apply id="S6.SS3.6.p6.1.m1.1.1.2.3.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.1.m1.1.1.2.3.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2.3">subscript</csymbol><ci id="S6.SS3.6.p6.1.m1.1.1.2.3.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.2.3.2">𝑘</ci><cn id="S6.SS3.6.p6.1.m1.1.1.2.3.3.cmml" type="integer" xref="S6.SS3.6.p6.1.m1.1.1.2.3.3">1</cn></apply></apply><apply id="S6.SS3.6.p6.1.m1.1.1.3.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3"><plus id="S6.SS3.6.p6.1.m1.1.1.3.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.1"></plus><apply id="S6.SS3.6.p6.1.m1.1.1.3.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.1.m1.1.1.3.2.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS3.6.p6.1.m1.1.1.3.2.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.2.2">𝑚</ci><ci id="S6.SS3.6.p6.1.m1.1.1.3.2.3.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.2.3">𝑛</ci></apply><apply id="S6.SS3.6.p6.1.m1.1.1.3.3.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.3"><divide id="S6.SS3.6.p6.1.m1.1.1.3.3.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.3"></divide><apply id="S6.SS3.6.p6.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2"><plus id="S6.SS3.6.p6.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS3.6.p6.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.6.p6.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.6.p6.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.6.p6.1.m1.1.1.3.3.3.cmml" xref="S6.SS3.6.p6.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.1.m1.1c">k_{0}+k_{1}>m_{n}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.1.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Let <math alttext="q" class="ltx_Math" display="inline" id="S6.SS3.6.p6.2.m2.1"><semantics id="S6.SS3.6.p6.2.m2.1a"><mi id="S6.SS3.6.p6.2.m2.1.1" xref="S6.SS3.6.p6.2.m2.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.2.m2.1b"><ci id="S6.SS3.6.p6.2.m2.1.1.cmml" xref="S6.SS3.6.p6.2.m2.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.2.m2.1c">q</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.2.m2.1d">italic_q</annotation></semantics></math> be the largest integer such that <math alttext="q+\frac{\gamma+1}{p}<k_{0}+k_{1}" class="ltx_Math" display="inline" id="S6.SS3.6.p6.3.m3.1"><semantics id="S6.SS3.6.p6.3.m3.1a"><mrow id="S6.SS3.6.p6.3.m3.1.1" xref="S6.SS3.6.p6.3.m3.1.1.cmml"><mrow id="S6.SS3.6.p6.3.m3.1.1.2" xref="S6.SS3.6.p6.3.m3.1.1.2.cmml"><mi id="S6.SS3.6.p6.3.m3.1.1.2.2" xref="S6.SS3.6.p6.3.m3.1.1.2.2.cmml">q</mi><mo id="S6.SS3.6.p6.3.m3.1.1.2.1" xref="S6.SS3.6.p6.3.m3.1.1.2.1.cmml">+</mo><mfrac id="S6.SS3.6.p6.3.m3.1.1.2.3" xref="S6.SS3.6.p6.3.m3.1.1.2.3.cmml"><mrow id="S6.SS3.6.p6.3.m3.1.1.2.3.2" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2.cmml"><mi id="S6.SS3.6.p6.3.m3.1.1.2.3.2.2" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2.2.cmml">γ</mi><mo id="S6.SS3.6.p6.3.m3.1.1.2.3.2.1" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2.1.cmml">+</mo><mn id="S6.SS3.6.p6.3.m3.1.1.2.3.2.3" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.6.p6.3.m3.1.1.2.3.3" xref="S6.SS3.6.p6.3.m3.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS3.6.p6.3.m3.1.1.1" xref="S6.SS3.6.p6.3.m3.1.1.1.cmml"><</mo><mrow id="S6.SS3.6.p6.3.m3.1.1.3" xref="S6.SS3.6.p6.3.m3.1.1.3.cmml"><msub id="S6.SS3.6.p6.3.m3.1.1.3.2" xref="S6.SS3.6.p6.3.m3.1.1.3.2.cmml"><mi id="S6.SS3.6.p6.3.m3.1.1.3.2.2" xref="S6.SS3.6.p6.3.m3.1.1.3.2.2.cmml">k</mi><mn id="S6.SS3.6.p6.3.m3.1.1.3.2.3" xref="S6.SS3.6.p6.3.m3.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS3.6.p6.3.m3.1.1.3.1" xref="S6.SS3.6.p6.3.m3.1.1.3.1.cmml">+</mo><msub id="S6.SS3.6.p6.3.m3.1.1.3.3" xref="S6.SS3.6.p6.3.m3.1.1.3.3.cmml"><mi id="S6.SS3.6.p6.3.m3.1.1.3.3.2" xref="S6.SS3.6.p6.3.m3.1.1.3.3.2.cmml">k</mi><mn id="S6.SS3.6.p6.3.m3.1.1.3.3.3" xref="S6.SS3.6.p6.3.m3.1.1.3.3.3.cmml">1</mn></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.3.m3.1b"><apply id="S6.SS3.6.p6.3.m3.1.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1"><lt id="S6.SS3.6.p6.3.m3.1.1.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1.1"></lt><apply id="S6.SS3.6.p6.3.m3.1.1.2.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2"><plus id="S6.SS3.6.p6.3.m3.1.1.2.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.1"></plus><ci id="S6.SS3.6.p6.3.m3.1.1.2.2.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.2">𝑞</ci><apply id="S6.SS3.6.p6.3.m3.1.1.2.3.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.3"><divide id="S6.SS3.6.p6.3.m3.1.1.2.3.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.3"></divide><apply id="S6.SS3.6.p6.3.m3.1.1.2.3.2.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2"><plus id="S6.SS3.6.p6.3.m3.1.1.2.3.2.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2.1"></plus><ci id="S6.SS3.6.p6.3.m3.1.1.2.3.2.2.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2.2">𝛾</ci><cn id="S6.SS3.6.p6.3.m3.1.1.2.3.2.3.cmml" type="integer" xref="S6.SS3.6.p6.3.m3.1.1.2.3.2.3">1</cn></apply><ci id="S6.SS3.6.p6.3.m3.1.1.2.3.3.cmml" xref="S6.SS3.6.p6.3.m3.1.1.2.3.3">𝑝</ci></apply></apply><apply id="S6.SS3.6.p6.3.m3.1.1.3.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3"><plus id="S6.SS3.6.p6.3.m3.1.1.3.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3.1"></plus><apply id="S6.SS3.6.p6.3.m3.1.1.3.2.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.3.m3.1.1.3.2.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3.2">subscript</csymbol><ci id="S6.SS3.6.p6.3.m3.1.1.3.2.2.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3.2.2">𝑘</ci><cn id="S6.SS3.6.p6.3.m3.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.6.p6.3.m3.1.1.3.2.3">0</cn></apply><apply id="S6.SS3.6.p6.3.m3.1.1.3.3.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.3.m3.1.1.3.3.1.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3.3">subscript</csymbol><ci id="S6.SS3.6.p6.3.m3.1.1.3.3.2.cmml" xref="S6.SS3.6.p6.3.m3.1.1.3.3.2">𝑘</ci><cn id="S6.SS3.6.p6.3.m3.1.1.3.3.3.cmml" type="integer" xref="S6.SS3.6.p6.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.3.m3.1c">q+\frac{\gamma+1}{p}<k_{0}+k_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.3.m3.1d">italic_q + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> and define <math alttext="\overline{q}=(0,1,\dots,q)" class="ltx_Math" display="inline" id="S6.SS3.6.p6.4.m4.4"><semantics id="S6.SS3.6.p6.4.m4.4a"><mrow id="S6.SS3.6.p6.4.m4.4.5" xref="S6.SS3.6.p6.4.m4.4.5.cmml"><mover accent="true" id="S6.SS3.6.p6.4.m4.4.5.2" xref="S6.SS3.6.p6.4.m4.4.5.2.cmml"><mi id="S6.SS3.6.p6.4.m4.4.5.2.2" xref="S6.SS3.6.p6.4.m4.4.5.2.2.cmml">q</mi><mo id="S6.SS3.6.p6.4.m4.4.5.2.1" xref="S6.SS3.6.p6.4.m4.4.5.2.1.cmml">¯</mo></mover><mo id="S6.SS3.6.p6.4.m4.4.5.1" xref="S6.SS3.6.p6.4.m4.4.5.1.cmml">=</mo><mrow id="S6.SS3.6.p6.4.m4.4.5.3.2" xref="S6.SS3.6.p6.4.m4.4.5.3.1.cmml"><mo id="S6.SS3.6.p6.4.m4.4.5.3.2.1" stretchy="false" xref="S6.SS3.6.p6.4.m4.4.5.3.1.cmml">(</mo><mn id="S6.SS3.6.p6.4.m4.1.1" xref="S6.SS3.6.p6.4.m4.1.1.cmml">0</mn><mo id="S6.SS3.6.p6.4.m4.4.5.3.2.2" xref="S6.SS3.6.p6.4.m4.4.5.3.1.cmml">,</mo><mn id="S6.SS3.6.p6.4.m4.2.2" xref="S6.SS3.6.p6.4.m4.2.2.cmml">1</mn><mo id="S6.SS3.6.p6.4.m4.4.5.3.2.3" xref="S6.SS3.6.p6.4.m4.4.5.3.1.cmml">,</mo><mi id="S6.SS3.6.p6.4.m4.3.3" mathvariant="normal" xref="S6.SS3.6.p6.4.m4.3.3.cmml">…</mi><mo id="S6.SS3.6.p6.4.m4.4.5.3.2.4" xref="S6.SS3.6.p6.4.m4.4.5.3.1.cmml">,</mo><mi id="S6.SS3.6.p6.4.m4.4.4" xref="S6.SS3.6.p6.4.m4.4.4.cmml">q</mi><mo id="S6.SS3.6.p6.4.m4.4.5.3.2.5" stretchy="false" xref="S6.SS3.6.p6.4.m4.4.5.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.4.m4.4b"><apply id="S6.SS3.6.p6.4.m4.4.5.cmml" xref="S6.SS3.6.p6.4.m4.4.5"><eq id="S6.SS3.6.p6.4.m4.4.5.1.cmml" xref="S6.SS3.6.p6.4.m4.4.5.1"></eq><apply id="S6.SS3.6.p6.4.m4.4.5.2.cmml" xref="S6.SS3.6.p6.4.m4.4.5.2"><ci id="S6.SS3.6.p6.4.m4.4.5.2.1.cmml" xref="S6.SS3.6.p6.4.m4.4.5.2.1">¯</ci><ci id="S6.SS3.6.p6.4.m4.4.5.2.2.cmml" xref="S6.SS3.6.p6.4.m4.4.5.2.2">𝑞</ci></apply><vector id="S6.SS3.6.p6.4.m4.4.5.3.1.cmml" xref="S6.SS3.6.p6.4.m4.4.5.3.2"><cn id="S6.SS3.6.p6.4.m4.1.1.cmml" type="integer" xref="S6.SS3.6.p6.4.m4.1.1">0</cn><cn id="S6.SS3.6.p6.4.m4.2.2.cmml" type="integer" xref="S6.SS3.6.p6.4.m4.2.2">1</cn><ci id="S6.SS3.6.p6.4.m4.3.3.cmml" xref="S6.SS3.6.p6.4.m4.3.3">…</ci><ci id="S6.SS3.6.p6.4.m4.4.4.cmml" xref="S6.SS3.6.p6.4.m4.4.4">𝑞</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.4.m4.4c">\overline{q}=(0,1,\dots,q)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.4.m4.4d">over¯ start_ARG italic_q end_ARG = ( 0 , 1 , … , italic_q )</annotation></semantics></math>. Moreover, by setting <math alttext="\overline{X}=(X,\dots,X)" class="ltx_Math" display="inline" id="S6.SS3.6.p6.5.m5.3"><semantics id="S6.SS3.6.p6.5.m5.3a"><mrow id="S6.SS3.6.p6.5.m5.3.4" xref="S6.SS3.6.p6.5.m5.3.4.cmml"><mover accent="true" id="S6.SS3.6.p6.5.m5.3.4.2" xref="S6.SS3.6.p6.5.m5.3.4.2.cmml"><mi id="S6.SS3.6.p6.5.m5.3.4.2.2" xref="S6.SS3.6.p6.5.m5.3.4.2.2.cmml">X</mi><mo id="S6.SS3.6.p6.5.m5.3.4.2.1" xref="S6.SS3.6.p6.5.m5.3.4.2.1.cmml">¯</mo></mover><mo id="S6.SS3.6.p6.5.m5.3.4.1" xref="S6.SS3.6.p6.5.m5.3.4.1.cmml">=</mo><mrow id="S6.SS3.6.p6.5.m5.3.4.3.2" xref="S6.SS3.6.p6.5.m5.3.4.3.1.cmml"><mo id="S6.SS3.6.p6.5.m5.3.4.3.2.1" stretchy="false" xref="S6.SS3.6.p6.5.m5.3.4.3.1.cmml">(</mo><mi id="S6.SS3.6.p6.5.m5.1.1" xref="S6.SS3.6.p6.5.m5.1.1.cmml">X</mi><mo id="S6.SS3.6.p6.5.m5.3.4.3.2.2" xref="S6.SS3.6.p6.5.m5.3.4.3.1.cmml">,</mo><mi id="S6.SS3.6.p6.5.m5.2.2" mathvariant="normal" xref="S6.SS3.6.p6.5.m5.2.2.cmml">…</mi><mo id="S6.SS3.6.p6.5.m5.3.4.3.2.3" xref="S6.SS3.6.p6.5.m5.3.4.3.1.cmml">,</mo><mi id="S6.SS3.6.p6.5.m5.3.3" xref="S6.SS3.6.p6.5.m5.3.3.cmml">X</mi><mo id="S6.SS3.6.p6.5.m5.3.4.3.2.4" stretchy="false" xref="S6.SS3.6.p6.5.m5.3.4.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.5.m5.3b"><apply id="S6.SS3.6.p6.5.m5.3.4.cmml" xref="S6.SS3.6.p6.5.m5.3.4"><eq id="S6.SS3.6.p6.5.m5.3.4.1.cmml" xref="S6.SS3.6.p6.5.m5.3.4.1"></eq><apply id="S6.SS3.6.p6.5.m5.3.4.2.cmml" xref="S6.SS3.6.p6.5.m5.3.4.2"><ci id="S6.SS3.6.p6.5.m5.3.4.2.1.cmml" xref="S6.SS3.6.p6.5.m5.3.4.2.1">¯</ci><ci id="S6.SS3.6.p6.5.m5.3.4.2.2.cmml" xref="S6.SS3.6.p6.5.m5.3.4.2.2">𝑋</ci></apply><vector id="S6.SS3.6.p6.5.m5.3.4.3.1.cmml" xref="S6.SS3.6.p6.5.m5.3.4.3.2"><ci id="S6.SS3.6.p6.5.m5.1.1.cmml" xref="S6.SS3.6.p6.5.m5.1.1">𝑋</ci><ci id="S6.SS3.6.p6.5.m5.2.2.cmml" xref="S6.SS3.6.p6.5.m5.2.2">…</ci><ci id="S6.SS3.6.p6.5.m5.3.3.cmml" xref="S6.SS3.6.p6.5.m5.3.3">𝑋</ci></vector></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.5.m5.3c">\overline{X}=(X,\dots,X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.5.m5.3d">over¯ start_ARG italic_X end_ARG = ( italic_X , … , italic_X )</annotation></semantics></math> with length <math alttext="q+1" class="ltx_Math" display="inline" id="S6.SS3.6.p6.6.m6.1"><semantics id="S6.SS3.6.p6.6.m6.1a"><mrow id="S6.SS3.6.p6.6.m6.1.1" xref="S6.SS3.6.p6.6.m6.1.1.cmml"><mi id="S6.SS3.6.p6.6.m6.1.1.2" xref="S6.SS3.6.p6.6.m6.1.1.2.cmml">q</mi><mo id="S6.SS3.6.p6.6.m6.1.1.1" xref="S6.SS3.6.p6.6.m6.1.1.1.cmml">+</mo><mn id="S6.SS3.6.p6.6.m6.1.1.3" xref="S6.SS3.6.p6.6.m6.1.1.3.cmml">1</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.6.m6.1b"><apply id="S6.SS3.6.p6.6.m6.1.1.cmml" xref="S6.SS3.6.p6.6.m6.1.1"><plus id="S6.SS3.6.p6.6.m6.1.1.1.cmml" xref="S6.SS3.6.p6.6.m6.1.1.1"></plus><ci id="S6.SS3.6.p6.6.m6.1.1.2.cmml" xref="S6.SS3.6.p6.6.m6.1.1.2">𝑞</ci><cn id="S6.SS3.6.p6.6.m6.1.1.3.cmml" type="integer" xref="S6.SS3.6.p6.6.m6.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.6.m6.1c">q+1</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.6.m6.1d">italic_q + 1</annotation></semantics></math>, we can define a normal boundary operator of type <math alttext="(p,k_{0}+k_{1},\gamma,\overline{q},\overline{X})" class="ltx_Math" display="inline" id="S6.SS3.6.p6.7.m7.5"><semantics id="S6.SS3.6.p6.7.m7.5a"><mrow id="S6.SS3.6.p6.7.m7.5.5.1" xref="S6.SS3.6.p6.7.m7.5.5.2.cmml"><mo id="S6.SS3.6.p6.7.m7.5.5.1.2" stretchy="false" xref="S6.SS3.6.p6.7.m7.5.5.2.cmml">(</mo><mi id="S6.SS3.6.p6.7.m7.1.1" xref="S6.SS3.6.p6.7.m7.1.1.cmml">p</mi><mo id="S6.SS3.6.p6.7.m7.5.5.1.3" xref="S6.SS3.6.p6.7.m7.5.5.2.cmml">,</mo><mrow id="S6.SS3.6.p6.7.m7.5.5.1.1" xref="S6.SS3.6.p6.7.m7.5.5.1.1.cmml"><msub id="S6.SS3.6.p6.7.m7.5.5.1.1.2" xref="S6.SS3.6.p6.7.m7.5.5.1.1.2.cmml"><mi id="S6.SS3.6.p6.7.m7.5.5.1.1.2.2" xref="S6.SS3.6.p6.7.m7.5.5.1.1.2.2.cmml">k</mi><mn id="S6.SS3.6.p6.7.m7.5.5.1.1.2.3" xref="S6.SS3.6.p6.7.m7.5.5.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.6.p6.7.m7.5.5.1.1.1" xref="S6.SS3.6.p6.7.m7.5.5.1.1.1.cmml">+</mo><msub id="S6.SS3.6.p6.7.m7.5.5.1.1.3" xref="S6.SS3.6.p6.7.m7.5.5.1.1.3.cmml"><mi id="S6.SS3.6.p6.7.m7.5.5.1.1.3.2" xref="S6.SS3.6.p6.7.m7.5.5.1.1.3.2.cmml">k</mi><mn id="S6.SS3.6.p6.7.m7.5.5.1.1.3.3" xref="S6.SS3.6.p6.7.m7.5.5.1.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.6.p6.7.m7.5.5.1.4" xref="S6.SS3.6.p6.7.m7.5.5.2.cmml">,</mo><mi id="S6.SS3.6.p6.7.m7.2.2" xref="S6.SS3.6.p6.7.m7.2.2.cmml">γ</mi><mo id="S6.SS3.6.p6.7.m7.5.5.1.5" xref="S6.SS3.6.p6.7.m7.5.5.2.cmml">,</mo><mover accent="true" id="S6.SS3.6.p6.7.m7.3.3" xref="S6.SS3.6.p6.7.m7.3.3.cmml"><mi id="S6.SS3.6.p6.7.m7.3.3.2" xref="S6.SS3.6.p6.7.m7.3.3.2.cmml">q</mi><mo id="S6.SS3.6.p6.7.m7.3.3.1" xref="S6.SS3.6.p6.7.m7.3.3.1.cmml">¯</mo></mover><mo id="S6.SS3.6.p6.7.m7.5.5.1.6" xref="S6.SS3.6.p6.7.m7.5.5.2.cmml">,</mo><mover accent="true" id="S6.SS3.6.p6.7.m7.4.4" xref="S6.SS3.6.p6.7.m7.4.4.cmml"><mi id="S6.SS3.6.p6.7.m7.4.4.2" xref="S6.SS3.6.p6.7.m7.4.4.2.cmml">X</mi><mo id="S6.SS3.6.p6.7.m7.4.4.1" xref="S6.SS3.6.p6.7.m7.4.4.1.cmml">¯</mo></mover><mo id="S6.SS3.6.p6.7.m7.5.5.1.7" stretchy="false" xref="S6.SS3.6.p6.7.m7.5.5.2.cmml">)</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.7.m7.5b"><vector id="S6.SS3.6.p6.7.m7.5.5.2.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1"><ci id="S6.SS3.6.p6.7.m7.1.1.cmml" xref="S6.SS3.6.p6.7.m7.1.1">𝑝</ci><apply id="S6.SS3.6.p6.7.m7.5.5.1.1.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1"><plus id="S6.SS3.6.p6.7.m7.5.5.1.1.1.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1.1"></plus><apply id="S6.SS3.6.p6.7.m7.5.5.1.1.2.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.7.m7.5.5.1.1.2.1.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1.2">subscript</csymbol><ci id="S6.SS3.6.p6.7.m7.5.5.1.1.2.2.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1.2.2">𝑘</ci><cn id="S6.SS3.6.p6.7.m7.5.5.1.1.2.3.cmml" type="integer" xref="S6.SS3.6.p6.7.m7.5.5.1.1.2.3">0</cn></apply><apply id="S6.SS3.6.p6.7.m7.5.5.1.1.3.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.7.m7.5.5.1.1.3.1.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1.3">subscript</csymbol><ci id="S6.SS3.6.p6.7.m7.5.5.1.1.3.2.cmml" xref="S6.SS3.6.p6.7.m7.5.5.1.1.3.2">𝑘</ci><cn id="S6.SS3.6.p6.7.m7.5.5.1.1.3.3.cmml" type="integer" xref="S6.SS3.6.p6.7.m7.5.5.1.1.3.3">1</cn></apply></apply><ci id="S6.SS3.6.p6.7.m7.2.2.cmml" xref="S6.SS3.6.p6.7.m7.2.2">𝛾</ci><apply id="S6.SS3.6.p6.7.m7.3.3.cmml" xref="S6.SS3.6.p6.7.m7.3.3"><ci id="S6.SS3.6.p6.7.m7.3.3.1.cmml" xref="S6.SS3.6.p6.7.m7.3.3.1">¯</ci><ci id="S6.SS3.6.p6.7.m7.3.3.2.cmml" xref="S6.SS3.6.p6.7.m7.3.3.2">𝑞</ci></apply><apply id="S6.SS3.6.p6.7.m7.4.4.cmml" xref="S6.SS3.6.p6.7.m7.4.4"><ci id="S6.SS3.6.p6.7.m7.4.4.1.cmml" xref="S6.SS3.6.p6.7.m7.4.4.1">¯</ci><ci id="S6.SS3.6.p6.7.m7.4.4.2.cmml" xref="S6.SS3.6.p6.7.m7.4.4.2">𝑋</ci></apply></vector></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.7.m7.5c">(p,k_{0}+k_{1},\gamma,\overline{q},\overline{X})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.7.m7.5d">( italic_p , italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_γ , over¯ start_ARG italic_q end_ARG , over¯ start_ARG italic_X end_ARG )</annotation></semantics></math> by</p> <table class="ltx_equationgroup ltx_eqn_table" id="S6.E9"> <tbody> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S6.E9X"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{C}" class="ltx_Math" display="inline" id="S6.E9X.2.1.1.m1.1"><semantics id="S6.E9X.2.1.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S6.E9X.2.1.1.m1.1.1" xref="S6.E9X.2.1.1.m1.1.1.cmml">𝒞</mi><annotation-xml encoding="MathML-Content" id="S6.E9X.2.1.1.m1.1b"><ci id="S6.E9X.2.1.1.m1.1.1.cmml" xref="S6.E9X.2.1.1.m1.1.1">𝒞</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.E9X.2.1.1.m1.1c">\displaystyle\mathcal{C}</annotation><annotation encoding="application/x-llamapun" id="S6.E9X.2.1.1.m1.1d">caligraphic_C</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=(\mathcal{C}^{0},\dots,\mathcal{C}^{q}):W^{k_{0}+k_{1},p}\to% \prod_{j=0}^{q}B^{k_{0}+k_{1}-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)," class="ltx_Math" display="inline" id="S6.E9X.3.2.2.m1.7"><semantics id="S6.E9X.3.2.2.m1.7a"><mrow id="S6.E9X.3.2.2.m1.7.7.1" xref="S6.E9X.3.2.2.m1.7.7.1.1.cmml"><mrow id="S6.E9X.3.2.2.m1.7.7.1.1" xref="S6.E9X.3.2.2.m1.7.7.1.1.cmml"><mrow id="S6.E9X.3.2.2.m1.7.7.1.1.2" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.cmml"><mi id="S6.E9X.3.2.2.m1.7.7.1.1.2.4" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.4.cmml"></mi><mo id="S6.E9X.3.2.2.m1.7.7.1.1.2.3" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.3.cmml">=</mo><mrow id="S6.E9X.3.2.2.m1.7.7.1.1.2.2.2" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.2.3.cmml"><mo id="S6.E9X.3.2.2.m1.7.7.1.1.2.2.2.3" stretchy="false" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.2.3.cmml">(</mo><msup id="S6.E9X.3.2.2.m1.7.7.1.1.1.1.1.1" xref="S6.E9X.3.2.2.m1.7.7.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.E9X.3.2.2.m1.7.7.1.1.1.1.1.1.2" xref="S6.E9X.3.2.2.m1.7.7.1.1.1.1.1.1.2.cmml">𝒞</mi><mn id="S6.E9X.3.2.2.m1.7.7.1.1.1.1.1.1.3" xref="S6.E9X.3.2.2.m1.7.7.1.1.1.1.1.1.3.cmml">0</mn></msup><mo id="S6.E9X.3.2.2.m1.7.7.1.1.2.2.2.4" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.2.3.cmml">,</mo><mi id="S6.E9X.3.2.2.m1.5.5" mathvariant="normal" xref="S6.E9X.3.2.2.m1.5.5.cmml">…</mi><mo id="S6.E9X.3.2.2.m1.7.7.1.1.2.2.2.5" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.2.3.cmml">,</mo><msup id="S6.E9X.3.2.2.m1.7.7.1.1.2.2.2.2" xref="S6.E9X.3.2.2.m1.7.7.1.1.2.2.2.2.cmml"><mi 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xref="S6.E9X.3.2.2.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.E9X.3.2.2.m1.2.2.2.2.1.2.3" xref="S6.E9X.3.2.2.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.E9X.3.2.2.m1.2.2.2.2.1.1" xref="S6.E9X.3.2.2.m1.2.2.2.2.1.1.cmml">+</mo><msub id="S6.E9X.3.2.2.m1.2.2.2.2.1.3" xref="S6.E9X.3.2.2.m1.2.2.2.2.1.3.cmml"><mi id="S6.E9X.3.2.2.m1.2.2.2.2.1.3.2" xref="S6.E9X.3.2.2.m1.2.2.2.2.1.3.2.cmml">k</mi><mn id="S6.E9X.3.2.2.m1.2.2.2.2.1.3.3" xref="S6.E9X.3.2.2.m1.2.2.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.E9X.3.2.2.m1.2.2.2.2.2" xref="S6.E9X.3.2.2.m1.2.2.2.3.cmml">,</mo><mi id="S6.E9X.3.2.2.m1.1.1.1.1" xref="S6.E9X.3.2.2.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.E9X.3.2.2.m1.7.7.1.1.3.2" stretchy="false" xref="S6.E9X.3.2.2.m1.7.7.1.1.3.2.cmml">→</mo><mrow id="S6.E9X.3.2.2.m1.7.7.1.1.3.1" xref="S6.E9X.3.2.2.m1.7.7.1.1.3.1.cmml"><mstyle displaystyle="true" id="S6.E9X.3.2.2.m1.7.7.1.1.3.1.2" xref="S6.E9X.3.2.2.m1.7.7.1.1.3.1.2.cmml"><munderover id="S6.E9X.3.2.2.m1.7.7.1.1.3.1.2a" 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0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="2"><span class="ltx_tag ltx_tag_equationgroup ltx_align_right">(6.9)</span></td> </tr> <tr class="ltx_equation ltx_eqn_row ltx_align_baseline" id="S6.E9Xa"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{C}^{j}" class="ltx_Math" display="inline" id="S6.E9Xa.2.1.1.m1.1"><semantics 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xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.1.1.1"><csymbol cd="ambiguous" id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.1.1.1.1.cmml" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.1.1.1">subscript</csymbol><ci id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.1.1.1.2.cmml" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.1.1.1.2">𝑚</ci><cn id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.1.1.1.3.cmml" type="integer" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.1.1.1.3">0</cn></apply><ci id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.1.cmml" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.1">…</ci><apply id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2.cmml" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2"><csymbol cd="ambiguous" id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2.1.cmml" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2">subscript</csymbol><ci id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2.2.cmml" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2.2">𝑚</ci><ci id="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2.3.cmml" xref="S6.E9.m1.4.4.4.4.4.4.4.4.2.1.mf.2.1.2.2.2.3">𝑛</ci></apply></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E9Xa.3.2.2.m1.1c">\displaystyle:=\begin{cases}\operatorname{Tr}_{j}&\mbox{if }j\notin\{m_{0},% \dots,m_{n}\},\\ (1-\pi_{j})\operatorname{Tr}_{j}+\widetilde{B}^{j}&\mbox{if }j\in\{m_{0},\dots% ,m_{n}\}.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S6.E9Xa.3.2.2.m1.1d">:= { start_ROW start_CELL roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_CELL start_CELL if italic_j ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , end_CELL end_ROW start_ROW start_CELL ( 1 - italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT + over~ start_ARG italic_B end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT end_CELL start_CELL if italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr> </tbody> </table> <p class="ltx_p" id="S6.SS3.6.p6.17">Note that by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E8" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.8</span></a>) we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.E10"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\pi_{j}\mathcal{C}^{j}=\pi_{j}\widetilde{\mathcal{B}}^{j}=\widetilde{\mathcal{% B}}^{j}\qquad\text{ for }j\in\{m_{0},\dots,m_{n}\}." class="ltx_Math" display="block" id="S6.E10.m1.2"><semantics id="S6.E10.m1.2a"><mrow id="S6.E10.m1.2.2.1"><mrow id="S6.E10.m1.2.2.1.1.2" xref="S6.E10.m1.2.2.1.1.3.cmml"><mrow id="S6.E10.m1.2.2.1.1.1.1" xref="S6.E10.m1.2.2.1.1.1.1.cmml"><mrow id="S6.E10.m1.2.2.1.1.1.1.2" xref="S6.E10.m1.2.2.1.1.1.1.2.cmml"><msub id="S6.E10.m1.2.2.1.1.1.1.2.2" xref="S6.E10.m1.2.2.1.1.1.1.2.2.cmml"><mi id="S6.E10.m1.2.2.1.1.1.1.2.2.2" xref="S6.E10.m1.2.2.1.1.1.1.2.2.2.cmml">π</mi><mi 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end_POSTSUPERSCRIPT = italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT = over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT for italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.10)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.6.p6.18">We claim that</p> <table class="ltx_equation ltx_eqn_table" id="S6.E11"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{C}v=0\quad\text{if and only if}\quad\overline{\operatorname{Tr}}_{q}v% =0\qquad\text{ for }v\in W^{k_{0}+k_{1},p}." class="ltx_Math" display="block" id="S6.E11.m1.5"><semantics id="S6.E11.m1.5a"><mrow id="S6.E11.m1.5.5.1"><mrow id="S6.E11.m1.5.5.1.1.2" xref="S6.E11.m1.5.5.1.1.3.cmml"><mrow id="S6.E11.m1.5.5.1.1.1.1" xref="S6.E11.m1.5.5.1.1.1.1.cmml"><mrow id="S6.E11.m1.5.5.1.1.1.1.2" xref="S6.E11.m1.5.5.1.1.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.E11.m1.5.5.1.1.1.1.2.2" xref="S6.E11.m1.5.5.1.1.1.1.2.2.cmml">𝒞</mi><mo id="S6.E11.m1.5.5.1.1.1.1.2.1" xref="S6.E11.m1.5.5.1.1.1.1.2.1.cmml">⁢</mo><mi id="S6.E11.m1.5.5.1.1.1.1.2.3" xref="S6.E11.m1.5.5.1.1.1.1.2.3.cmml">v</mi></mrow><mo id="S6.E11.m1.5.5.1.1.1.1.1" xref="S6.E11.m1.5.5.1.1.1.1.1.cmml">=</mo><mrow id="S6.E11.m1.5.5.1.1.1.1.3.2" xref="S6.E11.m1.5.5.1.1.1.1.3.1.cmml"><mn id="S6.E11.m1.3.3" xref="S6.E11.m1.3.3.cmml">0</mn><mspace id="S6.E11.m1.5.5.1.1.1.1.3.2.1" width="1em" xref="S6.E11.m1.5.5.1.1.1.1.3.1.cmml"></mspace><mtext id="S6.E11.m1.4.4" xref="S6.E11.m1.4.4a.cmml">if and only if</mtext></mrow></mrow><mspace id="S6.E11.m1.5.5.1.1.2.3" width="1.167em" xref="S6.E11.m1.5.5.1.1.3a.cmml"></mspace><mrow id="S6.E11.m1.5.5.1.1.2.2.2" xref="S6.E11.m1.5.5.1.1.2.2.3.cmml"><mrow id="S6.E11.m1.5.5.1.1.2.2.1.1" xref="S6.E11.m1.5.5.1.1.2.2.1.1.cmml"><mrow id="S6.E11.m1.5.5.1.1.2.2.1.1.2" xref="S6.E11.m1.5.5.1.1.2.2.1.1.2.cmml"><msub id="S6.E11.m1.5.5.1.1.2.2.1.1.2.2" xref="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.cmml"><mover accent="true" id="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.2" xref="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.2.cmml"><mi id="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.2.2" xref="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.2.2.cmml">Tr</mi><mo id="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.2.1" xref="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.2.1.cmml">¯</mo></mover><mi id="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.3" xref="S6.E11.m1.5.5.1.1.2.2.1.1.2.2.3.cmml">q</mi></msub><mo id="S6.E11.m1.5.5.1.1.2.2.1.1.2.1" lspace="0.167em" xref="S6.E11.m1.5.5.1.1.2.2.1.1.2.1.cmml">⁢</mo><mi 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id="S6.E11.m1.2.2.2.2.1.3.3.cmml" type="integer" xref="S6.E11.m1.2.2.2.2.1.3.3">1</cn></apply></apply><ci id="S6.E11.m1.1.1.1.1.cmml" xref="S6.E11.m1.1.1.1.1">𝑝</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E11.m1.5c">\mathcal{C}v=0\quad\text{if and only if}\quad\overline{\operatorname{Tr}}_{q}v% =0\qquad\text{ for }v\in W^{k_{0}+k_{1},p}.</annotation><annotation encoding="application/x-llamapun" id="S6.E11.m1.5d">caligraphic_C italic_v = 0 if and only if over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT italic_v = 0 for italic_v ∈ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.11)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.6.p6.11">Indeed, let <math alttext="v\in W^{k_{0}+k_{1},p}" class="ltx_Math" display="inline" id="S6.SS3.6.p6.8.m1.2"><semantics id="S6.SS3.6.p6.8.m1.2a"><mrow id="S6.SS3.6.p6.8.m1.2.3" xref="S6.SS3.6.p6.8.m1.2.3.cmml"><mi id="S6.SS3.6.p6.8.m1.2.3.2" xref="S6.SS3.6.p6.8.m1.2.3.2.cmml">v</mi><mo id="S6.SS3.6.p6.8.m1.2.3.1" xref="S6.SS3.6.p6.8.m1.2.3.1.cmml">∈</mo><msup id="S6.SS3.6.p6.8.m1.2.3.3" xref="S6.SS3.6.p6.8.m1.2.3.3.cmml"><mi id="S6.SS3.6.p6.8.m1.2.3.3.2" xref="S6.SS3.6.p6.8.m1.2.3.3.2.cmml">W</mi><mrow id="S6.SS3.6.p6.8.m1.2.2.2.2" xref="S6.SS3.6.p6.8.m1.2.2.2.3.cmml"><mrow id="S6.SS3.6.p6.8.m1.2.2.2.2.1" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.cmml"><msub id="S6.SS3.6.p6.8.m1.2.2.2.2.1.2" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.cmml"><mi id="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.2" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.3" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.6.p6.8.m1.2.2.2.2.1.1" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.1.cmml">+</mo><msub id="S6.SS3.6.p6.8.m1.2.2.2.2.1.3" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.cmml"><mi id="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.2" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.2.cmml">k</mi><mn id="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.3" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.6.p6.8.m1.2.2.2.2.2" xref="S6.SS3.6.p6.8.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS3.6.p6.8.m1.1.1.1.1" xref="S6.SS3.6.p6.8.m1.1.1.1.1.cmml">p</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.8.m1.2b"><apply id="S6.SS3.6.p6.8.m1.2.3.cmml" xref="S6.SS3.6.p6.8.m1.2.3"><in id="S6.SS3.6.p6.8.m1.2.3.1.cmml" xref="S6.SS3.6.p6.8.m1.2.3.1"></in><ci id="S6.SS3.6.p6.8.m1.2.3.2.cmml" xref="S6.SS3.6.p6.8.m1.2.3.2">𝑣</ci><apply id="S6.SS3.6.p6.8.m1.2.3.3.cmml" xref="S6.SS3.6.p6.8.m1.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.8.m1.2.3.3.1.cmml" xref="S6.SS3.6.p6.8.m1.2.3.3">superscript</csymbol><ci id="S6.SS3.6.p6.8.m1.2.3.3.2.cmml" xref="S6.SS3.6.p6.8.m1.2.3.3.2">𝑊</ci><list id="S6.SS3.6.p6.8.m1.2.2.2.3.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2"><apply id="S6.SS3.6.p6.8.m1.2.2.2.2.1.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1"><plus id="S6.SS3.6.p6.8.m1.2.2.2.2.1.1.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.1"></plus><apply id="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.1.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.2.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.2.3">0</cn></apply><apply id="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.1.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.3">subscript</csymbol><ci id="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.2.cmml" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.2">𝑘</ci><cn id="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.3.cmml" type="integer" xref="S6.SS3.6.p6.8.m1.2.2.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.6.p6.8.m1.1.1.1.1.cmml" xref="S6.SS3.6.p6.8.m1.1.1.1.1">𝑝</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.8.m1.2c">v\in W^{k_{0}+k_{1},p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.8.m1.2d">italic_v ∈ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> and <math alttext="\mathcal{C}v=0" class="ltx_Math" display="inline" id="S6.SS3.6.p6.9.m2.1"><semantics id="S6.SS3.6.p6.9.m2.1a"><mrow id="S6.SS3.6.p6.9.m2.1.1" xref="S6.SS3.6.p6.9.m2.1.1.cmml"><mrow id="S6.SS3.6.p6.9.m2.1.1.2" xref="S6.SS3.6.p6.9.m2.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.6.p6.9.m2.1.1.2.2" xref="S6.SS3.6.p6.9.m2.1.1.2.2.cmml">𝒞</mi><mo id="S6.SS3.6.p6.9.m2.1.1.2.1" xref="S6.SS3.6.p6.9.m2.1.1.2.1.cmml">⁢</mo><mi id="S6.SS3.6.p6.9.m2.1.1.2.3" xref="S6.SS3.6.p6.9.m2.1.1.2.3.cmml">v</mi></mrow><mo id="S6.SS3.6.p6.9.m2.1.1.1" xref="S6.SS3.6.p6.9.m2.1.1.1.cmml">=</mo><mn id="S6.SS3.6.p6.9.m2.1.1.3" xref="S6.SS3.6.p6.9.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.9.m2.1b"><apply id="S6.SS3.6.p6.9.m2.1.1.cmml" xref="S6.SS3.6.p6.9.m2.1.1"><eq id="S6.SS3.6.p6.9.m2.1.1.1.cmml" xref="S6.SS3.6.p6.9.m2.1.1.1"></eq><apply id="S6.SS3.6.p6.9.m2.1.1.2.cmml" xref="S6.SS3.6.p6.9.m2.1.1.2"><times id="S6.SS3.6.p6.9.m2.1.1.2.1.cmml" xref="S6.SS3.6.p6.9.m2.1.1.2.1"></times><ci id="S6.SS3.6.p6.9.m2.1.1.2.2.cmml" xref="S6.SS3.6.p6.9.m2.1.1.2.2">𝒞</ci><ci id="S6.SS3.6.p6.9.m2.1.1.2.3.cmml" xref="S6.SS3.6.p6.9.m2.1.1.2.3">𝑣</ci></apply><cn id="S6.SS3.6.p6.9.m2.1.1.3.cmml" type="integer" xref="S6.SS3.6.p6.9.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.9.m2.1c">\mathcal{C}v=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.9.m2.1d">caligraphic_C italic_v = 0</annotation></semantics></math>, then for <math alttext="0\leq\widetilde{q}\leq q" class="ltx_Math" display="inline" id="S6.SS3.6.p6.10.m3.1"><semantics id="S6.SS3.6.p6.10.m3.1a"><mrow id="S6.SS3.6.p6.10.m3.1.1" xref="S6.SS3.6.p6.10.m3.1.1.cmml"><mn id="S6.SS3.6.p6.10.m3.1.1.2" xref="S6.SS3.6.p6.10.m3.1.1.2.cmml">0</mn><mo id="S6.SS3.6.p6.10.m3.1.1.3" xref="S6.SS3.6.p6.10.m3.1.1.3.cmml">≤</mo><mover accent="true" id="S6.SS3.6.p6.10.m3.1.1.4" xref="S6.SS3.6.p6.10.m3.1.1.4.cmml"><mi id="S6.SS3.6.p6.10.m3.1.1.4.2" xref="S6.SS3.6.p6.10.m3.1.1.4.2.cmml">q</mi><mo id="S6.SS3.6.p6.10.m3.1.1.4.1" xref="S6.SS3.6.p6.10.m3.1.1.4.1.cmml">~</mo></mover><mo id="S6.SS3.6.p6.10.m3.1.1.5" xref="S6.SS3.6.p6.10.m3.1.1.5.cmml">≤</mo><mi id="S6.SS3.6.p6.10.m3.1.1.6" xref="S6.SS3.6.p6.10.m3.1.1.6.cmml">q</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.10.m3.1b"><apply id="S6.SS3.6.p6.10.m3.1.1.cmml" xref="S6.SS3.6.p6.10.m3.1.1"><and id="S6.SS3.6.p6.10.m3.1.1a.cmml" xref="S6.SS3.6.p6.10.m3.1.1"></and><apply id="S6.SS3.6.p6.10.m3.1.1b.cmml" xref="S6.SS3.6.p6.10.m3.1.1"><leq id="S6.SS3.6.p6.10.m3.1.1.3.cmml" xref="S6.SS3.6.p6.10.m3.1.1.3"></leq><cn id="S6.SS3.6.p6.10.m3.1.1.2.cmml" type="integer" xref="S6.SS3.6.p6.10.m3.1.1.2">0</cn><apply id="S6.SS3.6.p6.10.m3.1.1.4.cmml" xref="S6.SS3.6.p6.10.m3.1.1.4"><ci id="S6.SS3.6.p6.10.m3.1.1.4.1.cmml" xref="S6.SS3.6.p6.10.m3.1.1.4.1">~</ci><ci id="S6.SS3.6.p6.10.m3.1.1.4.2.cmml" xref="S6.SS3.6.p6.10.m3.1.1.4.2">𝑞</ci></apply></apply><apply id="S6.SS3.6.p6.10.m3.1.1c.cmml" xref="S6.SS3.6.p6.10.m3.1.1"><leq id="S6.SS3.6.p6.10.m3.1.1.5.cmml" xref="S6.SS3.6.p6.10.m3.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.6.p6.10.m3.1.1.4.cmml" id="S6.SS3.6.p6.10.m3.1.1d.cmml" xref="S6.SS3.6.p6.10.m3.1.1"></share><ci id="S6.SS3.6.p6.10.m3.1.1.6.cmml" xref="S6.SS3.6.p6.10.m3.1.1.6">𝑞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.10.m3.1c">0\leq\widetilde{q}\leq q</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.10.m3.1d">0 ≤ over~ start_ARG italic_q end_ARG ≤ italic_q</annotation></semantics></math> with <math alttext="\widetilde{q}\notin\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S6.SS3.6.p6.11.m4.3"><semantics id="S6.SS3.6.p6.11.m4.3a"><mrow id="S6.SS3.6.p6.11.m4.3.3" xref="S6.SS3.6.p6.11.m4.3.3.cmml"><mover accent="true" id="S6.SS3.6.p6.11.m4.3.3.4" xref="S6.SS3.6.p6.11.m4.3.3.4.cmml"><mi id="S6.SS3.6.p6.11.m4.3.3.4.2" xref="S6.SS3.6.p6.11.m4.3.3.4.2.cmml">q</mi><mo id="S6.SS3.6.p6.11.m4.3.3.4.1" xref="S6.SS3.6.p6.11.m4.3.3.4.1.cmml">~</mo></mover><mo id="S6.SS3.6.p6.11.m4.3.3.3" xref="S6.SS3.6.p6.11.m4.3.3.3.cmml">∉</mo><mrow id="S6.SS3.6.p6.11.m4.3.3.2.2" xref="S6.SS3.6.p6.11.m4.3.3.2.3.cmml"><mo id="S6.SS3.6.p6.11.m4.3.3.2.2.3" stretchy="false" xref="S6.SS3.6.p6.11.m4.3.3.2.3.cmml">{</mo><msub id="S6.SS3.6.p6.11.m4.2.2.1.1.1" xref="S6.SS3.6.p6.11.m4.2.2.1.1.1.cmml"><mi id="S6.SS3.6.p6.11.m4.2.2.1.1.1.2" xref="S6.SS3.6.p6.11.m4.2.2.1.1.1.2.cmml">m</mi><mn id="S6.SS3.6.p6.11.m4.2.2.1.1.1.3" xref="S6.SS3.6.p6.11.m4.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S6.SS3.6.p6.11.m4.3.3.2.2.4" xref="S6.SS3.6.p6.11.m4.3.3.2.3.cmml">,</mo><mi id="S6.SS3.6.p6.11.m4.1.1" mathvariant="normal" xref="S6.SS3.6.p6.11.m4.1.1.cmml">…</mi><mo id="S6.SS3.6.p6.11.m4.3.3.2.2.5" xref="S6.SS3.6.p6.11.m4.3.3.2.3.cmml">,</mo><msub id="S6.SS3.6.p6.11.m4.3.3.2.2.2" xref="S6.SS3.6.p6.11.m4.3.3.2.2.2.cmml"><mi id="S6.SS3.6.p6.11.m4.3.3.2.2.2.2" xref="S6.SS3.6.p6.11.m4.3.3.2.2.2.2.cmml">m</mi><mi id="S6.SS3.6.p6.11.m4.3.3.2.2.2.3" xref="S6.SS3.6.p6.11.m4.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S6.SS3.6.p6.11.m4.3.3.2.2.6" stretchy="false" xref="S6.SS3.6.p6.11.m4.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.11.m4.3b"><apply id="S6.SS3.6.p6.11.m4.3.3.cmml" xref="S6.SS3.6.p6.11.m4.3.3"><notin id="S6.SS3.6.p6.11.m4.3.3.3.cmml" xref="S6.SS3.6.p6.11.m4.3.3.3"></notin><apply id="S6.SS3.6.p6.11.m4.3.3.4.cmml" xref="S6.SS3.6.p6.11.m4.3.3.4"><ci id="S6.SS3.6.p6.11.m4.3.3.4.1.cmml" xref="S6.SS3.6.p6.11.m4.3.3.4.1">~</ci><ci id="S6.SS3.6.p6.11.m4.3.3.4.2.cmml" xref="S6.SS3.6.p6.11.m4.3.3.4.2">𝑞</ci></apply><set id="S6.SS3.6.p6.11.m4.3.3.2.3.cmml" xref="S6.SS3.6.p6.11.m4.3.3.2.2"><apply id="S6.SS3.6.p6.11.m4.2.2.1.1.1.cmml" xref="S6.SS3.6.p6.11.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.6.p6.11.m4.2.2.1.1.1.1.cmml" xref="S6.SS3.6.p6.11.m4.2.2.1.1.1">subscript</csymbol><ci id="S6.SS3.6.p6.11.m4.2.2.1.1.1.2.cmml" xref="S6.SS3.6.p6.11.m4.2.2.1.1.1.2">𝑚</ci><cn id="S6.SS3.6.p6.11.m4.2.2.1.1.1.3.cmml" type="integer" xref="S6.SS3.6.p6.11.m4.2.2.1.1.1.3">0</cn></apply><ci id="S6.SS3.6.p6.11.m4.1.1.cmml" xref="S6.SS3.6.p6.11.m4.1.1">…</ci><apply id="S6.SS3.6.p6.11.m4.3.3.2.2.2.cmml" xref="S6.SS3.6.p6.11.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.11.m4.3.3.2.2.2.1.cmml" xref="S6.SS3.6.p6.11.m4.3.3.2.2.2">subscript</csymbol><ci id="S6.SS3.6.p6.11.m4.3.3.2.2.2.2.cmml" xref="S6.SS3.6.p6.11.m4.3.3.2.2.2.2">𝑚</ci><ci id="S6.SS3.6.p6.11.m4.3.3.2.2.2.3.cmml" xref="S6.SS3.6.p6.11.m4.3.3.2.2.2.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.11.m4.3c">\widetilde{q}\notin\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.11.m4.3d">over~ start_ARG italic_q end_ARG ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math>, we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.E12"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{Tr}_{\widetilde{q}}v=\mathcal{C}^{q}v=0." class="ltx_Math" display="block" id="S6.E12.m1.1"><semantics id="S6.E12.m1.1a"><mrow id="S6.E12.m1.1.1.1" xref="S6.E12.m1.1.1.1.1.cmml"><mrow id="S6.E12.m1.1.1.1.1" xref="S6.E12.m1.1.1.1.1.cmml"><mrow id="S6.E12.m1.1.1.1.1.2" xref="S6.E12.m1.1.1.1.1.2.cmml"><msub id="S6.E12.m1.1.1.1.1.2.1" xref="S6.E12.m1.1.1.1.1.2.1.cmml"><mi id="S6.E12.m1.1.1.1.1.2.1.2" xref="S6.E12.m1.1.1.1.1.2.1.2.cmml">Tr</mi><mover accent="true" id="S6.E12.m1.1.1.1.1.2.1.3" xref="S6.E12.m1.1.1.1.1.2.1.3.cmml"><mi id="S6.E12.m1.1.1.1.1.2.1.3.2" xref="S6.E12.m1.1.1.1.1.2.1.3.2.cmml">q</mi><mo id="S6.E12.m1.1.1.1.1.2.1.3.1" xref="S6.E12.m1.1.1.1.1.2.1.3.1.cmml">~</mo></mover></msub><mo id="S6.E12.m1.1.1.1.1.2a" lspace="0.167em" xref="S6.E12.m1.1.1.1.1.2.cmml">⁡</mo><mi id="S6.E12.m1.1.1.1.1.2.2" xref="S6.E12.m1.1.1.1.1.2.2.cmml">v</mi></mrow><mo id="S6.E12.m1.1.1.1.1.3" xref="S6.E12.m1.1.1.1.1.3.cmml">=</mo><mrow id="S6.E12.m1.1.1.1.1.4" xref="S6.E12.m1.1.1.1.1.4.cmml"><msup id="S6.E12.m1.1.1.1.1.4.2" xref="S6.E12.m1.1.1.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.E12.m1.1.1.1.1.4.2.2" xref="S6.E12.m1.1.1.1.1.4.2.2.cmml">𝒞</mi><mi id="S6.E12.m1.1.1.1.1.4.2.3" xref="S6.E12.m1.1.1.1.1.4.2.3.cmml">q</mi></msup><mo id="S6.E12.m1.1.1.1.1.4.1" xref="S6.E12.m1.1.1.1.1.4.1.cmml">⁢</mo><mi id="S6.E12.m1.1.1.1.1.4.3" xref="S6.E12.m1.1.1.1.1.4.3.cmml">v</mi></mrow><mo id="S6.E12.m1.1.1.1.1.5" xref="S6.E12.m1.1.1.1.1.5.cmml">=</mo><mn id="S6.E12.m1.1.1.1.1.6" xref="S6.E12.m1.1.1.1.1.6.cmml">0</mn></mrow><mo id="S6.E12.m1.1.1.1.2" lspace="0em" xref="S6.E12.m1.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.E12.m1.1b"><apply id="S6.E12.m1.1.1.1.1.cmml" xref="S6.E12.m1.1.1.1"><and id="S6.E12.m1.1.1.1.1a.cmml" xref="S6.E12.m1.1.1.1"></and><apply id="S6.E12.m1.1.1.1.1b.cmml" xref="S6.E12.m1.1.1.1"><eq id="S6.E12.m1.1.1.1.1.3.cmml" xref="S6.E12.m1.1.1.1.1.3"></eq><apply id="S6.E12.m1.1.1.1.1.2.cmml" xref="S6.E12.m1.1.1.1.1.2"><apply id="S6.E12.m1.1.1.1.1.2.1.cmml" xref="S6.E12.m1.1.1.1.1.2.1"><csymbol cd="ambiguous" id="S6.E12.m1.1.1.1.1.2.1.1.cmml" xref="S6.E12.m1.1.1.1.1.2.1">subscript</csymbol><ci id="S6.E12.m1.1.1.1.1.2.1.2.cmml" xref="S6.E12.m1.1.1.1.1.2.1.2">Tr</ci><apply id="S6.E12.m1.1.1.1.1.2.1.3.cmml" xref="S6.E12.m1.1.1.1.1.2.1.3"><ci id="S6.E12.m1.1.1.1.1.2.1.3.1.cmml" xref="S6.E12.m1.1.1.1.1.2.1.3.1">~</ci><ci id="S6.E12.m1.1.1.1.1.2.1.3.2.cmml" xref="S6.E12.m1.1.1.1.1.2.1.3.2">𝑞</ci></apply></apply><ci id="S6.E12.m1.1.1.1.1.2.2.cmml" xref="S6.E12.m1.1.1.1.1.2.2">𝑣</ci></apply><apply id="S6.E12.m1.1.1.1.1.4.cmml" xref="S6.E12.m1.1.1.1.1.4"><times id="S6.E12.m1.1.1.1.1.4.1.cmml" xref="S6.E12.m1.1.1.1.1.4.1"></times><apply id="S6.E12.m1.1.1.1.1.4.2.cmml" xref="S6.E12.m1.1.1.1.1.4.2"><csymbol cd="ambiguous" id="S6.E12.m1.1.1.1.1.4.2.1.cmml" xref="S6.E12.m1.1.1.1.1.4.2">superscript</csymbol><ci id="S6.E12.m1.1.1.1.1.4.2.2.cmml" xref="S6.E12.m1.1.1.1.1.4.2.2">𝒞</ci><ci id="S6.E12.m1.1.1.1.1.4.2.3.cmml" xref="S6.E12.m1.1.1.1.1.4.2.3">𝑞</ci></apply><ci id="S6.E12.m1.1.1.1.1.4.3.cmml" xref="S6.E12.m1.1.1.1.1.4.3">𝑣</ci></apply></apply><apply id="S6.E12.m1.1.1.1.1c.cmml" xref="S6.E12.m1.1.1.1"><eq id="S6.E12.m1.1.1.1.1.5.cmml" xref="S6.E12.m1.1.1.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S6.E12.m1.1.1.1.1.4.cmml" id="S6.E12.m1.1.1.1.1d.cmml" xref="S6.E12.m1.1.1.1"></share><cn id="S6.E12.m1.1.1.1.1.6.cmml" type="integer" xref="S6.E12.m1.1.1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.E12.m1.1c">\operatorname{Tr}_{\widetilde{q}}v=\mathcal{C}^{q}v=0.</annotation><annotation encoding="application/x-llamapun" id="S6.E12.m1.1d">roman_Tr start_POSTSUBSCRIPT over~ start_ARG italic_q end_ARG end_POSTSUBSCRIPT italic_v = caligraphic_C start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT italic_v = 0 .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.12)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.6.p6.13">If <math alttext="\widetilde{q}=m_{i}\in\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S6.SS3.6.p6.12.m1.3"><semantics id="S6.SS3.6.p6.12.m1.3a"><mrow id="S6.SS3.6.p6.12.m1.3.3" xref="S6.SS3.6.p6.12.m1.3.3.cmml"><mover accent="true" id="S6.SS3.6.p6.12.m1.3.3.4" xref="S6.SS3.6.p6.12.m1.3.3.4.cmml"><mi id="S6.SS3.6.p6.12.m1.3.3.4.2" xref="S6.SS3.6.p6.12.m1.3.3.4.2.cmml">q</mi><mo id="S6.SS3.6.p6.12.m1.3.3.4.1" xref="S6.SS3.6.p6.12.m1.3.3.4.1.cmml">~</mo></mover><mo id="S6.SS3.6.p6.12.m1.3.3.5" xref="S6.SS3.6.p6.12.m1.3.3.5.cmml">=</mo><msub id="S6.SS3.6.p6.12.m1.3.3.6" xref="S6.SS3.6.p6.12.m1.3.3.6.cmml"><mi id="S6.SS3.6.p6.12.m1.3.3.6.2" xref="S6.SS3.6.p6.12.m1.3.3.6.2.cmml">m</mi><mi id="S6.SS3.6.p6.12.m1.3.3.6.3" xref="S6.SS3.6.p6.12.m1.3.3.6.3.cmml">i</mi></msub><mo id="S6.SS3.6.p6.12.m1.3.3.7" xref="S6.SS3.6.p6.12.m1.3.3.7.cmml">∈</mo><mrow id="S6.SS3.6.p6.12.m1.3.3.2.2" xref="S6.SS3.6.p6.12.m1.3.3.2.3.cmml"><mo id="S6.SS3.6.p6.12.m1.3.3.2.2.3" stretchy="false" xref="S6.SS3.6.p6.12.m1.3.3.2.3.cmml">{</mo><msub id="S6.SS3.6.p6.12.m1.2.2.1.1.1" xref="S6.SS3.6.p6.12.m1.2.2.1.1.1.cmml"><mi id="S6.SS3.6.p6.12.m1.2.2.1.1.1.2" xref="S6.SS3.6.p6.12.m1.2.2.1.1.1.2.cmml">m</mi><mn id="S6.SS3.6.p6.12.m1.2.2.1.1.1.3" xref="S6.SS3.6.p6.12.m1.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S6.SS3.6.p6.12.m1.3.3.2.2.4" xref="S6.SS3.6.p6.12.m1.3.3.2.3.cmml">,</mo><mi id="S6.SS3.6.p6.12.m1.1.1" mathvariant="normal" xref="S6.SS3.6.p6.12.m1.1.1.cmml">…</mi><mo id="S6.SS3.6.p6.12.m1.3.3.2.2.5" xref="S6.SS3.6.p6.12.m1.3.3.2.3.cmml">,</mo><msub id="S6.SS3.6.p6.12.m1.3.3.2.2.2" xref="S6.SS3.6.p6.12.m1.3.3.2.2.2.cmml"><mi id="S6.SS3.6.p6.12.m1.3.3.2.2.2.2" xref="S6.SS3.6.p6.12.m1.3.3.2.2.2.2.cmml">m</mi><mi id="S6.SS3.6.p6.12.m1.3.3.2.2.2.3" xref="S6.SS3.6.p6.12.m1.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S6.SS3.6.p6.12.m1.3.3.2.2.6" stretchy="false" xref="S6.SS3.6.p6.12.m1.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.12.m1.3b"><apply id="S6.SS3.6.p6.12.m1.3.3.cmml" xref="S6.SS3.6.p6.12.m1.3.3"><and id="S6.SS3.6.p6.12.m1.3.3a.cmml" xref="S6.SS3.6.p6.12.m1.3.3"></and><apply id="S6.SS3.6.p6.12.m1.3.3b.cmml" xref="S6.SS3.6.p6.12.m1.3.3"><eq id="S6.SS3.6.p6.12.m1.3.3.5.cmml" xref="S6.SS3.6.p6.12.m1.3.3.5"></eq><apply id="S6.SS3.6.p6.12.m1.3.3.4.cmml" xref="S6.SS3.6.p6.12.m1.3.3.4"><ci id="S6.SS3.6.p6.12.m1.3.3.4.1.cmml" xref="S6.SS3.6.p6.12.m1.3.3.4.1">~</ci><ci id="S6.SS3.6.p6.12.m1.3.3.4.2.cmml" xref="S6.SS3.6.p6.12.m1.3.3.4.2">𝑞</ci></apply><apply id="S6.SS3.6.p6.12.m1.3.3.6.cmml" xref="S6.SS3.6.p6.12.m1.3.3.6"><csymbol cd="ambiguous" id="S6.SS3.6.p6.12.m1.3.3.6.1.cmml" xref="S6.SS3.6.p6.12.m1.3.3.6">subscript</csymbol><ci id="S6.SS3.6.p6.12.m1.3.3.6.2.cmml" xref="S6.SS3.6.p6.12.m1.3.3.6.2">𝑚</ci><ci id="S6.SS3.6.p6.12.m1.3.3.6.3.cmml" xref="S6.SS3.6.p6.12.m1.3.3.6.3">𝑖</ci></apply></apply><apply id="S6.SS3.6.p6.12.m1.3.3c.cmml" xref="S6.SS3.6.p6.12.m1.3.3"><in id="S6.SS3.6.p6.12.m1.3.3.7.cmml" xref="S6.SS3.6.p6.12.m1.3.3.7"></in><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.6.p6.12.m1.3.3.6.cmml" id="S6.SS3.6.p6.12.m1.3.3d.cmml" xref="S6.SS3.6.p6.12.m1.3.3"></share><set id="S6.SS3.6.p6.12.m1.3.3.2.3.cmml" xref="S6.SS3.6.p6.12.m1.3.3.2.2"><apply id="S6.SS3.6.p6.12.m1.2.2.1.1.1.cmml" xref="S6.SS3.6.p6.12.m1.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.6.p6.12.m1.2.2.1.1.1.1.cmml" xref="S6.SS3.6.p6.12.m1.2.2.1.1.1">subscript</csymbol><ci id="S6.SS3.6.p6.12.m1.2.2.1.1.1.2.cmml" xref="S6.SS3.6.p6.12.m1.2.2.1.1.1.2">𝑚</ci><cn id="S6.SS3.6.p6.12.m1.2.2.1.1.1.3.cmml" type="integer" xref="S6.SS3.6.p6.12.m1.2.2.1.1.1.3">0</cn></apply><ci id="S6.SS3.6.p6.12.m1.1.1.cmml" xref="S6.SS3.6.p6.12.m1.1.1">…</ci><apply id="S6.SS3.6.p6.12.m1.3.3.2.2.2.cmml" xref="S6.SS3.6.p6.12.m1.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.12.m1.3.3.2.2.2.1.cmml" xref="S6.SS3.6.p6.12.m1.3.3.2.2.2">subscript</csymbol><ci id="S6.SS3.6.p6.12.m1.3.3.2.2.2.2.cmml" xref="S6.SS3.6.p6.12.m1.3.3.2.2.2.2">𝑚</ci><ci id="S6.SS3.6.p6.12.m1.3.3.2.2.2.3.cmml" xref="S6.SS3.6.p6.12.m1.3.3.2.2.2.3">𝑛</ci></apply></set></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.12.m1.3c">\widetilde{q}=m_{i}\in\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.12.m1.3d">over~ start_ARG italic_q end_ARG = italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math> for some <math alttext="0\leq i\leq n" class="ltx_Math" display="inline" id="S6.SS3.6.p6.13.m2.1"><semantics id="S6.SS3.6.p6.13.m2.1a"><mrow id="S6.SS3.6.p6.13.m2.1.1" xref="S6.SS3.6.p6.13.m2.1.1.cmml"><mn id="S6.SS3.6.p6.13.m2.1.1.2" xref="S6.SS3.6.p6.13.m2.1.1.2.cmml">0</mn><mo id="S6.SS3.6.p6.13.m2.1.1.3" xref="S6.SS3.6.p6.13.m2.1.1.3.cmml">≤</mo><mi id="S6.SS3.6.p6.13.m2.1.1.4" xref="S6.SS3.6.p6.13.m2.1.1.4.cmml">i</mi><mo id="S6.SS3.6.p6.13.m2.1.1.5" xref="S6.SS3.6.p6.13.m2.1.1.5.cmml">≤</mo><mi id="S6.SS3.6.p6.13.m2.1.1.6" xref="S6.SS3.6.p6.13.m2.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.13.m2.1b"><apply id="S6.SS3.6.p6.13.m2.1.1.cmml" xref="S6.SS3.6.p6.13.m2.1.1"><and id="S6.SS3.6.p6.13.m2.1.1a.cmml" xref="S6.SS3.6.p6.13.m2.1.1"></and><apply id="S6.SS3.6.p6.13.m2.1.1b.cmml" xref="S6.SS3.6.p6.13.m2.1.1"><leq id="S6.SS3.6.p6.13.m2.1.1.3.cmml" xref="S6.SS3.6.p6.13.m2.1.1.3"></leq><cn id="S6.SS3.6.p6.13.m2.1.1.2.cmml" type="integer" xref="S6.SS3.6.p6.13.m2.1.1.2">0</cn><ci id="S6.SS3.6.p6.13.m2.1.1.4.cmml" xref="S6.SS3.6.p6.13.m2.1.1.4">𝑖</ci></apply><apply id="S6.SS3.6.p6.13.m2.1.1c.cmml" xref="S6.SS3.6.p6.13.m2.1.1"><leq id="S6.SS3.6.p6.13.m2.1.1.5.cmml" xref="S6.SS3.6.p6.13.m2.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.6.p6.13.m2.1.1.4.cmml" id="S6.SS3.6.p6.13.m2.1.1d.cmml" xref="S6.SS3.6.p6.13.m2.1.1"></share><ci id="S6.SS3.6.p6.13.m2.1.1.6.cmml" xref="S6.SS3.6.p6.13.m2.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.13.m2.1c">0\leq i\leq n</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.13.m2.1d">0 ≤ italic_i ≤ italic_n</annotation></semantics></math>, then by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E8" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.8</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E12" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.12</span></a>) it holds that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx41"> <tbody id="S6.Ex57"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle 0=\mathcal{C}^{m_{i}}v" class="ltx_Math" display="inline" id="S6.Ex57.m1.1"><semantics id="S6.Ex57.m1.1a"><mrow id="S6.Ex57.m1.1.1" xref="S6.Ex57.m1.1.1.cmml"><mn id="S6.Ex57.m1.1.1.2" xref="S6.Ex57.m1.1.1.2.cmml">0</mn><mo id="S6.Ex57.m1.1.1.1" xref="S6.Ex57.m1.1.1.1.cmml">=</mo><mrow id="S6.Ex57.m1.1.1.3" xref="S6.Ex57.m1.1.1.3.cmml"><msup id="S6.Ex57.m1.1.1.3.2" xref="S6.Ex57.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex57.m1.1.1.3.2.2" xref="S6.Ex57.m1.1.1.3.2.2.cmml">𝒞</mi><msub id="S6.Ex57.m1.1.1.3.2.3" xref="S6.Ex57.m1.1.1.3.2.3.cmml"><mi id="S6.Ex57.m1.1.1.3.2.3.2" xref="S6.Ex57.m1.1.1.3.2.3.2.cmml">m</mi><mi id="S6.Ex57.m1.1.1.3.2.3.3" xref="S6.Ex57.m1.1.1.3.2.3.3.cmml">i</mi></msub></msup><mo id="S6.Ex57.m1.1.1.3.1" xref="S6.Ex57.m1.1.1.3.1.cmml">⁢</mo><mi id="S6.Ex57.m1.1.1.3.3" xref="S6.Ex57.m1.1.1.3.3.cmml">v</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex57.m1.1b"><apply id="S6.Ex57.m1.1.1.cmml" xref="S6.Ex57.m1.1.1"><eq id="S6.Ex57.m1.1.1.1.cmml" xref="S6.Ex57.m1.1.1.1"></eq><cn id="S6.Ex57.m1.1.1.2.cmml" type="integer" xref="S6.Ex57.m1.1.1.2">0</cn><apply id="S6.Ex57.m1.1.1.3.cmml" xref="S6.Ex57.m1.1.1.3"><times id="S6.Ex57.m1.1.1.3.1.cmml" xref="S6.Ex57.m1.1.1.3.1"></times><apply id="S6.Ex57.m1.1.1.3.2.cmml" xref="S6.Ex57.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Ex57.m1.1.1.3.2.1.cmml" xref="S6.Ex57.m1.1.1.3.2">superscript</csymbol><ci id="S6.Ex57.m1.1.1.3.2.2.cmml" xref="S6.Ex57.m1.1.1.3.2.2">𝒞</ci><apply id="S6.Ex57.m1.1.1.3.2.3.cmml" xref="S6.Ex57.m1.1.1.3.2.3"><csymbol cd="ambiguous" id="S6.Ex57.m1.1.1.3.2.3.1.cmml" xref="S6.Ex57.m1.1.1.3.2.3">subscript</csymbol><ci id="S6.Ex57.m1.1.1.3.2.3.2.cmml" xref="S6.Ex57.m1.1.1.3.2.3.2">𝑚</ci><ci id="S6.Ex57.m1.1.1.3.2.3.3.cmml" xref="S6.Ex57.m1.1.1.3.2.3.3">𝑖</ci></apply></apply><ci id="S6.Ex57.m1.1.1.3.3.cmml" xref="S6.Ex57.m1.1.1.3.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex57.m1.1c">\displaystyle 0=\mathcal{C}^{m_{i}}v</annotation><annotation encoding="application/x-llamapun" id="S6.Ex57.m1.1d">0 = caligraphic_C start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_v</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=(1-\pi_{m_{i}})\operatorname{Tr}_{m_{i}}v+\pi_{m_{i}}\widetilde{% \mathcal{B}}^{m_{i}}v" class="ltx_Math" display="inline" id="S6.Ex57.m2.1"><semantics id="S6.Ex57.m2.1a"><mrow id="S6.Ex57.m2.1.1" xref="S6.Ex57.m2.1.1.cmml"><mi id="S6.Ex57.m2.1.1.3" xref="S6.Ex57.m2.1.1.3.cmml"></mi><mo id="S6.Ex57.m2.1.1.2" xref="S6.Ex57.m2.1.1.2.cmml">=</mo><mrow id="S6.Ex57.m2.1.1.1" xref="S6.Ex57.m2.1.1.1.cmml"><mrow id="S6.Ex57.m2.1.1.1.1" xref="S6.Ex57.m2.1.1.1.1.cmml"><mrow id="S6.Ex57.m2.1.1.1.1.1.1" xref="S6.Ex57.m2.1.1.1.1.1.1.1.cmml"><mo id="S6.Ex57.m2.1.1.1.1.1.1.2" stretchy="false" xref="S6.Ex57.m2.1.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex57.m2.1.1.1.1.1.1.1" xref="S6.Ex57.m2.1.1.1.1.1.1.1.cmml"><mn id="S6.Ex57.m2.1.1.1.1.1.1.1.2" xref="S6.Ex57.m2.1.1.1.1.1.1.1.2.cmml">1</mn><mo id="S6.Ex57.m2.1.1.1.1.1.1.1.1" xref="S6.Ex57.m2.1.1.1.1.1.1.1.1.cmml">−</mo><msub id="S6.Ex57.m2.1.1.1.1.1.1.1.3" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex57.m2.1.1.1.1.1.1.1.3.2" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.2.cmml">π</mi><msub id="S6.Ex57.m2.1.1.1.1.1.1.1.3.3" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.cmml"><mi id="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.2" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.2.cmml">m</mi><mi id="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.3" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.3.cmml">i</mi></msub></msub></mrow><mo id="S6.Ex57.m2.1.1.1.1.1.1.3" stretchy="false" xref="S6.Ex57.m2.1.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.Ex57.m2.1.1.1.1.2" lspace="0.167em" xref="S6.Ex57.m2.1.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex57.m2.1.1.1.1.3" xref="S6.Ex57.m2.1.1.1.1.3.cmml"><msub id="S6.Ex57.m2.1.1.1.1.3.1" xref="S6.Ex57.m2.1.1.1.1.3.1.cmml"><mi id="S6.Ex57.m2.1.1.1.1.3.1.2" xref="S6.Ex57.m2.1.1.1.1.3.1.2.cmml">Tr</mi><msub id="S6.Ex57.m2.1.1.1.1.3.1.3" xref="S6.Ex57.m2.1.1.1.1.3.1.3.cmml"><mi id="S6.Ex57.m2.1.1.1.1.3.1.3.2" xref="S6.Ex57.m2.1.1.1.1.3.1.3.2.cmml">m</mi><mi id="S6.Ex57.m2.1.1.1.1.3.1.3.3" xref="S6.Ex57.m2.1.1.1.1.3.1.3.3.cmml">i</mi></msub></msub><mo id="S6.Ex57.m2.1.1.1.1.3a" lspace="0.167em" xref="S6.Ex57.m2.1.1.1.1.3.cmml">⁡</mo><mi id="S6.Ex57.m2.1.1.1.1.3.2" xref="S6.Ex57.m2.1.1.1.1.3.2.cmml">v</mi></mrow></mrow><mo id="S6.Ex57.m2.1.1.1.2" xref="S6.Ex57.m2.1.1.1.2.cmml">+</mo><mrow id="S6.Ex57.m2.1.1.1.3" xref="S6.Ex57.m2.1.1.1.3.cmml"><msub id="S6.Ex57.m2.1.1.1.3.2" xref="S6.Ex57.m2.1.1.1.3.2.cmml"><mi id="S6.Ex57.m2.1.1.1.3.2.2" xref="S6.Ex57.m2.1.1.1.3.2.2.cmml">π</mi><msub id="S6.Ex57.m2.1.1.1.3.2.3" xref="S6.Ex57.m2.1.1.1.3.2.3.cmml"><mi id="S6.Ex57.m2.1.1.1.3.2.3.2" xref="S6.Ex57.m2.1.1.1.3.2.3.2.cmml">m</mi><mi id="S6.Ex57.m2.1.1.1.3.2.3.3" xref="S6.Ex57.m2.1.1.1.3.2.3.3.cmml">i</mi></msub></msub><mo id="S6.Ex57.m2.1.1.1.3.1" xref="S6.Ex57.m2.1.1.1.3.1.cmml">⁢</mo><msup id="S6.Ex57.m2.1.1.1.3.3" xref="S6.Ex57.m2.1.1.1.3.3.cmml"><mover accent="true" id="S6.Ex57.m2.1.1.1.3.3.2" xref="S6.Ex57.m2.1.1.1.3.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex57.m2.1.1.1.3.3.2.2" xref="S6.Ex57.m2.1.1.1.3.3.2.2.cmml">ℬ</mi><mo id="S6.Ex57.m2.1.1.1.3.3.2.1" xref="S6.Ex57.m2.1.1.1.3.3.2.1.cmml">~</mo></mover><msub id="S6.Ex57.m2.1.1.1.3.3.3" xref="S6.Ex57.m2.1.1.1.3.3.3.cmml"><mi id="S6.Ex57.m2.1.1.1.3.3.3.2" xref="S6.Ex57.m2.1.1.1.3.3.3.2.cmml">m</mi><mi id="S6.Ex57.m2.1.1.1.3.3.3.3" xref="S6.Ex57.m2.1.1.1.3.3.3.3.cmml">i</mi></msub></msup><mo id="S6.Ex57.m2.1.1.1.3.1a" xref="S6.Ex57.m2.1.1.1.3.1.cmml">⁢</mo><mi id="S6.Ex57.m2.1.1.1.3.4" xref="S6.Ex57.m2.1.1.1.3.4.cmml">v</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex57.m2.1b"><apply id="S6.Ex57.m2.1.1.cmml" xref="S6.Ex57.m2.1.1"><eq id="S6.Ex57.m2.1.1.2.cmml" xref="S6.Ex57.m2.1.1.2"></eq><csymbol cd="latexml" id="S6.Ex57.m2.1.1.3.cmml" xref="S6.Ex57.m2.1.1.3">absent</csymbol><apply id="S6.Ex57.m2.1.1.1.cmml" xref="S6.Ex57.m2.1.1.1"><plus id="S6.Ex57.m2.1.1.1.2.cmml" xref="S6.Ex57.m2.1.1.1.2"></plus><apply id="S6.Ex57.m2.1.1.1.1.cmml" xref="S6.Ex57.m2.1.1.1.1"><times id="S6.Ex57.m2.1.1.1.1.2.cmml" xref="S6.Ex57.m2.1.1.1.1.2"></times><apply id="S6.Ex57.m2.1.1.1.1.1.1.1.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1"><minus id="S6.Ex57.m2.1.1.1.1.1.1.1.1.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.1"></minus><cn id="S6.Ex57.m2.1.1.1.1.1.1.1.2.cmml" type="integer" xref="S6.Ex57.m2.1.1.1.1.1.1.1.2">1</cn><apply id="S6.Ex57.m2.1.1.1.1.1.1.1.3.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.1.1.1.1.3.1.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.Ex57.m2.1.1.1.1.1.1.1.3.2.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.2">𝜋</ci><apply id="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.1.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.3">subscript</csymbol><ci id="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.2.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.2">𝑚</ci><ci id="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.3.cmml" xref="S6.Ex57.m2.1.1.1.1.1.1.1.3.3.3">𝑖</ci></apply></apply></apply><apply id="S6.Ex57.m2.1.1.1.1.3.cmml" xref="S6.Ex57.m2.1.1.1.1.3"><apply id="S6.Ex57.m2.1.1.1.1.3.1.cmml" xref="S6.Ex57.m2.1.1.1.1.3.1"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.1.3.1.1.cmml" xref="S6.Ex57.m2.1.1.1.1.3.1">subscript</csymbol><ci id="S6.Ex57.m2.1.1.1.1.3.1.2.cmml" xref="S6.Ex57.m2.1.1.1.1.3.1.2">Tr</ci><apply id="S6.Ex57.m2.1.1.1.1.3.1.3.cmml" xref="S6.Ex57.m2.1.1.1.1.3.1.3"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.1.3.1.3.1.cmml" xref="S6.Ex57.m2.1.1.1.1.3.1.3">subscript</csymbol><ci id="S6.Ex57.m2.1.1.1.1.3.1.3.2.cmml" xref="S6.Ex57.m2.1.1.1.1.3.1.3.2">𝑚</ci><ci id="S6.Ex57.m2.1.1.1.1.3.1.3.3.cmml" xref="S6.Ex57.m2.1.1.1.1.3.1.3.3">𝑖</ci></apply></apply><ci id="S6.Ex57.m2.1.1.1.1.3.2.cmml" xref="S6.Ex57.m2.1.1.1.1.3.2">𝑣</ci></apply></apply><apply id="S6.Ex57.m2.1.1.1.3.cmml" xref="S6.Ex57.m2.1.1.1.3"><times id="S6.Ex57.m2.1.1.1.3.1.cmml" xref="S6.Ex57.m2.1.1.1.3.1"></times><apply id="S6.Ex57.m2.1.1.1.3.2.cmml" xref="S6.Ex57.m2.1.1.1.3.2"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.3.2.1.cmml" xref="S6.Ex57.m2.1.1.1.3.2">subscript</csymbol><ci id="S6.Ex57.m2.1.1.1.3.2.2.cmml" xref="S6.Ex57.m2.1.1.1.3.2.2">𝜋</ci><apply id="S6.Ex57.m2.1.1.1.3.2.3.cmml" xref="S6.Ex57.m2.1.1.1.3.2.3"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.3.2.3.1.cmml" xref="S6.Ex57.m2.1.1.1.3.2.3">subscript</csymbol><ci id="S6.Ex57.m2.1.1.1.3.2.3.2.cmml" xref="S6.Ex57.m2.1.1.1.3.2.3.2">𝑚</ci><ci id="S6.Ex57.m2.1.1.1.3.2.3.3.cmml" xref="S6.Ex57.m2.1.1.1.3.2.3.3">𝑖</ci></apply></apply><apply id="S6.Ex57.m2.1.1.1.3.3.cmml" xref="S6.Ex57.m2.1.1.1.3.3"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.3.3.1.cmml" xref="S6.Ex57.m2.1.1.1.3.3">superscript</csymbol><apply id="S6.Ex57.m2.1.1.1.3.3.2.cmml" xref="S6.Ex57.m2.1.1.1.3.3.2"><ci id="S6.Ex57.m2.1.1.1.3.3.2.1.cmml" xref="S6.Ex57.m2.1.1.1.3.3.2.1">~</ci><ci id="S6.Ex57.m2.1.1.1.3.3.2.2.cmml" xref="S6.Ex57.m2.1.1.1.3.3.2.2">ℬ</ci></apply><apply id="S6.Ex57.m2.1.1.1.3.3.3.cmml" xref="S6.Ex57.m2.1.1.1.3.3.3"><csymbol cd="ambiguous" id="S6.Ex57.m2.1.1.1.3.3.3.1.cmml" xref="S6.Ex57.m2.1.1.1.3.3.3">subscript</csymbol><ci id="S6.Ex57.m2.1.1.1.3.3.3.2.cmml" xref="S6.Ex57.m2.1.1.1.3.3.3.2">𝑚</ci><ci id="S6.Ex57.m2.1.1.1.3.3.3.3.cmml" xref="S6.Ex57.m2.1.1.1.3.3.3.3">𝑖</ci></apply></apply><ci id="S6.Ex57.m2.1.1.1.3.4.cmml" xref="S6.Ex57.m2.1.1.1.3.4">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex57.m2.1c">\displaystyle=(1-\pi_{m_{i}})\operatorname{Tr}_{m_{i}}v+\pi_{m_{i}}\widetilde{% \mathcal{B}}^{m_{i}}v</annotation><annotation encoding="application/x-llamapun" id="S6.Ex57.m2.1d">= ( 1 - italic_π start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT ) roman_Tr start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_v + italic_π start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_v</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex58"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{Tr}_{m_{i}}v+\sum_{j=0}^{m_{i}-1}\widetilde{b}_{i,% j}\operatorname{Tr}_{j}v" class="ltx_Math" display="inline" id="S6.Ex58.m1.2"><semantics id="S6.Ex58.m1.2a"><mrow id="S6.Ex58.m1.2.3" xref="S6.Ex58.m1.2.3.cmml"><mi id="S6.Ex58.m1.2.3.2" xref="S6.Ex58.m1.2.3.2.cmml"></mi><mo id="S6.Ex58.m1.2.3.1" xref="S6.Ex58.m1.2.3.1.cmml">=</mo><mrow id="S6.Ex58.m1.2.3.3" xref="S6.Ex58.m1.2.3.3.cmml"><mrow id="S6.Ex58.m1.2.3.3.2" xref="S6.Ex58.m1.2.3.3.2.cmml"><msub id="S6.Ex58.m1.2.3.3.2.1" xref="S6.Ex58.m1.2.3.3.2.1.cmml"><mi id="S6.Ex58.m1.2.3.3.2.1.2" xref="S6.Ex58.m1.2.3.3.2.1.2.cmml">Tr</mi><msub id="S6.Ex58.m1.2.3.3.2.1.3" xref="S6.Ex58.m1.2.3.3.2.1.3.cmml"><mi id="S6.Ex58.m1.2.3.3.2.1.3.2" xref="S6.Ex58.m1.2.3.3.2.1.3.2.cmml">m</mi><mi id="S6.Ex58.m1.2.3.3.2.1.3.3" xref="S6.Ex58.m1.2.3.3.2.1.3.3.cmml">i</mi></msub></msub><mo id="S6.Ex58.m1.2.3.3.2a" lspace="0.167em" xref="S6.Ex58.m1.2.3.3.2.cmml">⁡</mo><mi id="S6.Ex58.m1.2.3.3.2.2" xref="S6.Ex58.m1.2.3.3.2.2.cmml">v</mi></mrow><mo id="S6.Ex58.m1.2.3.3.1" xref="S6.Ex58.m1.2.3.3.1.cmml">+</mo><mrow id="S6.Ex58.m1.2.3.3.3" xref="S6.Ex58.m1.2.3.3.3.cmml"><mstyle displaystyle="true" id="S6.Ex58.m1.2.3.3.3.1" xref="S6.Ex58.m1.2.3.3.3.1.cmml"><munderover id="S6.Ex58.m1.2.3.3.3.1a" xref="S6.Ex58.m1.2.3.3.3.1.cmml"><mo id="S6.Ex58.m1.2.3.3.3.1.2.2" movablelimits="false" xref="S6.Ex58.m1.2.3.3.3.1.2.2.cmml">∑</mo><mrow id="S6.Ex58.m1.2.3.3.3.1.2.3" xref="S6.Ex58.m1.2.3.3.3.1.2.3.cmml"><mi id="S6.Ex58.m1.2.3.3.3.1.2.3.2" xref="S6.Ex58.m1.2.3.3.3.1.2.3.2.cmml">j</mi><mo id="S6.Ex58.m1.2.3.3.3.1.2.3.1" xref="S6.Ex58.m1.2.3.3.3.1.2.3.1.cmml">=</mo><mn id="S6.Ex58.m1.2.3.3.3.1.2.3.3" xref="S6.Ex58.m1.2.3.3.3.1.2.3.3.cmml">0</mn></mrow><mrow id="S6.Ex58.m1.2.3.3.3.1.3" xref="S6.Ex58.m1.2.3.3.3.1.3.cmml"><msub id="S6.Ex58.m1.2.3.3.3.1.3.2" xref="S6.Ex58.m1.2.3.3.3.1.3.2.cmml"><mi id="S6.Ex58.m1.2.3.3.3.1.3.2.2" xref="S6.Ex58.m1.2.3.3.3.1.3.2.2.cmml">m</mi><mi id="S6.Ex58.m1.2.3.3.3.1.3.2.3" xref="S6.Ex58.m1.2.3.3.3.1.3.2.3.cmml">i</mi></msub><mo id="S6.Ex58.m1.2.3.3.3.1.3.1" xref="S6.Ex58.m1.2.3.3.3.1.3.1.cmml">−</mo><mn id="S6.Ex58.m1.2.3.3.3.1.3.3" xref="S6.Ex58.m1.2.3.3.3.1.3.3.cmml">1</mn></mrow></munderover></mstyle><mrow id="S6.Ex58.m1.2.3.3.3.2" xref="S6.Ex58.m1.2.3.3.3.2.cmml"><msub id="S6.Ex58.m1.2.3.3.3.2.2" xref="S6.Ex58.m1.2.3.3.3.2.2.cmml"><mover accent="true" id="S6.Ex58.m1.2.3.3.3.2.2.2" xref="S6.Ex58.m1.2.3.3.3.2.2.2.cmml"><mi id="S6.Ex58.m1.2.3.3.3.2.2.2.2" xref="S6.Ex58.m1.2.3.3.3.2.2.2.2.cmml">b</mi><mo id="S6.Ex58.m1.2.3.3.3.2.2.2.1" xref="S6.Ex58.m1.2.3.3.3.2.2.2.1.cmml">~</mo></mover><mrow id="S6.Ex58.m1.2.2.2.4" 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xref="S6.Ex58.m1.2.3.1"></eq><csymbol cd="latexml" id="S6.Ex58.m1.2.3.2.cmml" xref="S6.Ex58.m1.2.3.2">absent</csymbol><apply id="S6.Ex58.m1.2.3.3.cmml" xref="S6.Ex58.m1.2.3.3"><plus id="S6.Ex58.m1.2.3.3.1.cmml" xref="S6.Ex58.m1.2.3.3.1"></plus><apply id="S6.Ex58.m1.2.3.3.2.cmml" xref="S6.Ex58.m1.2.3.3.2"><apply id="S6.Ex58.m1.2.3.3.2.1.cmml" xref="S6.Ex58.m1.2.3.3.2.1"><csymbol cd="ambiguous" id="S6.Ex58.m1.2.3.3.2.1.1.cmml" xref="S6.Ex58.m1.2.3.3.2.1">subscript</csymbol><ci id="S6.Ex58.m1.2.3.3.2.1.2.cmml" xref="S6.Ex58.m1.2.3.3.2.1.2">Tr</ci><apply id="S6.Ex58.m1.2.3.3.2.1.3.cmml" xref="S6.Ex58.m1.2.3.3.2.1.3"><csymbol cd="ambiguous" id="S6.Ex58.m1.2.3.3.2.1.3.1.cmml" xref="S6.Ex58.m1.2.3.3.2.1.3">subscript</csymbol><ci id="S6.Ex58.m1.2.3.3.2.1.3.2.cmml" xref="S6.Ex58.m1.2.3.3.2.1.3.2">𝑚</ci><ci id="S6.Ex58.m1.2.3.3.2.1.3.3.cmml" xref="S6.Ex58.m1.2.3.3.2.1.3.3">𝑖</ci></apply></apply><ci id="S6.Ex58.m1.2.3.3.2.2.cmml" xref="S6.Ex58.m1.2.3.3.2.2">𝑣</ci></apply><apply 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xref="S6.Ex58.m1.2.3.3.3.2.2.2.2">𝑏</ci></apply><list id="S6.Ex58.m1.2.2.2.3.cmml" xref="S6.Ex58.m1.2.2.2.4"><ci id="S6.Ex58.m1.1.1.1.1.cmml" xref="S6.Ex58.m1.1.1.1.1">𝑖</ci><ci id="S6.Ex58.m1.2.2.2.2.cmml" xref="S6.Ex58.m1.2.2.2.2">𝑗</ci></list></apply><apply id="S6.Ex58.m1.2.3.3.3.2.3.cmml" xref="S6.Ex58.m1.2.3.3.3.2.3"><apply id="S6.Ex58.m1.2.3.3.3.2.3.1.cmml" xref="S6.Ex58.m1.2.3.3.3.2.3.1"><csymbol cd="ambiguous" id="S6.Ex58.m1.2.3.3.3.2.3.1.1.cmml" xref="S6.Ex58.m1.2.3.3.3.2.3.1">subscript</csymbol><ci id="S6.Ex58.m1.2.3.3.3.2.3.1.2.cmml" xref="S6.Ex58.m1.2.3.3.3.2.3.1.2">Tr</ci><ci id="S6.Ex58.m1.2.3.3.3.2.3.1.3.cmml" xref="S6.Ex58.m1.2.3.3.3.2.3.1.3">𝑗</ci></apply><ci id="S6.Ex58.m1.2.3.3.3.2.3.2.cmml" xref="S6.Ex58.m1.2.3.3.3.2.3.2">𝑣</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex58.m1.2c">\displaystyle=\operatorname{Tr}_{m_{i}}v+\sum_{j=0}^{m_{i}-1}\widetilde{b}_{i,% j}\operatorname{Tr}_{j}v</annotation><annotation encoding="application/x-llamapun" id="S6.Ex58.m1.2d">= roman_Tr start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_v + ∑ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT - 1 end_POSTSUPERSCRIPT over~ start_ARG italic_b end_ARG start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_v</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex59"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\operatorname{Tr}_{m_{i}}v+\sum_{j\in\{m_{0},\dots,m_{i-1}\}}% \widetilde{b}_{i,j}\operatorname{Tr}_{j}v." 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id="S6.Ex59.m1.4.4.1.1.cmml" xref="S6.Ex59.m1.4.4.1.1">𝑖</ci><ci id="S6.Ex59.m1.5.5.2.2.cmml" xref="S6.Ex59.m1.5.5.2.2">𝑗</ci></list></apply><apply id="S6.Ex59.m1.6.6.1.1.3.3.2.3.cmml" xref="S6.Ex59.m1.6.6.1.1.3.3.2.3"><apply id="S6.Ex59.m1.6.6.1.1.3.3.2.3.1.cmml" xref="S6.Ex59.m1.6.6.1.1.3.3.2.3.1"><csymbol cd="ambiguous" id="S6.Ex59.m1.6.6.1.1.3.3.2.3.1.1.cmml" xref="S6.Ex59.m1.6.6.1.1.3.3.2.3.1">subscript</csymbol><ci id="S6.Ex59.m1.6.6.1.1.3.3.2.3.1.2.cmml" xref="S6.Ex59.m1.6.6.1.1.3.3.2.3.1.2">Tr</ci><ci id="S6.Ex59.m1.6.6.1.1.3.3.2.3.1.3.cmml" xref="S6.Ex59.m1.6.6.1.1.3.3.2.3.1.3">𝑗</ci></apply><ci id="S6.Ex59.m1.6.6.1.1.3.3.2.3.2.cmml" xref="S6.Ex59.m1.6.6.1.1.3.3.2.3.2">𝑣</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex59.m1.6c">\displaystyle=\operatorname{Tr}_{m_{i}}v+\sum_{j\in\{m_{0},\dots,m_{i-1}\}}% \widetilde{b}_{i,j}\operatorname{Tr}_{j}v.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex59.m1.6d">= roman_Tr start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_v + ∑ start_POSTSUBSCRIPT italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_i - 1 end_POSTSUBSCRIPT } end_POSTSUBSCRIPT over~ start_ARG italic_b end_ARG start_POSTSUBSCRIPT italic_i , italic_j end_POSTSUBSCRIPT roman_Tr start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_v .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.6.p6.16">It follows that <math alttext="\operatorname{Tr}_{m_{0}}v=\mathcal{C}^{m_{0}}v=0" class="ltx_Math" display="inline" id="S6.SS3.6.p6.14.m1.1"><semantics id="S6.SS3.6.p6.14.m1.1a"><mrow id="S6.SS3.6.p6.14.m1.1.1" xref="S6.SS3.6.p6.14.m1.1.1.cmml"><mrow id="S6.SS3.6.p6.14.m1.1.1.2" xref="S6.SS3.6.p6.14.m1.1.1.2.cmml"><msub id="S6.SS3.6.p6.14.m1.1.1.2.1" xref="S6.SS3.6.p6.14.m1.1.1.2.1.cmml"><mi id="S6.SS3.6.p6.14.m1.1.1.2.1.2" xref="S6.SS3.6.p6.14.m1.1.1.2.1.2.cmml">Tr</mi><msub id="S6.SS3.6.p6.14.m1.1.1.2.1.3" xref="S6.SS3.6.p6.14.m1.1.1.2.1.3.cmml"><mi id="S6.SS3.6.p6.14.m1.1.1.2.1.3.2" xref="S6.SS3.6.p6.14.m1.1.1.2.1.3.2.cmml">m</mi><mn id="S6.SS3.6.p6.14.m1.1.1.2.1.3.3" xref="S6.SS3.6.p6.14.m1.1.1.2.1.3.3.cmml">0</mn></msub></msub><mo id="S6.SS3.6.p6.14.m1.1.1.2a" lspace="0.167em" xref="S6.SS3.6.p6.14.m1.1.1.2.cmml">⁡</mo><mi id="S6.SS3.6.p6.14.m1.1.1.2.2" xref="S6.SS3.6.p6.14.m1.1.1.2.2.cmml">v</mi></mrow><mo id="S6.SS3.6.p6.14.m1.1.1.3" xref="S6.SS3.6.p6.14.m1.1.1.3.cmml">=</mo><mrow id="S6.SS3.6.p6.14.m1.1.1.4" xref="S6.SS3.6.p6.14.m1.1.1.4.cmml"><msup id="S6.SS3.6.p6.14.m1.1.1.4.2" xref="S6.SS3.6.p6.14.m1.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.6.p6.14.m1.1.1.4.2.2" xref="S6.SS3.6.p6.14.m1.1.1.4.2.2.cmml">𝒞</mi><msub id="S6.SS3.6.p6.14.m1.1.1.4.2.3" xref="S6.SS3.6.p6.14.m1.1.1.4.2.3.cmml"><mi id="S6.SS3.6.p6.14.m1.1.1.4.2.3.2" xref="S6.SS3.6.p6.14.m1.1.1.4.2.3.2.cmml">m</mi><mn id="S6.SS3.6.p6.14.m1.1.1.4.2.3.3" xref="S6.SS3.6.p6.14.m1.1.1.4.2.3.3.cmml">0</mn></msub></msup><mo id="S6.SS3.6.p6.14.m1.1.1.4.1" xref="S6.SS3.6.p6.14.m1.1.1.4.1.cmml">⁢</mo><mi id="S6.SS3.6.p6.14.m1.1.1.4.3" xref="S6.SS3.6.p6.14.m1.1.1.4.3.cmml">v</mi></mrow><mo id="S6.SS3.6.p6.14.m1.1.1.5" xref="S6.SS3.6.p6.14.m1.1.1.5.cmml">=</mo><mn id="S6.SS3.6.p6.14.m1.1.1.6" xref="S6.SS3.6.p6.14.m1.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.14.m1.1b"><apply id="S6.SS3.6.p6.14.m1.1.1.cmml" xref="S6.SS3.6.p6.14.m1.1.1"><and id="S6.SS3.6.p6.14.m1.1.1a.cmml" xref="S6.SS3.6.p6.14.m1.1.1"></and><apply id="S6.SS3.6.p6.14.m1.1.1b.cmml" xref="S6.SS3.6.p6.14.m1.1.1"><eq id="S6.SS3.6.p6.14.m1.1.1.3.cmml" xref="S6.SS3.6.p6.14.m1.1.1.3"></eq><apply id="S6.SS3.6.p6.14.m1.1.1.2.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2"><apply id="S6.SS3.6.p6.14.m1.1.1.2.1.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2.1"><csymbol cd="ambiguous" id="S6.SS3.6.p6.14.m1.1.1.2.1.1.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2.1">subscript</csymbol><ci id="S6.SS3.6.p6.14.m1.1.1.2.1.2.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2.1.2">Tr</ci><apply id="S6.SS3.6.p6.14.m1.1.1.2.1.3.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.14.m1.1.1.2.1.3.1.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2.1.3">subscript</csymbol><ci id="S6.SS3.6.p6.14.m1.1.1.2.1.3.2.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2.1.3.2">𝑚</ci><cn id="S6.SS3.6.p6.14.m1.1.1.2.1.3.3.cmml" type="integer" xref="S6.SS3.6.p6.14.m1.1.1.2.1.3.3">0</cn></apply></apply><ci id="S6.SS3.6.p6.14.m1.1.1.2.2.cmml" xref="S6.SS3.6.p6.14.m1.1.1.2.2">𝑣</ci></apply><apply id="S6.SS3.6.p6.14.m1.1.1.4.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4"><times id="S6.SS3.6.p6.14.m1.1.1.4.1.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.1"></times><apply id="S6.SS3.6.p6.14.m1.1.1.4.2.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.14.m1.1.1.4.2.1.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.2">superscript</csymbol><ci id="S6.SS3.6.p6.14.m1.1.1.4.2.2.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.2.2">𝒞</ci><apply id="S6.SS3.6.p6.14.m1.1.1.4.2.3.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.2.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.14.m1.1.1.4.2.3.1.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.2.3">subscript</csymbol><ci id="S6.SS3.6.p6.14.m1.1.1.4.2.3.2.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.2.3.2">𝑚</ci><cn id="S6.SS3.6.p6.14.m1.1.1.4.2.3.3.cmml" type="integer" xref="S6.SS3.6.p6.14.m1.1.1.4.2.3.3">0</cn></apply></apply><ci id="S6.SS3.6.p6.14.m1.1.1.4.3.cmml" xref="S6.SS3.6.p6.14.m1.1.1.4.3">𝑣</ci></apply></apply><apply id="S6.SS3.6.p6.14.m1.1.1c.cmml" xref="S6.SS3.6.p6.14.m1.1.1"><eq id="S6.SS3.6.p6.14.m1.1.1.5.cmml" xref="S6.SS3.6.p6.14.m1.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.6.p6.14.m1.1.1.4.cmml" id="S6.SS3.6.p6.14.m1.1.1d.cmml" xref="S6.SS3.6.p6.14.m1.1.1"></share><cn id="S6.SS3.6.p6.14.m1.1.1.6.cmml" type="integer" xref="S6.SS3.6.p6.14.m1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.14.m1.1c">\operatorname{Tr}_{m_{0}}v=\mathcal{C}^{m_{0}}v=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.14.m1.1d">roman_Tr start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_v = caligraphic_C start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_v = 0</annotation></semantics></math> and successively we obtain <math alttext="\operatorname{Tr}_{m_{i}}v=\mathcal{C}^{m_{i}}v=0" class="ltx_Math" display="inline" id="S6.SS3.6.p6.15.m2.1"><semantics id="S6.SS3.6.p6.15.m2.1a"><mrow id="S6.SS3.6.p6.15.m2.1.1" xref="S6.SS3.6.p6.15.m2.1.1.cmml"><mrow id="S6.SS3.6.p6.15.m2.1.1.2" xref="S6.SS3.6.p6.15.m2.1.1.2.cmml"><msub id="S6.SS3.6.p6.15.m2.1.1.2.1" xref="S6.SS3.6.p6.15.m2.1.1.2.1.cmml"><mi id="S6.SS3.6.p6.15.m2.1.1.2.1.2" xref="S6.SS3.6.p6.15.m2.1.1.2.1.2.cmml">Tr</mi><msub id="S6.SS3.6.p6.15.m2.1.1.2.1.3" xref="S6.SS3.6.p6.15.m2.1.1.2.1.3.cmml"><mi id="S6.SS3.6.p6.15.m2.1.1.2.1.3.2" xref="S6.SS3.6.p6.15.m2.1.1.2.1.3.2.cmml">m</mi><mi id="S6.SS3.6.p6.15.m2.1.1.2.1.3.3" xref="S6.SS3.6.p6.15.m2.1.1.2.1.3.3.cmml">i</mi></msub></msub><mo id="S6.SS3.6.p6.15.m2.1.1.2a" lspace="0.167em" xref="S6.SS3.6.p6.15.m2.1.1.2.cmml">⁡</mo><mi id="S6.SS3.6.p6.15.m2.1.1.2.2" xref="S6.SS3.6.p6.15.m2.1.1.2.2.cmml">v</mi></mrow><mo id="S6.SS3.6.p6.15.m2.1.1.3" xref="S6.SS3.6.p6.15.m2.1.1.3.cmml">=</mo><mrow id="S6.SS3.6.p6.15.m2.1.1.4" xref="S6.SS3.6.p6.15.m2.1.1.4.cmml"><msup id="S6.SS3.6.p6.15.m2.1.1.4.2" xref="S6.SS3.6.p6.15.m2.1.1.4.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.6.p6.15.m2.1.1.4.2.2" xref="S6.SS3.6.p6.15.m2.1.1.4.2.2.cmml">𝒞</mi><msub id="S6.SS3.6.p6.15.m2.1.1.4.2.3" xref="S6.SS3.6.p6.15.m2.1.1.4.2.3.cmml"><mi id="S6.SS3.6.p6.15.m2.1.1.4.2.3.2" xref="S6.SS3.6.p6.15.m2.1.1.4.2.3.2.cmml">m</mi><mi id="S6.SS3.6.p6.15.m2.1.1.4.2.3.3" xref="S6.SS3.6.p6.15.m2.1.1.4.2.3.3.cmml">i</mi></msub></msup><mo id="S6.SS3.6.p6.15.m2.1.1.4.1" xref="S6.SS3.6.p6.15.m2.1.1.4.1.cmml">⁢</mo><mi id="S6.SS3.6.p6.15.m2.1.1.4.3" xref="S6.SS3.6.p6.15.m2.1.1.4.3.cmml">v</mi></mrow><mo id="S6.SS3.6.p6.15.m2.1.1.5" xref="S6.SS3.6.p6.15.m2.1.1.5.cmml">=</mo><mn id="S6.SS3.6.p6.15.m2.1.1.6" xref="S6.SS3.6.p6.15.m2.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.15.m2.1b"><apply id="S6.SS3.6.p6.15.m2.1.1.cmml" xref="S6.SS3.6.p6.15.m2.1.1"><and id="S6.SS3.6.p6.15.m2.1.1a.cmml" xref="S6.SS3.6.p6.15.m2.1.1"></and><apply id="S6.SS3.6.p6.15.m2.1.1b.cmml" xref="S6.SS3.6.p6.15.m2.1.1"><eq id="S6.SS3.6.p6.15.m2.1.1.3.cmml" xref="S6.SS3.6.p6.15.m2.1.1.3"></eq><apply id="S6.SS3.6.p6.15.m2.1.1.2.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2"><apply id="S6.SS3.6.p6.15.m2.1.1.2.1.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.1"><csymbol cd="ambiguous" id="S6.SS3.6.p6.15.m2.1.1.2.1.1.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.1">subscript</csymbol><ci id="S6.SS3.6.p6.15.m2.1.1.2.1.2.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.1.2">Tr</ci><apply id="S6.SS3.6.p6.15.m2.1.1.2.1.3.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.15.m2.1.1.2.1.3.1.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.1.3">subscript</csymbol><ci id="S6.SS3.6.p6.15.m2.1.1.2.1.3.2.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.1.3.2">𝑚</ci><ci id="S6.SS3.6.p6.15.m2.1.1.2.1.3.3.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.1.3.3">𝑖</ci></apply></apply><ci id="S6.SS3.6.p6.15.m2.1.1.2.2.cmml" xref="S6.SS3.6.p6.15.m2.1.1.2.2">𝑣</ci></apply><apply id="S6.SS3.6.p6.15.m2.1.1.4.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4"><times id="S6.SS3.6.p6.15.m2.1.1.4.1.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.1"></times><apply id="S6.SS3.6.p6.15.m2.1.1.4.2.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.2"><csymbol cd="ambiguous" id="S6.SS3.6.p6.15.m2.1.1.4.2.1.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.2">superscript</csymbol><ci id="S6.SS3.6.p6.15.m2.1.1.4.2.2.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.2.2">𝒞</ci><apply id="S6.SS3.6.p6.15.m2.1.1.4.2.3.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.2.3"><csymbol cd="ambiguous" id="S6.SS3.6.p6.15.m2.1.1.4.2.3.1.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.2.3">subscript</csymbol><ci id="S6.SS3.6.p6.15.m2.1.1.4.2.3.2.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.2.3.2">𝑚</ci><ci id="S6.SS3.6.p6.15.m2.1.1.4.2.3.3.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.2.3.3">𝑖</ci></apply></apply><ci id="S6.SS3.6.p6.15.m2.1.1.4.3.cmml" xref="S6.SS3.6.p6.15.m2.1.1.4.3">𝑣</ci></apply></apply><apply id="S6.SS3.6.p6.15.m2.1.1c.cmml" xref="S6.SS3.6.p6.15.m2.1.1"><eq id="S6.SS3.6.p6.15.m2.1.1.5.cmml" xref="S6.SS3.6.p6.15.m2.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.6.p6.15.m2.1.1.4.cmml" id="S6.SS3.6.p6.15.m2.1.1d.cmml" xref="S6.SS3.6.p6.15.m2.1.1"></share><cn id="S6.SS3.6.p6.15.m2.1.1.6.cmml" type="integer" xref="S6.SS3.6.p6.15.m2.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.15.m2.1c">\operatorname{Tr}_{m_{i}}v=\mathcal{C}^{m_{i}}v=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.15.m2.1d">roman_Tr start_POSTSUBSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_v = caligraphic_C start_POSTSUPERSCRIPT italic_m start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_v = 0</annotation></semantics></math> for <math alttext="0\leq i\leq n" class="ltx_Math" display="inline" id="S6.SS3.6.p6.16.m3.1"><semantics id="S6.SS3.6.p6.16.m3.1a"><mrow id="S6.SS3.6.p6.16.m3.1.1" xref="S6.SS3.6.p6.16.m3.1.1.cmml"><mn id="S6.SS3.6.p6.16.m3.1.1.2" xref="S6.SS3.6.p6.16.m3.1.1.2.cmml">0</mn><mo id="S6.SS3.6.p6.16.m3.1.1.3" xref="S6.SS3.6.p6.16.m3.1.1.3.cmml">≤</mo><mi id="S6.SS3.6.p6.16.m3.1.1.4" xref="S6.SS3.6.p6.16.m3.1.1.4.cmml">i</mi><mo id="S6.SS3.6.p6.16.m3.1.1.5" xref="S6.SS3.6.p6.16.m3.1.1.5.cmml">≤</mo><mi id="S6.SS3.6.p6.16.m3.1.1.6" xref="S6.SS3.6.p6.16.m3.1.1.6.cmml">n</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.6.p6.16.m3.1b"><apply id="S6.SS3.6.p6.16.m3.1.1.cmml" xref="S6.SS3.6.p6.16.m3.1.1"><and id="S6.SS3.6.p6.16.m3.1.1a.cmml" xref="S6.SS3.6.p6.16.m3.1.1"></and><apply id="S6.SS3.6.p6.16.m3.1.1b.cmml" xref="S6.SS3.6.p6.16.m3.1.1"><leq id="S6.SS3.6.p6.16.m3.1.1.3.cmml" xref="S6.SS3.6.p6.16.m3.1.1.3"></leq><cn id="S6.SS3.6.p6.16.m3.1.1.2.cmml" type="integer" xref="S6.SS3.6.p6.16.m3.1.1.2">0</cn><ci id="S6.SS3.6.p6.16.m3.1.1.4.cmml" xref="S6.SS3.6.p6.16.m3.1.1.4">𝑖</ci></apply><apply id="S6.SS3.6.p6.16.m3.1.1c.cmml" xref="S6.SS3.6.p6.16.m3.1.1"><leq id="S6.SS3.6.p6.16.m3.1.1.5.cmml" xref="S6.SS3.6.p6.16.m3.1.1.5"></leq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.6.p6.16.m3.1.1.4.cmml" id="S6.SS3.6.p6.16.m3.1.1d.cmml" xref="S6.SS3.6.p6.16.m3.1.1"></share><ci id="S6.SS3.6.p6.16.m3.1.1.6.cmml" xref="S6.SS3.6.p6.16.m3.1.1.6">𝑛</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.6.p6.16.m3.1c">0\leq i\leq n</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.6.p6.16.m3.1d">0 ≤ italic_i ≤ italic_n</annotation></semantics></math> as well. The converse statement is trivial, so we have proved the claim (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E11" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.11</span></a>).</p> </div> <div class="ltx_para" id="S6.SS3.7.p7"> <p class="ltx_p" id="S6.SS3.7.p7.1">From (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E11" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.11</span></a>) and the definition of <math alttext="W^{k_{0}+k_{1},p}_{0}" class="ltx_Math" display="inline" id="S6.SS3.7.p7.1.m1.2"><semantics id="S6.SS3.7.p7.1.m1.2a"><msubsup id="S6.SS3.7.p7.1.m1.2.3" xref="S6.SS3.7.p7.1.m1.2.3.cmml"><mi id="S6.SS3.7.p7.1.m1.2.3.2.2" xref="S6.SS3.7.p7.1.m1.2.3.2.2.cmml">W</mi><mn id="S6.SS3.7.p7.1.m1.2.3.3" xref="S6.SS3.7.p7.1.m1.2.3.3.cmml">0</mn><mrow id="S6.SS3.7.p7.1.m1.2.2.2.2" xref="S6.SS3.7.p7.1.m1.2.2.2.3.cmml"><mrow id="S6.SS3.7.p7.1.m1.2.2.2.2.1" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.cmml"><msub id="S6.SS3.7.p7.1.m1.2.2.2.2.1.2" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.cmml"><mi id="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.2" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.3" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.7.p7.1.m1.2.2.2.2.1.1" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.1.cmml">+</mo><msub id="S6.SS3.7.p7.1.m1.2.2.2.2.1.3" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.cmml"><mi id="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.2" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.2.cmml">k</mi><mn id="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.3" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.7.p7.1.m1.2.2.2.2.2" xref="S6.SS3.7.p7.1.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS3.7.p7.1.m1.1.1.1.1" xref="S6.SS3.7.p7.1.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><annotation-xml encoding="MathML-Content" id="S6.SS3.7.p7.1.m1.2b"><apply id="S6.SS3.7.p7.1.m1.2.3.cmml" xref="S6.SS3.7.p7.1.m1.2.3"><csymbol cd="ambiguous" id="S6.SS3.7.p7.1.m1.2.3.1.cmml" xref="S6.SS3.7.p7.1.m1.2.3">subscript</csymbol><apply id="S6.SS3.7.p7.1.m1.2.3.2.cmml" xref="S6.SS3.7.p7.1.m1.2.3"><csymbol cd="ambiguous" id="S6.SS3.7.p7.1.m1.2.3.2.1.cmml" xref="S6.SS3.7.p7.1.m1.2.3">superscript</csymbol><ci id="S6.SS3.7.p7.1.m1.2.3.2.2.cmml" xref="S6.SS3.7.p7.1.m1.2.3.2.2">𝑊</ci><list id="S6.SS3.7.p7.1.m1.2.2.2.3.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2"><apply id="S6.SS3.7.p7.1.m1.2.2.2.2.1.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1"><plus id="S6.SS3.7.p7.1.m1.2.2.2.2.1.1.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.1"></plus><apply id="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.1.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.2.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.2.3">0</cn></apply><apply id="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.1.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.3">subscript</csymbol><ci id="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.2.cmml" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.2">𝑘</ci><cn id="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.3.cmml" type="integer" xref="S6.SS3.7.p7.1.m1.2.2.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.7.p7.1.m1.1.1.1.1.cmml" xref="S6.SS3.7.p7.1.m1.1.1.1.1">𝑝</ci></list></apply><cn id="S6.SS3.7.p7.1.m1.2.3.3.cmml" type="integer" xref="S6.SS3.7.p7.1.m1.2.3.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.7.p7.1.m1.2c">W^{k_{0}+k_{1},p}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.7.p7.1.m1.2d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math>, it follows that</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex60"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math 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xref="S6.Ex60.m1.2.2.2.2.1.1.cmml">+</mo><msub id="S6.Ex60.m1.2.2.2.2.1.3" xref="S6.Ex60.m1.2.2.2.2.1.3.cmml"><mi id="S6.Ex60.m1.2.2.2.2.1.3.2" xref="S6.Ex60.m1.2.2.2.2.1.3.2.cmml">k</mi><mn id="S6.Ex60.m1.2.2.2.2.1.3.3" xref="S6.Ex60.m1.2.2.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.Ex60.m1.2.2.2.2.2" xref="S6.Ex60.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex60.m1.1.1.1.1" xref="S6.Ex60.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex60.m1.7.7.1.1.3" xref="S6.Ex60.m1.7.7.1.1.3.cmml">=</mo><msubsup id="S6.Ex60.m1.7.7.1.1.4" xref="S6.Ex60.m1.7.7.1.1.4.cmml"><mi id="S6.Ex60.m1.7.7.1.1.4.2.2" xref="S6.Ex60.m1.7.7.1.1.4.2.2.cmml">W</mi><msub id="S6.Ex60.m1.7.7.1.1.4.3" xref="S6.Ex60.m1.7.7.1.1.4.3.cmml"><mover accent="true" id="S6.Ex60.m1.7.7.1.1.4.3.2" xref="S6.Ex60.m1.7.7.1.1.4.3.2.cmml"><mi id="S6.Ex60.m1.7.7.1.1.4.3.2.2" xref="S6.Ex60.m1.7.7.1.1.4.3.2.2.cmml">Tr</mi><mo id="S6.Ex60.m1.7.7.1.1.4.3.2.1" xref="S6.Ex60.m1.7.7.1.1.4.3.2.1.cmml">¯</mo></mover><mi 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encoding="application/x-llamapun" id="S6.Ex60.m1.7d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_W start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.7.p7.4">By Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem6" title="Theorem 5.6. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.6</span></a> there exists a coretraction</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex61"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\operatorname{ext}_{\mathcal{C}}:\prod_{j=0}^{q}B^{k_{0}+k_{1}-j-\frac{\gamma+% 1}{p}}_{p,p}(\mathbb{R}^{d-1};X)\to W^{k_{0}+k_{1},p}" class="ltx_Math" display="block" id="S6.Ex61.m1.6"><semantics id="S6.Ex61.m1.6a"><mrow id="S6.Ex61.m1.6.6" xref="S6.Ex61.m1.6.6.cmml"><msub id="S6.Ex61.m1.6.6.3" xref="S6.Ex61.m1.6.6.3.cmml"><mi id="S6.Ex61.m1.6.6.3.2" xref="S6.Ex61.m1.6.6.3.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex61.m1.6.6.3.3" xref="S6.Ex61.m1.6.6.3.3.cmml">𝒞</mi></msub><mo id="S6.Ex61.m1.6.6.2" lspace="0.278em" rspace="0.111em" 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xref="S6.Ex61.m1.4.4.2.2.1.3"><csymbol cd="ambiguous" id="S6.Ex61.m1.4.4.2.2.1.3.1.cmml" xref="S6.Ex61.m1.4.4.2.2.1.3">subscript</csymbol><ci id="S6.Ex61.m1.4.4.2.2.1.3.2.cmml" xref="S6.Ex61.m1.4.4.2.2.1.3.2">𝑘</ci><cn id="S6.Ex61.m1.4.4.2.2.1.3.3.cmml" type="integer" xref="S6.Ex61.m1.4.4.2.2.1.3.3">1</cn></apply></apply><ci id="S6.Ex61.m1.3.3.1.1.cmml" xref="S6.Ex61.m1.3.3.1.1">𝑝</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex61.m1.6c">\operatorname{ext}_{\mathcal{C}}:\prod_{j=0}^{q}B^{k_{0}+k_{1}-j-\frac{\gamma+% 1}{p}}_{p,p}(\mathbb{R}^{d-1};X)\to W^{k_{0}+k_{1},p}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex61.m1.6d">roman_ext start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT : ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 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xref="S6.SS3.7.p7.3.m2.2.2.3.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.7.p7.3.m2.2.2.2" xref="S6.SS3.7.p7.3.m2.2.2.2.cmml">∈</mo><mrow id="S6.SS3.7.p7.3.m2.2.2.1.1" xref="S6.SS3.7.p7.3.m2.2.2.1.2.cmml"><mo id="S6.SS3.7.p7.3.m2.2.2.1.1.2" stretchy="false" xref="S6.SS3.7.p7.3.m2.2.2.1.2.cmml">(</mo><mrow id="S6.SS3.7.p7.3.m2.2.2.1.1.1" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.cmml"><mi id="S6.SS3.7.p7.3.m2.2.2.1.1.1.2" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.2.cmml">q</mi><mo id="S6.SS3.7.p7.3.m2.2.2.1.1.1.1" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.1.cmml">+</mo><mfrac id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.cmml"><mrow id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.cmml"><mi id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.2" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.2.cmml">γ</mi><mo id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.1" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.1.cmml">+</mo><mn id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.3" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.3" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS3.7.p7.3.m2.2.2.1.1.3" xref="S6.SS3.7.p7.3.m2.2.2.1.2.cmml">,</mo><mi id="S6.SS3.7.p7.3.m2.1.1" xref="S6.SS3.7.p7.3.m2.1.1.cmml">k</mi><mo id="S6.SS3.7.p7.3.m2.2.2.1.1.4" stretchy="false" xref="S6.SS3.7.p7.3.m2.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.7.p7.3.m2.2b"><apply id="S6.SS3.7.p7.3.m2.2.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2"><in id="S6.SS3.7.p7.3.m2.2.2.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.2"></in><apply id="S6.SS3.7.p7.3.m2.2.2.3.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3"><plus id="S6.SS3.7.p7.3.m2.2.2.3.1.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3.1"></plus><apply id="S6.SS3.7.p7.3.m2.2.2.3.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3.2"><csymbol cd="ambiguous" id="S6.SS3.7.p7.3.m2.2.2.3.2.1.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3.2">subscript</csymbol><ci id="S6.SS3.7.p7.3.m2.2.2.3.2.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3.2.2">𝑘</ci><cn id="S6.SS3.7.p7.3.m2.2.2.3.2.3.cmml" type="integer" xref="S6.SS3.7.p7.3.m2.2.2.3.2.3">0</cn></apply><apply id="S6.SS3.7.p7.3.m2.2.2.3.3.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.7.p7.3.m2.2.2.3.3.1.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3.3">subscript</csymbol><ci id="S6.SS3.7.p7.3.m2.2.2.3.3.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.3.3.2">𝑘</ci><cn id="S6.SS3.7.p7.3.m2.2.2.3.3.3.cmml" type="integer" xref="S6.SS3.7.p7.3.m2.2.2.3.3.3">1</cn></apply></apply><interval closure="open-closed" id="S6.SS3.7.p7.3.m2.2.2.1.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1"><apply id="S6.SS3.7.p7.3.m2.2.2.1.1.1.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1"><plus id="S6.SS3.7.p7.3.m2.2.2.1.1.1.1.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.1"></plus><ci id="S6.SS3.7.p7.3.m2.2.2.1.1.1.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.2">𝑞</ci><apply id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3"><divide id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.1.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3"></divide><apply id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2"><plus id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.1.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.1"></plus><ci id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.2.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.2">𝛾</ci><cn id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.2.3">1</cn></apply><ci id="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.3.cmml" xref="S6.SS3.7.p7.3.m2.2.2.1.1.1.3.3">𝑝</ci></apply></apply><ci id="S6.SS3.7.p7.3.m2.1.1.cmml" xref="S6.SS3.7.p7.3.m2.1.1">𝑘</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.7.p7.3.m2.2c">k_{0}+k_{1}\in(q+\frac{\gamma+1}{p},k]</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.7.p7.3.m2.2d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ ( italic_q + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG , italic_k ]</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS3.8.p8"> <p class="ltx_p" id="S6.SS3.8.p8.9">Let <math alttext="u\in W^{k_{0}+\ell,p}_{\mathcal{B}}=W^{k_{0}+\ell,p}_{\widetilde{\mathcal{B}}}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.1.m1.4"><semantics id="S6.SS3.8.p8.1.m1.4a"><mrow id="S6.SS3.8.p8.1.m1.4.5" xref="S6.SS3.8.p8.1.m1.4.5.cmml"><mi id="S6.SS3.8.p8.1.m1.4.5.2" xref="S6.SS3.8.p8.1.m1.4.5.2.cmml">u</mi><mo id="S6.SS3.8.p8.1.m1.4.5.3" xref="S6.SS3.8.p8.1.m1.4.5.3.cmml">∈</mo><msubsup id="S6.SS3.8.p8.1.m1.4.5.4" xref="S6.SS3.8.p8.1.m1.4.5.4.cmml"><mi id="S6.SS3.8.p8.1.m1.4.5.4.2.2" xref="S6.SS3.8.p8.1.m1.4.5.4.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.8.p8.1.m1.4.5.4.3" xref="S6.SS3.8.p8.1.m1.4.5.4.3.cmml">ℬ</mi><mrow id="S6.SS3.8.p8.1.m1.2.2.2.2" xref="S6.SS3.8.p8.1.m1.2.2.2.3.cmml"><mrow id="S6.SS3.8.p8.1.m1.2.2.2.2.1" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.cmml"><msub id="S6.SS3.8.p8.1.m1.2.2.2.2.1.2" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.cmml"><mi id="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.2" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.3" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.1.m1.2.2.2.2.1.1" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.SS3.8.p8.1.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.8.p8.1.m1.2.2.2.2.2" xref="S6.SS3.8.p8.1.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.1.m1.1.1.1.1" xref="S6.SS3.8.p8.1.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.SS3.8.p8.1.m1.4.5.5" xref="S6.SS3.8.p8.1.m1.4.5.5.cmml">=</mo><msubsup id="S6.SS3.8.p8.1.m1.4.5.6" xref="S6.SS3.8.p8.1.m1.4.5.6.cmml"><mi id="S6.SS3.8.p8.1.m1.4.5.6.2.2" xref="S6.SS3.8.p8.1.m1.4.5.6.2.2.cmml">W</mi><mover accent="true" id="S6.SS3.8.p8.1.m1.4.5.6.3" xref="S6.SS3.8.p8.1.m1.4.5.6.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.8.p8.1.m1.4.5.6.3.2" xref="S6.SS3.8.p8.1.m1.4.5.6.3.2.cmml">ℬ</mi><mo id="S6.SS3.8.p8.1.m1.4.5.6.3.1" xref="S6.SS3.8.p8.1.m1.4.5.6.3.1.cmml">~</mo></mover><mrow id="S6.SS3.8.p8.1.m1.4.4.2.2" xref="S6.SS3.8.p8.1.m1.4.4.2.3.cmml"><mrow id="S6.SS3.8.p8.1.m1.4.4.2.2.1" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.cmml"><msub id="S6.SS3.8.p8.1.m1.4.4.2.2.1.2" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.cmml"><mi id="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.2" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.3" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.1.m1.4.4.2.2.1.1" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.1.cmml">+</mo><mi id="S6.SS3.8.p8.1.m1.4.4.2.2.1.3" mathvariant="normal" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.8.p8.1.m1.4.4.2.2.2" xref="S6.SS3.8.p8.1.m1.4.4.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.1.m1.3.3.1.1" xref="S6.SS3.8.p8.1.m1.3.3.1.1.cmml">p</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.1.m1.4b"><apply id="S6.SS3.8.p8.1.m1.4.5.cmml" xref="S6.SS3.8.p8.1.m1.4.5"><and id="S6.SS3.8.p8.1.m1.4.5a.cmml" xref="S6.SS3.8.p8.1.m1.4.5"></and><apply id="S6.SS3.8.p8.1.m1.4.5b.cmml" xref="S6.SS3.8.p8.1.m1.4.5"><in id="S6.SS3.8.p8.1.m1.4.5.3.cmml" xref="S6.SS3.8.p8.1.m1.4.5.3"></in><ci id="S6.SS3.8.p8.1.m1.4.5.2.cmml" xref="S6.SS3.8.p8.1.m1.4.5.2">𝑢</ci><apply id="S6.SS3.8.p8.1.m1.4.5.4.cmml" xref="S6.SS3.8.p8.1.m1.4.5.4"><csymbol cd="ambiguous" id="S6.SS3.8.p8.1.m1.4.5.4.1.cmml" xref="S6.SS3.8.p8.1.m1.4.5.4">subscript</csymbol><apply id="S6.SS3.8.p8.1.m1.4.5.4.2.cmml" xref="S6.SS3.8.p8.1.m1.4.5.4"><csymbol cd="ambiguous" id="S6.SS3.8.p8.1.m1.4.5.4.2.1.cmml" xref="S6.SS3.8.p8.1.m1.4.5.4">superscript</csymbol><ci id="S6.SS3.8.p8.1.m1.4.5.4.2.2.cmml" xref="S6.SS3.8.p8.1.m1.4.5.4.2.2">𝑊</ci><list id="S6.SS3.8.p8.1.m1.2.2.2.3.cmml" xref="S6.SS3.8.p8.1.m1.2.2.2.2"><apply id="S6.SS3.8.p8.1.m1.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1"><plus id="S6.SS3.8.p8.1.m1.2.2.2.2.1.1.cmml" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.1"></plus><apply id="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.cmml" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.1.cmml" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.2.cmml" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.2.3">0</cn></apply><ci id="S6.SS3.8.p8.1.m1.2.2.2.2.1.3.cmml" xref="S6.SS3.8.p8.1.m1.2.2.2.2.1.3">ℓ</ci></apply><ci id="S6.SS3.8.p8.1.m1.1.1.1.1.cmml" xref="S6.SS3.8.p8.1.m1.1.1.1.1">𝑝</ci></list></apply><ci id="S6.SS3.8.p8.1.m1.4.5.4.3.cmml" xref="S6.SS3.8.p8.1.m1.4.5.4.3">ℬ</ci></apply></apply><apply id="S6.SS3.8.p8.1.m1.4.5c.cmml" xref="S6.SS3.8.p8.1.m1.4.5"><eq id="S6.SS3.8.p8.1.m1.4.5.5.cmml" xref="S6.SS3.8.p8.1.m1.4.5.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S6.SS3.8.p8.1.m1.4.5.4.cmml" id="S6.SS3.8.p8.1.m1.4.5d.cmml" xref="S6.SS3.8.p8.1.m1.4.5"></share><apply id="S6.SS3.8.p8.1.m1.4.5.6.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6"><csymbol cd="ambiguous" id="S6.SS3.8.p8.1.m1.4.5.6.1.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6">subscript</csymbol><apply id="S6.SS3.8.p8.1.m1.4.5.6.2.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6"><csymbol cd="ambiguous" id="S6.SS3.8.p8.1.m1.4.5.6.2.1.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6">superscript</csymbol><ci id="S6.SS3.8.p8.1.m1.4.5.6.2.2.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6.2.2">𝑊</ci><list id="S6.SS3.8.p8.1.m1.4.4.2.3.cmml" xref="S6.SS3.8.p8.1.m1.4.4.2.2"><apply id="S6.SS3.8.p8.1.m1.4.4.2.2.1.cmml" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1"><plus id="S6.SS3.8.p8.1.m1.4.4.2.2.1.1.cmml" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.1"></plus><apply id="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.cmml" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.1.cmml" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.2">subscript</csymbol><ci id="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.2.cmml" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.2.3">0</cn></apply><ci id="S6.SS3.8.p8.1.m1.4.4.2.2.1.3.cmml" xref="S6.SS3.8.p8.1.m1.4.4.2.2.1.3">ℓ</ci></apply><ci id="S6.SS3.8.p8.1.m1.3.3.1.1.cmml" xref="S6.SS3.8.p8.1.m1.3.3.1.1">𝑝</ci></list></apply><apply id="S6.SS3.8.p8.1.m1.4.5.6.3.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6.3"><ci id="S6.SS3.8.p8.1.m1.4.5.6.3.1.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6.3.1">~</ci><ci id="S6.SS3.8.p8.1.m1.4.5.6.3.2.cmml" xref="S6.SS3.8.p8.1.m1.4.5.6.3.2">ℬ</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.1.m1.4c">u\in W^{k_{0}+\ell,p}_{\mathcal{B}}=W^{k_{0}+\ell,p}_{\widetilde{\mathcal{B}}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.1.m1.4d">italic_u ∈ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over~ start_ARG caligraphic_B end_ARG end_POSTSUBSCRIPT</annotation></semantics></math> (recall (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E6" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.6</span></a>)) and we will prove that <math alttext="u\in[W^{k_{0},p},W^{k_{0}+k_{1},p}_{\mathcal{B}}]_{\theta}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.2.m2.6"><semantics id="S6.SS3.8.p8.2.m2.6a"><mrow id="S6.SS3.8.p8.2.m2.6.6" xref="S6.SS3.8.p8.2.m2.6.6.cmml"><mi id="S6.SS3.8.p8.2.m2.6.6.4" xref="S6.SS3.8.p8.2.m2.6.6.4.cmml">u</mi><mo id="S6.SS3.8.p8.2.m2.6.6.3" xref="S6.SS3.8.p8.2.m2.6.6.3.cmml">∈</mo><msub id="S6.SS3.8.p8.2.m2.6.6.2" xref="S6.SS3.8.p8.2.m2.6.6.2.cmml"><mrow id="S6.SS3.8.p8.2.m2.6.6.2.2.2" xref="S6.SS3.8.p8.2.m2.6.6.2.2.3.cmml"><mo id="S6.SS3.8.p8.2.m2.6.6.2.2.2.3" stretchy="false" xref="S6.SS3.8.p8.2.m2.6.6.2.2.3.cmml">[</mo><msup id="S6.SS3.8.p8.2.m2.5.5.1.1.1.1" xref="S6.SS3.8.p8.2.m2.5.5.1.1.1.1.cmml"><mi id="S6.SS3.8.p8.2.m2.5.5.1.1.1.1.2" xref="S6.SS3.8.p8.2.m2.5.5.1.1.1.1.2.cmml">W</mi><mrow id="S6.SS3.8.p8.2.m2.2.2.2.2" xref="S6.SS3.8.p8.2.m2.2.2.2.3.cmml"><msub id="S6.SS3.8.p8.2.m2.2.2.2.2.1" xref="S6.SS3.8.p8.2.m2.2.2.2.2.1.cmml"><mi id="S6.SS3.8.p8.2.m2.2.2.2.2.1.2" xref="S6.SS3.8.p8.2.m2.2.2.2.2.1.2.cmml">k</mi><mn id="S6.SS3.8.p8.2.m2.2.2.2.2.1.3" xref="S6.SS3.8.p8.2.m2.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.2.m2.2.2.2.2.2" xref="S6.SS3.8.p8.2.m2.2.2.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.2.m2.1.1.1.1" xref="S6.SS3.8.p8.2.m2.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.SS3.8.p8.2.m2.6.6.2.2.2.4" xref="S6.SS3.8.p8.2.m2.6.6.2.2.3.cmml">,</mo><msubsup id="S6.SS3.8.p8.2.m2.6.6.2.2.2.2" xref="S6.SS3.8.p8.2.m2.6.6.2.2.2.2.cmml"><mi id="S6.SS3.8.p8.2.m2.6.6.2.2.2.2.2.2" xref="S6.SS3.8.p8.2.m2.6.6.2.2.2.2.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.8.p8.2.m2.6.6.2.2.2.2.3" xref="S6.SS3.8.p8.2.m2.6.6.2.2.2.2.3.cmml">ℬ</mi><mrow id="S6.SS3.8.p8.2.m2.4.4.2.2" xref="S6.SS3.8.p8.2.m2.4.4.2.3.cmml"><mrow id="S6.SS3.8.p8.2.m2.4.4.2.2.1" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.cmml"><msub id="S6.SS3.8.p8.2.m2.4.4.2.2.1.2" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.2.cmml"><mi id="S6.SS3.8.p8.2.m2.4.4.2.2.1.2.2" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.8.p8.2.m2.4.4.2.2.1.2.3" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.2.m2.4.4.2.2.1.1" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.1.cmml">+</mo><msub id="S6.SS3.8.p8.2.m2.4.4.2.2.1.3" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.3.cmml"><mi id="S6.SS3.8.p8.2.m2.4.4.2.2.1.3.2" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.3.2.cmml">k</mi><mn id="S6.SS3.8.p8.2.m2.4.4.2.2.1.3.3" xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.8.p8.2.m2.4.4.2.2.2" xref="S6.SS3.8.p8.2.m2.4.4.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.2.m2.3.3.1.1" xref="S6.SS3.8.p8.2.m2.3.3.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.SS3.8.p8.2.m2.6.6.2.2.2.5" stretchy="false" xref="S6.SS3.8.p8.2.m2.6.6.2.2.3.cmml">]</mo></mrow><mi id="S6.SS3.8.p8.2.m2.6.6.2.4" xref="S6.SS3.8.p8.2.m2.6.6.2.4.cmml">θ</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.2.m2.6b"><apply id="S6.SS3.8.p8.2.m2.6.6.cmml" xref="S6.SS3.8.p8.2.m2.6.6"><in id="S6.SS3.8.p8.2.m2.6.6.3.cmml" xref="S6.SS3.8.p8.2.m2.6.6.3"></in><ci id="S6.SS3.8.p8.2.m2.6.6.4.cmml" xref="S6.SS3.8.p8.2.m2.6.6.4">𝑢</ci><apply id="S6.SS3.8.p8.2.m2.6.6.2.cmml" xref="S6.SS3.8.p8.2.m2.6.6.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.2.m2.6.6.2.3.cmml" xref="S6.SS3.8.p8.2.m2.6.6.2">subscript</csymbol><interval closure="closed" id="S6.SS3.8.p8.2.m2.6.6.2.2.3.cmml" xref="S6.SS3.8.p8.2.m2.6.6.2.2.2"><apply id="S6.SS3.8.p8.2.m2.5.5.1.1.1.1.cmml" xref="S6.SS3.8.p8.2.m2.5.5.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.2.m2.5.5.1.1.1.1.1.cmml" xref="S6.SS3.8.p8.2.m2.5.5.1.1.1.1">superscript</csymbol><ci id="S6.SS3.8.p8.2.m2.5.5.1.1.1.1.2.cmml" xref="S6.SS3.8.p8.2.m2.5.5.1.1.1.1.2">𝑊</ci><list id="S6.SS3.8.p8.2.m2.2.2.2.3.cmml" xref="S6.SS3.8.p8.2.m2.2.2.2.2"><apply 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xref="S6.SS3.8.p8.2.m2.4.4.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.8.p8.2.m2.3.3.1.1.cmml" xref="S6.SS3.8.p8.2.m2.3.3.1.1">𝑝</ci></list></apply><ci id="S6.SS3.8.p8.2.m2.6.6.2.2.2.2.3.cmml" xref="S6.SS3.8.p8.2.m2.6.6.2.2.2.2.3">ℬ</ci></apply></interval><ci id="S6.SS3.8.p8.2.m2.6.6.2.4.cmml" xref="S6.SS3.8.p8.2.m2.6.6.2.4">𝜃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.2.m2.6c">u\in[W^{k_{0},p},W^{k_{0}+k_{1},p}_{\mathcal{B}}]_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.2.m2.6d">italic_u ∈ [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\theta=\frac{\ell}{k_{1}}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.3.m3.1"><semantics id="S6.SS3.8.p8.3.m3.1a"><mrow id="S6.SS3.8.p8.3.m3.1.1" xref="S6.SS3.8.p8.3.m3.1.1.cmml"><mi id="S6.SS3.8.p8.3.m3.1.1.2" xref="S6.SS3.8.p8.3.m3.1.1.2.cmml">θ</mi><mo id="S6.SS3.8.p8.3.m3.1.1.1" xref="S6.SS3.8.p8.3.m3.1.1.1.cmml">=</mo><mfrac id="S6.SS3.8.p8.3.m3.1.1.3" xref="S6.SS3.8.p8.3.m3.1.1.3.cmml"><mi id="S6.SS3.8.p8.3.m3.1.1.3.2" mathvariant="normal" xref="S6.SS3.8.p8.3.m3.1.1.3.2.cmml">ℓ</mi><msub id="S6.SS3.8.p8.3.m3.1.1.3.3" xref="S6.SS3.8.p8.3.m3.1.1.3.3.cmml"><mi id="S6.SS3.8.p8.3.m3.1.1.3.3.2" xref="S6.SS3.8.p8.3.m3.1.1.3.3.2.cmml">k</mi><mn id="S6.SS3.8.p8.3.m3.1.1.3.3.3" xref="S6.SS3.8.p8.3.m3.1.1.3.3.3.cmml">1</mn></msub></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.3.m3.1b"><apply id="S6.SS3.8.p8.3.m3.1.1.cmml" xref="S6.SS3.8.p8.3.m3.1.1"><eq id="S6.SS3.8.p8.3.m3.1.1.1.cmml" xref="S6.SS3.8.p8.3.m3.1.1.1"></eq><ci id="S6.SS3.8.p8.3.m3.1.1.2.cmml" xref="S6.SS3.8.p8.3.m3.1.1.2">𝜃</ci><apply id="S6.SS3.8.p8.3.m3.1.1.3.cmml" xref="S6.SS3.8.p8.3.m3.1.1.3"><divide id="S6.SS3.8.p8.3.m3.1.1.3.1.cmml" xref="S6.SS3.8.p8.3.m3.1.1.3"></divide><ci id="S6.SS3.8.p8.3.m3.1.1.3.2.cmml" xref="S6.SS3.8.p8.3.m3.1.1.3.2">ℓ</ci><apply id="S6.SS3.8.p8.3.m3.1.1.3.3.cmml" xref="S6.SS3.8.p8.3.m3.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.3.m3.1.1.3.3.1.cmml" xref="S6.SS3.8.p8.3.m3.1.1.3.3">subscript</csymbol><ci id="S6.SS3.8.p8.3.m3.1.1.3.3.2.cmml" xref="S6.SS3.8.p8.3.m3.1.1.3.3.2">𝑘</ci><cn id="S6.SS3.8.p8.3.m3.1.1.3.3.3.cmml" type="integer" xref="S6.SS3.8.p8.3.m3.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.3.m3.1c">\theta=\frac{\ell}{k_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.3.m3.1d">italic_θ = divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>. This will be done by constructing a suitable <math alttext="v" class="ltx_Math" display="inline" id="S6.SS3.8.p8.4.m4.1"><semantics id="S6.SS3.8.p8.4.m4.1a"><mi id="S6.SS3.8.p8.4.m4.1.1" xref="S6.SS3.8.p8.4.m4.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.4.m4.1b"><ci id="S6.SS3.8.p8.4.m4.1.1.cmml" xref="S6.SS3.8.p8.4.m4.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.4.m4.1c">v</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.4.m4.1d">italic_v</annotation></semantics></math> such that <math alttext="u=v+(u-v)" class="ltx_Math" display="inline" id="S6.SS3.8.p8.5.m5.1"><semantics id="S6.SS3.8.p8.5.m5.1a"><mrow id="S6.SS3.8.p8.5.m5.1.1" xref="S6.SS3.8.p8.5.m5.1.1.cmml"><mi id="S6.SS3.8.p8.5.m5.1.1.3" xref="S6.SS3.8.p8.5.m5.1.1.3.cmml">u</mi><mo id="S6.SS3.8.p8.5.m5.1.1.2" xref="S6.SS3.8.p8.5.m5.1.1.2.cmml">=</mo><mrow id="S6.SS3.8.p8.5.m5.1.1.1" xref="S6.SS3.8.p8.5.m5.1.1.1.cmml"><mi id="S6.SS3.8.p8.5.m5.1.1.1.3" xref="S6.SS3.8.p8.5.m5.1.1.1.3.cmml">v</mi><mo id="S6.SS3.8.p8.5.m5.1.1.1.2" xref="S6.SS3.8.p8.5.m5.1.1.1.2.cmml">+</mo><mrow id="S6.SS3.8.p8.5.m5.1.1.1.1.1" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.cmml"><mo id="S6.SS3.8.p8.5.m5.1.1.1.1.1.2" stretchy="false" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.cmml"><mi id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.2" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.2.cmml">u</mi><mo id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.1" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.1.cmml">−</mo><mi id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.3" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S6.SS3.8.p8.5.m5.1.1.1.1.1.3" stretchy="false" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.5.m5.1b"><apply id="S6.SS3.8.p8.5.m5.1.1.cmml" xref="S6.SS3.8.p8.5.m5.1.1"><eq id="S6.SS3.8.p8.5.m5.1.1.2.cmml" xref="S6.SS3.8.p8.5.m5.1.1.2"></eq><ci id="S6.SS3.8.p8.5.m5.1.1.3.cmml" xref="S6.SS3.8.p8.5.m5.1.1.3">𝑢</ci><apply id="S6.SS3.8.p8.5.m5.1.1.1.cmml" xref="S6.SS3.8.p8.5.m5.1.1.1"><plus id="S6.SS3.8.p8.5.m5.1.1.1.2.cmml" xref="S6.SS3.8.p8.5.m5.1.1.1.2"></plus><ci id="S6.SS3.8.p8.5.m5.1.1.1.3.cmml" xref="S6.SS3.8.p8.5.m5.1.1.1.3">𝑣</ci><apply id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.cmml" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1"><minus id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.1.cmml" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.1"></minus><ci id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.2.cmml" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.2">𝑢</ci><ci id="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.3.cmml" xref="S6.SS3.8.p8.5.m5.1.1.1.1.1.1.3">𝑣</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.5.m5.1c">u=v+(u-v)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.5.m5.1d">italic_u = italic_v + ( italic_u - italic_v )</annotation></semantics></math> and all the traces of <math alttext="u-v" class="ltx_Math" display="inline" id="S6.SS3.8.p8.6.m6.1"><semantics id="S6.SS3.8.p8.6.m6.1a"><mrow id="S6.SS3.8.p8.6.m6.1.1" xref="S6.SS3.8.p8.6.m6.1.1.cmml"><mi id="S6.SS3.8.p8.6.m6.1.1.2" xref="S6.SS3.8.p8.6.m6.1.1.2.cmml">u</mi><mo id="S6.SS3.8.p8.6.m6.1.1.1" xref="S6.SS3.8.p8.6.m6.1.1.1.cmml">−</mo><mi id="S6.SS3.8.p8.6.m6.1.1.3" xref="S6.SS3.8.p8.6.m6.1.1.3.cmml">v</mi></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.6.m6.1b"><apply id="S6.SS3.8.p8.6.m6.1.1.cmml" xref="S6.SS3.8.p8.6.m6.1.1"><minus id="S6.SS3.8.p8.6.m6.1.1.1.cmml" xref="S6.SS3.8.p8.6.m6.1.1.1"></minus><ci id="S6.SS3.8.p8.6.m6.1.1.2.cmml" xref="S6.SS3.8.p8.6.m6.1.1.2">𝑢</ci><ci id="S6.SS3.8.p8.6.m6.1.1.3.cmml" xref="S6.SS3.8.p8.6.m6.1.1.3">𝑣</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.6.m6.1c">u-v</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.6.m6.1d">italic_u - italic_v</annotation></semantics></math> are zero. Let <math alttext="q_{1}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.7.m7.1"><semantics id="S6.SS3.8.p8.7.m7.1a"><msub id="S6.SS3.8.p8.7.m7.1.1" xref="S6.SS3.8.p8.7.m7.1.1.cmml"><mi id="S6.SS3.8.p8.7.m7.1.1.2" xref="S6.SS3.8.p8.7.m7.1.1.2.cmml">q</mi><mn id="S6.SS3.8.p8.7.m7.1.1.3" xref="S6.SS3.8.p8.7.m7.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.7.m7.1b"><apply id="S6.SS3.8.p8.7.m7.1.1.cmml" xref="S6.SS3.8.p8.7.m7.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.7.m7.1.1.1.cmml" xref="S6.SS3.8.p8.7.m7.1.1">subscript</csymbol><ci id="S6.SS3.8.p8.7.m7.1.1.2.cmml" xref="S6.SS3.8.p8.7.m7.1.1.2">𝑞</ci><cn id="S6.SS3.8.p8.7.m7.1.1.3.cmml" type="integer" xref="S6.SS3.8.p8.7.m7.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.7.m7.1c">q_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.7.m7.1d">italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> be the largest integer such that <math alttext="q_{1}+\frac{\gamma+1}{p}<k_{0}+\ell" class="ltx_Math" display="inline" id="S6.SS3.8.p8.8.m8.1"><semantics id="S6.SS3.8.p8.8.m8.1a"><mrow id="S6.SS3.8.p8.8.m8.1.1" xref="S6.SS3.8.p8.8.m8.1.1.cmml"><mrow id="S6.SS3.8.p8.8.m8.1.1.2" xref="S6.SS3.8.p8.8.m8.1.1.2.cmml"><msub id="S6.SS3.8.p8.8.m8.1.1.2.2" xref="S6.SS3.8.p8.8.m8.1.1.2.2.cmml"><mi id="S6.SS3.8.p8.8.m8.1.1.2.2.2" xref="S6.SS3.8.p8.8.m8.1.1.2.2.2.cmml">q</mi><mn id="S6.SS3.8.p8.8.m8.1.1.2.2.3" xref="S6.SS3.8.p8.8.m8.1.1.2.2.3.cmml">1</mn></msub><mo id="S6.SS3.8.p8.8.m8.1.1.2.1" xref="S6.SS3.8.p8.8.m8.1.1.2.1.cmml">+</mo><mfrac id="S6.SS3.8.p8.8.m8.1.1.2.3" xref="S6.SS3.8.p8.8.m8.1.1.2.3.cmml"><mrow id="S6.SS3.8.p8.8.m8.1.1.2.3.2" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2.cmml"><mi id="S6.SS3.8.p8.8.m8.1.1.2.3.2.2" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2.2.cmml">γ</mi><mo id="S6.SS3.8.p8.8.m8.1.1.2.3.2.1" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2.1.cmml">+</mo><mn id="S6.SS3.8.p8.8.m8.1.1.2.3.2.3" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.8.p8.8.m8.1.1.2.3.3" xref="S6.SS3.8.p8.8.m8.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS3.8.p8.8.m8.1.1.1" xref="S6.SS3.8.p8.8.m8.1.1.1.cmml"><</mo><mrow id="S6.SS3.8.p8.8.m8.1.1.3" xref="S6.SS3.8.p8.8.m8.1.1.3.cmml"><msub id="S6.SS3.8.p8.8.m8.1.1.3.2" xref="S6.SS3.8.p8.8.m8.1.1.3.2.cmml"><mi id="S6.SS3.8.p8.8.m8.1.1.3.2.2" xref="S6.SS3.8.p8.8.m8.1.1.3.2.2.cmml">k</mi><mn id="S6.SS3.8.p8.8.m8.1.1.3.2.3" xref="S6.SS3.8.p8.8.m8.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.8.m8.1.1.3.1" xref="S6.SS3.8.p8.8.m8.1.1.3.1.cmml">+</mo><mi id="S6.SS3.8.p8.8.m8.1.1.3.3" mathvariant="normal" xref="S6.SS3.8.p8.8.m8.1.1.3.3.cmml">ℓ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.8.m8.1b"><apply id="S6.SS3.8.p8.8.m8.1.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1"><lt id="S6.SS3.8.p8.8.m8.1.1.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1.1"></lt><apply id="S6.SS3.8.p8.8.m8.1.1.2.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2"><plus id="S6.SS3.8.p8.8.m8.1.1.2.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.1"></plus><apply id="S6.SS3.8.p8.8.m8.1.1.2.2.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.8.m8.1.1.2.2.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.2">subscript</csymbol><ci id="S6.SS3.8.p8.8.m8.1.1.2.2.2.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.2.2">𝑞</ci><cn id="S6.SS3.8.p8.8.m8.1.1.2.2.3.cmml" type="integer" xref="S6.SS3.8.p8.8.m8.1.1.2.2.3">1</cn></apply><apply id="S6.SS3.8.p8.8.m8.1.1.2.3.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.3"><divide id="S6.SS3.8.p8.8.m8.1.1.2.3.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.3"></divide><apply id="S6.SS3.8.p8.8.m8.1.1.2.3.2.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2"><plus id="S6.SS3.8.p8.8.m8.1.1.2.3.2.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2.1"></plus><ci id="S6.SS3.8.p8.8.m8.1.1.2.3.2.2.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2.2">𝛾</ci><cn id="S6.SS3.8.p8.8.m8.1.1.2.3.2.3.cmml" type="integer" xref="S6.SS3.8.p8.8.m8.1.1.2.3.2.3">1</cn></apply><ci id="S6.SS3.8.p8.8.m8.1.1.2.3.3.cmml" xref="S6.SS3.8.p8.8.m8.1.1.2.3.3">𝑝</ci></apply></apply><apply id="S6.SS3.8.p8.8.m8.1.1.3.cmml" xref="S6.SS3.8.p8.8.m8.1.1.3"><plus id="S6.SS3.8.p8.8.m8.1.1.3.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1.3.1"></plus><apply id="S6.SS3.8.p8.8.m8.1.1.3.2.cmml" xref="S6.SS3.8.p8.8.m8.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.8.m8.1.1.3.2.1.cmml" xref="S6.SS3.8.p8.8.m8.1.1.3.2">subscript</csymbol><ci id="S6.SS3.8.p8.8.m8.1.1.3.2.2.cmml" xref="S6.SS3.8.p8.8.m8.1.1.3.2.2">𝑘</ci><cn id="S6.SS3.8.p8.8.m8.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.8.p8.8.m8.1.1.3.2.3">0</cn></apply><ci id="S6.SS3.8.p8.8.m8.1.1.3.3.cmml" xref="S6.SS3.8.p8.8.m8.1.1.3.3">ℓ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.8.m8.1c">q_{1}+\frac{\gamma+1}{p}<k_{0}+\ell</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.8.m8.1d">italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ</annotation></semantics></math> and define for <math alttext="j\leq q_{1}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.9.m9.1"><semantics id="S6.SS3.8.p8.9.m9.1a"><mrow id="S6.SS3.8.p8.9.m9.1.1" xref="S6.SS3.8.p8.9.m9.1.1.cmml"><mi id="S6.SS3.8.p8.9.m9.1.1.2" xref="S6.SS3.8.p8.9.m9.1.1.2.cmml">j</mi><mo id="S6.SS3.8.p8.9.m9.1.1.1" xref="S6.SS3.8.p8.9.m9.1.1.1.cmml">≤</mo><msub id="S6.SS3.8.p8.9.m9.1.1.3" xref="S6.SS3.8.p8.9.m9.1.1.3.cmml"><mi id="S6.SS3.8.p8.9.m9.1.1.3.2" xref="S6.SS3.8.p8.9.m9.1.1.3.2.cmml">q</mi><mn id="S6.SS3.8.p8.9.m9.1.1.3.3" xref="S6.SS3.8.p8.9.m9.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.9.m9.1b"><apply id="S6.SS3.8.p8.9.m9.1.1.cmml" xref="S6.SS3.8.p8.9.m9.1.1"><leq id="S6.SS3.8.p8.9.m9.1.1.1.cmml" xref="S6.SS3.8.p8.9.m9.1.1.1"></leq><ci id="S6.SS3.8.p8.9.m9.1.1.2.cmml" xref="S6.SS3.8.p8.9.m9.1.1.2">𝑗</ci><apply id="S6.SS3.8.p8.9.m9.1.1.3.cmml" xref="S6.SS3.8.p8.9.m9.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.9.m9.1.1.3.1.cmml" xref="S6.SS3.8.p8.9.m9.1.1.3">subscript</csymbol><ci id="S6.SS3.8.p8.9.m9.1.1.3.2.cmml" xref="S6.SS3.8.p8.9.m9.1.1.3.2">𝑞</ci><cn id="S6.SS3.8.p8.9.m9.1.1.3.3.cmml" type="integer" xref="S6.SS3.8.p8.9.m9.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.9.m9.1c">j\leq q_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.9.m9.1d">italic_j ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math></p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex62"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{j}:=\begin{cases}\mathcal{C}^{j}u&\mbox{if 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id="S6.Ex62.m1.4.4.4.4.2.1.2.1.2.2.2.2.cmml" xref="S6.Ex62.m1.4.4.4.4.2.1.2.1.2.2.2.2">𝑚</ci><ci id="S6.Ex62.m1.4.4.4.4.2.1.2.1.2.2.2.3.cmml" xref="S6.Ex62.m1.4.4.4.4.2.1.2.1.2.2.2.3">𝑛</ci></apply></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex62.m1.4c">g_{j}:=\begin{cases}\mathcal{C}^{j}u&\mbox{if }j\notin\{m_{0},\dots,m_{n}\},\\ (1-\pi_{j})\mathcal{C}^{j}u&\mbox{if }j\in\{m_{0},\dots,m_{n}\}.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex62.m1.4d">italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT := { start_ROW start_CELL caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u end_CELL start_CELL if italic_j ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , end_CELL end_ROW start_ROW start_CELL ( 1 - italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u end_CELL start_CELL if italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.27">By Lemma <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S5.Thmtheorem2" title="Lemma 5.2. ‣ 5. Trace theorem for boundary operators ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">5.2</span></a> and <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib32" title="">32</a>, Theorem 14.4.30]</cite>, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex63"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="g_{j}\in B^{k_{0}+\ell-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)=\big{[}% B^{k_{0}-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X),B^{k_{0}+k_{1}-j-% \frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)\big{]}_{\theta}." class="ltx_Math" display="block" id="S6.Ex63.m1.10"><semantics id="S6.Ex63.m1.10a"><mrow id="S6.Ex63.m1.10.10.1" xref="S6.Ex63.m1.10.10.1.1.cmml"><mrow id="S6.Ex63.m1.10.10.1.1" 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id="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.2.2" xref="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.2.2.cmml">γ</mi><mo id="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.2.1" xref="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.2.1.cmml">+</mo><mn id="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.2.3" xref="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.2.3.cmml">1</mn></mrow><mi id="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.3" xref="S6.Ex63.m1.10.10.1.1.1.3.2.3.4.3.cmml">p</mi></mfrac></mrow></msubsup><mo id="S6.Ex63.m1.10.10.1.1.1.2" xref="S6.Ex63.m1.10.10.1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex63.m1.10.10.1.1.1.1.1" xref="S6.Ex63.m1.10.10.1.1.1.1.2.cmml"><mo id="S6.Ex63.m1.10.10.1.1.1.1.1.2" stretchy="false" xref="S6.Ex63.m1.10.10.1.1.1.1.2.cmml">(</mo><msup id="S6.Ex63.m1.10.10.1.1.1.1.1.1" xref="S6.Ex63.m1.10.10.1.1.1.1.1.1.cmml"><mi id="S6.Ex63.m1.10.10.1.1.1.1.1.1.2" xref="S6.Ex63.m1.10.10.1.1.1.1.1.1.2.cmml">ℝ</mi><mrow id="S6.Ex63.m1.10.10.1.1.1.1.1.1.3" xref="S6.Ex63.m1.10.10.1.1.1.1.1.1.3.cmml"><mi id="S6.Ex63.m1.10.10.1.1.1.1.1.1.3.2" 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xref="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.cmml"><mi id="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.2" xref="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.2.cmml">B</mi><mrow id="S6.Ex63.m1.4.4.2.4" xref="S6.Ex63.m1.4.4.2.3.cmml"><mi id="S6.Ex63.m1.3.3.1.1" xref="S6.Ex63.m1.3.3.1.1.cmml">p</mi><mo id="S6.Ex63.m1.4.4.2.4.1" xref="S6.Ex63.m1.4.4.2.3.cmml">,</mo><mi id="S6.Ex63.m1.4.4.2.2" xref="S6.Ex63.m1.4.4.2.2.cmml">p</mi></mrow><mrow id="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3" xref="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.cmml"><msub id="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.2" xref="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.2.cmml"><mi id="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.2.2" xref="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.2.2.cmml">k</mi><mn id="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.2.3" xref="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.2.3.cmml">0</mn></msub><mo id="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.1" xref="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.1.cmml">−</mo><mi id="S6.Ex63.m1.10.10.1.1.2.1.1.1.3.2.3.3" 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id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.2.2" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.2.2.cmml">k</mi><mn id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.2.3" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.2.3.cmml">0</mn></msub><mo id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.1" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.1.cmml">+</mo><msub id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.3" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.3.cmml"><mi id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.3.2" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.3.2.cmml">k</mi><mn id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.3.3" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.1" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.1.cmml">−</mo><mi id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.3" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.3.cmml">j</mi><mo id="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.1a" xref="S6.Ex63.m1.10.10.1.1.3.2.2.2.3.2.3.1.cmml">−</mo><mfrac 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\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)\big{]}_{\theta}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex63.m1.10d">italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ italic_B start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) = [ italic_B start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) , italic_B start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.28">Therefore, by definition of the complex interpolation method there exists a</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex64"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\widetilde{f}_{j}\in\mathscr{H}\big{(}B^{k_{0}-j-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};X),B^{k_{0}+k_{1}-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1% };X)\big{)}" class="ltx_Math" display="block" id="S6.Ex64.m1.8"><semantics id="S6.Ex64.m1.8a"><mrow 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xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.2.3.3">1</cn></apply></apply><ci id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.3.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.3">𝑗</ci><apply id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4"><divide id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.1.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4"></divide><apply id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2"><plus id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2.1.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2.1"></plus><ci id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2.2.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2.2">𝛾</ci><cn id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2.3.cmml" type="integer" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.2.3">1</cn></apply><ci id="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.3.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.3.2.3.4.3">𝑝</ci></apply></apply></apply><list id="S6.Ex64.m1.4.4.2.3.cmml" xref="S6.Ex64.m1.4.4.2.4"><ci id="S6.Ex64.m1.3.3.1.1.cmml" xref="S6.Ex64.m1.3.3.1.1">𝑝</ci><ci id="S6.Ex64.m1.4.4.2.2.cmml" xref="S6.Ex64.m1.4.4.2.2">𝑝</ci></list></apply><list id="S6.Ex64.m1.8.8.2.2.2.2.1.2.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1"><apply id="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.1.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1.1">superscript</csymbol><ci id="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.2.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.2">ℝ</ci><apply id="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3"><minus id="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3.1.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3.1"></minus><ci id="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3.2.cmml" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3.2">𝑑</ci><cn id="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3.3.cmml" type="integer" xref="S6.Ex64.m1.8.8.2.2.2.2.1.1.1.3.3">1</cn></apply></apply><ci id="S6.Ex64.m1.6.6.cmml" xref="S6.Ex64.m1.6.6">𝑋</ci></list></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex64.m1.8c">\widetilde{f}_{j}\in\mathscr{H}\big{(}B^{k_{0}-j-\frac{\gamma+1}{p}}_{p,p}(% \mathbb{R}^{d-1};X),B^{k_{0}+k_{1}-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1% };X)\big{)}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex64.m1.8d">over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ∈ script_H ( italic_B start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) , italic_B start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.12">such that <math alttext="\widetilde{f}_{j}(\theta)=g_{j}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.10.m1.1"><semantics id="S6.SS3.8.p8.10.m1.1a"><mrow id="S6.SS3.8.p8.10.m1.1.2" xref="S6.SS3.8.p8.10.m1.1.2.cmml"><mrow id="S6.SS3.8.p8.10.m1.1.2.2" xref="S6.SS3.8.p8.10.m1.1.2.2.cmml"><msub id="S6.SS3.8.p8.10.m1.1.2.2.2" xref="S6.SS3.8.p8.10.m1.1.2.2.2.cmml"><mover accent="true" id="S6.SS3.8.p8.10.m1.1.2.2.2.2" xref="S6.SS3.8.p8.10.m1.1.2.2.2.2.cmml"><mi id="S6.SS3.8.p8.10.m1.1.2.2.2.2.2" xref="S6.SS3.8.p8.10.m1.1.2.2.2.2.2.cmml">f</mi><mo id="S6.SS3.8.p8.10.m1.1.2.2.2.2.1" xref="S6.SS3.8.p8.10.m1.1.2.2.2.2.1.cmml">~</mo></mover><mi id="S6.SS3.8.p8.10.m1.1.2.2.2.3" xref="S6.SS3.8.p8.10.m1.1.2.2.2.3.cmml">j</mi></msub><mo id="S6.SS3.8.p8.10.m1.1.2.2.1" xref="S6.SS3.8.p8.10.m1.1.2.2.1.cmml">⁢</mo><mrow id="S6.SS3.8.p8.10.m1.1.2.2.3.2" xref="S6.SS3.8.p8.10.m1.1.2.2.cmml"><mo id="S6.SS3.8.p8.10.m1.1.2.2.3.2.1" stretchy="false" xref="S6.SS3.8.p8.10.m1.1.2.2.cmml">(</mo><mi id="S6.SS3.8.p8.10.m1.1.1" xref="S6.SS3.8.p8.10.m1.1.1.cmml">θ</mi><mo id="S6.SS3.8.p8.10.m1.1.2.2.3.2.2" stretchy="false" xref="S6.SS3.8.p8.10.m1.1.2.2.cmml">)</mo></mrow></mrow><mo id="S6.SS3.8.p8.10.m1.1.2.1" xref="S6.SS3.8.p8.10.m1.1.2.1.cmml">=</mo><msub id="S6.SS3.8.p8.10.m1.1.2.3" xref="S6.SS3.8.p8.10.m1.1.2.3.cmml"><mi id="S6.SS3.8.p8.10.m1.1.2.3.2" xref="S6.SS3.8.p8.10.m1.1.2.3.2.cmml">g</mi><mi id="S6.SS3.8.p8.10.m1.1.2.3.3" xref="S6.SS3.8.p8.10.m1.1.2.3.3.cmml">j</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.10.m1.1b"><apply id="S6.SS3.8.p8.10.m1.1.2.cmml" xref="S6.SS3.8.p8.10.m1.1.2"><eq id="S6.SS3.8.p8.10.m1.1.2.1.cmml" xref="S6.SS3.8.p8.10.m1.1.2.1"></eq><apply id="S6.SS3.8.p8.10.m1.1.2.2.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2"><times id="S6.SS3.8.p8.10.m1.1.2.2.1.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2.1"></times><apply id="S6.SS3.8.p8.10.m1.1.2.2.2.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.10.m1.1.2.2.2.1.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2.2">subscript</csymbol><apply id="S6.SS3.8.p8.10.m1.1.2.2.2.2.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2.2.2"><ci id="S6.SS3.8.p8.10.m1.1.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2.2.2.1">~</ci><ci id="S6.SS3.8.p8.10.m1.1.2.2.2.2.2.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2.2.2.2">𝑓</ci></apply><ci id="S6.SS3.8.p8.10.m1.1.2.2.2.3.cmml" xref="S6.SS3.8.p8.10.m1.1.2.2.2.3">𝑗</ci></apply><ci id="S6.SS3.8.p8.10.m1.1.1.cmml" xref="S6.SS3.8.p8.10.m1.1.1">𝜃</ci></apply><apply id="S6.SS3.8.p8.10.m1.1.2.3.cmml" xref="S6.SS3.8.p8.10.m1.1.2.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.10.m1.1.2.3.1.cmml" xref="S6.SS3.8.p8.10.m1.1.2.3">subscript</csymbol><ci id="S6.SS3.8.p8.10.m1.1.2.3.2.cmml" xref="S6.SS3.8.p8.10.m1.1.2.3.2">𝑔</ci><ci id="S6.SS3.8.p8.10.m1.1.2.3.3.cmml" xref="S6.SS3.8.p8.10.m1.1.2.3.3">𝑗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.10.m1.1c">\widetilde{f}_{j}(\theta)=g_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.10.m1.1d">over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( italic_θ ) = italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math> for <math alttext="j\leq q_{1}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.11.m2.1"><semantics id="S6.SS3.8.p8.11.m2.1a"><mrow id="S6.SS3.8.p8.11.m2.1.1" xref="S6.SS3.8.p8.11.m2.1.1.cmml"><mi id="S6.SS3.8.p8.11.m2.1.1.2" xref="S6.SS3.8.p8.11.m2.1.1.2.cmml">j</mi><mo id="S6.SS3.8.p8.11.m2.1.1.1" xref="S6.SS3.8.p8.11.m2.1.1.1.cmml">≤</mo><msub id="S6.SS3.8.p8.11.m2.1.1.3" xref="S6.SS3.8.p8.11.m2.1.1.3.cmml"><mi id="S6.SS3.8.p8.11.m2.1.1.3.2" xref="S6.SS3.8.p8.11.m2.1.1.3.2.cmml">q</mi><mn id="S6.SS3.8.p8.11.m2.1.1.3.3" xref="S6.SS3.8.p8.11.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.11.m2.1b"><apply id="S6.SS3.8.p8.11.m2.1.1.cmml" xref="S6.SS3.8.p8.11.m2.1.1"><leq id="S6.SS3.8.p8.11.m2.1.1.1.cmml" xref="S6.SS3.8.p8.11.m2.1.1.1"></leq><ci id="S6.SS3.8.p8.11.m2.1.1.2.cmml" xref="S6.SS3.8.p8.11.m2.1.1.2">𝑗</ci><apply id="S6.SS3.8.p8.11.m2.1.1.3.cmml" xref="S6.SS3.8.p8.11.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.11.m2.1.1.3.1.cmml" xref="S6.SS3.8.p8.11.m2.1.1.3">subscript</csymbol><ci id="S6.SS3.8.p8.11.m2.1.1.3.2.cmml" xref="S6.SS3.8.p8.11.m2.1.1.3.2">𝑞</ci><cn id="S6.SS3.8.p8.11.m2.1.1.3.3.cmml" type="integer" xref="S6.SS3.8.p8.11.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.11.m2.1c">j\leq q_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.11.m2.1d">italic_j ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. In addition, for <math alttext="j\leq q_{1}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.12.m3.1"><semantics id="S6.SS3.8.p8.12.m3.1a"><mrow id="S6.SS3.8.p8.12.m3.1.1" xref="S6.SS3.8.p8.12.m3.1.1.cmml"><mi id="S6.SS3.8.p8.12.m3.1.1.2" xref="S6.SS3.8.p8.12.m3.1.1.2.cmml">j</mi><mo id="S6.SS3.8.p8.12.m3.1.1.1" xref="S6.SS3.8.p8.12.m3.1.1.1.cmml">≤</mo><msub id="S6.SS3.8.p8.12.m3.1.1.3" xref="S6.SS3.8.p8.12.m3.1.1.3.cmml"><mi id="S6.SS3.8.p8.12.m3.1.1.3.2" xref="S6.SS3.8.p8.12.m3.1.1.3.2.cmml">q</mi><mn id="S6.SS3.8.p8.12.m3.1.1.3.3" xref="S6.SS3.8.p8.12.m3.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.12.m3.1b"><apply id="S6.SS3.8.p8.12.m3.1.1.cmml" xref="S6.SS3.8.p8.12.m3.1.1"><leq id="S6.SS3.8.p8.12.m3.1.1.1.cmml" xref="S6.SS3.8.p8.12.m3.1.1.1"></leq><ci id="S6.SS3.8.p8.12.m3.1.1.2.cmml" xref="S6.SS3.8.p8.12.m3.1.1.2">𝑗</ci><apply id="S6.SS3.8.p8.12.m3.1.1.3.cmml" xref="S6.SS3.8.p8.12.m3.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.12.m3.1.1.3.1.cmml" xref="S6.SS3.8.p8.12.m3.1.1.3">subscript</csymbol><ci id="S6.SS3.8.p8.12.m3.1.1.3.2.cmml" xref="S6.SS3.8.p8.12.m3.1.1.3.2">𝑞</ci><cn id="S6.SS3.8.p8.12.m3.1.1.3.3.cmml" type="integer" xref="S6.SS3.8.p8.12.m3.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.12.m3.1c">j\leq q_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.12.m3.1d">italic_j ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> we define</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex65"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="f_{j}:=\begin{cases}\widetilde{f}_{j}&\mbox{if }j\notin\{m_{0},\dots,m_{n}\},% \\ (1-\pi_{j})\widetilde{f}_{j}&\mbox{if }j\in\{m_{0},\dots,m_{n}\},\end{cases}" class="ltx_Math" display="block" 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cd="ambiguous" id="S6.Ex65.m1.4.4.4.4.2.1.2.1.1.1.1.1.cmml" xref="S6.Ex65.m1.4.4.4.4.2.1.2.1.1.1.1">subscript</csymbol><ci id="S6.Ex65.m1.4.4.4.4.2.1.2.1.1.1.1.2.cmml" xref="S6.Ex65.m1.4.4.4.4.2.1.2.1.1.1.1.2">𝑚</ci><cn id="S6.Ex65.m1.4.4.4.4.2.1.2.1.1.1.1.3.cmml" type="integer" xref="S6.Ex65.m1.4.4.4.4.2.1.2.1.1.1.1.3">0</cn></apply><ci id="S6.Ex65.m1.4.4.4.4.2.1.1.cmml" xref="S6.Ex65.m1.4.4.4.4.2.1.1">…</ci><apply id="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2.cmml" xref="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2"><csymbol cd="ambiguous" id="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2.1.cmml" xref="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2">subscript</csymbol><ci id="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2.2.cmml" xref="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2.2">𝑚</ci><ci id="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2.3.cmml" xref="S6.Ex65.m1.4.4.4.4.2.1.2.1.2.2.2.3">𝑛</ci></apply></set></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex65.m1.4c">f_{j}:=\begin{cases}\widetilde{f}_{j}&\mbox{if }j\notin\{m_{0},\dots,m_{n}\},% \\ (1-\pi_{j})\widetilde{f}_{j}&\mbox{if }j\in\{m_{0},\dots,m_{n}\},\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex65.m1.4d">italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT := { start_ROW start_CELL over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_CELL start_CELL if italic_j ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , end_CELL end_ROW start_ROW start_CELL ( 1 - italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT end_CELL start_CELL if italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.13">which satisfies the same properties as <math alttext="\widetilde{f}_{j}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.13.m1.1"><semantics id="S6.SS3.8.p8.13.m1.1a"><msub id="S6.SS3.8.p8.13.m1.1.1" xref="S6.SS3.8.p8.13.m1.1.1.cmml"><mover accent="true" id="S6.SS3.8.p8.13.m1.1.1.2" xref="S6.SS3.8.p8.13.m1.1.1.2.cmml"><mi id="S6.SS3.8.p8.13.m1.1.1.2.2" xref="S6.SS3.8.p8.13.m1.1.1.2.2.cmml">f</mi><mo id="S6.SS3.8.p8.13.m1.1.1.2.1" xref="S6.SS3.8.p8.13.m1.1.1.2.1.cmml">~</mo></mover><mi id="S6.SS3.8.p8.13.m1.1.1.3" xref="S6.SS3.8.p8.13.m1.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.13.m1.1b"><apply id="S6.SS3.8.p8.13.m1.1.1.cmml" xref="S6.SS3.8.p8.13.m1.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.13.m1.1.1.1.cmml" xref="S6.SS3.8.p8.13.m1.1.1">subscript</csymbol><apply id="S6.SS3.8.p8.13.m1.1.1.2.cmml" xref="S6.SS3.8.p8.13.m1.1.1.2"><ci id="S6.SS3.8.p8.13.m1.1.1.2.1.cmml" xref="S6.SS3.8.p8.13.m1.1.1.2.1">~</ci><ci id="S6.SS3.8.p8.13.m1.1.1.2.2.cmml" xref="S6.SS3.8.p8.13.m1.1.1.2.2">𝑓</ci></apply><ci id="S6.SS3.8.p8.13.m1.1.1.3.cmml" xref="S6.SS3.8.p8.13.m1.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.13.m1.1c">\widetilde{f}_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.13.m1.1d">over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>. Then</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex66"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="F:=\operatorname{ext}_{\mathcal{C}}(f_{0},\dots,f_{q_{1}},0,\dots,0)\in% \mathscr{H}(W^{k_{0},p},W^{k_{0}+k_{1},p})" class="ltx_Math" display="block" id="S6.Ex66.m1.13"><semantics id="S6.Ex66.m1.13a"><mrow id="S6.Ex66.m1.13.13" xref="S6.Ex66.m1.13.13.cmml"><mi id="S6.Ex66.m1.13.13.7" xref="S6.Ex66.m1.13.13.7.cmml">F</mi><mo id="S6.Ex66.m1.13.13.8" lspace="0.278em" rspace="0.278em" xref="S6.Ex66.m1.13.13.8.cmml">:=</mo><mrow id="S6.Ex66.m1.11.11.3.3" xref="S6.Ex66.m1.11.11.3.4.cmml"><msub id="S6.Ex66.m1.9.9.1.1.1" xref="S6.Ex66.m1.9.9.1.1.1.cmml"><mi id="S6.Ex66.m1.9.9.1.1.1.2" xref="S6.Ex66.m1.9.9.1.1.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex66.m1.9.9.1.1.1.3" xref="S6.Ex66.m1.9.9.1.1.1.3.cmml">𝒞</mi></msub><mo id="S6.Ex66.m1.11.11.3.3a" xref="S6.Ex66.m1.11.11.3.4.cmml">⁡</mo><mrow id="S6.Ex66.m1.11.11.3.3.3" xref="S6.Ex66.m1.11.11.3.4.cmml"><mo id="S6.Ex66.m1.11.11.3.3.3.3" stretchy="false" xref="S6.Ex66.m1.11.11.3.4.cmml">(</mo><msub id="S6.Ex66.m1.10.10.2.2.2.1" xref="S6.Ex66.m1.10.10.2.2.2.1.cmml"><mi id="S6.Ex66.m1.10.10.2.2.2.1.2" xref="S6.Ex66.m1.10.10.2.2.2.1.2.cmml">f</mi><mn id="S6.Ex66.m1.10.10.2.2.2.1.3" xref="S6.Ex66.m1.10.10.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.Ex66.m1.11.11.3.3.3.4" xref="S6.Ex66.m1.11.11.3.4.cmml">,</mo><mi id="S6.Ex66.m1.5.5" mathvariant="normal" xref="S6.Ex66.m1.5.5.cmml">…</mi><mo id="S6.Ex66.m1.11.11.3.3.3.5" xref="S6.Ex66.m1.11.11.3.4.cmml">,</mo><msub id="S6.Ex66.m1.11.11.3.3.3.2" xref="S6.Ex66.m1.11.11.3.3.3.2.cmml"><mi id="S6.Ex66.m1.11.11.3.3.3.2.2" xref="S6.Ex66.m1.11.11.3.3.3.2.2.cmml">f</mi><msub id="S6.Ex66.m1.11.11.3.3.3.2.3" xref="S6.Ex66.m1.11.11.3.3.3.2.3.cmml"><mi id="S6.Ex66.m1.11.11.3.3.3.2.3.2" 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italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_f start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , 0 , … , 0 ) ∈ script_H ( italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.29">and</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex67"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="v:=F(\theta)=\operatorname{ext}_{\mathcal{C}}\big{(}f_{0}(\theta),\dots,f_{q_{% 1}}(\theta),0,\dots,0\big{)}=\operatorname{ext}_{\mathcal{C}}(g_{0},\dots,g_{q% _{1}},0,\dots,0)." class="ltx_Math" display="block" id="S6.Ex67.m1.12"><semantics id="S6.Ex67.m1.12a"><mrow id="S6.Ex67.m1.12.12.1" xref="S6.Ex67.m1.12.12.1.1.cmml"><mrow id="S6.Ex67.m1.12.12.1.1" xref="S6.Ex67.m1.12.12.1.1.cmml"><mi id="S6.Ex67.m1.12.12.1.1.8" xref="S6.Ex67.m1.12.12.1.1.8.cmml">v</mi><mo id="S6.Ex67.m1.12.12.1.1.9" lspace="0.278em" rspace="0.278em" xref="S6.Ex67.m1.12.12.1.1.9.cmml">:=</mo><mrow id="S6.Ex67.m1.12.12.1.1.10" xref="S6.Ex67.m1.12.12.1.1.10.cmml"><mi id="S6.Ex67.m1.12.12.1.1.10.2" xref="S6.Ex67.m1.12.12.1.1.10.2.cmml">F</mi><mo id="S6.Ex67.m1.12.12.1.1.10.1" xref="S6.Ex67.m1.12.12.1.1.10.1.cmml">⁢</mo><mrow id="S6.Ex67.m1.12.12.1.1.10.3.2" xref="S6.Ex67.m1.12.12.1.1.10.cmml"><mo id="S6.Ex67.m1.12.12.1.1.10.3.2.1" stretchy="false" xref="S6.Ex67.m1.12.12.1.1.10.cmml">(</mo><mi id="S6.Ex67.m1.1.1" xref="S6.Ex67.m1.1.1.cmml">θ</mi><mo id="S6.Ex67.m1.12.12.1.1.10.3.2.2" 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xref="S6.Ex67.m1.12.12.1.1.6.3.3.2.3">subscript</csymbol><ci id="S6.Ex67.m1.12.12.1.1.6.3.3.2.3.2.cmml" xref="S6.Ex67.m1.12.12.1.1.6.3.3.2.3.2">𝑞</ci><cn id="S6.Ex67.m1.12.12.1.1.6.3.3.2.3.3.cmml" type="integer" xref="S6.Ex67.m1.12.12.1.1.6.3.3.2.3.3">1</cn></apply></apply><cn id="S6.Ex67.m1.9.9.cmml" type="integer" xref="S6.Ex67.m1.9.9">0</cn><ci id="S6.Ex67.m1.10.10.cmml" xref="S6.Ex67.m1.10.10">…</ci><cn id="S6.Ex67.m1.11.11.cmml" type="integer" xref="S6.Ex67.m1.11.11">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex67.m1.12c">v:=F(\theta)=\operatorname{ext}_{\mathcal{C}}\big{(}f_{0}(\theta),\dots,f_{q_{% 1}}(\theta),0,\dots,0\big{)}=\operatorname{ext}_{\mathcal{C}}(g_{0},\dots,g_{q% _{1}},0,\dots,0).</annotation><annotation encoding="application/x-llamapun" id="S6.Ex67.m1.12d">italic_v := italic_F ( italic_θ ) = roman_ext start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( italic_θ ) , … , italic_f start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ( italic_θ ) , 0 , … , 0 ) = roman_ext start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_g start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , 0 , … , 0 ) .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.19">Moreover, <math alttext="F" class="ltx_Math" display="inline" id="S6.SS3.8.p8.14.m1.1"><semantics id="S6.SS3.8.p8.14.m1.1a"><mi id="S6.SS3.8.p8.14.m1.1.1" xref="S6.SS3.8.p8.14.m1.1.1.cmml">F</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.14.m1.1b"><ci id="S6.SS3.8.p8.14.m1.1.1.cmml" xref="S6.SS3.8.p8.14.m1.1.1">𝐹</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.14.m1.1c">F</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.14.m1.1d">italic_F</annotation></semantics></math> is bounded and continuous on <math alttext="\{z\in\mathbb{C}:\operatorname{Re}z=1\}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.15.m2.2"><semantics id="S6.SS3.8.p8.15.m2.2a"><mrow id="S6.SS3.8.p8.15.m2.2.2.2" xref="S6.SS3.8.p8.15.m2.2.2.3.cmml"><mo id="S6.SS3.8.p8.15.m2.2.2.2.3" stretchy="false" xref="S6.SS3.8.p8.15.m2.2.2.3.1.cmml">{</mo><mrow id="S6.SS3.8.p8.15.m2.1.1.1.1" xref="S6.SS3.8.p8.15.m2.1.1.1.1.cmml"><mi id="S6.SS3.8.p8.15.m2.1.1.1.1.2" xref="S6.SS3.8.p8.15.m2.1.1.1.1.2.cmml">z</mi><mo id="S6.SS3.8.p8.15.m2.1.1.1.1.1" xref="S6.SS3.8.p8.15.m2.1.1.1.1.1.cmml">∈</mo><mi id="S6.SS3.8.p8.15.m2.1.1.1.1.3" xref="S6.SS3.8.p8.15.m2.1.1.1.1.3.cmml">ℂ</mi></mrow><mo id="S6.SS3.8.p8.15.m2.2.2.2.4" lspace="0.278em" rspace="0.278em" xref="S6.SS3.8.p8.15.m2.2.2.3.1.cmml">:</mo><mrow id="S6.SS3.8.p8.15.m2.2.2.2.2" xref="S6.SS3.8.p8.15.m2.2.2.2.2.cmml"><mrow id="S6.SS3.8.p8.15.m2.2.2.2.2.2" xref="S6.SS3.8.p8.15.m2.2.2.2.2.2.cmml"><mi id="S6.SS3.8.p8.15.m2.2.2.2.2.2.1" xref="S6.SS3.8.p8.15.m2.2.2.2.2.2.1.cmml">Re</mi><mo id="S6.SS3.8.p8.15.m2.2.2.2.2.2a" lspace="0.167em" xref="S6.SS3.8.p8.15.m2.2.2.2.2.2.cmml">⁡</mo><mi id="S6.SS3.8.p8.15.m2.2.2.2.2.2.2" xref="S6.SS3.8.p8.15.m2.2.2.2.2.2.2.cmml">z</mi></mrow><mo id="S6.SS3.8.p8.15.m2.2.2.2.2.1" xref="S6.SS3.8.p8.15.m2.2.2.2.2.1.cmml">=</mo><mn id="S6.SS3.8.p8.15.m2.2.2.2.2.3" xref="S6.SS3.8.p8.15.m2.2.2.2.2.3.cmml">1</mn></mrow><mo id="S6.SS3.8.p8.15.m2.2.2.2.5" stretchy="false" xref="S6.SS3.8.p8.15.m2.2.2.3.1.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.15.m2.2b"><apply id="S6.SS3.8.p8.15.m2.2.2.3.cmml" xref="S6.SS3.8.p8.15.m2.2.2.2"><csymbol cd="latexml" id="S6.SS3.8.p8.15.m2.2.2.3.1.cmml" xref="S6.SS3.8.p8.15.m2.2.2.2.3">conditional-set</csymbol><apply id="S6.SS3.8.p8.15.m2.1.1.1.1.cmml" xref="S6.SS3.8.p8.15.m2.1.1.1.1"><in id="S6.SS3.8.p8.15.m2.1.1.1.1.1.cmml" xref="S6.SS3.8.p8.15.m2.1.1.1.1.1"></in><ci id="S6.SS3.8.p8.15.m2.1.1.1.1.2.cmml" xref="S6.SS3.8.p8.15.m2.1.1.1.1.2">𝑧</ci><ci id="S6.SS3.8.p8.15.m2.1.1.1.1.3.cmml" xref="S6.SS3.8.p8.15.m2.1.1.1.1.3">ℂ</ci></apply><apply id="S6.SS3.8.p8.15.m2.2.2.2.2.cmml" xref="S6.SS3.8.p8.15.m2.2.2.2.2"><eq id="S6.SS3.8.p8.15.m2.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.15.m2.2.2.2.2.1"></eq><apply id="S6.SS3.8.p8.15.m2.2.2.2.2.2.cmml" xref="S6.SS3.8.p8.15.m2.2.2.2.2.2"><ci id="S6.SS3.8.p8.15.m2.2.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.15.m2.2.2.2.2.2.1">Re</ci><ci id="S6.SS3.8.p8.15.m2.2.2.2.2.2.2.cmml" xref="S6.SS3.8.p8.15.m2.2.2.2.2.2.2">𝑧</ci></apply><cn id="S6.SS3.8.p8.15.m2.2.2.2.2.3.cmml" type="integer" xref="S6.SS3.8.p8.15.m2.2.2.2.2.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.15.m2.2c">\{z\in\mathbb{C}:\operatorname{Re}z=1\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.15.m2.2d">{ italic_z ∈ blackboard_C : roman_Re italic_z = 1 }</annotation></semantics></math> with values in <math alttext="W^{k_{0}+k_{1},p}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.16.m3.2"><semantics id="S6.SS3.8.p8.16.m3.2a"><msup id="S6.SS3.8.p8.16.m3.2.3" xref="S6.SS3.8.p8.16.m3.2.3.cmml"><mi id="S6.SS3.8.p8.16.m3.2.3.2" xref="S6.SS3.8.p8.16.m3.2.3.2.cmml">W</mi><mrow id="S6.SS3.8.p8.16.m3.2.2.2.2" xref="S6.SS3.8.p8.16.m3.2.2.2.3.cmml"><mrow id="S6.SS3.8.p8.16.m3.2.2.2.2.1" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.cmml"><msub id="S6.SS3.8.p8.16.m3.2.2.2.2.1.2" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.cmml"><mi id="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.2" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.3" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.16.m3.2.2.2.2.1.1" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.1.cmml">+</mo><msub id="S6.SS3.8.p8.16.m3.2.2.2.2.1.3" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.cmml"><mi id="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.2" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.2.cmml">k</mi><mn id="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.3" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.8.p8.16.m3.2.2.2.2.2" xref="S6.SS3.8.p8.16.m3.2.2.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.16.m3.1.1.1.1" xref="S6.SS3.8.p8.16.m3.1.1.1.1.cmml">p</mi></mrow></msup><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.16.m3.2b"><apply id="S6.SS3.8.p8.16.m3.2.3.cmml" xref="S6.SS3.8.p8.16.m3.2.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.16.m3.2.3.1.cmml" xref="S6.SS3.8.p8.16.m3.2.3">superscript</csymbol><ci id="S6.SS3.8.p8.16.m3.2.3.2.cmml" xref="S6.SS3.8.p8.16.m3.2.3.2">𝑊</ci><list id="S6.SS3.8.p8.16.m3.2.2.2.3.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2"><apply id="S6.SS3.8.p8.16.m3.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1"><plus id="S6.SS3.8.p8.16.m3.2.2.2.2.1.1.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.1"></plus><apply id="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.1.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.2">subscript</csymbol><ci id="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.2.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.2.3">0</cn></apply><apply id="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.1.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.3">subscript</csymbol><ci id="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.2.cmml" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.2">𝑘</ci><cn id="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.3.cmml" type="integer" xref="S6.SS3.8.p8.16.m3.2.2.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.8.p8.16.m3.1.1.1.1.cmml" xref="S6.SS3.8.p8.16.m3.1.1.1.1">𝑝</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.16.m3.2c">W^{k_{0}+k_{1},p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.16.m3.2d">italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math> and by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E10" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.10</span></a>), properties of the coretraction <math alttext="\operatorname{ext}_{\mathcal{C}}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.17.m4.1"><semantics id="S6.SS3.8.p8.17.m4.1a"><msub id="S6.SS3.8.p8.17.m4.1.1" xref="S6.SS3.8.p8.17.m4.1.1.cmml"><mi id="S6.SS3.8.p8.17.m4.1.1.2" xref="S6.SS3.8.p8.17.m4.1.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.8.p8.17.m4.1.1.3" xref="S6.SS3.8.p8.17.m4.1.1.3.cmml">𝒞</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.17.m4.1b"><apply id="S6.SS3.8.p8.17.m4.1.1.cmml" xref="S6.SS3.8.p8.17.m4.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.17.m4.1.1.1.cmml" xref="S6.SS3.8.p8.17.m4.1.1">subscript</csymbol><ci id="S6.SS3.8.p8.17.m4.1.1.2.cmml" xref="S6.SS3.8.p8.17.m4.1.1.2">ext</ci><ci id="S6.SS3.8.p8.17.m4.1.1.3.cmml" xref="S6.SS3.8.p8.17.m4.1.1.3">𝒞</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.17.m4.1c">\operatorname{ext}_{\mathcal{C}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.17.m4.1d">roman_ext start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT</annotation></semantics></math> and the definition of <math alttext="f_{j}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.18.m5.1"><semantics id="S6.SS3.8.p8.18.m5.1a"><msub id="S6.SS3.8.p8.18.m5.1.1" xref="S6.SS3.8.p8.18.m5.1.1.cmml"><mi id="S6.SS3.8.p8.18.m5.1.1.2" xref="S6.SS3.8.p8.18.m5.1.1.2.cmml">f</mi><mi id="S6.SS3.8.p8.18.m5.1.1.3" xref="S6.SS3.8.p8.18.m5.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.18.m5.1b"><apply id="S6.SS3.8.p8.18.m5.1.1.cmml" xref="S6.SS3.8.p8.18.m5.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.18.m5.1.1.1.cmml" xref="S6.SS3.8.p8.18.m5.1.1">subscript</csymbol><ci id="S6.SS3.8.p8.18.m5.1.1.2.cmml" xref="S6.SS3.8.p8.18.m5.1.1.2">𝑓</ci><ci id="S6.SS3.8.p8.18.m5.1.1.3.cmml" xref="S6.SS3.8.p8.18.m5.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.18.m5.1c">f_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.18.m5.1d">italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, we have for <math alttext="j\in\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.19.m6.3"><semantics id="S6.SS3.8.p8.19.m6.3a"><mrow id="S6.SS3.8.p8.19.m6.3.3" xref="S6.SS3.8.p8.19.m6.3.3.cmml"><mi id="S6.SS3.8.p8.19.m6.3.3.4" xref="S6.SS3.8.p8.19.m6.3.3.4.cmml">j</mi><mo id="S6.SS3.8.p8.19.m6.3.3.3" xref="S6.SS3.8.p8.19.m6.3.3.3.cmml">∈</mo><mrow id="S6.SS3.8.p8.19.m6.3.3.2.2" xref="S6.SS3.8.p8.19.m6.3.3.2.3.cmml"><mo id="S6.SS3.8.p8.19.m6.3.3.2.2.3" stretchy="false" xref="S6.SS3.8.p8.19.m6.3.3.2.3.cmml">{</mo><msub id="S6.SS3.8.p8.19.m6.2.2.1.1.1" xref="S6.SS3.8.p8.19.m6.2.2.1.1.1.cmml"><mi id="S6.SS3.8.p8.19.m6.2.2.1.1.1.2" xref="S6.SS3.8.p8.19.m6.2.2.1.1.1.2.cmml">m</mi><mn id="S6.SS3.8.p8.19.m6.2.2.1.1.1.3" xref="S6.SS3.8.p8.19.m6.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.19.m6.3.3.2.2.4" xref="S6.SS3.8.p8.19.m6.3.3.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.19.m6.1.1" mathvariant="normal" xref="S6.SS3.8.p8.19.m6.1.1.cmml">…</mi><mo id="S6.SS3.8.p8.19.m6.3.3.2.2.5" xref="S6.SS3.8.p8.19.m6.3.3.2.3.cmml">,</mo><msub id="S6.SS3.8.p8.19.m6.3.3.2.2.2" xref="S6.SS3.8.p8.19.m6.3.3.2.2.2.cmml"><mi id="S6.SS3.8.p8.19.m6.3.3.2.2.2.2" xref="S6.SS3.8.p8.19.m6.3.3.2.2.2.2.cmml">m</mi><mi id="S6.SS3.8.p8.19.m6.3.3.2.2.2.3" xref="S6.SS3.8.p8.19.m6.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S6.SS3.8.p8.19.m6.3.3.2.2.6" stretchy="false" xref="S6.SS3.8.p8.19.m6.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.19.m6.3b"><apply id="S6.SS3.8.p8.19.m6.3.3.cmml" xref="S6.SS3.8.p8.19.m6.3.3"><in id="S6.SS3.8.p8.19.m6.3.3.3.cmml" xref="S6.SS3.8.p8.19.m6.3.3.3"></in><ci id="S6.SS3.8.p8.19.m6.3.3.4.cmml" xref="S6.SS3.8.p8.19.m6.3.3.4">𝑗</ci><set id="S6.SS3.8.p8.19.m6.3.3.2.3.cmml" xref="S6.SS3.8.p8.19.m6.3.3.2.2"><apply id="S6.SS3.8.p8.19.m6.2.2.1.1.1.cmml" xref="S6.SS3.8.p8.19.m6.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.19.m6.2.2.1.1.1.1.cmml" xref="S6.SS3.8.p8.19.m6.2.2.1.1.1">subscript</csymbol><ci id="S6.SS3.8.p8.19.m6.2.2.1.1.1.2.cmml" xref="S6.SS3.8.p8.19.m6.2.2.1.1.1.2">𝑚</ci><cn id="S6.SS3.8.p8.19.m6.2.2.1.1.1.3.cmml" type="integer" xref="S6.SS3.8.p8.19.m6.2.2.1.1.1.3">0</cn></apply><ci id="S6.SS3.8.p8.19.m6.1.1.cmml" xref="S6.SS3.8.p8.19.m6.1.1">…</ci><apply id="S6.SS3.8.p8.19.m6.3.3.2.2.2.cmml" xref="S6.SS3.8.p8.19.m6.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.19.m6.3.3.2.2.2.1.cmml" xref="S6.SS3.8.p8.19.m6.3.3.2.2.2">subscript</csymbol><ci id="S6.SS3.8.p8.19.m6.3.3.2.2.2.2.cmml" xref="S6.SS3.8.p8.19.m6.3.3.2.2.2.2">𝑚</ci><ci id="S6.SS3.8.p8.19.m6.3.3.2.2.2.3.cmml" xref="S6.SS3.8.p8.19.m6.3.3.2.2.2.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.19.m6.3c">j\in\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.19.m6.3d">italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math></p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx42"> <tbody id="S6.Ex68"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\widetilde{\mathcal{B}}^{j}F(1)" class="ltx_Math" display="inline" id="S6.Ex68.m1.1"><semantics id="S6.Ex68.m1.1a"><mrow id="S6.Ex68.m1.1.2" xref="S6.Ex68.m1.1.2.cmml"><msup id="S6.Ex68.m1.1.2.2" xref="S6.Ex68.m1.1.2.2.cmml"><mover accent="true" id="S6.Ex68.m1.1.2.2.2" xref="S6.Ex68.m1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex68.m1.1.2.2.2.2" xref="S6.Ex68.m1.1.2.2.2.2.cmml">ℬ</mi><mo id="S6.Ex68.m1.1.2.2.2.1" xref="S6.Ex68.m1.1.2.2.2.1.cmml">~</mo></mover><mi id="S6.Ex68.m1.1.2.2.3" xref="S6.Ex68.m1.1.2.2.3.cmml">j</mi></msup><mo id="S6.Ex68.m1.1.2.1" xref="S6.Ex68.m1.1.2.1.cmml">⁢</mo><mi id="S6.Ex68.m1.1.2.3" xref="S6.Ex68.m1.1.2.3.cmml">F</mi><mo id="S6.Ex68.m1.1.2.1a" xref="S6.Ex68.m1.1.2.1.cmml">⁢</mo><mrow id="S6.Ex68.m1.1.2.4.2" xref="S6.Ex68.m1.1.2.cmml"><mo id="S6.Ex68.m1.1.2.4.2.1" stretchy="false" xref="S6.Ex68.m1.1.2.cmml">(</mo><mn id="S6.Ex68.m1.1.1" xref="S6.Ex68.m1.1.1.cmml">1</mn><mo id="S6.Ex68.m1.1.2.4.2.2" stretchy="false" xref="S6.Ex68.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex68.m1.1b"><apply id="S6.Ex68.m1.1.2.cmml" xref="S6.Ex68.m1.1.2"><times id="S6.Ex68.m1.1.2.1.cmml" xref="S6.Ex68.m1.1.2.1"></times><apply id="S6.Ex68.m1.1.2.2.cmml" xref="S6.Ex68.m1.1.2.2"><csymbol cd="ambiguous" id="S6.Ex68.m1.1.2.2.1.cmml" xref="S6.Ex68.m1.1.2.2">superscript</csymbol><apply id="S6.Ex68.m1.1.2.2.2.cmml" xref="S6.Ex68.m1.1.2.2.2"><ci id="S6.Ex68.m1.1.2.2.2.1.cmml" xref="S6.Ex68.m1.1.2.2.2.1">~</ci><ci id="S6.Ex68.m1.1.2.2.2.2.cmml" xref="S6.Ex68.m1.1.2.2.2.2">ℬ</ci></apply><ci id="S6.Ex68.m1.1.2.2.3.cmml" xref="S6.Ex68.m1.1.2.2.3">𝑗</ci></apply><ci id="S6.Ex68.m1.1.2.3.cmml" xref="S6.Ex68.m1.1.2.3">𝐹</ci><cn id="S6.Ex68.m1.1.1.cmml" type="integer" xref="S6.Ex68.m1.1.1">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex68.m1.1c">\displaystyle\widetilde{\mathcal{B}}^{j}F(1)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex68.m1.1d">over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_F ( 1 )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\pi_{j}\mathcal{C}^{j}\operatorname{ext}_{\mathcal{C}}(f_{0}(1),% \dots,f_{q_{1}}(1),0,\dots,0)" class="ltx_Math" display="inline" id="S6.Ex68.m2.9"><semantics id="S6.Ex68.m2.9a"><mrow id="S6.Ex68.m2.9.9" xref="S6.Ex68.m2.9.9.cmml"><mi id="S6.Ex68.m2.9.9.5" xref="S6.Ex68.m2.9.9.5.cmml"></mi><mo id="S6.Ex68.m2.9.9.4" xref="S6.Ex68.m2.9.9.4.cmml">=</mo><mrow id="S6.Ex68.m2.9.9.3" xref="S6.Ex68.m2.9.9.3.cmml"><msub id="S6.Ex68.m2.9.9.3.5" xref="S6.Ex68.m2.9.9.3.5.cmml"><mi id="S6.Ex68.m2.9.9.3.5.2" xref="S6.Ex68.m2.9.9.3.5.2.cmml">π</mi><mi id="S6.Ex68.m2.9.9.3.5.3" xref="S6.Ex68.m2.9.9.3.5.3.cmml">j</mi></msub><mo id="S6.Ex68.m2.9.9.3.4" xref="S6.Ex68.m2.9.9.3.4.cmml">⁢</mo><msup id="S6.Ex68.m2.9.9.3.6" xref="S6.Ex68.m2.9.9.3.6.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex68.m2.9.9.3.6.2" xref="S6.Ex68.m2.9.9.3.6.2.cmml">𝒞</mi><mi id="S6.Ex68.m2.9.9.3.6.3" xref="S6.Ex68.m2.9.9.3.6.3.cmml">j</mi></msup><mo id="S6.Ex68.m2.9.9.3.4a" lspace="0.167em" xref="S6.Ex68.m2.9.9.3.4.cmml">⁢</mo><mrow id="S6.Ex68.m2.9.9.3.3.3" xref="S6.Ex68.m2.9.9.3.3.4.cmml"><msub id="S6.Ex68.m2.7.7.1.1.1.1" xref="S6.Ex68.m2.7.7.1.1.1.1.cmml"><mi id="S6.Ex68.m2.7.7.1.1.1.1.2" xref="S6.Ex68.m2.7.7.1.1.1.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex68.m2.7.7.1.1.1.1.3" xref="S6.Ex68.m2.7.7.1.1.1.1.3.cmml">𝒞</mi></msub><mo id="S6.Ex68.m2.9.9.3.3.3a" xref="S6.Ex68.m2.9.9.3.3.4.cmml">⁡</mo><mrow id="S6.Ex68.m2.9.9.3.3.3.3" xref="S6.Ex68.m2.9.9.3.3.4.cmml"><mo id="S6.Ex68.m2.9.9.3.3.3.3.3" stretchy="false" xref="S6.Ex68.m2.9.9.3.3.4.cmml">(</mo><mrow id="S6.Ex68.m2.8.8.2.2.2.2.1" xref="S6.Ex68.m2.8.8.2.2.2.2.1.cmml"><msub id="S6.Ex68.m2.8.8.2.2.2.2.1.2" xref="S6.Ex68.m2.8.8.2.2.2.2.1.2.cmml"><mi id="S6.Ex68.m2.8.8.2.2.2.2.1.2.2" xref="S6.Ex68.m2.8.8.2.2.2.2.1.2.2.cmml">f</mi><mn id="S6.Ex68.m2.8.8.2.2.2.2.1.2.3" xref="S6.Ex68.m2.8.8.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.Ex68.m2.8.8.2.2.2.2.1.1" xref="S6.Ex68.m2.8.8.2.2.2.2.1.1.cmml">⁢</mo><mrow id="S6.Ex68.m2.8.8.2.2.2.2.1.3.2" xref="S6.Ex68.m2.8.8.2.2.2.2.1.cmml"><mo id="S6.Ex68.m2.8.8.2.2.2.2.1.3.2.1" stretchy="false" xref="S6.Ex68.m2.8.8.2.2.2.2.1.cmml">(</mo><mn id="S6.Ex68.m2.1.1" xref="S6.Ex68.m2.1.1.cmml">1</mn><mo id="S6.Ex68.m2.8.8.2.2.2.2.1.3.2.2" stretchy="false" xref="S6.Ex68.m2.8.8.2.2.2.2.1.cmml">)</mo></mrow></mrow><mo id="S6.Ex68.m2.9.9.3.3.3.3.4" xref="S6.Ex68.m2.9.9.3.3.4.cmml">,</mo><mi id="S6.Ex68.m2.3.3" mathvariant="normal" xref="S6.Ex68.m2.3.3.cmml">…</mi><mo id="S6.Ex68.m2.9.9.3.3.3.3.5" xref="S6.Ex68.m2.9.9.3.3.4.cmml">,</mo><mrow id="S6.Ex68.m2.9.9.3.3.3.3.2" xref="S6.Ex68.m2.9.9.3.3.3.3.2.cmml"><msub id="S6.Ex68.m2.9.9.3.3.3.3.2.2" xref="S6.Ex68.m2.9.9.3.3.3.3.2.2.cmml"><mi id="S6.Ex68.m2.9.9.3.3.3.3.2.2.2" xref="S6.Ex68.m2.9.9.3.3.3.3.2.2.2.cmml">f</mi><msub id="S6.Ex68.m2.9.9.3.3.3.3.2.2.3" xref="S6.Ex68.m2.9.9.3.3.3.3.2.2.3.cmml"><mi id="S6.Ex68.m2.9.9.3.3.3.3.2.2.3.2" 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id="S6.Ex68.m2.9.9.3.3.3.3.2.2.3.3.cmml" type="integer" xref="S6.Ex68.m2.9.9.3.3.3.3.2.2.3.3">1</cn></apply></apply><cn id="S6.Ex68.m2.2.2.cmml" type="integer" xref="S6.Ex68.m2.2.2">1</cn></apply><cn id="S6.Ex68.m2.4.4.cmml" type="integer" xref="S6.Ex68.m2.4.4">0</cn><ci id="S6.Ex68.m2.5.5.cmml" xref="S6.Ex68.m2.5.5">…</ci><cn id="S6.Ex68.m2.6.6.cmml" type="integer" xref="S6.Ex68.m2.6.6">0</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex68.m2.9c">\displaystyle=\pi_{j}\mathcal{C}^{j}\operatorname{ext}_{\mathcal{C}}(f_{0}(1),% \dots,f_{q_{1}}(1),0,\dots,0)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex68.m2.9d">= italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_ext start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_f start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ( 1 ) , … , italic_f start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 1 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xref="S6.Ex69.m1.4.4.4.4.2.1.2.1.2.2.6">𝑞</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex69.m1.4c">\displaystyle=\begin{cases}\pi_{j}f_{j}(1)=0&\text{ if }j\in\{m_{0},\dots,m_{n% }\},\,j\leq q_{1},\\ 0&\text{ if }j\in\{m_{0},\dots,m_{n}\},\,q_{1}<j\leq q.\end{cases}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex69.m1.4d">= { start_ROW start_CELL italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT italic_f start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ( 1 ) = 0 end_CELL start_CELL if italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , italic_j ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL if italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_j ≤ italic_q . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.20">This implies <math alttext="F\in\mathscr{H}(W^{k_{0},p},W^{k_{0}+k_{1},p}_{\widetilde{\mathcal{B}}})" class="ltx_Math" display="inline" id="S6.SS3.8.p8.20.m1.6"><semantics id="S6.SS3.8.p8.20.m1.6a"><mrow id="S6.SS3.8.p8.20.m1.6.6" xref="S6.SS3.8.p8.20.m1.6.6.cmml"><mi id="S6.SS3.8.p8.20.m1.6.6.4" xref="S6.SS3.8.p8.20.m1.6.6.4.cmml">F</mi><mo id="S6.SS3.8.p8.20.m1.6.6.3" xref="S6.SS3.8.p8.20.m1.6.6.3.cmml">∈</mo><mrow id="S6.SS3.8.p8.20.m1.6.6.2" xref="S6.SS3.8.p8.20.m1.6.6.2.cmml"><mi class="ltx_font_mathscript" id="S6.SS3.8.p8.20.m1.6.6.2.4" xref="S6.SS3.8.p8.20.m1.6.6.2.4.cmml">ℋ</mi><mo id="S6.SS3.8.p8.20.m1.6.6.2.3" xref="S6.SS3.8.p8.20.m1.6.6.2.3.cmml">⁢</mo><mrow id="S6.SS3.8.p8.20.m1.6.6.2.2.2" xref="S6.SS3.8.p8.20.m1.6.6.2.2.3.cmml"><mo id="S6.SS3.8.p8.20.m1.6.6.2.2.2.3" 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id="S6.SS3.8.p8.20.m1.6.6.2.2.2.2.3.2.cmml" xref="S6.SS3.8.p8.20.m1.6.6.2.2.2.2.3.2">ℬ</ci></apply></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.20.m1.6c">F\in\mathscr{H}(W^{k_{0},p},W^{k_{0}+k_{1},p}_{\widetilde{\mathcal{B}}})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.20.m1.6d">italic_F ∈ script_H ( italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over~ start_ARG caligraphic_B end_ARG end_POSTSUBSCRIPT )</annotation></semantics></math> and thus by (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E6" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.6</span></a>) we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.E13"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="v=F(\theta)\in\big{[}W^{k_{0},p},W^{k_{0}+k_{1},p}_{\mathcal{B}}\big{]}_{% \theta}." class="ltx_Math" display="block" id="S6.E13.m1.6"><semantics id="S6.E13.m1.6a"><mrow id="S6.E13.m1.6.6.1" xref="S6.E13.m1.6.6.1.1.cmml"><mrow id="S6.E13.m1.6.6.1.1" xref="S6.E13.m1.6.6.1.1.cmml"><mi id="S6.E13.m1.6.6.1.1.4" xref="S6.E13.m1.6.6.1.1.4.cmml">v</mi><mo id="S6.E13.m1.6.6.1.1.5" xref="S6.E13.m1.6.6.1.1.5.cmml">=</mo><mrow id="S6.E13.m1.6.6.1.1.6" xref="S6.E13.m1.6.6.1.1.6.cmml"><mi id="S6.E13.m1.6.6.1.1.6.2" xref="S6.E13.m1.6.6.1.1.6.2.cmml">F</mi><mo id="S6.E13.m1.6.6.1.1.6.1" 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xref="S6.E13.m1.6.6.1.1.5"></eq><ci id="S6.E13.m1.6.6.1.1.4.cmml" xref="S6.E13.m1.6.6.1.1.4">𝑣</ci><apply id="S6.E13.m1.6.6.1.1.6.cmml" xref="S6.E13.m1.6.6.1.1.6"><times id="S6.E13.m1.6.6.1.1.6.1.cmml" xref="S6.E13.m1.6.6.1.1.6.1"></times><ci id="S6.E13.m1.6.6.1.1.6.2.cmml" xref="S6.E13.m1.6.6.1.1.6.2">𝐹</ci><ci id="S6.E13.m1.5.5.cmml" xref="S6.E13.m1.5.5">𝜃</ci></apply></apply><apply id="S6.E13.m1.6.6.1.1c.cmml" xref="S6.E13.m1.6.6.1"><in id="S6.E13.m1.6.6.1.1.7.cmml" xref="S6.E13.m1.6.6.1.1.7"></in><share href="https://arxiv.org/html/2503.14636v1#S6.E13.m1.6.6.1.1.6.cmml" id="S6.E13.m1.6.6.1.1d.cmml" xref="S6.E13.m1.6.6.1"></share><apply id="S6.E13.m1.6.6.1.1.2.cmml" xref="S6.E13.m1.6.6.1.1.2"><csymbol cd="ambiguous" id="S6.E13.m1.6.6.1.1.2.3.cmml" xref="S6.E13.m1.6.6.1.1.2">subscript</csymbol><interval closure="closed" id="S6.E13.m1.6.6.1.1.2.2.3.cmml" xref="S6.E13.m1.6.6.1.1.2.2.2"><apply id="S6.E13.m1.6.6.1.1.1.1.1.1.cmml" xref="S6.E13.m1.6.6.1.1.1.1.1.1"><csymbol cd="ambiguous" 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.</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.13)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.24">To continue, we claim that <math alttext="u-v\in W^{k_{0}+\ell,p}_{\mathcal{C}}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.21.m1.2"><semantics id="S6.SS3.8.p8.21.m1.2a"><mrow id="S6.SS3.8.p8.21.m1.2.3" xref="S6.SS3.8.p8.21.m1.2.3.cmml"><mrow id="S6.SS3.8.p8.21.m1.2.3.2" xref="S6.SS3.8.p8.21.m1.2.3.2.cmml"><mi id="S6.SS3.8.p8.21.m1.2.3.2.2" xref="S6.SS3.8.p8.21.m1.2.3.2.2.cmml">u</mi><mo id="S6.SS3.8.p8.21.m1.2.3.2.1" xref="S6.SS3.8.p8.21.m1.2.3.2.1.cmml">−</mo><mi id="S6.SS3.8.p8.21.m1.2.3.2.3" xref="S6.SS3.8.p8.21.m1.2.3.2.3.cmml">v</mi></mrow><mo id="S6.SS3.8.p8.21.m1.2.3.1" xref="S6.SS3.8.p8.21.m1.2.3.1.cmml">∈</mo><msubsup id="S6.SS3.8.p8.21.m1.2.3.3" xref="S6.SS3.8.p8.21.m1.2.3.3.cmml"><mi id="S6.SS3.8.p8.21.m1.2.3.3.2.2" xref="S6.SS3.8.p8.21.m1.2.3.3.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.8.p8.21.m1.2.3.3.3" xref="S6.SS3.8.p8.21.m1.2.3.3.3.cmml">𝒞</mi><mrow id="S6.SS3.8.p8.21.m1.2.2.2.2" xref="S6.SS3.8.p8.21.m1.2.2.2.3.cmml"><mrow id="S6.SS3.8.p8.21.m1.2.2.2.2.1" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.cmml"><msub id="S6.SS3.8.p8.21.m1.2.2.2.2.1.2" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.cmml"><mi id="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.2" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.2.cmml">k</mi><mn id="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.3" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.21.m1.2.2.2.2.1.1" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.1.cmml">+</mo><mi id="S6.SS3.8.p8.21.m1.2.2.2.2.1.3" mathvariant="normal" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.3.cmml">ℓ</mi></mrow><mo id="S6.SS3.8.p8.21.m1.2.2.2.2.2" xref="S6.SS3.8.p8.21.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.21.m1.1.1.1.1" xref="S6.SS3.8.p8.21.m1.1.1.1.1.cmml">p</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.21.m1.2b"><apply id="S6.SS3.8.p8.21.m1.2.3.cmml" xref="S6.SS3.8.p8.21.m1.2.3"><in id="S6.SS3.8.p8.21.m1.2.3.1.cmml" xref="S6.SS3.8.p8.21.m1.2.3.1"></in><apply id="S6.SS3.8.p8.21.m1.2.3.2.cmml" xref="S6.SS3.8.p8.21.m1.2.3.2"><minus id="S6.SS3.8.p8.21.m1.2.3.2.1.cmml" xref="S6.SS3.8.p8.21.m1.2.3.2.1"></minus><ci id="S6.SS3.8.p8.21.m1.2.3.2.2.cmml" xref="S6.SS3.8.p8.21.m1.2.3.2.2">𝑢</ci><ci id="S6.SS3.8.p8.21.m1.2.3.2.3.cmml" xref="S6.SS3.8.p8.21.m1.2.3.2.3">𝑣</ci></apply><apply id="S6.SS3.8.p8.21.m1.2.3.3.cmml" xref="S6.SS3.8.p8.21.m1.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.21.m1.2.3.3.1.cmml" xref="S6.SS3.8.p8.21.m1.2.3.3">subscript</csymbol><apply id="S6.SS3.8.p8.21.m1.2.3.3.2.cmml" xref="S6.SS3.8.p8.21.m1.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.21.m1.2.3.3.2.1.cmml" xref="S6.SS3.8.p8.21.m1.2.3.3">superscript</csymbol><ci id="S6.SS3.8.p8.21.m1.2.3.3.2.2.cmml" xref="S6.SS3.8.p8.21.m1.2.3.3.2.2">𝑊</ci><list id="S6.SS3.8.p8.21.m1.2.2.2.3.cmml" xref="S6.SS3.8.p8.21.m1.2.2.2.2"><apply id="S6.SS3.8.p8.21.m1.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1"><plus id="S6.SS3.8.p8.21.m1.2.2.2.2.1.1.cmml" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.1"></plus><apply id="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.cmml" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.1.cmml" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.2">subscript</csymbol><ci id="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.2.cmml" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.2.3">0</cn></apply><ci id="S6.SS3.8.p8.21.m1.2.2.2.2.1.3.cmml" xref="S6.SS3.8.p8.21.m1.2.2.2.2.1.3">ℓ</ci></apply><ci id="S6.SS3.8.p8.21.m1.1.1.1.1.cmml" xref="S6.SS3.8.p8.21.m1.1.1.1.1">𝑝</ci></list></apply><ci id="S6.SS3.8.p8.21.m1.2.3.3.3.cmml" xref="S6.SS3.8.p8.21.m1.2.3.3.3">𝒞</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.21.m1.2c">u-v\in W^{k_{0}+\ell,p}_{\mathcal{C}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.21.m1.2d">italic_u - italic_v ∈ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT</annotation></semantics></math>. Indeed, by definition of <math alttext="g_{j}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.22.m2.1"><semantics id="S6.SS3.8.p8.22.m2.1a"><msub id="S6.SS3.8.p8.22.m2.1.1" xref="S6.SS3.8.p8.22.m2.1.1.cmml"><mi id="S6.SS3.8.p8.22.m2.1.1.2" xref="S6.SS3.8.p8.22.m2.1.1.2.cmml">g</mi><mi id="S6.SS3.8.p8.22.m2.1.1.3" xref="S6.SS3.8.p8.22.m2.1.1.3.cmml">j</mi></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.22.m2.1b"><apply id="S6.SS3.8.p8.22.m2.1.1.cmml" xref="S6.SS3.8.p8.22.m2.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.22.m2.1.1.1.cmml" xref="S6.SS3.8.p8.22.m2.1.1">subscript</csymbol><ci id="S6.SS3.8.p8.22.m2.1.1.2.cmml" xref="S6.SS3.8.p8.22.m2.1.1.2">𝑔</ci><ci id="S6.SS3.8.p8.22.m2.1.1.3.cmml" xref="S6.SS3.8.p8.22.m2.1.1.3">𝑗</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.22.m2.1c">g_{j}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.22.m2.1d">italic_g start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT</annotation></semantics></math>, (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E10" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.10</span></a>) and the fact that <math alttext="\widetilde{\mathcal{B}}^{j}u=0" class="ltx_Math" display="inline" id="S6.SS3.8.p8.23.m3.1"><semantics id="S6.SS3.8.p8.23.m3.1a"><mrow id="S6.SS3.8.p8.23.m3.1.1" xref="S6.SS3.8.p8.23.m3.1.1.cmml"><mrow id="S6.SS3.8.p8.23.m3.1.1.2" xref="S6.SS3.8.p8.23.m3.1.1.2.cmml"><msup id="S6.SS3.8.p8.23.m3.1.1.2.2" xref="S6.SS3.8.p8.23.m3.1.1.2.2.cmml"><mover accent="true" id="S6.SS3.8.p8.23.m3.1.1.2.2.2" xref="S6.SS3.8.p8.23.m3.1.1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.SS3.8.p8.23.m3.1.1.2.2.2.2" xref="S6.SS3.8.p8.23.m3.1.1.2.2.2.2.cmml">ℬ</mi><mo id="S6.SS3.8.p8.23.m3.1.1.2.2.2.1" xref="S6.SS3.8.p8.23.m3.1.1.2.2.2.1.cmml">~</mo></mover><mi id="S6.SS3.8.p8.23.m3.1.1.2.2.3" xref="S6.SS3.8.p8.23.m3.1.1.2.2.3.cmml">j</mi></msup><mo id="S6.SS3.8.p8.23.m3.1.1.2.1" xref="S6.SS3.8.p8.23.m3.1.1.2.1.cmml">⁢</mo><mi id="S6.SS3.8.p8.23.m3.1.1.2.3" xref="S6.SS3.8.p8.23.m3.1.1.2.3.cmml">u</mi></mrow><mo id="S6.SS3.8.p8.23.m3.1.1.1" xref="S6.SS3.8.p8.23.m3.1.1.1.cmml">=</mo><mn id="S6.SS3.8.p8.23.m3.1.1.3" xref="S6.SS3.8.p8.23.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.23.m3.1b"><apply id="S6.SS3.8.p8.23.m3.1.1.cmml" xref="S6.SS3.8.p8.23.m3.1.1"><eq id="S6.SS3.8.p8.23.m3.1.1.1.cmml" xref="S6.SS3.8.p8.23.m3.1.1.1"></eq><apply id="S6.SS3.8.p8.23.m3.1.1.2.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2"><times id="S6.SS3.8.p8.23.m3.1.1.2.1.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.1"></times><apply id="S6.SS3.8.p8.23.m3.1.1.2.2.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.23.m3.1.1.2.2.1.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.2">superscript</csymbol><apply id="S6.SS3.8.p8.23.m3.1.1.2.2.2.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.2.2"><ci id="S6.SS3.8.p8.23.m3.1.1.2.2.2.1.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.2.2.1">~</ci><ci id="S6.SS3.8.p8.23.m3.1.1.2.2.2.2.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.2.2.2">ℬ</ci></apply><ci id="S6.SS3.8.p8.23.m3.1.1.2.2.3.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.2.3">𝑗</ci></apply><ci id="S6.SS3.8.p8.23.m3.1.1.2.3.cmml" xref="S6.SS3.8.p8.23.m3.1.1.2.3">𝑢</ci></apply><cn id="S6.SS3.8.p8.23.m3.1.1.3.cmml" type="integer" xref="S6.SS3.8.p8.23.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.23.m3.1c">\widetilde{\mathcal{B}}^{j}u=0</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.23.m3.1d">over~ start_ARG caligraphic_B end_ARG start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u = 0</annotation></semantics></math> for <math alttext="j\in\{m_{0},\dots,m_{n}\}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.24.m4.3"><semantics id="S6.SS3.8.p8.24.m4.3a"><mrow id="S6.SS3.8.p8.24.m4.3.3" xref="S6.SS3.8.p8.24.m4.3.3.cmml"><mi id="S6.SS3.8.p8.24.m4.3.3.4" xref="S6.SS3.8.p8.24.m4.3.3.4.cmml">j</mi><mo id="S6.SS3.8.p8.24.m4.3.3.3" xref="S6.SS3.8.p8.24.m4.3.3.3.cmml">∈</mo><mrow id="S6.SS3.8.p8.24.m4.3.3.2.2" xref="S6.SS3.8.p8.24.m4.3.3.2.3.cmml"><mo id="S6.SS3.8.p8.24.m4.3.3.2.2.3" stretchy="false" xref="S6.SS3.8.p8.24.m4.3.3.2.3.cmml">{</mo><msub id="S6.SS3.8.p8.24.m4.2.2.1.1.1" xref="S6.SS3.8.p8.24.m4.2.2.1.1.1.cmml"><mi id="S6.SS3.8.p8.24.m4.2.2.1.1.1.2" xref="S6.SS3.8.p8.24.m4.2.2.1.1.1.2.cmml">m</mi><mn id="S6.SS3.8.p8.24.m4.2.2.1.1.1.3" xref="S6.SS3.8.p8.24.m4.2.2.1.1.1.3.cmml">0</mn></msub><mo id="S6.SS3.8.p8.24.m4.3.3.2.2.4" xref="S6.SS3.8.p8.24.m4.3.3.2.3.cmml">,</mo><mi id="S6.SS3.8.p8.24.m4.1.1" mathvariant="normal" xref="S6.SS3.8.p8.24.m4.1.1.cmml">…</mi><mo id="S6.SS3.8.p8.24.m4.3.3.2.2.5" xref="S6.SS3.8.p8.24.m4.3.3.2.3.cmml">,</mo><msub id="S6.SS3.8.p8.24.m4.3.3.2.2.2" xref="S6.SS3.8.p8.24.m4.3.3.2.2.2.cmml"><mi id="S6.SS3.8.p8.24.m4.3.3.2.2.2.2" xref="S6.SS3.8.p8.24.m4.3.3.2.2.2.2.cmml">m</mi><mi id="S6.SS3.8.p8.24.m4.3.3.2.2.2.3" xref="S6.SS3.8.p8.24.m4.3.3.2.2.2.3.cmml">n</mi></msub><mo id="S6.SS3.8.p8.24.m4.3.3.2.2.6" stretchy="false" xref="S6.SS3.8.p8.24.m4.3.3.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.24.m4.3b"><apply id="S6.SS3.8.p8.24.m4.3.3.cmml" xref="S6.SS3.8.p8.24.m4.3.3"><in id="S6.SS3.8.p8.24.m4.3.3.3.cmml" xref="S6.SS3.8.p8.24.m4.3.3.3"></in><ci id="S6.SS3.8.p8.24.m4.3.3.4.cmml" xref="S6.SS3.8.p8.24.m4.3.3.4">𝑗</ci><set id="S6.SS3.8.p8.24.m4.3.3.2.3.cmml" xref="S6.SS3.8.p8.24.m4.3.3.2.2"><apply id="S6.SS3.8.p8.24.m4.2.2.1.1.1.cmml" xref="S6.SS3.8.p8.24.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.24.m4.2.2.1.1.1.1.cmml" xref="S6.SS3.8.p8.24.m4.2.2.1.1.1">subscript</csymbol><ci id="S6.SS3.8.p8.24.m4.2.2.1.1.1.2.cmml" xref="S6.SS3.8.p8.24.m4.2.2.1.1.1.2">𝑚</ci><cn id="S6.SS3.8.p8.24.m4.2.2.1.1.1.3.cmml" type="integer" xref="S6.SS3.8.p8.24.m4.2.2.1.1.1.3">0</cn></apply><ci id="S6.SS3.8.p8.24.m4.1.1.cmml" xref="S6.SS3.8.p8.24.m4.1.1">…</ci><apply id="S6.SS3.8.p8.24.m4.3.3.2.2.2.cmml" xref="S6.SS3.8.p8.24.m4.3.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.24.m4.3.3.2.2.2.1.cmml" xref="S6.SS3.8.p8.24.m4.3.3.2.2.2">subscript</csymbol><ci id="S6.SS3.8.p8.24.m4.3.3.2.2.2.2.cmml" xref="S6.SS3.8.p8.24.m4.3.3.2.2.2.2">𝑚</ci><ci id="S6.SS3.8.p8.24.m4.3.3.2.2.2.3.cmml" xref="S6.SS3.8.p8.24.m4.3.3.2.2.2.3">𝑛</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.24.m4.3c">j\in\{m_{0},\dots,m_{n}\}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.24.m4.3d">italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT }</annotation></semantics></math>, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx43"> <tbody id="S6.Ex70"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\mathcal{C}^{j}(u-v)" class="ltx_Math" display="inline" id="S6.Ex70.m1.1"><semantics id="S6.Ex70.m1.1a"><mrow id="S6.Ex70.m1.1.1" xref="S6.Ex70.m1.1.1.cmml"><msup id="S6.Ex70.m1.1.1.3" xref="S6.Ex70.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex70.m1.1.1.3.2" xref="S6.Ex70.m1.1.1.3.2.cmml">𝒞</mi><mi id="S6.Ex70.m1.1.1.3.3" xref="S6.Ex70.m1.1.1.3.3.cmml">j</mi></msup><mo id="S6.Ex70.m1.1.1.2" xref="S6.Ex70.m1.1.1.2.cmml">⁢</mo><mrow id="S6.Ex70.m1.1.1.1.1" xref="S6.Ex70.m1.1.1.1.1.1.cmml"><mo id="S6.Ex70.m1.1.1.1.1.2" stretchy="false" xref="S6.Ex70.m1.1.1.1.1.1.cmml">(</mo><mrow id="S6.Ex70.m1.1.1.1.1.1" xref="S6.Ex70.m1.1.1.1.1.1.cmml"><mi id="S6.Ex70.m1.1.1.1.1.1.2" xref="S6.Ex70.m1.1.1.1.1.1.2.cmml">u</mi><mo id="S6.Ex70.m1.1.1.1.1.1.1" xref="S6.Ex70.m1.1.1.1.1.1.1.cmml">−</mo><mi id="S6.Ex70.m1.1.1.1.1.1.3" xref="S6.Ex70.m1.1.1.1.1.1.3.cmml">v</mi></mrow><mo id="S6.Ex70.m1.1.1.1.1.3" stretchy="false" xref="S6.Ex70.m1.1.1.1.1.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.Ex70.m1.1b"><apply id="S6.Ex70.m1.1.1.cmml" xref="S6.Ex70.m1.1.1"><times id="S6.Ex70.m1.1.1.2.cmml" xref="S6.Ex70.m1.1.1.2"></times><apply id="S6.Ex70.m1.1.1.3.cmml" xref="S6.Ex70.m1.1.1.3"><csymbol cd="ambiguous" id="S6.Ex70.m1.1.1.3.1.cmml" xref="S6.Ex70.m1.1.1.3">superscript</csymbol><ci id="S6.Ex70.m1.1.1.3.2.cmml" xref="S6.Ex70.m1.1.1.3.2">𝒞</ci><ci id="S6.Ex70.m1.1.1.3.3.cmml" xref="S6.Ex70.m1.1.1.3.3">𝑗</ci></apply><apply id="S6.Ex70.m1.1.1.1.1.1.cmml" xref="S6.Ex70.m1.1.1.1.1"><minus id="S6.Ex70.m1.1.1.1.1.1.1.cmml" xref="S6.Ex70.m1.1.1.1.1.1.1"></minus><ci id="S6.Ex70.m1.1.1.1.1.1.2.cmml" xref="S6.Ex70.m1.1.1.1.1.1.2">𝑢</ci><ci id="S6.Ex70.m1.1.1.1.1.1.3.cmml" xref="S6.Ex70.m1.1.1.1.1.1.3">𝑣</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex70.m1.1c">\displaystyle\mathcal{C}^{j}(u-v)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex70.m1.1d">caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT ( italic_u - italic_v )</annotation></semantics></math></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{C}^{j}u-\mathcal{C}^{j}\operatorname{ext}_{\mathcal{C}}% (g_{0},\dots,g_{q_{1}},0,\dots,0)" class="ltx_Math" display="inline" id="S6.Ex70.m2.7"><semantics id="S6.Ex70.m2.7a"><mrow id="S6.Ex70.m2.7.7" xref="S6.Ex70.m2.7.7.cmml"><mi id="S6.Ex70.m2.7.7.5" xref="S6.Ex70.m2.7.7.5.cmml"></mi><mo id="S6.Ex70.m2.7.7.4" xref="S6.Ex70.m2.7.7.4.cmml">=</mo><mrow id="S6.Ex70.m2.7.7.3" xref="S6.Ex70.m2.7.7.3.cmml"><mrow id="S6.Ex70.m2.7.7.3.5" xref="S6.Ex70.m2.7.7.3.5.cmml"><msup id="S6.Ex70.m2.7.7.3.5.2" xref="S6.Ex70.m2.7.7.3.5.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex70.m2.7.7.3.5.2.2" xref="S6.Ex70.m2.7.7.3.5.2.2.cmml">𝒞</mi><mi id="S6.Ex70.m2.7.7.3.5.2.3" xref="S6.Ex70.m2.7.7.3.5.2.3.cmml">j</mi></msup><mo id="S6.Ex70.m2.7.7.3.5.1" xref="S6.Ex70.m2.7.7.3.5.1.cmml">⁢</mo><mi id="S6.Ex70.m2.7.7.3.5.3" xref="S6.Ex70.m2.7.7.3.5.3.cmml">u</mi></mrow><mo id="S6.Ex70.m2.7.7.3.4" xref="S6.Ex70.m2.7.7.3.4.cmml">−</mo><mrow id="S6.Ex70.m2.7.7.3.3" xref="S6.Ex70.m2.7.7.3.3.cmml"><msup id="S6.Ex70.m2.7.7.3.3.5" xref="S6.Ex70.m2.7.7.3.3.5.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex70.m2.7.7.3.3.5.2" xref="S6.Ex70.m2.7.7.3.3.5.2.cmml">𝒞</mi><mi id="S6.Ex70.m2.7.7.3.3.5.3" xref="S6.Ex70.m2.7.7.3.3.5.3.cmml">j</mi></msup><mo id="S6.Ex70.m2.7.7.3.3.4" lspace="0.167em" xref="S6.Ex70.m2.7.7.3.3.4.cmml">⁢</mo><mrow id="S6.Ex70.m2.7.7.3.3.3.3" xref="S6.Ex70.m2.7.7.3.3.3.4.cmml"><msub id="S6.Ex70.m2.5.5.1.1.1.1.1" xref="S6.Ex70.m2.5.5.1.1.1.1.1.cmml"><mi id="S6.Ex70.m2.5.5.1.1.1.1.1.2" xref="S6.Ex70.m2.5.5.1.1.1.1.1.2.cmml">ext</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex70.m2.5.5.1.1.1.1.1.3" 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xref="S6.Ex70.m2.4.4">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex70.m2.7c">\displaystyle=\mathcal{C}^{j}u-\mathcal{C}^{j}\operatorname{ext}_{\mathcal{C}}% (g_{0},\dots,g_{q_{1}},0,\dots,0)</annotation><annotation encoding="application/x-llamapun" id="S6.Ex70.m2.7d">= caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u - caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT roman_ext start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT ( italic_g start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_g start_POSTSUBSCRIPT italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , 0 , … , 0 )</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex71"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\mathcal{C}^{j}u-g_{j}=\begin{cases}g_{j}-g_{j}=0&\mbox{if }j% \notin\{m_{0},\dots,m_{n}\},\,j\leq q_{1},\\ \mathcal{C}^{j}u-(1-\pi_{j})\mathcal{C}^{j}u=0&\mbox{if }j\in\{m_{0},\dots,m_{% n}\},\,j\leq q_{1}.\end{cases}" class="ltx_Math" display="inline" id="S6.Ex71.m1.4"><semantics id="S6.Ex71.m1.4a"><mrow id="S6.Ex71.m1.4.5" xref="S6.Ex71.m1.4.5.cmml"><mi id="S6.Ex71.m1.4.5.2" xref="S6.Ex71.m1.4.5.2.cmml"></mi><mo id="S6.Ex71.m1.4.5.3" xref="S6.Ex71.m1.4.5.3.cmml">=</mo><mrow id="S6.Ex71.m1.4.5.4" xref="S6.Ex71.m1.4.5.4.cmml"><mrow id="S6.Ex71.m1.4.5.4.2" xref="S6.Ex71.m1.4.5.4.2.cmml"><msup id="S6.Ex71.m1.4.5.4.2.2" xref="S6.Ex71.m1.4.5.4.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex71.m1.4.5.4.2.2.2" xref="S6.Ex71.m1.4.5.4.2.2.2.cmml">𝒞</mi><mi id="S6.Ex71.m1.4.5.4.2.2.3" xref="S6.Ex71.m1.4.5.4.2.2.3.cmml">j</mi></msup><mo id="S6.Ex71.m1.4.5.4.2.1" xref="S6.Ex71.m1.4.5.4.2.1.cmml">⁢</mo><mi id="S6.Ex71.m1.4.5.4.2.3" xref="S6.Ex71.m1.4.5.4.2.3.cmml">u</mi></mrow><mo id="S6.Ex71.m1.4.5.4.1" xref="S6.Ex71.m1.4.5.4.1.cmml">−</mo><msub id="S6.Ex71.m1.4.5.4.3" xref="S6.Ex71.m1.4.5.4.3.cmml"><mi id="S6.Ex71.m1.4.5.4.3.2" xref="S6.Ex71.m1.4.5.4.3.2.cmml">g</mi><mi id="S6.Ex71.m1.4.5.4.3.3" xref="S6.Ex71.m1.4.5.4.3.3.cmml">j</mi></msub></mrow><mo id="S6.Ex71.m1.4.5.5" xref="S6.Ex71.m1.4.5.5.cmml">=</mo><mrow id="S6.Ex71.m1.4.4a" xref="S6.Ex71.m1.4.5.6.1.cmml"><mo id="S6.Ex71.m1.4.4a.5" xref="S6.Ex71.m1.4.5.6.1.1.cmml">{</mo><mtable columnspacing="5pt" id="S6.Ex71.m1.4.4.4a" rowspacing="0pt" xref="S6.Ex71.m1.4.5.6.1.cmml"><mtr id="S6.Ex71.m1.4.4.4aa" xref="S6.Ex71.m1.4.5.6.1.cmml"><mtd class="ltx_align_left" columnalign="left" id="S6.Ex71.m1.4.4.4ab" xref="S6.Ex71.m1.4.5.6.1.cmml"><mrow id="S6.Ex71.m1.1.1.1.1.1.1" xref="S6.Ex71.m1.1.1.1.1.1.1.cmml"><mrow id="S6.Ex71.m1.1.1.1.1.1.1.2" xref="S6.Ex71.m1.1.1.1.1.1.1.2.cmml"><msub id="S6.Ex71.m1.1.1.1.1.1.1.2.2" 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italic_j end_POSTSUBSCRIPT = 0 end_CELL start_CELL if italic_j ∉ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , italic_j ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u - ( 1 - italic_π start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) caligraphic_C start_POSTSUPERSCRIPT italic_j end_POSTSUPERSCRIPT italic_u = 0 end_CELL start_CELL if italic_j ∈ { italic_m start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , … , italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT } , italic_j ≤ italic_q start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT . end_CELL end_ROW</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.30">It follows from Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem2" title="Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.2</span></a><a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.I1.i1" title="item i ‣ Proposition 6.2. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">i</span></a> that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx44"> <tbody id="S6.E14"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle u-v\in W^{k_{0}+\ell,p}_{\mathcal{C}}=W^{k_{0}+\ell,p}_{0}=\big{% [}W^{k_{0},p}_{0},W^{k_{0}+k_{1},p}_{0}\big{]}_{\theta}\hookrightarrow\big{[}W% ^{k_{0},p},W^{k_{0}+k_{1},p}_{\mathcal{B}}\big{]}_{\theta}." class="ltx_Math" display="inline" id="S6.E14.m1.13"><semantics id="S6.E14.m1.13a"><mrow id="S6.E14.m1.13.13.1" xref="S6.E14.m1.13.13.1.1.cmml"><mrow id="S6.E14.m1.13.13.1.1" xref="S6.E14.m1.13.13.1.1.cmml"><mrow id="S6.E14.m1.13.13.1.1.6" xref="S6.E14.m1.13.13.1.1.6.cmml"><mi id="S6.E14.m1.13.13.1.1.6.2" xref="S6.E14.m1.13.13.1.1.6.2.cmml">u</mi><mo 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W^{k_{0}+\ell,p}_{\mathcal{C}}=W^{k_{0}+\ell,p}_{0}=\big{% [}W^{k_{0},p}_{0},W^{k_{0}+k_{1},p}_{0}\big{]}_{\theta}\hookrightarrow\big{[}W% ^{k_{0},p},W^{k_{0}+k_{1},p}_{\mathcal{B}}\big{]}_{\theta}.</annotation><annotation encoding="application/x-llamapun" id="S6.E14.m1.13d">italic_u - italic_v ∈ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.14)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.8.p8.26">Then (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E13" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.13</span></a>) and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E14" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. 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xref="S6.SS3.8.p8.25.m1.6.6.2.1.1.1.2">𝑊</ci><list id="S6.SS3.8.p8.25.m1.2.2.2.3.cmml" xref="S6.SS3.8.p8.25.m1.2.2.2.2"><apply id="S6.SS3.8.p8.25.m1.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.25.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.8.p8.25.m1.2.2.2.2.1.1.cmml" xref="S6.SS3.8.p8.25.m1.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.8.p8.25.m1.2.2.2.2.1.2.cmml" xref="S6.SS3.8.p8.25.m1.2.2.2.2.1.2">𝑘</ci><cn id="S6.SS3.8.p8.25.m1.2.2.2.2.1.3.cmml" type="integer" xref="S6.SS3.8.p8.25.m1.2.2.2.2.1.3">0</cn></apply><ci id="S6.SS3.8.p8.25.m1.1.1.1.1.cmml" xref="S6.SS3.8.p8.25.m1.1.1.1.1">𝑝</ci></list></apply><apply id="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.cmml" xref="S6.SS3.8.p8.25.m1.7.7.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.1.cmml" xref="S6.SS3.8.p8.25.m1.7.7.3.2.2.2">subscript</csymbol><apply id="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.2.cmml" xref="S6.SS3.8.p8.25.m1.7.7.3.2.2.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.2.1.cmml" xref="S6.SS3.8.p8.25.m1.7.7.3.2.2.2">superscript</csymbol><ci id="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.2.2.cmml" xref="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.2.2">𝑊</ci><list id="S6.SS3.8.p8.25.m1.4.4.2.3.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2"><apply id="S6.SS3.8.p8.25.m1.4.4.2.2.1.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1"><plus id="S6.SS3.8.p8.25.m1.4.4.2.2.1.1.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.1"></plus><apply id="S6.SS3.8.p8.25.m1.4.4.2.2.1.2.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.2"><csymbol cd="ambiguous" id="S6.SS3.8.p8.25.m1.4.4.2.2.1.2.1.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.2">subscript</csymbol><ci id="S6.SS3.8.p8.25.m1.4.4.2.2.1.2.2.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.2.2">𝑘</ci><cn id="S6.SS3.8.p8.25.m1.4.4.2.2.1.2.3.cmml" type="integer" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.2.3">0</cn></apply><apply id="S6.SS3.8.p8.25.m1.4.4.2.2.1.3.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.25.m1.4.4.2.2.1.3.1.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.3">subscript</csymbol><ci id="S6.SS3.8.p8.25.m1.4.4.2.2.1.3.2.cmml" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.3.2">𝑘</ci><cn id="S6.SS3.8.p8.25.m1.4.4.2.2.1.3.3.cmml" type="integer" xref="S6.SS3.8.p8.25.m1.4.4.2.2.1.3.3">1</cn></apply></apply><ci id="S6.SS3.8.p8.25.m1.3.3.1.1.cmml" xref="S6.SS3.8.p8.25.m1.3.3.1.1">𝑝</ci></list></apply><ci id="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.3.cmml" xref="S6.SS3.8.p8.25.m1.7.7.3.2.2.2.3">ℬ</ci></apply></interval><ci id="S6.SS3.8.p8.25.m1.7.7.3.4.cmml" xref="S6.SS3.8.p8.25.m1.7.7.3.4">𝜃</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.25.m1.7c">u=v+(u-v)\in[W^{k_{0},p},W^{k_{0}+k_{1},p}_{\mathcal{B}}]_{\theta}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.25.m1.7d">italic_u = italic_v + ( italic_u - italic_v ) ∈ [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\theta=\frac{\ell}{k_{1}}" class="ltx_Math" display="inline" id="S6.SS3.8.p8.26.m2.1"><semantics id="S6.SS3.8.p8.26.m2.1a"><mrow id="S6.SS3.8.p8.26.m2.1.1" xref="S6.SS3.8.p8.26.m2.1.1.cmml"><mi id="S6.SS3.8.p8.26.m2.1.1.2" xref="S6.SS3.8.p8.26.m2.1.1.2.cmml">θ</mi><mo id="S6.SS3.8.p8.26.m2.1.1.1" xref="S6.SS3.8.p8.26.m2.1.1.1.cmml">=</mo><mfrac id="S6.SS3.8.p8.26.m2.1.1.3" xref="S6.SS3.8.p8.26.m2.1.1.3.cmml"><mi id="S6.SS3.8.p8.26.m2.1.1.3.2" mathvariant="normal" xref="S6.SS3.8.p8.26.m2.1.1.3.2.cmml">ℓ</mi><msub id="S6.SS3.8.p8.26.m2.1.1.3.3" xref="S6.SS3.8.p8.26.m2.1.1.3.3.cmml"><mi id="S6.SS3.8.p8.26.m2.1.1.3.3.2" xref="S6.SS3.8.p8.26.m2.1.1.3.3.2.cmml">k</mi><mn id="S6.SS3.8.p8.26.m2.1.1.3.3.3" xref="S6.SS3.8.p8.26.m2.1.1.3.3.3.cmml">1</mn></msub></mfrac></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.8.p8.26.m2.1b"><apply id="S6.SS3.8.p8.26.m2.1.1.cmml" xref="S6.SS3.8.p8.26.m2.1.1"><eq id="S6.SS3.8.p8.26.m2.1.1.1.cmml" xref="S6.SS3.8.p8.26.m2.1.1.1"></eq><ci id="S6.SS3.8.p8.26.m2.1.1.2.cmml" xref="S6.SS3.8.p8.26.m2.1.1.2">𝜃</ci><apply id="S6.SS3.8.p8.26.m2.1.1.3.cmml" xref="S6.SS3.8.p8.26.m2.1.1.3"><divide id="S6.SS3.8.p8.26.m2.1.1.3.1.cmml" xref="S6.SS3.8.p8.26.m2.1.1.3"></divide><ci id="S6.SS3.8.p8.26.m2.1.1.3.2.cmml" xref="S6.SS3.8.p8.26.m2.1.1.3.2">ℓ</ci><apply id="S6.SS3.8.p8.26.m2.1.1.3.3.cmml" xref="S6.SS3.8.p8.26.m2.1.1.3.3"><csymbol cd="ambiguous" id="S6.SS3.8.p8.26.m2.1.1.3.3.1.cmml" xref="S6.SS3.8.p8.26.m2.1.1.3.3">subscript</csymbol><ci id="S6.SS3.8.p8.26.m2.1.1.3.3.2.cmml" xref="S6.SS3.8.p8.26.m2.1.1.3.3.2">𝑘</ci><cn id="S6.SS3.8.p8.26.m2.1.1.3.3.3.cmml" type="integer" xref="S6.SS3.8.p8.26.m2.1.1.3.3.3">1</cn></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.8.p8.26.m2.1c">\theta=\frac{\ell}{k_{1}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.8.p8.26.m2.1d">italic_θ = divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS3.9.p9"> <p class="ltx_p" id="S6.SS3.9.p9.3"><span class="ltx_text ltx_font_italic" id="S6.SS3.9.p9.3.1">Step 2b. </span>Assume that <math alttext="k_{0}+k_{1}<m_{n}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.9.p9.1.m1.1"><semantics id="S6.SS3.9.p9.1.m1.1a"><mrow id="S6.SS3.9.p9.1.m1.1.1" xref="S6.SS3.9.p9.1.m1.1.1.cmml"><mrow id="S6.SS3.9.p9.1.m1.1.1.2" xref="S6.SS3.9.p9.1.m1.1.1.2.cmml"><msub id="S6.SS3.9.p9.1.m1.1.1.2.2" xref="S6.SS3.9.p9.1.m1.1.1.2.2.cmml"><mi id="S6.SS3.9.p9.1.m1.1.1.2.2.2" xref="S6.SS3.9.p9.1.m1.1.1.2.2.2.cmml">k</mi><mn id="S6.SS3.9.p9.1.m1.1.1.2.2.3" xref="S6.SS3.9.p9.1.m1.1.1.2.2.3.cmml">0</mn></msub><mo id="S6.SS3.9.p9.1.m1.1.1.2.1" xref="S6.SS3.9.p9.1.m1.1.1.2.1.cmml">+</mo><msub id="S6.SS3.9.p9.1.m1.1.1.2.3" xref="S6.SS3.9.p9.1.m1.1.1.2.3.cmml"><mi id="S6.SS3.9.p9.1.m1.1.1.2.3.2" xref="S6.SS3.9.p9.1.m1.1.1.2.3.2.cmml">k</mi><mn id="S6.SS3.9.p9.1.m1.1.1.2.3.3" xref="S6.SS3.9.p9.1.m1.1.1.2.3.3.cmml">1</mn></msub></mrow><mo id="S6.SS3.9.p9.1.m1.1.1.1" xref="S6.SS3.9.p9.1.m1.1.1.1.cmml"><</mo><mrow id="S6.SS3.9.p9.1.m1.1.1.3" xref="S6.SS3.9.p9.1.m1.1.1.3.cmml"><msub id="S6.SS3.9.p9.1.m1.1.1.3.2" xref="S6.SS3.9.p9.1.m1.1.1.3.2.cmml"><mi id="S6.SS3.9.p9.1.m1.1.1.3.2.2" xref="S6.SS3.9.p9.1.m1.1.1.3.2.2.cmml">m</mi><mi id="S6.SS3.9.p9.1.m1.1.1.3.2.3" xref="S6.SS3.9.p9.1.m1.1.1.3.2.3.cmml">n</mi></msub><mo id="S6.SS3.9.p9.1.m1.1.1.3.1" xref="S6.SS3.9.p9.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.9.p9.1.m1.1.1.3.3" xref="S6.SS3.9.p9.1.m1.1.1.3.3.cmml"><mrow id="S6.SS3.9.p9.1.m1.1.1.3.3.2" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS3.9.p9.1.m1.1.1.3.3.2.2" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.9.p9.1.m1.1.1.3.3.2.1" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.9.p9.1.m1.1.1.3.3.2.3" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.9.p9.1.m1.1.1.3.3.3" xref="S6.SS3.9.p9.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.9.p9.1.m1.1b"><apply id="S6.SS3.9.p9.1.m1.1.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1"><lt id="S6.SS3.9.p9.1.m1.1.1.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.1"></lt><apply id="S6.SS3.9.p9.1.m1.1.1.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2"><plus id="S6.SS3.9.p9.1.m1.1.1.2.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2.1"></plus><apply id="S6.SS3.9.p9.1.m1.1.1.2.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2.2"><csymbol cd="ambiguous" id="S6.SS3.9.p9.1.m1.1.1.2.2.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2.2">subscript</csymbol><ci id="S6.SS3.9.p9.1.m1.1.1.2.2.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2.2.2">𝑘</ci><cn id="S6.SS3.9.p9.1.m1.1.1.2.2.3.cmml" type="integer" xref="S6.SS3.9.p9.1.m1.1.1.2.2.3">0</cn></apply><apply id="S6.SS3.9.p9.1.m1.1.1.2.3.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS3.9.p9.1.m1.1.1.2.3.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2.3">subscript</csymbol><ci id="S6.SS3.9.p9.1.m1.1.1.2.3.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.2.3.2">𝑘</ci><cn id="S6.SS3.9.p9.1.m1.1.1.2.3.3.cmml" type="integer" xref="S6.SS3.9.p9.1.m1.1.1.2.3.3">1</cn></apply></apply><apply id="S6.SS3.9.p9.1.m1.1.1.3.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3"><plus id="S6.SS3.9.p9.1.m1.1.1.3.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.1"></plus><apply id="S6.SS3.9.p9.1.m1.1.1.3.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.9.p9.1.m1.1.1.3.2.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS3.9.p9.1.m1.1.1.3.2.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.2.2">𝑚</ci><ci id="S6.SS3.9.p9.1.m1.1.1.3.2.3.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.2.3">𝑛</ci></apply><apply id="S6.SS3.9.p9.1.m1.1.1.3.3.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.3"><divide id="S6.SS3.9.p9.1.m1.1.1.3.3.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.3"></divide><apply id="S6.SS3.9.p9.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2"><plus id="S6.SS3.9.p9.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS3.9.p9.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.9.p9.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.9.p9.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.9.p9.1.m1.1.1.3.3.3.cmml" xref="S6.SS3.9.p9.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.9.p9.1.m1.1c">k_{0}+k_{1}<m_{n}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.9.p9.1.m1.1d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Fix <math alttext="q\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS3.9.p9.2.m2.1"><semantics id="S6.SS3.9.p9.2.m2.1a"><mrow id="S6.SS3.9.p9.2.m2.1.1" xref="S6.SS3.9.p9.2.m2.1.1.cmml"><mi id="S6.SS3.9.p9.2.m2.1.1.2" xref="S6.SS3.9.p9.2.m2.1.1.2.cmml">q</mi><mo id="S6.SS3.9.p9.2.m2.1.1.1" xref="S6.SS3.9.p9.2.m2.1.1.1.cmml">∈</mo><msub id="S6.SS3.9.p9.2.m2.1.1.3" xref="S6.SS3.9.p9.2.m2.1.1.3.cmml"><mi id="S6.SS3.9.p9.2.m2.1.1.3.2" xref="S6.SS3.9.p9.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS3.9.p9.2.m2.1.1.3.3" xref="S6.SS3.9.p9.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.9.p9.2.m2.1b"><apply id="S6.SS3.9.p9.2.m2.1.1.cmml" xref="S6.SS3.9.p9.2.m2.1.1"><in id="S6.SS3.9.p9.2.m2.1.1.1.cmml" xref="S6.SS3.9.p9.2.m2.1.1.1"></in><ci id="S6.SS3.9.p9.2.m2.1.1.2.cmml" xref="S6.SS3.9.p9.2.m2.1.1.2">𝑞</ci><apply id="S6.SS3.9.p9.2.m2.1.1.3.cmml" xref="S6.SS3.9.p9.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.9.p9.2.m2.1.1.3.1.cmml" xref="S6.SS3.9.p9.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS3.9.p9.2.m2.1.1.3.2.cmml" xref="S6.SS3.9.p9.2.m2.1.1.3.2">ℕ</ci><cn id="S6.SS3.9.p9.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS3.9.p9.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.9.p9.2.m2.1c">q\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.9.p9.2.m2.1d">italic_q ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="k_{0}+q\in(m_{n}+\frac{\gamma+1}{p},k]" class="ltx_Math" display="inline" id="S6.SS3.9.p9.3.m3.2"><semantics id="S6.SS3.9.p9.3.m3.2a"><mrow id="S6.SS3.9.p9.3.m3.2.2" xref="S6.SS3.9.p9.3.m3.2.2.cmml"><mrow id="S6.SS3.9.p9.3.m3.2.2.3" xref="S6.SS3.9.p9.3.m3.2.2.3.cmml"><msub id="S6.SS3.9.p9.3.m3.2.2.3.2" xref="S6.SS3.9.p9.3.m3.2.2.3.2.cmml"><mi id="S6.SS3.9.p9.3.m3.2.2.3.2.2" xref="S6.SS3.9.p9.3.m3.2.2.3.2.2.cmml">k</mi><mn id="S6.SS3.9.p9.3.m3.2.2.3.2.3" xref="S6.SS3.9.p9.3.m3.2.2.3.2.3.cmml">0</mn></msub><mo id="S6.SS3.9.p9.3.m3.2.2.3.1" xref="S6.SS3.9.p9.3.m3.2.2.3.1.cmml">+</mo><mi id="S6.SS3.9.p9.3.m3.2.2.3.3" xref="S6.SS3.9.p9.3.m3.2.2.3.3.cmml">q</mi></mrow><mo id="S6.SS3.9.p9.3.m3.2.2.2" xref="S6.SS3.9.p9.3.m3.2.2.2.cmml">∈</mo><mrow id="S6.SS3.9.p9.3.m3.2.2.1.1" xref="S6.SS3.9.p9.3.m3.2.2.1.2.cmml"><mo id="S6.SS3.9.p9.3.m3.2.2.1.1.2" stretchy="false" xref="S6.SS3.9.p9.3.m3.2.2.1.2.cmml">(</mo><mrow id="S6.SS3.9.p9.3.m3.2.2.1.1.1" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.cmml"><msub id="S6.SS3.9.p9.3.m3.2.2.1.1.1.2" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.cmml"><mi id="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.2" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.2.cmml">m</mi><mi id="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.3" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.SS3.9.p9.3.m3.2.2.1.1.1.1" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.1.cmml">+</mo><mfrac id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.cmml"><mrow id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.cmml"><mi id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.2" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.2.cmml">γ</mi><mo id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.1" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.1.cmml">+</mo><mn id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.3" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.3" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS3.9.p9.3.m3.2.2.1.1.3" xref="S6.SS3.9.p9.3.m3.2.2.1.2.cmml">,</mo><mi id="S6.SS3.9.p9.3.m3.1.1" xref="S6.SS3.9.p9.3.m3.1.1.cmml">k</mi><mo id="S6.SS3.9.p9.3.m3.2.2.1.1.4" stretchy="false" xref="S6.SS3.9.p9.3.m3.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.9.p9.3.m3.2b"><apply id="S6.SS3.9.p9.3.m3.2.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2"><in id="S6.SS3.9.p9.3.m3.2.2.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.2"></in><apply id="S6.SS3.9.p9.3.m3.2.2.3.cmml" xref="S6.SS3.9.p9.3.m3.2.2.3"><plus id="S6.SS3.9.p9.3.m3.2.2.3.1.cmml" xref="S6.SS3.9.p9.3.m3.2.2.3.1"></plus><apply id="S6.SS3.9.p9.3.m3.2.2.3.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.3.2"><csymbol cd="ambiguous" id="S6.SS3.9.p9.3.m3.2.2.3.2.1.cmml" xref="S6.SS3.9.p9.3.m3.2.2.3.2">subscript</csymbol><ci id="S6.SS3.9.p9.3.m3.2.2.3.2.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.3.2.2">𝑘</ci><cn id="S6.SS3.9.p9.3.m3.2.2.3.2.3.cmml" type="integer" xref="S6.SS3.9.p9.3.m3.2.2.3.2.3">0</cn></apply><ci id="S6.SS3.9.p9.3.m3.2.2.3.3.cmml" xref="S6.SS3.9.p9.3.m3.2.2.3.3">𝑞</ci></apply><interval closure="open-closed" id="S6.SS3.9.p9.3.m3.2.2.1.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1"><apply id="S6.SS3.9.p9.3.m3.2.2.1.1.1.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1"><plus id="S6.SS3.9.p9.3.m3.2.2.1.1.1.1.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.1"></plus><apply id="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.1.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.2">subscript</csymbol><ci id="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.2">𝑚</ci><ci id="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.3.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.2.3">𝑛</ci></apply><apply id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3"><divide id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.1.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3"></divide><apply id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2"><plus id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.1.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.1"></plus><ci id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.2.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.2">𝛾</ci><cn id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.2.3">1</cn></apply><ci id="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.3.cmml" xref="S6.SS3.9.p9.3.m3.2.2.1.1.1.3.3">𝑝</ci></apply></apply><ci id="S6.SS3.9.p9.3.m3.1.1.cmml" xref="S6.SS3.9.p9.3.m3.1.1">𝑘</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.9.p9.3.m3.2c">k_{0}+q\in(m_{n}+\frac{\gamma+1}{p},k]</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.9.p9.3.m3.2d">italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_q ∈ ( italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG , italic_k ]</annotation></semantics></math>. Then by reiteration for the complex interpolation method (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib9" title="">9</a>, Theorem 4.6.1]</cite>) and Step 2a twice, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex72"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}W^{k_{0},p},W^{k_{0}+k_{1},p}_{\mathcal{B}}\big{]}_{\frac{\ell}{k_{1}}}% =\big{[}W^{k_{0},p},[W^{k_{0},p},W^{k_{0}+q,p}_{\mathcal{B}}]_{\frac{k_{1}}{q}% }\big{]}_{\frac{\ell}{k_{1}}}=\big{[}W^{k_{0},p},W^{k_{0}+q,p}_{\mathcal{B}}% \big{]}_{\frac{\ell}{q}}=W^{k_{0}+\ell,p}_{\mathcal{B}}." class="ltx_Math" display="block" id="S6.Ex72.m1.17"><semantics id="S6.Ex72.m1.17a"><mrow id="S6.Ex72.m1.17.17.1" xref="S6.Ex72.m1.17.17.1.1.cmml"><mrow id="S6.Ex72.m1.17.17.1.1" xref="S6.Ex72.m1.17.17.1.1.cmml"><msub id="S6.Ex72.m1.17.17.1.1.2" xref="S6.Ex72.m1.17.17.1.1.2.cmml"><mrow id="S6.Ex72.m1.17.17.1.1.2.2.2" xref="S6.Ex72.m1.17.17.1.1.2.2.3.cmml"><mo id="S6.Ex72.m1.17.17.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex72.m1.17.17.1.1.2.2.3.cmml">[</mo><msup id="S6.Ex72.m1.17.17.1.1.1.1.1.1" xref="S6.Ex72.m1.17.17.1.1.1.1.1.1.cmml"><mi id="S6.Ex72.m1.17.17.1.1.1.1.1.1.2" xref="S6.Ex72.m1.17.17.1.1.1.1.1.1.2.cmml">W</mi><mrow id="S6.Ex72.m1.2.2.2.2" xref="S6.Ex72.m1.2.2.2.3.cmml"><msub id="S6.Ex72.m1.2.2.2.2.1" xref="S6.Ex72.m1.2.2.2.2.1.cmml"><mi id="S6.Ex72.m1.2.2.2.2.1.2" xref="S6.Ex72.m1.2.2.2.2.1.2.cmml">k</mi><mn id="S6.Ex72.m1.2.2.2.2.1.3" xref="S6.Ex72.m1.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.Ex72.m1.2.2.2.2.2" xref="S6.Ex72.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex72.m1.1.1.1.1" xref="S6.Ex72.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.Ex72.m1.17.17.1.1.2.2.2.4" xref="S6.Ex72.m1.17.17.1.1.2.2.3.cmml">,</mo><msubsup id="S6.Ex72.m1.17.17.1.1.2.2.2.2" 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italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT = [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_q , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT divide start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG start_ARG italic_q end_ARG end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT = [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_q , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_q end_ARG end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.9.p9.4">Combining Steps 1 and 2 completes the proof of (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E4" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a>).</p> </div> <div class="ltx_para" id="S6.SS3.10.p10"> <p class="ltx_p" id="S6.SS3.10.p10.4"><span class="ltx_text ltx_font_italic" id="S6.SS3.10.p10.4.1">Step 3. </span>It remains to prove that</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx45"> <tbody id="S6.Ex73"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle W^{k+\ell,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big% {[}W^{k_{0},p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{\mathcal{B}}^% {k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}." class="ltx_Math" display="inline" id="S6.Ex73.m1.10"><semantics id="S6.Ex73.m1.10a"><mrow id="S6.Ex73.m1.10.10.1" 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id="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.4"><csymbol cd="ambiguous" id="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.1.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.4">superscript</csymbol><apply id="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.4"><csymbol cd="ambiguous" id="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.2.1.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.4">subscript</csymbol><ci id="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.2.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.2.2">𝑊</ci><ci id="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.2.3.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.4.2.3">ℬ</ci></apply><list id="S6.Ex73.m1.6.6.2.3.cmml" xref="S6.Ex73.m1.6.6.2.2"><apply id="S6.Ex73.m1.6.6.2.2.1.cmml" xref="S6.Ex73.m1.6.6.2.2.1"><plus id="S6.Ex73.m1.6.6.2.2.1.1.cmml" xref="S6.Ex73.m1.6.6.2.2.1.1"></plus><apply id="S6.Ex73.m1.6.6.2.2.1.2.cmml" xref="S6.Ex73.m1.6.6.2.2.1.2"><csymbol cd="ambiguous" id="S6.Ex73.m1.6.6.2.2.1.2.1.cmml" 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id="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.2.1.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1">superscript</csymbol><ci id="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.2.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.2.2">ℝ</ci><ci id="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.2.3.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.2.3">𝑑</ci></apply><plus id="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.3.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.1.1.1.3"></plus></apply><apply id="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2"><csymbol cd="ambiguous" id="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2.1.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2">subscript</csymbol><ci id="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2.2">𝑤</ci><ci id="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2.3.cmml" xref="S6.Ex73.m1.10.10.1.1.4.2.2.2.2.2.2.3">𝛾</ci></apply><ci id="S6.Ex73.m1.9.9.cmml" xref="S6.Ex73.m1.9.9">𝑋</ci></vector></apply></interval><apply id="S6.Ex73.m1.10.10.1.1.4.4.cmml" xref="S6.Ex73.m1.10.10.1.1.4.4"><divide id="S6.Ex73.m1.10.10.1.1.4.4.1.cmml" xref="S6.Ex73.m1.10.10.1.1.4.4"></divide><ci id="S6.Ex73.m1.10.10.1.1.4.4.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.4.2">ℓ</ci><apply id="S6.Ex73.m1.10.10.1.1.4.4.3.cmml" xref="S6.Ex73.m1.10.10.1.1.4.4.3"><csymbol cd="ambiguous" id="S6.Ex73.m1.10.10.1.1.4.4.3.1.cmml" xref="S6.Ex73.m1.10.10.1.1.4.4.3">subscript</csymbol><ci id="S6.Ex73.m1.10.10.1.1.4.4.3.2.cmml" xref="S6.Ex73.m1.10.10.1.1.4.4.3.2">𝑘</ci><cn id="S6.Ex73.m1.10.10.1.1.4.4.3.3.cmml" type="integer" xref="S6.Ex73.m1.10.10.1.1.4.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex73.m1.10c">\displaystyle W^{k+\ell,p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big% {[}W^{k_{0},p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X),W_{\mathcal{B}}^% {k_{0}+k_{1},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\frac{\ell}{k_{1}}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex73.m1.10d">italic_W start_POSTSUPERSCRIPT italic_k + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_W start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.10.p10.3">Fix <math alttext="q\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS3.10.p10.1.m1.1"><semantics id="S6.SS3.10.p10.1.m1.1a"><mrow id="S6.SS3.10.p10.1.m1.1.1" xref="S6.SS3.10.p10.1.m1.1.1.cmml"><mi id="S6.SS3.10.p10.1.m1.1.1.2" xref="S6.SS3.10.p10.1.m1.1.1.2.cmml">q</mi><mo id="S6.SS3.10.p10.1.m1.1.1.1" xref="S6.SS3.10.p10.1.m1.1.1.1.cmml">∈</mo><msub id="S6.SS3.10.p10.1.m1.1.1.3" xref="S6.SS3.10.p10.1.m1.1.1.3.cmml"><mi id="S6.SS3.10.p10.1.m1.1.1.3.2" xref="S6.SS3.10.p10.1.m1.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS3.10.p10.1.m1.1.1.3.3" xref="S6.SS3.10.p10.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.10.p10.1.m1.1b"><apply id="S6.SS3.10.p10.1.m1.1.1.cmml" xref="S6.SS3.10.p10.1.m1.1.1"><in id="S6.SS3.10.p10.1.m1.1.1.1.cmml" xref="S6.SS3.10.p10.1.m1.1.1.1"></in><ci id="S6.SS3.10.p10.1.m1.1.1.2.cmml" xref="S6.SS3.10.p10.1.m1.1.1.2">𝑞</ci><apply id="S6.SS3.10.p10.1.m1.1.1.3.cmml" xref="S6.SS3.10.p10.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.10.p10.1.m1.1.1.3.1.cmml" xref="S6.SS3.10.p10.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS3.10.p10.1.m1.1.1.3.2.cmml" xref="S6.SS3.10.p10.1.m1.1.1.3.2">ℕ</ci><cn id="S6.SS3.10.p10.1.m1.1.1.3.3.cmml" type="integer" xref="S6.SS3.10.p10.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.10.p10.1.m1.1c">q\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.10.p10.1.m1.1d">italic_q ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="q<k_{0}" class="ltx_Math" display="inline" id="S6.SS3.10.p10.2.m2.1"><semantics id="S6.SS3.10.p10.2.m2.1a"><mrow id="S6.SS3.10.p10.2.m2.1.1" xref="S6.SS3.10.p10.2.m2.1.1.cmml"><mi id="S6.SS3.10.p10.2.m2.1.1.2" xref="S6.SS3.10.p10.2.m2.1.1.2.cmml">q</mi><mo id="S6.SS3.10.p10.2.m2.1.1.1" xref="S6.SS3.10.p10.2.m2.1.1.1.cmml"><</mo><msub id="S6.SS3.10.p10.2.m2.1.1.3" xref="S6.SS3.10.p10.2.m2.1.1.3.cmml"><mi id="S6.SS3.10.p10.2.m2.1.1.3.2" xref="S6.SS3.10.p10.2.m2.1.1.3.2.cmml">k</mi><mn id="S6.SS3.10.p10.2.m2.1.1.3.3" xref="S6.SS3.10.p10.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.10.p10.2.m2.1b"><apply id="S6.SS3.10.p10.2.m2.1.1.cmml" xref="S6.SS3.10.p10.2.m2.1.1"><lt id="S6.SS3.10.p10.2.m2.1.1.1.cmml" xref="S6.SS3.10.p10.2.m2.1.1.1"></lt><ci id="S6.SS3.10.p10.2.m2.1.1.2.cmml" xref="S6.SS3.10.p10.2.m2.1.1.2">𝑞</ci><apply id="S6.SS3.10.p10.2.m2.1.1.3.cmml" xref="S6.SS3.10.p10.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.10.p10.2.m2.1.1.3.1.cmml" xref="S6.SS3.10.p10.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS3.10.p10.2.m2.1.1.3.2.cmml" xref="S6.SS3.10.p10.2.m2.1.1.3.2">𝑘</ci><cn id="S6.SS3.10.p10.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS3.10.p10.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.10.p10.2.m2.1c">q<k_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.10.p10.2.m2.1d">italic_q < italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> and define <math alttext="\theta_{1}:=(k_{0}-q)/(k_{0}+k_{1}-q)" class="ltx_Math" display="inline" id="S6.SS3.10.p10.3.m3.2"><semantics id="S6.SS3.10.p10.3.m3.2a"><mrow id="S6.SS3.10.p10.3.m3.2.2" xref="S6.SS3.10.p10.3.m3.2.2.cmml"><msub id="S6.SS3.10.p10.3.m3.2.2.4" xref="S6.SS3.10.p10.3.m3.2.2.4.cmml"><mi id="S6.SS3.10.p10.3.m3.2.2.4.2" xref="S6.SS3.10.p10.3.m3.2.2.4.2.cmml">θ</mi><mn id="S6.SS3.10.p10.3.m3.2.2.4.3" xref="S6.SS3.10.p10.3.m3.2.2.4.3.cmml">1</mn></msub><mo id="S6.SS3.10.p10.3.m3.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.SS3.10.p10.3.m3.2.2.3.cmml">:=</mo><mrow id="S6.SS3.10.p10.3.m3.2.2.2" xref="S6.SS3.10.p10.3.m3.2.2.2.cmml"><mrow id="S6.SS3.10.p10.3.m3.1.1.1.1.1" xref="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.cmml"><mo id="S6.SS3.10.p10.3.m3.1.1.1.1.1.2" stretchy="false" xref="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS3.10.p10.3.m3.1.1.1.1.1.1" xref="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.cmml"><msub id="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.2" xref="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.2.cmml"><mi id="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.2.2" xref="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.2.2.cmml">k</mi><mn id="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.2.3" xref="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.10.p10.3.m3.1.1.1.1.1.1.1" 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xref="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.2.2">𝑘</ci><cn id="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.2.3.cmml" type="integer" xref="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.2.3">0</cn></apply><apply id="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3.cmml" xref="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3"><csymbol cd="ambiguous" id="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3.1.cmml" xref="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3">subscript</csymbol><ci id="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3.2.cmml" xref="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3.2">𝑘</ci><cn id="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3.3.cmml" type="integer" xref="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.2.3.3">1</cn></apply></apply><ci id="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.3.cmml" xref="S6.SS3.10.p10.3.m3.2.2.2.2.1.1.3">𝑞</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.10.p10.3.m3.2c">\theta_{1}:=(k_{0}-q)/(k_{0}+k_{1}-q)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.10.p10.3.m3.2d">italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := ( italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_q ) / ( italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_q )</annotation></semantics></math>. Then by reiteration and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E4" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a>) twice, we obtain</p> <table class="ltx_equationgroup ltx_eqn_align ltx_eqn_table" id="S6.EGx46"> <tbody id="S6.Ex74"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_align_right ltx_eqn_cell"><math alttext="\displaystyle\big{[}W^{k_{0},p}_{\mathcal{B}},W^{k_{0}+k_{1},p}_{\mathcal{B}}% \big{]}_{\frac{\ell}{k_{1}}}" class="ltx_Math" display="inline" id="S6.Ex74.m1.6"><semantics id="S6.Ex74.m1.6a"><msub id="S6.Ex74.m1.6.6" xref="S6.Ex74.m1.6.6.cmml"><mrow id="S6.Ex74.m1.6.6.2.2" xref="S6.Ex74.m1.6.6.2.3.cmml"><mo id="S6.Ex74.m1.6.6.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex74.m1.6.6.2.3.cmml">[</mo><msubsup id="S6.Ex74.m1.5.5.1.1.1" xref="S6.Ex74.m1.5.5.1.1.1.cmml"><mi id="S6.Ex74.m1.5.5.1.1.1.2.2" xref="S6.Ex74.m1.5.5.1.1.1.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex74.m1.5.5.1.1.1.3" xref="S6.Ex74.m1.5.5.1.1.1.3.cmml">ℬ</mi><mrow id="S6.Ex74.m1.2.2.2.2" xref="S6.Ex74.m1.2.2.2.3.cmml"><msub id="S6.Ex74.m1.2.2.2.2.1" xref="S6.Ex74.m1.2.2.2.2.1.cmml"><mi id="S6.Ex74.m1.2.2.2.2.1.2" xref="S6.Ex74.m1.2.2.2.2.1.2.cmml">k</mi><mn id="S6.Ex74.m1.2.2.2.2.1.3" xref="S6.Ex74.m1.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.Ex74.m1.2.2.2.2.2" xref="S6.Ex74.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex74.m1.1.1.1.1" xref="S6.Ex74.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex74.m1.6.6.2.2.4" xref="S6.Ex74.m1.6.6.2.3.cmml">,</mo><msubsup id="S6.Ex74.m1.6.6.2.2.2" xref="S6.Ex74.m1.6.6.2.2.2.cmml"><mi id="S6.Ex74.m1.6.6.2.2.2.2.2" xref="S6.Ex74.m1.6.6.2.2.2.2.2.cmml">W</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex74.m1.6.6.2.2.2.3" xref="S6.Ex74.m1.6.6.2.2.2.3.cmml">ℬ</mi><mrow id="S6.Ex74.m1.4.4.2.2" xref="S6.Ex74.m1.4.4.2.3.cmml"><mrow id="S6.Ex74.m1.4.4.2.2.1" xref="S6.Ex74.m1.4.4.2.2.1.cmml"><msub 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xref="S6.Ex74.m2.8.8.2.4.2">ℓ</ci><apply id="S6.Ex74.m2.8.8.2.4.3.cmml" xref="S6.Ex74.m2.8.8.2.4.3"><csymbol cd="ambiguous" id="S6.Ex74.m2.8.8.2.4.3.1.cmml" xref="S6.Ex74.m2.8.8.2.4.3">subscript</csymbol><ci id="S6.Ex74.m2.8.8.2.4.3.2.cmml" xref="S6.Ex74.m2.8.8.2.4.3.2">𝑘</ci><cn id="S6.Ex74.m2.8.8.2.4.3.3.cmml" type="integer" xref="S6.Ex74.m2.8.8.2.4.3.3">1</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex74.m2.8c">\displaystyle=\big{[}[W^{q,p},W^{k_{0}+k_{1},p}_{\mathcal{B}}]_{\theta_{1}},W^% {k_{0}+k_{1},p}_{\mathcal{B}}\big{]}_{\frac{\ell}{k_{1}}}</annotation><annotation encoding="application/x-llamapun" id="S6.Ex74.m2.8d">= [ [ italic_W start_POSTSUPERSCRIPT italic_q , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> <tbody id="S6.Ex75"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_td ltx_eqn_cell"></td> <td class="ltx_td ltx_align_left ltx_eqn_cell"><math alttext="\displaystyle=\big{[}W^{q,p},W^{k_{0}+k_{1},p}_{\mathcal{B}}\big{]}_{(1-\frac{% \ell}{k_{1}})\theta_{1}+\frac{\ell}{k_{1}}}=W^{k+\ell,p}_{\mathcal{B}}." class="ltx_Math" display="inline" id="S6.Ex75.m1.8"><semantics 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xref="S6.Ex75.m1.7.7.2.2.1.3">ℓ</ci></apply><ci id="S6.Ex75.m1.6.6.1.1.cmml" xref="S6.Ex75.m1.6.6.1.1">𝑝</ci></list></apply><ci id="S6.Ex75.m1.8.8.1.1.7.3.cmml" xref="S6.Ex75.m1.8.8.1.1.7.3">ℬ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex75.m1.8c">\displaystyle=\big{[}W^{q,p},W^{k_{0}+k_{1},p}_{\mathcal{B}}\big{]}_{(1-\frac{% \ell}{k_{1}})\theta_{1}+\frac{\ell}{k_{1}}}=W^{k+\ell,p}_{\mathcal{B}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex75.m1.8d">= [ italic_W start_POSTSUPERSCRIPT italic_q , italic_p end_POSTSUPERSCRIPT , italic_W start_POSTSUPERSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT ( 1 - divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + divide start_ARG roman_ℓ end_ARG start_ARG italic_k start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG end_POSTSUBSCRIPT = italic_W start_POSTSUPERSCRIPT italic_k + roman_ℓ , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.10.p10.5">This completes the proof of the theorem. ∎</p> </div> </div> <div class="ltx_para" id="S6.SS3.p2"> <p class="ltx_p" id="S6.SS3.p2.1">We conclude with the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4" title="Theorem 6.4 (Complex interpolation of weighted Bessel potential spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a>.</p> </div> <div class="ltx_proof" id="S6.SS3.18"> <h6 class="ltx_title ltx_runin ltx_font_italic ltx_title_proof">Proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem4" title="Theorem 6.4 (Complex interpolation of weighted Bessel potential spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.4</span></a>.</h6> <div class="ltx_para" id="S6.SS3.11.p1"> <p class="ltx_p" id="S6.SS3.11.p1.2">We first prove in Steps 1 and 2 that</p> <table class="ltx_equation ltx_eqn_table" id="S6.E15"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H_{\mathcal{B}}^{s_{\theta},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big{[}H^{s_{0% },p}(\mathbb{R}^{d}_{+},w_{\gamma};X),H_{\mathcal{B}}^{s_{1},p}(\mathbb{R}^{d}% _{+},w_{\gamma};X)\big{]}_{\theta}." class="ltx_Math" display="block" id="S6.E15.m1.10"><semantics id="S6.E15.m1.10a"><mrow id="S6.E15.m1.10.10.1" xref="S6.E15.m1.10.10.1.1.cmml"><mrow id="S6.E15.m1.10.10.1.1" xref="S6.E15.m1.10.10.1.1.cmml"><mrow id="S6.E15.m1.10.10.1.1.2" xref="S6.E15.m1.10.10.1.1.2.cmml"><msubsup 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id="S6.E15.m1.10c">H_{\mathcal{B}}^{s_{\theta},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big{[}H^{s_{0% },p}(\mathbb{R}^{d}_{+},w_{\gamma};X),H_{\mathcal{B}}^{s_{1},p}(\mathbb{R}^{d}% _{+},w_{\gamma};X)\big{]}_{\theta}.</annotation><annotation encoding="application/x-llamapun" id="S6.E15.m1.10d">italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> <td class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right" rowspan="1"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6.15)</span></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.11.p1.1">For notational convenience we write <math alttext="H^{s,p}:=H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)" class="ltx_Math" display="inline" id="S6.SS3.11.p1.1.m1.7"><semantics id="S6.SS3.11.p1.1.m1.7a"><mrow id="S6.SS3.11.p1.1.m1.7.7" xref="S6.SS3.11.p1.1.m1.7.7.cmml"><msup id="S6.SS3.11.p1.1.m1.7.7.4" xref="S6.SS3.11.p1.1.m1.7.7.4.cmml"><mi 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xref="S6.SS3.11.p1.1.m1.7.7.2.2.2.2.3">𝛾</ci></apply><ci id="S6.SS3.11.p1.1.m1.5.5.cmml" xref="S6.SS3.11.p1.1.m1.5.5">𝑋</ci></vector></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.11.p1.1.m1.7c">H^{s,p}:=H^{s,p}(\mathbb{R}^{d}_{+},w_{\gamma};X)</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.11.p1.1.m1.7d">italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT := italic_H start_POSTSUPERSCRIPT italic_s , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X )</annotation></semantics></math>.</p> </div> <div class="ltx_para" id="S6.SS3.12.p2"> <p class="ltx_p" id="S6.SS3.12.p2.1"><span class="ltx_text ltx_font_italic" id="S6.SS3.12.p2.1.1">Step 1: the embedding “<math alttext="\hookleftarrow" class="ltx_Math" display="inline" id="S6.SS3.12.p2.1.1.m1.1"><semantics 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xref="S6.Ex76.m1.8.8.2.2.1">subscript</csymbol><ci id="S6.Ex76.m1.8.8.2.2.1.2.cmml" xref="S6.Ex76.m1.8.8.2.2.1.2">𝑠</ci><cn id="S6.Ex76.m1.8.8.2.2.1.3.cmml" type="integer" xref="S6.Ex76.m1.8.8.2.2.1.3">1</cn></apply><ci id="S6.Ex76.m1.7.7.1.1.cmml" xref="S6.Ex76.m1.7.7.1.1">𝑝</ci></list></apply></interval><ci id="S6.Ex76.m1.11.11.1.1.4.4.cmml" xref="S6.Ex76.m1.11.11.1.1.4.4">𝜃</ci></apply></apply><apply id="S6.Ex76.m1.11.11.1.1c.cmml" xref="S6.Ex76.m1.11.11.1"><eq id="S6.Ex76.m1.11.11.1.1.7.cmml" xref="S6.Ex76.m1.11.11.1.1.7"></eq><share href="https://arxiv.org/html/2503.14636v1#S6.Ex76.m1.11.11.1.1.4.cmml" id="S6.Ex76.m1.11.11.1.1d.cmml" xref="S6.Ex76.m1.11.11.1"></share><apply id="S6.Ex76.m1.11.11.1.1.8.cmml" xref="S6.Ex76.m1.11.11.1.1.8"><csymbol cd="ambiguous" id="S6.Ex76.m1.11.11.1.1.8.1.cmml" xref="S6.Ex76.m1.11.11.1.1.8">superscript</csymbol><ci id="S6.Ex76.m1.11.11.1.1.8.2.cmml" xref="S6.Ex76.m1.11.11.1.1.8.2">𝐻</ci><list id="S6.Ex76.m1.10.10.2.3.cmml" xref="S6.Ex76.m1.10.10.2.2"><apply id="S6.Ex76.m1.10.10.2.2.1.cmml" xref="S6.Ex76.m1.10.10.2.2.1"><csymbol cd="ambiguous" id="S6.Ex76.m1.10.10.2.2.1.1.cmml" xref="S6.Ex76.m1.10.10.2.2.1">subscript</csymbol><ci id="S6.Ex76.m1.10.10.2.2.1.2.cmml" xref="S6.Ex76.m1.10.10.2.2.1.2">𝑠</ci><ci id="S6.Ex76.m1.10.10.2.2.1.3.cmml" xref="S6.Ex76.m1.10.10.2.2.1.3">𝜃</ci></apply><ci id="S6.Ex76.m1.9.9.1.1.cmml" xref="S6.Ex76.m1.9.9.1.1">𝑝</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex76.m1.11c">\big{[}H^{s_{0},p},H_{\mathcal{B}}^{s_{1},p}\big{]}_{\theta}\hookrightarrow% \big{[}H^{s_{0},p},H^{s_{1},p}\big{]}_{\theta}=H^{s_{\theta},p}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex76.m1.11d">[ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.12.p2.2">Thus the embedding <math alttext="[H^{s_{0},p},H_{\mathcal{B}}^{s_{1},p}]_{\theta}\hookrightarrow H^{s_{\theta},% p}_{\mathcal{B}}" class="ltx_Math" display="inline" id="S6.SS3.12.p2.2.m1.8"><semantics id="S6.SS3.12.p2.2.m1.8a"><mrow id="S6.SS3.12.p2.2.m1.8.8" xref="S6.SS3.12.p2.2.m1.8.8.cmml"><msub id="S6.SS3.12.p2.2.m1.8.8.2" 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xref="S6.SS3.12.p2.2.m1.8.8.4">superscript</csymbol><ci id="S6.SS3.12.p2.2.m1.8.8.4.2.2.cmml" xref="S6.SS3.12.p2.2.m1.8.8.4.2.2">𝐻</ci><list id="S6.SS3.12.p2.2.m1.6.6.2.3.cmml" xref="S6.SS3.12.p2.2.m1.6.6.2.2"><apply id="S6.SS3.12.p2.2.m1.6.6.2.2.1.cmml" xref="S6.SS3.12.p2.2.m1.6.6.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.12.p2.2.m1.6.6.2.2.1.1.cmml" xref="S6.SS3.12.p2.2.m1.6.6.2.2.1">subscript</csymbol><ci id="S6.SS3.12.p2.2.m1.6.6.2.2.1.2.cmml" xref="S6.SS3.12.p2.2.m1.6.6.2.2.1.2">𝑠</ci><ci id="S6.SS3.12.p2.2.m1.6.6.2.2.1.3.cmml" xref="S6.SS3.12.p2.2.m1.6.6.2.2.1.3">𝜃</ci></apply><ci id="S6.SS3.12.p2.2.m1.5.5.1.1.cmml" xref="S6.SS3.12.p2.2.m1.5.5.1.1">𝑝</ci></list></apply><ci id="S6.SS3.12.p2.2.m1.8.8.4.3.cmml" xref="S6.SS3.12.p2.2.m1.8.8.4.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.12.p2.2.m1.8c">[H^{s_{0},p},H_{\mathcal{B}}^{s_{1},p}]_{\theta}\hookrightarrow H^{s_{\theta},% p}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.12.p2.2.m1.8d">[ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math> can be proved similarly as in Step 1 of the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>.</p> </div> <div class="ltx_para" id="S6.SS3.13.p3"> <p class="ltx_p" id="S6.SS3.13.p3.1"><span class="ltx_text ltx_font_italic" id="S6.SS3.13.p3.1.1">Step 2: the embedding “<math alttext="\hookrightarrow" class="ltx_Math" display="inline" id="S6.SS3.13.p3.1.1.m1.1"><semantics id="S6.SS3.13.p3.1.1.m1.1a"><mo id="S6.SS3.13.p3.1.1.m1.1.1" stretchy="false" xref="S6.SS3.13.p3.1.1.m1.1.1.cmml">↪</mo><annotation-xml encoding="MathML-Content" id="S6.SS3.13.p3.1.1.m1.1b"><ci id="S6.SS3.13.p3.1.1.m1.1.1.cmml" xref="S6.SS3.13.p3.1.1.m1.1.1">↪</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.13.p3.1.1.m1.1c">\hookrightarrow</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.13.p3.1.1.m1.1d">↪</annotation></semantics></math>”.</span> We adopt the same notation as in Step 2 of the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>.</p> </div> <div class="ltx_para" id="S6.SS3.14.p4"> <p class="ltx_p" id="S6.SS3.14.p4.3"><span class="ltx_text ltx_font_italic" id="S6.SS3.14.p4.3.1">Step 2a. </span>Assume that <math alttext="s_{1}>m_{n}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.14.p4.1.m1.1"><semantics id="S6.SS3.14.p4.1.m1.1a"><mrow id="S6.SS3.14.p4.1.m1.1.1" xref="S6.SS3.14.p4.1.m1.1.1.cmml"><msub id="S6.SS3.14.p4.1.m1.1.1.2" xref="S6.SS3.14.p4.1.m1.1.1.2.cmml"><mi id="S6.SS3.14.p4.1.m1.1.1.2.2" xref="S6.SS3.14.p4.1.m1.1.1.2.2.cmml">s</mi><mn id="S6.SS3.14.p4.1.m1.1.1.2.3" xref="S6.SS3.14.p4.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS3.14.p4.1.m1.1.1.1" xref="S6.SS3.14.p4.1.m1.1.1.1.cmml">></mo><mrow id="S6.SS3.14.p4.1.m1.1.1.3" xref="S6.SS3.14.p4.1.m1.1.1.3.cmml"><msub id="S6.SS3.14.p4.1.m1.1.1.3.2" xref="S6.SS3.14.p4.1.m1.1.1.3.2.cmml"><mi id="S6.SS3.14.p4.1.m1.1.1.3.2.2" xref="S6.SS3.14.p4.1.m1.1.1.3.2.2.cmml">m</mi><mi id="S6.SS3.14.p4.1.m1.1.1.3.2.3" xref="S6.SS3.14.p4.1.m1.1.1.3.2.3.cmml">n</mi></msub><mo id="S6.SS3.14.p4.1.m1.1.1.3.1" xref="S6.SS3.14.p4.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.14.p4.1.m1.1.1.3.3" xref="S6.SS3.14.p4.1.m1.1.1.3.3.cmml"><mrow id="S6.SS3.14.p4.1.m1.1.1.3.3.2" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS3.14.p4.1.m1.1.1.3.3.2.2" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.14.p4.1.m1.1.1.3.3.2.1" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.14.p4.1.m1.1.1.3.3.2.3" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.14.p4.1.m1.1.1.3.3.3" xref="S6.SS3.14.p4.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.14.p4.1.m1.1b"><apply id="S6.SS3.14.p4.1.m1.1.1.cmml" xref="S6.SS3.14.p4.1.m1.1.1"><gt id="S6.SS3.14.p4.1.m1.1.1.1.cmml" xref="S6.SS3.14.p4.1.m1.1.1.1"></gt><apply 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xref="S6.SS3.14.p4.1.m1.1.1.3.3"></divide><apply id="S6.SS3.14.p4.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2"><plus id="S6.SS3.14.p4.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS3.14.p4.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.14.p4.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.14.p4.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.14.p4.1.m1.1.1.3.3.3.cmml" xref="S6.SS3.14.p4.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.14.p4.1.m1.1c">s_{1}>m_{n}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.14.p4.1.m1.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and <math alttext="s_{1}\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}" 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id="S6.SS3.14.p4.2.m2.1.1.3.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3"><plus id="S6.SS3.14.p4.2.m2.1.1.3.1.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.1"></plus><apply id="S6.SS3.14.p4.2.m2.1.1.3.2.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.14.p4.2.m2.1.1.3.2.1.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.2">subscript</csymbol><ci id="S6.SS3.14.p4.2.m2.1.1.3.2.2.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.2.2">ℕ</ci><cn id="S6.SS3.14.p4.2.m2.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.14.p4.2.m2.1.1.3.2.3">0</cn></apply><apply id="S6.SS3.14.p4.2.m2.1.1.3.3.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.3"><divide id="S6.SS3.14.p4.2.m2.1.1.3.3.1.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.3"></divide><apply id="S6.SS3.14.p4.2.m2.1.1.3.3.2.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.3.2"><plus id="S6.SS3.14.p4.2.m2.1.1.3.3.2.1.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.3.2.1"></plus><ci id="S6.SS3.14.p4.2.m2.1.1.3.3.2.2.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.14.p4.2.m2.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.14.p4.2.m2.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.14.p4.2.m2.1.1.3.3.3.cmml" xref="S6.SS3.14.p4.2.m2.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.14.p4.2.m2.1c">s_{1}\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.14.p4.2.m2.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∉ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Note that the second condition on <math alttext="s_{1}" class="ltx_Math" display="inline" id="S6.SS3.14.p4.3.m3.1"><semantics id="S6.SS3.14.p4.3.m3.1a"><msub id="S6.SS3.14.p4.3.m3.1.1" xref="S6.SS3.14.p4.3.m3.1.1.cmml"><mi id="S6.SS3.14.p4.3.m3.1.1.2" xref="S6.SS3.14.p4.3.m3.1.1.2.cmml">s</mi><mn id="S6.SS3.14.p4.3.m3.1.1.3" xref="S6.SS3.14.p4.3.m3.1.1.3.cmml">1</mn></msub><annotation-xml encoding="MathML-Content" id="S6.SS3.14.p4.3.m3.1b"><apply id="S6.SS3.14.p4.3.m3.1.1.cmml" xref="S6.SS3.14.p4.3.m3.1.1"><csymbol cd="ambiguous" id="S6.SS3.14.p4.3.m3.1.1.1.cmml" xref="S6.SS3.14.p4.3.m3.1.1">subscript</csymbol><ci id="S6.SS3.14.p4.3.m3.1.1.2.cmml" xref="S6.SS3.14.p4.3.m3.1.1.2">𝑠</ci><cn id="S6.SS3.14.p4.3.m3.1.1.3.cmml" type="integer" xref="S6.SS3.14.p4.3.m3.1.1.3">1</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.14.p4.3.m3.1c">s_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.14.p4.3.m3.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math> is new compared to the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a>. This condition will be removed in Step 2b.</p> </div> <div class="ltx_para" id="S6.SS3.15.p5"> <p class="ltx_p" id="S6.SS3.15.p5.2">Let <math alttext="q" class="ltx_Math" display="inline" id="S6.SS3.15.p5.1.m1.1"><semantics id="S6.SS3.15.p5.1.m1.1a"><mi id="S6.SS3.15.p5.1.m1.1.1" xref="S6.SS3.15.p5.1.m1.1.1.cmml">q</mi><annotation-xml encoding="MathML-Content" id="S6.SS3.15.p5.1.m1.1b"><ci id="S6.SS3.15.p5.1.m1.1.1.cmml" xref="S6.SS3.15.p5.1.m1.1.1">𝑞</ci></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.15.p5.1.m1.1c">q</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.15.p5.1.m1.1d">italic_q</annotation></semantics></math> be the largest integer such that <math alttext="q+\frac{\gamma+1}{p}<s_{1}" class="ltx_Math" display="inline" id="S6.SS3.15.p5.2.m2.1"><semantics id="S6.SS3.15.p5.2.m2.1a"><mrow id="S6.SS3.15.p5.2.m2.1.1" xref="S6.SS3.15.p5.2.m2.1.1.cmml"><mrow id="S6.SS3.15.p5.2.m2.1.1.2" xref="S6.SS3.15.p5.2.m2.1.1.2.cmml"><mi id="S6.SS3.15.p5.2.m2.1.1.2.2" xref="S6.SS3.15.p5.2.m2.1.1.2.2.cmml">q</mi><mo id="S6.SS3.15.p5.2.m2.1.1.2.1" xref="S6.SS3.15.p5.2.m2.1.1.2.1.cmml">+</mo><mfrac id="S6.SS3.15.p5.2.m2.1.1.2.3" xref="S6.SS3.15.p5.2.m2.1.1.2.3.cmml"><mrow id="S6.SS3.15.p5.2.m2.1.1.2.3.2" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2.cmml"><mi id="S6.SS3.15.p5.2.m2.1.1.2.3.2.2" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2.2.cmml">γ</mi><mo id="S6.SS3.15.p5.2.m2.1.1.2.3.2.1" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2.1.cmml">+</mo><mn id="S6.SS3.15.p5.2.m2.1.1.2.3.2.3" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.15.p5.2.m2.1.1.2.3.3" xref="S6.SS3.15.p5.2.m2.1.1.2.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS3.15.p5.2.m2.1.1.1" xref="S6.SS3.15.p5.2.m2.1.1.1.cmml"><</mo><msub id="S6.SS3.15.p5.2.m2.1.1.3" xref="S6.SS3.15.p5.2.m2.1.1.3.cmml"><mi id="S6.SS3.15.p5.2.m2.1.1.3.2" xref="S6.SS3.15.p5.2.m2.1.1.3.2.cmml">s</mi><mn id="S6.SS3.15.p5.2.m2.1.1.3.3" xref="S6.SS3.15.p5.2.m2.1.1.3.3.cmml">1</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.15.p5.2.m2.1b"><apply id="S6.SS3.15.p5.2.m2.1.1.cmml" xref="S6.SS3.15.p5.2.m2.1.1"><lt id="S6.SS3.15.p5.2.m2.1.1.1.cmml" xref="S6.SS3.15.p5.2.m2.1.1.1"></lt><apply id="S6.SS3.15.p5.2.m2.1.1.2.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2"><plus id="S6.SS3.15.p5.2.m2.1.1.2.1.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.1"></plus><ci id="S6.SS3.15.p5.2.m2.1.1.2.2.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.2">𝑞</ci><apply id="S6.SS3.15.p5.2.m2.1.1.2.3.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.3"><divide id="S6.SS3.15.p5.2.m2.1.1.2.3.1.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.3"></divide><apply id="S6.SS3.15.p5.2.m2.1.1.2.3.2.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2"><plus id="S6.SS3.15.p5.2.m2.1.1.2.3.2.1.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2.1"></plus><ci id="S6.SS3.15.p5.2.m2.1.1.2.3.2.2.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2.2">𝛾</ci><cn id="S6.SS3.15.p5.2.m2.1.1.2.3.2.3.cmml" type="integer" xref="S6.SS3.15.p5.2.m2.1.1.2.3.2.3">1</cn></apply><ci id="S6.SS3.15.p5.2.m2.1.1.2.3.3.cmml" xref="S6.SS3.15.p5.2.m2.1.1.2.3.3">𝑝</ci></apply></apply><apply id="S6.SS3.15.p5.2.m2.1.1.3.cmml" xref="S6.SS3.15.p5.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.15.p5.2.m2.1.1.3.1.cmml" xref="S6.SS3.15.p5.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS3.15.p5.2.m2.1.1.3.2.cmml" xref="S6.SS3.15.p5.2.m2.1.1.3.2">𝑠</ci><cn id="S6.SS3.15.p5.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS3.15.p5.2.m2.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.15.p5.2.m2.1c">q+\frac{\gamma+1}{p}<s_{1}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.15.p5.2.m2.1d">italic_q + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG < italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT</annotation></semantics></math>. The operator</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex77"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\mathcal{C}=(\mathcal{C}^{0},\dots,\mathcal{C}^{q}):H^{s_{1},p}\to\prod_{j=0}^% {q}B^{s_{1}-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X)," class="ltx_Math" display="block" id="S6.Ex77.m1.7"><semantics id="S6.Ex77.m1.7a"><mrow id="S6.Ex77.m1.7.7.1" xref="S6.Ex77.m1.7.7.1.1.cmml"><mrow id="S6.Ex77.m1.7.7.1.1" xref="S6.Ex77.m1.7.7.1.1.cmml"><mrow id="S6.Ex77.m1.7.7.1.1.2" xref="S6.Ex77.m1.7.7.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S6.Ex77.m1.7.7.1.1.2.4" xref="S6.Ex77.m1.7.7.1.1.2.4.cmml">𝒞</mi><mo id="S6.Ex77.m1.7.7.1.1.2.3" xref="S6.Ex77.m1.7.7.1.1.2.3.cmml">=</mo><mrow id="S6.Ex77.m1.7.7.1.1.2.2.2" xref="S6.Ex77.m1.7.7.1.1.2.2.3.cmml"><mo id="S6.Ex77.m1.7.7.1.1.2.2.2.3" stretchy="false" 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xref="S6.Ex77.m1.7.7.1.1.3.1.1.3.2.3.4.3">𝑝</ci></apply></apply></apply><list id="S6.Ex77.m1.4.4.2.3.cmml" xref="S6.Ex77.m1.4.4.2.4"><ci id="S6.Ex77.m1.3.3.1.1.cmml" xref="S6.Ex77.m1.3.3.1.1">𝑝</ci><ci id="S6.Ex77.m1.4.4.2.2.cmml" xref="S6.Ex77.m1.4.4.2.2">𝑝</ci></list></apply><list id="S6.Ex77.m1.7.7.1.1.3.1.1.1.2.cmml" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1"><apply id="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.cmml" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1"><csymbol cd="ambiguous" id="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.1.cmml" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1">superscript</csymbol><ci id="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.2.cmml" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.2">ℝ</ci><apply id="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3.cmml" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3"><minus id="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3.1.cmml" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3.1"></minus><ci id="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3.2.cmml" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3.2">𝑑</ci><cn id="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.Ex77.m1.7.7.1.1.3.1.1.1.1.1.3.3">1</cn></apply></apply><ci id="S6.Ex77.m1.6.6.cmml" xref="S6.Ex77.m1.6.6">𝑋</ci></list></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex77.m1.7c">\mathcal{C}=(\mathcal{C}^{0},\dots,\mathcal{C}^{q}):H^{s_{1},p}\to\prod_{j=0}^% {q}B^{s_{1}-j-\frac{\gamma+1}{p}}_{p,p}(\mathbb{R}^{d-1};X),</annotation><annotation encoding="application/x-llamapun" id="S6.Ex77.m1.7d">caligraphic_C = ( caligraphic_C start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT , … , caligraphic_C start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT ) : italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT → ∏ start_POSTSUBSCRIPT italic_j = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_q end_POSTSUPERSCRIPT italic_B start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_j - divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_p , italic_p end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d - 1 end_POSTSUPERSCRIPT ; italic_X ) ,</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.15.p5.3">as defined in (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E9" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.9</span></a>) satisfies (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E11" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.11</span></a>) for <math alttext="v\in H^{s_{1},p}" class="ltx_Math" display="inline" id="S6.SS3.15.p5.3.m1.2"><semantics id="S6.SS3.15.p5.3.m1.2a"><mrow id="S6.SS3.15.p5.3.m1.2.3" xref="S6.SS3.15.p5.3.m1.2.3.cmml"><mi id="S6.SS3.15.p5.3.m1.2.3.2" xref="S6.SS3.15.p5.3.m1.2.3.2.cmml">v</mi><mo id="S6.SS3.15.p5.3.m1.2.3.1" xref="S6.SS3.15.p5.3.m1.2.3.1.cmml">∈</mo><msup id="S6.SS3.15.p5.3.m1.2.3.3" xref="S6.SS3.15.p5.3.m1.2.3.3.cmml"><mi id="S6.SS3.15.p5.3.m1.2.3.3.2" xref="S6.SS3.15.p5.3.m1.2.3.3.2.cmml">H</mi><mrow id="S6.SS3.15.p5.3.m1.2.2.2.2" xref="S6.SS3.15.p5.3.m1.2.2.2.3.cmml"><msub id="S6.SS3.15.p5.3.m1.2.2.2.2.1" xref="S6.SS3.15.p5.3.m1.2.2.2.2.1.cmml"><mi id="S6.SS3.15.p5.3.m1.2.2.2.2.1.2" xref="S6.SS3.15.p5.3.m1.2.2.2.2.1.2.cmml">s</mi><mn id="S6.SS3.15.p5.3.m1.2.2.2.2.1.3" xref="S6.SS3.15.p5.3.m1.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S6.SS3.15.p5.3.m1.2.2.2.2.2" xref="S6.SS3.15.p5.3.m1.2.2.2.3.cmml">,</mo><mi id="S6.SS3.15.p5.3.m1.1.1.1.1" xref="S6.SS3.15.p5.3.m1.1.1.1.1.cmml">p</mi></mrow></msup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.15.p5.3.m1.2b"><apply id="S6.SS3.15.p5.3.m1.2.3.cmml" xref="S6.SS3.15.p5.3.m1.2.3"><in id="S6.SS3.15.p5.3.m1.2.3.1.cmml" xref="S6.SS3.15.p5.3.m1.2.3.1"></in><ci id="S6.SS3.15.p5.3.m1.2.3.2.cmml" xref="S6.SS3.15.p5.3.m1.2.3.2">𝑣</ci><apply id="S6.SS3.15.p5.3.m1.2.3.3.cmml" xref="S6.SS3.15.p5.3.m1.2.3.3"><csymbol cd="ambiguous" id="S6.SS3.15.p5.3.m1.2.3.3.1.cmml" xref="S6.SS3.15.p5.3.m1.2.3.3">superscript</csymbol><ci id="S6.SS3.15.p5.3.m1.2.3.3.2.cmml" xref="S6.SS3.15.p5.3.m1.2.3.3.2">𝐻</ci><list id="S6.SS3.15.p5.3.m1.2.2.2.3.cmml" xref="S6.SS3.15.p5.3.m1.2.2.2.2"><apply id="S6.SS3.15.p5.3.m1.2.2.2.2.1.cmml" xref="S6.SS3.15.p5.3.m1.2.2.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.15.p5.3.m1.2.2.2.2.1.1.cmml" xref="S6.SS3.15.p5.3.m1.2.2.2.2.1">subscript</csymbol><ci id="S6.SS3.15.p5.3.m1.2.2.2.2.1.2.cmml" xref="S6.SS3.15.p5.3.m1.2.2.2.2.1.2">𝑠</ci><cn id="S6.SS3.15.p5.3.m1.2.2.2.2.1.3.cmml" type="integer" xref="S6.SS3.15.p5.3.m1.2.2.2.2.1.3">1</cn></apply><ci id="S6.SS3.15.p5.3.m1.1.1.1.1.cmml" xref="S6.SS3.15.p5.3.m1.1.1.1.1">𝑝</ci></list></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.15.p5.3.m1.2c">v\in H^{s_{1},p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.15.p5.3.m1.2d">italic_v ∈ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT</annotation></semantics></math>. Therefore, we have</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex78"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{s_{1},p}_{\mathcal{C}}=H^{s_{1},p}_{\overline{\operatorname{Tr}}_{q}}=H_{0}% ^{s_{1},p}." class="ltx_Math" display="block" id="S6.Ex78.m1.7"><semantics id="S6.Ex78.m1.7a"><mrow id="S6.Ex78.m1.7.7.1" xref="S6.Ex78.m1.7.7.1.1.cmml"><mrow id="S6.Ex78.m1.7.7.1.1" xref="S6.Ex78.m1.7.7.1.1.cmml"><msubsup id="S6.Ex78.m1.7.7.1.1.2" xref="S6.Ex78.m1.7.7.1.1.2.cmml"><mi id="S6.Ex78.m1.7.7.1.1.2.2.2" xref="S6.Ex78.m1.7.7.1.1.2.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex78.m1.7.7.1.1.2.3" xref="S6.Ex78.m1.7.7.1.1.2.3.cmml">𝒞</mi><mrow id="S6.Ex78.m1.2.2.2.2" xref="S6.Ex78.m1.2.2.2.3.cmml"><msub id="S6.Ex78.m1.2.2.2.2.1" xref="S6.Ex78.m1.2.2.2.2.1.cmml"><mi id="S6.Ex78.m1.2.2.2.2.1.2" xref="S6.Ex78.m1.2.2.2.2.1.2.cmml">s</mi><mn id="S6.Ex78.m1.2.2.2.2.1.3" xref="S6.Ex78.m1.2.2.2.2.1.3.cmml">1</mn></msub><mo id="S6.Ex78.m1.2.2.2.2.2" xref="S6.Ex78.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex78.m1.1.1.1.1" xref="S6.Ex78.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex78.m1.7.7.1.1.3" xref="S6.Ex78.m1.7.7.1.1.3.cmml">=</mo><msubsup id="S6.Ex78.m1.7.7.1.1.4" xref="S6.Ex78.m1.7.7.1.1.4.cmml"><mi id="S6.Ex78.m1.7.7.1.1.4.2.2" xref="S6.Ex78.m1.7.7.1.1.4.2.2.cmml">H</mi><msub id="S6.Ex78.m1.7.7.1.1.4.3" xref="S6.Ex78.m1.7.7.1.1.4.3.cmml"><mover accent="true" id="S6.Ex78.m1.7.7.1.1.4.3.2" xref="S6.Ex78.m1.7.7.1.1.4.3.2.cmml"><mi id="S6.Ex78.m1.7.7.1.1.4.3.2.2" xref="S6.Ex78.m1.7.7.1.1.4.3.2.2.cmml">Tr</mi><mo id="S6.Ex78.m1.7.7.1.1.4.3.2.1" xref="S6.Ex78.m1.7.7.1.1.4.3.2.1.cmml">¯</mo></mover><mi id="S6.Ex78.m1.7.7.1.1.4.3.3" xref="S6.Ex78.m1.7.7.1.1.4.3.3.cmml">q</mi></msub><mrow id="S6.Ex78.m1.4.4.2.2" xref="S6.Ex78.m1.4.4.2.3.cmml"><msub id="S6.Ex78.m1.4.4.2.2.1" xref="S6.Ex78.m1.4.4.2.2.1.cmml"><mi id="S6.Ex78.m1.4.4.2.2.1.2" xref="S6.Ex78.m1.4.4.2.2.1.2.cmml">s</mi><mn id="S6.Ex78.m1.4.4.2.2.1.3" xref="S6.Ex78.m1.4.4.2.2.1.3.cmml">1</mn></msub><mo id="S6.Ex78.m1.4.4.2.2.2" xref="S6.Ex78.m1.4.4.2.3.cmml">,</mo><mi id="S6.Ex78.m1.3.3.1.1" xref="S6.Ex78.m1.3.3.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex78.m1.7.7.1.1.5" xref="S6.Ex78.m1.7.7.1.1.5.cmml">=</mo><msubsup id="S6.Ex78.m1.7.7.1.1.6" xref="S6.Ex78.m1.7.7.1.1.6.cmml"><mi id="S6.Ex78.m1.7.7.1.1.6.2.2" xref="S6.Ex78.m1.7.7.1.1.6.2.2.cmml">H</mi><mn id="S6.Ex78.m1.7.7.1.1.6.2.3" xref="S6.Ex78.m1.7.7.1.1.6.2.3.cmml">0</mn><mrow id="S6.Ex78.m1.6.6.2.2" xref="S6.Ex78.m1.6.6.2.3.cmml"><msub id="S6.Ex78.m1.6.6.2.2.1" xref="S6.Ex78.m1.6.6.2.2.1.cmml"><mi id="S6.Ex78.m1.6.6.2.2.1.2" xref="S6.Ex78.m1.6.6.2.2.1.2.cmml">s</mi><mn id="S6.Ex78.m1.6.6.2.2.1.3" xref="S6.Ex78.m1.6.6.2.2.1.3.cmml">1</mn></msub><mo id="S6.Ex78.m1.6.6.2.2.2" xref="S6.Ex78.m1.6.6.2.3.cmml">,</mo><mi id="S6.Ex78.m1.5.5.1.1" 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xref="S6.Ex78.m1.7.7.1.1.5"></eq><share href="https://arxiv.org/html/2503.14636v1#S6.Ex78.m1.7.7.1.1.4.cmml" id="S6.Ex78.m1.7.7.1.1d.cmml" xref="S6.Ex78.m1.7.7.1"></share><apply id="S6.Ex78.m1.7.7.1.1.6.cmml" xref="S6.Ex78.m1.7.7.1.1.6"><csymbol cd="ambiguous" id="S6.Ex78.m1.7.7.1.1.6.1.cmml" xref="S6.Ex78.m1.7.7.1.1.6">superscript</csymbol><apply id="S6.Ex78.m1.7.7.1.1.6.2.cmml" xref="S6.Ex78.m1.7.7.1.1.6"><csymbol cd="ambiguous" id="S6.Ex78.m1.7.7.1.1.6.2.1.cmml" xref="S6.Ex78.m1.7.7.1.1.6">subscript</csymbol><ci id="S6.Ex78.m1.7.7.1.1.6.2.2.cmml" xref="S6.Ex78.m1.7.7.1.1.6.2.2">𝐻</ci><cn id="S6.Ex78.m1.7.7.1.1.6.2.3.cmml" type="integer" xref="S6.Ex78.m1.7.7.1.1.6.2.3">0</cn></apply><list id="S6.Ex78.m1.6.6.2.3.cmml" xref="S6.Ex78.m1.6.6.2.2"><apply id="S6.Ex78.m1.6.6.2.2.1.cmml" xref="S6.Ex78.m1.6.6.2.2.1"><csymbol cd="ambiguous" id="S6.Ex78.m1.6.6.2.2.1.1.cmml" xref="S6.Ex78.m1.6.6.2.2.1">subscript</csymbol><ci id="S6.Ex78.m1.6.6.2.2.1.2.cmml" xref="S6.Ex78.m1.6.6.2.2.1.2">𝑠</ci><cn id="S6.Ex78.m1.6.6.2.2.1.3.cmml" type="integer" xref="S6.Ex78.m1.6.6.2.2.1.3">1</cn></apply><ci id="S6.Ex78.m1.5.5.1.1.cmml" xref="S6.Ex78.m1.5.5.1.1">𝑝</ci></list></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.Ex78.m1.7c">H^{s_{1},p}_{\mathcal{C}}=H^{s_{1},p}_{\overline{\operatorname{Tr}}_{q}}=H_{0}% ^{s_{1},p}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex78.m1.7d">italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_C end_POSTSUBSCRIPT = italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT over¯ start_ARG roman_Tr end_ARG start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.15.p5.4">We can now proceed similarly as in Step 2a of the proof of Theorem <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem5" title="Theorem 6.5 (Complex interpolation of weighted Sobolev spaces). ‣ 6.2. Main results about complex interpolation of weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.5</span></a> using Proposition <a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.Thmtheorem1" title="Proposition 6.1. ‣ 6.1. Interpolation results for weighted spaces ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.1</span></a> to obtain (cf. (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E14" title="In Proof of Theorem 6.5. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.14</span></a>))</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex79"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H^{s_{\theta},p}_{\mathcal{C}}=H_{0}^{s_{\theta},p}=\big{[}H_{0}^{s_{0},p},H_{% 0}^{s_{1},p}\big{]}_{\theta}\hookrightarrow\big{[}H^{s_{0},p},H^{s_{1},p}_{% \mathcal{B}}\big{]}_{\theta}." class="ltx_Math" display="block" id="S6.Ex79.m1.13"><semantics id="S6.Ex79.m1.13a"><mrow id="S6.Ex79.m1.13.13.1" xref="S6.Ex79.m1.13.13.1.1.cmml"><mrow id="S6.Ex79.m1.13.13.1.1" xref="S6.Ex79.m1.13.13.1.1.cmml"><msubsup id="S6.Ex79.m1.13.13.1.1.6" xref="S6.Ex79.m1.13.13.1.1.6.cmml"><mi id="S6.Ex79.m1.13.13.1.1.6.2.2" xref="S6.Ex79.m1.13.13.1.1.6.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" 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italic_p end_POSTSUPERSCRIPT = [ italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT ↪ [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S6.SS3.16.p6"> <p class="ltx_p" id="S6.SS3.16.p6.6"><span class="ltx_text ltx_font_italic" id="S6.SS3.16.p6.6.1">Step 2b. </span>Assume that <math alttext="s_{1}>m_{n}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.16.p6.1.m1.1"><semantics id="S6.SS3.16.p6.1.m1.1a"><mrow id="S6.SS3.16.p6.1.m1.1.1" xref="S6.SS3.16.p6.1.m1.1.1.cmml"><msub id="S6.SS3.16.p6.1.m1.1.1.2" xref="S6.SS3.16.p6.1.m1.1.1.2.cmml"><mi id="S6.SS3.16.p6.1.m1.1.1.2.2" xref="S6.SS3.16.p6.1.m1.1.1.2.2.cmml">s</mi><mn id="S6.SS3.16.p6.1.m1.1.1.2.3" xref="S6.SS3.16.p6.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS3.16.p6.1.m1.1.1.1" xref="S6.SS3.16.p6.1.m1.1.1.1.cmml">></mo><mrow id="S6.SS3.16.p6.1.m1.1.1.3" xref="S6.SS3.16.p6.1.m1.1.1.3.cmml"><msub id="S6.SS3.16.p6.1.m1.1.1.3.2" xref="S6.SS3.16.p6.1.m1.1.1.3.2.cmml"><mi id="S6.SS3.16.p6.1.m1.1.1.3.2.2" xref="S6.SS3.16.p6.1.m1.1.1.3.2.2.cmml">m</mi><mi id="S6.SS3.16.p6.1.m1.1.1.3.2.3" xref="S6.SS3.16.p6.1.m1.1.1.3.2.3.cmml">n</mi></msub><mo id="S6.SS3.16.p6.1.m1.1.1.3.1" xref="S6.SS3.16.p6.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.16.p6.1.m1.1.1.3.3" xref="S6.SS3.16.p6.1.m1.1.1.3.3.cmml"><mrow id="S6.SS3.16.p6.1.m1.1.1.3.3.2" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS3.16.p6.1.m1.1.1.3.3.2.2" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.16.p6.1.m1.1.1.3.3.2.1" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.16.p6.1.m1.1.1.3.3.2.3" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.16.p6.1.m1.1.1.3.3.3" xref="S6.SS3.16.p6.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.16.p6.1.m1.1b"><apply id="S6.SS3.16.p6.1.m1.1.1.cmml" xref="S6.SS3.16.p6.1.m1.1.1"><gt id="S6.SS3.16.p6.1.m1.1.1.1.cmml" xref="S6.SS3.16.p6.1.m1.1.1.1"></gt><apply id="S6.SS3.16.p6.1.m1.1.1.2.cmml" xref="S6.SS3.16.p6.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.16.p6.1.m1.1.1.2.1.cmml" xref="S6.SS3.16.p6.1.m1.1.1.2">subscript</csymbol><ci id="S6.SS3.16.p6.1.m1.1.1.2.2.cmml" xref="S6.SS3.16.p6.1.m1.1.1.2.2">𝑠</ci><cn id="S6.SS3.16.p6.1.m1.1.1.2.3.cmml" type="integer" xref="S6.SS3.16.p6.1.m1.1.1.2.3">1</cn></apply><apply id="S6.SS3.16.p6.1.m1.1.1.3.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3"><plus id="S6.SS3.16.p6.1.m1.1.1.3.1.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.1"></plus><apply id="S6.SS3.16.p6.1.m1.1.1.3.2.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.16.p6.1.m1.1.1.3.2.1.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS3.16.p6.1.m1.1.1.3.2.2.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.2.2">𝑚</ci><ci id="S6.SS3.16.p6.1.m1.1.1.3.2.3.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.2.3">𝑛</ci></apply><apply id="S6.SS3.16.p6.1.m1.1.1.3.3.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.3"><divide id="S6.SS3.16.p6.1.m1.1.1.3.3.1.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.3"></divide><apply id="S6.SS3.16.p6.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2"><plus id="S6.SS3.16.p6.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS3.16.p6.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.16.p6.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.16.p6.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.16.p6.1.m1.1.1.3.3.3.cmml" xref="S6.SS3.16.p6.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.16.p6.1.m1.1c">s_{1}>m_{n}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.16.p6.1.m1.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT > italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> and there exists an <math alttext="\ell\in\mathbb{N}_{0}" class="ltx_Math" display="inline" id="S6.SS3.16.p6.2.m2.1"><semantics id="S6.SS3.16.p6.2.m2.1a"><mrow id="S6.SS3.16.p6.2.m2.1.1" xref="S6.SS3.16.p6.2.m2.1.1.cmml"><mi id="S6.SS3.16.p6.2.m2.1.1.2" mathvariant="normal" xref="S6.SS3.16.p6.2.m2.1.1.2.cmml">ℓ</mi><mo id="S6.SS3.16.p6.2.m2.1.1.1" xref="S6.SS3.16.p6.2.m2.1.1.1.cmml">∈</mo><msub id="S6.SS3.16.p6.2.m2.1.1.3" xref="S6.SS3.16.p6.2.m2.1.1.3.cmml"><mi id="S6.SS3.16.p6.2.m2.1.1.3.2" xref="S6.SS3.16.p6.2.m2.1.1.3.2.cmml">ℕ</mi><mn id="S6.SS3.16.p6.2.m2.1.1.3.3" xref="S6.SS3.16.p6.2.m2.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.16.p6.2.m2.1b"><apply id="S6.SS3.16.p6.2.m2.1.1.cmml" xref="S6.SS3.16.p6.2.m2.1.1"><in id="S6.SS3.16.p6.2.m2.1.1.1.cmml" xref="S6.SS3.16.p6.2.m2.1.1.1"></in><ci id="S6.SS3.16.p6.2.m2.1.1.2.cmml" xref="S6.SS3.16.p6.2.m2.1.1.2">ℓ</ci><apply id="S6.SS3.16.p6.2.m2.1.1.3.cmml" xref="S6.SS3.16.p6.2.m2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.16.p6.2.m2.1.1.3.1.cmml" xref="S6.SS3.16.p6.2.m2.1.1.3">subscript</csymbol><ci id="S6.SS3.16.p6.2.m2.1.1.3.2.cmml" xref="S6.SS3.16.p6.2.m2.1.1.3.2">ℕ</ci><cn id="S6.SS3.16.p6.2.m2.1.1.3.3.cmml" type="integer" xref="S6.SS3.16.p6.2.m2.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.16.p6.2.m2.1c">\ell\in\mathbb{N}_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.16.p6.2.m2.1d">roman_ℓ ∈ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> such that <math alttext="s_{1}=\ell+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.16.p6.3.m3.1"><semantics id="S6.SS3.16.p6.3.m3.1a"><mrow id="S6.SS3.16.p6.3.m3.1.1" xref="S6.SS3.16.p6.3.m3.1.1.cmml"><msub id="S6.SS3.16.p6.3.m3.1.1.2" xref="S6.SS3.16.p6.3.m3.1.1.2.cmml"><mi id="S6.SS3.16.p6.3.m3.1.1.2.2" xref="S6.SS3.16.p6.3.m3.1.1.2.2.cmml">s</mi><mn id="S6.SS3.16.p6.3.m3.1.1.2.3" xref="S6.SS3.16.p6.3.m3.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS3.16.p6.3.m3.1.1.1" xref="S6.SS3.16.p6.3.m3.1.1.1.cmml">=</mo><mrow id="S6.SS3.16.p6.3.m3.1.1.3" xref="S6.SS3.16.p6.3.m3.1.1.3.cmml"><mi id="S6.SS3.16.p6.3.m3.1.1.3.2" mathvariant="normal" xref="S6.SS3.16.p6.3.m3.1.1.3.2.cmml">ℓ</mi><mo id="S6.SS3.16.p6.3.m3.1.1.3.1" xref="S6.SS3.16.p6.3.m3.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.16.p6.3.m3.1.1.3.3" xref="S6.SS3.16.p6.3.m3.1.1.3.3.cmml"><mrow id="S6.SS3.16.p6.3.m3.1.1.3.3.2" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2.cmml"><mi id="S6.SS3.16.p6.3.m3.1.1.3.3.2.2" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.16.p6.3.m3.1.1.3.3.2.1" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.16.p6.3.m3.1.1.3.3.2.3" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.16.p6.3.m3.1.1.3.3.3" xref="S6.SS3.16.p6.3.m3.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.16.p6.3.m3.1b"><apply id="S6.SS3.16.p6.3.m3.1.1.cmml" xref="S6.SS3.16.p6.3.m3.1.1"><eq id="S6.SS3.16.p6.3.m3.1.1.1.cmml" xref="S6.SS3.16.p6.3.m3.1.1.1"></eq><apply id="S6.SS3.16.p6.3.m3.1.1.2.cmml" xref="S6.SS3.16.p6.3.m3.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.16.p6.3.m3.1.1.2.1.cmml" xref="S6.SS3.16.p6.3.m3.1.1.2">subscript</csymbol><ci id="S6.SS3.16.p6.3.m3.1.1.2.2.cmml" xref="S6.SS3.16.p6.3.m3.1.1.2.2">𝑠</ci><cn id="S6.SS3.16.p6.3.m3.1.1.2.3.cmml" type="integer" xref="S6.SS3.16.p6.3.m3.1.1.2.3">1</cn></apply><apply id="S6.SS3.16.p6.3.m3.1.1.3.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3"><plus id="S6.SS3.16.p6.3.m3.1.1.3.1.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.1"></plus><ci id="S6.SS3.16.p6.3.m3.1.1.3.2.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.2">ℓ</ci><apply id="S6.SS3.16.p6.3.m3.1.1.3.3.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.3"><divide id="S6.SS3.16.p6.3.m3.1.1.3.3.1.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.3"></divide><apply id="S6.SS3.16.p6.3.m3.1.1.3.3.2.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2"><plus id="S6.SS3.16.p6.3.m3.1.1.3.3.2.1.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2.1"></plus><ci id="S6.SS3.16.p6.3.m3.1.1.3.3.2.2.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.16.p6.3.m3.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.16.p6.3.m3.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.16.p6.3.m3.1.1.3.3.3.cmml" xref="S6.SS3.16.p6.3.m3.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.16.p6.3.m3.1c">s_{1}=\ell+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.16.p6.3.m3.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = roman_ℓ + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Fix <math alttext="t\in(s_{1},s]" class="ltx_Math" display="inline" id="S6.SS3.16.p6.4.m4.2"><semantics id="S6.SS3.16.p6.4.m4.2a"><mrow id="S6.SS3.16.p6.4.m4.2.2" xref="S6.SS3.16.p6.4.m4.2.2.cmml"><mi id="S6.SS3.16.p6.4.m4.2.2.3" xref="S6.SS3.16.p6.4.m4.2.2.3.cmml">t</mi><mo id="S6.SS3.16.p6.4.m4.2.2.2" xref="S6.SS3.16.p6.4.m4.2.2.2.cmml">∈</mo><mrow id="S6.SS3.16.p6.4.m4.2.2.1.1" xref="S6.SS3.16.p6.4.m4.2.2.1.2.cmml"><mo id="S6.SS3.16.p6.4.m4.2.2.1.1.2" stretchy="false" xref="S6.SS3.16.p6.4.m4.2.2.1.2.cmml">(</mo><msub id="S6.SS3.16.p6.4.m4.2.2.1.1.1" xref="S6.SS3.16.p6.4.m4.2.2.1.1.1.cmml"><mi id="S6.SS3.16.p6.4.m4.2.2.1.1.1.2" xref="S6.SS3.16.p6.4.m4.2.2.1.1.1.2.cmml">s</mi><mn id="S6.SS3.16.p6.4.m4.2.2.1.1.1.3" xref="S6.SS3.16.p6.4.m4.2.2.1.1.1.3.cmml">1</mn></msub><mo id="S6.SS3.16.p6.4.m4.2.2.1.1.3" xref="S6.SS3.16.p6.4.m4.2.2.1.2.cmml">,</mo><mi id="S6.SS3.16.p6.4.m4.1.1" xref="S6.SS3.16.p6.4.m4.1.1.cmml">s</mi><mo id="S6.SS3.16.p6.4.m4.2.2.1.1.4" stretchy="false" xref="S6.SS3.16.p6.4.m4.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.16.p6.4.m4.2b"><apply id="S6.SS3.16.p6.4.m4.2.2.cmml" xref="S6.SS3.16.p6.4.m4.2.2"><in id="S6.SS3.16.p6.4.m4.2.2.2.cmml" xref="S6.SS3.16.p6.4.m4.2.2.2"></in><ci id="S6.SS3.16.p6.4.m4.2.2.3.cmml" xref="S6.SS3.16.p6.4.m4.2.2.3">𝑡</ci><interval closure="open-closed" id="S6.SS3.16.p6.4.m4.2.2.1.2.cmml" xref="S6.SS3.16.p6.4.m4.2.2.1.1"><apply id="S6.SS3.16.p6.4.m4.2.2.1.1.1.cmml" xref="S6.SS3.16.p6.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.16.p6.4.m4.2.2.1.1.1.1.cmml" xref="S6.SS3.16.p6.4.m4.2.2.1.1.1">subscript</csymbol><ci id="S6.SS3.16.p6.4.m4.2.2.1.1.1.2.cmml" xref="S6.SS3.16.p6.4.m4.2.2.1.1.1.2">𝑠</ci><cn id="S6.SS3.16.p6.4.m4.2.2.1.1.1.3.cmml" type="integer" xref="S6.SS3.16.p6.4.m4.2.2.1.1.1.3">1</cn></apply><ci id="S6.SS3.16.p6.4.m4.1.1.cmml" xref="S6.SS3.16.p6.4.m4.1.1">𝑠</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.16.p6.4.m4.2c">t\in(s_{1},s]</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.16.p6.4.m4.2d">italic_t ∈ ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_s ]</annotation></semantics></math> such that <math alttext="t\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.16.p6.5.m5.1"><semantics id="S6.SS3.16.p6.5.m5.1a"><mrow id="S6.SS3.16.p6.5.m5.1.1" xref="S6.SS3.16.p6.5.m5.1.1.cmml"><mi id="S6.SS3.16.p6.5.m5.1.1.2" xref="S6.SS3.16.p6.5.m5.1.1.2.cmml">t</mi><mo id="S6.SS3.16.p6.5.m5.1.1.1" xref="S6.SS3.16.p6.5.m5.1.1.1.cmml">∉</mo><mrow id="S6.SS3.16.p6.5.m5.1.1.3" xref="S6.SS3.16.p6.5.m5.1.1.3.cmml"><msub id="S6.SS3.16.p6.5.m5.1.1.3.2" xref="S6.SS3.16.p6.5.m5.1.1.3.2.cmml"><mi id="S6.SS3.16.p6.5.m5.1.1.3.2.2" xref="S6.SS3.16.p6.5.m5.1.1.3.2.2.cmml">ℕ</mi><mn id="S6.SS3.16.p6.5.m5.1.1.3.2.3" xref="S6.SS3.16.p6.5.m5.1.1.3.2.3.cmml">0</mn></msub><mo id="S6.SS3.16.p6.5.m5.1.1.3.1" xref="S6.SS3.16.p6.5.m5.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.16.p6.5.m5.1.1.3.3" xref="S6.SS3.16.p6.5.m5.1.1.3.3.cmml"><mrow id="S6.SS3.16.p6.5.m5.1.1.3.3.2" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2.cmml"><mi id="S6.SS3.16.p6.5.m5.1.1.3.3.2.2" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.16.p6.5.m5.1.1.3.3.2.1" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.16.p6.5.m5.1.1.3.3.2.3" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.16.p6.5.m5.1.1.3.3.3" xref="S6.SS3.16.p6.5.m5.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.16.p6.5.m5.1b"><apply id="S6.SS3.16.p6.5.m5.1.1.cmml" xref="S6.SS3.16.p6.5.m5.1.1"><notin id="S6.SS3.16.p6.5.m5.1.1.1.cmml" xref="S6.SS3.16.p6.5.m5.1.1.1"></notin><ci id="S6.SS3.16.p6.5.m5.1.1.2.cmml" xref="S6.SS3.16.p6.5.m5.1.1.2">𝑡</ci><apply id="S6.SS3.16.p6.5.m5.1.1.3.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3"><plus id="S6.SS3.16.p6.5.m5.1.1.3.1.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.1"></plus><apply id="S6.SS3.16.p6.5.m5.1.1.3.2.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.16.p6.5.m5.1.1.3.2.1.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.2">subscript</csymbol><ci id="S6.SS3.16.p6.5.m5.1.1.3.2.2.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.2.2">ℕ</ci><cn id="S6.SS3.16.p6.5.m5.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.16.p6.5.m5.1.1.3.2.3">0</cn></apply><apply id="S6.SS3.16.p6.5.m5.1.1.3.3.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.3"><divide id="S6.SS3.16.p6.5.m5.1.1.3.3.1.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.3"></divide><apply id="S6.SS3.16.p6.5.m5.1.1.3.3.2.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2"><plus id="S6.SS3.16.p6.5.m5.1.1.3.3.2.1.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2.1"></plus><ci id="S6.SS3.16.p6.5.m5.1.1.3.3.2.2.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.16.p6.5.m5.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.16.p6.5.m5.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.16.p6.5.m5.1.1.3.3.3.cmml" xref="S6.SS3.16.p6.5.m5.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.16.p6.5.m5.1c">t\notin\mathbb{N}_{0}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.16.p6.5.m5.1d">italic_t ∉ blackboard_N start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Furthermore, let <math alttext="\theta_{1}:=(s_{1}-s_{0})/(t-s_{0})" class="ltx_Math" display="inline" id="S6.SS3.16.p6.6.m6.2"><semantics id="S6.SS3.16.p6.6.m6.2a"><mrow id="S6.SS3.16.p6.6.m6.2.2" xref="S6.SS3.16.p6.6.m6.2.2.cmml"><msub id="S6.SS3.16.p6.6.m6.2.2.4" xref="S6.SS3.16.p6.6.m6.2.2.4.cmml"><mi id="S6.SS3.16.p6.6.m6.2.2.4.2" xref="S6.SS3.16.p6.6.m6.2.2.4.2.cmml">θ</mi><mn id="S6.SS3.16.p6.6.m6.2.2.4.3" xref="S6.SS3.16.p6.6.m6.2.2.4.3.cmml">1</mn></msub><mo id="S6.SS3.16.p6.6.m6.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.SS3.16.p6.6.m6.2.2.3.cmml">:=</mo><mrow id="S6.SS3.16.p6.6.m6.2.2.2" xref="S6.SS3.16.p6.6.m6.2.2.2.cmml"><mrow id="S6.SS3.16.p6.6.m6.1.1.1.1.1" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.cmml"><mo id="S6.SS3.16.p6.6.m6.1.1.1.1.1.2" stretchy="false" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.cmml"><msub id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.cmml"><mi id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.2" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.2.cmml">s</mi><mn id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.3" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.1" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.1.cmml">−</mo><msub id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.cmml"><mi id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.2" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.2.cmml">s</mi><mn id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.3" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.SS3.16.p6.6.m6.1.1.1.1.1.3" stretchy="false" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS3.16.p6.6.m6.2.2.2.3" xref="S6.SS3.16.p6.6.m6.2.2.2.3.cmml">/</mo><mrow id="S6.SS3.16.p6.6.m6.2.2.2.2.1" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.cmml"><mo id="S6.SS3.16.p6.6.m6.2.2.2.2.1.2" stretchy="false" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.cmml">(</mo><mrow id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.cmml"><mi id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.2" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.2.cmml">t</mi><mo id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.1" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.1.cmml">−</mo><msub id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.cmml"><mi id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.2" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.2.cmml">s</mi><mn id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.3" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.SS3.16.p6.6.m6.2.2.2.2.1.3" stretchy="false" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.16.p6.6.m6.2b"><apply id="S6.SS3.16.p6.6.m6.2.2.cmml" xref="S6.SS3.16.p6.6.m6.2.2"><csymbol cd="latexml" id="S6.SS3.16.p6.6.m6.2.2.3.cmml" xref="S6.SS3.16.p6.6.m6.2.2.3">assign</csymbol><apply id="S6.SS3.16.p6.6.m6.2.2.4.cmml" xref="S6.SS3.16.p6.6.m6.2.2.4"><csymbol cd="ambiguous" id="S6.SS3.16.p6.6.m6.2.2.4.1.cmml" xref="S6.SS3.16.p6.6.m6.2.2.4">subscript</csymbol><ci id="S6.SS3.16.p6.6.m6.2.2.4.2.cmml" xref="S6.SS3.16.p6.6.m6.2.2.4.2">𝜃</ci><cn id="S6.SS3.16.p6.6.m6.2.2.4.3.cmml" type="integer" xref="S6.SS3.16.p6.6.m6.2.2.4.3">1</cn></apply><apply id="S6.SS3.16.p6.6.m6.2.2.2.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2"><divide id="S6.SS3.16.p6.6.m6.2.2.2.3.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2.3"></divide><apply id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1"><minus id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.1.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.1"></minus><apply id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.1.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.2.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.2">𝑠</ci><cn id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.1.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.2.cmml" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.2">𝑠</ci><cn id="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.SS3.16.p6.6.m6.1.1.1.1.1.1.3.3">0</cn></apply></apply><apply id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1"><minus id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.1.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.1"></minus><ci id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.2.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.2">𝑡</ci><apply id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.1.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3">subscript</csymbol><ci id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.2.cmml" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.2">𝑠</ci><cn id="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S6.SS3.16.p6.6.m6.2.2.2.2.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.16.p6.6.m6.2c">\theta_{1}:=(s_{1}-s_{0})/(t-s_{0})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.16.p6.6.m6.2d">italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) / ( italic_t - italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. Then by reiteration for the complex interpolation method (see <cite class="ltx_cite ltx_citemacro_cite">[<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#bib.bib9" title="">9</a>, Theorem 4.6.1]</cite>) and Step 2a twice, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex80"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}H^{s_{0},p},H^{s_{1},p}_{\mathcal{B}}\big{]}_{\theta}=\big{[}H^{s_{0},p% },[H^{s_{0},p},H^{t,p}_{\mathcal{B}}]_{\theta_{1}}\big{]}_{\theta}=\big{[}H^{s% _{0},p},H^{t,p}_{\mathcal{B}}\big{]}_{\theta\theta_{1}}=H^{s_{\theta},p}_{% \mathcal{B}}." class="ltx_Math" display="block" id="S6.Ex80.m1.17"><semantics id="S6.Ex80.m1.17a"><mrow id="S6.Ex80.m1.17.17.1" xref="S6.Ex80.m1.17.17.1.1.cmml"><mrow id="S6.Ex80.m1.17.17.1.1" xref="S6.Ex80.m1.17.17.1.1.cmml"><msub id="S6.Ex80.m1.17.17.1.1.2" xref="S6.Ex80.m1.17.17.1.1.2.cmml"><mrow id="S6.Ex80.m1.17.17.1.1.2.2.2" xref="S6.Ex80.m1.17.17.1.1.2.2.3.cmml"><mo id="S6.Ex80.m1.17.17.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex80.m1.17.17.1.1.2.2.3.cmml">[</mo><msup id="S6.Ex80.m1.17.17.1.1.1.1.1.1" xref="S6.Ex80.m1.17.17.1.1.1.1.1.1.cmml"><mi id="S6.Ex80.m1.17.17.1.1.1.1.1.1.2" xref="S6.Ex80.m1.17.17.1.1.1.1.1.1.2.cmml">H</mi><mrow id="S6.Ex80.m1.2.2.2.2" xref="S6.Ex80.m1.2.2.2.3.cmml"><msub id="S6.Ex80.m1.2.2.2.2.1" xref="S6.Ex80.m1.2.2.2.2.1.cmml"><mi id="S6.Ex80.m1.2.2.2.2.1.2" xref="S6.Ex80.m1.2.2.2.2.1.2.cmml">s</mi><mn id="S6.Ex80.m1.2.2.2.2.1.3" xref="S6.Ex80.m1.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.Ex80.m1.2.2.2.2.2" xref="S6.Ex80.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex80.m1.1.1.1.1" xref="S6.Ex80.m1.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.Ex80.m1.17.17.1.1.2.2.2.4" xref="S6.Ex80.m1.17.17.1.1.2.2.3.cmml">,</mo><msubsup id="S6.Ex80.m1.17.17.1.1.2.2.2.2" xref="S6.Ex80.m1.17.17.1.1.2.2.2.2.cmml"><mi 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},[H^{s_{0},p},H^{t,p}_{\mathcal{B}}]_{\theta_{1}}\big{]}_{\theta}=\big{[}H^{s% _{0},p},H^{t,p}_{\mathcal{B}}\big{]}_{\theta\theta_{1}}=H^{s_{\theta},p}_{% \mathcal{B}}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex80.m1.17d">[ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_t , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_t , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT = italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> </div> <div class="ltx_para" id="S6.SS3.17.p7"> <p class="ltx_p" id="S6.SS3.17.p7.4"><span class="ltx_text ltx_font_italic" id="S6.SS3.17.p7.4.1">Step 2c. </span>Assume that <math alttext="s_{1}<m_{n}+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.17.p7.1.m1.1"><semantics id="S6.SS3.17.p7.1.m1.1a"><mrow id="S6.SS3.17.p7.1.m1.1.1" xref="S6.SS3.17.p7.1.m1.1.1.cmml"><msub id="S6.SS3.17.p7.1.m1.1.1.2" xref="S6.SS3.17.p7.1.m1.1.1.2.cmml"><mi id="S6.SS3.17.p7.1.m1.1.1.2.2" xref="S6.SS3.17.p7.1.m1.1.1.2.2.cmml">s</mi><mn id="S6.SS3.17.p7.1.m1.1.1.2.3" xref="S6.SS3.17.p7.1.m1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS3.17.p7.1.m1.1.1.1" xref="S6.SS3.17.p7.1.m1.1.1.1.cmml"><</mo><mrow id="S6.SS3.17.p7.1.m1.1.1.3" xref="S6.SS3.17.p7.1.m1.1.1.3.cmml"><msub id="S6.SS3.17.p7.1.m1.1.1.3.2" xref="S6.SS3.17.p7.1.m1.1.1.3.2.cmml"><mi id="S6.SS3.17.p7.1.m1.1.1.3.2.2" xref="S6.SS3.17.p7.1.m1.1.1.3.2.2.cmml">m</mi><mi id="S6.SS3.17.p7.1.m1.1.1.3.2.3" xref="S6.SS3.17.p7.1.m1.1.1.3.2.3.cmml">n</mi></msub><mo id="S6.SS3.17.p7.1.m1.1.1.3.1" xref="S6.SS3.17.p7.1.m1.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.17.p7.1.m1.1.1.3.3" xref="S6.SS3.17.p7.1.m1.1.1.3.3.cmml"><mrow id="S6.SS3.17.p7.1.m1.1.1.3.3.2" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2.cmml"><mi id="S6.SS3.17.p7.1.m1.1.1.3.3.2.2" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.17.p7.1.m1.1.1.3.3.2.1" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.17.p7.1.m1.1.1.3.3.2.3" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.17.p7.1.m1.1.1.3.3.3" xref="S6.SS3.17.p7.1.m1.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.17.p7.1.m1.1b"><apply id="S6.SS3.17.p7.1.m1.1.1.cmml" xref="S6.SS3.17.p7.1.m1.1.1"><lt id="S6.SS3.17.p7.1.m1.1.1.1.cmml" xref="S6.SS3.17.p7.1.m1.1.1.1"></lt><apply id="S6.SS3.17.p7.1.m1.1.1.2.cmml" xref="S6.SS3.17.p7.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.17.p7.1.m1.1.1.2.1.cmml" xref="S6.SS3.17.p7.1.m1.1.1.2">subscript</csymbol><ci id="S6.SS3.17.p7.1.m1.1.1.2.2.cmml" xref="S6.SS3.17.p7.1.m1.1.1.2.2">𝑠</ci><cn id="S6.SS3.17.p7.1.m1.1.1.2.3.cmml" type="integer" xref="S6.SS3.17.p7.1.m1.1.1.2.3">1</cn></apply><apply id="S6.SS3.17.p7.1.m1.1.1.3.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3"><plus id="S6.SS3.17.p7.1.m1.1.1.3.1.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.1"></plus><apply id="S6.SS3.17.p7.1.m1.1.1.3.2.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S6.SS3.17.p7.1.m1.1.1.3.2.1.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.2">subscript</csymbol><ci id="S6.SS3.17.p7.1.m1.1.1.3.2.2.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.2.2">𝑚</ci><ci id="S6.SS3.17.p7.1.m1.1.1.3.2.3.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.2.3">𝑛</ci></apply><apply id="S6.SS3.17.p7.1.m1.1.1.3.3.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.3"><divide id="S6.SS3.17.p7.1.m1.1.1.3.3.1.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.3"></divide><apply id="S6.SS3.17.p7.1.m1.1.1.3.3.2.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2"><plus id="S6.SS3.17.p7.1.m1.1.1.3.3.2.1.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2.1"></plus><ci id="S6.SS3.17.p7.1.m1.1.1.3.3.2.2.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.17.p7.1.m1.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.17.p7.1.m1.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.17.p7.1.m1.1.1.3.3.3.cmml" xref="S6.SS3.17.p7.1.m1.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.17.p7.1.m1.1c">s_{1}<m_{n}+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.17.p7.1.m1.1d">italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT < italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math>. Fix <math alttext="t\in(m_{n}+\frac{\gamma+1}{p},s]" class="ltx_Math" display="inline" id="S6.SS3.17.p7.2.m2.2"><semantics id="S6.SS3.17.p7.2.m2.2a"><mrow id="S6.SS3.17.p7.2.m2.2.2" xref="S6.SS3.17.p7.2.m2.2.2.cmml"><mi id="S6.SS3.17.p7.2.m2.2.2.3" xref="S6.SS3.17.p7.2.m2.2.2.3.cmml">t</mi><mo id="S6.SS3.17.p7.2.m2.2.2.2" xref="S6.SS3.17.p7.2.m2.2.2.2.cmml">∈</mo><mrow id="S6.SS3.17.p7.2.m2.2.2.1.1" xref="S6.SS3.17.p7.2.m2.2.2.1.2.cmml"><mo id="S6.SS3.17.p7.2.m2.2.2.1.1.2" stretchy="false" xref="S6.SS3.17.p7.2.m2.2.2.1.2.cmml">(</mo><mrow id="S6.SS3.17.p7.2.m2.2.2.1.1.1" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.cmml"><msub id="S6.SS3.17.p7.2.m2.2.2.1.1.1.2" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.cmml"><mi id="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.2" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.2.cmml">m</mi><mi id="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.3" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.3.cmml">n</mi></msub><mo id="S6.SS3.17.p7.2.m2.2.2.1.1.1.1" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.1.cmml">+</mo><mfrac id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.cmml"><mrow id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.cmml"><mi id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.2" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.2.cmml">γ</mi><mo id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.1" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.1.cmml">+</mo><mn id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.3" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.3" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.3.cmml">p</mi></mfrac></mrow><mo id="S6.SS3.17.p7.2.m2.2.2.1.1.3" xref="S6.SS3.17.p7.2.m2.2.2.1.2.cmml">,</mo><mi id="S6.SS3.17.p7.2.m2.1.1" xref="S6.SS3.17.p7.2.m2.1.1.cmml">s</mi><mo id="S6.SS3.17.p7.2.m2.2.2.1.1.4" stretchy="false" xref="S6.SS3.17.p7.2.m2.2.2.1.2.cmml">]</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.17.p7.2.m2.2b"><apply id="S6.SS3.17.p7.2.m2.2.2.cmml" xref="S6.SS3.17.p7.2.m2.2.2"><in id="S6.SS3.17.p7.2.m2.2.2.2.cmml" xref="S6.SS3.17.p7.2.m2.2.2.2"></in><ci id="S6.SS3.17.p7.2.m2.2.2.3.cmml" xref="S6.SS3.17.p7.2.m2.2.2.3">𝑡</ci><interval closure="open-closed" id="S6.SS3.17.p7.2.m2.2.2.1.2.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1"><apply id="S6.SS3.17.p7.2.m2.2.2.1.1.1.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1"><plus id="S6.SS3.17.p7.2.m2.2.2.1.1.1.1.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.1"></plus><apply id="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.1.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.2">subscript</csymbol><ci id="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.2.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.2">𝑚</ci><ci id="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.3.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.2.3">𝑛</ci></apply><apply id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3"><divide id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.1.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3"></divide><apply id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2"><plus id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.1.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.1"></plus><ci id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.2.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.2">𝛾</ci><cn id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.3.cmml" type="integer" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.2.3">1</cn></apply><ci id="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.3.cmml" xref="S6.SS3.17.p7.2.m2.2.2.1.1.1.3.3">𝑝</ci></apply></apply><ci id="S6.SS3.17.p7.2.m2.1.1.cmml" xref="S6.SS3.17.p7.2.m2.1.1">𝑠</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.17.p7.2.m2.2c">t\in(m_{n}+\frac{\gamma+1}{p},s]</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.17.p7.2.m2.2d">italic_t ∈ ( italic_m start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG , italic_s ]</annotation></semantics></math> and let <math alttext="\theta_{1}:=(s_{1}-s_{0})/(t-s_{0})" class="ltx_Math" display="inline" id="S6.SS3.17.p7.3.m3.2"><semantics id="S6.SS3.17.p7.3.m3.2a"><mrow id="S6.SS3.17.p7.3.m3.2.2" xref="S6.SS3.17.p7.3.m3.2.2.cmml"><msub id="S6.SS3.17.p7.3.m3.2.2.4" xref="S6.SS3.17.p7.3.m3.2.2.4.cmml"><mi id="S6.SS3.17.p7.3.m3.2.2.4.2" xref="S6.SS3.17.p7.3.m3.2.2.4.2.cmml">θ</mi><mn id="S6.SS3.17.p7.3.m3.2.2.4.3" xref="S6.SS3.17.p7.3.m3.2.2.4.3.cmml">1</mn></msub><mo id="S6.SS3.17.p7.3.m3.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.SS3.17.p7.3.m3.2.2.3.cmml">:=</mo><mrow id="S6.SS3.17.p7.3.m3.2.2.2" xref="S6.SS3.17.p7.3.m3.2.2.2.cmml"><mrow id="S6.SS3.17.p7.3.m3.1.1.1.1.1" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.cmml"><mo id="S6.SS3.17.p7.3.m3.1.1.1.1.1.2" stretchy="false" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.cmml"><msub id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.cmml"><mi id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.2" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.2.cmml">s</mi><mn id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.3" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.1" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.1.cmml">−</mo><msub id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.cmml"><mi id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.2" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.2.cmml">s</mi><mn id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.3" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.SS3.17.p7.3.m3.1.1.1.1.1.3" stretchy="false" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS3.17.p7.3.m3.2.2.2.3" xref="S6.SS3.17.p7.3.m3.2.2.2.3.cmml">/</mo><mrow id="S6.SS3.17.p7.3.m3.2.2.2.2.1" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.cmml"><mo id="S6.SS3.17.p7.3.m3.2.2.2.2.1.2" stretchy="false" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.cmml">(</mo><mrow id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.cmml"><mi id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.2" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.2.cmml">t</mi><mo id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.1" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.1.cmml">−</mo><msub id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.cmml"><mi id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.2" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.2.cmml">s</mi><mn id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.3" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.3.cmml">0</mn></msub></mrow><mo id="S6.SS3.17.p7.3.m3.2.2.2.2.1.3" stretchy="false" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.17.p7.3.m3.2b"><apply id="S6.SS3.17.p7.3.m3.2.2.cmml" xref="S6.SS3.17.p7.3.m3.2.2"><csymbol cd="latexml" id="S6.SS3.17.p7.3.m3.2.2.3.cmml" xref="S6.SS3.17.p7.3.m3.2.2.3">assign</csymbol><apply id="S6.SS3.17.p7.3.m3.2.2.4.cmml" xref="S6.SS3.17.p7.3.m3.2.2.4"><csymbol cd="ambiguous" id="S6.SS3.17.p7.3.m3.2.2.4.1.cmml" xref="S6.SS3.17.p7.3.m3.2.2.4">subscript</csymbol><ci id="S6.SS3.17.p7.3.m3.2.2.4.2.cmml" xref="S6.SS3.17.p7.3.m3.2.2.4.2">𝜃</ci><cn id="S6.SS3.17.p7.3.m3.2.2.4.3.cmml" type="integer" xref="S6.SS3.17.p7.3.m3.2.2.4.3">1</cn></apply><apply id="S6.SS3.17.p7.3.m3.2.2.2.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2"><divide id="S6.SS3.17.p7.3.m3.2.2.2.3.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2.3"></divide><apply id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1"><minus id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.1.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.1"></minus><apply id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.1.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.2.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.2">𝑠</ci><cn id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.2.3">1</cn></apply><apply id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.1.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3">subscript</csymbol><ci id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.2.cmml" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.2">𝑠</ci><cn id="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.3.cmml" type="integer" xref="S6.SS3.17.p7.3.m3.1.1.1.1.1.1.3.3">0</cn></apply></apply><apply id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1"><minus id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.1.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.1"></minus><ci id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.2.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.2">𝑡</ci><apply id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.1.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3">subscript</csymbol><ci id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.2.cmml" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.2">𝑠</ci><cn id="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.3.cmml" type="integer" xref="S6.SS3.17.p7.3.m3.2.2.2.2.1.1.3.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.17.p7.3.m3.2c">\theta_{1}:=(s_{1}-s_{0})/(t-s_{0})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.17.p7.3.m3.2d">italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ) / ( italic_t - italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT )</annotation></semantics></math>. Then again by reiteration and Step 2a, we obtain <math alttext="[H^{s_{0},p},H^{s_{1},p}_{\mathcal{B}}]_{\theta}=H^{s_{\theta},p}_{\mathcal{B}}" class="ltx_Math" display="inline" id="S6.SS3.17.p7.4.m4.8"><semantics id="S6.SS3.17.p7.4.m4.8a"><mrow id="S6.SS3.17.p7.4.m4.8.8" xref="S6.SS3.17.p7.4.m4.8.8.cmml"><msub id="S6.SS3.17.p7.4.m4.8.8.2" xref="S6.SS3.17.p7.4.m4.8.8.2.cmml"><mrow id="S6.SS3.17.p7.4.m4.8.8.2.2.2" xref="S6.SS3.17.p7.4.m4.8.8.2.2.3.cmml"><mo id="S6.SS3.17.p7.4.m4.8.8.2.2.2.3" stretchy="false" xref="S6.SS3.17.p7.4.m4.8.8.2.2.3.cmml">[</mo><msup id="S6.SS3.17.p7.4.m4.7.7.1.1.1.1" xref="S6.SS3.17.p7.4.m4.7.7.1.1.1.1.cmml"><mi id="S6.SS3.17.p7.4.m4.7.7.1.1.1.1.2" xref="S6.SS3.17.p7.4.m4.7.7.1.1.1.1.2.cmml">H</mi><mrow id="S6.SS3.17.p7.4.m4.2.2.2.2" xref="S6.SS3.17.p7.4.m4.2.2.2.3.cmml"><msub id="S6.SS3.17.p7.4.m4.2.2.2.2.1" xref="S6.SS3.17.p7.4.m4.2.2.2.2.1.cmml"><mi id="S6.SS3.17.p7.4.m4.2.2.2.2.1.2" xref="S6.SS3.17.p7.4.m4.2.2.2.2.1.2.cmml">s</mi><mn id="S6.SS3.17.p7.4.m4.2.2.2.2.1.3" xref="S6.SS3.17.p7.4.m4.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.SS3.17.p7.4.m4.2.2.2.2.2" xref="S6.SS3.17.p7.4.m4.2.2.2.3.cmml">,</mo><mi id="S6.SS3.17.p7.4.m4.1.1.1.1" xref="S6.SS3.17.p7.4.m4.1.1.1.1.cmml">p</mi></mrow></msup><mo id="S6.SS3.17.p7.4.m4.8.8.2.2.2.4" xref="S6.SS3.17.p7.4.m4.8.8.2.2.3.cmml">,</mo><msubsup id="S6.SS3.17.p7.4.m4.8.8.2.2.2.2" xref="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.cmml"><mi id="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.2.2" xref="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.3" xref="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.3.cmml">ℬ</mi><mrow id="S6.SS3.17.p7.4.m4.4.4.2.2" xref="S6.SS3.17.p7.4.m4.4.4.2.3.cmml"><msub id="S6.SS3.17.p7.4.m4.4.4.2.2.1" xref="S6.SS3.17.p7.4.m4.4.4.2.2.1.cmml"><mi id="S6.SS3.17.p7.4.m4.4.4.2.2.1.2" xref="S6.SS3.17.p7.4.m4.4.4.2.2.1.2.cmml">s</mi><mn id="S6.SS3.17.p7.4.m4.4.4.2.2.1.3" xref="S6.SS3.17.p7.4.m4.4.4.2.2.1.3.cmml">1</mn></msub><mo id="S6.SS3.17.p7.4.m4.4.4.2.2.2" xref="S6.SS3.17.p7.4.m4.4.4.2.3.cmml">,</mo><mi id="S6.SS3.17.p7.4.m4.3.3.1.1" xref="S6.SS3.17.p7.4.m4.3.3.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.SS3.17.p7.4.m4.8.8.2.2.2.5" stretchy="false" xref="S6.SS3.17.p7.4.m4.8.8.2.2.3.cmml">]</mo></mrow><mi id="S6.SS3.17.p7.4.m4.8.8.2.4" xref="S6.SS3.17.p7.4.m4.8.8.2.4.cmml">θ</mi></msub><mo id="S6.SS3.17.p7.4.m4.8.8.3" xref="S6.SS3.17.p7.4.m4.8.8.3.cmml">=</mo><msubsup id="S6.SS3.17.p7.4.m4.8.8.4" xref="S6.SS3.17.p7.4.m4.8.8.4.cmml"><mi id="S6.SS3.17.p7.4.m4.8.8.4.2.2" xref="S6.SS3.17.p7.4.m4.8.8.4.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.SS3.17.p7.4.m4.8.8.4.3" xref="S6.SS3.17.p7.4.m4.8.8.4.3.cmml">ℬ</mi><mrow id="S6.SS3.17.p7.4.m4.6.6.2.2" xref="S6.SS3.17.p7.4.m4.6.6.2.3.cmml"><msub id="S6.SS3.17.p7.4.m4.6.6.2.2.1" xref="S6.SS3.17.p7.4.m4.6.6.2.2.1.cmml"><mi id="S6.SS3.17.p7.4.m4.6.6.2.2.1.2" xref="S6.SS3.17.p7.4.m4.6.6.2.2.1.2.cmml">s</mi><mi id="S6.SS3.17.p7.4.m4.6.6.2.2.1.3" xref="S6.SS3.17.p7.4.m4.6.6.2.2.1.3.cmml">θ</mi></msub><mo id="S6.SS3.17.p7.4.m4.6.6.2.2.2" xref="S6.SS3.17.p7.4.m4.6.6.2.3.cmml">,</mo><mi id="S6.SS3.17.p7.4.m4.5.5.1.1" xref="S6.SS3.17.p7.4.m4.5.5.1.1.cmml">p</mi></mrow></msubsup></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.17.p7.4.m4.8b"><apply id="S6.SS3.17.p7.4.m4.8.8.cmml" xref="S6.SS3.17.p7.4.m4.8.8"><eq id="S6.SS3.17.p7.4.m4.8.8.3.cmml" xref="S6.SS3.17.p7.4.m4.8.8.3"></eq><apply id="S6.SS3.17.p7.4.m4.8.8.2.cmml" xref="S6.SS3.17.p7.4.m4.8.8.2"><csymbol cd="ambiguous" id="S6.SS3.17.p7.4.m4.8.8.2.3.cmml" xref="S6.SS3.17.p7.4.m4.8.8.2">subscript</csymbol><interval closure="closed" id="S6.SS3.17.p7.4.m4.8.8.2.2.3.cmml" xref="S6.SS3.17.p7.4.m4.8.8.2.2.2"><apply id="S6.SS3.17.p7.4.m4.7.7.1.1.1.1.cmml" xref="S6.SS3.17.p7.4.m4.7.7.1.1.1.1"><csymbol cd="ambiguous" id="S6.SS3.17.p7.4.m4.7.7.1.1.1.1.1.cmml" xref="S6.SS3.17.p7.4.m4.7.7.1.1.1.1">superscript</csymbol><ci id="S6.SS3.17.p7.4.m4.7.7.1.1.1.1.2.cmml" 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xref="S6.SS3.17.p7.4.m4.8.8.2.2.2.2">superscript</csymbol><ci id="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.2.2.cmml" xref="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.2.2">𝐻</ci><list id="S6.SS3.17.p7.4.m4.4.4.2.3.cmml" xref="S6.SS3.17.p7.4.m4.4.4.2.2"><apply id="S6.SS3.17.p7.4.m4.4.4.2.2.1.cmml" xref="S6.SS3.17.p7.4.m4.4.4.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.17.p7.4.m4.4.4.2.2.1.1.cmml" xref="S6.SS3.17.p7.4.m4.4.4.2.2.1">subscript</csymbol><ci id="S6.SS3.17.p7.4.m4.4.4.2.2.1.2.cmml" xref="S6.SS3.17.p7.4.m4.4.4.2.2.1.2">𝑠</ci><cn id="S6.SS3.17.p7.4.m4.4.4.2.2.1.3.cmml" type="integer" xref="S6.SS3.17.p7.4.m4.4.4.2.2.1.3">1</cn></apply><ci id="S6.SS3.17.p7.4.m4.3.3.1.1.cmml" xref="S6.SS3.17.p7.4.m4.3.3.1.1">𝑝</ci></list></apply><ci id="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.3.cmml" xref="S6.SS3.17.p7.4.m4.8.8.2.2.2.2.3">ℬ</ci></apply></interval><ci id="S6.SS3.17.p7.4.m4.8.8.2.4.cmml" xref="S6.SS3.17.p7.4.m4.8.8.2.4">𝜃</ci></apply><apply id="S6.SS3.17.p7.4.m4.8.8.4.cmml" xref="S6.SS3.17.p7.4.m4.8.8.4"><csymbol cd="ambiguous" id="S6.SS3.17.p7.4.m4.8.8.4.1.cmml" xref="S6.SS3.17.p7.4.m4.8.8.4">subscript</csymbol><apply id="S6.SS3.17.p7.4.m4.8.8.4.2.cmml" xref="S6.SS3.17.p7.4.m4.8.8.4"><csymbol cd="ambiguous" id="S6.SS3.17.p7.4.m4.8.8.4.2.1.cmml" xref="S6.SS3.17.p7.4.m4.8.8.4">superscript</csymbol><ci id="S6.SS3.17.p7.4.m4.8.8.4.2.2.cmml" xref="S6.SS3.17.p7.4.m4.8.8.4.2.2">𝐻</ci><list id="S6.SS3.17.p7.4.m4.6.6.2.3.cmml" xref="S6.SS3.17.p7.4.m4.6.6.2.2"><apply id="S6.SS3.17.p7.4.m4.6.6.2.2.1.cmml" xref="S6.SS3.17.p7.4.m4.6.6.2.2.1"><csymbol cd="ambiguous" id="S6.SS3.17.p7.4.m4.6.6.2.2.1.1.cmml" xref="S6.SS3.17.p7.4.m4.6.6.2.2.1">subscript</csymbol><ci id="S6.SS3.17.p7.4.m4.6.6.2.2.1.2.cmml" xref="S6.SS3.17.p7.4.m4.6.6.2.2.1.2">𝑠</ci><ci id="S6.SS3.17.p7.4.m4.6.6.2.2.1.3.cmml" xref="S6.SS3.17.p7.4.m4.6.6.2.2.1.3">𝜃</ci></apply><ci id="S6.SS3.17.p7.4.m4.5.5.1.1.cmml" xref="S6.SS3.17.p7.4.m4.5.5.1.1">𝑝</ci></list></apply><ci id="S6.SS3.17.p7.4.m4.8.8.4.3.cmml" xref="S6.SS3.17.p7.4.m4.8.8.4.3">ℬ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.17.p7.4.m4.8c">[H^{s_{0},p},H^{s_{1},p}_{\mathcal{B}}]_{\theta}=H^{s_{\theta},p}_{\mathcal{B}}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.17.p7.4.m4.8d">[ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT</annotation></semantics></math>. This completes the proof of (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E15" title="In Proof of Theorem 6.4. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.15</span></a>).</p> </div> <div class="ltx_para" id="S6.SS3.18.p8"> <p class="ltx_p" id="S6.SS3.18.p8.6"><span class="ltx_text ltx_font_italic" id="S6.SS3.18.p8.6.1">Step 3. </span>It remains to prove</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex81"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="H_{\mathcal{B}}^{s_{\theta},p}(\mathbb{R}^{d}_{+},w_{\gamma};X)=\big{[}H^{s_{0% },p}_{\mathcal{B}}(\mathbb{R}^{d}_{+},w_{\gamma};X),H_{\mathcal{B}}^{s_{1},p}(% \mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\theta}." class="ltx_Math" display="block" id="S6.Ex81.m1.10"><semantics id="S6.Ex81.m1.10a"><mrow id="S6.Ex81.m1.10.10.1" xref="S6.Ex81.m1.10.10.1.1.cmml"><mrow id="S6.Ex81.m1.10.10.1.1" xref="S6.Ex81.m1.10.10.1.1.cmml"><mrow id="S6.Ex81.m1.10.10.1.1.2" xref="S6.Ex81.m1.10.10.1.1.2.cmml"><msubsup id="S6.Ex81.m1.10.10.1.1.2.4" xref="S6.Ex81.m1.10.10.1.1.2.4.cmml"><mi id="S6.Ex81.m1.10.10.1.1.2.4.2.2" xref="S6.Ex81.m1.10.10.1.1.2.4.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex81.m1.10.10.1.1.2.4.2.3" xref="S6.Ex81.m1.10.10.1.1.2.4.2.3.cmml">ℬ</mi><mrow id="S6.Ex81.m1.2.2.2.2" xref="S6.Ex81.m1.2.2.2.3.cmml"><msub id="S6.Ex81.m1.2.2.2.2.1" xref="S6.Ex81.m1.2.2.2.2.1.cmml"><mi id="S6.Ex81.m1.2.2.2.2.1.2" xref="S6.Ex81.m1.2.2.2.2.1.2.cmml">s</mi><mi id="S6.Ex81.m1.2.2.2.2.1.3" xref="S6.Ex81.m1.2.2.2.2.1.3.cmml">θ</mi></msub><mo id="S6.Ex81.m1.2.2.2.2.2" xref="S6.Ex81.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex81.m1.1.1.1.1" xref="S6.Ex81.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex81.m1.10.10.1.1.2.3" xref="S6.Ex81.m1.10.10.1.1.2.3.cmml">⁢</mo><mrow id="S6.Ex81.m1.10.10.1.1.2.2.2" xref="S6.Ex81.m1.10.10.1.1.2.2.3.cmml"><mo id="S6.Ex81.m1.10.10.1.1.2.2.2.3" stretchy="false" 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\mathbb{R}^{d}_{+},w_{\gamma};X)\big{]}_{\theta}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex81.m1.10d">italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) = [ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) , italic_H start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT ( blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT start_POSTSUBSCRIPT + end_POSTSUBSCRIPT , italic_w start_POSTSUBSCRIPT italic_γ end_POSTSUBSCRIPT ; italic_X ) ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.18.p8.5">Fix <math alttext="\widetilde{s}<s_{0}" class="ltx_Math" display="inline" id="S6.SS3.18.p8.1.m1.1"><semantics id="S6.SS3.18.p8.1.m1.1a"><mrow id="S6.SS3.18.p8.1.m1.1.1" xref="S6.SS3.18.p8.1.m1.1.1.cmml"><mover accent="true" id="S6.SS3.18.p8.1.m1.1.1.2" xref="S6.SS3.18.p8.1.m1.1.1.2.cmml"><mi id="S6.SS3.18.p8.1.m1.1.1.2.2" xref="S6.SS3.18.p8.1.m1.1.1.2.2.cmml">s</mi><mo id="S6.SS3.18.p8.1.m1.1.1.2.1" xref="S6.SS3.18.p8.1.m1.1.1.2.1.cmml">~</mo></mover><mo id="S6.SS3.18.p8.1.m1.1.1.1" xref="S6.SS3.18.p8.1.m1.1.1.1.cmml"><</mo><msub id="S6.SS3.18.p8.1.m1.1.1.3" xref="S6.SS3.18.p8.1.m1.1.1.3.cmml"><mi id="S6.SS3.18.p8.1.m1.1.1.3.2" xref="S6.SS3.18.p8.1.m1.1.1.3.2.cmml">s</mi><mn id="S6.SS3.18.p8.1.m1.1.1.3.3" xref="S6.SS3.18.p8.1.m1.1.1.3.3.cmml">0</mn></msub></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.18.p8.1.m1.1b"><apply id="S6.SS3.18.p8.1.m1.1.1.cmml" xref="S6.SS3.18.p8.1.m1.1.1"><lt id="S6.SS3.18.p8.1.m1.1.1.1.cmml" xref="S6.SS3.18.p8.1.m1.1.1.1"></lt><apply id="S6.SS3.18.p8.1.m1.1.1.2.cmml" xref="S6.SS3.18.p8.1.m1.1.1.2"><ci id="S6.SS3.18.p8.1.m1.1.1.2.1.cmml" xref="S6.SS3.18.p8.1.m1.1.1.2.1">~</ci><ci id="S6.SS3.18.p8.1.m1.1.1.2.2.cmml" xref="S6.SS3.18.p8.1.m1.1.1.2.2">𝑠</ci></apply><apply id="S6.SS3.18.p8.1.m1.1.1.3.cmml" xref="S6.SS3.18.p8.1.m1.1.1.3"><csymbol cd="ambiguous" id="S6.SS3.18.p8.1.m1.1.1.3.1.cmml" xref="S6.SS3.18.p8.1.m1.1.1.3">subscript</csymbol><ci id="S6.SS3.18.p8.1.m1.1.1.3.2.cmml" xref="S6.SS3.18.p8.1.m1.1.1.3.2">𝑠</ci><cn id="S6.SS3.18.p8.1.m1.1.1.3.3.cmml" type="integer" xref="S6.SS3.18.p8.1.m1.1.1.3.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.18.p8.1.m1.1c">\widetilde{s}<s_{0}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.18.p8.1.m1.1d">over~ start_ARG italic_s end_ARG < italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT</annotation></semantics></math> with <math alttext="\widetilde{s}>0" class="ltx_Math" display="inline" id="S6.SS3.18.p8.2.m2.1"><semantics id="S6.SS3.18.p8.2.m2.1a"><mrow id="S6.SS3.18.p8.2.m2.1.1" xref="S6.SS3.18.p8.2.m2.1.1.cmml"><mover accent="true" id="S6.SS3.18.p8.2.m2.1.1.2" xref="S6.SS3.18.p8.2.m2.1.1.2.cmml"><mi id="S6.SS3.18.p8.2.m2.1.1.2.2" xref="S6.SS3.18.p8.2.m2.1.1.2.2.cmml">s</mi><mo id="S6.SS3.18.p8.2.m2.1.1.2.1" xref="S6.SS3.18.p8.2.m2.1.1.2.1.cmml">~</mo></mover><mo id="S6.SS3.18.p8.2.m2.1.1.1" xref="S6.SS3.18.p8.2.m2.1.1.1.cmml">></mo><mn id="S6.SS3.18.p8.2.m2.1.1.3" xref="S6.SS3.18.p8.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.18.p8.2.m2.1b"><apply id="S6.SS3.18.p8.2.m2.1.1.cmml" xref="S6.SS3.18.p8.2.m2.1.1"><gt id="S6.SS3.18.p8.2.m2.1.1.1.cmml" xref="S6.SS3.18.p8.2.m2.1.1.1"></gt><apply id="S6.SS3.18.p8.2.m2.1.1.2.cmml" xref="S6.SS3.18.p8.2.m2.1.1.2"><ci id="S6.SS3.18.p8.2.m2.1.1.2.1.cmml" xref="S6.SS3.18.p8.2.m2.1.1.2.1">~</ci><ci id="S6.SS3.18.p8.2.m2.1.1.2.2.cmml" xref="S6.SS3.18.p8.2.m2.1.1.2.2">𝑠</ci></apply><cn id="S6.SS3.18.p8.2.m2.1.1.3.cmml" type="integer" xref="S6.SS3.18.p8.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.18.p8.2.m2.1c">\widetilde{s}>0</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.18.p8.2.m2.1d">over~ start_ARG italic_s end_ARG > 0</annotation></semantics></math> if <math alttext="s_{0}>0" class="ltx_Math" display="inline" id="S6.SS3.18.p8.3.m3.1"><semantics id="S6.SS3.18.p8.3.m3.1a"><mrow id="S6.SS3.18.p8.3.m3.1.1" xref="S6.SS3.18.p8.3.m3.1.1.cmml"><msub id="S6.SS3.18.p8.3.m3.1.1.2" xref="S6.SS3.18.p8.3.m3.1.1.2.cmml"><mi id="S6.SS3.18.p8.3.m3.1.1.2.2" xref="S6.SS3.18.p8.3.m3.1.1.2.2.cmml">s</mi><mn id="S6.SS3.18.p8.3.m3.1.1.2.3" xref="S6.SS3.18.p8.3.m3.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.18.p8.3.m3.1.1.1" xref="S6.SS3.18.p8.3.m3.1.1.1.cmml">></mo><mn id="S6.SS3.18.p8.3.m3.1.1.3" xref="S6.SS3.18.p8.3.m3.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.18.p8.3.m3.1b"><apply id="S6.SS3.18.p8.3.m3.1.1.cmml" xref="S6.SS3.18.p8.3.m3.1.1"><gt id="S6.SS3.18.p8.3.m3.1.1.1.cmml" xref="S6.SS3.18.p8.3.m3.1.1.1"></gt><apply id="S6.SS3.18.p8.3.m3.1.1.2.cmml" xref="S6.SS3.18.p8.3.m3.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.18.p8.3.m3.1.1.2.1.cmml" xref="S6.SS3.18.p8.3.m3.1.1.2">subscript</csymbol><ci id="S6.SS3.18.p8.3.m3.1.1.2.2.cmml" xref="S6.SS3.18.p8.3.m3.1.1.2.2">𝑠</ci><cn id="S6.SS3.18.p8.3.m3.1.1.2.3.cmml" type="integer" xref="S6.SS3.18.p8.3.m3.1.1.2.3">0</cn></apply><cn id="S6.SS3.18.p8.3.m3.1.1.3.cmml" type="integer" xref="S6.SS3.18.p8.3.m3.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.18.p8.3.m3.1c">s_{0}>0</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.18.p8.3.m3.1d">italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT > 0</annotation></semantics></math> and <math alttext="\widetilde{s}>-1+\frac{\gamma+1}{p}" class="ltx_Math" display="inline" id="S6.SS3.18.p8.4.m4.1"><semantics id="S6.SS3.18.p8.4.m4.1a"><mrow id="S6.SS3.18.p8.4.m4.1.1" xref="S6.SS3.18.p8.4.m4.1.1.cmml"><mover accent="true" id="S6.SS3.18.p8.4.m4.1.1.2" xref="S6.SS3.18.p8.4.m4.1.1.2.cmml"><mi id="S6.SS3.18.p8.4.m4.1.1.2.2" xref="S6.SS3.18.p8.4.m4.1.1.2.2.cmml">s</mi><mo id="S6.SS3.18.p8.4.m4.1.1.2.1" xref="S6.SS3.18.p8.4.m4.1.1.2.1.cmml">~</mo></mover><mo id="S6.SS3.18.p8.4.m4.1.1.1" xref="S6.SS3.18.p8.4.m4.1.1.1.cmml">></mo><mrow id="S6.SS3.18.p8.4.m4.1.1.3" xref="S6.SS3.18.p8.4.m4.1.1.3.cmml"><mrow id="S6.SS3.18.p8.4.m4.1.1.3.2" xref="S6.SS3.18.p8.4.m4.1.1.3.2.cmml"><mo id="S6.SS3.18.p8.4.m4.1.1.3.2a" xref="S6.SS3.18.p8.4.m4.1.1.3.2.cmml">−</mo><mn id="S6.SS3.18.p8.4.m4.1.1.3.2.2" xref="S6.SS3.18.p8.4.m4.1.1.3.2.2.cmml">1</mn></mrow><mo id="S6.SS3.18.p8.4.m4.1.1.3.1" xref="S6.SS3.18.p8.4.m4.1.1.3.1.cmml">+</mo><mfrac id="S6.SS3.18.p8.4.m4.1.1.3.3" xref="S6.SS3.18.p8.4.m4.1.1.3.3.cmml"><mrow id="S6.SS3.18.p8.4.m4.1.1.3.3.2" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2.cmml"><mi id="S6.SS3.18.p8.4.m4.1.1.3.3.2.2" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2.2.cmml">γ</mi><mo id="S6.SS3.18.p8.4.m4.1.1.3.3.2.1" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2.1.cmml">+</mo><mn id="S6.SS3.18.p8.4.m4.1.1.3.3.2.3" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2.3.cmml">1</mn></mrow><mi id="S6.SS3.18.p8.4.m4.1.1.3.3.3" xref="S6.SS3.18.p8.4.m4.1.1.3.3.3.cmml">p</mi></mfrac></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.18.p8.4.m4.1b"><apply id="S6.SS3.18.p8.4.m4.1.1.cmml" xref="S6.SS3.18.p8.4.m4.1.1"><gt id="S6.SS3.18.p8.4.m4.1.1.1.cmml" xref="S6.SS3.18.p8.4.m4.1.1.1"></gt><apply id="S6.SS3.18.p8.4.m4.1.1.2.cmml" xref="S6.SS3.18.p8.4.m4.1.1.2"><ci id="S6.SS3.18.p8.4.m4.1.1.2.1.cmml" xref="S6.SS3.18.p8.4.m4.1.1.2.1">~</ci><ci id="S6.SS3.18.p8.4.m4.1.1.2.2.cmml" xref="S6.SS3.18.p8.4.m4.1.1.2.2">𝑠</ci></apply><apply id="S6.SS3.18.p8.4.m4.1.1.3.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3"><plus id="S6.SS3.18.p8.4.m4.1.1.3.1.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.1"></plus><apply id="S6.SS3.18.p8.4.m4.1.1.3.2.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.2"><minus id="S6.SS3.18.p8.4.m4.1.1.3.2.1.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.2"></minus><cn id="S6.SS3.18.p8.4.m4.1.1.3.2.2.cmml" type="integer" xref="S6.SS3.18.p8.4.m4.1.1.3.2.2">1</cn></apply><apply id="S6.SS3.18.p8.4.m4.1.1.3.3.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.3"><divide id="S6.SS3.18.p8.4.m4.1.1.3.3.1.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.3"></divide><apply id="S6.SS3.18.p8.4.m4.1.1.3.3.2.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2"><plus id="S6.SS3.18.p8.4.m4.1.1.3.3.2.1.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2.1"></plus><ci id="S6.SS3.18.p8.4.m4.1.1.3.3.2.2.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2.2">𝛾</ci><cn id="S6.SS3.18.p8.4.m4.1.1.3.3.2.3.cmml" type="integer" xref="S6.SS3.18.p8.4.m4.1.1.3.3.2.3">1</cn></apply><ci id="S6.SS3.18.p8.4.m4.1.1.3.3.3.cmml" xref="S6.SS3.18.p8.4.m4.1.1.3.3.3">𝑝</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.18.p8.4.m4.1c">\widetilde{s}>-1+\frac{\gamma+1}{p}</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.18.p8.4.m4.1d">over~ start_ARG italic_s end_ARG > - 1 + divide start_ARG italic_γ + 1 end_ARG start_ARG italic_p end_ARG</annotation></semantics></math> otherwise. Let <math alttext="\theta_{1}:=(s_{0}-\widetilde{s})/(s_{1}-\widetilde{s})" class="ltx_Math" display="inline" id="S6.SS3.18.p8.5.m5.2"><semantics id="S6.SS3.18.p8.5.m5.2a"><mrow id="S6.SS3.18.p8.5.m5.2.2" xref="S6.SS3.18.p8.5.m5.2.2.cmml"><msub id="S6.SS3.18.p8.5.m5.2.2.4" xref="S6.SS3.18.p8.5.m5.2.2.4.cmml"><mi id="S6.SS3.18.p8.5.m5.2.2.4.2" xref="S6.SS3.18.p8.5.m5.2.2.4.2.cmml">θ</mi><mn id="S6.SS3.18.p8.5.m5.2.2.4.3" xref="S6.SS3.18.p8.5.m5.2.2.4.3.cmml">1</mn></msub><mo id="S6.SS3.18.p8.5.m5.2.2.3" lspace="0.278em" rspace="0.278em" xref="S6.SS3.18.p8.5.m5.2.2.3.cmml">:=</mo><mrow id="S6.SS3.18.p8.5.m5.2.2.2" xref="S6.SS3.18.p8.5.m5.2.2.2.cmml"><mrow id="S6.SS3.18.p8.5.m5.1.1.1.1.1" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.cmml"><mo id="S6.SS3.18.p8.5.m5.1.1.1.1.1.2" stretchy="false" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.cmml">(</mo><mrow id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.cmml"><msub id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.cmml"><mi id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.2" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.2.cmml">s</mi><mn id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.3" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.3.cmml">0</mn></msub><mo id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.1" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.1.cmml">−</mo><mover accent="true" id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.cmml"><mi id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.2" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.2.cmml">s</mi><mo id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.1" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.1.cmml">~</mo></mover></mrow><mo id="S6.SS3.18.p8.5.m5.1.1.1.1.1.3" stretchy="false" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.cmml">)</mo></mrow><mo id="S6.SS3.18.p8.5.m5.2.2.2.3" xref="S6.SS3.18.p8.5.m5.2.2.2.3.cmml">/</mo><mrow id="S6.SS3.18.p8.5.m5.2.2.2.2.1" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.cmml"><mo id="S6.SS3.18.p8.5.m5.2.2.2.2.1.2" stretchy="false" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.cmml">(</mo><mrow id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.cmml"><msub id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.cmml"><mi id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.2" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.2.cmml">s</mi><mn id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.3" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.3.cmml">1</mn></msub><mo id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.1" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.1.cmml">−</mo><mover accent="true" id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.cmml"><mi id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.2" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.2.cmml">s</mi><mo id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.1" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.1.cmml">~</mo></mover></mrow><mo id="S6.SS3.18.p8.5.m5.2.2.2.2.1.3" stretchy="false" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S6.SS3.18.p8.5.m5.2b"><apply id="S6.SS3.18.p8.5.m5.2.2.cmml" xref="S6.SS3.18.p8.5.m5.2.2"><csymbol cd="latexml" id="S6.SS3.18.p8.5.m5.2.2.3.cmml" xref="S6.SS3.18.p8.5.m5.2.2.3">assign</csymbol><apply id="S6.SS3.18.p8.5.m5.2.2.4.cmml" xref="S6.SS3.18.p8.5.m5.2.2.4"><csymbol cd="ambiguous" id="S6.SS3.18.p8.5.m5.2.2.4.1.cmml" xref="S6.SS3.18.p8.5.m5.2.2.4">subscript</csymbol><ci id="S6.SS3.18.p8.5.m5.2.2.4.2.cmml" xref="S6.SS3.18.p8.5.m5.2.2.4.2">𝜃</ci><cn id="S6.SS3.18.p8.5.m5.2.2.4.3.cmml" type="integer" xref="S6.SS3.18.p8.5.m5.2.2.4.3">1</cn></apply><apply id="S6.SS3.18.p8.5.m5.2.2.2.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2"><divide id="S6.SS3.18.p8.5.m5.2.2.2.3.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.3"></divide><apply id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1"><minus id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.1.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.1"></minus><apply id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.1.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2">subscript</csymbol><ci id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.2.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.2">𝑠</ci><cn id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.3.cmml" type="integer" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.2.3">0</cn></apply><apply id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3"><ci id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.1.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.1">~</ci><ci id="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.2.cmml" xref="S6.SS3.18.p8.5.m5.1.1.1.1.1.1.3.2">𝑠</ci></apply></apply><apply id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1"><minus id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.1.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.1"></minus><apply id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2"><csymbol cd="ambiguous" id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.1.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2">subscript</csymbol><ci id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.2.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.2">𝑠</ci><cn id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.3.cmml" type="integer" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.2.3">1</cn></apply><apply id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3"><ci id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.1.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.1">~</ci><ci id="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.2.cmml" xref="S6.SS3.18.p8.5.m5.2.2.2.2.1.1.3.2">𝑠</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS3.18.p8.5.m5.2c">\theta_{1}:=(s_{0}-\widetilde{s})/(s_{1}-\widetilde{s})</annotation><annotation encoding="application/x-llamapun" id="S6.SS3.18.p8.5.m5.2d">italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT := ( italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - over~ start_ARG italic_s end_ARG ) / ( italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - over~ start_ARG italic_s end_ARG )</annotation></semantics></math>. Then by reiteration and (<a class="ltx_ref" href="https://arxiv.org/html/2503.14636v1#S6.E15" title="In Proof of Theorem 6.4. ‣ 6.3. The proofs of Theorems 6.4 and 6.5 ‣ 6. Complex interpolation ‣ Complex interpolation of weighted Sobolev spaces with boundary conditions"><span class="ltx_text ltx_ref_tag">6.15</span></a>) twice, we obtain</p> <table class="ltx_equation ltx_eqn_table" id="S6.Ex82"> <tbody><tr class="ltx_equation ltx_eqn_row ltx_align_baseline"> <td class="ltx_eqn_cell ltx_eqn_center_padleft"></td> <td class="ltx_eqn_cell ltx_align_center"><math alttext="\big{[}H^{s_{0},p}_{\mathcal{B}},H^{s_{1},p}_{\mathcal{B}}\big{]}_{\theta}=% \big{[}[H^{\widetilde{s},p},H^{s_{1},p}_{\mathcal{B}}]_{\theta_{1}},H^{s_{1},p% }_{\mathcal{B}}\big{]}_{\theta}=\big{[}H^{\widetilde{s},p},H^{s_{1},p}_{% \mathcal{B}}\big{]}_{(1-\theta)\theta_{1}+\theta}=H^{s_{\theta},p}_{\mathcal{B% }}." class="ltx_Math" display="block" id="S6.Ex82.m1.18"><semantics id="S6.Ex82.m1.18a"><mrow id="S6.Ex82.m1.18.18.1" xref="S6.Ex82.m1.18.18.1.1.cmml"><mrow id="S6.Ex82.m1.18.18.1.1" xref="S6.Ex82.m1.18.18.1.1.cmml"><msub id="S6.Ex82.m1.18.18.1.1.2" xref="S6.Ex82.m1.18.18.1.1.2.cmml"><mrow id="S6.Ex82.m1.18.18.1.1.2.2.2" xref="S6.Ex82.m1.18.18.1.1.2.2.3.cmml"><mo id="S6.Ex82.m1.18.18.1.1.2.2.2.3" maxsize="120%" minsize="120%" xref="S6.Ex82.m1.18.18.1.1.2.2.3.cmml">[</mo><msubsup id="S6.Ex82.m1.18.18.1.1.1.1.1.1" xref="S6.Ex82.m1.18.18.1.1.1.1.1.1.cmml"><mi id="S6.Ex82.m1.18.18.1.1.1.1.1.1.2.2" xref="S6.Ex82.m1.18.18.1.1.1.1.1.1.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex82.m1.18.18.1.1.1.1.1.1.3" xref="S6.Ex82.m1.18.18.1.1.1.1.1.1.3.cmml">ℬ</mi><mrow id="S6.Ex82.m1.2.2.2.2" xref="S6.Ex82.m1.2.2.2.3.cmml"><msub id="S6.Ex82.m1.2.2.2.2.1" xref="S6.Ex82.m1.2.2.2.2.1.cmml"><mi id="S6.Ex82.m1.2.2.2.2.1.2" xref="S6.Ex82.m1.2.2.2.2.1.2.cmml">s</mi><mn id="S6.Ex82.m1.2.2.2.2.1.3" xref="S6.Ex82.m1.2.2.2.2.1.3.cmml">0</mn></msub><mo id="S6.Ex82.m1.2.2.2.2.2" xref="S6.Ex82.m1.2.2.2.3.cmml">,</mo><mi id="S6.Ex82.m1.1.1.1.1" xref="S6.Ex82.m1.1.1.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex82.m1.18.18.1.1.2.2.2.4" xref="S6.Ex82.m1.18.18.1.1.2.2.3.cmml">,</mo><msubsup id="S6.Ex82.m1.18.18.1.1.2.2.2.2" xref="S6.Ex82.m1.18.18.1.1.2.2.2.2.cmml"><mi id="S6.Ex82.m1.18.18.1.1.2.2.2.2.2.2" xref="S6.Ex82.m1.18.18.1.1.2.2.2.2.2.2.cmml">H</mi><mi class="ltx_font_mathcaligraphic" id="S6.Ex82.m1.18.18.1.1.2.2.2.2.3" xref="S6.Ex82.m1.18.18.1.1.2.2.2.2.3.cmml">ℬ</mi><mrow id="S6.Ex82.m1.4.4.2.2" xref="S6.Ex82.m1.4.4.2.3.cmml"><msub id="S6.Ex82.m1.4.4.2.2.1" xref="S6.Ex82.m1.4.4.2.2.1.cmml"><mi id="S6.Ex82.m1.4.4.2.2.1.2" xref="S6.Ex82.m1.4.4.2.2.1.2.cmml">s</mi><mn id="S6.Ex82.m1.4.4.2.2.1.3" xref="S6.Ex82.m1.4.4.2.2.1.3.cmml">1</mn></msub><mo id="S6.Ex82.m1.4.4.2.2.2" xref="S6.Ex82.m1.4.4.2.3.cmml">,</mo><mi id="S6.Ex82.m1.3.3.1.1" xref="S6.Ex82.m1.3.3.1.1.cmml">p</mi></mrow></msubsup><mo id="S6.Ex82.m1.18.18.1.1.2.2.2.5" maxsize="120%" minsize="120%" xref="S6.Ex82.m1.18.18.1.1.2.2.3.cmml">]</mo></mrow><mi id="S6.Ex82.m1.18.18.1.1.2.4" xref="S6.Ex82.m1.18.18.1.1.2.4.cmml">θ</mi></msub><mo 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\mathcal{B}}\big{]}_{(1-\theta)\theta_{1}+\theta}=H^{s_{\theta},p}_{\mathcal{B% }}.</annotation><annotation encoding="application/x-llamapun" id="S6.Ex82.m1.18d">[ italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = [ [ italic_H start_POSTSUPERSCRIPT over~ start_ARG italic_s end_ARG , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT = [ italic_H start_POSTSUPERSCRIPT over~ start_ARG italic_s end_ARG , italic_p end_POSTSUPERSCRIPT , italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT ] start_POSTSUBSCRIPT ( 1 - italic_θ ) italic_θ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_θ end_POSTSUBSCRIPT = italic_H start_POSTSUPERSCRIPT italic_s start_POSTSUBSCRIPT italic_θ end_POSTSUBSCRIPT , italic_p end_POSTSUPERSCRIPT start_POSTSUBSCRIPT caligraphic_B end_POSTSUBSCRIPT .</annotation></semantics></math></td> <td class="ltx_eqn_cell ltx_eqn_center_padright"></td> </tr></tbody> </table> <p class="ltx_p" id="S6.SS3.18.p8.7">This completes the proof of the theorem. ∎</p> </div> </div> </section> </section> <section class="ltx_bibliography" id="bib"> <h2 class="ltx_title 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