CINXE.COM

<!DOCTYPE html> <html lang="en" class="no-js"> <head> <meta charset="UTF-8"> <meta http-equiv="X-UA-Compatible" content="IE=edge"> <meta name="applicable-device" content="pc,mobile"> <meta name="viewport" content="width=device-width, initial-scale=1"> <meta name="robots" content="max-image-preview:large"> <meta name="access" content="Yes"> <meta name="360-site-verification" content="1268d79b5e96aecf3ff2a7dac04ad990" /> <title>Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator | Annali di Matematica Pura ed Applicata (1923 -) </title> <meta name="twitter:site" content="@SpringerLink"/> <meta name="twitter:card" content="summary_large_image"/> <meta name="twitter:image:alt" content="Content cover image"/> <meta name="twitter:title" content="Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator"/> <meta name="twitter:description" content="Annali di Matematica Pura ed Applicata (1923 -) - In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega..."/> <meta name="twitter:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/10231"/> <meta name="journal_id" content="10231"/> <meta name="dc.title" content="Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator"/> <meta name="dc.source" content="Annali di Matematica Pura ed Applicata (1923 -) 2023 202:5"/> <meta name="dc.format" content="text/html"/> <meta name="dc.publisher" content="Springer"/> <meta name="dc.date" content="2023-03-27"/> <meta name="dc.type" content="OriginalPaper"/> <meta name="dc.language" content="En"/> <meta name="dc.copyright" content="2023 The Author(s)"/> <meta name="dc.rights" content="2023 The Author(s)"/> <meta name="dc.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="dc.description" content="In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ and $$\text {id}_\tau : {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ , where $$\Omega \subset {{{\mathbb {R}}}^d}$$ is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $$\text {id}_\tau $$ . This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov’s linear, bounded universal extension operator for these spaces."/> <meta name="prism.issn" content="1618-1891"/> <meta name="prism.publicationName" content="Annali di Matematica Pura ed Applicata (1923 -)"/> <meta name="prism.publicationDate" content="2023-03-27"/> <meta name="prism.volume" content="202"/> <meta name="prism.number" content="5"/> <meta name="prism.section" content="OriginalPaper"/> <meta name="prism.startingPage" content="2481"/> <meta name="prism.endingPage" content="2516"/> <meta name="prism.copyright" content="2023 The Author(s)"/> <meta name="prism.rightsAgent" content="journalpermissions@springernature.com"/> <meta name="prism.url" content="https://link.springer.com/article/10.1007/s10231-023-01327-w"/> <meta name="prism.doi" content="doi:10.1007/s10231-023-01327-w"/> <meta name="citation_pdf_url" content="https://link.springer.com/content/pdf/10.1007/s10231-023-01327-w.pdf"/> <meta name="citation_fulltext_html_url" content="https://link.springer.com/article/10.1007/s10231-023-01327-w"/> <meta name="citation_journal_title" content="Annali di Matematica Pura ed Applicata (1923 -)"/> <meta name="citation_journal_abbrev" content="Annali di Matematica"/> <meta name="citation_publisher" content="Springer Berlin Heidelberg"/> <meta name="citation_issn" content="1618-1891"/> <meta name="citation_title" content="Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator"/> <meta name="citation_volume" content="202"/> <meta name="citation_issue" content="5"/> <meta name="citation_publication_date" content="2023/10"/> <meta name="citation_online_date" content="2023/03/27"/> <meta name="citation_firstpage" content="2481"/> <meta name="citation_lastpage" content="2516"/> <meta name="citation_article_type" content="Article"/> <meta name="citation_fulltext_world_readable" content=""/> <meta name="citation_language" content="en"/> <meta name="dc.identifier" content="doi:10.1007/s10231-023-01327-w"/> <meta name="DOI" content="10.1007/s10231-023-01327-w"/> <meta name="size" content="2446886"/> <meta name="citation_doi" content="10.1007/s10231-023-01327-w"/> <meta name="citation_springer_api_url" content="http://api.springer.com/xmldata/jats?q=doi:10.1007/s10231-023-01327-w&api_key="/> <meta name="description" content="In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookr"/> <meta name="dc.creator" content="Gonçalves, Helena F."/> <meta name="dc.creator" content="Haroske, Dorothee D."/> <meta name="dc.creator" content="Skrzypczak, Leszek"/> <meta name="dc.subject" content="Mathematics, general"/> <meta name="citation_reference" content="citation_journal_title=Studia Math.; citation_title=A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces; citation_author=H-Q Bui, M Paluszyński, MH Taibleson; citation_volume=119; citation_publication_date=1996; citation_pages=219-246; citation_id=CR1"/> <meta name="citation_reference" content="citation_title=Function Spaces, Entropy Numbers, Differential Operators; citation_publication_date=1996; citation_id=CR2; citation_author=DE Edmunds; citation_author=H Triebel; citation_publisher=Cambridge Univ. Press"/> <meta name="citation_reference" content="citation_journal_title=Electron. J. Differential Equat.; citation_title=An embedding theorem for Campanato spaces; citation_author=A Baraka; citation_volume=66; citation_publication_date=2002; citation_pages=1-17; citation_id=CR3"/> <meta name="citation_reference" content="El Baraka, A.: Function spaces of BMO and Campanato type, pp.109-115, in: Proc. of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. 9, Southwest Texas State Univ., San Marcos, TX (2002)"/> <meta name="citation_reference" content="citation_journal_title=J. Funct. Spaces Appl.; citation_title=Littlewood-Paley characterization for Campanato spaces; citation_author=A Baraka; citation_volume=4; citation_publication_date=2006; citation_pages=193-220; citation_id=CR5"/> <meta name="citation_reference" content="citation_journal_title=Banach J. Math. Anal; citation_title=Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces; citation_author=H F Gonçalves; citation_volume=15; citation_issue=31; citation_publication_date=2021; citation_pages=50; citation_id=CR6"/> <meta name="citation_reference" content="citation_journal_title=Math. Nachr.; citation_title=Characterization of Triebel-Lizorkin-type spaces with variable exponents via maximal functions, local means and non-smooth atomic decompositions; citation_author=HF Gonçalves, SD Moura; citation_volume=291; citation_publication_date=2018; citation_pages=2024-2044; citation_id=CR7"/> <meta name="citation_reference" content="citation_journal_title=Rev. Mat. Complut.; citation_title=Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains; citation_author=HF Gonçalves, DD Haroske, L Skrzypczak; citation_volume=34; citation_publication_date=2021; citation_pages=761-795; citation_id=CR8"/> <meta name="citation_reference" content="citation_journal_title=Nonlinear Anal Series A: Theory, Methods Appl; citation_title=Smoothness Morrey Spaces of regular distributions, and some unboundedness properties; citation_author=DD Haroske, SD Moura, L Skrzypczak; citation_volume=139; citation_publication_date=2016; citation_pages=218-244; citation_id=CR9"/> <meta name="citation_reference" content="citation_journal_title=Acta Math. Sin. (Engl. Ser.); citation_title=Continuous embeddings of Besov-Morrey function spaces; citation_author=D. D Haroske, L Skrzypczak; citation_volume=28; citation_publication_date=2012; citation_pages=1307-1328; citation_id=CR10"/> <meta name="citation_reference" content="citation_journal_title=Studia Math.; citation_title=Embeddings of Besov-Morrey spaces on bounded domains; citation_author=DD Haroske, L Skrzypczak; citation_volume=218; citation_publication_date=2013; citation_pages=119-144; citation_id=CR11"/> <meta name="citation_reference" content="citation_journal_title=Rev. Mat. Complut.; citation_title=On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces; citation_author=DD Haroske, L Skrzypczak; citation_volume=27; citation_publication_date=2014; citation_pages=541-573; citation_id=CR12"/> <meta name="citation_reference" content="Haroske, D. D., Skrzypczak, L.: Some quantitative result on compact embeddings in smoothness Morrey spaces on bounded domains; an approach via interpolation pp.181–191, in: Function Spaces XII, Banach Center Publ, vol. 119, Polish Acad. Sci., Warsaw (2019)"/> <meta name="citation_reference" content="Haroske, D. D., Skrzypczak, L.: Entropy numbers of compact embeddings of smoothness Morrey spaces on bounded domains. J. Approx. Theory 256, 24 pp. (2020)"/> <meta name="citation_reference" content="citation_journal_title=Comm. Partial Different. Equat.; citation_title=Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data; citation_author=H Kozono, M Yamazaki; citation_volume=19; citation_publication_date=1994; citation_pages=959-1014; citation_id=CR15"/> <meta name="citation_reference" content="citation_journal_title=Dissertationes Math. (Rozprawy Mat.); citation_title=A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces; citation_author=Y Liang, D Yang, W Yuan, Y Sawano, T Ullrich; citation_volume=489; citation_publication_date=2013; citation_pages=1-114; citation_id=CR16"/> <meta name="citation_reference" content="citation_journal_title=Trans. Amer. Math. Soc.; citation_title=Besov-Morrey spaces: function space theory and applications to non-linear PDE; citation_author=AL Mazzucato; citation_volume=355; citation_publication_date=2003; citation_pages=1297-1364; citation_id=CR17"/> <meta name="citation_reference" content="citation_title=Wavelets and Operators; citation_publication_date=1992; citation_id=CR18; citation_author=Y Meyer; citation_publisher=Cambridge Univ. Press"/> <meta name="citation_reference" content="citation_journal_title=Trans. Amer. Math. Soc.; citation_title=On the solutions of quasi-linear elliptic partial differential equations; citation_author=CB Morrey; citation_volume=43; citation_publication_date=1938; citation_pages=126-166; citation_id=CR19"/> <meta name="citation_reference" content="citation_journal_title=Nonlinear Anal.; citation_title=Traces and extensions of generalized smoothness Morrey spaces on domains; citation_author=SD Moura, JS Neves, C Schneider; citation_volume=181; citation_publication_date=2019; citation_pages=311-339; citation_id=CR20"/> <meta name="citation_reference" content="citation_journal_title=J. Funct. Anal.; citation_title=On the theory of spaces; citation_author=J Peetre; citation_volume=4; citation_publication_date=1969; citation_pages=71-87; citation_id=CR21"/> <meta name="citation_reference" content="citation_journal_title=Math. Nachr.; citation_title=Local means, wavelet bases, representations, and isomorphisms in Besov-Morrey and Triebel-Lizorkin-Morrey spaces; citation_author=M Rosenthal; citation_volume=286; citation_publication_date=2013; citation_pages=59-87; citation_id=CR22"/> <meta name="citation_reference" content="citation_journal_title=J. London Math. Soc.; citation_title=On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains; citation_author=VS Rychkov; citation_volume=60; citation_publication_date=1999; citation_pages=237-257; citation_id=CR23"/> <meta name="citation_reference" content="citation_journal_title=Funct. Approx. Comment. Math.; citation_title=Wavelet characterizations of Besov-Morrey and Triebel-Lizorkin-Morrey spaces; citation_author=Y Sawano; citation_volume=38; citation_publication_date=2008; citation_pages=93-107; citation_id=CR24"/> <meta name="citation_reference" content="citation_journal_title=Sin. (Engl. Ser.); citation_title=A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Acta Math; citation_author=Y Sawano; citation_volume=25; citation_publication_date=2009; citation_pages=1223-1242; citation_id=CR25"/> <meta name="citation_reference" content="Sawano, Y., Di Fazio, G., Hakim, D. I.: Morrey spaces. Introduction and Applications to Integral Operators and PDE’s. Vol. I, Monographs and Research Notes in Mathematics. Chapman & Hall CRC Press, Boca Raton, FL (2020)"/> <meta name="citation_reference" content="Sawano, Y., Di Fazio, G., Hakim, D. I.: Morrey spaces. Introduction and Applications to Integral Operators and PDE’s. Vol. II, Monographs and Research Notes in Mathematics. Chapman & Hall CRC Press, Boca Raton, FL (2020)"/> <meta name="citation_reference" content="citation_journal_title=Math. Z.; citation_title=Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces; citation_author=Y Sawano, H Tanaka; citation_volume=257; citation_publication_date=2007; citation_pages=871-905; citation_id=CR28"/> <meta name="citation_reference" content="citation_journal_title=Math. Nachr.; citation_title=Sawano, Y., Tanaka. H; citation_author=; citation_volume=282; citation_publication_date=2009; citation_pages=1788-1810; citation_id=CR29"/> <meta name="citation_reference" content="citation_journal_title=Math. Nachr.; citation_title=Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces on domains; citation_author=Y Sawano; citation_volume=283; citation_publication_date=2010; citation_pages=1456-1487; citation_id=CR30"/> <meta name="citation_reference" content="citation_journal_title=Eurasian Math. J.; citation_title=Smoothness spaces related to Morrey spaces - a survey. I; citation_author=W Sickel; citation_volume=3; citation_publication_date=2012; citation_pages=110-149; citation_id=CR31"/> <meta name="citation_reference" content="citation_journal_title=Eurasian Math. J.; citation_title=Smoothness spaces related to Morrey spaces - a survey. II; citation_author=W Sickel; citation_volume=4; citation_publication_date=2013; citation_pages=82-124; citation_id=CR32"/> <meta name="citation_reference" content="citation_journal_title=Math. Nachr.; citation_title=Some properties of Morrey type Besov-Triebel spaces; citation_author=L Tang, J Xu; citation_volume=278; citation_publication_date=2005; citation_pages=904-917; citation_id=CR33"/> <meta name="citation_reference" content="citation_title=Theory of Function Spaces; citation_publication_date=1983; citation_id=CR34; citation_author=H Triebel; citation_publisher=Birkhäuser"/> <meta name="citation_reference" content="citation_title=Theory of Function Spaces; citation_publication_date=1992; citation_id=CR35; citation_author=H Triebel; citation_publisher=II"/> <meta name="citation_reference" content="citation_title=Theory of Function Spaces; citation_publication_date=2006; citation_id=CR36; citation_author=H Triebel; citation_publisher=III"/> <meta name="citation_reference" content="citation_title=Local Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics 20, European Mathematical Society; citation_publication_date=2013; citation_id=CR37; citation_author=H Triebel; citation_publisher=EMS)"/> <meta name="citation_reference" content="citation_title=Hybrid Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics 24, European Mathematical Society; citation_publication_date=2015; citation_id=CR38; citation_author=H Triebel; citation_publisher=EMS)"/> <meta name="citation_reference" content="citation_title=Theory of Function Spaces; citation_publication_date=2020; citation_id=CR39; citation_author=H Triebel; citation_publisher=IV"/> <meta name="citation_reference" content="citation_title=A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts 37; citation_publication_date=1997; citation_id=CR40; citation_author=P Wojtaszczyk; citation_publisher=Cambridge Univ. Press"/> <meta name="citation_reference" content="citation_journal_title=J. Funct. Anal.; citation_title=A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces; citation_author=D Yang, W Yuan; citation_volume=255; citation_publication_date=2008; citation_pages=2760-2809; citation_id=CR41"/> <meta name="citation_reference" content="citation_journal_title=Math. Z.; citation_title=New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces; citation_author=D Yang, W Yuan; citation_volume=265; citation_publication_date=2010; citation_pages=451-480; citation_id=CR42"/> <meta name="citation_reference" content="citation_journal_title=Nonlinear Anal.; citation_title=Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means; citation_author=D Yang, W Yuan; citation_volume=73; citation_publication_date=2010; citation_pages=3805-3820; citation_id=CR43"/> <meta name="citation_reference" content="citation_journal_title=Appl. Anal.; citation_title=Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces; citation_author=D Yang, W Yuan; citation_volume=92; citation_publication_date=2013; citation_pages=549-561; citation_id=CR44"/> <meta name="citation_reference" content="citation_journal_title=J. Approx. Theory; citation_title=Limiting embeddings in smoothness Morrey spaces, continuity envelopes and applications; citation_author=W Yuan, DD Haroske, SD Moura, L Skrzypczak, D Yang; citation_volume=192; citation_publication_date=2015; citation_pages=306-335; citation_id=CR45"/> <meta name="citation_reference" content="citation_journal_title=Appl. Anal.; citation_title=Embedding properties of Besov-type spaces; citation_author=W Yuan, DD Haroske, L Skrzypczak, D Yang; citation_volume=94; citation_publication_date=2015; citation_pages=318-340; citation_id=CR46"/> <meta name="citation_reference" content="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)"/> <meta name="citation_reference" content="citation_journal_title=Sci. China Math.; citation_title=Interpolation of Morrey-Campanato and related smoothness spaces; citation_author=W Yuan, W Sickel, D Yang; citation_volume=58; citation_publication_date=2015; citation_pages=1835-1908; citation_id=CR48"/> <meta name="citation_reference" content="citation_journal_title=Banach J. Math. Anal.; citation_title=Triebel-Lizorkin type spaces with variable exponent; citation_author=D Yang, C Zhuo, W Yuan; citation_volume=9; citation_publication_date=2015; citation_pages=146-202; citation_id=CR49"/> <meta name="citation_reference" content="citation_journal_title=J. Funct. Anal.; citation_title=Besov-type spaces with variable smoothness and integrability; citation_author=D Yang, C Zhuo, W Yuan; citation_volume=269; citation_publication_date=2015; citation_pages=1840-1898; citation_id=CR50"/> <meta name="citation_reference" content="citation_journal_title=Z. Anal. Anwend.; citation_title=Complex interpolation of Besov-type spaces on domains; citation_author=C Zhuo; citation_volume=40; citation_publication_date=2021; citation_pages=313-347; citation_id=CR51"/> <meta name="citation_reference" content="citation_journal_title=Anal. Geom. Metric Spaces; citation_title=Complex interpolation of Lizorkin-Triebel-Morrey Spaces on Domains; citation_author=C Zhuo, M Hovemann, W Sickel; citation_volume=8; citation_publication_date=2020; citation_pages=268-304; citation_id=CR52"/> <meta name="citation_author" content="Gonçalves, Helena F."/> <meta name="citation_author_email" content="helena.goncalves@uni-jena.de"/> <meta name="citation_author_institution" content="Institute of Mathematics, Friedrich Schiller University Jena, Jena, Germany"/> <meta name="citation_author" content="Haroske, Dorothee D."/> <meta name="citation_author_email" content="dorothee.haroske@uni-jena.de"/> <meta name="citation_author_institution" content="Institute of Mathematics, Friedrich Schiller University Jena, Jena, Germany"/> <meta name="citation_author" content="Skrzypczak, Leszek"/> <meta name="citation_author_email" content="leszek.skrzypczak@amu.edu.pl"/> <meta name="citation_author_institution" content="Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland"/> <meta name="format-detection" content="telephone=no"/> <meta name="citation_cover_date" content="2023/10/01"/> <meta property="og:url" content="https://link.springer.com/article/10.1007/s10231-023-01327-w"/> <meta property="og:type" content="article"/> <meta property="og:site_name" content="SpringerLink"/> <meta property="og:title" content="Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator - Annali di Matematica Pura ed Applicata (1923 -)"/> <meta property="og:description" content="In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, $$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ id τ : B p 1 , q 1 s 1 , τ 1 ( Ω ) ↪ B p 2 , q 2 s 2 , τ 2 ( Ω ) and $$\text {id}_\tau : {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$$ id τ : F p 1 , q 1 s 1 , τ 1 ( Ω ) ↪ F p 2 , q 2 s 2 , τ 2 ( Ω ) , where $$\Omega \subset {{{\mathbb {R}}}^d}$$ Ω ⊂ R d is a bounded domain, obtaining necessary and sufficient conditions for the continuity of $$\text {id}_\tau $$ id τ . This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov’s linear, bounded universal extension operator for these spaces."/> <meta property="og:image" content="https://media.springernature.com/full/springer-static/cover-hires/journal/10231"/> <meta name="format-detection" content="telephone=no"> <link rel="apple-touch-icon" sizes="180x180" href=/oscar-static/img/favicons/darwin/apple-touch-icon-92e819bf8a.png> <link rel="icon" type="image/png" sizes="192x192" href=/oscar-static/img/favicons/darwin/android-chrome-192x192-6f081ca7e5.png> <link rel="icon" type="image/png" sizes="32x32" href=/oscar-static/img/favicons/darwin/favicon-32x32-1435da3e82.png> <link rel="icon" type="image/png" sizes="16x16" href=/oscar-static/img/favicons/darwin/favicon-16x16-ed57f42bd2.png> <link rel="shortcut icon" data-test="shortcut-icon" href=/oscar-static/img/favicons/darwin/favicon-c6d59aafac.ico> <meta name="theme-color" content="#e6e6e6">   <link rel="stylesheet" media="print" href=/oscar-static/app-springerlink/css/print-b8af42253b.css> <style> html{text-size-adjust:100%;line-height:1.15}body{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;margin:0}details,main{display:block}h1{font-size:2em;margin:.67em 0}a{background-color:transparent;color:#025e8d}sub{bottom:-.25em;font-size:75%;line-height:0;position:relative;vertical-align:baseline}img{border:0;height:auto;max-width:100%;vertical-align:middle}button,input{font-family:inherit;font-size:100%;line-height:1.15;margin:0;overflow:visible}button{text-transform:none}[type=button],[type=submit],button{-webkit-appearance:button}[type=search]{-webkit-appearance:textfield;outline-offset:-2px}summary{display:list-item}[hidden]{display:none}button{cursor:pointer}svg{height:1rem;width:1rem} </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { body{background:#fff;color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;line-height:1.8;min-height:100%}a{color:#025e8d;text-decoration:underline;text-decoration-skip-ink:auto}button{cursor:pointer}img{border:0;height:auto;max-width:100%;vertical-align:middle}html{box-sizing:border-box;font-size:100%;height:100%;overflow-y:scroll}h1{font-size:2.25rem}h2{font-size:1.75rem}h1,h2,h4{font-weight:700;line-height:1.2}h4{font-size:1.25rem}body{font-size:1.125rem}*{box-sizing:inherit}p{margin-bottom:2rem;margin-top:0}p:last-of-type{margin-bottom:0}.c-ad{text-align:center}@media only screen and (min-width:480px){.c-ad{padding:8px}}.c-ad--728x90{display:none}.c-ad--728x90 .c-ad__inner{min-height:calc(1.5em + 94px)}@media only screen and (min-width:876px){.js .c-ad--728x90{display:none}}.c-ad__label{color:#333;font-size:.875rem;font-weight:400;line-height:1.5;margin-bottom:4px}.c-ad__label,.c-status-message{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-status-message{align-items:center;box-sizing:border-box;display:flex;position:relative;width:100%}.c-status-message :last-child{margin-bottom:0}.c-status-message--boxed{background-color:#fff;border:1px solid #ccc;line-height:1.4;padding:16px}.c-status-message__heading{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700}.c-status-message__icon{fill:currentcolor;display:inline-block;flex:0 0 auto;height:1.5em;margin-right:8px;transform:translate(0);vertical-align:text-top;width:1.5em}.c-status-message__icon--top{align-self:flex-start}.c-status-message--info .c-status-message__icon{color:#003f8d}.c-status-message--boxed.c-status-message--info{border-bottom:4px solid #003f8d}.c-status-message--error .c-status-message__icon{color:#c40606}.c-status-message--boxed.c-status-message--error{border-bottom:4px solid #c40606}.c-status-message--success .c-status-message__icon{color:#00b8b0}.c-status-message--boxed.c-status-message--success{border-bottom:4px solid #00b8b0}.c-status-message--warning .c-status-message__icon{color:#edbc53}.c-status-message--boxed.c-status-message--warning{border-bottom:4px solid #edbc53}.eds-c-header{background-color:#fff;border-bottom:2px solid #01324b;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;line-height:1.5;padding:8px 0 0}.eds-c-header__container{align-items:center;display:flex;flex-wrap:nowrap;gap:8px 16px;justify-content:space-between;margin:0 auto 8px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav{border-top:2px solid #c5e0f4;padding-top:4px;position:relative}.eds-c-header__nav-container{align-items:center;display:flex;flex-wrap:wrap;margin:0 auto 4px;max-width:1280px;padding:0 8px;position:relative}.eds-c-header__nav-container>:not(:last-child){margin-right:32px}.eds-c-header__link-container{align-items:center;display:flex;flex:1 0 auto;gap:8px 16px;justify-content:space-between}.eds-c-header__list{list-style:none;margin:0;padding:0}.eds-c-header__list-item{font-weight:700;margin:0 auto;max-width:1280px;padding:8px}.eds-c-header__list-item:not(:last-child){border-bottom:2px solid #c5e0f4}.eds-c-header__item{color:inherit}@media only screen and (min-width:768px){.eds-c-header__item--menu{display:none;visibility:hidden}.eds-c-header__item--menu:first-child+*{margin-block-start:0}}.eds-c-header__item--inline-links{display:none;visibility:hidden}@media only screen and (min-width:768px){.eds-c-header__item--inline-links{display:flex;gap:16px 16px;visibility:visible}}.eds-c-header__item--divider:before{border-left:2px solid #c5e0f4;content:"";height:calc(100% - 16px);margin-left:-15px;position:absolute;top:8px}.eds-c-header__brand{padding:16px 8px}.eds-c-header__brand a{display:block;line-height:1;text-decoration:none}.eds-c-header__brand img{height:1.5rem;width:auto}.eds-c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.eds-c-header__icon{fill:currentcolor;display:inline-block;font-size:1.5rem;height:1em;transform:translate(0);vertical-align:bottom;width:1em}.eds-c-header__icon+*{margin-left:8px}.eds-c-header__expander{background-color:#f0f7fc}.eds-c-header__search{display:block;padding:24px 0}@media only screen and (min-width:768px){.eds-c-header__search{max-width:70%}}.eds-c-header__search-container{position:relative}.eds-c-header__search-label{color:inherit;display:inline-block;font-weight:700;margin-bottom:8px}.eds-c-header__search-input{background-color:#fff;border:1px solid #000;padding:8px 48px 8px 8px;width:100%}.eds-c-header__search-button{background-color:transparent;border:0;color:inherit;height:100%;padding:0 8px;position:absolute;right:0}.has-tethered.eds-c-header__expander{border-bottom:2px solid #01324b;left:0;margin-top:-2px;top:100%;width:100%;z-index:10}@media only screen and (min-width:768px){.has-tethered.eds-c-header__expander--menu{display:none;visibility:hidden}}.has-tethered .eds-c-header__heading{display:none;visibility:hidden}.has-tethered .eds-c-header__heading:first-child+*{margin-block-start:0}.has-tethered .eds-c-header__search{margin:auto}.eds-c-header__heading{margin:0 auto;max-width:1280px;padding:16px 16px 0}.eds-c-pagination{align-items:center;display:flex;flex-wrap:wrap;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;gap:16px 0;justify-content:center;line-height:1.4;list-style:none;margin:0;padding:32px 0}@media only screen and (min-width:480px){.eds-c-pagination{padding:32px 16px}}.eds-c-pagination__item{margin-right:8px}.eds-c-pagination__item--prev{margin-right:16px}.eds-c-pagination__item--next .eds-c-pagination__link,.eds-c-pagination__item--prev .eds-c-pagination__link{padding:16px 8px}.eds-c-pagination__item--next{margin-left:8px}.eds-c-pagination__item:last-child{margin-right:0}.eds-c-pagination__link{align-items:center;color:#222;cursor:pointer;display:inline-block;font-size:1rem;margin:0;padding:16px 24px;position:relative;text-align:center;transition:all .2s ease 0s}.eds-c-pagination__link:visited{color:#222}.eds-c-pagination__link--disabled{border-color:#555;color:#555;cursor:default}.eds-c-pagination__link--active{background-color:#01324b;background-image:none;border-radius:8px;color:#fff}.eds-c-pagination__link--active:focus,.eds-c-pagination__link--active:hover,.eds-c-pagination__link--active:visited{color:#fff}.eds-c-pagination__link-container{align-items:center;display:flex}.eds-c-pagination__icon{fill:#222;height:1.5rem;width:1.5rem}.eds-c-pagination__icon--disabled{fill:#555}.eds-c-pagination__visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.c-breadcrumbs{color:#333;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;list-style:none;margin:0;padding:0}.c-breadcrumbs>li{display:inline}svg.c-breadcrumbs__chevron{fill:#333;height:10px;margin:0 .25rem;width:10px}.c-breadcrumbs--contrast,.c-breadcrumbs--contrast .c-breadcrumbs__link{color:#fff}.c-breadcrumbs--contrast svg.c-breadcrumbs__chevron{fill:#fff}@media only screen and (max-width:479px){.c-breadcrumbs .c-breadcrumbs__item{display:none}.c-breadcrumbs .c-breadcrumbs__item:last-child,.c-breadcrumbs .c-breadcrumbs__item:nth-last-child(2){display:inline}}.c-skip-link{background:#01324b;bottom:auto;color:#fff;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;padding:8px;position:absolute;text-align:center;transform:translateY(-100%);width:100%;z-index:9999}@media (prefers-reduced-motion:reduce){.c-skip-link{transition:top .3s ease-in-out 0s}}@media print{.c-skip-link{display:none}}.c-skip-link:active,.c-skip-link:hover,.c-skip-link:link,.c-skip-link:visited{color:#fff}.c-skip-link:focus{transform:translateY(0)}.l-with-sidebar{display:flex;flex-wrap:wrap}.l-with-sidebar>*{margin:0}.l-with-sidebar__sidebar{flex-basis:var(--with-sidebar--basis,400px);flex-grow:1}.l-with-sidebar>:not(.l-with-sidebar__sidebar){flex-basis:0px;flex-grow:999;min-width:var(--with-sidebar--min,53%)}.l-with-sidebar>:first-child{padding-right:4rem}@supports (gap:1em){.l-with-sidebar>:first-child{padding-right:0}.l-with-sidebar{gap:var(--with-sidebar--gap,4rem)}}.c-header__link{color:inherit;display:inline-block;font-weight:700;padding:16px 8px;position:relative;text-decoration-color:transparent;white-space:nowrap;word-break:normal}.app-masthead__colour-4{--background-color:#ff9500;--gradient-light:rgba(0,0,0,.5);--gradient-dark:rgba(0,0,0,.8)}.app-masthead{background:var(--background-color,#0070a8);position:relative}.app-masthead:after{background:radial-gradient(circle at top right,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)));bottom:0;content:"";left:0;position:absolute;right:0;top:0}@media only screen and (max-width:479px){.app-masthead:after{background:linear-gradient(225deg,var(--gradient-light,rgba(0,0,0,.4)),var(--gradient-dark,rgba(0,0,0,.7)))}}.app-masthead__container{color:var(--masthead-color,#fff);margin:0 auto;max-width:1280px;padding:0 16px;position:relative;z-index:1}.u-button{align-items:center;background-color:#01324b;background-image:none;border:4px solid transparent;border-radius:32px;cursor:pointer;display:inline-flex;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;font-weight:700;justify-content:center;line-height:1.3;margin:0;padding:16px 32px;position:relative;transition:all .2s ease 0s;width:auto}.u-button svg,.u-button--contrast svg,.u-button--primary svg,.u-button--secondary svg,.u-button--tertiary svg{fill:currentcolor}.u-button,.u-button:visited{color:#fff}.u-button,.u-button:hover{box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button:hover{border:4px solid #fff}.u-button:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button:focus,.u-button:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--primary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover svg path,.u-button--primary:focus svg path,.u-button--primary:hover svg path,.u-button:focus svg path,.u-button:hover svg path{fill:#01324b}.u-button--primary{background-color:#01324b;background-image:none;border:4px solid transparent;box-shadow:0 0 0 1px #01324b;color:#fff;font-weight:700}.u-button--primary:visited{color:#fff}.u-button--primary:hover{border:4px solid #fff;box-shadow:0 0 0 1px #01324b;text-decoration:none}.u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.u-button--primary:focus,.u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.u-button--secondary{background-color:#fff;border:4px solid #fff;color:#01324b;font-weight:700}.u-button--secondary:visited{color:#01324b}.u-button--secondary:hover{border:4px solid #01324b;box-shadow:none}.u-button--secondary:focus,.u-button--secondary:hover{background-color:#01324b;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus svg path,.app-masthead--pastel .c-pdf-download .u-button--secondary:hover svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus svg path,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover svg path,.u-button--secondary:focus svg path,.u-button--secondary:hover svg path,.u-button--tertiary:focus svg path,.u-button--tertiary:hover svg path{fill:#fff}.u-button--tertiary{background-color:#ebf1f5;border:4px solid transparent;box-shadow:none;color:#666;font-weight:700}.u-button--tertiary:visited{color:#666}.u-button--tertiary:hover{border:4px solid #01324b;box-shadow:none}.u-button--tertiary:focus,.u-button--tertiary:hover{background-color:#01324b;color:#fff}.u-button--contrast{background-color:transparent;background-image:none;color:#fff;font-weight:400}.u-button--contrast:visited{color:#fff}.u-button--contrast,.u-button--contrast:focus,.u-button--contrast:hover{border:4px solid #fff}.u-button--contrast:focus,.u-button--contrast:hover{background-color:#fff;background-image:none;color:#000}.u-button--contrast:focus svg path,.u-button--contrast:hover svg path{fill:#000}.u-button--disabled,.u-button:disabled{background-color:transparent;background-image:none;border:4px solid #ccc;color:#000;cursor:default;font-weight:400;opacity:.7}.u-button--disabled svg,.u-button:disabled svg{fill:currentcolor}.u-button--disabled:visited,.u-button:disabled:visited{color:#000}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{border:4px solid #ccc;text-decoration:none}.u-button--disabled:focus,.u-button--disabled:hover,.u-button:disabled:focus,.u-button:disabled:hover{background-color:transparent;background-image:none;color:#000}.u-button--disabled:focus svg path,.u-button--disabled:hover svg path,.u-button:disabled:focus svg path,.u-button:disabled:hover svg path{fill:#000}.u-button--small,.u-button--xsmall{font-size:.875rem;padding:2px 8px}.u-button--small{padding:8px 16px}.u-button--large{font-size:1.125rem;padding:10px 35px}.u-button--full-width{display:flex;width:100%}.u-button--icon-left svg{margin-right:8px}.u-button--icon-right svg{margin-left:8px}.u-clear-both{clear:both}.u-container{margin:0 auto;max-width:1280px;padding:0 16px}.u-justify-content-space-between{justify-content:space-between}.u-display-none{display:none}.js .u-js-hide,.u-hide{display:none;visibility:hidden}.u-visually-hidden{clip:rect(0,0,0,0);border:0;clip-path:inset(50%);height:1px;overflow:hidden;padding:0;position:absolute!important;white-space:nowrap;width:1px}.u-icon{fill:currentcolor;display:inline-block;height:1em;transform:translate(0);vertical-align:text-top;width:1em}.u-list-reset{list-style:none;margin:0;padding:0}.u-ma-16{margin:16px}.u-mt-0{margin-top:0}.u-mt-24{margin-top:24px}.u-mt-32{margin-top:32px}.u-mb-8{margin-bottom:8px}.u-mb-32{margin-bottom:32px}.u-button-reset{background-color:transparent;border:0;padding:0}.u-sans-serif{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.u-serif{font-family:Merriweather,serif}h1,h2,h4{-webkit-font-smoothing:antialiased}p{overflow-wrap:break-word;word-break:break-word}.u-h4{font-size:1.25rem;font-weight:700;line-height:1.2}.u-mbs-0{margin-block-start:0!important}.c-article-header{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-identifiers{color:#6f6f6f;display:flex;flex-wrap:wrap;font-size:1rem;line-height:1.3;list-style:none;margin:0 0 8px;padding:0}.c-article-identifiers__item{border-right:1px solid #6f6f6f;list-style:none;margin-right:8px;padding-right:8px}.c-article-identifiers__item:last-child{border-right:0;margin-right:0;padding-right:0}@media only screen and (min-width:876px){.c-article-title{font-size:1.875rem;line-height:1.2}}.c-article-author-list{display:inline;font-size:1rem;list-style:none;margin:0 8px 0 0;padding:0;width:100%}.c-article-author-list__item{display:inline;padding-right:0}.c-article-author-list__show-more{display:none;margin-right:4px}.c-article-author-list__button,.js .c-article-author-list__item--hide,.js .c-article-author-list__show-more{display:none}.js .c-article-author-list--long .c-article-author-list__show-more,.js .c-article-author-list--long+.c-article-author-list__button{display:inline}@media only screen and (max-width:767px){.js .c-article-author-list__item--hide-small-screen{display:none}.js .c-article-author-list--short .c-article-author-list__show-more,.js .c-article-author-list--short+.c-article-author-list__button{display:inline}}#uptodate-client,.js .c-article-author-list--expanded .c-article-author-list__show-more{display:none!important}.js .c-article-author-list--expanded .c-article-author-list__item--hide-small-screen{display:inline!important}.c-article-author-list__button,.c-button-author-list{background:#ebf1f5;border:4px solid #ebf1f5;border-radius:20px;color:#666;font-size:.875rem;line-height:1.4;padding:2px 11px 2px 8px;text-decoration:none}.c-article-author-list__button svg,.c-button-author-list svg{margin:1px 4px 0 0}.c-article-author-list__button:hover,.c-button-author-list:hover{background:#025e8d;border-color:transparent;color:#fff}.c-article-body .c-article-access-provider{padding:8px 16px}.c-article-body .c-article-access-provider,.c-notes{border:1px solid #d5d5d5;border-image:initial;border-left:none;border-right:none;margin:24px 0}.c-article-body .c-article-access-provider__text{color:#555}.c-article-body .c-article-access-provider__text,.c-notes__text{font-size:1rem;margin-bottom:0;padding-bottom:2px;padding-top:2px;text-align:center}.c-article-body .c-article-author-affiliation__address{color:inherit;font-weight:700;margin:0}.c-article-body .c-article-author-affiliation__authors-list{list-style:none;margin:0;padding:0}.c-article-body .c-article-author-affiliation__authors-item{display:inline;margin-left:0}.c-article-authors-search{margin-bottom:24px;margin-top:0}.c-article-authors-search__item,.c-article-authors-search__title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-article-authors-search__title{color:#626262;font-size:1.05rem;font-weight:700;margin:0;padding:0}.c-article-authors-search__item{font-size:1rem}.c-article-authors-search__text{margin:0}.c-code-block{border:1px solid #fff;font-family:monospace;margin:0 0 24px;padding:20px}.c-code-block__heading{font-weight:400;margin-bottom:16px}.c-code-block__line{display:block;overflow-wrap:break-word;white-space:pre-wrap}.c-article-share-box{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;margin-bottom:24px}.c-article-share-box__description{font-size:1rem;margin-bottom:8px}.c-article-share-box__no-sharelink-info{font-size:.813rem;font-weight:700;margin-bottom:24px;padding-top:4px}.c-article-share-box__only-read-input{border:1px solid #d5d5d5;box-sizing:content-box;display:inline-block;font-size:.875rem;font-weight:700;height:24px;margin-bottom:8px;padding:8px 10px}.c-article-share-box__additional-info{color:#626262;font-size:.813rem}.c-article-share-box__button{background:#fff;box-sizing:content-box;text-align:center}.c-article-share-box__button--link-like{background-color:transparent;border:0;color:#025e8d;cursor:pointer;font-size:.875rem;margin-bottom:8px;margin-left:10px}.c-article-associated-content__container .c-article-associated-content__collection-label{font-size:.875rem;line-height:1.4}.c-article-associated-content__container .c-article-associated-content__collection-title{line-height:1.3}.c-reading-companion{clear:both;min-height:389px}.c-reading-companion__figures-list,.c-reading-companion__references-list{list-style:none;min-height:389px;padding:0}.c-reading-companion__references-list--numeric{list-style:decimal inside}.c-reading-companion__figure-item{border-top:1px solid #d5d5d5;font-size:1rem;padding:16px 8px 16px 0}.c-reading-companion__figure-item:first-child{border-top:none;padding-top:8px}.c-reading-companion__reference-item{font-size:1rem}.c-reading-companion__reference-item:first-child{border-top:none}.c-reading-companion__reference-item a{word-break:break-word}.c-reading-companion__reference-citation{display:inline}.c-reading-companion__reference-links{font-size:.813rem;font-weight:700;list-style:none;margin:8px 0 0;padding:0;text-align:right}.c-reading-companion__reference-links>a{display:inline-block;padding-left:8px}.c-reading-companion__reference-links>a:first-child{display:inline-block;padding-left:0}.c-reading-companion__figure-title{display:block;font-size:1.25rem;font-weight:700;line-height:1.2;margin:0 0 8px}.c-reading-companion__figure-links{display:flex;justify-content:space-between;margin:8px 0 0}.c-reading-companion__figure-links>a{align-items:center;display:flex}.c-article-section__figure-caption{display:block;margin-bottom:8px;word-break:break-word}.c-article-section__figure .video,p.app-article-masthead__access--above-download{margin:0 0 16px}.c-article-section__figure-description{font-size:1rem}.c-article-section__figure-description>*{margin-bottom:0}.c-cod{display:block;font-size:1rem;width:100%}.c-cod__form{background:#ebf0f3}.c-cod__prompt{font-size:1.125rem;line-height:1.3;margin:0 0 24px}.c-cod__label{display:block;margin:0 0 4px}.c-cod__row{display:flex;margin:0 0 16px}.c-cod__row:last-child{margin:0}.c-cod__input{border:1px solid #d5d5d5;border-radius:2px;flex-shrink:0;margin:0;padding:13px}.c-cod__input--submit{background-color:#025e8d;border:1px solid #025e8d;color:#fff;flex-shrink:1;margin-left:8px;transition:background-color .2s ease-out 0s,color .2s ease-out 0s}.c-cod__input--submit-single{flex-basis:100%;flex-shrink:0;margin:0}.c-cod__input--submit:focus,.c-cod__input--submit:hover{background-color:#fff;color:#025e8d}.save-data .c-article-author-institutional-author__sub-division,.save-data .c-article-equation__number,.save-data .c-article-figure-description,.save-data .c-article-fullwidth-content,.save-data .c-article-main-column,.save-data .c-article-satellite-article-link,.save-data .c-article-satellite-subtitle,.save-data .c-article-table-container,.save-data .c-blockquote__body,.save-data .c-code-block__heading,.save-data .c-reading-companion__figure-title,.save-data .c-reading-companion__reference-citation,.save-data .c-site-messages--nature-briefing-email-variant .serif,.save-data .c-site-messages--nature-briefing-email-variant.serif,.save-data .serif,.save-data .u-serif,.save-data h1,.save-data h2,.save-data h3{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-pdf-download__link{display:flex;flex:1 1 0%;padding:13px 24px}.c-pdf-download__link:hover{text-decoration:none}@media only screen and (min-width:768px){.c-context-bar--sticky .c-pdf-download__link{align-items:center;flex:1 1 183px}}@media only screen and (max-width:320px){.c-context-bar--sticky .c-pdf-download__link{padding:16px}}.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{display:flex;flex-direction:row;gap:16px 16px;margin:0;max-width:100%;padding:16px 0 0}.c-article-body .c-article-recommendations-list__item,.c-book-body .c-article-recommendations-list__item{flex:1 1 0%}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-list,.c-book-body .c-article-recommendations-list{flex-direction:column}}.c-article-body .c-article-recommendations-card__authors{display:none;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:.875rem;line-height:1.5;margin:0 0 8px}@media only screen and (max-width:767px){.c-article-body .c-article-recommendations-card__authors{display:block;margin:0}}.c-article-body .c-article-history{margin-top:24px}.app-article-metrics-bar p{margin:0}.app-article-masthead{display:flex;flex-direction:column;gap:16px 16px;padding:16px 0 24px}.app-article-masthead__info{display:flex;flex-direction:column;flex-grow:1}.app-article-masthead__brand{border-top:1px solid hsla(0,0%,100%,.8);display:flex;flex-direction:column;flex-shrink:0;gap:8px 8px;min-height:96px;padding:16px 0 0}.app-article-masthead__brand img{border:1px solid #fff;border-radius:8px;box-shadow:0 4px 15px 0 hsla(0,0%,50%,.25);height:auto;left:0;position:absolute;width:72px}.app-article-masthead__journal-link{display:block;font-size:1.125rem;font-weight:700;margin:0 0 8px;max-width:400px;padding:0 0 0 88px;position:relative}.app-article-masthead__journal-title{-webkit-box-orient:vertical;-webkit-line-clamp:3;display:-webkit-box;overflow:hidden}.app-article-masthead__submission-link{align-items:center;display:flex;font-size:1rem;gap:4px 4px;margin:0 0 0 88px}.app-article-masthead__access{align-items:center;display:flex;flex-wrap:wrap;font-size:.875rem;font-weight:300;gap:4px 4px;margin:0}.app-article-masthead__buttons{display:flex;flex-flow:column wrap;gap:16px 16px}.app-article-masthead__access svg,.app-masthead--pastel .c-pdf-download .u-button--primary svg,.app-masthead--pastel .c-pdf-download .u-button--secondary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary svg,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary svg{fill:currentcolor}.app-article-masthead a{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary{background-color:#025e8d;background-image:none;border:2px solid transparent;box-shadow:none;color:#fff;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--primary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:visited{color:#fff}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus{border:4px solid #fc0;box-shadow:none;outline:0;text-decoration:none}.app-masthead--pastel .c-pdf-download .u-button--primary:focus,.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background-color:#fff;background-image:none;color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--primary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--primary:hover{background:0 0;border:2px solid #025e8d;box-shadow:none;color:#025e8d}.app-masthead--pastel .c-pdf-download .u-button--secondary,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary{background:0 0;border:2px solid #025e8d;color:#025e8d;font-weight:700}.app-masthead--pastel .c-pdf-download .u-button--secondary:visited,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:visited{color:#01324b}.app-masthead--pastel .c-pdf-download .u-button--secondary:hover,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:hover{background-color:#01324b;background-color:#025e8d;border:2px solid transparent;box-shadow:none;color:#fff}.app-masthead--pastel .c-pdf-download .u-button--secondary:focus,.c-context-bar--sticky .c-context-bar__container .c-pdf-download .u-button--secondary:focus{background-color:#fff;background-image:none;border:4px solid #fc0;color:#01324b}@media only screen and (min-width:768px){.app-article-masthead{flex-direction:row;gap:64px 64px;padding:24px 0}.app-article-masthead__brand{border:0;padding:0}.app-article-masthead__brand img{height:auto;position:static;width:auto}.app-article-masthead__buttons{align-items:center;flex-direction:row;margin-top:auto}.app-article-masthead__journal-link{display:flex;flex-direction:column;gap:24px 24px;margin:0 0 8px;padding:0}.app-article-masthead__submission-link{margin:0}}@media only screen and (min-width:1024px){.app-article-masthead__brand{flex-basis:400px}}.app-article-masthead .c-article-identifiers{font-size:.875rem;font-weight:300;line-height:1;margin:0 0 8px;overflow:hidden;padding:0}.app-article-masthead .c-article-identifiers--cite-list{margin:0 0 16px}.app-article-masthead .c-article-identifiers *{color:#fff}.app-article-masthead .c-cod{display:none}.app-article-masthead .c-article-identifiers__item{border-left:1px solid #fff;border-right:0;margin:0 17px 8px -9px;padding:0 0 0 8px}.app-article-masthead .c-article-identifiers__item--cite{border-left:0}.app-article-metrics-bar{display:flex;flex-wrap:wrap;font-size:1rem;padding:16px 0 0;row-gap:24px}.app-article-metrics-bar__item{padding:0 16px 0 0}.app-article-metrics-bar__count{font-weight:700}.app-article-metrics-bar__label{font-weight:400;padding-left:4px}.app-article-metrics-bar__icon{height:auto;margin-right:4px;margin-top:-4px;width:auto}.app-article-metrics-bar__arrow-icon{margin:4px 0 0 4px}.app-article-metrics-bar a{color:#000}.app-article-metrics-bar .app-article-metrics-bar__item--metrics{padding-right:0}.app-overview-section .c-article-author-list,.app-overview-section__authors{line-height:2}.app-article-metrics-bar{margin-top:8px}.c-book-toc-pagination+.c-book-section__back-to-top{margin-top:0}.c-article-body .c-article-access-provider__text--chapter{color:#222;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;padding:20px 0}.c-article-body .c-article-access-provider__text--chapter svg.c-status-message__icon{fill:#003f8d;vertical-align:middle}.c-article-body-section__content--separator{padding-top:40px}.c-pdf-download__link{max-height:44px}.app-article-access .u-button--primary,.app-article-access .u-button--primary:visited{color:#fff}.c-article-sidebar{display:none}@media only screen and (min-width:1024px){.c-article-sidebar{display:block}}.c-cod__form{border-radius:12px}.c-cod__label{font-size:.875rem}.c-cod .c-status-message{align-items:center;justify-content:center;margin-bottom:16px;padding-bottom:16px}@media only screen and (min-width:1024px){.c-cod .c-status-message{align-items:inherit}}.c-cod .c-status-message__icon{margin-top:4px}.c-cod .c-cod__prompt{font-size:1rem;margin-bottom:16px}.c-article-body .app-article-access,.c-book-body .app-article-access{display:block}@media only screen and (min-width:1024px){.c-article-body .app-article-access,.c-book-body .app-article-access{display:none}}.c-article-body .app-card-service{margin-bottom:32px}@media only screen and (min-width:1024px){.c-article-body .app-card-service{display:none}}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary,.c-cod__row .u-button--primary{background-color:#025e8d;border:2px solid #025e8d;box-shadow:none;font-size:1rem;font-weight:700;gap:8px 8px;justify-content:center;line-height:1.5;padding:8px 24px}.app-article-access .buybox__buy .u-button--secondary,.app-article-access .u-button--primary:hover,.c-cod__row .u-button--primary:hover{background-color:#fff;color:#025e8d}.app-article-access .buybox__buy .u-button--secondary:hover{background-color:#025e8d;color:#fff}.buybox__buy .c-notes__text{color:#666;font-size:.875rem;padding:0 16px 8px}.c-cod__input{flex-basis:auto;width:100%}.c-article-title{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:2.25rem;font-weight:700;line-height:1.2;margin:12px 0}.c-reading-companion__figure-item figure{margin:0}@media only screen and (min-width:768px){.c-article-title{margin:16px 0}}.app-article-access{border:1px solid #c5e0f4;border-radius:12px}.app-article-access__heading{border-bottom:1px solid #c5e0f4;font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1.125rem;font-weight:700;margin:0;padding:16px;text-align:center}.app-article-access .buybox__info svg{vertical-align:middle}.c-article-body .app-article-access p{margin-bottom:0}.app-article-access .buybox__info{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif;font-size:1rem;margin:0}.app-article-access{margin:0 0 32px}@media only screen and (min-width:1024px){.app-article-access{margin:0 0 24px}}.c-status-message{font-size:1rem}.c-article-body{font-size:1.125rem}.c-article-body dl,.c-article-body ol,.c-article-body p,.c-article-body ul{margin-bottom:32px;margin-top:0}.c-article-access-provider__text:last-of-type,.c-article-body .c-notes__text:last-of-type{margin-bottom:0}.c-article-body ol p,.c-article-body ul p{margin-bottom:16px}.c-article-section__figure-caption{font-family:Merriweather Sans,Helvetica Neue,Helvetica,Arial,sans-serif}.c-reading-companion__figure-item{border-top-color:#c5e0f4}.c-reading-companion__sticky{max-width:400px}.c-article-section .c-article-section__figure-description>*{font-size:1rem;margin-bottom:16px}.c-reading-companion__reference-item{border-top:1px solid #d5d5d5;padding:16px 0}.c-reading-companion__reference-item:first-child{padding-top:0}.c-article-share-box__button,.js .c-article-authors-search__item .c-article-button{background:0 0;border:2px solid #025e8d;border-radius:32px;box-shadow:none;color:#025e8d;font-size:1rem;font-weight:700;line-height:1.5;margin:0;padding:8px 24px;transition:all .2s ease 0s}.c-article-authors-search__item .c-article-button{width:100%}.c-pdf-download .u-button{background-color:#fff;border:2px solid #fff;color:#01324b;justify-content:center}.c-context-bar__container .c-pdf-download .u-button svg,.c-pdf-download .u-button svg{fill:currentcolor}.c-pdf-download .u-button:visited{color:#01324b}.c-pdf-download .u-button:hover{border:4px solid #01324b;box-shadow:none}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background-color:#01324b}.c-pdf-download .u-button:focus svg path,.c-pdf-download .u-button:hover svg path{fill:#fff}.c-context-bar__container .c-pdf-download .u-button{background-image:none;border:2px solid;color:#fff}.c-context-bar__container .c-pdf-download .u-button:visited{color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus{box-shadow:none;outline:0;text-decoration:none}.c-context-bar__container .c-pdf-download .u-button:focus,.c-context-bar__container .c-pdf-download .u-button:hover{background-color:#fff;background-image:none;color:#01324b}.c-context-bar__container .c-pdf-download .u-button:focus svg path,.c-context-bar__container .c-pdf-download .u-button:hover svg path{fill:#01324b}.c-context-bar__container .c-pdf-download .u-button,.c-pdf-download .u-button{box-shadow:none;font-size:1rem;font-weight:700;line-height:1.5;padding:8px 24px}.c-context-bar__container .c-pdf-download .u-button{background-color:#025e8d}.c-pdf-download .u-button:hover{border:2px solid #fff}.c-pdf-download .u-button:focus,.c-pdf-download .u-button:hover{background:0 0;box-shadow:none;color:#fff}.c-context-bar__container .c-pdf-download .u-button:hover{border:2px solid #025e8d;box-shadow:none;color:#025e8d}.c-context-bar__container .c-pdf-download .u-button:focus,.c-pdf-download .u-button:focus{border:2px solid #025e8d}.c-article-share-box__button:focus:focus,.c-article__pill-button:focus:focus,.c-context-bar__container .c-pdf-download .u-button:focus:focus,.c-pdf-download .u-button:focus:focus{outline:3px solid #08c;will-change:transform}.c-pdf-download__link .u-icon{padding-top:0}.c-bibliographic-information__column button{margin-bottom:16px}.c-article-body .c-article-author-affiliation__list p,.c-article-body .c-article-author-information__list p,figure{margin:0}.c-article-share-box__button{margin-right:16px}.c-status-message--boxed{border-radius:12px}.c-article-associated-content__collection-title{font-size:1rem}.app-card-service__description,.c-article-body .app-card-service__description{color:#222;margin-bottom:0;margin-top:8px}.app-article-access__subscriptions a,.app-article-access__subscriptions a:visited,.app-book-series-listing__item a,.app-book-series-listing__item a:hover,.app-book-series-listing__item a:visited,.c-article-author-list a,.c-article-author-list a:visited,.c-article-buy-box a,.c-article-buy-box a:visited,.c-article-peer-review a,.c-article-peer-review a:visited,.c-article-satellite-subtitle a,.c-article-satellite-subtitle a:visited,.c-breadcrumbs__link,.c-breadcrumbs__link:hover,.c-breadcrumbs__link:visited{color:#000}.c-article-author-list svg{height:24px;margin:0 0 0 6px;width:24px}.c-article-header{margin-bottom:32px}@media only screen and (min-width:876px){.js .c-ad--conditional{display:block}}.u-lazy-ad-wrapper{background-color:#fff;display:none;min-height:149px}@media only screen and (min-width:876px){.u-lazy-ad-wrapper{display:block}}p.c-ad__label{margin-bottom:4px}.c-ad--728x90{background-color:#fff;border-bottom:2px solid #cedbe0} } </style> <style>@media only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark) { .eds-c-header__brand img{height:24px;width:203px}.app-article-masthead__journal-link img{height:93px;width:72px}@media only screen and (min-width:769px){.app-article-masthead__journal-link img{height:161px;width:122px}} } </style> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href=/oscar-static/app-springerlink/css/core-darwin-9fe647df8f.css media="print" onload="this.media='all';this.onload=null"> <link rel="stylesheet" data-test="critical-css-handler" data-inline-css-source="critical-css" href="/oscar-static/app-springerlink/css/enhanced-darwin-article-2a2a17cc99.css" media="print" onload="this.media='only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)';this.onload=null"> <script type="text/javascript"> config = { env: 'live', site: '10231.springer.com', siteWithPath: '10231.springer.com' + window.location.pathname, twitterHashtag: '10231', cmsPrefix: 'https://studio-cms.springernature.com/studio/', publisherBrand: 'Springer', mustardcut: false }; </script> <script> window.dataLayer = [{"GA Key":"UA-26408784-1","DOI":"10.1007/s10231-023-01327-w","Page":"article","springerJournal":true,"Publishing Model":"Hybrid Access","Country":"SG","japan":false,"doi":"10.1007-s10231-023-01327-w","Journal Id":10231,"Journal Title":"Annali di Matematica Pura ed Applicata (1923 -)","imprint":"Springer","Keywords":"Besov-type space, Triebel-Lizorkin-type spaces, Smoothness Morrey spaces on domains, Limiting embeddings, Extension operator., 46E35, 42B35","kwrd":["Besov-type_space","Triebel-Lizorkin-type_spaces","Smoothness_Morrey_spaces_on_domains","Limiting_embeddings","Extension_operator.","46E35","42B35"],"Labs":"Y","ksg":"Krux.segments","kuid":"Krux.uid","Has Body":"Y","Features":[],"Open Access":"Y","hasAccess":"Y","bypassPaywall":"N","user":{"license":{"businessPartnerID":[],"businessPartnerIDString":""}},"Access Type":"open","Bpids":"","Bpnames":"","BPID":["1"],"VG Wort Identifier":"vgzm.415900-10.1007-s10231-023-01327-w","Full HTML":"Y","Subject Codes":["SCM","SCM00009"],"pmc":["M","M00009"],"session":{"authentication":{"loginStatus":"N"},"attributes":{"edition":"academic"}},"content":{"serial":{"eissn":"1618-1891","pissn":"0373-3114"},"type":"Article","category":{"pmc":{"primarySubject":"Mathematics","primarySubjectCode":"M","secondarySubjects":{"1":"Mathematics, general"},"secondarySubjectCodes":{"1":"M00009"}},"sucode":"SC10","articleType":"Article"},"attributes":{"deliveryPlatform":"oscar"}},"page":{"attributes":{"environment":"live"},"category":{"pageType":"article"}},"Event Category":"Article"}]; </script> <script data-test="springer-link-article-datalayer"> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ ga4MeasurementId: 'G-B3E4QL2TPR', ga360TrackingId: 'UA-26408784-1', twitterId: 'o47a7', baiduId: 'aef3043f025ccf2305af8a194652d70b', ga4ServerUrl: 'https://collect.springer.com', imprint: 'springerlink', page: { attributes:{ featureFlags: [{ name: 'darwin-orion', active: true }], darwinAvailable: true } } }); </script> <script> (function(w, d) { w.config = w.config || {}; w.config.mustardcut = false; if (w.matchMedia && w.matchMedia('only print, only all and (prefers-color-scheme: no-preference), only all and (prefers-color-scheme: light), only all and (prefers-color-scheme: dark)').matches) { w.config.mustardcut = true; d.classList.add('js'); d.classList.remove('grade-c'); d.classList.remove('no-js'); } })(window, document.documentElement); </script> <script class="js-entry"> if (window.config.mustardcut) { (function(w, d) { window.Component = {}; window.suppressShareButton = false; window.onArticlePage = true; var currentScript = d.currentScript || d.head.querySelector('script.js-entry'); function catchNoModuleSupport() { var scriptEl = d.createElement('script'); return (!('noModule' in scriptEl) && 'onbeforeload' in scriptEl) } var headScripts = [ {'src': '/oscar-static/js/polyfill-es5-bundle-572d4fec60.js', 'async': false} ]; var bodyScripts = [ {'src': '/oscar-static/js/global-article-es5-bundle-f28c859359.js', 'async': false, 'module': false}, {'src': '/oscar-static/js/global-article-es6-bundle-9e329e4cbc.js', 'async': false, 'module': true} ]; function createScript(script) { var scriptEl = d.createElement('script'); scriptEl.src = script.src; scriptEl.async = script.async; if (script.module === true) { scriptEl.type = "module"; if (catchNoModuleSupport()) { scriptEl.src = ''; } } else if (script.module === false) { scriptEl.setAttribute('nomodule', true) } if (script.charset) { scriptEl.setAttribute('charset', script.charset); } return scriptEl; } for (var i = 0; i < headScripts.length; ++i) { var scriptEl = createScript(headScripts[i]); currentScript.parentNode.insertBefore(scriptEl, currentScript.nextSibling); } d.addEventListener('DOMContentLoaded', function() { for (var i = 0; i < bodyScripts.length; ++i) { var scriptEl = createScript(bodyScripts[i]); d.body.appendChild(scriptEl); } }); // Webfont repeat view var config = w.config; if (config && config.publisherBrand && sessionStorage.fontsLoaded === 'true') { d.documentElement.className += ' webfonts-loaded'; } })(window, document); } </script> <script data-src="https://cdn.optimizely.com/js/27195530232.js" data-cc-script="C03"></script> <script data-test="gtm-head"> window.initGTM = function() { if (window.config.mustardcut) { (function (w, d, s, l, i) { w[l] = w[l] || []; w[l].push({'gtm.start': new Date().getTime(), event: 'gtm.js'}); var f = d.getElementsByTagName(s)[0], j = d.createElement(s), dl = l != 'dataLayer' ? '&l=' + l : ''; j.async = true; j.src = 'https://www.googletagmanager.com/gtm.js?id=' + i + dl; f.parentNode.insertBefore(j, f); })(window, document, 'script', 'dataLayer', 'GTM-MRVXSHQ'); } } </script> <script> (function (w, d, t) { function cc() { var h = w.location.hostname; var e = d.createElement(t), s = d.getElementsByTagName(t)[0]; if (h.indexOf('springer.com') > -1 && h.indexOf('biomedcentral.com') === -1 && h.indexOf('springeropen.com') === -1) { if (h.indexOf('link-qa.springer.com') > -1 || h.indexOf('test-www.springer.com') > -1) { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-54.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springer.com/production_live/en/consent-bundle-17-54.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('biomedcentral.com') > -1) { if (h.indexOf('biomedcentral.com.qa') > -1) { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-39.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.biomedcentral.com/production_live/en/consent-bundle-15-39.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springeropen.com') > -1) { if (h.indexOf('springeropen.com.qa') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-36.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-16-36.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-MRVXSHQ')"); } } else if (h.indexOf('springernature.com') > -1) { if (h.indexOf('beta-qa.springernature.com') > -1) { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } else { e.src = 'https://cmp.springernature.com/production_live/en/consent-bundle-49-43.js'; e.setAttribute('onload', "initGTM(window,document,'script','dataLayer','GTM-NK22KLS')"); } } else { e.src = '/oscar-static/js/cookie-consent-es5-bundle-cb57c2c98a.js'; e.setAttribute('data-consent', h); } s.insertAdjacentElement('afterend', e); } cc(); })(window, document, 'script'); </script> <link rel="canonical" href="https://link.springer.com/article/10.1007/s10231-023-01327-w"/> <script type="application/ld+json">{"mainEntity":{"headline":"Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator","description":"In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, \n \n \n \n $$\\text {id}_\\tau : {B}_{p_1,q_1}^{s_1,\\tau _1}(\\Omega ) \\hookrightarrow {B}_{p_2,q_2}^{s_2,\\tau _2}(\\Omega )$$\n \n and \n \n \n \n $$\\text {id}_\\tau : {F}_{p_1,q_1}^{s_1,\\tau _1}(\\Omega ) \\hookrightarrow {F}_{p_2,q_2}^{s_2,\\tau _2}(\\Omega )$$\n \n , where \n \n \n \n $$\\Omega \\subset {{{\\mathbb {R}}}^d}$$\n \n is a bounded domain, obtaining necessary and sufficient conditions for the continuity of \n \n \n \n $$\\text {id}_\\tau $$\n \n . This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov’s linear, bounded universal extension operator for these spaces.","datePublished":"2023-03-27T00:00:00Z","dateModified":"2023-03-27T00:00:00Z","pageStart":"2481","pageEnd":"2516","license":"http://creativecommons.org/licenses/by/4.0/","sameAs":"https://doi.org/10.1007/s10231-023-01327-w","keywords":["Besov-type space","Triebel-Lizorkin-type spaces","Smoothness Morrey spaces on domains","Limiting embeddings","Extension operator.","46E35","42B35","Mathematics","general"],"image":[],"isPartOf":{"name":"Annali di Matematica Pura ed Applicata (1923 -)","issn":["1618-1891","0373-3114"],"volumeNumber":"202","@type":["Periodical","PublicationVolume"]},"publisher":{"name":"Springer Berlin Heidelberg","logo":{"url":"https://www.springernature.com/app-sn/public/images/logo-springernature.png","@type":"ImageObject"},"@type":"Organization"},"author":[{"name":"Helena F. Gonçalves","url":"http://orcid.org/0000-0001-5608-9102","affiliation":[{"name":"Friedrich Schiller University Jena","address":{"name":"Institute of Mathematics, Friedrich Schiller University Jena, Jena, Germany","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Dorothee D. Haroske","url":"http://orcid.org/0000-0002-2576-0300","affiliation":[{"name":"Friedrich Schiller University Jena","address":{"name":"Institute of Mathematics, Friedrich Schiller University Jena, Jena, Germany","@type":"PostalAddress"},"@type":"Organization"}],"@type":"Person"},{"name":"Leszek Skrzypczak","url":"http://orcid.org/0000-0002-7484-2900","affiliation":[{"name":"Adam Mickiewicz University","address":{"name":"Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań, Poland","@type":"PostalAddress"},"@type":"Organization"}],"email":"leszek.skrzypczak@amu.edu.pl","@type":"Person"}],"isAccessibleForFree":true,"@type":"ScholarlyArticle"},"@context":"https://schema.org","@type":"WebPage"}</script> </head> <body class="" >  <noscript> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>   <noscript data-test="gtm-body"> <iframe src="https://www.googletagmanager.com/ns.html?id=GTM-MRVXSHQ" height="0" width="0" style="display:none;visibility:hidden"></iframe> </noscript>  <div class="u-visually-hidden" aria-hidden="true" data-test="darwin-icons"> <?xml version="1.0" encoding="UTF-8"?><!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd"><svg xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><symbol id="icon-eds-i-accesses-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H15a1 1 0 0 1 0-2h4.455a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM8 13c2.052 0 4.66 1.61 6.36 3.4l.124.141c.333.41.516.925.516 1.459 0 .6-.232 1.178-.64 1.599C12.666 21.388 10.054 23 8 23c-2.052 0-4.66-1.61-6.353-3.393A2.31 2.31 0 0 1 1 18c0-.6.232-1.178.64-1.6C3.34 14.61 5.948 13 8 13Zm0 2c-1.369 0-3.552 1.348-4.917 2.785A.31.31 0 0 0 3 18c0 .083.031.161.09.222C4.447 19.652 6.631 21 8 21c1.37 0 3.556-1.35 4.917-2.785A.31.31 0 0 0 13 18a.32.32 0 0 0-.048-.17l-.042-.052C11.553 16.348 9.369 15 8 15Zm0 1a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-altmetric-medium" viewBox="0 0 24 24"><path d="M12 1c5.978 0 10.843 4.77 10.996 10.712l.004.306-.002.022-.002.248C22.843 18.23 17.978 23 12 23 5.925 23 1 18.075 1 12S5.925 1 12 1Zm-1.726 9.246L8.848 12.53a1 1 0 0 1-.718.461L8.003 13l-4.947.014a9.001 9.001 0 0 0 17.887-.001L16.553 13l-2.205 3.53a1 1 0 0 1-1.735-.068l-.05-.11-2.289-6.106ZM12 3a9.001 9.001 0 0 0-8.947 8.013l4.391-.012L9.652 7.47a1 1 0 0 1 1.784.179l2.288 6.104 1.428-2.283a1 1 0 0 1 .722-.462l.129-.008 4.943.012A9.001 9.001 0 0 0 12 3Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-medium" viewBox="0 0 24 24"><path d="m11.852 20.989.058.007L12 21l.075-.003.126-.017.111-.03.111-.044.098-.052.104-.074.082-.073 6-6a1 1 0 0 0-1.414-1.414L13 17.585v-12.2C13 4.075 11.964 3 10.667 3H4a1 1 0 1 0 0 2h6.667c.175 0 .333.164.333.385v12.2l-4.293-4.292a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l6 6c.035.036.073.068.112.097l.11.071.114.054.105.035.118.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-down-small" viewBox="0 0 16 16"><path d="M1 2a1 1 0 0 0 1 1h5v8.585L3.707 8.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414l5 5 .063.059.093.069.081.048.105.048.104.035.105.022.096.01h.136l.122-.018.113-.03.103-.04.1-.053.102-.07.052-.043 5.04-5.037a1 1 0 1 0-1.415-1.414L9 11.583V3a2 2 0 0 0-2-2H2a1 1 0 0 0-1 1Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-medium" viewBox="0 0 24 24"><path d="m11.852 3.011.058-.007L12 3l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 6 6a1 1 0 1 1-1.414 1.414L13 6.415v12.2C13 19.925 11.964 21 10.667 21H4a1 1 0 0 1 0-2h6.667c.175 0 .333-.164.333-.385v-12.2l-4.293 4.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l6-6c.035-.036.073-.068.112-.097l.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-arrow-bend-up-small" viewBox="0 0 16 16"><path d="M1 13.998a1 1 0 0 1 1-1h5V4.413L3.707 7.705a1 1 0 0 1-1.32.084l-.094-.084a1 1 0 0 1 0-1.414l5-5 .063-.059.093-.068.081-.05.105-.047.104-.035.105-.022L7.94 1l.136.001.122.017.113.03.103.04.1.053.102.07.052.043 5.04 5.037a1 1 0 1 1-1.415 1.414L9 4.415v8.583a2 2 0 0 1-2 2H2a1 1 0 0 1-1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-medium" viewBox="0 0 24 24"><path d="M14 3h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L21 4v6a1 1 0 0 1-2 0V6.414l-4.293 4.293a1 1 0 0 1-1.414-1.414L17.584 5H14a1 1 0 0 1-.993-.883L13 4a1 1 0 0 1 1-1ZM4 13a1 1 0 0 1 1 1v3.584l4.293-4.291a1 1 0 1 1 1.414 1.414L6.414 19H10a1 1 0 0 1 .993.883L11 20a1 1 0 0 1-1 1l-6.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.01 1.01 0 0 1-.097-.112l-.071-.11-.054-.114-.035-.105-.025-.118-.007-.058L3 20v-6a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-arrow-diagonal-small" viewBox="0 0 16 16"><path d="m2 15-.082-.004-.119-.016-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08a1.008 1.008 0 0 1-.097-.112l-.071-.11-.031-.062-.034-.081-.024-.076-.025-.118-.007-.058L1 14.02V9a1 1 0 1 1 2 0v2.584l2.793-2.791a1 1 0 1 1 1.414 1.414L4.414 13H7a1 1 0 0 1 .993.883L8 14a1 1 0 0 1-1 1H2ZM14 1l.081.003.12.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.031.062.034.081.024.076.03.148L15 2v5a1 1 0 0 1-2 0V4.414l-2.96 2.96A1 1 0 1 1 8.626 5.96L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1h5Z"/></symbol><symbol id="icon-eds-i-arrow-down-medium" viewBox="0 0 24 24"><path d="m20.707 12.728-7.99 7.98a.996.996 0 0 1-.561.281l-.157.011a.998.998 0 0 1-.788-.384l-7.918-7.908a1 1 0 0 1 1.414-1.416L11 17.576V4a1 1 0 0 1 2 0v13.598l6.293-6.285a1 1 0 0 1 1.32-.082l.095.083a1 1 0 0 1-.001 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-down-small" viewBox="0 0 16 16"><path d="m1.293 8.707 6 6 .063.059.093.069.081.048.105.049.104.034.056.013.118.017L8 15l.076-.003.122-.017.113-.03.085-.032.063-.03.098-.058.06-.043.05-.043 6.04-6.037a1 1 0 0 0-1.414-1.414L9 11.583V2a1 1 0 1 0-2 0v9.585L2.707 7.293a1 1 0 0 0-1.32-.083l-.094.083a1 1 0 0 0 0 1.414Z"/></symbol><symbol id="icon-eds-i-arrow-left-medium" viewBox="0 0 24 24"><path d="m11.272 3.293-7.98 7.99a.996.996 0 0 0-.281.561L3 12.001c0 .32.15.605.384.788l7.908 7.918a1 1 0 0 0 1.416-1.414L6.424 13H20a1 1 0 0 0 0-2H6.402l6.285-6.293a1 1 0 0 0 .082-1.32l-.083-.095a1 1 0 0 0-1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-left-small" viewBox="0 0 16 16"><path d="m7.293 1.293-6 6-.059.063-.069.093-.048.081-.049.105-.034.104-.013.056-.017.118L1 8l.003.076.017.122.03.113.032.085.03.063.058.098.043.06.043.05 6.037 6.04a1 1 0 0 0 1.414-1.414L4.417 9H14a1 1 0 0 0 0-2H4.415l4.292-4.293a1 1 0 0 0 .083-1.32l-.083-.094a1 1 0 0 0-1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-right-medium" viewBox="0 0 24 24"><path d="m12.728 3.293 7.98 7.99a.996.996 0 0 1 .281.561l.011.157c0 .32-.15.605-.384.788l-7.908 7.918a1 1 0 0 1-1.416-1.414L17.576 13H4a1 1 0 0 1 0-2h13.598l-6.285-6.293a1 1 0 0 1-.082-1.32l.083-.095a1 1 0 0 1 1.414.001Z"/></symbol><symbol id="icon-eds-i-arrow-right-small" viewBox="0 0 16 16"><path d="m8.707 1.293 6 6 .059.063.069.093.048.081.049.105.034.104.013.056.017.118L15 8l-.003.076-.017.122-.03.113-.032.085-.03.063-.058.098-.043.06-.043.05-6.037 6.04a1 1 0 0 1-1.414-1.414L11.583 9H2a1 1 0 1 1 0-2h9.585L7.293 2.707a1 1 0 0 1-.083-1.32l.083-.094a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-arrow-up-medium" viewBox="0 0 24 24"><path d="m3.293 11.272 7.99-7.98a.996.996 0 0 1 .561-.281L12.001 3c.32 0 .605.15.788.384l7.918 7.908a1 1 0 0 1-1.414 1.416L13 6.424V20a1 1 0 0 1-2 0V6.402l-6.293 6.285a1 1 0 0 1-1.32.082l-.095-.083a1 1 0 0 1 .001-1.414Z"/></symbol><symbol id="icon-eds-i-arrow-up-small" viewBox="0 0 16 16"><path d="m1.293 7.293 6-6 .063-.059.093-.069.081-.048.105-.049.104-.034.056-.013.118-.017L8 1l.076.003.122.017.113.03.085.032.063.03.098.058.06.043.05.043 6.04 6.037a1 1 0 0 1-1.414 1.414L9 4.417V14a1 1 0 0 1-2 0V4.415L2.707 8.707a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414Z"/></symbol><symbol id="icon-eds-i-article-medium" viewBox="0 0 24 24"><path d="M8 7a1 1 0 0 0 0 2h4a1 1 0 1 0 0-2H8ZM8 11a1 1 0 1 0 0 2h8a1 1 0 1 0 0-2H8ZM7 16a1 1 0 0 1 1-1h8a1 1 0 1 1 0 2H8a1 1 0 0 1-1-1Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V3.5A2.5 2.5 0 0 0 18.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3H18.5a.5.5 0 0 1 .5.5v16.962c0 .293-.24.538-.546.538H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-book-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v12c0 1.16-.79 2.135-1.86 2.418l-.14.031V21h1a1 1 0 0 1 .993.883L21 22a1 1 0 0 1-1 1H6.5A3.5 3.5 0 0 1 3 19.5v-15A3.5 3.5 0 0 1 6.5 1h12ZM17 18H6.5a1.5 1.5 0 0 0-1.493 1.356L5 19.5A1.5 1.5 0 0 0 6.5 21H17v-3Zm1.5-15h-12A1.5 1.5 0 0 0 5 4.5v11.837l.054-.025a3.481 3.481 0 0 1 1.254-.307L6.5 16h12a.5.5 0 0 0 .492-.41L19 15.5v-12a.5.5 0 0 0-.5-.5ZM15 6a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-book-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M1 3.786C1 2.759 1.857 2 2.82 2H6.18c.964 0 1.82.759 1.82 1.786V4h3.168c.668 0 1.298.364 1.616.938.158-.109.333-.195.523-.252l3.216-.965c.923-.277 1.962.204 2.257 1.187l4.146 13.82c.296.984-.307 1.957-1.23 2.234l-3.217.965c-.923.277-1.962-.203-2.257-1.187L13 10.005v10.21c0 1.04-.878 1.785-1.834 1.785H7.833c-.291 0-.575-.07-.83-.195A1.849 1.849 0 0 1 6.18 22H2.821C1.857 22 1 21.241 1 20.214V3.786ZM3 4v11h3V4H3Zm0 16v-3h3v3H3Zm15.075-.04-.814-2.712 2.874-.862.813 2.712-2.873.862Zm1.485-5.49-2.874.862-2.634-8.782 2.873-.862 2.635 8.782ZM8 20V6h3v14H8Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-calendar-acceptance-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-.534 7.747a1 1 0 0 1 .094 1.412l-4.846 5.538a1 1 0 0 1-1.352.141l-2.77-2.076a1 1 0 0 1 1.2-1.6l2.027 1.519 4.236-4.84a1 1 0 0 1 1.411-.094ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-date-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1ZM8 15a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm-4-4a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2Zm4 0a1 1 0 1 1 0 2 1 1 0 0 1 0-2ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-decision-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-2.935 8.246 2.686 2.645c.34.335.34.883 0 1.218l-2.686 2.645a.858.858 0 0 1-1.213-.009.854.854 0 0 1 .009-1.21l1.05-1.035H7.984a.992.992 0 0 1-.984-1c0-.552.44-1 .984-1h5.928l-1.051-1.036a.854.854 0 0 1-.085-1.121l.076-.088a.858.858 0 0 1 1.213-.009ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-calendar-impact-factor-medium" viewBox="0 0 24 24"><path d="M17 2a1 1 0 0 1 1 1v1h1.5C20.817 4 22 5.183 22 6.5v13c0 1.317-1.183 2.5-2.5 2.5h-15C3.183 22 2 20.817 2 19.5v-13C2 5.183 3.183 4 4.5 4a1 1 0 1 1 0 2c-.212 0-.5.288-.5.5v13c0 .212.288.5.5.5h15c.212 0 .5-.288.5-.5v-13c0-.212-.288-.5-.5-.5H18v1a1 1 0 0 1-2 0V3a1 1 0 0 1 1-1Zm-3.2 6.924a.48.48 0 0 1 .125.544l-1.52 3.283h2.304c.27 0 .491.215.491.483a.477.477 0 0 1-.13.327l-4.18 4.484a.498.498 0 0 1-.69.031.48.48 0 0 1-.125-.544l1.52-3.284H9.291a.487.487 0 0 1-.491-.482c0-.121.047-.238.13-.327l4.18-4.484a.498.498 0 0 1 .69-.031ZM7.5 2a1 1 0 0 1 1 1v1H14a1 1 0 0 1 0 2H8.5v1a1 1 0 1 1-2 0V3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-call-papers-medium" viewBox="0 0 24 24"><g><path d="m20.707 2.883-1.414 1.414a1 1 0 0 0 1.414 1.414l1.414-1.414a1 1 0 0 0-1.414-1.414Z"/><path d="M6 16.054c0 2.026 1.052 2.943 3 2.943a1 1 0 1 1 0 2c-2.996 0-5-1.746-5-4.943v-1.227a4.068 4.068 0 0 1-1.83-1.189 4.553 4.553 0 0 1-.87-1.455 4.868 4.868 0 0 1-.3-1.686c0-1.17.417-2.298 1.17-3.14.38-.426.834-.767 1.338-1 .51-.237 1.06-.36 1.617-.36L6.632 6H7l7.932-2.895A2.363 2.363 0 0 1 18 5.36v9.28a2.36 2.36 0 0 1-3.069 2.25l.084.03L7 14.997H6v1.057Zm9.637-11.057a.415.415 0 0 0-.083.008L8 7.638v5.536l7.424 1.786.104.02c.035.01.072.02.109.02.2 0 .363-.16.363-.36V5.36c0-.2-.163-.363-.363-.363Zm-9.638 3h-.874a1.82 1.82 0 0 0-.625.111l-.15.063a2.128 2.128 0 0 0-.689.517c-.42.47-.661 1.123-.661 1.81 0 .34.06.678.176.992.114.308.28.585.485.816.4.447.925.691 1.464.691h.874v-5Z" clip-rule="evenodd"/><path d="M20 8.997h2a1 1 0 1 1 0 2h-2a1 1 0 1 1 0-2ZM20.707 14.293l1.414 1.414a1 1 0 0 1-1.414 1.414l-1.414-1.414a1 1 0 0 1 1.414-1.414Z"/></g></symbol><symbol id="icon-eds-i-card-medium" viewBox="0 0 24 24"><path d="M19.615 2c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23Zm0 2H4.385c-.213 0-.265.034-.317.14A.71.71 0 0 0 4 4.385v15.23c0 .213.034.265.14.317a.71.71 0 0 0 .245.068h15.23c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM17 16a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm0-3a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h10Zm-.5-7A1.5 1.5 0 0 1 18 7.5v3a1.5 1.5 0 0 1-1.5 1.5h-9A1.5 1.5 0 0 1 6 10.5v-3A1.5 1.5 0 0 1 7.5 6h9ZM16 8H8v2h8V8Z"/></symbol><symbol id="icon-eds-i-cart-medium" viewBox="0 0 24 24"><path d="M5.76 1a1 1 0 0 1 .994.902L7.155 6h13.34c.18 0 .358.02.532.057l.174.045a2.5 2.5 0 0 1 1.693 3.103l-2.069 7.03c-.36 1.099-1.398 1.823-2.49 1.763H8.65c-1.272.015-2.352-.927-2.546-2.244L4.852 3H2a1 1 0 0 1-.993-.883L1 2a1 1 0 0 1 1-1h3.76Zm2.328 14.51a.555.555 0 0 0 .55.488l9.751.001a.533.533 0 0 0 .527-.357l2.059-7a.5.5 0 0 0-.48-.642H7.351l.737 7.51ZM18 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4ZM8 19a2 2 0 1 1 0 4 2 2 0 0 1 0-4Z"/></symbol><symbol id="icon-eds-i-check-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm5.125 4.72a1 1 0 0 1 .156 1.405l-6 7.5a1 1 0 0 1-1.421.143l-3-2.5a1 1 0 0 1 1.28-1.536l2.217 1.846 5.362-6.703a1 1 0 0 1 1.406-.156Z"/></symbol><symbol id="icon-eds-i-check-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm5.125 6.72a1 1 0 0 0-1.406.155l-5.362 6.703-2.217-1.846a1 1 0 1 0-1.28 1.536l3 2.5a1 1 0 0 0 1.42-.143l6-7.5a1 1 0 0 0-.155-1.406Z"/></symbol><symbol id="icon-eds-i-chevron-down-medium" viewBox="0 0 24 24"><path d="M3.305 8.28a1 1 0 0 0-.024 1.415l7.495 7.762c.314.345.757.543 1.224.543.467 0 .91-.198 1.204-.522l7.515-7.783a1 1 0 1 0-1.438-1.39L12 15.845l-7.28-7.54A1 1 0 0 0 3.4 8.2l-.096.082Z"/></symbol><symbol id="icon-eds-i-chevron-down-small" viewBox="0 0 16 16"><path d="M13.692 5.278a1 1 0 0 1 .03 1.414L9.103 11.51a1.491 1.491 0 0 1-2.188.019L2.278 6.692a1 1 0 0 1 1.444-1.384L8 9.771l4.278-4.463a1 1 0 0 1 1.318-.111l.096.081Z"/></symbol><symbol id="icon-eds-i-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.72 3.305a1 1 0 0 0-1.415-.024l-7.762 7.495A1.655 1.655 0 0 0 6 12c0 .467.198.91.522 1.204l7.783 7.515a1 1 0 1 0 1.39-1.438L8.155 12l7.54-7.28A1 1 0 0 0 15.8 3.4l-.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-left-small" viewBox="0 0 16 16"><path d="M10.722 2.308a1 1 0 0 0-1.414-.03L4.49 6.897a1.491 1.491 0 0 0-.019 2.188l4.838 4.637a1 1 0 1 0 1.384-1.444L6.229 8l4.463-4.278a1 1 0 0 0 .111-1.318l-.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28 3.305a1 1 0 0 1 1.415-.024l7.762 7.495c.345.314.543.757.543 1.224 0 .467-.198.91-.522 1.204l-7.783 7.515a1 1 0 1 1-1.39-1.438L15.845 12l-7.54-7.28A1 1 0 0 1 8.2 3.4l.082-.096Z"/></symbol><symbol id="icon-eds-i-chevron-right-small" viewBox="0 0 16 16"><path d="M5.278 2.308a1 1 0 0 1 1.414-.03l4.819 4.619a1.491 1.491 0 0 1 .019 2.188l-4.838 4.637a1 1 0 1 1-1.384-1.444L9.771 8 5.308 3.722a1 1 0 0 1-.111-1.318l.081-.096Z"/></symbol><symbol id="icon-eds-i-chevron-up-medium" viewBox="0 0 24 24"><path d="M20.695 15.72a1 1 0 0 0 .024-1.415l-7.495-7.762A1.655 1.655 0 0 0 12 6c-.467 0-.91.198-1.204.522l-7.515 7.783a1 1 0 1 0 1.438 1.39L12 8.155l7.28 7.54a1 1 0 0 0 1.319.106l.096-.082Z"/></symbol><symbol id="icon-eds-i-chevron-up-small" viewBox="0 0 16 16"><path d="M13.692 10.722a1 1 0 0 0 .03-1.414L9.103 4.49a1.491 1.491 0 0 0-2.188-.019L2.278 9.308a1 1 0 0 0 1.444 1.384L8 6.229l4.278 4.463a1 1 0 0 0 1.318.111l.096-.081Z"/></symbol><symbol id="icon-eds-i-citations-medium" viewBox="0 0 24 24"><path d="M15.59 1a1 1 0 0 1 .706.291l5.41 5.385a1 1 0 0 1 .294.709v13.077c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742h-5.843a1 1 0 1 1 0-2h5.843a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.8L15.178 3H5.545a.543.543 0 0 0-.538.451L5 3.538v8.607a1 1 0 0 1-2 0V3.538A2.542 2.542 0 0 1 5.545 1h10.046ZM5.483 14.35c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Zm5 0c.197.26.17.62-.049.848l-.095.083-.016.011c-.36.24-.628.45-.804.634-.393.409-.59.93-.59 1.562.077-.019.192-.028.345-.028.442 0 .84.158 1.195.474.355.316.532.716.532 1.2 0 .501-.173.9-.518 1.198-.345.298-.767.446-1.266.446-.672 0-1.209-.195-1.612-.585-.403-.39-.604-.976-.604-1.757 0-.744.11-1.39.33-1.938.222-.549.49-1.009.807-1.38a4.28 4.28 0 0 1 .992-.88c.07-.043.148-.087.232-.133a.881.881 0 0 1 1.121.245Z"/></symbol><symbol id="icon-eds-i-clipboard-check-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-1.909 4.205a1 1 0 0 1 .19 1.401l-5.334 7a1 1 0 0 1-1.344.23l-2.667-1.75a1 1 0 1 1 1.098-1.672l1.887 1.238 4.769-6.258a1 1 0 0 1 1.401-.19ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-clipboard-report-medium" viewBox="0 0 24 24"><path d="M14.4 1c1.238 0 2.274.865 2.536 2.024L18.5 3C19.886 3 21 4.14 21 5.535v14.93C21 21.86 19.886 23 18.5 23h-13C4.114 23 3 21.86 3 20.465V5.535C3 4.14 4.114 3 5.5 3h1.57c.27-1.147 1.3-2 2.53-2h4.8Zm4.115 4-1.59.024A2.601 2.601 0 0 1 14.4 7H9.6c-1.23 0-2.26-.853-2.53-2H5.5c-.27 0-.5.234-.5.535v14.93c0 .3.23.535.5.535h13c.27 0 .5-.234.5-.535V5.535c0-.3-.23-.535-.485-.535Zm-2.658 10.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857Zm0-3.929a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h7.857ZM14.4 3H9.6a.6.6 0 0 0-.6.6v.8a.6.6 0 0 0 .6.6h4.8a.6.6 0 0 0 .6-.6v-.8a.6.6 0 0 0-.6-.6Z"/></symbol><symbol id="icon-eds-i-close-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM8.707 7.293 12 10.585l3.293-3.292a1 1 0 0 1 1.414 1.414L13.415 12l3.292 3.293a1 1 0 0 1-1.414 1.414L12 13.415l-3.293 3.292a1 1 0 1 1-1.414-1.414L10.585 12 7.293 8.707a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-cloud-upload-medium" viewBox="0 0 24 24"><path d="m12.852 10.011.028-.004L13 10l.075.003.126.017.086.022.136.052.098.052.104.074.082.073 3 3a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L14 13.416V20a1 1 0 0 1-2 0v-6.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l3-3 .112-.097.11-.071.114-.054.105-.035.118-.025Zm.587-7.962c3.065.362 5.497 2.662 5.992 5.562l.013.085.207.073c2.117.782 3.496 2.845 3.337 5.097l-.022.226c-.297 2.561-2.503 4.491-5.124 4.502a1 1 0 1 1-.009-2c1.619-.007 2.967-1.186 3.147-2.733.179-1.542-.86-2.979-2.487-3.353-.512-.149-.894-.579-.981-1.165-.21-2.237-2-4.035-4.308-4.308-2.31-.273-4.497 1.06-5.25 3.19l-.049.113c-.234.468-.718.756-1.176.743-1.418.057-2.689.857-3.32 2.084a3.668 3.668 0 0 0 .262 3.798c.796 1.136 2.169 1.764 3.583 1.635a1 1 0 1 1 .182 1.992c-2.125.194-4.193-.753-5.403-2.48a5.668 5.668 0 0 1-.403-5.86c.85-1.652 2.449-2.79 4.323-3.092l.287-.039.013-.028c1.207-2.741 4.125-4.404 7.186-4.042Z"/></symbol><symbol id="icon-eds-i-collection-medium" viewBox="0 0 24 24"><path d="M21 7a1 1 0 0 1 1 1v12.5a2.5 2.5 0 0 1-2.5 2.5H8a1 1 0 0 1 0-2h11.5a.5.5 0 0 0 .5-.5V8a1 1 0 0 1 1-1Zm-5.5-5A2.5 2.5 0 0 1 18 4.5v12a2.5 2.5 0 0 1-2.5 2.5h-11A2.5 2.5 0 0 1 2 16.5v-12A2.5 2.5 0 0 1 4.5 2h11Zm0 2h-11a.5.5 0 0 0-.5.5v12a.5.5 0 0 0 .5.5h11a.5.5 0 0 0 .5-.5v-12a.5.5 0 0 0-.5-.5ZM13 13a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6Zm0-3.5a1 1 0 0 1 0 2H7a1 1 0 0 1 0-2h6ZM13 6a1 1 0 0 1 0 2H7a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-conference-series-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M4.5 2A2.5 2.5 0 0 0 2 4.5v11A2.5 2.5 0 0 0 4.5 18h2.37l-2.534 2.253a1 1 0 0 0 1.328 1.494L9.88 18H11v3a1 1 0 1 0 2 0v-3h1.12l4.216 3.747a1 1 0 0 0 1.328-1.494L17.13 18h2.37a2.5 2.5 0 0 0 2.5-2.5v-11A2.5 2.5 0 0 0 19.5 2h-15ZM20 6V4.5a.5.5 0 0 0-.5-.5h-15a.5.5 0 0 0-.5.5V6h16ZM4 8v7.5a.5.5 0 0 0 .5.5h15a.5.5 0 0 0 .5-.5V8H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-delivery-medium" viewBox="0 0 24 24"><path d="M8.51 20.598a3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 4.161 19L3.5 19A2.5 2.5 0 0 1 1 16.5v-11A2.5 2.5 0 0 1 3.5 3h10a2.5 2.5 0 0 1 2.45 2.004L16 5h2.527c.976 0 1.855.585 2.27 1.49l2.112 4.62a1 1 0 0 1 .091.416v4.856C23 17.814 21.889 19 20.484 19h-.523a1.01 1.01 0 0 1-.121-.007 2.96 2.96 0 0 1-1.33 1.605 3.037 3.037 0 0 1-3.02 0A2.968 2.968 0 0 1 14.161 19H9.838a2.968 2.968 0 0 1-1.327 1.597Zm-2.024-3.462a.955.955 0 0 0-.481.73L5.999 18l.001.022a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0A.97.97 0 0 0 8 17.978a.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0Zm10 0a.955.955 0 0 0-.481.73l-.005.156a.944.944 0 0 0 .388.777l.098.065c.316.181.712.181 1.028 0a.97.97 0 0 0 .486-.886.95.95 0 0 0-.486-.842 1.037 1.037 0 0 0-1.028 0ZM21 12h-5v3.17a3.038 3.038 0 0 1 2.51.232 2.993 2.993 0 0 1 1.277 1.45l.058.155.058-.005.581-.002c.27 0 .516-.263.516-.618V12Zm-7.5-7h-10a.5.5 0 0 0-.5.5v11a.5.5 0 0 0 .5.5h.662a2.964 2.964 0 0 1 1.155-1.491l.172-.107a3.037 3.037 0 0 1 3.022 0A2.987 2.987 0 0 1 9.843 17H13.5a.5.5 0 0 0 .5-.5v-11a.5.5 0 0 0-.5-.5Zm5.027 2H16v3h4.203l-1.224-2.677a.532.532 0 0 0-.375-.316L18.527 7Z"/></symbol><symbol id="icon-eds-i-download-medium" viewBox="0 0 24 24"><path d="M22 18.5a3.5 3.5 0 0 1-3.5 3.5h-13A3.5 3.5 0 0 1 2 18.5V18a1 1 0 0 1 2 0v.5A1.5 1.5 0 0 0 5.5 20h13a1.5 1.5 0 0 0 1.5-1.5V18a1 1 0 0 1 2 0v.5Zm-3.293-7.793-6 6-.063.059-.093.069-.081.048-.105.049-.104.034-.056.013-.118.017L12 17l-.076-.003-.122-.017-.113-.03-.085-.032-.063-.03-.098-.058-.06-.043-.05-.043-6.04-6.037a1 1 0 0 1 1.414-1.414l4.294 4.29L11 3a1 1 0 0 1 2 0l.001 10.585 4.292-4.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414Z"/></symbol><symbol id="icon-eds-i-edit-medium" viewBox="0 0 24 24"><path d="M17.149 2a2.38 2.38 0 0 1 1.699.711l2.446 2.46a2.384 2.384 0 0 1 .005 3.38L10.01 19.906a1 1 0 0 1-.434.257l-6.3 1.8a1 1 0 0 1-1.237-1.237l1.8-6.3a1 1 0 0 1 .257-.434L15.443 2.718A2.385 2.385 0 0 1 17.15 2Zm-3.874 5.689-7.586 7.536-1.234 4.319 4.318-1.234 7.54-7.582-3.038-3.039ZM17.149 4a.395.395 0 0 0-.286.126L14.695 6.28l3.029 3.029 2.162-2.173a.384.384 0 0 0 .106-.197L20 6.864c0-.103-.04-.2-.119-.278l-2.457-2.47A.385.385 0 0 0 17.149 4Z"/></symbol><symbol id="icon-eds-i-education-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M12.41 2.088a1 1 0 0 0-.82 0l-10 4.5a1 1 0 0 0 0 1.824L3 9.047v7.124A3.001 3.001 0 0 0 4 22a3 3 0 0 0 1-5.83V9.948l1 .45V14.5a1 1 0 0 0 .087.408L7 14.5c-.913.408-.912.41-.912.41l.001.003.003.006.007.015a1.988 1.988 0 0 0 .083.16c.054.097.131.225.236.373.21.297.53.68.993 1.057C8.351 17.292 9.824 18 12 18c2.176 0 3.65-.707 4.589-1.476.463-.378.783-.76.993-1.057a4.162 4.162 0 0 0 .319-.533l.007-.015.003-.006v-.003h.002s0-.002-.913-.41l.913.408A1 1 0 0 0 18 14.5v-4.103l4.41-1.985a1 1 0 0 0 0-1.824l-10-4.5ZM16 11.297l-3.59 1.615a1 1 0 0 1-.82 0L8 11.297v2.94a3.388 3.388 0 0 0 .677.739C9.267 15.457 10.294 16 12 16s2.734-.543 3.323-1.024a3.388 3.388 0 0 0 .677-.739v-2.94ZM4.437 7.5 12 4.097 19.563 7.5 12 10.903 4.437 7.5ZM3 19a1 1 0 1 1 2 0 1 1 0 0 1-2 0Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-error-diamond-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008Zm0 2a.646.646 0 0 0-.38.123l-.093.08-8.34 8.34a.646.646 0 0 0-.18.355L3 12c0 .171.068.336.19.457l8.353 8.354a.646.646 0 0 0 .914 0l8.354-8.354a.646.646 0 0 0-.001-.914l-8.351-8.354A.646.646 0 0 0 12.002 3ZM12 14.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-error-filled-medium" viewBox="0 0 24 24"><path d="M12.002 1c.702 0 1.375.279 1.871.775l8.35 8.353a2.646 2.646 0 0 1 .001 3.744l-8.353 8.353a2.646 2.646 0 0 1-3.742 0l-8.353-8.353a2.646 2.646 0 0 1 0-3.744l8.353-8.353.156-.142c.424-.362.952-.58 1.507-.625l.21-.008ZM12 14.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-eds-i-external-link-medium" viewBox="0 0 24 24"><path d="M9 2a1 1 0 1 1 0 2H4.6c-.371 0-.6.209-.6.5v15c0 .291.229.5.6.5h14.8c.371 0 .6-.209.6-.5V15a1 1 0 0 1 2 0v4.5c0 1.438-1.162 2.5-2.6 2.5H4.6C3.162 22 2 20.938 2 19.5v-15C2 3.062 3.162 2 4.6 2H9Zm6 0h6l.075.003.126.017.111.03.111.044.098.052.096.067.09.08c.036.035.068.073.097.112l.071.11.054.114.035.105.03.148L22 3v6a1 1 0 0 1-2 0V5.414l-6.693 6.693a1 1 0 0 1-1.414-1.414L18.584 4H15a1 1 0 0 1-.993-.883L14 3a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-external-link-small" viewBox="0 0 16 16"><path d="M5 1a1 1 0 1 1 0 2l-2-.001V13L13 13v-2a1 1 0 0 1 2 0v2c0 1.15-.93 2-2.067 2H3.067C1.93 15 1 14.15 1 13V3c0-1.15.93-2 2.067-2H5Zm4 0h5l.075.003.126.017.111.03.111.044.098.052.096.067.09.08.044.047.073.093.051.083.054.113.035.105.03.148L15 2v5a1 1 0 0 1-2 0V4.414L9.107 8.307a1 1 0 0 1-1.414-1.414L11.584 3H9a1 1 0 0 1-.993-.883L8 2a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-download-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM12 7a1 1 0 0 1 1 1v6.585l2.293-2.292a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-4 4a1.008 1.008 0 0 1-.112.097l-.11.071-.114.054-.105.035-.149.03L12 18l-.075-.003-.126-.017-.111-.03-.111-.044-.098-.052-.096-.067-.09-.08-4-4a1 1 0 0 1 1.414-1.414L11 14.585V8a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-file-report-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962c0 .674-.269 1.32-.747 1.796a2.549 2.549 0 0 1-1.798.742H5.545c-.674 0-1.32-.267-1.798-.742A2.535 2.535 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .142.057.278.158.379.102.102.242.159.387.159h12.91a.549.549 0 0 0 .387-.16.535.535 0 0 0 .158-.378V7.915L14.085 3ZM16 17a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-3a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-4.793-6.207L13 9.585l1.793-1.792a1 1 0 0 1 1.32-.083l.094.083a1 1 0 0 1 0 1.414l-2.5 2.5a1 1 0 0 1-1.414 0L10.5 9.915l-1.793 1.792a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l2.5-2.5a1 1 0 0 1 1.414 0Z"/></symbol><symbol id="icon-eds-i-file-text-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3ZM16 15a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm0-4a1 1 0 0 1 0 2H8a1 1 0 0 1 0-2h8Zm-5-4a1 1 0 0 1 0 2H8a1 1 0 1 1 0-2h3Z"/></symbol><symbol id="icon-eds-i-file-upload-medium" viewBox="0 0 24 24"><path d="M14.5 1a1 1 0 0 1 .707.293l5.5 5.5A1 1 0 0 1 21 7.5v12.962A2.542 2.542 0 0 1 18.455 23H5.545A2.542 2.542 0 0 1 3 20.462V3.538A2.542 2.542 0 0 1 5.545 1H14.5Zm-.415 2h-8.54A.542.542 0 0 0 5 3.538v16.924c0 .296.243.538.545.538h12.91a.542.542 0 0 0 .545-.538V7.915L14.085 3Zm-2.233 4.011.058-.007L12 7l.075.003.126.017.111.03.111.044.098.052.104.074.082.073 4 4a1 1 0 0 1 0 1.414l-.094.083a1 1 0 0 1-1.32-.083L13 10.415V17a1 1 0 0 1-2 0v-6.585l-2.293 2.292a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l4-4 .112-.097.11-.071.114-.054.105-.035.118-.025Z"/></symbol><symbol id="icon-eds-i-filter-medium" viewBox="0 0 24 24"><path d="M21 2a1 1 0 0 1 .82 1.573L15 13.314V18a1 1 0 0 1-.31.724l-.09.076-4 3A1 1 0 0 1 9 21v-7.684L2.18 3.573a1 1 0 0 1 .707-1.567L3 2h18Zm-1.921 2H4.92l5.9 8.427a1 1 0 0 1 .172.45L11 13v6l2-1.5V13a1 1 0 0 1 .117-.469l.064-.104L19.079 4Z"/></symbol><symbol id="icon-eds-i-funding-medium" viewBox="0 0 24 24"><path fill-rule="evenodd" d="M23 8A7 7 0 1 0 9 8a7 7 0 0 0 14 0ZM9.006 12.225A4.07 4.07 0 0 0 6.12 11.02H2a.979.979 0 1 0 0 1.958h4.12c.558 0 1.094.222 1.489.617l2.207 2.288c.27.27.27.687.012.944a.656.656 0 0 1-.928 0L7.744 15.67a.98.98 0 0 0-1.386 1.384l1.157 1.158c.535.536 1.244.791 1.946.765l.041.002h6.922c.874 0 1.597.748 1.597 1.688 0 .203-.146.354-.309.354H7.755c-.487 0-.96-.178-1.339-.504L2.64 17.259a.979.979 0 0 0-1.28 1.482L5.137 22c.733.631 1.66.979 2.618.979h9.957c1.26 0 2.267-1.043 2.267-2.312 0-2.006-1.584-3.646-3.555-3.646h-4.529a2.617 2.617 0 0 0-.681-2.509l-2.208-2.287ZM16 3a5 5 0 1 0 0 10 5 5 0 0 0 0-10Zm.979 3.5a.979.979 0 1 0-1.958 0v3a.979.979 0 1 0 1.958 0v-3Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-hashtag-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18ZM9.52 18.189a1 1 0 1 1-1.964-.378l.437-2.274H6a1 1 0 1 1 0-2h2.378l.592-3.076H6a1 1 0 0 1 0-2h3.354l.51-2.65a1 1 0 1 1 1.964.378l-.437 2.272h3.04l.51-2.65a1 1 0 1 1 1.964.378l-.438 2.272H18a1 1 0 0 1 0 2h-1.917l-.592 3.076H18a1 1 0 0 1 0 2h-2.893l-.51 2.652a1 1 0 1 1-1.964-.378l.437-2.274h-3.04l-.51 2.652Zm.895-4.652h3.04l.591-3.076h-3.04l-.591 3.076Z"/></symbol><symbol id="icon-eds-i-home-medium" viewBox="0 0 24 24"><path d="M5 22a1 1 0 0 1-1-1v-8.586l-1.293 1.293a1 1 0 0 1-1.32.083l-.094-.083a1 1 0 0 1 0-1.414l10-10a1 1 0 0 1 1.414 0l10 10a1 1 0 0 1-1.414 1.414L20 12.415V21a1 1 0 0 1-1 1H5Zm7-17.585-6 5.999V20h5v-4a1 1 0 0 1 2 0v4h5v-9.585l-6-6Z"/></symbol><symbol id="icon-eds-i-image-medium" viewBox="0 0 24 24"><path d="M19.615 2A2.385 2.385 0 0 1 22 4.385v15.23A2.385 2.385 0 0 1 19.615 22H4.385A2.385 2.385 0 0 1 2 19.615V4.385A2.385 2.385 0 0 1 4.385 2h15.23Zm0 2H4.385A.385.385 0 0 0 4 4.385v15.23c0 .213.172.385.385.385h1.244l10.228-8.76a1 1 0 0 1 1.254-.037L20 13.392V4.385A.385.385 0 0 0 19.615 4Zm-3.07 9.283L8.703 20h10.912a.385.385 0 0 0 .385-.385v-3.713l-3.455-2.619ZM9.5 6a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-impact-factor-medium" viewBox="0 0 24 24"><path d="M16.49 2.672c.74.694.986 1.765.632 2.712l-.04.1-1.549 3.54h1.477a2.496 2.496 0 0 1 2.485 2.34l.005.163c0 .618-.23 1.21-.642 1.675l-7.147 7.961a2.48 2.48 0 0 1-3.554.165 2.512 2.512 0 0 1-.633-2.712l.042-.103L9.108 15H7.46c-1.393 0-2.379-1.11-2.455-2.369L5 12.473c0-.593.142-1.145.628-1.692l7.307-7.944a2.48 2.48 0 0 1 3.555-.165ZM14.43 4.164l-7.33 7.97c-.083.093-.101.214-.101.34 0 .277.19.526.46.526h4.163l.097-.009c.015 0 .03.003.046.009.181.078.264.32.186.5l-2.554 5.817a.512.512 0 0 0 .127.552.48.48 0 0 0 .69-.033l7.155-7.97a.513.513 0 0 0 .13-.34.497.497 0 0 0-.49-.502h-3.988a.355.355 0 0 1-.328-.497l2.555-5.844a.512.512 0 0 0-.127-.552.48.48 0 0 0-.69.033Z"/></symbol><symbol id="icon-eds-i-info-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 7a1 1 0 0 1 1 1v5h1.5a1 1 0 0 1 0 2h-5a1 1 0 0 1 0-2H11v-4h-.5a1 1 0 0 1-.993-.883L9.5 11a1 1 0 0 1 1-1H12Zm0-4.5a1.5 1.5 0 0 1 .144 2.993L12 8.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-info-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 9h-1.5a1 1 0 0 0-1 1l.007.117A1 1 0 0 0 10.5 12h.5v4H9.5a1 1 0 0 0 0 2h5a1 1 0 0 0 0-2H13v-5a1 1 0 0 0-1-1Zm0-4.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 5.5Z"/></symbol><symbol id="icon-eds-i-journal-medium" viewBox="0 0 24 24"><path d="M18.5 1A2.5 2.5 0 0 1 21 3.5v14a2.5 2.5 0 0 1-2.5 2.5h-13a.5.5 0 1 0 0 1H20a1 1 0 0 1 0 2H5.5A2.5 2.5 0 0 1 3 20.5v-17A2.5 2.5 0 0 1 5.5 1h13ZM7 3H5.5a.5.5 0 0 0-.5.5v14.549l.016-.002c.104-.02.211-.035.32-.042L5.5 18H7V3Zm11.5 0H9v15h9.5a.5.5 0 0 0 .5-.5v-14a.5.5 0 0 0-.5-.5ZM16 5a1 1 0 0 1 1 1v4a1 1 0 0 1-1 1h-5a1 1 0 0 1-1-1V6a1 1 0 0 1 1-1h5Zm-1 2h-3v2h3V7Z"/></symbol><symbol id="icon-eds-i-mail-medium" viewBox="0 0 24 24"><path d="M20.462 3C21.875 3 23 4.184 23 5.619v12.762C23 19.816 21.875 21 20.462 21H3.538C2.125 21 1 19.816 1 18.381V5.619C1 4.184 2.125 3 3.538 3h16.924ZM21 8.158l-7.378 6.258a2.549 2.549 0 0 1-3.253-.008L3 8.16v10.222c0 .353.253.619.538.619h16.924c.285 0 .538-.266.538-.619V8.158ZM20.462 5H3.538c-.264 0-.5.228-.534.542l8.65 7.334c.2.165.492.165.684.007l8.656-7.342-.001-.025c-.044-.3-.274-.516-.531-.516Z"/></symbol><symbol id="icon-eds-i-mail-send-medium" viewBox="0 0 24 24"><path d="M20.444 5a2.562 2.562 0 0 1 2.548 2.37l.007.078.001.123v7.858A2.564 2.564 0 0 1 20.444 18H9.556A2.564 2.564 0 0 1 7 15.429l.001-7.977.007-.082A2.561 2.561 0 0 1 9.556 5h10.888ZM21 9.331l-5.46 3.51a1 1 0 0 1-1.08 0L9 9.332v6.097c0 .317.251.571.556.571h10.888a.564.564 0 0 0 .556-.571V9.33ZM20.444 7H9.556a.543.543 0 0 0-.32.105l5.763 3.706 5.766-3.706a.543.543 0 0 0-.32-.105ZM4.308 5a1 1 0 1 1 0 2H2a1 1 0 1 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Zm0 5.5a1 1 0 0 1 0 2H2a1 1 0 0 1 0-2h2.308Z"/></symbol><symbol id="icon-eds-i-mentions-medium" viewBox="0 0 24 24"><path d="m9.452 1.293 5.92 5.92 2.92-2.92a1 1 0 0 1 1.415 1.414l-2.92 2.92 5.92 5.92a1 1 0 0 1 0 1.415 10.371 10.371 0 0 1-10.378 2.584l.652 3.258A1 1 0 0 1 12 23H2a1 1 0 0 1-.874-1.486l4.789-8.62C4.194 9.074 4.9 4.43 8.038 1.292a1 1 0 0 1 1.414 0Zm-2.355 13.59L3.699 21h7.081l-.689-3.442a10.392 10.392 0 0 1-2.775-2.396l-.22-.28Zm1.69-11.427-.07.09a8.374 8.374 0 0 0 11.737 11.737l.089-.071L8.787 3.456Z"/></symbol><symbol id="icon-eds-i-menu-medium" viewBox="0 0 24 24"><path d="M21 4a1 1 0 0 1 0 2H3a1 1 0 1 1 0-2h18Zm-4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h14Zm4 7a1 1 0 0 1 0 2H3a1 1 0 0 1 0-2h18Z"/></symbol><symbol id="icon-eds-i-metrics-medium" viewBox="0 0 24 24"><path d="M3 22a1 1 0 0 1-1-1V3a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v7h4V8a1 1 0 0 1 1-1h6a1 1 0 0 1 1 1v13a1 1 0 0 1-.883.993L21 22H3Zm17-2V9h-4v11h4Zm-6-8h-4v8h4v-8ZM8 4H4v16h4V4Z"/></symbol><symbol id="icon-eds-i-news-medium" viewBox="0 0 24 24"><path d="M17.384 3c.975 0 1.77.787 1.77 1.762v13.333c0 .462.354.846.815.899l.107.006.109-.006a.915.915 0 0 0 .809-.794l.006-.105V8.19a1 1 0 0 1 2 0v9.905A2.914 2.914 0 0 1 20.077 21H3.538a2.547 2.547 0 0 1-1.644-.601l-.147-.135A2.516 2.516 0 0 1 1 18.476V4.762C1 3.787 1.794 3 2.77 3h14.614Zm-.231 2H3v13.476c0 .11.035.216.1.304l.054.063c.101.1.24.157.384.157l13.761-.001-.026-.078a2.88 2.88 0 0 1-.115-.655l-.004-.17L17.153 5ZM14 15.021a.979.979 0 1 1 0 1.958H6a.979.979 0 1 1 0-1.958h8Zm0-8c.54 0 .979.438.979.979v4c0 .54-.438.979-.979.979H6A.979.979 0 0 1 5.021 12V8c0-.54.438-.979.979-.979h8Zm-.98 1.958H6.979v2.041h6.041V8.979Z"/></symbol><symbol id="icon-eds-i-newsletter-medium" viewBox="0 0 24 24"><path d="M21 10a1 1 0 0 1 1 1v9.5a2.5 2.5 0 0 1-2.5 2.5h-15A2.5 2.5 0 0 1 2 20.5V11a1 1 0 0 1 2 0v.439l8 4.888 8-4.889V11a1 1 0 0 1 1-1Zm-1 3.783-7.479 4.57a1 1 0 0 1-1.042 0l-7.48-4.57V20.5a.5.5 0 0 0 .501.5h15a.5.5 0 0 0 .5-.5v-6.717ZM15 9a1 1 0 0 1 0 2H9a1 1 0 0 1 0-2h6Zm2.5-8A2.5 2.5 0 0 1 20 3.5V9a1 1 0 0 1-2 0V3.5a.5.5 0 0 0-.5-.5h-11a.5.5 0 0 0-.5.5V9a1 1 0 1 1-2 0V3.5A2.5 2.5 0 0 1 6.5 1h11ZM15 5a1 1 0 0 1 0 2H9a1 1 0 1 1 0-2h6Z"/></symbol><symbol id="icon-eds-i-notifcation-medium" viewBox="0 0 24 24"><path d="M14 20a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM3 18l-.133-.007c-1.156-.124-1.156-1.862 0-1.986l.3-.012C4.32 15.923 5 15.107 5 14V9.5C5 5.368 8.014 2 12 2s7 3.368 7 7.5V14c0 1.107.68 1.923 1.832 1.995l.301.012c1.156.124 1.156 1.862 0 1.986L21 18H3Zm9-14C9.17 4 7 6.426 7 9.5V14c0 .671-.146 1.303-.416 1.858L6.51 16h10.979l-.073-.142a4.192 4.192 0 0 1-.412-1.658L17 14V9.5C17 6.426 14.83 4 12 4Z"/></symbol><symbol id="icon-eds-i-publish-medium" viewBox="0 0 24 24"><g><path d="M16.296 1.291A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V13a1 1 0 1 0 2 0V3.538l.007-.087A.543.543 0 0 1 5.545 3h9.633L20 7.8v12.662a.534.534 0 0 1-.158.379.548.548 0 0 1-.387.159H11a1 1 0 1 0 0 2h8.455c.674 0 1.32-.267 1.798-.742A2.534 2.534 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385Z"/><path d="M10.762 16.647a1 1 0 0 0-1.525-1.294l-4.472 5.271-2.153-1.665a1 1 0 1 0-1.224 1.582l2.91 2.25a1 1 0 0 0 1.374-.144l5.09-6ZM16 10a1 1 0 1 1 0 2H8a1 1 0 1 1 0-2h8ZM12 7a1 1 0 0 0-1-1H8a1 1 0 1 0 0 2h3a1 1 0 0 0 1-1Z"/></g></symbol><symbol id="icon-eds-i-refresh-medium" viewBox="0 0 24 24"><g><path d="M7.831 5.636H6.032A8.76 8.76 0 0 1 9 3.631 8.549 8.549 0 0 1 12.232 3c.603 0 1.192.063 1.76.182C17.979 4.017 21 7.632 21 12a1 1 0 1 0 2 0c0-5.296-3.674-9.746-8.591-10.776A10.61 10.61 0 0 0 5 3.851V2.805a1 1 0 0 0-.987-1H4a1 1 0 0 0-1 1v3.831a1 1 0 0 0 1 1h3.831a1 1 0 0 0 .013-2h-.013ZM17.968 18.364c-1.59 1.632-3.784 2.636-6.2 2.636C6.948 21 3 16.993 3 12a1 1 0 1 0-2 0c0 6.053 4.799 11 10.768 11 2.788 0 5.324-1.082 7.232-2.85v1.045a1 1 0 1 0 2 0v-3.831a1 1 0 0 0-1-1h-3.831a1 1 0 0 0 0 2h1.799Z"/></g></symbol><symbol id="icon-eds-i-search-medium" viewBox="0 0 24 24"><path d="M11 1c5.523 0 10 4.477 10 10 0 2.4-.846 4.604-2.256 6.328l3.963 3.965a1 1 0 0 1-1.414 1.414l-3.965-3.963A9.959 9.959 0 0 1 11 21C5.477 21 1 16.523 1 11S5.477 1 11 1Zm0 2a8 8 0 1 0 0 16 8 8 0 0 0 0-16Z"/></symbol><symbol id="icon-eds-i-settings-medium" viewBox="0 0 24 24"><path d="M11.382 1h1.24a2.508 2.508 0 0 1 2.334 1.63l.523 1.378 1.59.933 1.444-.224c.954-.132 1.89.3 2.422 1.101l.095.155.598 1.066a2.56 2.56 0 0 1-.195 2.848l-.894 1.161v1.896l.92 1.163c.6.768.707 1.812.295 2.674l-.09.17-.606 1.08a2.504 2.504 0 0 1-2.531 1.25l-1.428-.223-1.589.932-.523 1.378a2.512 2.512 0 0 1-2.155 1.625L12.65 23h-1.27a2.508 2.508 0 0 1-2.334-1.63l-.524-1.379-1.59-.933-1.443.225c-.954.132-1.89-.3-2.422-1.101l-.095-.155-.598-1.066a2.56 2.56 0 0 1 .195-2.847l.891-1.161v-1.898l-.919-1.162a2.562 2.562 0 0 1-.295-2.674l.09-.17.606-1.08a2.504 2.504 0 0 1 2.531-1.25l1.43.223 1.618-.938.524-1.375.07-.167A2.507 2.507 0 0 1 11.382 1Zm.003 2a.509.509 0 0 0-.47.338l-.65 1.71a1 1 0 0 1-.434.51L7.6 6.85a1 1 0 0 1-.655.123l-1.762-.275a.497.497 0 0 0-.498.252l-.61 1.088a.562.562 0 0 0 .04.619l1.13 1.43a1 1 0 0 1 .216.62v2.585a1 1 0 0 1-.207.61L4.15 15.339a.568.568 0 0 0-.036.634l.601 1.072a.494.494 0 0 0 .484.26l1.78-.278a1 1 0 0 1 .66.126l2.2 1.292a1 1 0 0 1 .43.507l.648 1.71a.508.508 0 0 0 .467.338h1.263a.51.51 0 0 0 .47-.34l.65-1.708a1 1 0 0 1 .428-.507l2.201-1.292a1 1 0 0 1 .66-.126l1.763.275a.497.497 0 0 0 .498-.252l.61-1.088a.562.562 0 0 0-.04-.619l-1.13-1.43a1 1 0 0 1-.216-.62v-2.585a1 1 0 0 1 .207-.61l1.105-1.437a.568.568 0 0 0 .037-.634l-.601-1.072a.494.494 0 0 0-.484-.26l-1.78.278a1 1 0 0 1-.66-.126l-2.2-1.292a1 1 0 0 1-.43-.507l-.649-1.71A.508.508 0 0 0 12.62 3h-1.234ZM12 8a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-shipping-medium" viewBox="0 0 24 24"><path d="M16.515 2c1.406 0 2.706.728 3.352 1.902l2.02 3.635.02.042.036.089.031.105.012.058.01.073.004.075v11.577c0 .64-.244 1.255-.683 1.713a2.356 2.356 0 0 1-1.701.731H4.386a2.356 2.356 0 0 1-1.702-.731 2.476 2.476 0 0 1-.683-1.713V7.948c.01-.217.083-.43.22-.6L4.2 3.905C4.833 2.755 6.089 2.032 7.486 2h9.029ZM20 9H4v10.556a.49.49 0 0 0 .075.26l.053.07a.356.356 0 0 0 .257.114h15.23c.094 0 .186-.04.258-.115a.477.477 0 0 0 .127-.33V9Zm-2 7.5a1 1 0 0 1 0 2h-4a1 1 0 0 1 0-2h4ZM16.514 4H13v3h6.3l-1.183-2.13c-.288-.522-.908-.87-1.603-.87ZM11 3.999H7.51c-.679.017-1.277.36-1.566.887L4.728 7H11V3.999Z"/></symbol><symbol id="icon-eds-i-step-guide-medium" viewBox="0 0 24 24"><path d="M11.394 9.447a1 1 0 1 0-1.788-.894l-.88 1.759-.019-.02a1 1 0 1 0-1.414 1.415l1 1a1 1 0 0 0 1.601-.26l1.5-3ZM12 11a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM12 17a1 1 0 0 1 1-1h3a1 1 0 1 1 0 2h-3a1 1 0 0 1-1-1ZM10.947 14.105a1 1 0 0 1 .447 1.342l-1.5 3a1 1 0 0 1-1.601.26l-1-1a1 1 0 1 1 1.414-1.414l.02.019.879-1.76a1 1 0 0 1 1.341-.447Z"/><path d="M5.545 1A2.542 2.542 0 0 0 3 3.538v16.924A2.542 2.542 0 0 0 5.545 23h12.91A2.542 2.542 0 0 0 21 20.462V7.5a1 1 0 0 0-.293-.707l-5.5-5.5A1 1 0 0 0 14.5 1H5.545ZM5 3.538C5 3.245 5.24 3 5.545 3h8.54L19 7.914v12.547c0 .294-.24.539-.546.539H5.545A.542.542 0 0 1 5 20.462V3.538Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-submission-medium" viewBox="0 0 24 24"><g><path d="M5 3.538C5 3.245 5.24 3 5.545 3h9.633L20 7.8v12.662a.535.535 0 0 1-.158.379.549.549 0 0 1-.387.159H6a1 1 0 0 1-1-1v-2.5a1 1 0 1 0-2 0V20a3 3 0 0 0 3 3h13.455c.673 0 1.32-.266 1.798-.742A2.535 2.535 0 0 0 22 20.462V7.385a1 1 0 0 0-.294-.709l-5.41-5.385A1 1 0 0 0 15.591 1H5.545A2.542 2.542 0 0 0 3 3.538V7a1 1 0 0 0 2 0V3.538Z"/><path d="m13.707 13.707-4 4a1 1 0 0 1-1.414 0l-.083-.094a1 1 0 0 1 .083-1.32L10.585 14 2 14a1 1 0 1 1 0-2l8.583.001-2.29-2.294a1 1 0 0 1 1.414-1.414l4.037 4.04.043.05.043.06.059.098.03.063.031.085.03.113.017.122L14 13l-.004.087-.017.118-.013.056-.034.104-.049.105-.048.081-.07.093-.058.063Z"/></g></symbol><symbol id="icon-eds-i-table-1-medium" viewBox="0 0 24 24"><path d="M4.385 22a2.56 2.56 0 0 1-1.14-.279C2.485 21.341 2 20.614 2 19.615V4.385c0-.315.067-.716.279-1.14C2.659 2.485 3.386 2 4.385 2h15.23c.315 0 .716.067 1.14.279.76.38 1.245 1.107 1.245 2.106v15.23c0 .315-.067.716-.279 1.14-.38.76-1.107 1.245-2.106 1.245H4.385ZM4 19.615c0 .213.034.265.14.317a.71.71 0 0 0 .245.068H8v-4H4v3.615ZM20 16H10v4h9.615c.213 0 .265-.034.317-.14a.71.71 0 0 0 .068-.245V16Zm0-2v-4H10v4h10ZM4 14h4v-4H4v4ZM19.615 4H10v4h10V4.385c0-.213-.034-.265-.14-.317A.71.71 0 0 0 19.615 4ZM8 4H4.385l-.082.002c-.146.01-.19.047-.235.138A.71.71 0 0 0 4 4.385V8h4V4Z"/></symbol><symbol id="icon-eds-i-table-2-medium" viewBox="0 0 24 24"><path d="M4.384 22A2.384 2.384 0 0 1 2 19.616V4.384A2.384 2.384 0 0 1 4.384 2h15.232A2.384 2.384 0 0 1 22 4.384v15.232A2.384 2.384 0 0 1 19.616 22H4.384ZM10 15H4v4.616c0 .212.172.384.384.384H10v-5Zm5 0h-3v5h3v-5Zm5 0h-3v5h2.616a.384.384 0 0 0 .384-.384V15ZM10 9H4v4h6V9Zm5 0h-3v4h3V9Zm5 0h-3v4h3V9Zm-.384-5H4.384A.384.384 0 0 0 4 4.384V7h16V4.384A.384.384 0 0 0 19.616 4Z"/></symbol><symbol id="icon-eds-i-tag-medium" viewBox="0 0 24 24"><path d="m12.621 1.998.127.004L20.496 2a1.5 1.5 0 0 1 1.497 1.355L22 3.5l-.005 7.669c.038.456-.133.905-.447 1.206l-9.02 9.018a2.075 2.075 0 0 1-2.932 0l-6.99-6.99a2.075 2.075 0 0 1 .001-2.933L11.61 2.47c.246-.258.573-.418.881-.46l.131-.011Zm.286 2-8.885 8.886a.075.075 0 0 0 0 .106l6.987 6.988c.03.03.077.03.106 0l8.883-8.883L19.999 4l-7.092-.002ZM16 6.5a1.5 1.5 0 0 1 .144 2.993L16 9.5a1.5 1.5 0 0 1 0-3Z"/></symbol><symbol id="icon-eds-i-trash-medium" viewBox="0 0 24 24"><path d="M12 1c2.717 0 4.913 2.232 4.997 5H21a1 1 0 0 1 0 2h-1v12.5c0 1.389-1.152 2.5-2.556 2.5H6.556C5.152 23 4 21.889 4 20.5V8H3a1 1 0 1 1 0-2h4.003l.001-.051C7.114 3.205 9.3 1 12 1Zm6 7H6v12.5c0 .238.19.448.454.492l.102.008h10.888c.315 0 .556-.232.556-.5V8Zm-4 3a1 1 0 0 1 1 1v6.005a1 1 0 0 1-2 0V12a1 1 0 0 1 1-1Zm-4 0a1 1 0 0 1 1 1v6a1 1 0 0 1-2 0v-6a1 1 0 0 1 1-1Zm2-8c-1.595 0-2.914 1.32-2.996 3h5.991v-.02C14.903 4.31 13.589 3 12 3Z"/></symbol><symbol id="icon-eds-i-user-account-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 16c-1.806 0-3.52.994-4.664 2.698A8.947 8.947 0 0 0 12 21a8.958 8.958 0 0 0 4.664-1.301C15.52 17.994 13.806 17 12 17Zm0-14a9 9 0 0 0-6.25 15.476C7.253 16.304 9.54 15 12 15s4.747 1.304 6.25 3.475A9 9 0 0 0 12 3Zm0 3a4 4 0 1 1 0 8 4 4 0 0 1 0-8Zm0 2a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"/></symbol><symbol id="icon-eds-i-user-add-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a1 1 0 0 1 1 1v3h3a1 1 0 0 1 0 2h-3v3a1 1 0 0 1-2 0v-3h-3a1 1 0 0 1 0-2h3v-3a1 1 0 0 1 1-1Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Z"/></symbol><symbol id="icon-eds-i-user-assign-medium" viewBox="0 0 24 24"><path d="M16.226 13.298a1 1 0 0 1 1.414-.01l.084.093a1 1 0 0 1-.073 1.32L15.39 17H22a1 1 0 0 1 0 2h-6.611l2.262 2.298a1 1 0 0 1-1.425 1.404l-3.939-4a1 1 0 0 1 0-1.404l3.94-4Zm-3.771-.449a1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 10.5 20a1 1 0 0 1 .993.883L11.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-block-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm9 10a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm-5.545-.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM15 18a3 3 0 0 0 4.294 2.707l-4.001-4c-.188.391-.293.83-.293 1.293Zm3-3c-.463 0-.902.105-1.294.293l4.001 4A3 3 0 0 0 18 15Z"/></symbol><symbol id="icon-eds-i-user-check-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm13.647 12.237a1 1 0 0 1 .116 1.41l-5.091 6a1 1 0 0 1-1.375.144l-2.909-2.25a1 1 0 1 1 1.224-1.582l2.153 1.665 4.472-5.271a1 1 0 0 1 1.41-.116Zm-8.139-.977c.22.214.428.44.622.678a1 1 0 1 1-1.548 1.266 6.025 6.025 0 0 0-1.795-1.49.86.86 0 0 1-.163-.048l-.079-.036a5.721 5.721 0 0 0-2.62-.63l-.194.006c-2.76.134-5.022 2.177-5.592 4.864l-.035.175-.035.213c-.03.201-.05.405-.06.61L3.003 20 10 20a1 1 0 0 1 .993.883L11 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876l.005-.223.02-.356.02-.222.03-.248.022-.15c.02-.133.044-.265.071-.397.44-2.178 1.725-4.105 3.595-5.301a7.75 7.75 0 0 1 3.755-1.215l.12-.004a7.908 7.908 0 0 1 5.87 2.252Z"/></symbol><symbol id="icon-eds-i-user-delete-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6ZM4.763 13.227a7.713 7.713 0 0 1 7.692-.378 1 1 0 1 1-.91 1.781 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20H11.5a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897Zm11.421 1.543 2.554 2.553 2.555-2.553a1 1 0 0 1 1.414 1.414l-2.554 2.554 2.554 2.555a1 1 0 0 1-1.414 1.414l-2.555-2.554-2.554 2.554a1 1 0 0 1-1.414-1.414l2.553-2.555-2.553-2.554a1 1 0 0 1 1.414-1.414Z"/></symbol><symbol id="icon-eds-i-user-edit-medium" viewBox="0 0 24 24"><path d="m19.876 10.77 2.831 2.83a1 1 0 0 1 0 1.415l-7.246 7.246a1 1 0 0 1-.572.284l-3.277.446a1 1 0 0 1-1.125-1.13l.461-3.277a1 1 0 0 1 .283-.567l7.23-7.246a1 1 0 0 1 1.415-.001Zm-7.421 2.08a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 7.5 20a1 1 0 0 1 .993.883L8.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378Zm6.715.042-6.29 6.3-.23 1.639 1.633-.222 6.302-6.302-1.415-1.415ZM9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Z"/></symbol><symbol id="icon-eds-i-user-linked-medium" viewBox="0 0 24 24"><path d="M15.65 6c.31 0 .706.066 1.122.274C17.522 6.65 18 7.366 18 8.35v12.3c0 .31-.066.706-.274 1.122-.375.75-1.092 1.228-2.076 1.228H3.35a2.52 2.52 0 0 1-1.122-.274C1.478 22.35 1 21.634 1 20.65V8.35c0-.31.066-.706.274-1.122C1.65 6.478 2.366 6 3.35 6h12.3Zm0 2-12.376.002c-.134.007-.17.04-.21.12A.672.672 0 0 0 3 8.35v12.3c0 .198.028.24.122.287.09.044.2.063.228.063h.887c.788-2.269 2.814-3.5 5.263-3.5 2.45 0 4.475 1.231 5.263 3.5h.887c.198 0 .24-.028.287-.122.044-.09.063-.2.063-.228V8.35c0-.198-.028-.24-.122-.287A.672.672 0 0 0 15.65 8ZM9.5 19.5c-1.36 0-2.447.51-3.06 1.5h6.12c-.613-.99-1.7-1.5-3.06-1.5ZM20.65 1A2.35 2.35 0 0 1 23 3.348V15.65A2.35 2.35 0 0 1 20.65 18H20a1 1 0 0 1 0-2h.65a.35.35 0 0 0 .35-.35V3.348A.35.35 0 0 0 20.65 3H8.35a.35.35 0 0 0-.35.348V4a1 1 0 1 1-2 0v-.652A2.35 2.35 0 0 1 8.35 1h12.3ZM9.5 10a3.5 3.5 0 1 1 0 7 3.5 3.5 0 0 1 0-7Zm0 2a1.5 1.5 0 1 0 0 3 1.5 1.5 0 0 0 0-3Z"/></symbol><symbol id="icon-eds-i-user-multiple-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm6 0a5 5 0 0 1 0 10 1 1 0 0 1-.117-1.993L15 9a3 3 0 0 0 0-6 1 1 0 0 1 0-2ZM9 3a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm8.857 9.545a7.99 7.99 0 0 1 2.651 1.715A8.31 8.31 0 0 1 23 20.134V21a1 1 0 0 1-1 1h-3a1 1 0 0 1 0-2h1.995l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209a5.99 5.99 0 0 0-1.988-1.287 1 1 0 1 1 .732-1.861Zm-3.349 1.715A8.31 8.31 0 0 1 17 20.134V21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.877c.044-4.343 3.387-7.908 7.638-8.115a7.908 7.908 0 0 1 5.87 2.252ZM9.016 14l-.285.006c-3.104.15-5.58 2.718-5.725 5.9L3.004 20h11.991l-.005-.153a6.307 6.307 0 0 0-1.673-3.945l-.204-.209A5.924 5.924 0 0 0 9.3 14.008L9.016 14Z"/></symbol><symbol id="icon-eds-i-user-notify-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm10 18v1a1 1 0 0 1-2 0v-1h-3a1 1 0 0 1 0-2v-2.818C14 13.885 15.777 12 18 12s4 1.885 4 4.182V19a1 1 0 0 1 0 2h-3Zm-6.545-8.15a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM18 14c-1.091 0-2 .964-2 2.182V19h4v-2.818c0-1.165-.832-2.098-1.859-2.177L18 14Z"/></symbol><symbol id="icon-eds-i-user-remove-medium" viewBox="0 0 24 24"><path d="M9 1a5 5 0 1 1 0 10A5 5 0 0 1 9 1Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm3.455 9.85a1 1 0 1 1-.91 1.78 5.713 5.713 0 0 0-5.705.282c-1.67 1.068-2.728 2.927-2.832 4.956L3.004 20 11.5 20a1 1 0 0 1 .993.883L12.5 21a1 1 0 0 1-1 1H2a1 1 0 0 1-1-1v-.876c.028-2.812 1.446-5.416 3.763-6.897a7.713 7.713 0 0 1 7.692-.378ZM22 17a1 1 0 0 1 0 2h-8a1 1 0 0 1 0-2h8Z"/></symbol><symbol id="icon-eds-i-user-single-medium" viewBox="0 0 24 24"><path d="M12 1a5 5 0 1 1 0 10 5 5 0 0 1 0-10Zm0 2a3 3 0 1 0 0 6 3 3 0 0 0 0-6Zm-.406 9.008a8.965 8.965 0 0 1 6.596 2.494A9.161 9.161 0 0 1 21 21.025V22a1 1 0 0 1-1 1H4a1 1 0 0 1-1-1v-.985c.05-4.825 3.815-8.777 8.594-9.007Zm.39 1.992-.299.006c-3.63.175-6.518 3.127-6.678 6.775L5 21h13.998l-.009-.268a7.157 7.157 0 0 0-1.97-4.573l-.214-.213A6.967 6.967 0 0 0 11.984 14Z"/></symbol><symbol id="icon-eds-i-warning-circle-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 2a9 9 0 1 0 0 18 9 9 0 0 0 0-18Zm0 11.5a1.5 1.5 0 0 1 .144 2.993L12 17.5a1.5 1.5 0 0 1 0-3ZM12 6a1 1 0 0 1 1 1v5a1 1 0 0 1-2 0V7a1 1 0 0 1 1-1Z"/></symbol><symbol id="icon-eds-i-warning-filled-medium" viewBox="0 0 24 24"><path d="M12 1c6.075 0 11 4.925 11 11s-4.925 11-11 11S1 18.075 1 12 5.925 1 12 1Zm0 13.5a1.5 1.5 0 0 0 0 3l.144-.007A1.5 1.5 0 0 0 12 14.5ZM12 6a1 1 0 0 0-1 1v5a1 1 0 0 0 2 0V7a1 1 0 0 0-1-1Z"/></symbol><symbol id="icon-chevron-left-medium" viewBox="0 0 24 24"><path d="M15.7194 3.3054C15.3358 2.90809 14.7027 2.89699 14.3054 3.28061L6.54342 10.7757C6.19804 11.09 6 11.5335 6 12C6 12.4665 6.19804 12.91 6.5218 13.204L14.3054 20.7194C14.7027 21.103 15.3358 21.0919 15.7194 20.6946C16.103 20.2973 16.0919 19.6642 15.6946 19.2806L8.155 12L15.6946 4.71939C16.0614 4.36528 16.099 3.79863 15.8009 3.40105L15.7194 3.3054Z"/></symbol><symbol id="icon-chevron-right-medium" viewBox="0 0 24 24"><path d="M8.28061 3.3054C8.66423 2.90809 9.29729 2.89699 9.6946 3.28061L17.4566 10.7757C17.802 11.09 18 11.5335 18 12C18 12.4665 17.802 12.91 17.4782 13.204L9.6946 20.7194C9.29729 21.103 8.66423 21.0919 8.28061 20.6946C7.89699 20.2973 7.90809 19.6642 8.3054 19.2806L15.845 12L8.3054 4.71939C7.93865 4.36528 7.90098 3.79863 8.19908 3.40105L8.28061 3.3054Z"/></symbol><symbol id="icon-eds-alerts" viewBox="0 0 32 32"><path d="M28 12.667c.736 0 1.333.597 1.333 1.333v13.333A3.333 3.333 0 0 1 26 30.667H6a3.333 3.333 0 0 1-3.333-3.334V14a1.333 1.333 0 1 1 2.666 0v1.252L16 21.769l10.667-6.518V14c0-.736.597-1.333 1.333-1.333Zm-1.333 5.71-9.972 6.094c-.427.26-.963.26-1.39 0l-9.972-6.094v8.956c0 .368.299.667.667.667h20a.667.667 0 0 0 .667-.667v-8.956ZM19.333 12a1.333 1.333 0 1 1 0 2.667h-6.666a1.333 1.333 0 1 1 0-2.667h6.666Zm4-10.667a3.333 3.333 0 0 1 3.334 3.334v6.666a1.333 1.333 0 1 1-2.667 0V4.667A.667.667 0 0 0 23.333 4H8.667A.667.667 0 0 0 8 4.667v6.666a1.333 1.333 0 1 1-2.667 0V4.667a3.333 3.333 0 0 1 3.334-3.334h14.666Zm-4 5.334a1.333 1.333 0 0 1 0 2.666h-6.666a1.333 1.333 0 1 1 0-2.666h6.666Z"/></symbol><symbol id="icon-eds-arrow-up" viewBox="0 0 24 24"><path fill-rule="evenodd" d="m13.002 7.408 4.88 4.88a.99.99 0 0 0 1.32.08l.09-.08c.39-.39.39-1.03 0-1.42l-6.58-6.58a1.01 1.01 0 0 0-1.42 0l-6.58 6.58a1 1 0 0 0-.09 1.32l.08.1a1 1 0 0 0 1.42-.01l4.88-4.87v11.59a.99.99 0 0 0 .88.99l.12.01c.55 0 1-.45 1-1V7.408z" class="layer"/></symbol><symbol id="icon-eds-checklist" viewBox="0 0 32 32"><path d="M19.2 1.333a3.468 3.468 0 0 1 3.381 2.699L24.667 4C26.515 4 28 5.52 28 7.38v19.906c0 1.86-1.485 3.38-3.333 3.38H7.333c-1.848 0-3.333-1.52-3.333-3.38V7.38C4 5.52 5.485 4 7.333 4h2.093A3.468 3.468 0 0 1 12.8 1.333h6.4ZM9.426 6.667H7.333c-.36 0-.666.312-.666.713v19.906c0 .401.305.714.666.714h17.334c.36 0 .666-.313.666-.714V7.38c0-.4-.305-.713-.646-.714l-2.121.033A3.468 3.468 0 0 1 19.2 9.333h-6.4a3.468 3.468 0 0 1-3.374-2.666Zm12.715 5.606c.586.446.7 1.283.253 1.868l-7.111 9.334a1.333 1.333 0 0 1-1.792.306l-3.556-2.333a1.333 1.333 0 1 1 1.463-2.23l2.517 1.651 6.358-8.344a1.333 1.333 0 0 1 1.868-.252ZM19.2 4h-6.4a.8.8 0 0 0-.8.8v1.067a.8.8 0 0 0 .8.8h6.4a.8.8 0 0 0 .8-.8V4.8a.8.8 0 0 0-.8-.8Z"/></symbol><symbol id="icon-eds-citation" viewBox="0 0 36 36"><path d="M23.25 1.5a1.5 1.5 0 0 1 1.06.44l8.25 8.25a1.5 1.5 0 0 1 .44 1.06v19.5c0 2.105-1.645 3.75-3.75 3.75H18a1.5 1.5 0 0 1 0-3h11.25c.448 0 .75-.302.75-.75V11.873L22.628 4.5H8.31a.811.811 0 0 0-.8.68l-.011.13V16.5a1.5 1.5 0 0 1-3 0V5.31A3.81 3.81 0 0 1 8.31 1.5h14.94ZM8.223 20.358a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878C3.302 28.536 3 27.657 3 26.486c0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Zm7.5 0a.984.984 0 0 1-.192 1.378l-.048.034c-.54.36-.942.676-1.206.951-.59.614-.885 1.395-.885 2.343.115-.028.288-.042.518-.042.662 0 1.26.237 1.791.711.533.474.799 1.074.799 1.799 0 .753-.259 1.352-.777 1.799-.518.446-1.151.669-1.9.669-1.006 0-1.812-.293-2.417-.878-.604-.586-.906-1.465-.906-2.636 0-1.115.165-2.085.496-2.907.331-.823.734-1.513 1.209-2.071.475-.558.971-.997 1.49-1.318a6.01 6.01 0 0 1 .347-.2 1.321 1.321 0 0 1 1.681.368Z"/></symbol><symbol id="icon-eds-i-access-indicator" viewBox="0 0 16 16"><circle cx="4.5" cy="11.5" r="3.5" style="fill:currentColor"/><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702v7.846c0 .505-.197.993-.554 1.354a1.902 1.902 0 0 1-1.355.569H10a1 1 0 1 1 0-2h2V5.64L9.4 3H4Z" clip-rule="evenodd" style="fill:#222"/></symbol><symbol id="icon-eds-i-copy-link" viewBox="0 0 24 24"><path fill-rule="evenodd" clip-rule="evenodd" d="M19.4594 8.57015C19.0689 8.17963 19.0689 7.54646 19.4594 7.15594L20.2927 6.32261C20.2927 6.32261 20.2927 6.32261 20.2927 6.32261C21.0528 5.56252 21.0528 4.33019 20.2928 3.57014C19.5327 2.81007 18.3004 2.81007 17.5404 3.57014L16.7071 4.40347C16.3165 4.794 15.6834 4.794 15.2928 4.40348C14.9023 4.01296 14.9023 3.3798 15.2928 2.98927L16.1262 2.15594C17.6673 0.614803 20.1659 0.614803 21.707 2.15593C23.2481 3.69705 23.248 6.19569 21.707 7.7368L20.8737 8.57014C20.4831 8.96067 19.85 8.96067 19.4594 8.57015Z"/><path fill-rule="evenodd" clip-rule="evenodd" d="M18.0944 5.90592C18.4849 6.29643 18.4849 6.9296 18.0944 7.32013L16.4278 8.9868C16.0373 9.37733 15.4041 9.37734 15.0136 8.98682C14.6231 8.59631 14.6231 7.96314 15.0136 7.57261L16.6802 5.90594C17.0707 5.51541 17.7039 5.5154 18.0944 5.90592Z"/><path fill-rule="evenodd" clip-rule="evenodd" d="M13.5113 6.32243C13.9018 6.71295 13.9018 7.34611 13.5113 7.73664L12.678 8.56997C12.678 8.56997 12.678 8.56997 12.678 8.56997C11.9179 9.33006 11.9179 10.5624 12.6779 11.3224C13.438 12.0825 14.6703 12.0825 15.4303 11.3224L16.2636 10.4891C16.6542 10.0986 17.2873 10.0986 17.6779 10.4891C18.0684 10.8796 18.0684 11.5128 17.6779 11.9033L16.8445 12.7366C15.3034 14.2778 12.8048 14.2778 11.2637 12.7366C9.72262 11.1955 9.72266 8.69689 11.2637 7.15578L12.097 6.32244C12.4876 5.93191 13.1207 5.93191 13.5113 6.32243Z"/><path d="M8 20V22H19.4619C20.136 22 20.7822 21.7311 21.2582 21.2529C21.7333 20.7757 22 20.1289 22 19.4549V15C22 14.4477 21.5523 14 21 14C20.4477 14 20 14.4477 20 15V19.4549C20 19.6004 19.9426 19.7397 19.8408 19.842C19.7399 19.9433 19.6037 20 19.4619 20H8Z"/><path d="M4 13H2V19.4619C2 20.136 2.26889 20.7822 2.74705 21.2582C3.22434 21.7333 3.87105 22 4.5451 22H9C9.55228 22 10 21.5523 10 21C10 20.4477 9.55228 20 9 20H4.5451C4.39957 20 4.26028 19.9426 4.15804 19.8408C4.05668 19.7399 4 19.6037 4 19.4619V13Z"/><path d="M4 13H2V4.53808C2 3.86398 2.26889 3.21777 2.74705 2.74178C3.22434 2.26666 3.87105 2 4.5451 2H9C9.55228 2 10 2.44772 10 3C10 3.55228 9.55228 4 9 4H4.5451C4.39957 4 4.26028 4.05743 4.15804 4.15921C4.05668 4.26011 4 4.39633 4 4.53808V13Z"/></symbol><symbol id="icon-eds-i-github-medium" viewBox="0 0 24 24"><path d="M 11.964844 0 C 5.347656 0 0 5.269531 0 11.792969 C 0 17.003906 3.425781 21.417969 8.179688 22.976562 C 8.773438 23.09375 8.992188 22.722656 8.992188 22.410156 C 8.992188 22.136719 8.972656 21.203125 8.972656 20.226562 C 5.644531 20.929688 4.953125 18.820312 4.953125 18.820312 C 4.417969 17.453125 3.625 17.101562 3.625 17.101562 C 2.535156 16.378906 3.703125 16.378906 3.703125 16.378906 C 4.914062 16.457031 5.546875 17.589844 5.546875 17.589844 C 6.617188 19.386719 8.339844 18.878906 9.03125 18.566406 C 9.132812 17.804688 9.449219 17.277344 9.785156 16.984375 C 7.132812 16.710938 4.339844 15.695312 4.339844 11.167969 C 4.339844 9.878906 4.8125 8.824219 5.566406 8.003906 C 5.445312 7.710938 5.03125 6.5 5.683594 4.878906 C 5.683594 4.878906 6.695312 4.566406 8.972656 6.089844 C 9.949219 5.832031 10.953125 5.703125 11.964844 5.699219 C 12.972656 5.699219 14.003906 5.835938 14.957031 6.089844 C 17.234375 4.566406 18.242188 4.878906 18.242188 4.878906 C 18.898438 6.5 18.480469 7.710938 18.363281 8.003906 C 19.136719 8.824219 19.589844 9.878906 19.589844 11.167969 C 19.589844 15.695312 16.796875 16.691406 14.125 16.984375 C 14.558594 17.355469 14.933594 18.058594 14.933594 19.171875 C 14.933594 20.753906 14.914062 22.019531 14.914062 22.410156 C 14.914062 22.722656 15.132812 23.09375 15.726562 22.976562 C 20.480469 21.414062 23.910156 17.003906 23.910156 11.792969 C 23.929688 5.269531 18.558594 0 11.964844 0 Z M 11.964844 0 "/></symbol><symbol id="icon-eds-i-institution-medium" viewBox="0 0 24 24"><g><path fill-rule="evenodd" clip-rule="evenodd" d="M11.9967 1C11.6364 1 11.279 1.0898 10.961 1.2646C10.9318 1.28061 10.9035 1.29806 10.8761 1.31689L2.79765 6.87C2.46776 7.08001 2.20618 7.38466 2.07836 7.76668C1.94823 8.15561 1.98027 8.55648 2.12665 8.90067C2.42086 9.59246 3.12798 10 3.90107 10H4.99994V16H4.49994C3.11923 16 1.99994 17.1193 1.99994 18.5V19.5C1.99994 20.8807 3.11923 22 4.49994 22H19.4999C20.8807 22 21.9999 20.8807 21.9999 19.5V18.5C21.9999 17.1193 20.8807 16 19.4999 16H18.9999V10H20.0922C20.8653 10 21.5725 9.59252 21.8667 8.90065C22.0131 8.55642 22.0451 8.15553 21.9149 7.7666C21.7871 7.38459 21.5255 7.07997 21.1956 6.86998L13.1172 1.31689C13.0898 1.29806 13.0615 1.28061 13.0324 1.2646C12.7143 1.0898 12.357 1 11.9967 1ZM4.6844 8L11.9472 3.00755C11.9616 3.00295 11.9783 3 11.9967 3C12.015 3 12.0318 3.00295 12.0461 3.00755L19.3089 8H4.6844ZM16.9999 16V10H14.9999V16H16.9999ZM12.9999 16V10H10.9999V16H12.9999ZM8.99994 16V10H6.99994V16H8.99994ZM3.99994 18.5C3.99994 18.2239 4.2238 18 4.49994 18H19.4999C19.7761 18 19.9999 18.2239 19.9999 18.5V19.5C19.9999 19.7761 19.7761 20 19.4999 20H4.49994C4.2238 20 3.99994 19.7761 3.99994 19.5V18.5Z"/></g></symbol><symbol id="icon-eds-i-limited-access" viewBox="0 0 16 16"><path fill-rule="evenodd" d="M4 3v3a1 1 0 0 1-2 0V2.923C2 1.875 2.84 1 3.909 1h5.909a1 1 0 0 1 .713.298l3.181 3.231a1 1 0 0 1 .288.702V6a1 1 0 1 1-2 0v-.36L9.4 3H4ZM3 8a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm10 0a1 1 0 0 1 1 1v1a1 1 0 1 1-2 0V9a1 1 0 0 1 1-1Zm-3.5 6a1 1 0 0 1-1 1h-1a1 1 0 1 1 0-2h1a1 1 0 0 1 1 1Zm2.441-1a1 1 0 0 1 2 0c0 .73-.246 1.306-.706 1.664a1.61 1.61 0 0 1-.876.334l-.032.002H11.5a1 1 0 1 1 0-2h.441ZM4 13a1 1 0 0 0-2 0c0 .73.247 1.306.706 1.664a1.609 1.609 0 0 0 .876.334l.032.002H4.5a1 1 0 1 0 0-2H4Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-rss" viewBox="0 0 22 22"><path d="M1.96094 1C1.96094 0.447715 2.40865 0 2.96094 0C5.46109 0 7.93678 0.492038 10.2467 1.44806C12.5565 2.40407 14.6554 3.80534 16.4234 5.57189C18.1913 7.33843 19.5939 9.4357 20.5508 11.744C21.5077 14.0522 22.0001 16.5263 22.0001 19.0247C22.0001 19.577 21.5524 20.0247 21.0001 20.0247C20.4478 20.0247 20.0001 19.577 20.0001 19.0247C20.0001 16.7891 19.5595 14.5753 18.7033 12.5098C17.8471 10.4444 16.5919 8.56762 15.0097 6.98666C13.4275 5.40575 11.5492 4.15167 9.48182 3.29604C7.41447 2.4404 5.19868 2 2.96094 2C2.40865 2 1.96094 1.55228 1.96094 1Z"/><path fill-rule="evenodd" clip-rule="evenodd" d="M0 18.649C0 16.7974 1.50196 15.298 3.35294 15.298C5.20392 15.298 6.70588 16.7974 6.70588 18.649C6.70588 20.5003 5.20397 22 3.35294 22C1.50191 22 0 20.5003 0 18.649ZM3.35294 17.298C2.60493 17.298 2 17.9036 2 18.649C2 19.3943 2.60498 20 3.35294 20C4.1009 20 4.70588 19.3943 4.70588 18.649C4.70588 17.9036 4.10095 17.298 3.35294 17.298Z"/><path d="M3.3374 7.46115C2.78512 7.46115 2.3374 7.90887 2.3374 8.46115C2.3374 9.01344 2.78512 9.46115 3.3374 9.46115C4.54515 9.46115 5.74107 9.69885 6.85684 10.1606C7.97262 10.6224 8.98639 11.2993 9.84028 12.1525C10.6942 13.0057 11.3715 14.0185 11.8336 15.1332C12.2956 16.2478 12.5335 17.4424 12.5335 18.649C12.5335 19.2013 12.9812 19.649 13.5335 19.649C14.0858 19.649 14.5335 19.2013 14.5335 18.649C14.5335 17.1796 14.2438 15.7247 13.6811 14.3673C13.1184 13.0099 12.2936 11.7765 11.2539 10.7377C10.2142 9.69885 8.97999 8.87484 7.62168 8.31266C6.26337 7.75049 4.80757 7.46115 3.3374 7.46115Z"/></symbol><symbol id="icon-eds-i-search-category-medium" viewBox="0 0 32 32"><path fill-rule="evenodd" d="M2 5.306A3.306 3.306 0 0 1 5.306 2h5.833a3.306 3.306 0 0 1 3.306 3.306v5.833a3.306 3.306 0 0 1-3.306 3.305H5.306A3.306 3.306 0 0 1 2 11.14V5.306Zm3.306-.584a.583.583 0 0 0-.584.584v5.833c0 .322.261.583.584.583h5.833a.583.583 0 0 0 .583-.583V5.306a.583.583 0 0 0-.583-.584H5.306Zm15.555 8.945a7.194 7.194 0 1 0 4.034 13.153l2.781 2.781a1.361 1.361 0 1 0 1.925-1.925l-2.781-2.781a7.194 7.194 0 0 0-5.958-11.228Zm3.173 10.346a4.472 4.472 0 1 0-.021.021l.01-.01.011-.011Zm-5.117-19.29a.583.583 0 0 0-.584.583v5.833a1.361 1.361 0 0 1-2.722 0V5.306A3.306 3.306 0 0 1 18.917 2h5.833a3.306 3.306 0 0 1 3.306 3.306v5.833c0 .6-.161 1.166-.443 1.654a1.361 1.361 0 1 1-2.357-1.363.575.575 0 0 0 .078-.291V5.306a.583.583 0 0 0-.584-.584h-5.833ZM2 18.916a3.306 3.306 0 0 1 3.306-3.306h5.833a1.361 1.361 0 1 1 0 2.722H5.306a.583.583 0 0 0-.584.584v5.833c0 .322.261.583.584.583h5.833a.574.574 0 0 0 .29-.077 1.361 1.361 0 1 1 1.364 2.356 3.296 3.296 0 0 1-1.654.444H5.306A3.306 3.306 0 0 1 2 24.75v-5.833Z" clip-rule="evenodd"/></symbol><symbol id="icon-eds-i-subjects-medium" viewBox="0 0 24 24"><g id="icon-subjects-copy" stroke="none" stroke-width="1" fill-rule="evenodd"><path d="M13.3846154,2 C14.7015971,2 15.7692308,3.06762994 15.7692308,4.38461538 L15.7692308,7.15384615 C15.7692308,8.47082629 14.7015955,9.53846154 13.3846154,9.53846154 L13.1038388,9.53925278 C13.2061091,9.85347965 13.3815528,10.1423885 13.6195822,10.3804178 C13.9722182,10.7330539 14.436524,10.9483278 14.9293854,10.9918129 L15.1153846,11 C16.2068332,11 17.2535347,11.433562 18.0254647,12.2054189 C18.6411944,12.8212361 19.0416785,13.6120766 19.1784166,14.4609738 L19.6153846,14.4615385 C20.932386,14.4615385 22,15.5291672 22,16.8461538 L22,19.6153846 C22,20.9323924 20.9323924,22 19.6153846,22 L16.8461538,22 C15.5291672,22 14.4615385,20.932386 14.4615385,19.6153846 L14.4615385,16.8461538 C14.4615385,15.5291737 15.5291737,14.4615385 16.8461538,14.4615385 L17.126925,14.460779 C17.0246537,14.1465537 16.8492179,13.857633 16.6112344,13.6196157 C16.2144418,13.2228606 15.6764136,13 15.1153846,13 C14.0239122,13 12.9771569,12.5664197 12.2053686,11.7946314 C12.1335167,11.7227795 12.0645962,11.6485444 11.9986839,11.5721119 C11.9354038,11.6485444 11.8664833,11.7227795 11.7946314,11.7946314 C11.0228431,12.5664197 9.97608778,13 8.88461538,13 C8.323576,13 7.78552852,13.2228666 7.38881294,13.6195822 C7.15078359,13.8576115 6.97533988,14.1465203 6.8730696,14.4607472 L7.15384615,14.4615385 C8.47082629,14.4615385 9.53846154,15.5291737 9.53846154,16.8461538 L9.53846154,19.6153846 C9.53846154,20.932386 8.47083276,22 7.15384615,22 L4.38461538,22 C3.06762347,22 2,20.9323876 2,19.6153846 L2,16.8461538 C2,15.5291721 3.06762994,14.4615385 4.38461538,14.4615385 L4.8215823,14.4609378 C4.95831893,13.6120029 5.3588057,12.8211623 5.97459937,12.2053686 C6.69125996,11.488708 7.64500941,11.0636656 8.6514968,11.0066017 L8.88461538,11 C9.44565477,11 9.98370225,10.7771334 10.3804178,10.3804178 C10.6184472,10.1423885 10.7938909,9.85347965 10.8961612,9.53925278 L10.6153846,9.53846154 C9.29840448,9.53846154 8.23076923,8.47082629 8.23076923,7.15384615 L8.23076923,4.38461538 C8.23076923,3.06762994 9.29840286,2 10.6153846,2 L13.3846154,2 Z M7.15384615,16.4615385 L4.38461538,16.4615385 C4.17220099,16.4615385 4,16.63374 4,16.8461538 L4,19.6153846 C4,19.8278134 4.17218833,20 4.38461538,20 L7.15384615,20 C7.36626945,20 7.53846154,19.8278103 7.53846154,19.6153846 L7.53846154,16.8461538 C7.53846154,16.6337432 7.36625679,16.4615385 7.15384615,16.4615385 Z M19.6153846,16.4615385 L16.8461538,16.4615385 C16.6337432,16.4615385 16.4615385,16.6337432 16.4615385,16.8461538 L16.4615385,19.6153846 C16.4615385,19.8278103 16.6337306,20 16.8461538,20 L19.6153846,20 C19.8278229,20 20,19.8278229 20,19.6153846 L20,16.8461538 C20,16.6337306 19.8278103,16.4615385 19.6153846,16.4615385 Z M13.3846154,4 L10.6153846,4 C10.4029708,4 10.2307692,4.17220099 10.2307692,4.38461538 L10.2307692,7.15384615 C10.2307692,7.36625679 10.402974,7.53846154 10.6153846,7.53846154 L13.3846154,7.53846154 C13.597026,7.53846154 13.7692308,7.36625679 13.7692308,7.15384615 L13.7692308,4.38461538 C13.7692308,4.17220099 13.5970292,4 13.3846154,4 Z" id="Shape" fill-rule="nonzero"/></g></symbol><symbol id="icon-eds-small-arrow-left" viewBox="0 0 16 17"><path stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2" d="M14 8.092H2m0 0L8 2M2 8.092l6 6.035"/></symbol><symbol id="icon-eds-small-arrow-right" viewBox="0 0 16 16"><g fill-rule="evenodd" stroke="currentColor" stroke-linecap="round" stroke-linejoin="round" stroke-width="2"><path d="M2 8.092h12M8 2l6 6.092M8 14.127l6-6.035"/></g></symbol><symbol id="icon-orcid-logo" viewBox="0 0 40 40"><path fill-rule="evenodd" d="M12.281 10.453c.875 0 1.578-.719 1.578-1.578 0-.86-.703-1.578-1.578-1.578-.875 0-1.578.703-1.578 1.578 0 .86.703 1.578 1.578 1.578Zm-1.203 18.641h2.406V12.359h-2.406v16.735Z"/><path fill-rule="evenodd" d="M17.016 12.36h6.5c6.187 0 8.906 4.421 8.906 8.374 0 4.297-3.36 8.375-8.875 8.375h-6.531V12.36Zm6.234 14.578h-3.828V14.53h3.703c4.688 0 6.828 2.844 6.828 6.203 0 2.063-1.25 6.203-6.703 6.203Z" clip-rule="evenodd"/></symbol></svg> </div> <a class="c-skip-link" href="#main">Skip to main content</a> <header class="eds-c-header" data-eds-c-header> <div class="eds-c-header__container" data-eds-c-header-expander-anchor> <div class="eds-c-header__brand"> <a href="https://link.springer.com" data-test=springerlink-logo data-track="click_imprint_logo" data-track-context="unified header" data-track-action="click logo link" data-track-category="unified header" data-track-label="link" > <img src="/oscar-static/images/darwin/header/img/logo-springer-nature-link-3149409f62.svg" alt="Springer Nature Link"> </a> </div> <a class="c-header__link eds-c-header__link" id="identity-account-widget" data-track="click_login" data-track-context="header" href='https://idp.springer.com/auth/personal/springernature?redirect_uri=https://link.springer.com/article/10.1007/s10231-023-01327-w?'><span class="eds-c-header__widget-fragment-title">Log in</span></a> </div> <nav class="eds-c-header__nav" aria-label="header navigation"> <div class="eds-c-header__nav-container"> <div class="eds-c-header__item eds-c-header__item--menu"> <a href="#eds-c-header-nav" class="eds-c-header__link" data-eds-c-header-expander> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-menu-medium"></use> </svg><span>Menu</span> </a> </div> <div class="eds-c-header__item eds-c-header__item--inline-links"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </div> <div class="eds-c-header__link-container"> <div class="eds-c-header__item eds-c-header__item--divider"> <a href="#eds-c-header-popup-search" class="eds-c-header__link" data-eds-c-header-expander data-eds-c-header-test-search-btn> <svg class="eds-c-header__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg><span>Search</span> </a> </div> <div id="ecommerce-header-cart-icon-link" class="eds-c-header__item ecommerce-cart" style="display:inline-block"> <a class="eds-c-header__link" href="https://order.springer.com/public/cart" style="appearance:none;border:none;background:none;color:inherit;position:relative"> <svg id="eds-i-cart" class="eds-c-header__icon" xmlns="http://www.w3.org/2000/svg" height="24" width="24" viewBox="0 0 24 24" aria-hidden="true" focusable="false"> <path fill="currentColor" fill-rule="nonzero" d="M2 1a1 1 0 0 0 0 2l1.659.001 2.257 12.808a2.599 2.599 0 0 0 2.435 2.185l.167.004 9.976-.001a2.613 2.613 0 0 0 2.61-1.748l.03-.106 1.755-7.82.032-.107a2.546 2.546 0 0 0-.311-1.986l-.108-.157a2.604 2.604 0 0 0-2.197-1.076L6.042 5l-.56-3.17a1 1 0 0 0-.864-.82l-.12-.007L2.001 1ZM20.35 6.996a.63.63 0 0 1 .54.26.55.55 0 0 1 .082.505l-.028.1L19.2 15.63l-.022.05c-.094.177-.282.299-.526.317l-10.145.002a.61.61 0 0 1-.618-.515L6.394 6.999l13.955-.003ZM18 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4ZM8 19a2 2 0 1 0 0 4 2 2 0 0 0 0-4Z"></path> </svg><span>Cart</span><span class="cart-info" style="display:none;position:absolute;top:10px;right:45px;background-color:#C65301;color:#fff;width:18px;height:18px;font-size:11px;border-radius:50%;line-height:17.5px;text-align:center"></span></a> <script>(function () { var exports = {}; if (window.fetch) { "use strict"; Object.defineProperty(exports, "__esModule", { value: true }); exports.headerWidgetClientInit = void 0; var headerWidgetClientInit = function (getCartInfo) { document.body.addEventListener("updatedCart", function () { updateCartIcon(); }, false); return updateCartIcon(); function updateCartIcon() { return getCartInfo() .then(function (res) { return res.json(); }) .then(refreshCartState) .catch(function (_) { }); } function refreshCartState(json) { var indicator = document.querySelector("#ecommerce-header-cart-icon-link .cart-info"); /* istanbul ignore else */ if (indicator && json.itemCount) { indicator.style.display = 'block'; indicator.textContent = json.itemCount > 9 ? '9+' : json.itemCount.toString(); var moreThanOneItem = json.itemCount > 1; indicator.setAttribute('title', "there ".concat(moreThanOneItem ? "are" : "is", " ").concat(json.itemCount, " item").concat(moreThanOneItem ? "s" : "", " in your cart")); } return json; } }; exports.headerWidgetClientInit = headerWidgetClientInit; headerWidgetClientInit( function () { return window.fetch("https://cart.springer.com/cart-info", { credentials: "include", headers: { Accept: "application/json" } }) } ) }})()</script> </div> </div> </div> </nav> </header> <article lang="en" id="main" class="app-masthead__colour-6"> <section class="app-masthead " aria-label="article masthead"> <div class="app-masthead__container"> <div class="app-article-masthead u-sans-serif js-context-bar-sticky-point-masthead" data-track-component="article" data-test="masthead-component"> <div class="app-article-masthead__info"> <nav aria-label="breadcrumbs" data-test="breadcrumbs"> <ol class="c-breadcrumbs c-breadcrumbs--contrast" itemscope itemtype="https://schema.org/BreadcrumbList"> <li class="c-breadcrumbs__item" id="breadcrumb0" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb1"><span itemprop="name">Home</span></a><meta itemprop="position" content="1"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb1" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <a href="/journal/10231" class="c-breadcrumbs__link" itemprop="item" data-track="click_breadcrumb" data-track-context="article page" data-track-category="article" data-track-action="breadcrumbs" data-track-label="breadcrumb2"><span itemprop="name">Annali di Matematica Pura ed Applicata (1923 -)</span></a><meta itemprop="position" content="2"> <svg class="c-breadcrumbs__chevron" role="img" aria-hidden="true" focusable="false" width="10" height="10" viewBox="0 0 10 10"> <path d="m5.96738168 4.70639573 2.39518594-2.41447274c.37913917-.38219212.98637524-.38972225 1.35419292-.01894278.37750606.38054586.37784436.99719163-.00013556 1.37821513l-4.03074001 4.06319683c-.37758093.38062133-.98937525.38100976-1.367372-.00003075l-4.03091981-4.06337806c-.37759778-.38063832-.38381821-.99150444-.01600053-1.3622839.37750607-.38054587.98772445-.38240057 1.37006824.00302197l2.39538588 2.4146743.96295325.98624457z" fill-rule="evenodd" transform="matrix(0 -1 1 0 0 10)"/> </svg> </li> <li class="c-breadcrumbs__item" id="breadcrumb2" itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem"> <span itemprop="name">Article</span><meta itemprop="position" content="3"> </li> </ol> </nav> <h1 class="c-article-title" data-test="article-title" data-article-title="">Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator</h1> <ul class="c-article-identifiers"> <li class="c-article-identifiers__item"> <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link" class="u-color-open-access" data-test="open-access">Open access</a> </li> <li class="c-article-identifiers__item"> Published: <time datetime="2023-03-27">27 March 2023</time> </li> </ul> <ul class="c-article-identifiers c-article-identifiers--cite-list"> <li class="c-article-identifiers__item"> <span data-test="journal-volume">Volume 202</span>, pages 2481–2516, (<span data-test="article-publication-year">2023</span>) </li> <li class="c-article-identifiers__item c-article-identifiers__item--cite"> <a href="#citeas" data-track="click" data-track-action="cite this article" data-track-category="article body" data-track-label="link">Cite this article</a> </li> </ul> <div class="app-article-masthead__buttons" data-test="download-article-link-wrapper" data-track-context="masthead"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s10231-023-01327-w.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> <p class="app-article-masthead__access"> <svg width="16" height="16" focusable="false" role="img" aria-hidden="true"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-check-filled-medium"></use></svg> You have full access to this <a href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="click" data-track-action="open access" data-track-label="link">open access</a> article</p> </div> </div> <div class="app-article-masthead__brand"> <a href="/journal/10231" class="app-article-masthead__journal-link" data-track="click_journal_home" data-track-action="journal homepage" data-track-context="article page" data-track-label="link"> <picture> <source type="image/webp" media="(min-width: 768px)" width="120" height="159" srcset="https://media.springernature.com/w120/springer-static/cover-hires/journal/10231?as=webp, https://media.springernature.com/w316/springer-static/cover-hires/journal/10231?as=webp 2x"> <img width="72" height="95" src="https://media.springernature.com/w72/springer-static/cover-hires/journal/10231?as=webp" srcset="https://media.springernature.com/w144/springer-static/cover-hires/journal/10231?as=webp 2x" alt=""> </picture> <span class="app-article-masthead__journal-title">Annali di Matematica Pura ed Applicata (1923 -)</span> </a> <a href="https://link.springer.com/journal/10231/aims-and-scope" class="app-article-masthead__submission-link" data-track="click_aims_and_scope" data-track-action="aims and scope" data-track-context="article page" data-track-label="link"> Aims and scope <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> <a href="https://www.editorialmanager.com/ampa" class="app-article-masthead__submission-link" data-track="click_submit_manuscript" data-track-context="article masthead on springerlink article page" data-track-action="submit manuscript" data-track-label="link"> Submit manuscript <svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-arrow-right-medium"></use></svg> </a> </div> </div> </div> </section> <div class="c-article-main u-container u-mt-24 u-mb-32 l-with-sidebar" id="main-content" data-component="article-container"> <main class="u-serif js-main-column" data-track-component="article body"> <div class="c-context-bar u-hide" data-test="context-bar" data-context-bar aria-hidden="true"> <div class="c-context-bar__container u-container"> <div class="c-context-bar__title"> Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator </div> <div data-test="inCoD" data-track-context="sticky banner"> <div class="c-pdf-container"> <div class="c-pdf-download u-clear-both u-mb-16"> <a href="/content/pdf/10.1007/s10231-023-01327-w.pdf" class="u-button u-button--full-width u-button--primary u-justify-content-space-between c-pdf-download__link" data-article-pdf="true" data-readcube-pdf-url="true" data-test="pdf-link" data-draft-ignore="true" data-track="content_download" data-track-type="article pdf download" data-track-action="download pdf" data-track-label="button" data-track-external download> <span class="c-pdf-download__text">Download PDF</span> <svg aria-hidden="true" focusable="false" width="16" height="16" class="u-icon"><use xlink:href="#icon-eds-i-download-medium"/></svg> </a> </div> </div> </div> </div> </div> <div class="c-article-header"> <header> <ul class="c-article-author-list c-article-author-list--short" data-test="authors-list" data-component-authors-activator="authors-list"><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Helena_F_-Gon_alves-Aff1" data-author-popup="auth-Helena_F_-Gon_alves-Aff1" data-author-search="Gonçalves, Helena F.">Helena F. Gonçalves</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0001-5608-9102"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0001-5608-9102</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup>, </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Dorothee_D_-Haroske-Aff1" data-author-popup="auth-Dorothee_D_-Haroske-Aff1" data-author-search="Haroske, Dorothee D.">Dorothee D. Haroske</a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-2576-0300"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-2576-0300</a></span><sup class="u-js-hide"><a href="#Aff1">1</a></sup> & </li><li class="c-article-author-list__item"><a data-test="author-name" data-track="click" data-track-action="open author" data-track-label="link" href="#auth-Leszek-Skrzypczak-Aff2" data-author-popup="auth-Leszek-Skrzypczak-Aff2" data-author-search="Skrzypczak, Leszek" data-corresp-id="c1">Leszek Skrzypczak<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-mail-medium"></use></svg></a><span class="u-js-hide"> <a class="js-orcid" href="http://orcid.org/0000-0002-7484-2900"><span class="u-visually-hidden">ORCID: </span>orcid.org/0000-0002-7484-2900</a></span><sup class="u-js-hide"><a href="#Aff2">2</a></sup> </li></ul> <div data-test="article-metrics"> <ul class="app-article-metrics-bar u-list-reset"> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-accesses-medium"></use> </svg>1521 <span class="app-article-metrics-bar__label">Accesses</span></p> </li> <li class="app-article-metrics-bar__item"> <p class="app-article-metrics-bar__count"><svg class="u-icon app-article-metrics-bar__icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-citations-medium"></use> </svg>4 <span class="app-article-metrics-bar__label">Citations</span></p> </li> <li class="app-article-metrics-bar__item app-article-metrics-bar__item--metrics"> <p class="app-article-metrics-bar__details"><a href="/article/10.1007/s10231-023-01327-w/metrics" data-track="click" data-track-action="view metrics" data-track-label="link" rel="nofollow">Explore all metrics <svg class="u-icon app-article-metrics-bar__arrow-icon" width="24" height="24" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-arrow-right-medium"></use> </svg></a></p> </li> </ul> </div> <div class="u-mt-32"> </div> </header> </div> <div data-article-body="true" data-track-component="article body" class="c-article-body"> <section aria-labelledby="Abs1" data-title="Abstract" lang="en"><div class="c-article-section" id="Abs1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Abs1">Abstract</h2><div class="c-article-section__content" id="Abs1-content"><p>In this paper, we study limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces, <span class="mathjax-tex">$\text {id}_\tau : {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$</span> and <span class="mathjax-tex">$\text {id}_\tau : {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$</span>, where <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span> is a bounded domain, obtaining necessary and sufficient conditions for the continuity of <span class="mathjax-tex">$\text {id}_\tau $</span>. This can also be seen as the continuation of our previous studies of compactness of the embeddings in the non-limiting case. Moreover, we also construct Rychkov’s linear, bounded universal extension operator for these spaces.</p></div></div></section> <div data-test="cobranding-download"> </div> <section aria-labelledby="inline-recommendations" data-title="Inline Recommendations" class="c-article-recommendations" data-track-component="inline-recommendations"> <h3 class="c-article-recommendations-title" id="inline-recommendations">Similar content being viewed by others</h3> <div class="c-article-recommendations-list"> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s13163-020-00365-9?fromPaywallRec=false" data-track="select_recommendations_1" data-track-context="inline recommendations" data-track-action="click recommendations inline - 1" data-track-label="10.1007/s13163-020-00365-9">Compact embeddings in Besov-type and Triebel–Lizorkin-type spaces on bounded domains </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__access-type">Open access</span> <span class="c-article-meta-recommendations__date">16 July 2020</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s40840-021-01177-w?fromPaywallRec=false" data-track="select_recommendations_2" data-track-context="inline recommendations" data-track-action="click recommendations inline - 2" data-track-label="10.1007/s40840-021-01177-w">Extension of Variable Triebel–Lizorkin-Type Space on Domains </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">08 September 2021</span> </div> </div> </article> </div> <div class="c-article-recommendations-list__item"> <article class="c-article-recommendations-card" itemscope itemtype="http://schema.org/ScholarlyArticle"> <div class="c-article-recommendations-card__img"><img src="https://media.springernature.com/w215h120/springer-static/image/art%3Aplaceholder%2Fimages/placeholder-figure-springernature.png" loading="lazy" alt=""></div> <div class="c-article-recommendations-card__main"> <h3 class="c-article-recommendations-card__heading" itemprop="name headline"> <a class="c-article-recommendations-card__link" itemprop="url" href="https://link.springer.com/10.1007/s00041-019-09709-6?fromPaywallRec=false" data-track="select_recommendations_3" data-track-context="inline recommendations" data-track-action="click recommendations inline - 3" data-track-label="10.1007/s00041-019-09709-6">Nuclear Embeddings of Besov Spaces into Zygmund Spaces </a> </h3> <div class="c-article-meta-recommendations" data-test="recommendation-info"> <span class="c-article-meta-recommendations__item-type">Article</span> <span class="c-article-meta-recommendations__date">16 January 2020</span> </div> </div> </article> </div> </div> </section> <script> window.dataLayer = window.dataLayer || []; window.dataLayer.push({ recommendations: { recommender: 'semantic', model: 'specter', policy_id: 'NA', timestamp: 1742925240, embedded_user: 'null' } }); </script> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=10231" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div class="main-content"> <section data-title="Introduction"><div class="c-article-section" id="Sec1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec1"><span class="c-article-section__title-number">1 </span>Introduction</h2><div class="c-article-section__content" id="Sec1-content"><p>Besov-type spaces <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> and Triebel-Lizorkin-type spaces <span class="mathjax-tex">${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span>, <span class="mathjax-tex">$0< p <\infty $</span> (or <span class="mathjax-tex">$p=\infty $</span> in the <i>B</i>-case), <span class="mathjax-tex">$0 < q \le \infty $</span>, <span class="mathjax-tex">$\tau \ge 0$</span>, <span class="mathjax-tex">$s \in {{\mathbb {R}}}$</span>, are part of a class of function spaces built upon Morrey spaces <span class="mathjax-tex">${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$</span>, <span class="mathjax-tex">$0< p \le u < \infty $</span>. They are regularly called in the literature as <i>smoothness spaces of Morrey type</i> or, shortly, <i>smoothness Morrey spaces</i>, and they have been increasingly studied in the last decades, motivated firstly by possible applications.</p><p>The classical Morrey spaces <span class="mathjax-tex">${{\mathcal {M}}}_{u,p}$</span>, <span class="mathjax-tex">$0< p \le u < \infty $</span>, were introduced by Morrey in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938)" href="/article/10.1007/s10231-023-01327-w#ref-CR19" id="ref-link-section-d74823138e1148">19</a>] and are part of a wider class of Morrey-Campanato spaces, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Peetre, J.: On the theory of 
 
 
 
 $${{\cal{L} }}_{p,\lambda }$$
 
 
 L
 
 p
 ,
 λ
 
 
 
 spaces. J. Funct. Anal. 4, 71–87 (1969)" href="/article/10.1007/s10231-023-01327-w#ref-CR21" id="ref-link-section-d74823138e1151">21</a>]. They can be seen as a complement to <span class="mathjax-tex">$L_p$</span> spaces, since <span class="mathjax-tex">${{\mathcal {M}}}_{p,p}({{{\mathbb {R}}}^d})= L_p({{{\mathbb {R}}}^d})$</span>.</p><p>The (inhomogeneous) Besov-type and Triebel-Lizorkin-type spaces we work here with were introduced and intensively studied in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e1262">47</a>] by Yuan, Sickel and Yang. Their homogeneous versions were previously investigated by El Baraka in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="El Baraka, A.: An embedding theorem for Campanato spaces. Electron. J. Differential Equat. 66, 1–17 (2002)" href="#ref-CR3" id="ref-link-section-d74823138e1265">3</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="El Baraka, A.: Function spaces of BMO and Campanato type, pp.109-115, in: Proc. of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. 9, Southwest Texas State Univ., San Marcos, TX (2002)" href="#ref-CR4" id="ref-link-section-d74823138e1265_1">4</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="El Baraka, A.: Littlewood-Paley characterization for Campanato spaces. J. Funct. Spaces Appl. 4, 193–220 (2006)" href="/article/10.1007/s10231-023-01327-w#ref-CR5" id="ref-link-section-d74823138e1268">5</a>], and also by Yuan and Yang [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 41" title="Yang, D., Yuan, W.: A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J. Funct. Anal. 255, 2760–2809 (2008)" href="/article/10.1007/s10231-023-01327-w#ref-CR41" id="ref-link-section-d74823138e1271">41</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 42" title="Yang, D., Yuan, W.: New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces. Math. Z. 265, 451–480 (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR42" id="ref-link-section-d74823138e1274">42</a>]. Considering <span class="mathjax-tex">$\tau =0$</span>, one recovers the classical Besov and Triebel-Lizorkin spaces. Moreover, they are also closely connected with Besov-Morrey spaces <span class="mathjax-tex">${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span> and Triebel-Lizorkin-Morrey spaces <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span>, <span class="mathjax-tex">$0< p \le u < \infty $</span>, <span class="mathjax-tex">$0 < q \le \infty $</span>, <span class="mathjax-tex">$s \in {{\mathbb {R}}}$</span>, which are also included in the class of <i>smoothness Morrey spaces</i>. Namely, when <span class="mathjax-tex">$p\le u$</span>, <span class="mathjax-tex">$\tau =\frac{1}{p}-\frac{1}{u}$</span>, then the Triebel-Lizorkin-type space <span class="mathjax-tex">${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> coincides with <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span>, and it is also known that the Besov-Morrey space <span class="mathjax-tex">${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span> is a proper subspace of <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> with <span class="mathjax-tex">$\tau =\frac{1}{p}-\frac{1}{u}$</span>, <span class="mathjax-tex">$p<u$</span> and <span class="mathjax-tex">$q<\infty $</span>. The Besov-Morrey spaces were introduced by Kozono and Yamazaki in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Different. Equat. 19, 959–1014 (1994)" href="/article/10.1007/s10231-023-01327-w#ref-CR15" id="ref-link-section-d74823138e1911">15</a>] and used by them and later on by Mazzucato [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Mazzucato, A.L.: Besov-Morrey spaces: function space theory and applications to non-linear PDE. Trans. Amer. Math. Soc. 355, 1297–1364 (2003)" href="/article/10.1007/s10231-023-01327-w#ref-CR17" id="ref-link-section-d74823138e1914">17</a>] in the study of Navier–Stokes equations. In [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Tang, L., Xu, J.: Some properties of Morrey type Besov-Triebel spaces. Math. Nachr. 278, 904–917 (2005)" href="/article/10.1007/s10231-023-01327-w#ref-CR33" id="ref-link-section-d74823138e1918">33</a>] Tang and Xu introduced the corresponding Triebel-Lizorkin-Morrey spaces, thanks to establishing the Morrey version of the Fefferman-Stein vector-valued inequality. Some properties of these spaces including their wavelet characterisations were later described in the papers by Sawano [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Sawano, Y.: Wavelet characterizations of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Funct. Approx. Comment. Math. 38, 93–107 (2008)" href="/article/10.1007/s10231-023-01327-w#ref-CR24" id="ref-link-section-d74823138e1921">24</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Sawano, Y.: A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Acta Math. Sin. (Engl. Ser.) 25, 1223–1242 (2009)" href="/article/10.1007/s10231-023-01327-w#ref-CR25" id="ref-link-section-d74823138e1924">25</a>], Sawano and Tanaka [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Sawano, Y., Tanaka, H.: Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z. 257, 871–905 (2007)" href="/article/10.1007/s10231-023-01327-w#ref-CR28" id="ref-link-section-d74823138e1927">28</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces for non-doubling measures: Sawano, Y., Tanaka. H. Math. Nachr. 282, 1788–1810 (2009)" href="/article/10.1007/s10231-023-01327-w#ref-CR29" id="ref-link-section-d74823138e1930">29</a>] and Rosenthal [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 22" title="Rosenthal, M.: Local means, wavelet bases, representations, and isomorphisms in Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Math. Nachr. 286, 59–87 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR22" id="ref-link-section-d74823138e1933">22</a>]. The surveys [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. I. Eurasian Math. J. 3, 110–149 (2012)" href="/article/10.1007/s10231-023-01327-w#ref-CR31" id="ref-link-section-d74823138e1937">31</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. II. Eurasian Math. J. 4, 82–124 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR32" id="ref-link-section-d74823138e1940">32</a>] by Sickel are also worth of being consulted when studying these scales. Recently, some limiting embedding properties of these spaces were investigated in a series of papers [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Haroske, D.. D., Skrzypczak, L.: Continuous embeddings of Besov-Morrey function spaces. Acta Math. Sin. (Engl. Ser.) 28, 1307–1328 (2012)" href="#ref-CR10" id="ref-link-section-d74823138e1943">10</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="#ref-CR11" id="ref-link-section-d74823138e1943_1">11</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Haroske, D.D., Skrzypczak, L.: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces. Rev. Mat. Complut. 27, 541–573 (2014)" href="#ref-CR12" id="ref-link-section-d74823138e1943_2">12</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Haroske, D. D., Skrzypczak, L.: Some quantitative result on compact embeddings in smoothness Morrey spaces on bounded domains; an approach via interpolation pp.181–191, in: Function Spaces XII, Banach Center Publ, vol. 119, Polish Acad. Sci., Warsaw (2019)" href="#ref-CR13" id="ref-link-section-d74823138e1943_3">13</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 14" title="Haroske, D. D., Skrzypczak, L.: Entropy numbers of compact embeddings of smoothness Morrey spaces on bounded domains. J. Approx. Theory 256, 24 pp. (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR14" id="ref-link-section-d74823138e1946">14</a>]. As for the Besov-type and Triebel-Lizorkin-type spaces, also embedding properties have been recently studied in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. 34, 761–795 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR8" id="ref-link-section-d74823138e1949">8</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 45" title="Yuan, W., Haroske, D.D., Moura, S.D., Skrzypczak, L., Yang, D.: Limiting embeddings in smoothness Morrey spaces, continuity envelopes and applications. J. Approx. Theory 192, 306–335 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR45" id="ref-link-section-d74823138e1952">45</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e1956">46</a>].</p><p>Undoubtedly the question of necessary and sufficient conditions for continuous embeddings of certain function spaces is a natural and classical one. Beyond that, this paper should essentially be understood as the continuation of our earlier studies in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. 34, 761–795 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR8" id="ref-link-section-d74823138e1962">8</a>]. Proceeding contrary to the usual, there we started by studying compactness of the embeddings of Besov-type (<span class="mathjax-tex">$A=B$</span>) and Triebel-Lizorkin-type (<span class="mathjax-tex">$A=F$</span>) spaces,</p><div id="Equ93" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}\text {id}_\tau ~:~{A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )~\hookrightarrow ~{A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ),\end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span> is a bounded domain. Now we finally deal with the continuity of such embeddings, obtaining sufficient and necessary conditions on the parameters under which they hold true. According to the results obtained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. 34, 761–795 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR8" id="ref-link-section-d74823138e2212">8</a>], the embedding <span class="mathjax-tex">$\text {id}_\tau $</span> is compact if, and only if,</p><div id="Equ94" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} > \max \left\{ \left( \tau _2-\frac{1}{p_2} \right) _+- \left( \tau _1-\frac{1}{p_1} \right) _+, \frac{1}{p_1}-\tau _1 -\min \left\{ \frac{1}{p_2}-\tau _2, \frac{1}{p_2}(1-p_1\tau _1)_+\right\} \right\} =: \gamma , \end{aligned}$$</span></div></div><p>with <span class="mathjax-tex">$a_+{:=} \max \{a,0\}$</span>, and there is no continuous embedding <span class="mathjax-tex">$\text {id}_\tau $</span> when <span class="mathjax-tex">$s_1-s_2<d \, \gamma $</span>. Consequently, only the case when</p><div id="Equ95" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma \end{aligned}$$</span></div></div><p>is of interest to us here. In what follows, we call this setting ‘limiting situation’, giving meaning to the expression ‘limiting embedding’. In that way we complement earlier results in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR11" id="ref-link-section-d74823138e2643">11</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haroske, D.D., Skrzypczak, L.: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces. Rev. Mat. Complut. 27, 541–573 (2014)" href="/article/10.1007/s10231-023-01327-w#ref-CR12" id="ref-link-section-d74823138e2646">12</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e2649">46</a>] in related settings. Our main results are stated in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a>, Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a>. In the propositions we consider the situation when one of the spaces, the source one or the target one, coincides with some classical Besov space <span class="mathjax-tex">$B^\sigma _{\infty ,\infty }(\Omega )$</span>. The outcome for the spaces that do not satisfy this assumption can be found in the theorem. In almost all cases we prove the sharp sufficient and necessary conditions. Only in one case we have a small gap between them, cf. Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar42">4.10</a>.</p><p>Of independent interest is also the extension theorem we are able to prove for the spaces under consideration. The first extension operators for Besov-type and Triebel-Lizorkin-type spaces were constructed by Sickel, Yang and Yuan in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e2715">47</a>], cf. Theorem 6.11 and 6.13 ibidem. However it is assumed there that domains are <span class="mathjax-tex">$C^\infty $</span> smooth and the extension operators were not universal. Another not universal construction for the smooth domains was given by Moura, Neves and Schneider in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 20" title="Moura, S.D., Neves, J.S., Schneider, C.: Traces and extensions of generalized smoothness Morrey spaces on domains. Nonlinear Anal. 181, 311–339 (2019)" href="/article/10.1007/s10231-023-01327-w#ref-CR20" id="ref-link-section-d74823138e2740">20</a>]. The Rychkov universal extension operator for the Triebel-Lizorkin-type spaces defined on Lipschitz domains was recently constructed by Zhou, Hovemann and Sickel in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title="Zhuo, C., Hovemann, M., Sickel, W.: Complex interpolation of Lizorkin-Triebel-Morrey Spaces on Domains. Anal. Geom. Metric Spaces 8, 268–304 (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR52" id="ref-link-section-d74823138e2743">52</a>] with additional assumptions <span class="mathjax-tex">$p,q \in [1, \infty )$</span>. Here we considered all admissible parameters <i>p</i> and <i>q</i>. We concentrate on the Besov-type spaces, that are not a real interpolation space of Triebel-Lizorkin-type spaces, in contrast to the classical case, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey-Campanato and related smoothness spaces. Sci. China Math. 58, 1835–1908 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR48" id="ref-link-section-d74823138e2792">48</a>]. The extension theorem will not only help us to obtain the results about the continuity, but also will allow us to improve some necessary conditions of such embeddings on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>. We follow Rychkov’s approach from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e2821">23</a>] and construct such an operator, for all possible values of <span class="mathjax-tex">$p,q\in (0, \infty ]$</span>. We learned only recently that in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Zhuo, C.: Complex interpolation of Besov-type spaces on domains. Z. Anal. Anwend. 40, 313–347 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR51" id="ref-link-section-d74823138e2864">51</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title="Zhuo, C., Hovemann, M., Sickel, W.: Complex interpolation of Lizorkin-Triebel-Morrey Spaces on Domains. Anal. Geom. Metric Spaces 8, 268–304 (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR52" id="ref-link-section-d74823138e2867">52</a>] the authors followed a similar approach to construct such an extension operator adapted to their purposes, that is, for <span class="mathjax-tex">$p,q\ge 1$</span>. For the convenience of the reader we keep our argument for the full range of parameters here.</p><p>This paper is organized as follows. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10231-023-01327-w#Sec2">2</a> we recall the definition, on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> and on bounded domains <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span>, of the spaces considered in the paper and collect some basic properties, among them the wavelet characterisations. In Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10231-023-01327-w#Sec6">3</a> we deal with the construction of a universal linear bounded extension operator for the spaces <span class="mathjax-tex">${A}_{p,q}^{s,\tau }$</span>. Section <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10231-023-01327-w#Sec7">4</a> is, finally, devoted to the study of continuity properties of limiting embeddings of the spaces <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span>. Moreover, we make use of these results and the extension theorem from Section 3 to improve prior results on the continuity of embeddings of the corresponding spaces on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e3088">46</a>].</p></div></div></section><section data-title="Preliminaries"><div class="c-article-section" id="Sec2-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec2"><span class="c-article-section__title-number">2 </span>Preliminaries</h2><div class="c-article-section__content" id="Sec2-content"><p>First we fix some notation. By <span class="mathjax-tex">${{\mathbb {N}}}$</span> we denote the <i>set of natural numbers</i>, by <span class="mathjax-tex">${{\mathbb {N}}}_0$</span> the set <span class="mathjax-tex">${{\mathbb {N}}}\cup \{0\}$</span>, and by <span class="mathjax-tex">${{{\mathbb {Z}}}^d}$</span> the <i>set of all lattice points in </i><span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span><i>having integer components</i>. Let <span class="mathjax-tex">${{\mathbb {N}}}_0^d$</span>, where <span class="mathjax-tex">$d \in {{\mathbb {N}}}$</span>, be the set of all multi-indices, <span class="mathjax-tex">$\alpha =(\alpha _1,..., \alpha _d)$</span> with <span class="mathjax-tex">$\alpha _j \in {{\mathbb {N}}}_0$</span> and <span class="mathjax-tex">$|\alpha |{:=}\sum _{j=1}^d\alpha _j$</span>. If <span class="mathjax-tex">$x=(x_1,..., x_d) \in {{{\mathbb {R}}}^d}$</span> and <span class="mathjax-tex">$\alpha =(\alpha _1,..., \alpha _d) \in {{\mathbb {N}}}_0^d$</span>, then we put <span class="mathjax-tex">$x^\alpha =x_1^{\alpha _1} \cdot \cdot \cdot x_d^{\alpha _d}$</span>. For <span class="mathjax-tex">$a\in {{\mathbb {R}}}$</span>, let <span class="mathjax-tex">$\left\lfloor a \right\rfloor {:=}\max \{k\in {{\mathbb {Z}}}: k\le a\}$</span> and <span class="mathjax-tex">$a_+{:=}\max \{a,0\}$</span>. All unimportant positive constants will be denoted by <i>C</i>, occasionally with subscripts. By the notation <span class="mathjax-tex">$A \lesssim B$</span>, we mean that there exists a positive constant <i>C</i> such that <span class="mathjax-tex">$A \le C \,B$</span>, whereas the symbol <span class="mathjax-tex">$A \sim B$</span> stands for <span class="mathjax-tex">$A \lesssim B \lesssim A$</span>. We denote by <span class="mathjax-tex">$B(x,r) {:=} \{y\in {{{\mathbb {R}}}^d}: |x-y|<r\}$</span> the ball centred at <span class="mathjax-tex">$x\in {{{\mathbb {R}}}^d}$</span> with radius <span class="mathjax-tex">$r>0$</span>, and <span class="mathjax-tex">$|\cdot |$</span> denotes the Lebesgue measure when applied to measurable subsets of <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>.</p><p>Given two (quasi-)Banach spaces <i>X</i> and <i>Y</i>, we write <span class="mathjax-tex">$X\hookrightarrow Y$</span> if <span class="mathjax-tex">$X\subset Y$</span> and the natural embedding of <i>X</i> into <i>Y</i> is continuous.</p><h3 class="c-article__sub-heading" id="Sec3"><span class="c-article-section__title-number">2.1 </span>Smoothness spaces of Morrey type on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> </h3><p>Let <span class="mathjax-tex">$\mathcal {S}({{{\mathbb {R}}}^d})$</span> be the set of all <i>Schwartz functions</i> on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, endowed with the usual topology, and denote by <span class="mathjax-tex">$\mathcal {S}'({{{\mathbb {R}}}^d})$</span> its <i>topological dual</i>, namely, the space of all bounded linear functionals on <span class="mathjax-tex">$\mathcal {S}({{{\mathbb {R}}}^d})$</span> endowed with the weak <span class="mathjax-tex">$*$</span>-topology. For all <span class="mathjax-tex">$f\in {\mathcal {S}}({{{\mathbb {R}}}^d})$</span> or <span class="mathjax-tex">${\mathcal {S}}'({{{\mathbb {R}}}^d})$</span>, we use <span class="mathjax-tex">${\widehat{f}}$</span> to denote its <i>Fourier transform</i>, and <span class="mathjax-tex">$f^\vee $</span> for its inverse. Let <span class="mathjax-tex">$\mathcal {Q}$</span> be the collection of all <i>dyadic cubes</i> in <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, namely, <span class="mathjax-tex">$ \mathcal {Q}{:=} \{Q_{j,k}{:=} 2^{-j}([0,1)^d+k):\ j\in {{\mathbb {Z}}},\ k\in {{{\mathbb {Z}}}^d}\}. $</span> The symbol <span class="mathjax-tex">$\ell (Q)$</span> denotes the side-length of the cube <i>Q</i> and <span class="mathjax-tex">$j_Q{:=}-\log _2\ell (Q)$</span>. Moreover, we denote by <span class="mathjax-tex">$\chi _{Q_{j,m}}$</span> the characteristic function of the cube <span class="mathjax-tex">$Q_{j,m}$</span>.</p><p>Let <span class="mathjax-tex">$\varphi _0,$</span> <span class="mathjax-tex">$\varphi \in \mathcal {S}({{{\mathbb {R}}}^d})$</span>. We say that <span class="mathjax-tex">$(\varphi _0, \varphi )$</span> is an <i>admissible pair</i> if</p><div id="Equ1" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathop {\text {supp}}\nolimits }\,\widehat{\varphi _0}\subset \{\xi \in {{{\mathbb {R}}}^d}:\,|\xi |\le 2\}\,, \qquad |\widehat{\varphi _0}(\xi )|\ge C\ \text{ if }\ |\xi |\le 5/3, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.1) </div></div><p>and</p><div id="Equ2" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {\mathop {\text {supp}}\nolimits }\,\widehat{\varphi }\subset \{\xi \in {{{\mathbb {R}}}^d}: 1/2\le |\xi |\le 2\}\quad \text{ and }\quad |\widehat{\varphi }(\xi )|\ge C\ \text{ if }\ 3/5\le |\xi |\le 5/3, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.2) </div></div><p>where <i>C</i> is a positive constant. In what follows, for all <span class="mathjax-tex">$\varphi \in {\mathcal {S}}({{{\mathbb {R}}}^d})$</span> and <span class="mathjax-tex">$j\in {{\mathbb {N}}}$</span>, <span class="mathjax-tex">$\varphi _j(\cdot ){:=}2^{jd}\varphi (2^j\cdot )$</span>.</p> <h3 class="c-article__sub-heading" id="FPar1">Definition 2.1</h3> <p>Let <span class="mathjax-tex">$s\in {{{\mathbb {R}}}}$</span>, <span class="mathjax-tex">$\tau \in [0,\infty )$</span>, <span class="mathjax-tex">$q \in (0,\infty ]$</span> and <span class="mathjax-tex">$(\varphi _0,\varphi )$</span> be an admissible pair. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$p\in (0,\infty ]$</span>. The <i>Besov-type space</i> <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> is defined to be the collection of all <span class="mathjax-tex">$f\in \mathcal {S}'({{{\mathbb {R}}}^d})$</span> such that </p><div id="Equ96" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert f \mid {{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \sum _{j=\max \{j_P,0\}}^\infty \!\! 2^{js q}\left[ \int \limits _P |\varphi _j*f(x)|^p\;\textrm{d}x\right] ^{\frac{q}{p}}\right\} ^{\frac{1}{q}}<\infty \end{aligned}$$</span></div></div><p> with the usual modifications made in case of <span class="mathjax-tex">$p=\infty $</span> and/or <span class="mathjax-tex">$q=\infty $</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$p\in (0,\infty )$</span>. The <i>Triebel-Lizorkin-type space</i> <span class="mathjax-tex">${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> is defined to be the collection of all <span class="mathjax-tex">$f\in \mathcal {S}'({{{\mathbb {R}}}^d})$</span> such that </p><div id="Equ97" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert f \mid {{F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \int \limits _P\left[ \sum _{j=\max \{j_P,0\}}^\infty \!\! 2^{js q} |\varphi _j*f(x)|^q\right] ^{\frac{p}{q}}\;\textrm{d}x\right\} ^{\frac{1}{p}}<\infty \end{aligned}$$</span></div></div><p> with the usual modification made in case of <span class="mathjax-tex">$q=\infty $</span>.</p> </dd></dl> <h3 class="c-article__sub-heading" id="FPar2">Remark 2.2</h3> <p>These spaces were introduced in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e6524">47</a>] and proved therein to be quasi-Banach spaces. In the Banach case the scale of Nikol’skij-Besov type spaces <span class="mathjax-tex">${{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}$</span> had already been introduced and investigated in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="El Baraka, A.: An embedding theorem for Campanato spaces. Electron. J. Differential Equat. 66, 1–17 (2002)" href="#ref-CR3" id="ref-link-section-d74823138e6588">3</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="El Baraka, A.: Function spaces of BMO and Campanato type, pp.109-115, in: Proc. of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. 9, Southwest Texas State Univ., San Marcos, TX (2002)" href="#ref-CR4" id="ref-link-section-d74823138e6588_1">4</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 5" title="El Baraka, A.: Littlewood-Paley characterization for Campanato spaces. J. Funct. Spaces Appl. 4, 193–220 (2006)" href="/article/10.1007/s10231-023-01327-w#ref-CR5" id="ref-link-section-d74823138e6591">5</a>]. It is easy to see that, when <span class="mathjax-tex">$\tau =0$</span>, then <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> and <span class="mathjax-tex">${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> coincide with the classical Besov space <span class="mathjax-tex">$B^s_{p,q}({{{\mathbb {R}}}^d})$</span> and Triebel-Lizorkin space <span class="mathjax-tex">$F^s_{p,q}({{{\mathbb {R}}}^d})$</span>, respectively. In case of <span class="mathjax-tex">$\tau <0$</span> the spaces are trivial, <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})={F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})=\{0\}$</span>, <span class="mathjax-tex">$\tau <0$</span>. There exists extensive literature on such spaces; we refer, in particular, to the series of monographs [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)" href="#ref-CR34" id="ref-link-section-d74823138e7018">34</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" title="Triebel, H.: Theory of Function Spaces. II, Birkhäuser, Basel (1992)" href="#ref-CR35" id="ref-link-section-d74823138e7018_1">35</a>,<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Triebel, H.: Theory of Function Spaces. III, Birkhäuser, Basel (2006)" href="/article/10.1007/s10231-023-01327-w#ref-CR36" id="ref-link-section-d74823138e7021">36</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Triebel, H.: Theory of Function Spaces. IV, Birkhäuser, Basel (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR39" id="ref-link-section-d74823138e7024">39</a>] for a comprehensive treatment.</p> <p><i>Convention.</i> We adopt the nowadays usual custom to write <span class="mathjax-tex">$A^s_{p,q}({{{\mathbb {R}}}^d})$</span> instead of <span class="mathjax-tex">$B^s_{p,q}({{{\mathbb {R}}}^d})$</span> or <span class="mathjax-tex">$F^s_{p,q}({{{\mathbb {R}}}^d})$</span>, and <span class="mathjax-tex">${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> instead of <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> or <span class="mathjax-tex">${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span>, respectively, when both scales of spaces are meant simultaneously in some context.</p><p>We have elementary embeddings within this scale of spaces (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e7383">47</a>, Proposition 2.1]),</p><div id="Equ3" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} A^{s+\varepsilon ,\tau }_{p,r}({{{\mathbb {R}}}^d}) \hookrightarrow {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}) \qquad \text {if}\quad \varepsilon \in (0,\infty ), \quad r,\,q\in (0,\infty ], \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.3) </div></div><p>and</p><div id="Equ4" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {A}^{s,\tau }_{p,q_1}({{{\mathbb {R}}}^d}) \hookrightarrow {A}^{s,\tau }_{p,q_2}({{{\mathbb {R}}}^d})\quad \text {if} \quad q_1\le q_2, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.4) </div></div><p>as well as</p><div id="Equ5" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B^{s,\tau }_{p,\min \{p,q\}}({{{\mathbb {R}}}^d})\, \hookrightarrow \, {F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\, \hookrightarrow \, B^{s,\tau }_{p,\max \{p,q\}}({{{\mathbb {R}}}^d}), \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.5) </div></div><p>which directly extends the well-known classical case from <span class="mathjax-tex">$\tau =0$</span> to <span class="mathjax-tex">$\tau \in [0,\infty )$</span>, <span class="mathjax-tex">$p\in (0,\infty )$</span>, <span class="mathjax-tex">$q\in (0,\infty ]$</span> and <span class="mathjax-tex">$s\in {{\mathbb {R}}}$</span>.</p><p>It is also known from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e8098">47</a>, Proposition 2.6] that</p><div id="Equ6" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} A^{s,\tau }_{p,q}({{{\mathbb {R}}}^d}) \hookrightarrow B^{s+d(\tau -\frac{1}{p})}_{\infty ,\infty }({{{\mathbb {R}}}^d}). \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.6) </div></div><p>The following remarkable feature was proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 44" title="Yang, D., Yuan, W.: Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces. Appl. Anal. 92, 549–561 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR44" id="ref-link-section-d74823138e8243">44</a>].</p> <h3 class="c-article__sub-heading" id="FPar3">Proposition 2.3</h3> <p>Let <span class="mathjax-tex">$s\in {{{\mathbb {R}}}}$</span>, <span class="mathjax-tex">$\tau \in [0,\infty )$</span> and <span class="mathjax-tex">$p,\,q\in (0,\infty ]$</span> (with <span class="mathjax-tex">$p<\infty $</span> in the <i>F</i>-case). If either <span class="mathjax-tex">$\tau >\frac{1}{p}$</span> or <span class="mathjax-tex">$\tau =\frac{1}{p}$</span> and <span class="mathjax-tex">$q=\infty $</span>, then <span class="mathjax-tex">$A^{s,\tau }_{p,q}({{{\mathbb {R}}}^d}) = B^{s+d(\tau -\frac{1}{p})}_{\infty ,\infty }({{{\mathbb {R}}}^d})$</span>.</p> <p>Now we come to smoothness spaces of Morrey type <span class="mathjax-tex">${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span> and <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span>. Recall first that the <i>Morrey space</i> <span class="mathjax-tex">${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$</span>, <span class="mathjax-tex">$0<p\le u<\infty $</span>, is defined to be the set of all locally <i>p</i>-integrable functions <span class="mathjax-tex">$f\in L_p^{\textrm{loc}}({{{\mathbb {R}}}^d})$</span> such that</p><div id="Equ98" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert f \mid {{{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})}\Vert {:=}\, \sup _{x\in {{{\mathbb {R}}}^d}, R>0} R^{\frac{d}{u}-\frac{d}{p}} \left[ \int _{B(x,R)} |f(y)|^p \;\textrm{d}y \right] ^{\frac{1}{p}}\, <\, \infty \,. \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar4">Remark 2.4</h3> <p>The spaces <span class="mathjax-tex">${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})$</span> are quasi-Banach spaces (Banach spaces for <span class="mathjax-tex">$p \ge 1$</span>). They originated from Morrey’s study on PDE (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 19" title="Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938)" href="/article/10.1007/s10231-023-01327-w#ref-CR19" id="ref-link-section-d74823138e9176">19</a>]) and are part of the wider class of Morrey-Campanato spaces; cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 21" title="Peetre, J.: On the theory of 
 
 
 
 $${{\cal{L} }}_{p,\lambda }$$
 
 
 L
 
 p
 ,
 λ
 
 
 
 spaces. J. Funct. Anal. 4, 71–87 (1969)" href="/article/10.1007/s10231-023-01327-w#ref-CR21" id="ref-link-section-d74823138e9179">21</a>]. They can be considered as a complement to <span class="mathjax-tex">$L_p$</span> spaces. As a matter of fact, <span class="mathjax-tex">${{\mathcal {M}}}_{p,p}({{{\mathbb {R}}}^d}) = L_p({{{\mathbb {R}}}^d})$</span> with <span class="mathjax-tex">$p\in (0,\infty )$</span>. To extend this relation, we put <span class="mathjax-tex">${{\mathcal {M}}}_{\infty ,\infty }({{{\mathbb {R}}}^d}) = L_\infty ({{{\mathbb {R}}}^d})$</span>. One can easily see that <span class="mathjax-tex">${{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})=\{0\}$</span> for <span class="mathjax-tex">$u<p$</span>, and that for <span class="mathjax-tex">$0<p_2 \le p_1 \le u < \infty $</span>,</p><div id="Equ7" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} L_u({{{\mathbb {R}}}^d})= {{\mathcal {M}}}_{u,u}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {M}}}_{u,p_1}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {M}}}_{u,p_2}({{{\mathbb {R}}}^d}). \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.7) </div></div><p>In an analogous way, one can define the spaces <span class="mathjax-tex">${{\mathcal {M}}}_{\infty ,p}({{{\mathbb {R}}}^d})$</span>, <span class="mathjax-tex">$p\in (0, \infty )$</span>, but using the Lebesgue differentiation theorem, one can easily prove that <span class="mathjax-tex">${{\mathcal {M}}}_{\infty , p}({{{\mathbb {R}}}^d}) = L_\infty ({{{\mathbb {R}}}^d})$</span>.</p> <h3 class="c-article__sub-heading" id="FPar5">Definition 2.5</h3> <p>Let <span class="mathjax-tex">$0<p\le u<\infty $</span> or <span class="mathjax-tex">$p=u=\infty $</span>. Let <span class="mathjax-tex">$q\in (0,\infty ]$</span>, <span class="mathjax-tex">$s\in {{\mathbb {R}}}$</span> and <span class="mathjax-tex">$\varphi _0$</span>, <span class="mathjax-tex">$\varphi \in {\mathcal {S}}({{{\mathbb {R}}}^d})$</span> be as in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ1">2.1</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ2">2.2</a>), respectively. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>The <i>Besov-Morrey space</i> <span class="mathjax-tex">${{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span> is defined to be the set of all distributions <span class="mathjax-tex">$f\in \mathcal {S}'({{{\mathbb {R}}}^d})$</span> such that </p><div id="Equ8" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \big \Vert f\mid {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\big \Vert = \bigg [\sum _{j=0}^{\infty }2^{jsq}\big \Vert \varphi _j *f\mid {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\big \Vert ^q \bigg ]^{1/q} < \infty \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.8) </div></div><p> with the usual modification made in case of <span class="mathjax-tex">$q=\infty $</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$u\in (0,\infty )$</span>. The <i>Triebel-Lizorkin-Morrey space</i> <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span> is defined to be the set of all distributions <span class="mathjax-tex">$f\in \mathcal {S}'({{{\mathbb {R}}}^d})$</span> such that </p><div id="Equ9" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \big \Vert f \mid {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\big \Vert =\bigg \Vert \bigg [\sum _{j=0}^{\infty }2^{jsq} | (\varphi _j*f)(\cdot )|^q\bigg ]^{1/q} \mid {{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d})\bigg \Vert <\infty \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.9) </div></div><p> with the usual modification made in case of <span class="mathjax-tex">$q=\infty $</span>.</p> </dd></dl> <p><i>Convention.</i> Again we adopt the usual custom to write <span class="mathjax-tex">${{\mathcal {A}}}^{s}_{u,p,q}$</span> instead of <span class="mathjax-tex">${{\mathcal {N}}}^{s}_{u,p,q}$</span> or <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}$</span>, when both scales of spaces are meant simultaneously in some context.</p> <h3 class="c-article__sub-heading" id="FPar6">Remark 2.6</h3> <p>The spaces <span class="mathjax-tex">${{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span> are independent of the particular choices of <span class="mathjax-tex">$\varphi _0$</span>, <span class="mathjax-tex">$\varphi $</span> appearing in their definitions. They are quasi-Banach spaces (Banach spaces for <span class="mathjax-tex">$p,\,q\ge 1$</span>), and <span class="mathjax-tex">$\mathcal {S}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\hookrightarrow \mathcal {S}'({{{\mathbb {R}}}^d})$</span>. Moreover, for <span class="mathjax-tex">$u=p$</span> we re-obtain the usual Besov and Triebel-Lizorkin spaces,</p><div id="Equ10" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathcal {A}}}^{s}_{p,p,q}({{{\mathbb {R}}}^d}) = A^s_{p,q}({{{\mathbb {R}}}^d}) = A^{s,0}_{p,q}({{{\mathbb {R}}}^d}). \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.10) </div></div><p>Besov-Morrey spaces were introduced by Kozono and Yamazaki in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 15" title="Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Different. Equat. 19, 959–1014 (1994)" href="/article/10.1007/s10231-023-01327-w#ref-CR15" id="ref-link-section-d74823138e11490">15</a>]. They studied semi-linear heat equations and Navier–Stokes equations with initial data belonging to Besov-Morrey spaces. The investigations were continued by Mazzucato [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Mazzucato, A.L.: Besov-Morrey spaces: function space theory and applications to non-linear PDE. Trans. Amer. Math. Soc. 355, 1297–1364 (2003)" href="/article/10.1007/s10231-023-01327-w#ref-CR17" id="ref-link-section-d74823138e11493">17</a>], where one can find the atomic decomposition of some spaces. The Triebel-Lizorkin-Morrey spaces were later introduced by Tang and Xu [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Tang, L., Xu, J.: Some properties of Morrey type Besov-Triebel spaces. Math. Nachr. 278, 904–917 (2005)" href="/article/10.1007/s10231-023-01327-w#ref-CR33" id="ref-link-section-d74823138e11496">33</a>]. We follow the ideas of Tang and Xu [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 33" title="Tang, L., Xu, J.: Some properties of Morrey type Besov-Triebel spaces. Math. Nachr. 278, 904–917 (2005)" href="/article/10.1007/s10231-023-01327-w#ref-CR33" id="ref-link-section-d74823138e11499">33</a>], where a somewhat different definition is proposed. The ideas were further developed by Sawano and Tanaka [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Sawano, Y.: Wavelet characterizations of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Funct. Approx. Comment. Math. 38, 93–107 (2008)" href="/article/10.1007/s10231-023-01327-w#ref-CR24" id="ref-link-section-d74823138e11503">24</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 25" title="Sawano, Y.: A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Acta Math. Sin. (Engl. Ser.) 25, 1223–1242 (2009)" href="/article/10.1007/s10231-023-01327-w#ref-CR25" id="ref-link-section-d74823138e11506">25</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 28" title="Sawano, Y., Tanaka, H.: Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z. 257, 871–905 (2007)" href="/article/10.1007/s10231-023-01327-w#ref-CR28" id="ref-link-section-d74823138e11509">28</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 29" title="Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces for non-doubling measures: Sawano, Y., Tanaka. H. Math. Nachr. 282, 1788–1810 (2009)" href="/article/10.1007/s10231-023-01327-w#ref-CR29" id="ref-link-section-d74823138e11512">29</a>]. The most systematic and general approach to the spaces of this type can be found in the monograph [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e11515">47</a>] or in the survey papers by Sickel [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. I. Eurasian Math. J. 3, 110–149 (2012)" href="/article/10.1007/s10231-023-01327-w#ref-CR31" id="ref-link-section-d74823138e11518">31</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. II. Eurasian Math. J. 4, 82–124 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR32" id="ref-link-section-d74823138e11522">32</a>], which we also recommend for further up-to-date references on this subject. We refer to the recent monographs [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 26" title="Sawano, Y., Di Fazio, G., Hakim, D. I.: Morrey spaces. Introduction and Applications to Integral Operators and PDE’s. Vol. I, Monographs and Research Notes in Mathematics. Chapman & Hall CRC Press, Boca Raton, FL (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR26" id="ref-link-section-d74823138e11525">26</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 27" title="Sawano, Y., Di Fazio, G., Hakim, D. I.: Morrey spaces. Introduction and Applications to Integral Operators and PDE’s. Vol. II, Monographs and Research Notes in Mathematics. Chapman & Hall CRC Press, Boca Raton, FL (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR27" id="ref-link-section-d74823138e11528">27</a>] for applications.</p> <p>It turned out that many of the results from the classical situation have their counterparts for the spaces <span class="mathjax-tex">$\mathcal {A}^s_{u,p,q}({{{\mathbb {R}}}^d})$</span>, e. g.,</p><div id="Equ11" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathcal {A}}}^{s+\varepsilon }_{u,p,r}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {A}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\qquad \text {if}\quad \varepsilon >0, \quad r\in (0,\infty ], \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.11) </div></div><p>and <span class="mathjax-tex">${{\mathcal {A}}}^{s}_{u,p,q_1}({{{\mathbb {R}}}^d}) \hookrightarrow {{\mathcal {A}}}^{s}_{u,p,q_2}({{{\mathbb {R}}}^d})$</span> if <span class="mathjax-tex">$q_1\le q_2$</span>. However, there also exist some differences. Sawano proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 24" title="Sawano, Y.: Wavelet characterizations of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Funct. Approx. Comment. Math. 38, 93–107 (2008)" href="/article/10.1007/s10231-023-01327-w#ref-CR24" id="ref-link-section-d74823138e11919">24</a>] that, for <span class="mathjax-tex">$s\in {{\mathbb {R}}}$</span> and <span class="mathjax-tex">$0<p< u<\infty $</span>,</p><div id="Equ12" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathcal {N}}}^s_{u,p,\min \{p,q\}}({{{\mathbb {R}}}^d})\, \hookrightarrow \, {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\, \hookrightarrow \,{{\mathcal {N}}}^s_{u,p,\infty }({{{\mathbb {R}}}^d}), \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.12) </div></div><p>where, for the latter embedding, <span class="mathjax-tex">$r=\infty $</span> cannot be improved – unlike in case of <span class="mathjax-tex">$u=p$</span> (see (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ5">2.5</a>) with <span class="mathjax-tex">$\tau =0$</span>). More precisely,</p><div id="Equ99" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\hookrightarrow {{\mathcal {N}}}^s_{u,p,r}({{{\mathbb {R}}}^d})\quad \text {if, and only if,}\quad r=\infty ~~ \text {or} ~~ u=p\ \text {and}\ r\ge \max \{p,\,q\}. \end{aligned}$$</span></div></div><p>On the other hand, Mazzucato has shown in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 17" title="Mazzucato, A.L.: Besov-Morrey spaces: function space theory and applications to non-linear PDE. Trans. Amer. Math. Soc. 355, 1297–1364 (2003)" href="/article/10.1007/s10231-023-01327-w#ref-CR17" id="ref-link-section-d74823138e12439">17</a>, Proposition 4.1] that</p><div id="Equ100" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathcal {E}^0_{u,p,2}({{{\mathbb {R}}}^d})={{\mathcal {M}}}_{u,p}({{{\mathbb {R}}}^d}),\quad 1<p\le u<\infty , \end{aligned}$$</span></div></div><p>in particular,</p><div id="Equ13" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathcal {E}^0_{p,p,2}({{{\mathbb {R}}}^d})=L_p({{{\mathbb {R}}}^d})=F^0_{p,2}({{{\mathbb {R}}}^d}),\quad p\in (1,\infty ). \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.13) </div></div> <h3 class="c-article__sub-heading" id="FPar7">Remark 2.7</h3> <p>Let <i>s</i>, <i>u</i>, <i>p</i> and <i>q</i> be as in Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar5">2.5</a> and <span class="mathjax-tex">$\tau \in [0,\infty )$</span>. It is known that</p><div id="Equ14" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {{\mathcal {N}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d}) \hookrightarrow {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}) \qquad \text {with}\qquad \tau ={1}/{p}- {1}/{u}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.14) </div></div><p>cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e12955">47</a>, Corollary 3.3]. Moreover, the above embedding is proper if <span class="mathjax-tex">$\tau >0$</span> and <span class="mathjax-tex">$q<\infty $</span>. If <span class="mathjax-tex">$\tau =0$</span> or <span class="mathjax-tex">$q=\infty $</span>, then both spaces coincide with each other, in particular,</p><div id="Equ15" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathcal {N}^{s}_{u,p,\infty }({{{\mathbb {R}}}^d}) = B^{s,\frac{1}{p}- \frac{1}{u}}_{p,\infty }({{{\mathbb {R}}}^d}). \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.15) </div></div><p>As for the <i>F</i>-spaces, if <span class="mathjax-tex">$0\le \tau <{1}/{p}$</span>, then</p><div id="Equ16" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\, = \, {{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})\quad \text {with }\quad \tau = {1}/{p}-{1}/{u}\,,\quad 0< p\le u < \infty \,; \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.16) </div></div><p>cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e13412">47</a>, Corollary 3.3]. Moreover, if <span class="mathjax-tex">$p\in (0,\infty )$</span> and <span class="mathjax-tex">$q\in (0,\infty )$</span>, then</p><div id="Equ17" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F^{s,\, \frac{1}{p} }_{p\,,\,q}({{{\mathbb {R}}}^d}) \, = \, F^{s}_{\infty ,\,q}({{{\mathbb {R}}}^d})\, = \, B^{s,\, \frac{1}{q} }_{q\,,\,q}({{{\mathbb {R}}}^d}) \,; \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.17) </div></div><p>cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. I. Eurasian Math. J. 3, 110–149 (2012)" href="/article/10.1007/s10231-023-01327-w#ref-CR31" id="ref-link-section-d74823138e13682">31</a>, Propositions 3.4, 3.5] and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 32" title="Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. II. Eurasian Math. J. 4, 82–124 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR32" id="ref-link-section-d74823138e13685">32</a>, Remark 10].</p> <h3 class="c-article__sub-heading" id="FPar8">Remark 2.8</h3> <p>Recall that the space <span class="mathjax-tex">$\textrm{bmo}({{{\mathbb {R}}}^d})$</span> is covered by the above scale. More precisely, consider the local (non-homogeneous) space of functions of bounded mean oscillation, <span class="mathjax-tex">$\textrm{bmo}({{{\mathbb {R}}}^d})$</span>, consisting of all locally integrable functions <span class="mathjax-tex">$\ f\in L_1^{\textrm{loc}}({{{\mathbb {R}}}^d}) $</span> satisfying that</p><div id="Equ101" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\| f \right\| _{\textrm{bmo}}{:=} \sup _{|Q|\le 1}\; \frac{1}{|Q|} \int \limits _Q |f(x)-f_Q| \;\textrm{d}x + \sup _{|Q|> 1}\; \frac{1}{|Q|} \int \limits _Q |f(x)| \;\textrm{d}x<\infty , \end{aligned}$$</span></div></div><p>where <i>Q</i> appearing in the above definition runs over all cubes in <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, and <span class="mathjax-tex">$ f_Q $</span> denotes the mean value of <i>f</i> with respect to <i>Q</i>, namely, <span class="mathjax-tex">$ f_Q {:=} \frac{1}{|Q|} \;\int _Q f(x)\;\textrm{d}x$</span>, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)" href="/article/10.1007/s10231-023-01327-w#ref-CR34" id="ref-link-section-d74823138e14178">34</a>, 2.2.2 (viii)]. The space <span class="mathjax-tex">$\textrm{bmo}({{{\mathbb {R}}}^d})$</span> coincides with <span class="mathjax-tex">$F^{0}_{\infty , 2}({{{\mathbb {R}}}^d})$</span>, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 34" title="Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)" href="/article/10.1007/s10231-023-01327-w#ref-CR34" id="ref-link-section-d74823138e14273">34</a>, Theorem 2.5.8/2]. Hence the above result (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ17">2.17</a>) implies, in particular,</p><div id="Equ18" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textrm{bmo}({{{\mathbb {R}}}^d})= F^{0}_{\infty ,2}({{{\mathbb {R}}}^d})= F^{0, 1/p}_{p, 2}({{{\mathbb {R}}}^d})= {B^{0, 1/2}_{2, 2}({{{\mathbb {R}}}^d})}, \quad 0<p<\infty . \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.18) </div></div> <h3 class="c-article__sub-heading" id="FPar9">Remark 2.9</h3> <p>In contrast to this approach, Triebel followed the original Morrey-Campanato ideas to develop local spaces <span class="mathjax-tex">$\mathcal {L}^rA^s_{p,q}({{{\mathbb {R}}}^d})$</span> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 37" title="Triebel, H.: Local Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics 20, European Mathematical Society. EMS), Zürich (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR37" id="ref-link-section-d74823138e14566">37</a>], and so-called ‘hybrid’ spaces <span class="mathjax-tex">$L^rA^s_{p,q}({{{\mathbb {R}}}^d})$</span> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Triebel, H.: Hybrid Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics 24, European Mathematical Society. EMS), Zürich (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR38" id="ref-link-section-d74823138e14630">38</a>], where <span class="mathjax-tex">$0<p<\infty $</span>, <span class="mathjax-tex">$0<q\le \infty $</span>, <span class="mathjax-tex">$s\in {{\mathbb {R}}}$</span>, and <span class="mathjax-tex">$-\frac{d}{p}\le r<\infty $</span>. This construction is based on wavelet decompositions and also combines local and global elements as in Definitions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar1">2.1</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar5">2.5</a>. However, Triebel proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 38" title="Triebel, H.: Hybrid Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics 24, European Mathematical Society. EMS), Zürich (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR38" id="ref-link-section-d74823138e14757">38</a>, Theorem 3.38] that</p><div id="Equ19" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} L^rA^s_{p,q}({{{\mathbb {R}}}^d}) = {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}), \qquad \tau =\frac{1}{p}+\frac{r}{d}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.19) </div></div><p>in all admitted cases. We return to this coincidence below.</p> <p>As mentioned previously, in this paper we are interested in studying embeddings of type</p><div id="Equ102" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_\tau : {A}_{p_1,q_1}^{s_1,\tau _1}\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}\end{aligned}$$</span></div></div><p>on domains. To do so, we will strongly rely on the corresponding results for these spaces on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, obtained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e15067">46</a>]. In order to make the reading easier, we recall those results here. We begin with the situation of Besov-type spaces where the results can be found in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e15070">46</a>, Theorems 2.4, 2.5].</p> <h3 class="c-article__sub-heading" id="FPar10">Theorem 2.10</h3> <p>( [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e15080">46</a>]) Let <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0<p_i\le \infty $</span> and <span class="mathjax-tex">$\tau _i\ge 0$</span>, <span class="mathjax-tex">$i=1,2$</span>. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$\tau _2 > \frac{1}{p_2}$</span> or <span class="mathjax-tex">$\tau _2=\frac{1}{p_2}$</span> and <span class="mathjax-tex">$q_2=\infty $</span>. Then the embedding </p><div id="Equ20" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {B}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.20) </div></div><p> holds if, and only if, <span class="mathjax-tex">$\quad \displaystyle \frac{s_1-s_2}{d}\ge \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2$</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$\tau _1 > \frac{1}{p_1}$</span> or <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span> and <span class="mathjax-tex">$q_1=\infty $</span>. Then the embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ20">2.20</a>) holds if, and only if, </p><div id="Equ103" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\qquad \frac{s_1-s_2}{d}> \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \quad \text{ and } \quad \tau _2 \ge \frac{1}{p_2}\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \quad \text{ and } \quad \tau _2 > \frac{1}{p_2}\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \quad \text{ and } \quad \tau _2 = \frac{1}{p_2} \quad \text{ and } \quad q_2=\infty . \end{aligned}$$</span></div></div> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(iii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Assume that <span class="mathjax-tex">$\tau _i < \frac{1}{p_i}$</span> or <span class="mathjax-tex">$\tau _i=\frac{1}{p_i}$</span> and <span class="mathjax-tex">$q_i<\infty $</span>, <span class="mathjax-tex">$i=1,2$</span>.</p> </dd></dl><ol class="u-list-style-none"> <li> <span class="u-custom-list-number"> (a)</span> <p>The embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ20">2.20</a>) holds true if </p><div id="Equ104" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\ge 0, \qquad \frac{\tau _1}{p_2}\le \frac{\tau _2}{p_1} \end{aligned}$$</span></div></div><p> and </p><div id="Equ105" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\qquad \frac{s_1-s_2}{d}> \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \\&\qquad \text{ with } \\&\qquad \qquad \quad (s_1-s_2)(\tau _1-\tau _2)\ne 0, \quad \frac{\tau _1}{p_2}< \frac{\tau _2}{p_1}\\&\qquad \text{ or } \qquad (s_1-s_2)(\tau _1-\tau _2)\ne 0, \quad \frac{\tau _1}{p_2}= \frac{\tau _2}{p_1}, \quad \frac{\tau _1}{q_2}\le \frac{\tau _2}{q_1}\\&\qquad \text{ or } \qquad (s_1-s_2)(\tau _1-\tau _2)=0, \quad q_1 \le q_2,\\&\qquad \qquad \quad \text{ and } \quad p_1\ge p_2 \quad \text{ if } \quad s_1=s_2 \quad \text{ and } \quad \tau _1 p_1 = \tau _2 p_2 =1. \end{aligned}$$</span></div></div> </li> <li> <span class="u-custom-list-number"> (b)</span> <p>The conditions <span class="mathjax-tex">$ \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\ge 0$</span> and <span class="mathjax-tex">$\frac{\tau _1}{p_2}\le \frac{\tau _2}{p_1}$</span>, and <span class="mathjax-tex">$s_1-s_2\ge \frac{d}{p_1}- d\tau _1 - \frac{d}{p_2}+d\tau _2$</span> as well as <span class="mathjax-tex">$q_1 \le q_2$</span> if <span class="mathjax-tex">$s_1=s_2$</span> are also necessary for the embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ20">2.20</a>).</p> </li> </ol> <p>The counterpart for <i>F</i>-spaces reads as follows, we refer to [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e17460">46</a>, Corollaries 5.8, 5.9] for details.</p> <h3 class="c-article__sub-heading" id="FPar11">Theorem 2.11</h3> <p>( [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 46" title="Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. 94, 318–340 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR46" id="ref-link-section-d74823138e17470">46</a>]) Let <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0<p_i< \infty $</span> and <span class="mathjax-tex">$\tau _i\ge 0$</span>, <span class="mathjax-tex">$i=1,2$</span>. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$\tau _i > \frac{1}{p_i}$</span> or <span class="mathjax-tex">$\tau _i=\frac{1}{p_i}$</span> and <span class="mathjax-tex">$q_i=\infty $</span>, <span class="mathjax-tex">$i=1,2$</span>. Then the embedding </p><div id="Equ21" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {F}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.21) </div></div><p> holds if, and only if, <span class="mathjax-tex">$\quad \displaystyle \frac{s_1-s_2}{d}\ge \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2$</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Assume that <span class="mathjax-tex">$\tau _i < \frac{1}{p_i}$</span>, <span class="mathjax-tex">$i=1,2$</span>. Then the embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ21">2.21</a>) holds if, and only if, </p><div id="Equ106" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2\ge 0, \qquad \frac{\tau _1}{p_2}\le \frac{\tau _2}{p_1} \end{aligned}$$</span></div></div><p> and </p><div id="Equ107" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\qquad \frac{s_1-s_2}{d}>\frac{1}{p_1} - \tau _1 - \frac{1}{p_2}+\tau _2\\\ \text{ or }&\qquad \frac{s_1-s_2}{d}= \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2 \\ \text{ or }&\qquad s_1=s_2 \quad \text{ and } \quad q_1 \le q_2. \end{aligned}$$</span></div></div> </dd></dl> <h3 class="c-article__sub-heading" id="Sec4"><span class="c-article-section__title-number">2.2 </span>Spaces on domains</h3><p>Let <span class="mathjax-tex">$\Omega $</span> denote an open, nontrivial subset of <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>. We consider smoothness Morrey spaces on <span class="mathjax-tex">$\Omega $</span> defined by restriction. Let <span class="mathjax-tex">${{\mathcal {D}}}(\Omega )$</span> be the set of all infinitely differentiable functions supported in <span class="mathjax-tex">$\Omega $</span> and denote by <span class="mathjax-tex">${{\mathcal {D}}}'(\Omega )$</span> its dual. Since we are able to define the extension operator <span class="mathjax-tex">$\textrm{ext}: {{\mathcal {D}}}(\Omega ) \rightarrow \mathcal {S}({{{\mathbb {R}}}^d})$</span>, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 30" title="Sawano, Y.: Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces on domains. Math. Nachr. 283, 1456–1487 (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR30" id="ref-link-section-d74823138e18734">30</a>], the restriction operator <span class="mathjax-tex">$\textrm{re}: \mathcal {S}'({{{\mathbb {R}}}^d})\rightarrow {{\mathcal {D}}}'(\Omega )$</span> can be defined naturally as an adjoint operator</p><div id="Equ108" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \langle \textrm{re}(f), \varphi \rangle = \langle f, \textrm{ext}(\varphi )\rangle , \quad f\in \mathcal {S}'({{{\mathbb {R}}}^d}), \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\varphi \in \mathcal {D}(\Omega )$</span>. We will write <span class="mathjax-tex">$f\vert _{\Omega }=\textrm{re } (f)$</span>.</p> <h3 class="c-article__sub-heading" id="FPar12">Definition 2.12</h3> <p>Let <span class="mathjax-tex">$s\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$\tau \in [0,\infty )$</span>, <span class="mathjax-tex">$q\in (0,\infty ]$</span> and <span class="mathjax-tex">$p\in (0,\infty ]$</span> (with <span class="mathjax-tex">$p<\infty $</span> in the case of <span class="mathjax-tex">${A}_{p,q}^{s,\tau }={F}_{p,q}^{s,\tau }$</span>). Then <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> is defined by</p><div id="Equ109" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {A}_{p,q}^{s,\tau }(\Omega ){:=}\big \{f\in {{\mathcal {D}}}'(\Omega ): f=g\vert _{\Omega } \text { for some } g\in {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\big \} \end{aligned}$$</span></div></div><p>endowed with the quasi-norm</p><div id="Equ110" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \big \Vert f\mid {A}_{p,q}^{s,\tau }(\Omega )\big \Vert {:=} \inf \big \{ \Vert g\mid {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Vert : f=g\vert _{\Omega }, \; g\in {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\big \}. \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar13">Remark 2.13</h3> <p>The spaces <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> are quasi-Banach spaces (Banach spaces for <span class="mathjax-tex">$p,q\ge 1$</span>). When <span class="mathjax-tex">$\tau =0$</span> we re-obtain the usual Besov and Triebel-Lizorkin spaces defined on domains. For the particular case of <span class="mathjax-tex">$\Omega $</span> being a bounded <span class="mathjax-tex">$C^{\infty }$</span> domain in <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, some properties were studied in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e19913">47</a>, Section 6.4.2]. In particular, according to [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e19916">47</a>, Theorem 6.13], for such a domain <span class="mathjax-tex">$\Omega $</span>, there exists a linear and bounded extension operator</p><div id="Equ22" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textrm{ext}: {A}_{p,q}^{s,\tau }(\Omega )\rightarrow {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d}),\quad \text {where}\quad 1\le p<\infty , 0<q\le \infty , s\in {{\mathbb {R}}}, \tau \ge 0, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.22) </div></div><p>such that</p><div id="Equ23" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textrm{re} \circ \textrm{ext}= \text {id}\quad \text {in}\quad {A}_{p,q}^{s,\tau }(\Omega ), \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.23) </div></div><p>where <span class="mathjax-tex">$\textrm{re}: {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\rightarrow {A}_{p,q}^{s,\tau }(\Omega )$</span> is the restriction operator as above.</p> <p>Moreover, in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 9" title="Haroske, D.D., Moura, S.D., Skrzypczak, L.: Smoothness Morrey Spaces of regular distributions, and some unboundedness properties. Nonlinear Anal Series A: Theory, Methods Appl 139, 218–244 (2016)" href="/article/10.1007/s10231-023-01327-w#ref-CR9" id="ref-link-section-d74823138e20302">9</a>] we studied the question under what assumptions these spaces consist of regular distributions only.</p> <h3 class="c-article__sub-heading" id="FPar14">Remark 2.14</h3> <p>Let us mention that we have the counterparts of many continuous embeddings stated in the previous subsection for spaces on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> when dealing with spaces restricted to bounded domains. We recall them in further detail if appropriate and necessary for our arguments below.</p> <p>Later we shall mainly deal with Lipschitz domains. Therefore we recall the concept for convenience. By a Lipschitz domain we mean either a special or bounded Lipschitz domain. A special Lipschitz domain is defined as an open set <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span> lying above a graph of a Lipschitz function <span class="mathjax-tex">$\omega : {{\mathbb {R}}}^{d-1} \rightarrow {{\mathbb {R}}}$</span>. More precisely,</p><div id="Equ111" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Omega = \{(x', x_d) \in {{{\mathbb {R}}}^d}: x_d > \omega (x') \}, \end{aligned}$$</span></div></div><p>where</p><div id="Equ24" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} |\omega (x')-\omega (y')|\le A \, |x'-y'|, \quad x', y' \in {{\mathbb {R}}}^{d-1}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.24) </div></div><p>A bounded Lipschitz domain is a bounded domain <span class="mathjax-tex">$\Omega $</span> whose boundary <span class="mathjax-tex">$\partial \Omega $</span> can be covered by a finite number of open balls <span class="mathjax-tex">$B_k$</span> so that, possibly after an appropriate rotation, <span class="mathjax-tex">$\partial \Omega \cap B_k$</span> for each <i>k</i> is a part of the graph of a Lipschitz function. Let <span class="mathjax-tex">$\Omega $</span> be a special Lipschitz domain defined by a Lipschitz function <span class="mathjax-tex">$\omega $</span> that satisfies the condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ24">2.24</a>). We put</p><div id="Equ112" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} K{:=} \{ (x', x_n) \in {{{\mathbb {R}}}^d}: |x'|<A^{-1} x_n\} \end{aligned}$$</span></div></div><p>and <span class="mathjax-tex">$-K {:=}\{-x: x\in K\}$</span>. Then <i>K</i> has the property that <span class="mathjax-tex">$x+K \subset \Omega $</span> for any <span class="mathjax-tex">$x \in \Omega $</span>. Moreover let <span class="mathjax-tex">$\mathcal {S}'(\Omega )$</span> denote the subspace of <span class="mathjax-tex">$\mathcal {D}'(\Omega )$</span> consisting of all distributions of finite order and of at most polynomial growth at infinity, that is, <span class="mathjax-tex">$f\in \mathcal {S}'(\Omega )$</span> if, and only if, the estimate</p><div id="Equ113" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} |\langle f,\eta \rangle | \le c \sup _{x\in \Omega , |\alpha |\le M}|\;\textrm{D}^\alpha \eta (x)|(1+|x|)^M,\qquad \eta \in \mathcal {D}(\Omega ), \end{aligned}$$</span></div></div><p>holds with some constants <i>c</i> and <i>M</i> depending on <i>f</i>.</p> <h3 class="c-article__sub-heading" id="FPar15">Remark 2.15</h3> <p>As already mentioned, we shall study continuous embeddings of function spaces on domains. For convenience let us recall what is well known for the classical function spaces <span class="mathjax-tex">$A^s_{p,q}(\Omega )$</span>. Let <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<p_i, q_i\le \infty $</span>, <span class="mathjax-tex">$i=1,2$</span>, and <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span> a bounded Lipschitz domain. Then</p><div id="Equ114" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}^B_\Omega : {B}_{p_1,q_1}^{s_1}(\Omega ) \hookrightarrow {B}_{p_2,q_2}^{s_2}(\Omega ) \end{aligned}$$</span></div></div><p>is continuous if, and only if,</p><div id="Equ25" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {either}\qquad&\frac{s_1-s_2}{d} > \max \left\{ \frac{1}{p_1}-\frac{1}{p_2}, 0\right\} \nonumber \\ \text {or}\qquad&\frac{s_1-s_2}{d} = \max \left\{ \frac{1}{p_1}-\frac{1}{p_2}, 0\right\} \quad \text {and}\quad q_1\le q_2. \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.25) </div></div><p>Assume, in addition, <span class="mathjax-tex">$p_i<\infty $</span>, <span class="mathjax-tex">$i=1,2$</span>. Then</p><div id="Equ115" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}^F_\Omega : {F}_{p_1,q_1}^{s_1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2}(\Omega ) \end{aligned}$$</span></div></div><p>is continuous if, and only if, either (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ25">2.25</a>) is satisfied,</p><div id="Equ116" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {or}\qquad&\frac{s_1-s_2}{d} = \max \left\{ \frac{1}{p_1}-\frac{1}{p_2},0\right\} =\frac{1}{p_1}-\frac{1}{p_2}>0,\\ \text {or}\qquad&\frac{s_1-s_2}{d} = \max \left\{ \frac{1}{p_1}-\frac{1}{p_2}, 0\right\} =0 \quad \text {and}\quad q_1\le q_2. \end{aligned}$$</span></div></div><p>For (partial) results we refer to [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 2" title="Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge Univ. Press, Cambridge (1996)" href="/article/10.1007/s10231-023-01327-w#ref-CR2" id="ref-link-section-d74823138e22347">2</a>, Section 2.5.1], [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Triebel, H.: Theory of Function Spaces. III, Birkhäuser, Basel (2006)" href="/article/10.1007/s10231-023-01327-w#ref-CR36" id="ref-link-section-d74823138e22350">36</a>, p. 60], and, quite recently, the extension to spaces <span class="mathjax-tex">$F^s_{\infty ,q}(\Omega )$</span> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 39" title="Triebel, H.: Theory of Function Spaces. IV, Birkhäuser, Basel (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR39" id="ref-link-section-d74823138e22399">39</a>, Section 2.6.5]; cf. also our results in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR11" id="ref-link-section-d74823138e22402">11</a>, Theorem 3.1] and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haroske, D.D., Skrzypczak, L.: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces. Rev. Mat. Complut. 27, 541–573 (2014)" href="/article/10.1007/s10231-023-01327-w#ref-CR12" id="ref-link-section-d74823138e22406">12</a>, Theorem 5.2]. However, in most of the above cases, the domain <span class="mathjax-tex">$\Omega $</span> is there assumed to be smooth.</p> <h3 class="c-article__sub-heading" id="Sec5"><span class="c-article-section__title-number">2.3 </span>Wavelet decomposition in Besov-type and Triebel-Lizorkin-type spaces</h3><p>In this section we describe the wavelet characterisation of Besov-type and Triebel-Lizorkin-type spaces proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Liang, Y., Yang, D., Yuan, W., Sawano, Y., Ullrich, T.: A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math. (Rozprawy Mat.) 489, 1–114 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR16" id="ref-link-section-d74823138e22437">16</a>]. This is a key tool when studying embedding properties of function spaces, since it allows one to transfer the problem to the corresponding sequence spaces.</p><p>Let <span class="mathjax-tex">$\widetilde{\phi }$</span> be a scaling function on <span class="mathjax-tex">${{\mathbb {R}}}$</span> with compact support and of sufficiently high regularity, and <span class="mathjax-tex">$\widetilde{\psi }$</span> the corresponding orthonormal wavelet. Then we can extend these wavelets from <span class="mathjax-tex">${{\mathbb {R}}}$</span> to <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> by the usual tensor procedure, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Wojtaszczyk, P.: A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts 37. Cambridge Univ. Press, Cambridge (1997)" href="/article/10.1007/s10231-023-01327-w#ref-CR40" id="ref-link-section-d74823138e22552">40</a>, Section 5.1]. This yields a scaling function <span class="mathjax-tex">$\phi $</span> and associated wavelets <span class="mathjax-tex">$\psi _1, \dots , \psi _{2^d-1}$</span>, all defined on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>; see, e.g. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Wojtaszczyk, P.: A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts 37. Cambridge Univ. Press, Cambridge (1997)" href="/article/10.1007/s10231-023-01327-w#ref-CR40" id="ref-link-section-d74823138e22648">40</a>, Proposition 5.2]. We suppose that</p><div id="Equ117" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \phi , \psi _i \in C^{N_1}({{{\mathbb {R}}}^d}) \qquad \text { and } \qquad {\mathop {\text {supp}}\nolimits }\,\phi , {\mathop {\text {supp}}\nolimits }\,\psi _i \subset [-N_2, N_2]^d,\quad i=1, \dots , 2^d-1, \end{aligned}$$</span></div></div><p>for some <span class="mathjax-tex">$N_1, N_2 \in {{\mathbb {N}}}$</span>.</p><p>For <span class="mathjax-tex">$k\in {{{\mathbb {Z}}}^d}$</span>, <span class="mathjax-tex">$j\in {{\mathbb {N}}}_0$</span> and <span class="mathjax-tex">$i\in \{1,\ldots ,2^d-1\}$</span>, define</p><div id="Equ118" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \phi _{j,k}(x){:=} 2^{jd/2} \phi (2^jx-k){}\;{\text {and}}\;{} \psi _{i,j,k}(x){:=} 2^{jd/2} \psi _i(2^jx-k),x\in {{{\mathbb {R}}}^d}. \end{aligned}$$</span></div></div><p>It is well known that</p><div id="Equ119" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \int \limits _{{{\mathbb {R}}}^d}\psi _{i,j,k}(x)\, x^\gamma \;\textrm{d}x = 0 \qquad \text{ if } \qquad |\gamma |\le N_1 \end{aligned}$$</span></div></div><p>(see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 40" title="Wojtaszczyk, P.: A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts 37. Cambridge Univ. Press, Cambridge (1997)" href="/article/10.1007/s10231-023-01327-w#ref-CR40" id="ref-link-section-d74823138e23326">40</a>, Proposition 3.1]), and</p><div id="Equ120" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \{\phi _{0,k}: \ k\in {{{\mathbb {Z}}}^d}\}\, \cup \, \{\psi _{i,j,k}:\ k\in {{{\mathbb {Z}}}^d},\ j\in {{\mathbb {N}}}_0,\ i\in \{1,\ldots ,2^d-1\}\} \end{aligned}$$</span></div></div><p>forms an <i>orthonormal basis</i> of <span class="mathjax-tex">$L_2({{{\mathbb {R}}}^d})$</span> (see, e. g., [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 18" title="Meyer, Y.: Wavelets and Operators. Cambridge Univ. Press, Cambridge (1992)" href="/article/10.1007/s10231-023-01327-w#ref-CR18" id="ref-link-section-d74823138e23558">18</a>, Section 3.9] or [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 36" title="Triebel, H.: Theory of Function Spaces. III, Birkhäuser, Basel (2006)" href="/article/10.1007/s10231-023-01327-w#ref-CR36" id="ref-link-section-d74823138e23561">36</a>, Section 3.1]). Hence</p><div id="Equ26" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} f=\sum _{k\in {{{\mathbb {Z}}}^d}}\, \lambda _k \, \phi _{0,k}+\sum _{i=1}^{2^d-1} \sum _{j=0}^\infty \sum _{k\in {{{\mathbb {Z}}}^d}}\, \lambda _{i,j,k}\, \psi _{i,j,k}\, \end{aligned}$$</span></div><div class="c-article-equation__number"> (2.26) </div></div><p>in <span class="mathjax-tex">$L_2({{{\mathbb {R}}}^d})$</span>, where <span class="mathjax-tex">$\lambda _k {:=} \langle f,\,\phi _{0,k}\rangle $</span> and <span class="mathjax-tex">$\lambda _{i,j,k}{:=} \langle f,\,\psi _{i,j,k}\rangle $</span>. We will denote by <span class="mathjax-tex">$\lambda (f)$</span> the following sequence:</p><div id="Equ121" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}\lambda (f) {:=} \left( \lambda _k,\lambda _{i,j,k}\right) = \big ( \langle f,\,\phi _{0,k}\rangle , \langle f,\,\psi _{i,j,k}\rangle \big ). \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar16">Definition 2.16</h3> <p>Let <span class="mathjax-tex">$s\in {{{\mathbb {R}}}}$</span>, <span class="mathjax-tex">$\tau \in [0,\infty )$</span> and <span class="mathjax-tex">$q\in (0,\infty ]$</span>. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$p \in (0, \infty ]$</span>. The <i>sequence space</i> <span class="mathjax-tex">${{b}_{p,q}^{s,\tau }}{:=}{{b}_{p,q}^{s,\tau }}({{{\mathbb {R}}}^d})$</span> is defined to be the space of all complex-valued sequences <span class="mathjax-tex">$t{:=}\{t_{i,j,m}:\ i\in \{1,\ldots ,2^d-1\}, j\in {{\mathbb {N}}}_0, m\in {{{\mathbb {Z}}}^d}\}$</span> such that <span class="mathjax-tex">$\Vert t\mid {{b}_{p,q}^{s,\tau }}\Vert <\infty $</span>, where </p><div id="Equ122" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t\mid {{b}_{p,q}^{s,\tau }}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s+\frac{d}{2}-\frac{d}{p})q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^p\right] ^{\frac{q}{p}}\right\} ^{\frac{1}{q}}, \end{aligned}$$</span></div></div><p> with the usual modification when <span class="mathjax-tex">$p=\infty $</span> or <span class="mathjax-tex">$q=\infty $</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$p \in (0, \infty )$</span>. The <i>sequence space</i> <span class="mathjax-tex">${{f}_{p,q}^{s,\tau }}{:=}{{f}_{p,q}^{s,\tau }}({{{\mathbb {R}}}^d})$</span> is defined to be the space of all complex-valued sequences <span class="mathjax-tex">$t{:=}\{t_{i,j,m}:\ i\in \{1,\ldots ,2^d-1\}, j\in {{\mathbb {N}}}_0, m\in {{{\mathbb {Z}}}^d}\}$</span> such that <span class="mathjax-tex">$\Vert t\mid {{f}_{p,q}^{s,\tau }}\Vert <\infty $</span>, where </p><div id="Equ123" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t\mid {{f}_{p,q}^{s,\tau }}\Vert {:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }}\left\{ \int _P \left[ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s+\frac{d}{2})q} \sum _{i=1}^{2^d-1} \sum _{m\in {{{\mathbb {Z}}}^d}} |t_{i,j,m}|^q \, \chi _{Q_{j,m}}(x)\right] ^{\frac{p}{q}} \;\textrm{d}x \right\} ^{\frac{1}{p}}, \end{aligned}$$</span></div></div><p> with the usual modification when <span class="mathjax-tex">$q=\infty $</span>.</p> </dd></dl> <p>As a special case of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Liang, Y., Yang, D., Yuan, W., Sawano, Y., Ullrich, T.: A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math. (Rozprawy Mat.) 489, 1–114 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR16" id="ref-link-section-d74823138e25614">16</a>, Theorem 4.12], we have the following wavelet characterisation of <span class="mathjax-tex">${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span>.</p> <h3 class="c-article__sub-heading" id="FPar17">Proposition 2.17</h3> <p>( [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Liang, Y., Yang, D., Yuan, W., Sawano, Y., Ullrich, T.: A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math. (Rozprawy Mat.) 489, 1–114 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR16" id="ref-link-section-d74823138e25685">16</a>]) Let <span class="mathjax-tex">$s\in {{{\mathbb {R}}}}$</span>, <span class="mathjax-tex">$\tau \in [0,\infty )$</span>, <span class="mathjax-tex">$q\in (0,\infty ]$</span>. Moreover, let <span class="mathjax-tex">$N_1 \in {{\mathbb {N}}}_0$</span> be such that, when <span class="mathjax-tex">$A=B$</span> and <span class="mathjax-tex">$p\in (0,\infty ]$</span>,</p><div id="Equ124" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} N_1+1>\max \left\{ d+\frac{d}{p}-d\tau -s, \frac{2 d}{\min \{p,1\}}+d\tau +1, d+\frac{d}{p}+\frac{d}{2}, d+s, \frac{d}{p} -s, {s+d\tau }\right\} , \end{aligned}$$</span></div></div><p>and when <span class="mathjax-tex">$A=F$</span> and <span class="mathjax-tex">$p\in (0,\infty )$</span>,</p><div id="Equ125" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {N_1+1>\max \left\{ d+\frac{d}{p}-d\tau -s, \frac{2 d}{\min \{p,{q},1\}}+d\tau +1, d+\frac{d}{p}+\frac{d}{2}, d+s, \frac{d}{p} -s, {s+d\tau }\right\} .} \end{aligned}$$</span></div></div><p>Let <span class="mathjax-tex">$f\in {\mathcal {S}}'({{{\mathbb {R}}}^d})$</span>. Then <span class="mathjax-tex">$f\in {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> if, and only if, <i>f</i> can be represented as (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ26">2.26</a>) in <span class="mathjax-tex">${\mathcal {S}}'({{{\mathbb {R}}}^d})$</span> and</p><div id="Equ126" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \lambda (f)\mid {a}_{p,q}^{s,\tau }\Vert ^*{:=} \sup _{P\in \mathcal {Q}}\frac{1}{|P|^{\tau }} \left\{ \sum _{m:\ Q_{0,m}\subset P} | \langle f,\,\phi _{0,m}\rangle |^p\right\} ^{\frac{1}{p}} + \Vert \{\langle f,\,\psi _{i,j,m}\rangle \}_{i,j,m}\mid {a}_{p,q}^{s,\tau }\Vert <\infty . \end{aligned}$$</span></div></div><p>Moreover, <span class="mathjax-tex">$\Vert f \mid {A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Vert $</span> is equivalent to <span class="mathjax-tex">$\Vert \lambda (f)\mid {a}_{p,q}^{s,\tau }\Vert ^*.$</span></p> </div></div></section><section data-title="Extension Operator"><div class="c-article-section" id="Sec6-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec6"><span class="c-article-section__title-number">3 </span>Extension Operator</h2><div class="c-article-section__content" id="Sec6-content"><p>As mentioned in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar13">2.13</a> above, if <span class="mathjax-tex">$\Omega $</span> is a bounded <span class="mathjax-tex">$C^{\infty }$</span> domain in <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, an extension theorem for the spaces <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> was stated in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e27072">47</a>], but with the assumption that <span class="mathjax-tex">$p \in [1, \infty )$</span>. It is our aim in this section to establish an extended result, which holds for all <span class="mathjax-tex">$p \in (0, \infty ]$</span> (<span class="mathjax-tex">$p\in (0,\infty )$</span> in the <i>F</i>-case). Similarly to what was done in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey-Campanato and related smoothness spaces. Sci. China Math. 58, 1835–1908 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR48" id="ref-link-section-d74823138e27183">48</a>, Proposition 4.13] for Besov-Morrey spaces <span class="mathjax-tex">${{\mathcal {N}}}^{s}_{u,p,q}$</span>, we will follow Rychkov [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e27226">23</a>, Theorem 2.2] in the construction of such an operator. We start with some preparation.</p><p>Let <span class="mathjax-tex">$q \in (0, \infty ]$</span> and <span class="mathjax-tex">$\tau \in [0, \infty )$</span>. Denote by <span class="mathjax-tex">$\ell _q(L_p^\tau )$</span> the set of all sequences <span class="mathjax-tex">$\{g_j\}_{j \in {{\mathbb {N}}}_0}$</span> of measurable functions on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> such that</p><div id="Equ27" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \{g_j\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert {:=} \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j = \max \{j_P, 0\}}^{\infty } \Vert g_j \mid L_p(P)\Vert ^q\right\} ^{1/q} < \infty . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.1) </div></div><p>Similarly, <span class="mathjax-tex">$L_p^\tau (\ell _q)$</span> with <span class="mathjax-tex">$p\in (0, \infty )$</span> denotes the set of all sequences <span class="mathjax-tex">$\{g_j\}_{j \in {{\mathbb {N}}}_0}$</span> of measurable functions on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> such that</p><div id="Equ28" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \{g_j\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert {:=} \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\| \left\{ \sum _{j = \max \{j_P, 0\}}^{\infty } |g_j|^q\right\} ^{1/q} \mid L_p(P)\right\| < \infty . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.2) </div></div><p>The first auxiliary lemma can be seen as a particular case of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Liang, Y., Yang, D., Yuan, W., Sawano, Y., Ullrich, T.: A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math. (Rozprawy Mat.) 489, 1–114 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR16" id="ref-link-section-d74823138e28077">16</a>, Lemma 2.9], where the authors considered more general quasi-norms.</p> <h3 class="c-article__sub-heading" id="FPar18">Lemma 3.1</h3> <p>Let <span class="mathjax-tex">$q\in (0, \infty ]$</span>, <span class="mathjax-tex">$\tau \in [0, \infty )$</span>. Let <span class="mathjax-tex">$D_1, D_2 \in (0, \infty )$</span> with <span class="mathjax-tex">$D_2 >d \tau $</span>. For any sequence <span class="mathjax-tex">$\{g_\nu \}_{\nu \in {{\mathbb {N}}}_0}$</span> of measurable functions on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, consider</p><div id="Equ127" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} G_j(x){:=} \sum _{\nu =0}^j 2^{-(j -\nu )D_2} g_\nu (x) + \sum _{\nu =j+1}^\infty 2^{-(\nu -j)D_1}g_\nu (x), \quad j \in {{\mathbb {N}}}_0, \quad x \in {{{\mathbb {R}}}^d}. \end{aligned}$$</span></div></div><p>Then there exists a positive constant <i>C</i>, independent of <span class="mathjax-tex">$\{g_\nu \}_{\nu \in {{\mathbb {N}}}_0}$</span>, such that, for all <span class="mathjax-tex">$p \in (0, \infty ]$</span>,</p><div id="Equ128" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \{ G_j\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert \le C\, \Vert \{g_\nu \}_{\nu \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert , \end{aligned}$$</span></div></div><p>and, for all <span class="mathjax-tex">$p \in (0, \infty )$</span>,</p><div id="Equ129" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \{ G_j\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert \le C\, \Vert \{g_\nu \}_{\nu \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert . \end{aligned}$$</span></div></div> <p>Let <span class="mathjax-tex">$(\varphi _0,\varphi )$</span> be an admissible pair and <span class="mathjax-tex">$f \in \mathcal {S}'({{{\mathbb {R}}}^d})$</span>. For all <span class="mathjax-tex">$j \in {{\mathbb {N}}}_0$</span>, <span class="mathjax-tex">$a \in (0, \infty )$</span> and <span class="mathjax-tex">$x \in {{{\mathbb {R}}}^d}$</span>, we denote</p><div id="Equ130" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \varphi _j^{*, a}f(x) {:=} \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|\varphi _j *f(y)|}{(1+2^j|x-y|)^a}. \end{aligned}$$</span></div></div><p>Another tool we will need later is the characterisation of the spaces via Peetre maximal functions. For the homogeneous version of the spaces, this result was proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 43" title="Yang, D., Yuan, W.: Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means. Nonlinear Anal. 73, 3805–3820 (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR43" id="ref-link-section-d74823138e29406">43</a>, Theorem 1.1]. Here we use the results proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 50" title="Yang, D., Zhuo, C., Yuan, W.: Besov-type spaces with variable smoothness and integrability. J. Funct. Anal. 269, 1840–1898 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR50" id="ref-link-section-d74823138e29409">50</a>, Theorem 5.1] and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 49" title="Yang, D., Zhuo, C., Yuan, W.: Triebel-Lizorkin type spaces with variable exponent. Banach J. Math. Anal. 9, 146–202 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR49" id="ref-link-section-d74823138e29412">49</a>, Theorem 3.11] for the more general scale of Besov-type spaces and Triebel-Lizorkin-type spaces with variable exponents, respectively, which in particular cover our spaces. Adapted to our case, the results read as follows.</p> <h3 class="c-article__sub-heading" id="FPar19">Theorem 3.2</h3> <p>Let <span class="mathjax-tex">$s \in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$\tau \in [0, \infty )$</span>, <span class="mathjax-tex">$q \in (0, \infty ]$</span> and <span class="mathjax-tex">$(\varphi _0,\varphi )$</span> be an admissible pair. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$p\in (0, \infty ]$</span> and <span class="mathjax-tex">$a > \displaystyle \frac{d}{p}+d\tau $</span>. Then </p><div id="Equ131" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert f \mid {{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}\varphi _j^{*, a}f\}_{j \in {{\mathbb {N}}}_0}\mid \ell _q(L_p^\tau )\Vert . \end{aligned}$$</span></div></div> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$p\in (0, \infty )$</span> and <span class="mathjax-tex">$a > \displaystyle \frac{d}{\min \{p,q\}}+d\tau $</span>. Then </p><div id="Equ132" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert f \mid {{F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}\varphi _j^{*, a}f\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert . \end{aligned}$$</span></div></div> </dd></dl> <h3 class="c-article__sub-heading" id="FPar20">Remark 3.3</h3> <p>Note that in the above theorem the sequence <span class="mathjax-tex">$\{\varphi _j\}_{j \in {{\mathbb {N}}}_0}$</span> is built upon an admissible pair <span class="mathjax-tex">$(\varphi _0, \varphi )$</span>, as in Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar1">2.1</a>. Hence <span class="mathjax-tex">$\Vert f | {{B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}(\varphi _j *f)\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert \ $</span> and <span class="mathjax-tex">$\Vert f | {{F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})}\Vert \sim \Vert \{2^{js}(\varphi _j *f)\}_{j \in {{\mathbb {N}}}_0}\mid L_p^\tau (\ell _q)\Vert $</span>. However, in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 6" title="Gonçalves, H. F.: Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces. Banach J. Math. Anal 15(31), 50 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR6" id="ref-link-section-d74823138e30586">6</a>] and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 7" title="Gonçalves, H.F., Moura, S.D.: Characterization of Triebel-Lizorkin-type spaces with variable exponents via maximal functions, local means and non-smooth atomic decompositions. Math. Nachr. 291, 2024–2044 (2018)" href="/article/10.1007/s10231-023-01327-w#ref-CR7" id="ref-link-section-d74823138e30589">7</a>] the authors proved that, for Besov-type and Triebel-Lizorkin-type spaces with variable exponents, one can consider more general pairs of functions not only in this result, but also in the definition of the spaces.</p> <p>Following Rychkov [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e30596">23</a>], we will prove the existence of a universal linear bounded extension operator if <span class="mathjax-tex">$\Omega $</span> is a Lipschitz domain, recall the explanations given at the end of Sect. <a data-track="click" data-track-label="link" data-track-action="section anchor" href="/article/10.1007/s10231-023-01327-w#Sec4">2.2</a>. We first recall two results of Rychkov we shall need in our argument below.</p> <h3 class="c-article__sub-heading" id="FPar21">Lemma 3.4</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e30627">23</a>, Proposition 3.1]) Let <span class="mathjax-tex">$\Omega $</span> be a special Lipschitz domain. The distribution <span class="mathjax-tex">$f\in \mathcal {D}'(\Omega )$</span> belongs to <span class="mathjax-tex">$\mathcal {S}'(\Omega )$</span> if, and only if, there exist <span class="mathjax-tex">$g\in \mathcal {S}'({{{\mathbb {R}}}^d})$</span> such that <span class="mathjax-tex">$f=g|_\Omega $</span>.</p> <h3 class="c-article__sub-heading" id="FPar22">Lemma 3.5</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e30829">23</a>, Theorem 4.1 (a)]) Let <span class="mathjax-tex">$\Omega $</span> be a special Lipschitz domain and <i>K</i> its associated cone. Let <span class="mathjax-tex">$-K {:=}\{-x: x\in K\}$</span> be a ‘reflected’ cone. Then there exist functions <span class="mathjax-tex">$\phi _0,\phi , \psi _0, \psi \in \mathcal {S}({{{\mathbb {R}}}^d})$</span> supported in <span class="mathjax-tex">$-K$</span> such that</p><div id="Equ29" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \int _{{{{\mathbb {R}}}^d}}x^\alpha \phi (x) \;\textrm{d}x&= \int _{{{{\mathbb {R}}}^d}}x^\alpha \psi (x) \;\textrm{d}x = 0 \quad \text {for all multi-indices}\;\alpha , \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.3) </div></div><p>and</p><div id="Equ30" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} f&= \sum _{j \in {{\mathbb {N}}}_0} \psi _j *\phi _j *f \quad \text{ in } \quad {{\mathcal {D}}}'(\Omega ), \quad \text {for any} \; f\in \mathcal {S}'(\Omega ), \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.4) </div></div><p>where <span class="mathjax-tex">$\phi _j(\cdot )= 2^{jd}\phi (2^j\cdot )$</span> and <span class="mathjax-tex">$\psi _j(\cdot )= 2^{jd}\psi (2^j\cdot )$</span>, <span class="mathjax-tex">$j\in {{\mathbb {N}}}$</span>.</p> <p>We can now state the main result of this section.</p> <h3 class="c-article__sub-heading" id="FPar23">Theorem 3.6</h3> <p>Let <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span> be a special or bounded Lipschitz domain if <span class="mathjax-tex">$d\ge 2$</span>, or an interval if <span class="mathjax-tex">$d=1$</span>. Then there exists a linear bounded operator <span class="mathjax-tex">$\textrm{Ext}$</span> which maps <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> into <span class="mathjax-tex">${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> for all <span class="mathjax-tex">$s \in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$\tau \in [0, \infty )$</span>, <span class="mathjax-tex">$p \in (0, \infty ]$</span> (<span class="mathjax-tex">$p<\infty $</span> in the <i>F</i>-case) and <span class="mathjax-tex">$q \in (0, \infty ]$</span>, such that, for all <span class="mathjax-tex">$f \in {{\mathcal {D}}}'(\Omega )$</span>, <span class="mathjax-tex">$\textrm{Ext}f|_{\Omega }=f$</span> in <span class="mathjax-tex">${{\mathcal {D}}}'(\Omega )$</span>.</p> <h3 class="c-article__sub-heading" id="FPar24">Proof</h3> <p>We concentrate now on Besov-type spaces with <span class="mathjax-tex">$d\ge 2$</span> and give some details on the Triebel-Lizorkin scale at the end of the proof.</p> <p>We apply the extension operator constructed by V. Rychkov in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e31999">23</a>] and follow the main ideas of his proof. By a standard procedure (see [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e32002">23</a>, Subsection 1.2]), we only need to consider the case when <span class="mathjax-tex">$\Omega $</span> is a special Lipschitz domain.</p> <p>Let <span class="mathjax-tex">$\Omega $</span> be a special Lipschitz domain. The spaces <span class="mathjax-tex">${B}_{p,q}^{s,\tau }(\Omega )$</span> are defined by restriction therefore it follows from Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar21">3.4</a> and Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar22">3.5</a> that any distribution <span class="mathjax-tex">$f\in {B}_{p,q}^{s,\tau }(\Omega )$</span> can be represented in the form (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ30">3.4</a>) with the functions <span class="mathjax-tex">$\phi $</span> and <span class="mathjax-tex">$\psi $</span> satisfying (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ29">3.3</a>).</p> <p>For any distribution <span class="mathjax-tex">$f \in {{\mathcal {S}}}'(\Omega )$</span>, we define the mapping</p><div id="Equ31" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {ext}\,f \,{:=} \sum _{j \in {{\mathbb {N}}}_0} \psi _j *(\phi _j *f)_\Omega . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.5) </div></div><p>Here we use the notation <span class="mathjax-tex">$g_\Omega $</span> to denote the extension of a function <span class="mathjax-tex">$g:\Omega \rightarrow {{\mathbb {R}}}$</span> from <span class="mathjax-tex">$\Omega $</span> to <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> by setting</p><div id="Equ133" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} g_\Omega (x) = {\left\{ \begin{array}{ll} g(x), &{} \text{ if } x \in \Omega , \\ 0, &{} \text{ if } x \in {{{\mathbb {R}}}^d}\setminus \Omega . \end{array}\right. } \end{aligned}$$</span></div></div><p><i>Step 1.</i> Let <span class="mathjax-tex">$\{g_j\}_{j \in {{\mathbb {N}}}_0}$</span> be a sequence of measurable functions. Moreover, let <span class="mathjax-tex">$\mathcal {M}_{N}^{g_j}(x)$</span> denote the Peetre maximal function of <span class="mathjax-tex">$g_j$</span>, namely,</p><div id="Equ134" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathcal {M}_{N}^{g_j}(x){:=} \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|g_j(y)|}{(1+2^j|x-y|)^N} \end{aligned}$$</span></div></div><p>for all <span class="mathjax-tex">$x \in {{{\mathbb {R}}}^d}$</span> and <span class="mathjax-tex">$N \in {{\mathbb {N}}}\cap \left( \frac{d}{\min \{1,p\}}, \infty \right) $</span>. We prove that if <span class="mathjax-tex">$\{2^{js}\, \mathcal {M}_{N}^{g_j}\}_{j \in {{\mathbb {N}}}_0} \in \ell _q(L_p^\tau )$</span>, then the series <span class="mathjax-tex">$\sum _{j \in {{\mathbb {N}}}_0} \psi _j *g_j$</span> converges in <span class="mathjax-tex">$\mathcal {S}'({{{\mathbb {R}}}^d})$</span> and the <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> norm of its sum can be estimated by <span class="mathjax-tex">$\Vert \{2^{js}\, \mathcal {M}_{N}^{g_j}\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert $</span>.</p> <p>We start with the following elementary inequality</p><div id="Equ32" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} |\phi _\ell *\psi _j *g_j(x)| \le \mathcal {M}_{N}^{g_j}(x) \int |\phi _\ell *\psi _j(y)|(1+2^j|y|)^N \;\textrm{d}y\; . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.6) </div></div><p>Bui, Paluszyński and Taibleson proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Bui, H.-Q., Paluszyński, M., Taibleson, M.H.: A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math. 119, 219–246 (1996)" href="/article/10.1007/s10231-023-01327-w#ref-CR1" id="ref-link-section-d74823138e33593">1</a>] that</p><div id="Equ33" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \int |\phi _\ell *\psi _j(y)|(1+2^j|y|)^N \;\textrm{d}y \le C_{M,N} 2^{-|\ell -j|M}\qquad \text {for all} \; M>0, \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.7) </div></div><p>cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 1" title="Bui, H.-Q., Paluszyński, M., Taibleson, M.H.: A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math. 119, 219–246 (1996)" href="/article/10.1007/s10231-023-01327-w#ref-CR1" id="ref-link-section-d74823138e33768">1</a>, Lemma 2.1] or [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e33771">23</a>, proof of Theorem 4.1]. We take <span class="mathjax-tex">$M>|s|-d\tau $</span> and put <span class="mathjax-tex">$\sigma =M- |s|$</span>. Then</p><div id="Equ135" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 2^{\ell s}|\phi _\ell *\psi _j *g_j(x)| \lesssim 2^{-|\ell -j|\sigma }2^{js} \mathcal {M}^{g_j}_{N}(x), \quad x \in {{{\mathbb {R}}}^d}, \quad \ell \in {{\mathbb {N}}}_0. \end{aligned}$$</span></div></div><p>If the sequence <span class="mathjax-tex">$\{g_j\}_{j \in {{\mathbb {N}}}_0}$</span> is such that <span class="mathjax-tex">$\Vert \{2^{js}\,\mathcal {M}^{g_j}_{N}\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\Vert <\infty $</span>, then there exists a constant <i>c</i> such that for any dyadic cube <i>P</i> we have <span class="mathjax-tex">$\Vert \mathcal {M}^{g_j}_{N}|L_p(P)\Vert \le c |P|^\tau $</span>. In consequence, any function <span class="mathjax-tex">$g_j$</span> is a tempered distribution and <span class="mathjax-tex">$\psi _j *g_j\in \mathcal {S}'({{{\mathbb {R}}}^d})$</span>, <span class="mathjax-tex">$j \in {{\mathbb {N}}}_0$</span>.</p> <p>It holds true that</p><div id="Equ34" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \big \Vert \psi _j * g_j \mid B^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d})\big \Vert&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{\ell =\max \{j_P, 0\}}^{\infty } 2^{-(\ell 2 \sigma + |\ell -j|\sigma )q}\right\} ^{1/q} \bigg \Vert 2^{js} \mathcal {M}_N^{g_j}\mid L_p(P)\bigg \Vert \nonumber \\&\lesssim 2^{-j \sigma }\, \left\| \{2^{ks}\, \mathcal {M}_N^{g_k}\}_{k \in {{\mathbb {N}}}_0}\mid \ell _q(L_p^\tau )\right\| , \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.8) </div></div><p>using that <span class="mathjax-tex">$|\ell -j|\ge j-\ell $</span>. Therefore <span class="mathjax-tex">$\sum _{j \in {{\mathbb {N}}}_0} \psi _j *g_j$</span> converges in <span class="mathjax-tex">$B^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d})$</span> and hence in <span class="mathjax-tex">$\mathcal {S}'({{{\mathbb {R}}}^d})$</span>, since <span class="mathjax-tex">$B^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d}) \hookrightarrow \mathcal {S}'({{{\mathbb {R}}}^d})$</span>. In this way, we further have</p><div id="Equ136" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 2^{\ell s}\left| \phi _\ell *\left( \sum _{j=0}^\infty \psi _j *g_j \right) (x)\right| \lesssim \sum _{j=0}^\infty 2^{-|\ell -j|\sigma } 2^{js}\mathcal {M}_{N}^{g_j}(x), \quad x \in {{{\mathbb {R}}}^d}, \quad \ell \in {{\mathbb {N}}}_0. \end{aligned}$$</span></div></div><p>Applying this, we see that</p><div id="Equ137" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\bigg \Vert \sum _{j=0}^\infty \psi _j *g_j \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\bigg \Vert \\&\quad = \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{\ell =\max \{j_P, 0\}}^{\infty } 2^{\ell sq} \left[ \int _P \Big | \phi _\ell *\left( \sum _{j=0}^\infty \psi _j *g_j \right) (x)\Big |^p {\;\textrm{d}x} \right] ^{q/p} \right\} ^{1/q}\\&\quad \lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{\ell =\max \{j_P, 0\}}^{\infty } \bigg \Vert \sum _{j=0}^\infty 2^{-|\ell -j|\sigma } \, 2^{js}\,\mathcal {M}^{g_j}_{N} \mid L_p(P)\bigg \Vert ^{q} \right\} ^{1/q}. \end{aligned}$$</span></div></div><p>Now we can apply Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar18">3.1</a> since <span class="mathjax-tex">$\sigma >d\tau $</span>. We get</p><div id="Equ35" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Big \Vert \sum _{j=0}^\infty \psi _j *g_j \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Big \Vert&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j=\max \{j_P, 0\}}^{\infty } \big \Vert 2^{js}\,\mathcal {M}^{g_j}_{N} \mid L_p(P) \big \Vert ^{q} \right\} ^{1/q}\nonumber \\&= \big \Vert \{ 2^{js}\, \mathcal {M}_{N}^{g_j}\}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\big \Vert . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.9) </div></div><p><i>Step 2.</i> Let <span class="mathjax-tex">$f \in {B}_{p,q}^{s,\tau }(\Omega )$</span>. Then, for any <span class="mathjax-tex">$\varepsilon \in (0, \infty )$</span>, there exists an <span class="mathjax-tex">$h \in {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> such that <span class="mathjax-tex">$h|_{\Omega }=f$</span> in <span class="mathjax-tex">${{\mathcal {D}}}'(\Omega )$</span> and</p><div id="Equ36" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\| h \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\| \le \left\| f \mid {B}_{p,q}^{s,\tau }(\Omega )\right\| + \varepsilon . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.10) </div></div><p>We have <span class="mathjax-tex">$\phi _j *f(y)= \phi _j *h(y)$</span> if <span class="mathjax-tex">$y\in \Omega $</span> since <span class="mathjax-tex">${\mathop {\textrm{supp}}\nolimits }\phi _j\subset -K$</span> and <span class="mathjax-tex">$y+K\subset \Omega $</span> for any point <span class="mathjax-tex">$y\in \Omega $</span>. In consequence</p><div id="Equ37" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sup _{y \in \Omega } \frac{|\phi _j *f(y)|}{(1+2^j|x-y|)^N}\, \lesssim \displaystyle \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|\phi _j *h(y)|}{(1+2^j|{\tilde{x}}-y|)^N}, \quad&x \not \in \overline{\Omega }, \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.11) </div></div><p>where <span class="mathjax-tex">${\tilde{x}}{:=}(x', 2\omega (x')-x_n) \in \Omega $</span> is the symmetric point to <span class="mathjax-tex">$x=(x',x_n) \not \in \overline{\Omega }$</span> with respect to <span class="mathjax-tex">$\partial \Omega $</span>.</p> <p>Let <span class="mathjax-tex">$g_j {:=} (\phi _j *f)_{\Omega }$</span> for all <span class="mathjax-tex">$j \in {{\mathbb {N}}}_0$</span>. It was proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 23" title="Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. 60, 237–257 (1999)" href="/article/10.1007/s10231-023-01327-w#ref-CR23" id="ref-link-section-d74823138e37517">23</a>, p. 248] that</p><div id="Equ38" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sup _{y \in \Omega } \frac{| g_j(y)|}{(1+2^j|x-y|)^N}\,&{\left\{ \begin{array}{ll}\displaystyle \,= \sup _{y \in \Omega } \frac{|\phi _j *f(y)|}{(1+2^j|x-y|)^N}, &{} x \in \Omega , \\ \, \lesssim \displaystyle \sup _{y \in \Omega } \frac{|\phi _j *f(y)|}{(1+2^j|{\tilde{x}}-y|)^N}, &{} x \not \in \overline{\Omega }. \end{array}\right. } \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.12) </div></div><p>Now, we conclude from (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ35">3.9</a>), (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ37">3.11</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ38">3.12</a>) that</p><div id="Equ138" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\| \textrm{ext}f\mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\|&= \Big \Vert \sum _{j=0}^{\infty } \psi _j *(\phi _j *f)_\Omega \mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\Big \Vert \lesssim \left\| \{2^{js}\,\mathcal {M}_N^{g_j} \}_{j \in {{\mathbb {N}}}_0} \mid \ell _q(L_p^\tau )\right\| \\&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j=\max \{j_P, 0\}}^{\infty } 2^{jsq} \Big \Vert \sup _{y \in \Omega } \frac{|\phi _j *h(y)|}{(1+2^j|\cdot -y|)^N} \mid L_p(P)\Big \Vert ^q\right\} ^{1/q}\\&\lesssim \sup _{P \in \mathcal {Q}} \frac{1}{|P|^\tau } \left\{ \sum _{j=\max \{j_P, 0\}}^{\infty } 2^{jsq} \Big \Vert \sup _{y \in {{{\mathbb {R}}}^d}} \frac{|\phi _j *h(y)|}{(1+2^j|\cdot -y|)^N} \mid L_p(P)\Big \Vert ^q\right\} ^{1/q} . \end{aligned}$$</span></div></div><p>Thus the last inequalities, the characterisation of <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> via the Peetre maximal functions stated in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar19">3.2</a> (i) and the choice of <i>g</i> imply that</p><div id="Equ39" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\| \textrm{ext}f\mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\| \lesssim \left\| h\mid {B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})\right\| \lesssim \left\| f\mid {B}_{p,q}^{s,\tau }(\Omega )\right\| +\varepsilon . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.13) </div></div><p>Letting <span class="mathjax-tex">$\varepsilon \rightarrow 0$</span>, we then know that <span class="mathjax-tex">$\textrm{ext}$</span> is a bounded linear operator from <span class="mathjax-tex">${B}_{p,q}^{s,\tau }(\Omega )$</span> into <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span>.</p> <p>Finally, since the supports of <span class="mathjax-tex">$\psi _0$</span> and <span class="mathjax-tex">$\psi $</span> lie in <span class="mathjax-tex">$-K$</span>, it follows that</p><div id="Equ139" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textrm{ext}f|_\Omega = \sum _{j=0}^{\infty } \psi _j *\phi _j *f =f\quad \text{ in } {{\mathcal {D}}}'(\Omega ). \end{aligned}$$</span></div></div><p>Therefore, <span class="mathjax-tex">$\textrm{ext}$</span> is the desired extension operator from <span class="mathjax-tex">${B}_{p,q}^{s,\tau }(\Omega )$</span> into <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span>, which concludes the proof for the Besov-type spaces.</p> <p><i>Step 3.</i> The proof for the Triebel-Lizorkin-type spaces follows similarly. We can define the extension operator by the formula (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ31">3.5</a>) and use once more the Peetre maximal function <span class="mathjax-tex">$\mathcal {M}^{g_j}_N$</span> as the main tool for estimations. Therefore, we point out the differences without giving the details. The estimation of the norm <span class="mathjax-tex">$\big \Vert \psi _j *g_j \mid F^{s-2\sigma , \tau }_{p,q}({{{\mathbb {R}}}^d})\big \Vert $</span> is the first difference, but it can be done similarly as in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ34">3.8</a>), now with the <span class="mathjax-tex">$L_p^\tau (\ell _q)$</span> norm instead of <span class="mathjax-tex">$\ell _q(L_p^\tau )$</span> norm on the right hand side of the inequality. Afterwards, the counterpart of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ35">3.9</a>) can be obtained using again Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar18">3.1</a>, but now the estimate related to the spaces <span class="mathjax-tex">$L_p^\tau (\ell _q)$</span>. Finally, the characterisation of <span class="mathjax-tex">${F}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> via the Peetre maximal functions stated in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar19">3.2</a> (ii) leads us to obtain a similar estimate as (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ39">3.13</a>). <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar25">Remark 3.7</h3> <p>As mentioned above essentially the same proof for Triebel-Lizorkin-type spaces can be found in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title="Zhuo, C., Hovemann, M., Sickel, W.: Complex interpolation of Lizorkin-Triebel-Morrey Spaces on Domains. Anal. Geom. Metric Spaces 8, 268–304 (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR52" id="ref-link-section-d74823138e39883">52</a>] with additional restrictions <span class="mathjax-tex">$p,q\ge 1$</span>. Note that, in view of the coincidence (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ16">2.16</a>), also the Triebel-Lizorkin-Morrey spaces <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}$</span> are covered by our theorem. This complements the corresponding result obtained in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 48" title="Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey-Campanato and related smoothness spaces. Sci. China Math. 58, 1835–1908 (2015)" href="/article/10.1007/s10231-023-01327-w#ref-CR48" id="ref-link-section-d74823138e39956">48</a>, Proposition 4.13] for the class of Besov-Morrey spaces <span class="mathjax-tex">${{\mathcal {N}}}^{s}_{u,p,q}$</span>. Very recently similar arguments were used in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 51" title="Zhuo, C.: Complex interpolation of Besov-type spaces on domains. Z. Anal. Anwend. 40, 313–347 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR51" id="ref-link-section-d74823138e39999">51</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 52" title="Zhuo, C., Hovemann, M., Sickel, W.: Complex interpolation of Lizorkin-Triebel-Morrey Spaces on Domains. Anal. Geom. Metric Spaces 8, 268–304 (2020)" href="/article/10.1007/s10231-023-01327-w#ref-CR52" id="ref-link-section-d74823138e40002">52</a>] for the construction of the extension operator, but restricted to the case <span class="mathjax-tex">$p,q\ge 1$</span>. For the sake of completeness, we briefly sketched our proof here.</p> <h3 class="c-article__sub-heading" id="FPar26">Corollary 3.8</h3> <p>Let <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span> be an interval if <span class="mathjax-tex">$d=1$</span> or a Lipschitz domain if <span class="mathjax-tex">$d\ge 2$</span>. Then there exists a linear bounded operator <span class="mathjax-tex">$\textrm{ext}$</span> which maps <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}(\Omega )$</span> into <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}({{{\mathbb {R}}}^d})$</span> for all <span class="mathjax-tex">$s \in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$q \in (0, \infty ]$</span> and <span class="mathjax-tex">$0<p\le u< \infty $</span>, such that, for all <span class="mathjax-tex">$f \in {{\mathcal {D}}}'(\Omega )$</span>, <span class="mathjax-tex">$\textrm{ext}f|_{\Omega }=f$</span> in <span class="mathjax-tex">${{\mathcal {D}}}'(\Omega )$</span>.</p> <h3 class="c-article__sub-heading" id="FPar27">Proof</h3> <p>This follows immediately from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a> and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ16">2.16</a>). <span class="mathjax-tex">$\square $</span></p> <p>Using Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a> and the wavelet decomposition of the spaces <span class="mathjax-tex">${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span>, cf. Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar17">2.17</a>, we can now prove a result on the monotonicity of the spaces <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> regarding the parameter <span class="mathjax-tex">$\tau $</span>, which in fact does not hold when considering the spaces on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>.</p> <h3 class="c-article__sub-heading" id="FPar28">Proposition 3.9</h3> <p>Let <span class="mathjax-tex">$0< p\le \infty $</span> (<span class="mathjax-tex">$p<\infty $</span> in the F-case), <span class="mathjax-tex">$s\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q\le \infty $</span>, <span class="mathjax-tex">$0\le \tau _2\le \tau _1$</span>. Let <span class="mathjax-tex">$\Omega \subset {{{\mathbb {R}}}^d}$</span> be a bounded interval if <span class="mathjax-tex">$d=1$</span> or a bounded Lipschitz domain if <span class="mathjax-tex">$d\ge 2$</span>. Then</p><div id="Equ140" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: A^{s,\tau _1}_{p,q}(\Omega )\hookrightarrow A^{s,\tau _2}_{p,q}(\Omega ). \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar29">Proof</h3> <p>Let <span class="mathjax-tex">$\widetilde{Q}_0$</span> be a dyadic cube that contains <span class="mathjax-tex">$\overline{\Omega }$</span> in its interior and let <span class="mathjax-tex">$\widetilde{Q}$</span> be a (fixed) dyadic cube that contains the supports of all the functions <span class="mathjax-tex">$\psi _{i,j,k}$</span> and <span class="mathjax-tex">$\phi _{0,k}$</span> with non-empty intersection with <span class="mathjax-tex">$\widetilde{Q}_0$</span>. Let <span class="mathjax-tex">$f\in A^{s,\tau _1}_{p,q}(\Omega )$</span>. The compactly supported smooth functions are pointwise multipliers in <span class="mathjax-tex">$ {A}_{p,q}^{s,\tau }({{\mathbb {R}}}^d)$</span>, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 31" title="Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. I. Eurasian Math. J. 3, 110–149 (2012)" href="/article/10.1007/s10231-023-01327-w#ref-CR31" id="ref-link-section-d74823138e41329">31</a>] or [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e41332">47</a>, Theorem 6.1] for <span class="mathjax-tex">$\tau \le \frac{1}{p}$</span> and Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar3">2.3</a> for <span class="mathjax-tex">$\tau >\frac{1}{p}$</span>, therefore</p><div id="Equ40" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert f \mid {A}_{p,q}^{s,\tau }(\Omega )\Vert \sim \inf \{ \Vert g \mid {A}_{p,q}^{s,\tau }({{\mathbb {R}}}^d )\Vert :\quad g\in {A}_{p,q}^{s,\tau }({{\mathbb {R}}}^d )\quad \text {and}\quad {\mathop {\textrm{supp}}\nolimits }g\subset \widetilde{Q}_0\}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.14) </div></div><p>There exists a dyadic cube <span class="mathjax-tex">${\widetilde{Q}}_1$</span> such that <span class="mathjax-tex">$\langle g,\psi _{i,j,m}\rangle = \langle g,\phi _{0,m}\rangle =0$</span> for any <span class="mathjax-tex">$g\in A^{s,\tau _1}_{p,q}({{\mathbb {R}}}^d)$</span> with <span class="mathjax-tex">${\mathop {\textrm{supp}}\nolimits }g\subset \widetilde{Q}_0$</span> and <span class="mathjax-tex">$\phi _{0,\ell }$</span>, <span class="mathjax-tex">$\psi _{i,j,m}$</span> such that <span class="mathjax-tex">$ {\mathop {\textrm{supp}}\nolimits }\phi _{0,\ell }\nsubseteq {\widetilde{Q}}_1$</span>, <span class="mathjax-tex">$ {\mathop {\textrm{supp}}\nolimits }\psi _{i,j,m}\nsubseteq {\widetilde{Q}}_1$</span>. Moreover there exists a positive constant <i>C</i> such that</p><div id="Equ141" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{ 1}{|P|^{\tau _2}} \le C\, \frac{ 1}{|P|^{\tau _1}}\quad \text {for }\quad 0\le \tau _2 \le \tau _1\ \end{aligned}$$</span></div></div><p>if <span class="mathjax-tex">$P\subset \widetilde{Q}_1$</span>. Therefore</p><div id="Equ41" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \lambda (g)\mid&a^{s,\tau _2}_{p,q}\Vert ^*\le c \Vert \lambda (g)\mid a^{s,\tau _1}_{p,q}\Vert ^*. \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.15) </div></div><p>By Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a> there exists a linear and bounded extension operator <span class="mathjax-tex">$\textrm{ext}$</span> from <span class="mathjax-tex">$A^{s, \tau _1}_{p,q}(\Omega )$</span> into <span class="mathjax-tex">$A^{s, \tau _1}_{p,q}({{\mathbb {R}}}^d)$</span>. So, using also the wavelet decomposition of these spaces, if <span class="mathjax-tex">$\varphi \in C^\infty _0({{\mathbb {R}}}^d)$</span> is supported in <span class="mathjax-tex">$\widetilde{Q}_0$</span> and equals 1 on <span class="mathjax-tex">$\Omega $</span>, then by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ41">3.15</a>)</p><div id="Equ42" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert f \mid A^{s, \tau _2}_{p,q}(\Omega )\Vert&\le C\, \Vert \lambda (\varphi \textrm{ext}(f))\mid a^{s, \tau _2}_{p,q}\Vert ^*\le C\, \Vert \lambda (\varphi \textrm{ext}(f))\mid a^{s, \tau _1}_{p,q}\Vert ^*\nonumber \\&\le C\, \Vert \varphi \textrm{ext}(f)\mid A^{s, \tau _1}_{p,q}({{\mathbb {R}}}^d)\Vert \le C \, \Vert f \mid A^{s, \tau _1}_{p,q}(\Omega )\Vert . \end{aligned}$$</span></div><div class="c-article-equation__number"> (3.16) </div></div><p>The proof is complete. <span class="mathjax-tex">$\square $</span></p> </div></div></section><section data-title="Limiting embeddings"><div class="c-article-section" id="Sec7-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Sec7"><span class="c-article-section__title-number">4 </span>Limiting embeddings</h2><div class="c-article-section__content" id="Sec7-content"><p>We shall always assume in the sequel that <span class="mathjax-tex">$\Omega $</span> is a bounded Lipschitz domain in <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>. As already mentioned, we shall deal – differently from the standard approach – with <i>continuous</i> embeddings of type</p><div id="Equ142" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_\tau : {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div></div><p>only after we studied their <i>compactness</i> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. 34, 761–795 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR8" id="ref-link-section-d74823138e43218">8</a>].</p><p>But it will turn out that only the limiting cases are of particular interest. So we collect first what is more or less obvious. Note that we use the above notation always with the understanding that either both, source and target space, are of Besov-type (<span class="mathjax-tex">$A=B$</span>), or both are of Triebel-Lizorkin-type (<span class="mathjax-tex">$A=F$</span>).</p><p>For convenience we use the following abbreviation:</p><div id="Equ43" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \gamma (\tau _1,\tau _2,p_1,p_2)&:= \max \left\{ \left( \tau _2-\frac{1}{p_2}\right) _+ -\left( \tau _1-\frac{1}{p_1}\right) _+, \frac{1}{p_1} -\tau _1 - \frac{1}{p_2}+\tau _2,\right. \nonumber \\&\left. \frac{1}{p_1}-\tau _1 - \min \left\{ \frac{1}{p_2}-\tau _2, \frac{1}{p_2}(1-p_1\tau _1)_+\right\} \right\} \nonumber \\&\quad = {\left\{ \begin{array}{ll} \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2, &{} \text {if}\quad \tau _2\ge \frac{1}{p_2}, \\ \frac{1}{p_1}-\tau _1, &{}\text {if}\quad \tau _1\ge \frac{1}{p_1}, \ \tau _2< \frac{1}{p_2}, \\ \max \{0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \{\tau _2,\frac{p_1}{p_2}\tau _1\}\}, &{}\text {if}\quad \tau _1< \frac{1}{p_1}, \ \tau _2< \frac{1}{p_2}. \end{array}\right. } \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.1) </div></div><p>Here and in the sequel we put <span class="mathjax-tex">$p_i\tau _i=1$</span> in case of <span class="mathjax-tex">$p_i=\infty $</span> and <span class="mathjax-tex">$\tau _i=0$</span>. Similarly we shall understand <span class="mathjax-tex">$\frac{p_i}{p_k}=1$</span> if <span class="mathjax-tex">$p_i=p_k=\infty $</span>.</p> <h3 class="c-article__sub-heading" id="FPar30">Theorem 4.1</h3> <p>([<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. 34, 761–795 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR8" id="ref-link-section-d74823138e44156">8</a>]) Let <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0<p_i\le \infty $</span> (with <span class="mathjax-tex">$p_i<\infty $</span> in case of <span class="mathjax-tex">$A=F$</span>), <span class="mathjax-tex">$\tau _i\ge 0$</span>, <span class="mathjax-tex">$i=1,2$</span>. </p><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>The embedding </p><div id="Equ143" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div></div><p> is compact if, and only if, </p><div id="Equ144" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} > \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$</span></div></div> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>There is no continuous embedding </p><div id="Equ145" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div></div><p> if </p><div id="Equ44" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} < \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.2) </div></div> </dd></dl> <p>This result was proved in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. 34, 761–795 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR8" id="ref-link-section-d74823138e44913">8</a>] and shows us that we are indeed left to deal with the limiting case</p><div id="Equ146" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$</span></div></div><p>First we prove the following lemma that extends [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR11" id="ref-link-section-d74823138e45020">11</a>, Theorem 3.1] to the case <span class="mathjax-tex">$u=p=\infty $</span>. We recall that <span class="mathjax-tex">$\mathcal {N}^{s}_{\infty ,\infty ,q}({{{\mathbb {R}}}^d})= B^{s}_{\infty ,q}({{{\mathbb {R}}}^d}) $</span>.</p> <h3 class="c-article__sub-heading" id="FPar31">Lemma 4.2</h3> <p>Let <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0<p_i\le u_i< \infty $</span> or <span class="mathjax-tex">$p_i=u_i=\infty $</span>, <span class="mathjax-tex">$i=1,2$</span>. Then</p><div id="Equ45" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&{{\mathcal {N}}}^{s_1}_{u_1,p_1,q_1}(\Omega ) \hookrightarrow \mathcal {N}^{s_2}_{\infty ,\infty ,q_2}(\Omega ) \quad \text{ if, } \text{ and } \text{ only } \text{ if, }\nonumber \\&\frac{s_1-s_2}{d}> \frac{1}{u_1}\quad \text {or}\quad \frac{s_1-s_2}{d}= \frac{1}{u_1}\quad \text {and}\quad q_1\le q_2 , \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.3) </div></div><p>and</p><div id="Equ46" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathcal {N}^{s_1}_{\infty ,\infty ,q_1}(\Omega ) \hookrightarrow {{\mathcal {N}}}^{s_2}_{u_2,p_2,q_2}(\Omega )&\quad \text{ if, } \text{ and } \text{ only } \text{ if, } \quad s_1> s_2 \quad \text {or}\quad s_1=s_2 \quad \text {and}\quad q_1\le q_2 . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.4) </div></div> <h3 class="c-article__sub-heading" id="FPar32">Proof</h3> <p>The necessity of the conditions in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ45">4.3</a>) follows easily by the following chain of embeddings</p><div id="Equ147" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B^{s_1}_{u_1,q_1}(\Omega )\hookrightarrow {{\mathcal {N}}}^{s_1}_{u_1,p_1,q_1}(\Omega ) \hookrightarrow \mathcal {N}^{s_2}_{\infty ,\infty ,q_2}(\Omega ) = B^{s_2}_{\infty ,q_2}(\Omega ) \end{aligned}$$</span></div></div><p>and the properties of embeddings of classical Besov spaces. Whereas the sufficiency can be proved in the same way as in the proof of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR11" id="ref-link-section-d74823138e46091">11</a>, Theorem 3.1].</p> <p>To prove the second embedding it is sufficient to note that</p><div id="Equ148" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \mathcal {N}^{s_1}_{\infty ,\infty ,q_1}(\Omega ) = B^{s_1}_{\infty ,q_1}(\Omega ) \hookrightarrow B^{s_2}_{u_2,q_2}(\Omega ) \hookrightarrow {{\mathcal {N}}}^{s_2}_{u_2,p_2,q_2}(\Omega ). \end{aligned}$$</span></div></div><p>On the other hand it follows from (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ45">4.3</a>) that</p><div id="Equ149" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B^{s_1+\frac{d}{u_1}}_{u_1,q_1}(\Omega )\hookrightarrow \mathcal {N}^{s_1}_{\infty ,\infty ,q_1}(\Omega ) \hookrightarrow {{\mathcal {N}}}^{s_2}_{u_2,p_2,q_2}(\Omega ).\end{aligned}$$</span></div></div><p>So if the last embedding holds, then it follows from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR11" id="ref-link-section-d74823138e46526">11</a>, Corollary 3.7] that <span class="mathjax-tex">$s_1>s_2$</span>, or <span class="mathjax-tex">$s_1=s_2$</span> and <span class="mathjax-tex">$q_1\le q_2$</span>. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar33">Proposition 4.3</h3> <p>Let <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0<p_i\le \infty $</span> (with <span class="mathjax-tex">$p_i<\infty $</span> in case of <span class="mathjax-tex">$A=F$</span>), <span class="mathjax-tex">$\tau _i\ge 0$</span>, <span class="mathjax-tex">$i=1,2$</span>. Assume</p><div id="Equ150" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \tau _2\ge \frac{1}{p_2}\quad \text {with}\quad q_2=\infty \quad \text {if}\quad \tau _2=\frac{1}{p_2}. \end{aligned}$$</span></div></div><p>Then the embedding</p><div id="Equ151" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div></div><p>is continuous if, and only if,</p><div id="Equ152" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} \ge \gamma (\tau _1,\tau _2,p_1,p_2). \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar34">Proof</h3> <p>Note that by Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar30">4.1</a> we are left to deal with the limiting case <span class="mathjax-tex">$s_1-s_2 = d \gamma (\tau _1,\tau _2,p_1,p_2)$</span> only. In view of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar3">2.3</a> we always have <span class="mathjax-tex">${A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )=B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega )$</span> now. Assume first <span class="mathjax-tex">$\tau _1\ge \frac{1}{p_1}$</span> with <span class="mathjax-tex">$q_1=\infty $</span> if <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span>, then by the same result also <span class="mathjax-tex">${A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )=B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )$</span> such that <span class="mathjax-tex">$\text {id}_\tau $</span> is continuous if, and only if,</p><div id="Equ153" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}:B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega ) \hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ). \end{aligned}$$</span></div></div><p>But this is always true if <span class="mathjax-tex">$\frac{s_1-s_2}{d}=\gamma (\tau _1,\tau _2,p_1,p_2)= \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2$</span>, recall Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar15">2.15</a>.</p> <p>Assume next <span class="mathjax-tex">$0\le \tau _1<\frac{1}{p_1}$</span>. We put <span class="mathjax-tex">$\frac{1}{u_1}=\frac{1}{p_1}-\tau _1$</span>. We first show the sufficiency of <span class="mathjax-tex">$s_1-s_2= d\gamma (\tau _1,\tau _2,p_1,p_2)$</span> for the continuity of <span class="mathjax-tex">$\text {id}_\tau $</span>. We use (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ15">2.15</a>), (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ5">2.5</a>), Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar3">2.3</a> and Lemma <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar31">4.2</a> to obtain</p><div id="Equ154" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}{A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow B^{s_1,\tau _1}_{p_1,\infty } (\Omega )=\mathcal {N}^{s_1}_{u_1,p_1,\infty }(\Omega )\hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega )={A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ). \end{aligned}$$</span></div></div><p>On the other hand, for the necessity,</p><div id="Equ155" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}\mathcal {N}^{s_1}_{u_1,p_1,\min \{p_1,q_1\}}(\Omega )\hookrightarrow {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )=B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ) \end{aligned}$$</span></div></div><p>and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ45">4.3</a>) implies <span class="mathjax-tex">$s_1-s_2 \ = \ d\gamma (\tau _1,\tau _2,p_1,p_2)$</span> if <span class="mathjax-tex">$\text {id}_\tau $</span> is continuous.</p> <p>It remains to consider the case <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span>, <span class="mathjax-tex">$q_1<\infty $</span>. Now we benefit from the following chains of embeddings</p><div id="Equ47" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )\hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ) = {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.5) </div></div><p>for the sufficiency of the condition <span class="mathjax-tex">$s_1-s_2= d\gamma (\tau _1,\tau _2,p_1,p_2)$</span>, and</p><div id="Equ48" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B^{s_1}_{\infty ,\min (p_1,q_1)}(\Omega ) \hookrightarrow {{A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )} = B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty ,\infty }(\Omega ), \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.6) </div></div><p>for its necessity, cf. (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ5">2.5</a>) and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e49750">47</a>, Proposition 2.4]. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar35">Remark 4.4</h3> <p>Note that the above result is the direct counterpart of our result for spaces on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span> obtained in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> (i), since <span class="mathjax-tex">$\gamma (\tau _1,\tau _2,p_1,p_2)=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2 $</span> in the above setting.</p> <h3 class="c-article__sub-heading" id="FPar36">Remark 4.5</h3> <p>Recall the definition of the spaces <span class="mathjax-tex">$\textrm{bmo}({{{\mathbb {R}}}^d})$</span> in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar8">2.8</a> and define <span class="mathjax-tex">$\textrm{bmo}(\Omega )$</span> by restriction, that is, in analogy to Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar12">2.12</a>. Then</p><div id="Equ49" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \textrm{bmo}(\Omega )=F^{0,1/p}_{p,2}(\Omega ) = {B^{0,1/2}_{2,2}(\Omega ), \quad 0<p<\infty }, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.7) </div></div><p>extending (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ18">2.18</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ17">2.17</a>) to domains <span class="mathjax-tex">$\Omega $</span>. Taking <span class="mathjax-tex">$\textrm{bmo}(\Omega )$</span> as the source space, that is, <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span>, <span class="mathjax-tex">$s_1=0$</span>, then Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> implies that for <span class="mathjax-tex">$s\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<p,q\le \infty $</span>, and <span class="mathjax-tex">$\tau \ge \frac{1}{p}$</span> with <span class="mathjax-tex">$q=\infty $</span> if <span class="mathjax-tex">$\tau =\frac{1}{p}$</span>, then</p><div id="Equ156" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_\tau : \textrm{bmo}(\Omega ) \hookrightarrow {A}_{p,q}^{s,\tau }(\Omega ) \end{aligned}$$</span></div></div><p>is continuous if, and only if, <span class="mathjax-tex">$s\le -d (\tau -\frac{1}{p})\le 0$</span>. If <span class="mathjax-tex">$\textrm{bmo}(\Omega )$</span> was the target space, then Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> cannot be applied since <span class="mathjax-tex">$q_2=2<\infty $</span>.</p> <p>In view of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar30">4.1</a> and Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> we are left to study the situation</p><div id="Equ50" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)\quad \text {and}\quad \tau _2\le \frac{1}{p_2}\quad \text {with}\quad q_2<\infty \quad \text {if}\quad \tau _2=\frac{1}{p_2} \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.8) </div></div><p>in the sequel. Next we give some counterpart of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> dealing with the case when in the source space the parameter <span class="mathjax-tex">$\tau _1$</span> is large.</p> <h3 class="c-article__sub-heading" id="FPar37">Proposition 4.6</h3> <p>Let <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0<p_i\le \infty $</span>, <span class="mathjax-tex">$\tau _i\ge 0$</span>, <span class="mathjax-tex">$i=1,2$</span>. Assume that (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ50">4.8</a>) is satisfied and</p><div id="Equ157" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \tau _1\ge \frac{1}{p_1}\quad \text {with}\quad q_1=\infty \quad \text {if}\quad \tau _1=\frac{1}{p_1}. \end{aligned}$$</span></div></div><p>Then the embedding</p><div id="Equ158" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div></div><p>is continuous if, and only if,</p><div id="Equ159" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} q_2=\infty . \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar38">Proof</h3> <p>First note that <span class="mathjax-tex">$\gamma (\tau _1,\tau _2,p_1,p_2)= \frac{1}{p_1}-\tau _1$</span> by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ43">4.1</a>) and thus <span class="mathjax-tex">${A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )=B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )=B^{s_2}_{\infty ,\infty }(\Omega )$</span> in view of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar3">2.3</a>.</p> <p>We first deal with the case <span class="mathjax-tex">$A=B$</span> and start with the sufficiency of <span class="mathjax-tex">$q_2=\infty $</span>. Then Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> covers the case <span class="mathjax-tex">$\tau _2 =\frac{1}{p_2}$</span> and we may assume <span class="mathjax-tex">$\tau _2<\frac{1}{p_2}$</span>, recall (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ50">4.8</a>). But in view of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ6">2.6</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ15">2.15</a>) we get</p><div id="Equ160" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega ) \hookrightarrow \mathcal {N}^{s_2}_{u_2,p_2,\infty }(\Omega )= B^{s_2,\tau _2}_{p_2,\infty }(\Omega ). \end{aligned}$$</span></div></div><p>Now assume <span class="mathjax-tex">$A=F$</span> and again <span class="mathjax-tex">$\tau _2<\frac{1}{p_2}$</span>. If <span class="mathjax-tex">$f\in {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) =B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )$</span>, recall Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar3">2.3</a>, then there exists some <span class="mathjax-tex">$g\in B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }({{{\mathbb {R}}}^d})$</span> such that <span class="mathjax-tex">$f=g|_\Omega $</span> We can choose <i>g</i> such that it can be represented as in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ26">2.26</a>) with the summation over <span class="mathjax-tex">$k\in {{{\mathbb {Z}}}^d}$</span> restricted to the indices <i>k</i> such that <span class="mathjax-tex">$|k|\le K$</span> for some fixed <i>K</i> since <span class="mathjax-tex">$\Omega $</span> is a bounded domain. Moreover we can choose <i>g</i> in such a way that</p><div id="Equ161" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \lambda _k \mid \ell _\infty ({{{\mathbb {Z}}}^d})\Vert + \sup _{j\in {{\mathbb {N}}}} 2^{j(s_1+d(\tau _1-\frac{1}{p_1})+\frac{d}{2})} \sup _{i=1,\ldots ,2^d-1;\;k\in {{{\mathbb {Z}}}^d}} |\lambda _{i,j,k}| \le C \Vert f\mid B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )\Vert ,\end{aligned}$$</span></div></div><p>for some constant <span class="mathjax-tex">$C>0$</span> independent of <i>f</i>. We have to show that <span class="mathjax-tex">$f\in F^{s_2,\tau _2}_{p_2,\infty }(\Omega )$</span> and <span class="mathjax-tex">$\Vert f \mid F^{s_2,\tau _2}_{p_2,\infty }(\Omega ) \Vert \le c \Vert f\mid B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty }(\Omega )\Vert $</span>. It is sufficient to note that for any <span class="mathjax-tex">$i=1,\ldots , 2^d-1$</span>,</p><div id="Equ51" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\left\| \sup _{j,k} 2^{j(s_2-\frac{d}{u_2}+\frac{d}{2})} |\lambda _{i,j,k}|{2^{j\frac{d}{u_2}}\chi _{j,k}(\cdot )}\mid {\mathcal {M}_{u_2,p_2}}{({{{\mathbb {R}}}^d})}\right\| \nonumber \\&\quad \le C \sup _{j\in {{\mathbb {N}}}} 2^{j(s_1+d(\tau _1-\frac{1}{p_1})+\frac{d}{2})} \sup _{i=1,\ldots ,2^d-1;\;k\in {{{\mathbb {Z}}}^d}} |\lambda _{i,j,k}| . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.9) </div></div><p>The rest follows from the wavelet characterisation of <span class="mathjax-tex">$\mathcal {E}^{s_2}_{u_2,p_2,\infty }({{{\mathbb {R}}}^d})={F^{s_2,\tau _2}_{p_2,\infty }}({{{\mathbb {R}}}^d})$</span>, <span class="mathjax-tex">$\frac{1}{u_2}=\frac{1}{p_2}-\tau _2$</span>, cf. [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 16" title="Liang, Y., Yang, D., Yuan, W., Sawano, Y., Ullrich, T.: A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math. (Rozprawy Mat.) 489, 1–114 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR16" id="ref-link-section-d74823138e53688">16</a>]. But (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ51">4.9</a>) follows easily from the identities <span class="mathjax-tex">$s_2=s_1+d(\tau _1-\frac{1}{p_1})$</span> and <span class="mathjax-tex">$\Vert {2^{j\frac{d}{u_2}}\chi _{j,k}(\cdot )}\mid \mathcal {M}_{u_2,p_2}{({{{\mathbb {R}}}^d})}\Vert =1$</span>.</p> <p>Now we prove the necessity and assume that <span class="mathjax-tex">$\text {id}_\tau : {A}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$</span> is continuous. We start with the case <span class="mathjax-tex">$A=B$</span>. If <span class="mathjax-tex">$p_2=\infty $</span>, then <span class="mathjax-tex">$\tau =0$</span> by assumption (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ50">4.8</a>) and <span class="mathjax-tex">$B^{s_2,\tau _2}_{\infty ,q_2}(\Omega )= B^{s_2}_{\infty ,q_2}(\Omega )$</span>. So both the source and the target space are classical Besov spaces and it is well-known that, in that case, <span class="mathjax-tex">$q_2=\infty $</span>, recall Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar15">2.15</a>. So it remains to consider the case <span class="mathjax-tex">$p_2<\infty $</span>. To simplify the notation we assume that the support of the wavelets <span class="mathjax-tex">$\psi _{i,j,m}$</span> such that <span class="mathjax-tex">$Q_{j,m}\subset Q_{0,0}$</span> are contained in <span class="mathjax-tex">$\Omega $</span>. If it is not true one can easily rescale the argument.</p> <p>We take a sequence <span class="mathjax-tex">$\lambda =(\lambda _{i,j,m}){_{i,j,m}, i=1,...,2^d-1, j\in {{\mathbb {N}}}_0,m\in {{{\mathbb {Z}}}^d},}$</span> defined by the formula</p><div id="Equ162" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \lambda _{i,j,m}= {\left\{ \begin{array}{ll} 2^{-j(s_2+\frac{d}{2})} &{} \textrm{if }\quad i=1 \quad \textrm{and}\quad Q_{j,m}\subset Q_{0,0},\\ 0 &{} \textrm{otherwise}. \end{array}\right. } \end{aligned}$$</span></div></div><p>Then, using the sequence space version of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar3">2.3</a>,</p><div id="Equ163" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \lambda \mid {b}_{p_1,q_1}^{s_1,\tau _1}\Vert \sim \Vert \lambda \mid b^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty ,\infty } \Vert = 1 \end{aligned}$$</span></div></div><p>since <span class="mathjax-tex">$s_2= s_1+d(\tau _1-\frac{1}{p_1})$</span>. Here we used the notation <span class="mathjax-tex">$b^\sigma _{p,q} = b^{\sigma ,0}_{p,q}$</span>. On the other hand, for any dyadic cube <span class="mathjax-tex">$P\subset Q_{0,0}$</span> and any <span class="mathjax-tex">$j\ge j_P$</span>, we have</p><div id="Equ164" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})}\left( \sum _{Q_{j,m}\subset P}|\lambda _{i,j,m}|^{p_2}\right) ^{\frac{1}{p_2}} = 2^{-j_P\frac{1}{p_2}}. \end{aligned}$$</span></div></div><p>So if <span class="mathjax-tex">$q_2<\infty $</span>, then</p><div id="Equ165" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert \lambda \mid b^{s_2,\tau _2}_{p_2,q_2} \Vert = \infty . \end{aligned}$$</span></div></div><p>Therefore, if <span class="mathjax-tex">$q_2<\infty $</span>, the function</p><div id="Equ166" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} f=\sum _{i,j,m} \lambda _{i,j,m} \psi _{i,j,m} \end{aligned}$$</span></div></div><p>belongs to <span class="mathjax-tex">$B^{s_1,\tau _1}_{p_1,q_1}(\Omega )$</span> but not to <span class="mathjax-tex">$B^{s_2,\tau _2}_{p_2,q_2}(\Omega )$</span>, which contradicts our assumption and thus finishes the proof of the necessity for the <i>B</i>-case.</p> <p>The case <span class="mathjax-tex">$A=F$</span> follows by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ5">2.5</a>) and by what we just proved for the Besov-type spaces. Note that, for the <i>F</i>-spaces, we always have <span class="mathjax-tex">$p<\infty $</span>. Therefore, by the following chain of embeddings</p><div id="Equ167" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B^{s_1, \tau _1}_{p_1, \min \{p_1, q_1\}}(\Omega ) \hookrightarrow {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \hookrightarrow B^{s_2, \tau _2}_{p_2, \max \{p_2,q_2\}}(\Omega ), \end{aligned}$$</span></div></div><p>we obtain the necessity of the condition <span class="mathjax-tex">$\max \{p_2,q_2\}=\infty $</span>, which here reads as <span class="mathjax-tex">$q_2=\infty $</span>. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar39">Remark 4.7</h3> <p>Note that the above result differs from its <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>-counterpart in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> (ii). In that case, there is never a continuous embedding in the setting of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a>, that is, when conditions (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ50">4.8</a>) and <span class="mathjax-tex">$\tau _1\ge \frac{1}{p_1}$</span> with <span class="mathjax-tex">$q_1=\infty $</span> when <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span> are satisfied.</p> <h3 class="c-article__sub-heading" id="FPar40">Remark 4.8</h3> <p>Again we return to the special case when the source or target space of <span class="mathjax-tex">$\text {id}_\tau $</span> coincides with <span class="mathjax-tex">$\textrm{bmo}(\Omega )$</span>. Parallel to Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar36">4.5</a> we cannot apply Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a> in case of <span class="mathjax-tex">${A}_{p_1,q_1}^{s_1,\tau _1}(\Omega )=\textrm{bmo}(\Omega )$</span>. Otherwise, if <span class="mathjax-tex">${A}_{p_2,q_2}^{s_2,\tau _2}(\Omega )=\textrm{bmo}(\Omega )$</span>, then Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a> implies that there is never a continuous embedding of type</p><div id="Equ168" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}\text {id}_\tau : {A}_{p,q}^{s,\tau }(\Omega ) \hookrightarrow \textrm{bmo}(\Omega )\end{aligned}$$</span></div></div><p>in the limiting case (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ50">4.8</a>) which reads here as <span class="mathjax-tex">$0<p\le \infty $</span> (with <span class="mathjax-tex">$p<\infty $</span> in case of <span class="mathjax-tex">$A=F$</span>), <span class="mathjax-tex">$0<q\le \infty $</span>, <span class="mathjax-tex">$\tau \ge 0$</span> and <span class="mathjax-tex">$s = d(\frac{1}{p}-\tau )$</span>. Moreover, there is no such continuous embedding whenever <span class="mathjax-tex">$\tau _2=\frac{1}{p_2}$</span> and <span class="mathjax-tex">$q_2<\infty $</span> in the limiting case (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ50">4.8</a>).</p> <p>For the rest of this section we shall now assume that</p><div id="Equ52" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 0\le \tau _i\le \frac{1}{p_i}\quad \text {with}\quad q_i<\infty \quad \text {if}\quad \tau _i=\frac{1}{p_i}, \quad i=1,2, \quad \textrm{and}\quad \tau _1+\tau _2>0, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.10) </div></div><p>and thus</p><div id="Equ53" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)= \max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.11) </div></div><p>In view of the embeddings and coincidences (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ14">2.14</a>), (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ15">2.15</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ16">2.16</a>), together with our previous findings for the spaces <span class="mathjax-tex">${{\mathcal {A}}}^{s}_{u,p,q}(\Omega )$</span> in [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR11" id="ref-link-section-d74823138e57267">11</a>, <a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haroske, D.D., Skrzypczak, L.: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces. Rev. Mat. Complut. 27, 541–573 (2014)" href="/article/10.1007/s10231-023-01327-w#ref-CR12" id="ref-link-section-d74823138e57270">12</a>] (as well as some <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>-counterparts of <span class="mathjax-tex">${A}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span> in Theorems <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar11">2.11</a>), we expect some <i>q</i>-dependence now. For the moment, we restrict ourselves to the case of <span class="mathjax-tex">${B}_{p,q}^{s,\tau }$</span> spaces.</p> <h3 class="c-article__sub-heading" id="FPar41">Theorem 4.9</h3> <p>Let <span class="mathjax-tex">$0< p_1,p_2\le \infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0\le \tau _i\le \frac{1}{p_i}$</span>, <span class="mathjax-tex">$i=1,2$</span>. Assume that the conditions (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ52">4.10</a>) hold and that</p><div id="Equ169" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)=\max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$</span></div></div><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>The embedding </p><div id="Equ54" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.12) </div></div><p> is continuous if one of the following conditions holds: </p><div id="Equ55" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\frac{s_1-s_2}{d} = \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0, \quad \text {and}\quad p_1\tau _1<p_2\tau _2, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.13) </div></div><div id="Equ56" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text{ or } \qquad&\frac{s_1-s_2}{d}=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\frac{p_1}{p_2}\tau _1>0 \quad \text {and}\quad q_1\le \frac{p_1}{p_2}q_2 , \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.14) </div></div><div id="Equ57" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text{ or } \qquad&s_1=s_2 \quad \text {and}\quad q_1\le \min \left\{ 1, \frac{p_1}{p_2}\right\} q_2. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.15) </div></div> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>If the embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ54">4.12</a>) is continuous and one of the following conditions holds </p><div id="Equ58" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\gamma (\tau _1,\tau _2,p_1,p_2)= 0, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.16) </div></div><div id="Equ59" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text{ or }\qquad&\gamma (\tau _1,\tau _2,p_1,p_2)= \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\frac{p_1}{p_2}\tau _1>0 , \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.17) </div></div><p> then <span class="mathjax-tex">$q_1\le q_2$</span>. If the embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ54">4.12</a>) is continuous for any <span class="mathjax-tex">$q_1$</span> and <span class="mathjax-tex">$q_2$</span> with fixed <span class="mathjax-tex">$s_1,s_2,p_1,p_2,\tau _1,\tau _2$</span>, then (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ55">4.13</a>) holds.</p> </dd></dl> <h3 class="c-article__sub-heading" id="FPar42">Remark 4.10</h3> <p>As mentioned above, we are left to consider the embedding in the limiting case (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ53">4.11</a>) when (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ52">4.10</a>) is satisfied. However, in case of <span class="mathjax-tex">$\tau _i=\frac{1}{p_i}$</span>, <span class="mathjax-tex">$q_i=\infty $</span>, for <span class="mathjax-tex">$i=1$</span> or <span class="mathjax-tex">$i=2$</span>, the above Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> coincides with Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> or <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a>, respectively. So in fact situation (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ52">4.10</a>) is the only interesting one now.</p> <p>We have always <span class="mathjax-tex">$p_1<p_2$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ56">4.14</a>) so we have a small gap between sufficient and necessary conditions on <span class="mathjax-tex">$q_i$</span> here. We meet a similar situation if <span class="mathjax-tex">$s_1=s_2$</span>, <span class="mathjax-tex">$p_1<p_2$</span>, <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span> and <span class="mathjax-tex">$q_1<\infty $</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ57">4.15</a>). In all other cases the result is sharp.</p> <h3 class="c-article__sub-heading" id="FPar43">Proof (of Theorem 4.9)</h3> <p><i>Step 1.</i> We start by proving part (i). For this, we use an argument similar to the one used in the proof of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar28">3.9</a>, based on the extension operator from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a> and the wavelet decomposition of the spaces <span class="mathjax-tex">${B}_{p,q}^{s,\tau }({{{\mathbb {R}}}^d})$</span>, cf. Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar17">2.17</a>. We use the same notation as there. Let us denote by <span class="mathjax-tex">$\widetilde{{b}_{p,q}^{s,\tau }}(\widetilde{Q}_0)$</span> the sequence space defined by</p><div id="Equ170" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widetilde{{b}_{p,q}^{s,\tau }}(\widetilde{Q}_0){:=}&\left\{ t= \{t_{i,j,m}\}_{i,j,m}: t_{i,j,m} \in {{\mathbb {C}}},\, j \in {{\mathbb {N}}}_0, i=1, \dots , 2^d-1, \right. \\&\qquad \qquad \left. m\in {{{\mathbb {Z}}}^d}, Q_{j,m} \subset \widetilde{Q}_0,\, \Vert t \mid \widetilde{{b}_{p,q}^{s,\tau }}\Vert <\infty \right\} , \end{aligned}$$</span></div></div><p>where</p><div id="Equ60" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p,q}^{s,\tau }}\Vert {:=} \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}_0} \frac{ 1}{|P|^{\tau }}\left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s+\frac{d}{2}-\frac{d}{p})q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p}\right] ^{\frac{ q}{p}}\right\} ^{\frac{1}{q}}.\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.18) </div></div><p>Then, we have to prove that for some <span class="mathjax-tex">$C>0$</span></p><div id="Equ61" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert \le C \, \Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.19) </div></div><p>holds true for all <span class="mathjax-tex">$t\in \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}$</span>. Here and in the sequel we assume for convenience that <span class="mathjax-tex">$p_i, q_i<\infty $</span>, otherwise the modifications are obvious. Please note, once more, that the assumption <span class="mathjax-tex">$P\subset \widetilde{Q}_{0}$</span> implies that</p><div id="Equ62" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{ 1}{|P|^{a}} \le C \frac{ 1}{|P|^{b}}\quad \text {if }\quad a\le b \quad \text {and}\quad \#\{m:\;Q_{j,m}\subset P \} \sim 2^{jd}\min \{1, 2^{-j_P d}\}\quad \text {if} \quad j\ge j_P.\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.20) </div></div><p>Moreover, if <span class="mathjax-tex">$1\le |P|\le | \widetilde{Q}_0|$</span>, then</p><div id="Equ63" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{ 1}{|P|^{\tau _2}} \sim \frac{ 1}{|P|^{\tau _1}} \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.21) </div></div><p>for any <span class="mathjax-tex">$\tau _1$</span> and <span class="mathjax-tex">$\tau _2$</span>. To shorten the notation we put <span class="mathjax-tex">$\gamma = \gamma (\tau _1,\tau _2,p_1,p_2)$</span>.</p> <p><i>Substep 1.1.</i> If <span class="mathjax-tex">$\gamma =\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0$</span>, then <span class="mathjax-tex">$\frac{p_1}{p_2}\tau _1\le \tau _2$</span>. In this case the statement follows from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a> and Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> (iii) if <span class="mathjax-tex">$\tau _1\not = \tau _2$</span>.</p> <p>Indeed Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a> and Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar25">3.7</a> imply that there exists a common bounded extension operator <span class="mathjax-tex">$\textrm{ext}$</span> for the spaces <span class="mathjax-tex">${B}_{p_1,q_1}^{s_1,\tau _1}(\Omega )$</span> and <span class="mathjax-tex">${B}_{p_2,q_2}^{s_2,\tau _2}(\Omega )$</span> and we thus have the following commutative diagram</p><div id="Equ206" class="c-article-equation"><div class="c-article-equation__content"><img src="//media.springernature.com/lw206/springer-static/image/art%3A10.1007%2Fs10231-023-01327-w/MediaObjects/10231_2023_1327_Equ206_HTML.png" class="u-display-block" alt=""></div></div><p>Now the case (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ55">4.13</a>) follows from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> (iii), as well as the situation when</p><div id="Equ64" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d}=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0,\quad \tau _1\not = \tau _2,\quad \frac{p_1}{p_2}=\frac{\tau _2}{\tau _1}\quad \text {and}\quad q_1\le \frac{p_1}{p_2}q_2. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.22) </div></div><p><i>Substep 1.2.</i> Let <span class="mathjax-tex">$\gamma =\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _2>0$</span> and <span class="mathjax-tex">$\tau _1=\tau _2$</span>. In that case <span class="mathjax-tex">$p_1<p_2$</span> and <span class="mathjax-tex">$s_1-\frac{d}{p_1}=s_2-\frac{d}{p_2}$</span>. Let <span class="mathjax-tex">$t = \{t_{i,j,m}\}_{i,j,m}\in \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}$</span> and let <span class="mathjax-tex">$\Vert t\mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert =1$</span>. To simplify the notation we put <span class="mathjax-tex">$\lambda _{i,j,m}= 2^{-j(s_1+\frac{d}{2}-\frac{d}{p_1})}t_{i,j,m}$</span> and <span class="mathjax-tex">$\tau =\tau _1=\tau _2$</span>.</p> <p>For any <i>i</i>, <i>j</i> and <i>m</i> we have</p><div id="Equ171" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} |\lambda _{i,j,m}|\le 2^{-jd\tau }, \end{aligned}$$</span></div></div><p>and in consequence</p><div id="Equ65" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} |\lambda _{i,j,m}|^{p_2}\le |\lambda _{i,j,m}|^{p_1} 2^{-jd\tau (p_2-p_1)}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.23) </div></div><p>In a parallel way, for any dyadic cube <span class="mathjax-tex">$P\subset \widetilde{Q}_0$</span> we have</p><div id="Equ66" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_1} \le 2^{-j_Pd\tau p_1}, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.24) </div></div><p>so in consequence</p><div id="Equ67" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_2}\right) ^{\frac{q_2}{p_2}}&\le \left( \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_1}\right) ^{\frac{q_2}{p_2}} 2^{-jd\tau (p_2-p_1)\frac{q_2}{p_2}} \nonumber \\&\le \ c\ 2^{-j_Pd\tau \frac{p_1}{p_2}q_2} 2^{-jd\tau (1-\frac{p_1}{p_2})q_2} \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.25) </div></div><p>for any <span class="mathjax-tex">$i=1,\dots , 2^d-1$</span>. Summing up over <i>j</i> we get</p><div id="Equ68" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{j=j_P}^\infty \sum _{i=1}^{2^d-1} \left( \sum _{Q_{j,m}\subset P} |\lambda _{i,j,m}|^{p_2}\right) ^{\frac{q_2}{p_2}} \le c\ 2^d 2^{-j_Pd\tau \frac{p_1}{p_2}q_2} \sum _{j=j_P}^\infty 2^{-jd\tau (1-\frac{p_1}{p_2})q_2} = C 2^{-j_Pd\tau q_2}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.26) </div></div><p>This proves that <span class="mathjax-tex">$t = \{t_{i,j,m}\}_{i,j,m}\in \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}$</span> and <span class="mathjax-tex">$\Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert \le C$</span>.</p> <p><i>Substep 1.3.</i> Let <span class="mathjax-tex">$\gamma =0$</span>, i.e., <span class="mathjax-tex">$s_1=s_2$</span>. Then <span class="mathjax-tex">$p_2\le p_1$</span> and <span class="mathjax-tex">$ \frac{1}{p_1}-\frac{1}{p_2} \le \tau _1-\tau _2$</span> or <span class="mathjax-tex">$p_2 > p_1$</span> and <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span>. First we assume that <span class="mathjax-tex">$p_2\le p_1$</span> and <span class="mathjax-tex">$ \frac{1}{p_1}-\frac{1}{p_2} \le \tau _1-\tau _2$</span>. We conclude by Hölder’s inequality for any <span class="mathjax-tex">$i=1, \dots , 2^d-1$</span>, that</p><div id="Equ69" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{1}{p_2}} \le 2^{d(j-j_P)(\frac{1}{p_2}-\frac{1}{p_1})} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{1}{p_1}} \, . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.27) </div></div><p>In consequence, for any <span class="mathjax-tex">$q\in (0,\infty ]$</span>,</p><div id="Equ70" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{{p_2}}\right] ^{\frac{q}{p_2}} \right\} ^{\frac{1}{q}} \,\le 2^{j_P d(\frac{1}{p_1}-\frac{1}{p_2})} \nonumber \\ \times \left\{ \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q} 2^{j(s_2-s_1)q} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{q}{p_1}} \right\} ^{\frac{1}{q}}\, . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.28) </div></div><p>If <span class="mathjax-tex">$\gamma (\tau _1,\tau _2,p_1,p_2)=0$</span>, then <span class="mathjax-tex">$s_1=s_2$</span> and <span class="mathjax-tex">$a=\frac{1}{p_1}-\frac{1}{p_2}+\tau _2\le \tau _1=b$</span>. So (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ61">4.19</a>) follows from (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ70">4.28</a>) and (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ62">4.20</a>)-(<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ63">4.21</a>) for any <span class="mathjax-tex">$q_1\le q_2$</span>.</p> <p>Now let <span class="mathjax-tex">$\gamma =0$</span>, <span class="mathjax-tex">$p_2 > p_1$</span> and <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span>. First we consider the case <span class="mathjax-tex">$\tau _2=\frac{1}{p_2}$</span>. Let</p><div id="Equ71" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert =1, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.29) </div></div><p>which implies that, for every cube <span class="mathjax-tex">$P \in \mathcal {Q}, P \subset \widetilde{Q}_{0}$</span> and for every <span class="mathjax-tex">$j \ge \max \{j_P,0\}$</span>, we have</p><div id="Equ172" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{ 1}{|P|^{\tau _1 q_1}} \, 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{q_1}{p_1}} \le 1. \end{aligned}$$</span></div></div><p>In particular, we know that for every cube <span class="mathjax-tex">$Q_{\nu ,m} \in \mathcal {Q}, Q_{\nu ,m} \subset \widetilde{Q}_{0}$</span>, with <span class="mathjax-tex">$\nu \ge 0$</span> and for every <span class="mathjax-tex">$i=1,..., 2^d-1$</span>, we have</p><div id="Equ173" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 2^{\nu (s_1 + \frac{d}{2}- \frac{d}{p_1}+d \tau _1)}|t_{i,\nu ,m}|= 2^{\nu (s_1 + \frac{d}{2})}|t_{i,\nu ,m}| \le 1. \end{aligned}$$</span></div></div><p>So the condition <span class="mathjax-tex">$p_1< p_2$</span> implies</p><div id="Equ72" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} 2^{\nu ( s_1 + \frac{d}{2})p_2}|t_{i,\nu ,m}|^{p_2} \le 2^{\nu ( s_1 + \frac{d}{2})p_1}|t_{i,\nu ,m}|^{p_1}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.30) </div></div><p>We have to prove that <span class="mathjax-tex">$\Vert t \mid {\widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}}\Vert \lesssim 1$</span>. Let us fix a cube <span class="mathjax-tex">$P\in \mathcal {Q}, P \subset \widetilde{Q}_{0}$</span>. Thus, by the inequality (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ72">4.30</a>), it follows that for <span class="mathjax-tex">$q_1=\frac{p_1}{p_2}q_2$</span> we have</p><div id="Equ174" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\\&\quad = \sum _{j=\max \{j_P,0\}}^\infty 2^{-j \frac{d}{p_2}q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P}2^{j(s_1+\frac{d}{2})p_2} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\\&\quad \le \sum _{j=\max \{j_P,0\}}^\infty 2^{-j\frac{d}{p_1} q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P}2^{j(s_1+\frac{d}{2})p_1} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\\&\quad = \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\\&\quad \le \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert ^{q_1}\, |P|^{\tau _1 q_1}. \end{aligned}$$</span></div></div><p>In consequence for any cube <span class="mathjax-tex">$P\subset \widetilde{Q}_{0}$</span> we have</p><div id="Equ175" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{ 1}{|P|^{\tau _2}}&\left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\right) ^\frac{1}{q_2}\le |P|^{\frac{\tau _1 q_1}{q_2}-\tau _2 } \le 1\\ \end{aligned}$$</span></div></div><p>since <span class="mathjax-tex">$\tau _1 q_1 -\tau _2q_2 = 0$</span>. So by monotonicity if <span class="mathjax-tex">$\tau _2=\frac{1}{p_2}$</span>, then for any <span class="mathjax-tex">$q_1$</span> and <span class="mathjax-tex">$q_2$</span> such that <span class="mathjax-tex">$q_1 \le \frac{p_1}{p_2}q_2$</span>,</p><div id="Equ176" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Big \Vert t \mid {\widetilde{b^{s_2,\frac{1}{p_2}}_{p_2,q_2}}}\Big \Vert \le \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert . \end{aligned}$$</span></div></div><p>If <span class="mathjax-tex">$\gamma =0$</span> and <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span>, then <span class="mathjax-tex">$\tau _2\le \frac{1}{p_2}$</span>. If <span class="mathjax-tex">$\tau _2<\frac{1}{p_2}$</span>, then it follows from Substep 1.2 that</p><div id="Equ177" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid {\widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}}\Vert \le C\, \Big \Vert t \mid {\widetilde{ b^{s_2,\frac{1}{p_2}}_{p_2,q_2}}}\Big \Vert , \end{aligned}$$</span></div></div><p>so the final statement follows from the last two inequalities.</p> <p><i>Substep 1.4.</i> Now let <span class="mathjax-tex">$\gamma = \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\tau _1\frac{p_1}{p_2} > 0$</span> and <span class="mathjax-tex">$0\le \tau _1\le \frac{1}{p_1}$</span>. Please note that this assumption implies <span class="mathjax-tex">$\tau _1\frac{p_1}{p_2} \ge \tau _2$</span>, <span class="mathjax-tex">$p_1\tau _1<1$</span> and <span class="mathjax-tex">$p_1<p_2$</span>.</p> <p>The case <span class="mathjax-tex">$\tau _1\frac{p_1}{p_2}= \tau _2$</span> is covered by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ64">4.22</a>) since <span class="mathjax-tex">$\tau _2=\tau _1\frac{p_1}{p_2} < \tau _1$</span>. Let <span class="mathjax-tex">$\gamma >0$</span> and <span class="mathjax-tex">$\tau _2 < \frac{p_1}{p_2}\tau _1$</span>. We take <span class="mathjax-tex">$\tau _0$</span> such that <span class="mathjax-tex">$\tau _0=\frac{p_1}{p_2}\tau _1$</span>. The above considerations show that</p><div id="Equ178" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid {\widetilde{b^{s_2,\tau _0}_{p_2,q_2}}}\Vert \le C \, \Vert t \mid {\widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}}\Vert \end{aligned}$$</span></div></div><p>if <span class="mathjax-tex">$q_1\le \frac{p_1}{p_2}q_2$</span>. Since <span class="mathjax-tex">$\tau _2< \tau _0$</span> it follows from Substep 1.2 that</p><div id="Equ179" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid {\widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}}\Vert \le C \,\Vert t \mid {\widetilde{b^{s_2,\tau _0}_{p_2,q_2}}}\Vert . \end{aligned}$$</span></div></div><p><i>Step 2.</i> Now we come to the necessity. We do some preparation first.</p> <p>By the diffeomorphic properties of Besov-type spaces, using translations and dilations if necessary we can assume that the domain <span class="mathjax-tex">$\Omega $</span> satisfies the following conditions: there exists some number <span class="mathjax-tex">$\nu _0\in {{\mathbb {Z}}}$</span> such that</p><ul class="u-list-style-bullet"> <li> <p><span class="mathjax-tex">$Q_{\nu _0,0} \subset \Omega $</span>,</p> </li> <li> <p>if <span class="mathjax-tex">$Q_{j,m} \subset Q_{\nu _0,0}, \quad j\ge 0, \quad $</span> then <span class="mathjax-tex">$\quad {\mathop {\textrm{supp}}\nolimits }\psi _{i,j,m} \subset \Omega $</span>,</p> </li> <li> <p>if <span class="mathjax-tex">$Q_{0,m} \subset Q_{\nu _0,0}, \quad $</span> then <span class="mathjax-tex">$\quad {\mathop {\textrm{supp}}\nolimits }\phi _{0,m} \subset \Omega $</span>.</p> </li> </ul><p>Due to the isomorphism resulting from the wavelet decomposition between function and sequence spaces, and similar to the explanation given in Substep 2.1 of the proof of [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 11" title="Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. 218, 119–144 (2013)" href="/article/10.1007/s10231-023-01327-w#ref-CR11" id="ref-link-section-d74823138e69538">11</a>, Theorem 3.1], one can equivalently prove the necessary conditions for the embedding</p><div id="Equ180" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}(Q_{\nu _0,0}) \hookrightarrow \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}(Q_{\nu _0,0}), \end{aligned}$$</span></div></div><p>with <span class="mathjax-tex">$\nu _0<0$</span>. For convenience, let us denote <span class="mathjax-tex">$\widetilde{Q}=Q_{\nu _0,0}$</span>.</p> <p><i>Substep 2.1.</i> We show that <span class="mathjax-tex">$q_1\le q_2$</span> is necessary when <span class="mathjax-tex">$s_1=s_2$</span>. We assume <span class="mathjax-tex">$q_1 >q_2$</span>. Then we can choose a sequence of positive numbers <span class="mathjax-tex">$\{\gamma _j\}_{j\in {{\mathbb {N}}}_0} \in \ell _{q_1}({{\mathbb {N}}}_0){\setminus }\ell _{q_2}({{\mathbb {N}}}_0)$</span>. Let us define the sequence <span class="mathjax-tex">$t=\{t_{i,j,m}\}_{i,j,m}$</span>, <span class="mathjax-tex">$i=1,...,2^d-1, j\in {{\mathbb {N}}}_0, m \in {{{\mathbb {Z}}}^d},$</span> by</p><div id="Equ73" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} t_{i,j,m} {:=} {\left\{ \begin{array}{ll} 2^{-j(s_1 + \frac{d}{2})} \gamma _j \quad &{}\text{ if }\quad i=1 \quad \text{ and } \quad Q_{j,m}\subset [0,1)^d,\\ 0 &{}\text{ otherwise }. \end{array}\right. } \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.31) </div></div><p>Then,</p><div id="Equ181" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert = \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\right) ^\frac{1}{q_1}\\&\quad = \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2})q_1} \sum _{i=1}^{2^d-1} \left[ \int _P \left( \sum _{m \in {{{\mathbb {Z}}}^d}} |t_{i,j,m}|\chi _{j,m}(x)\right) ^{p_1}\;\textrm{d}x \right] ^{\frac{ q_1}{p_1}}\right) ^\frac{1}{q_1}\\&\quad = \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2})q_1} 2^{-j(s_1 + \frac{d}{2})q_1} |\gamma _j|^{q_1} |P\cap [0,1)^{d}|^{\frac{q_1}{p_1}}\right) ^\frac{1}{q_1}\\&\quad = \Vert \{\gamma _j\}_{{j\in {{\mathbb {N}}}_0}} \mid \ell _{q_1}\Vert < \infty , \end{aligned}$$</span></div></div><p>where the last equality holds because <span class="mathjax-tex">$\tau _1 \le \frac{1}{p_1}$</span>. On the other hand, we obtain similarly that</p><div id="Equ182" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert&= \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}}\frac{ 1}{|P|^{\tau _2}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_2}\right] ^{\frac{ q_2}{p_2}}\right) ^\frac{1}{q_2}\\&=\Vert \{\gamma _j\}_{{j\in {{\mathbb {N}}}_0}} \mid \ell _{q_2}\Vert = \infty , \end{aligned}$$</span></div></div><p>which contradicts the embedding.</p> <p><i>Substep 2.2.</i> Now we show that the condition <span class="mathjax-tex">$q_1\le q_2$</span> is also necessary when <span class="mathjax-tex">$\frac{s_1-s_2}{d}= \frac{1}{p_1}-\tau _1 - \frac{1}{p_2}+ \frac{p_1}{p_2}\tau _1>0$</span>. Let us assume <span class="mathjax-tex">$q_1>q_2$</span>. We adapt the counter-example used in Substep 2.4 of the proof [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 10" title="Haroske, D.. D., Skrzypczak, L.: Continuous embeddings of Besov-Morrey function spaces. Acta Math. Sin. (Engl. Ser.) 28, 1307–1328 (2012)" href="/article/10.1007/s10231-023-01327-w#ref-CR10" id="ref-link-section-d74823138e72194">10</a>, Theorem 3.2]. For any <span class="mathjax-tex">$0>\nu \ge \nu _0$</span>, we put</p><div id="Equ183" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} k_\nu {:=} \lfloor 2^{d|\nu |p_1 \tau _1}\rfloor , \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\lfloor x \rfloor =\max \{l \in {{\mathbb {Z}}}: l\le x\}$</span>. Then <span class="mathjax-tex">$1\le k_\nu <2^{d|\nu |}$</span> and</p><div id="Equ74" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} k_\nu \le c_{p_1, \tau _1} \, 2^{d(\mu - \nu )}\,k_{\mu }, \quad \text{ if }\quad \nu \le \mu <0. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.32) </div></div><p>For convenience, we assume <span class="mathjax-tex">$c_{p_1, \tau _1}=1$</span> (otherwise the proper modifications have to be done). As there, we define a sequence <span class="mathjax-tex">$t^{(\nu )}= \{t_{i,j,m}^{(\nu )}\}_{i,j,m}$</span>, <span class="mathjax-tex">$i=1,...,2^d-1, j\in {{\mathbb {N}}}_0, m \in {{{\mathbb {Z}}}^d},$</span> in the following way: we assume that <span class="mathjax-tex">$k_\nu $</span> elements of the sequence equal 1 and the rest equals 0. If <span class="mathjax-tex">$j\ne 0$</span>, <span class="mathjax-tex">$i\ne 1$</span> or <span class="mathjax-tex">$Q_{0,m} \nsubseteq Q_{\nu ,0}$</span>, then <span class="mathjax-tex">$t_{i,j,m}^{(\nu )}=0$</span>. Because of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ74">4.32</a>), we can choose the elements that equal 1 in such a way that the following property holds:</p><div id="Equ184" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text{ if } \quad Q_{\mu , l}\subseteq Q_{\nu ,0} \quad \text{ and } \quad Q_{\mu ,l}= \bigcup _{i=1}^{2^{-d \mu }} Q_{0,m_i}, \quad \text{ then } \text{ at } \text{ most } k_\mu \text{ elements } t_{1,0,m_i}^{(\nu )} \text{ equal } 1. \end{aligned}$$</span></div></div><p>Now we define a new sequence <span class="mathjax-tex">$t=\{t_{i,j,m}\}_{i,j,m} \in \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}$</span> by</p><div id="Equ185" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} t_{i,j,m}= \gamma _j \, t_{i,0,m}^{(\nu )}, \quad \text{ if }\quad j=\nu -\nu _0 \quad \text{ and } \quad Q_{0,m}\subset Q_{\nu ,0}, \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$\{\gamma _j\}_{j \in {{\mathbb {N}}}_0}$</span> is a sequence of positive numbers with <span class="mathjax-tex">$\{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+\tau _1)}\, \gamma _j\}_{j \in {{\mathbb {N}}}_0} \in \ell _{q_1}({{\mathbb {N}}}_0){\setminus }\ell _{q_2}({{\mathbb {N}}}_0)$</span>. If <span class="mathjax-tex">$Q_{\mu ,l}\subset Q_{\nu _0, 0}$</span>, then for fixed <span class="mathjax-tex">$j \ge \mu $</span>, there are at most <span class="mathjax-tex">$k_{\mu -l}$</span> non-zero elements <span class="mathjax-tex">$t_{i,j,m}$</span> such that <span class="mathjax-tex">$Q_{j,m} \subset Q_{\mu ,l}$</span>. Thus</p><div id="Equ186" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \sum _{m: Q_{j,m} \subset Q_{\mu ,l}} |t_{i,j,m}|^{p_1} \le \gamma _j^{p_1}\, 2^{d(j-\mu )\tau _1 p_1} \end{aligned}$$</span></div></div><p>and the last sum is <span class="mathjax-tex">$k_{\mu -j}\gamma _j^{p_1}$</span> if <span class="mathjax-tex">$\mu =\nu _0$</span>. Therefore,</p><div id="Equ75" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert&= \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}} \frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1} \sum _{i=1}^{2^d-1} \left[ \sum _{m:\ Q_{j,m}\subset P} |t_{i,j,m}|^{p_1}\right] ^{\frac{ q_1}{p_1}}\right) ^\frac{1}{q_1}\nonumber \\&\le \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}} \frac{ 1}{|P|^{\tau _1}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1})q_1}\, \gamma _j^{q_1}\, 2^{d(j-j_P)\tau _1q_1} \right) ^\frac{1}{q_1}\nonumber \\&= \sup _{P\in \mathcal {Q};\; P\subset \widetilde{Q}} \left( \sum _{j=\max \{j_P,0\}}^\infty 2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+d\tau _1)q_1} \,\gamma _j^{q_1} \right) ^\frac{1}{q_1}\nonumber \\&= \Vert \{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+d\tau _1)}\, \gamma _j\}_{{j\in {{\mathbb {N}}}_0}} \mid \ell _{q_1}\Vert <\infty . \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.33) </div></div><p>Similarly we obtain</p><div id="Equ187" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( \sum _{m:\ Q_{j,m}\subset Q_{\mu ,l}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} \le \left( \gamma _j^{p_2}\, k_{\mu -j}\right) ^{\frac{q_2}{p_2}}\le \gamma _j^{q_2}\, 2^{d(j-\mu )\tau _1 p_1\frac{q_2}{p_2}}, \end{aligned}$$</span></div></div><p>and when <span class="mathjax-tex">$\mu =\nu _0$</span></p><div id="Equ188" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} =\left( \gamma _j^{p_2}\, k_{\nu _0-j}\right) ^{\frac{q_2}{p_2}}\ge C\, \gamma _j^{q_2} \,2^{d(j-\nu _0)\tau _1 p_1\frac{q_2}{p_2}}, \end{aligned}$$</span></div></div><p>for some constant <i>C</i> independent of <span class="mathjax-tex">$\gamma $</span>. Then,</p><div id="Equ189" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} C\, \gamma _j^{q_2}\, 2^{\nu _0 d \frac{p_1 \tau _1}{p_2}q_2} \le \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} 2^{-j d\frac{p_1 \tau _1}{p_2}q_2}, \end{aligned}$$</span></div></div><p>which yields</p><div id="Equ190" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\Vert \{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+d\tau _1)}\, \gamma _j\}_{j \in {{\mathbb {N}}}_0} \mid \ell _{q_2}\Vert \\&\quad \le C \left\{ \sum _{j=0}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \, 2^{\nu _0 d\frac{p_1 \tau _1}{p_2}q_2} \sum _{i=1}^{2^d-1} \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} \right\} ^\frac{1}{q_2}\\&\quad \sim 2^{\nu _0 d\frac{p_1 \tau _1}{p_2}} \left\{ \sum _{j=\max \{0,j_P\}}^\infty 2^{j(s_2+\frac{d}{2}-\frac{d}{p_2})q_2} \sum _{i=1}^{2^d-1} \left( \sum _{m:\ Q_{j,m}\subset Q_{\nu _0,0}} |t_{i,j,m}|^{p_2} \right) ^{\frac{q_2}{p_2}} \right\} ^\frac{1}{q_2}\\&\quad \le \Vert t \mid \widetilde{{b}_{p_2,q_2}^{s_2,\tau _2}}\Vert \lesssim \Vert t \mid \widetilde{{b}_{p_1,q_1}^{s_1,\tau _1}}\Vert <\infty , \end{aligned}$$</span></div></div><p>using (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ75">4.33</a>) in the last step and [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 47" title="Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)" href="/article/10.1007/s10231-023-01327-w#ref-CR47" id="ref-link-section-d74823138e76852">47</a>, Lemma 3.3] in the second, since <span class="mathjax-tex">$\tau _2<\frac{1}{p_2}$</span> here. This contradicts our assumption on the sequence <span class="mathjax-tex">$\{2^{j(s_1+\frac{d}{2}-\frac{d}{p_1}+\tau _1)}\, \gamma _j\}_{j \in {{\mathbb {N}}}_0}$</span>, and completes the proof in this case. <span class="mathjax-tex">$\square $</span></p> <p>Before we turn our interest to Triebel-Lizorkin-type spaces, we shall discuss some special case and compare it with the classical result as recalled in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar15">2.15</a>. We concentrate on the limiting case (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ53">4.11</a>) under the assumptions (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ52">4.10</a>) again. Let us assume now <span class="mathjax-tex">$\tau _1=\tau _2=:\tau $</span>, i.e., <span class="mathjax-tex">$0<\tau \le \min \{\frac{1}{p_1},\frac{1}{p_2}\}$</span> with <span class="mathjax-tex">$q_i<\infty $</span> if <span class="mathjax-tex">$\tau =\frac{1}{p_i}$</span>, <span class="mathjax-tex">$i=1,2$</span>. In that case we find that (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ53">4.11</a>) reads as <span class="mathjax-tex">$s_1-s_2 = d \max \{0, \frac{1}{p_1}-\frac{1}{p_2}\}$</span>. Then Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> implies the following.</p> <h3 class="c-article__sub-heading" id="FPar44">Corollary 4.11</h3> <p>Let <span class="mathjax-tex">$0< p_i\le \infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0< \tau \le \min \{\frac{1}{p_1}, \frac{1}{p_2}\}$</span>, with <span class="mathjax-tex">$q_i<\infty $</span> if <span class="mathjax-tex">$\tau =\frac{1}{p_i}=\min \{\frac{1}{p_1},\frac{1}{p_2}\}$</span>, <span class="mathjax-tex">$i=1,2$</span>. Assume that</p><div id="Equ191" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau ,\tau ,p_1,p_2) =\max \left\{ 0, \frac{1}{p_1}-\frac{1}{p_2}\right\} . \end{aligned}$$</span></div></div><p>Then the embedding</p><div id="Equ76" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: B^{s_1,\tau }_{p_1,q_1}(\Omega )\hookrightarrow B^{s_2,\tau }_{p_2,q_2}(\Omega ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.34) </div></div><p>is continuous if, and only if, either <span class="mathjax-tex">$p_1<p_2$</span>, or <span class="mathjax-tex">$p_1\ge p_2$</span> with <span class="mathjax-tex">$ q_1\le q_2$</span>.</p> <h3 class="c-article__sub-heading" id="FPar45">Proof</h3> <p>The sufficiency follows from (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ55">4.13</a>) in case of <span class="mathjax-tex">$p_1<p_2$</span>, and from (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ57">4.15</a>) for <span class="mathjax-tex">$p_1\ge p_2$</span>. Note that the case (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ56">4.14</a>) is not applicable in this situation. The necessity is implied by (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ58">4.16</a>) in case of <span class="mathjax-tex">$p_1\ge p_2$</span>, and the last statement in (ii) if <span class="mathjax-tex">$p_1<p_2$</span>. Again, (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ59">4.17</a>) is not possible in this context. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar46">Remark 4.12</h3> <p>Let us explicitly comment on the difference between the above result for <span class="mathjax-tex">$\tau >0$</span> and the classical one for <span class="mathjax-tex">$\tau =0$</span> as recalled in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar15">2.15</a>. Only in case of embeddings of spaces with the same smoothness <span class="mathjax-tex">$s_1=s_2$</span> (and thus <span class="mathjax-tex">$p_1\ge p_2$</span>) we have an influence of the fine parameters <span class="mathjax-tex">$q_i$</span>, that is, <span class="mathjax-tex">$q_1\le q_2$</span>. This is parallel to the classical case <span class="mathjax-tex">$\tau =0$</span> and could thus be expected. However, what is far more surprising, is the outcome for <span class="mathjax-tex">$s_1>s_2$</span> and <span class="mathjax-tex">$p_1<p_2$</span>: in contrast to the classical setting for <span class="mathjax-tex">$\tau =0$</span> we do not have any <i>q</i>-dependence here as long as <span class="mathjax-tex">$\tau >0$</span> (and small enough, such that we are still in the new Morrey-type situation, unlike in Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a>). Again this explains the special rôle of the hybrid parameter <span class="mathjax-tex">$\tau $</span> which influences both smoothness and integrability.</p> <h3 class="c-article__sub-heading" id="FPar47">Remark 4.13</h3> <p>Note that Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar28">3.9</a> can be obtained also as an immediate consequence of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> and the Propositions <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a>.</p> <p> We collect now the counterpart of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> for the Triebel-Lizorkin-type spaces. When <span class="mathjax-tex">$\tau _i<\frac{1}{p_i}$</span>, <span class="mathjax-tex">$i=1,2$</span>, the result follows immediately from [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 12" title="Haroske, D.D., Skrzypczak, L.: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces. Rev. Mat. Complut. 27, 541–573 (2014)" href="/article/10.1007/s10231-023-01327-w#ref-CR12" id="ref-link-section-d74823138e78617">12</a>, Theorem 5.2] and the coincidence of <span class="mathjax-tex">${F}_{p,q}^{s,\tau }$</span> and <span class="mathjax-tex">${{\mathcal {E}}}^{s}_{u,p,q}$</span> spaces if <span class="mathjax-tex">$\tau =\frac{1}{p}-\frac{1}{u}$</span>, and it reads as follows.</p> <h3 class="c-article__sub-heading" id="FPar48">Corollary 4.14</h3> <p>Let <span class="mathjax-tex">$0< p_1,p_2<\infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0\le \tau _i< \frac{1}{p_i}$</span>, <span class="mathjax-tex">$i=1,2$</span>. Assume that</p><div id="Equ192" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)=\max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$</span></div></div><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>The embedding </p><div id="Equ77" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: {F}_{p_1,q_1}^{s_1,\tau _1}(\Omega )\hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}(\Omega ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.35) </div></div><p> is continuous if one of the following conditions holds: </p><div id="Equ78" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\frac{1}{p_1}-\frac{1}{p_2}> \tau _1-\tau _2 \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.36) </div></div><div id="Equ79" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {or}\qquad&\frac{1}{p_1}-\frac{1}{p_2}\le \tau _1-\tau _2\quad \text {and}\quad q_1\le \min \bigg \{1,\frac{p_1}{p_2}\bigg \}q_2. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.37) </div></div> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>If there is a continuous embedding <span class="mathjax-tex">$\text {id}_{\tau }$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ77">4.35</a>), then the parameters satisfy the condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ78">4.36</a>), or (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ79">4.37</a>) holds with <span class="mathjax-tex">$q_1\le q_2$</span>.</p> </dd></dl> <p>We return to the situation <span class="mathjax-tex">$\tau _1=\tau _2>0$</span> studied in Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar44">4.11</a>, but now in case of <i>F</i>-spaces.</p> <h3 class="c-article__sub-heading" id="FPar49">Corollary 4.15</h3> <p>Let <span class="mathjax-tex">$0< p_i<\infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$i=1,2$</span>, and <span class="mathjax-tex">$0\le \tau < \min \{\frac{1}{p_1}, \frac{1}{p_2}\}$</span>. Assume that</p><div id="Equ193" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau ,\tau ,p_1,p_2) =\max \left\{ 0, \frac{1}{p_1}-\frac{1}{p_2}\right\} . \end{aligned}$$</span></div></div><p>Then the embedding</p><div id="Equ80" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }: F^{s_1,\tau }_{p_1,q_1}(\Omega )\hookrightarrow F^{s_2,\tau }_{p_2,q_2}(\Omega ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.38) </div></div><p>is continuous if, and only if, either <span class="mathjax-tex">$p_1<p_2$</span>, or <span class="mathjax-tex">$p_1\ge p_2$</span> with <span class="mathjax-tex">$ q_1\le q_2$</span>.</p> <h3 class="c-article__sub-heading" id="FPar50">Proof</h3> <p>This is an immediate consequence of Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar48">4.14</a> when <span class="mathjax-tex">$\tau >0$</span> and of the classical situation when <span class="mathjax-tex">$\tau =0$</span>, cf. Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar46">4.12</a>. <span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar51">Remark 4.16</h3> <p>In contrast to Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar46">4.12</a> concerning Besov-type spaces, we thus obtain the natural counterpart of the well-known classical situation (<span class="mathjax-tex">$\tau =0$</span>) to the situation <span class="mathjax-tex">$\tau >0$</span>, recall Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar15">2.15</a>.</p> <p>We study now some more possible situations regarding Triebel-Lizorkin-type spaces.</p> <h3 class="c-article__sub-heading" id="FPar52">Corollary 4.17</h3> <p>Let <span class="mathjax-tex">$0< p_1,p_2<\infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0\le \tau _i\le \frac{1}{p_i}$</span>, with <span class="mathjax-tex">$q_i<\infty $</span> if <span class="mathjax-tex">$\tau _i=\frac{1}{p_i}$</span> <span class="mathjax-tex">$i=1,2$</span>. Assume that</p><div id="Equ194" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\tau _2,p_1,p_2)=\max \left\{ 0, \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+\max \left\{ \tau _2,\frac{p_1}{p_2}\tau _1\right\} \right\} . \end{aligned}$$</span></div></div><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>The embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ77">4.35</a>) is continuous if one of the following conditions holds: </p><div id="Equ81" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned}&\tau _1=\frac{1}{p_1},\quad {\tau _2 \le \frac{1}{p_2}}\quad \text {and}\quad q_1\le q_2, \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.39) </div></div><div id="Equ82" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {or}\qquad&\tau _1<\frac{1}{p_1}\quad \text {and}\quad \tau _2=\frac{1}{p_2}. \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.40) </div></div> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>If the embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ77">4.35</a>) is continuous and <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2=\frac{1}{p_2}$</span>, then <span class="mathjax-tex">$q_1\le q_2$</span>. Moreover, if <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2<\frac{1}{p_2}$</span>, then the continuity of the embedding (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ77">4.35</a>) implies <span class="mathjax-tex">$q_1 \le \max \{p_2,q_2\}$</span>.</p> </dd></dl> <h3 class="c-article__sub-heading" id="FPar53">Proof</h3> <p>The case <span class="mathjax-tex">$\tau _1 = \frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2 = \frac{1}{p_2}$</span> can be reduced to Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> due to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ17">2.17</a>). So we are left with the cases <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2<\frac{1}{p_2}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ81">4.39</a>), and <span class="mathjax-tex">$\tau _1<\frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2=\frac{1}{p_2}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ82">4.40</a>). In both cases we can use the coincidence (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ17">2.17</a>). To prove that the condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ82">4.40</a>) is sufficient we first take a sufficiently small number <span class="mathjax-tex">$q_3$</span> such that <span class="mathjax-tex">$\tau _1<\frac{1}{q_3}$</span>. We consider the following factorisation</p><div id="Equ83" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F^{s_1,\tau _1}_{p_1,q_1}(\Omega )\hookrightarrow B^{s_1,\tau _1}_{p_1,\infty }(\Omega ) \hookrightarrow B^{s_2,1/q_3}_{q_3,q_3}(\Omega ) = F^{s_2,1/p_2}_{p_2,q_3}(\Omega ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.41) </div></div><p>since</p><div id="Equ195" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d} = \gamma (\tau _1,\frac{1}{p_2}, p_1,p_2)= \frac{1}{p_1}-\tau _1= \gamma (\tau _1,\frac{1}{q_3}, p_1,q_3) > 0. \end{aligned}$$</span></div></div><p>Please note that the continuity of the second embedding follows from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> since the condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ55">4.13</a>) is satisfied. By elementary embeddings the statement holds for any <span class="mathjax-tex">$q_2\ge q_3$</span>.</p> <p>Regarding the case <span class="mathjax-tex">$\tau _1 = \frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2< \frac{1}{p_2}$</span>, we have now <span class="mathjax-tex">$\gamma (\tau _1,\tau _2,p_1,p_2)=0$</span>, so <span class="mathjax-tex">$s_1=s_2$</span>. Then</p><div id="Equ84" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} F^{s_1,1/p_1}_{p_1,q_1}(\Omega ) = B^{s_1,1/q_1}_{q_1,q_1}(\Omega )\hookrightarrow B^{s_2,\frac{1}{q_2}}_{q_2,q_2}(\Omega )= F^{s_2,\frac{1}{p_2}}_{p_2,q_2}(\Omega ) \hookrightarrow F^{s_2,\tau _2}_{p_2,q_2}(\Omega ), \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.42) </div></div><p>where we made use of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar28">3.9</a> in the last embedding. Moreover,</p><div id="Equ196" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \gamma \left( \frac{1}{q_1}, \frac{1}{q_2}, q_1,q_2\right) = 0, \end{aligned}$$</span></div></div><p>so the statement follows once more from (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ17">2.17</a>) and Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a>.</p> <p>Now we come to the necessity part of our result. The necessity of the condition <span class="mathjax-tex">$q_1\le q_2$</span> in the case of <span class="mathjax-tex">$\tau _1=\frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2=\frac{1}{p_2}$</span> follows from the second part of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a>. So we are left with the case <span class="mathjax-tex">$\tau _1 = \frac{1}{p_1}$</span> and <span class="mathjax-tex">$\tau _2< \frac{1}{p_2}$</span>. Here we can use the following factorisation</p><div id="Equ85" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} B^{s_1,1/q_1}_{q_1,q_1}(\Omega )= F^{s_1,1/p_1}_{p_1,q_1}(\Omega ) \hookrightarrow F^{s_2,\tau _2}_{p_2,q_2}(\Omega ) \hookrightarrow B^{s_2,\tau _2}_{p_2,\max \{p_2,q_2\}}(\Omega ). \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.43) </div></div><p>Now Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> implies <span class="mathjax-tex">$q_1\le \max \{p_2,q_2\}$</span> since</p><div id="Equ197" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \gamma \left( \frac{1}{p_1}, \tau _2, p_1,p_2\right) = \gamma \left( \frac{1}{q_1},\tau _2, q_1,p_2\right) =0. \end{aligned}$$</span></div></div><p><span class="mathjax-tex">$\square $</span></p> <h3 class="c-article__sub-heading" id="FPar54">Remark 4.18</h3> <p>We return to the special situation of (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ49">4.7</a>), i.e., to <span class="mathjax-tex">$\textrm{bmo}(\Omega )$</span> as a source or target space of <span class="mathjax-tex">$\text {id}_\tau $</span>. In continuation of Remarks <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar36">4.5</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar40">4.8</a> we now concentrate on the situation covered by Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> and Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar52">4.17</a>, that is, we assume in this remark that <span class="mathjax-tex">$\tau \le \frac{1}{p}$</span> with <span class="mathjax-tex">$q<\infty $</span> if <span class="mathjax-tex">$\tau =\frac{1}{p}$</span> for the target space <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ86">4.44</a>) and the source space <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ87">4.45</a>). First we deal with</p><div id="Equ86" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_\tau : \textrm{bmo}(\Omega )\hookrightarrow {A}_{p,q}^{s,\tau }(\Omega ) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.44) </div></div><p>such that the limiting situation reads as <span class="mathjax-tex">$s = 0$</span> in that case. Then Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> and Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar52">4.17</a> imply that <span class="mathjax-tex">$\text {id}_\tau $</span> is continuous if</p><div id="Equ198" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} s=0, \quad \tau \le \frac{1}{p}, \quad \text {and}\quad {\left\{ \begin{array}{ll} q\ge \max \{p,2\}, &{} A=B, \\ q\ge 2, &{} A=F. \end{array}\right. } \end{aligned}$$</span></div></div><p>Regarding the necessity, when <span class="mathjax-tex">$A=B$</span>, the condition <span class="mathjax-tex">$q\ge 2$</span> is necessary for the continuity of <span class="mathjax-tex">$\text {id}_\tau $</span>. In addition, when <span class="mathjax-tex">$A=F$</span>, then <span class="mathjax-tex">$q\ge 2$</span> is also necessary when <span class="mathjax-tex">$\tau =\frac{1}{p}$</span>, while for <span class="mathjax-tex">$\tau <\frac{1}{p}$</span> the condition <span class="mathjax-tex">$\max \{p,q\}\ge 2$</span> is necessary. Note that one can also use (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ49">4.7</a>) and Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar28">3.9</a> for the sufficiency argument. In the second setting,</p><div id="Equ87" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_\tau : {A}_{p,q}^{s,\tau }(\Omega )\hookrightarrow \textrm{bmo}(\Omega ), \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.45) </div></div><p>the limiting case means <span class="mathjax-tex">$s=d(\frac{1}{p}-\tau )$</span>. Then for the continuity of <span class="mathjax-tex">$\text {id}_\tau $</span> it is sufficient that</p><div id="Equ199" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {either}\quad \tau <\frac{1}{p}, \qquad \text {or}\qquad \tau =\frac{1}{p}\quad \text {and}\quad {\left\{ \begin{array}{ll} q\le \min \{p,2\}, &{} A=B, \\ q\le 2, &{} A=F, \end{array}\right. } \end{aligned}$$</span></div></div><p>where for <span class="mathjax-tex">$\tau =\frac{1}{p}$</span> the condition <span class="mathjax-tex">$q\le 2$</span> is also necessary for the continuity of <span class="mathjax-tex">$\text {id}_\tau $</span>.</p> <h3 class="c-article__sub-heading" id="FPar55">Remark 4.19</h3> <p>Note that one can formulate counterparts of our above embedding results for spaces of type <span class="mathjax-tex">${A}_{p,q}^{s,\tau }(\Omega )$</span> in terms of the hybrid spaces <span class="mathjax-tex">$L^rA^s_{p,q}(\Omega )$</span> as introduced in Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar9">2.9</a> using the coincidence (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ19">2.19</a>). We define the spaces <span class="mathjax-tex">$L^rA^s_{p,q}(\Omega )$</span> by restriction, parallel to the approach in Definition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar12">2.12</a>. In [<a data-track="click" data-track-action="reference anchor" data-track-label="link" data-test="citation-ref" aria-label="Reference 8" title="Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. 34, 761–795 (2021)" href="/article/10.1007/s10231-023-01327-w#ref-CR8" id="ref-link-section-d74823138e84787">8</a>, Remark 3.4] we have explicated the compactness condition for the embedding</p><div id="Equ200" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_L: L^{r_1}{A}_{p_1,q_1}^{s_1}(\Omega ) \hookrightarrow L^{r_2}{A}_{p_2,q_2}^{s_2}(\Omega ), \end{aligned}$$</span></div></div><p>which in the special case <span class="mathjax-tex">$r_1=r_2=r$</span> reads as</p><div id="Equ201" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} s_1-s_2 > {\left\{ \begin{array}{ll} 0, &{} r\ge 0, \\ r\, \min \{\frac{p_1}{p_2}-1,0\}, &{} r<0,\end{array}\right. } \end{aligned}$$</span></div></div><p>where <span class="mathjax-tex">$0<p_i<\infty $</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$i=1,2$</span>, and <span class="mathjax-tex">$-d \min \{\frac{1}{p_1}, \frac{1}{p_2}\} \le r<\infty $</span> is always assumed. For convenience we only discuss this special setting <span class="mathjax-tex">$r_1=r_2=r$</span> and <span class="mathjax-tex">$A=B$</span> here, but the other cases can be done in a parallel way.</p> <p>So the limiting case for the continuity of the embedding is just</p><div id="Equ88" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} s_1-s_2 = {\left\{ \begin{array}{ll} 0, &{} r\ge 0, \\ r\, \min \{\frac{p_1}{p_2}-1,0\}, &{} r<0.\end{array}\right. } \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.46) </div></div> <h3 class="c-article__sub-heading" id="FPar56">Corollary 4.20</h3> <p>Let <span class="mathjax-tex">$0< p_i<\infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$i=1,2$</span>, and <span class="mathjax-tex">$-d \min \left\{ \frac{1}{p_1}, \frac{1}{p_2}\right\}< r< \infty $</span>. Let</p><div id="Equ89" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_L: L^r{B}_{p_1,q_1}^{s_1}(\Omega )\hookrightarrow L^r{B}_{p_2,q_2}^{s_2}(\Omega ). \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.47) </div></div><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(i)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>If <span class="mathjax-tex">$r>0$</span>, then <span class="mathjax-tex">$\text {id}_L$</span> is continuous if, and only if, <span class="mathjax-tex">$s_1\ge s_2$</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$r=0$</span>.</p> </dd></dl><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii</b> <span class="mathjax-tex">$_a$</span>):</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>If <span class="mathjax-tex">$q_2=\infty $</span>, then <span class="mathjax-tex">$\text {id}_L$</span> is continuous if, and only if, <span class="mathjax-tex">$s_1\ge s_2$</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii</b> <span class="mathjax-tex">$_b$</span>):</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>If <span class="mathjax-tex">$q_2<\infty $</span> and <span class="mathjax-tex">$q_1=\infty $</span>, then <span class="mathjax-tex">$\text {id}_L$</span> is continuous if, and only if, <span class="mathjax-tex">$s_1>s_2$</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(ii</b> <span class="mathjax-tex">$_c$</span>):</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>If <span class="mathjax-tex">$q_i<\infty $</span>, <span class="mathjax-tex">$i=1,2$</span>, then <span class="mathjax-tex">$\text {id}_L$</span> is continuous if <span class="mathjax-tex">$s_1>s_2$</span> or <span class="mathjax-tex">$s_1=s_2$</span> and <span class="mathjax-tex">$q_1\le \min \{1, \frac{p_1}{p_2}\}q_2$</span>. Conversely, if <span class="mathjax-tex">$\text {id}_L$</span> is continuous, then either <span class="mathjax-tex">$s_1>s_2$</span> or <span class="mathjax-tex">$s_1=s_2$</span> and <span class="mathjax-tex">$q_1\le q_2$</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(iii)</b>:</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Let <span class="mathjax-tex">$r<0$</span>.</p> </dd></dl><dl class="c-abbreviation_list"><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(iii</b> <span class="mathjax-tex">$_a$</span>):</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Assume that <span class="mathjax-tex">$p_1\ge p_2$</span>. Then <span class="mathjax-tex">$\text {id}_L$</span> is continuous if, and only if, <span class="mathjax-tex">$s_1>s_2$</span> or <span class="mathjax-tex">$s_1=s_2$</span> and <span class="mathjax-tex">$q_1\le q_2$</span>.</p> </dd><dt class="c-abbreviation_list__term u-text-bold u-float-left u-pr-16" style="min-width:50px;"><dfn><b>(iii</b> <span class="mathjax-tex">$_b$</span>):</dfn></dt><dd class="c-abbreviation_list__description u-mb-24"> <p>Assume <span class="mathjax-tex">$p_1< p_2$</span>. Then <span class="mathjax-tex">$\text {id}_L$</span> is continuous if <span class="mathjax-tex">$s_1-s_2>r(\frac{p_1}{p_2}-1)$</span>, or <span class="mathjax-tex">$s_1-s_2=r(\frac{p_1}{p_2}-1)$</span> and <span class="mathjax-tex">$q_1\le \frac{p_1}{p_2} q_2$</span>. Conversely, the continuity of <span class="mathjax-tex">$\text {id}_L$</span> implies <span class="mathjax-tex">$s_1-s_2>r(\frac{p_1}{p_2}-1)$</span>, or <span class="mathjax-tex">$s_1-s_2=r(\frac{p_1}{p_2}-1)$</span> and <span class="mathjax-tex">$q_1\le q_2$</span>.</p> </dd></dl> <h3 class="c-article__sub-heading" id="FPar57">Proof</h3> <p>In view of Remark <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar55">4.19</a> we only need to consider the limiting case (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ88">4.46</a>), the rest is covered by Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar30">4.1</a> and the coincidence (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ19">2.19</a>), extended to spaces on domains. Then (i) and (ii<span class="mathjax-tex">$_a$</span>) are covered by Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> together with <span class="mathjax-tex">$r=d(\tau _i-\frac{1}{p_i})$</span>, <span class="mathjax-tex">$i=1,2$</span>. Likewise (ii<span class="mathjax-tex">$_b$</span>) is a consequence of Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar37">4.6</a> since there is no continuous embedding in the limiting case <span class="mathjax-tex">$s_1=s_2$</span>. Part (ii<span class="mathjax-tex">$_c$</span>) follows from Theorems <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar30">4.1</a>, as well as part (iii). <span class="mathjax-tex">$\square $</span></p> <p>We finish our paper by collecting some immediate extensions of Theorems <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> and <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar11">2.11</a> regarding the embeddings on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>. The first result improves part (b) of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> (iii), and it follows from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a> and Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a>.</p> <h3 class="c-article__sub-heading" id="FPar58">Corollary 4.21</h3> <p>Let <span class="mathjax-tex">$0< p_1,p_2\le \infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0\le \tau _i\le \frac{1}{p_i}$</span>, with <span class="mathjax-tex">$q_i<\infty $</span> if <span class="mathjax-tex">$\tau _i=\frac{1}{p_i}$</span>, <span class="mathjax-tex">$i=1,2$</span>. If</p><div id="Equ90" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }^{{{{\mathbb {R}}}^d}}: {B}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {B}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.48) </div></div><p>is continuous and <span class="mathjax-tex">$\frac{s_1-s_2}{d}=\frac{1}{p_1}-\tau _1-\frac{1}{p_2}+ \frac{p_1}{p_2}\tau _1>0$</span>, then</p><div id="Equ202" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} q_1\le q_2. \end{aligned}$$</span></div></div> <h3 class="c-article__sub-heading" id="FPar59">Proof</h3> <p>By Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a>, we have</p><div id="Equ203" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {B}_{p_1,q_1}^{s_1,\tau _1}(\Omega ) \xrightarrow {\textrm{ext}} {B}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \xrightarrow {\text {id}_\tau ^{{{{\mathbb {R}}}^d}}} {B}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \xrightarrow {\textrm{re}} {B}_{p_2,q_2}^{s_2,\tau _2}(\Omega ), \end{aligned}$$</span></div></div><p>i.e., <span class="mathjax-tex">$\text {id}_\tau ^\Omega = \textrm{re}\circ \text {id}_\tau ^{{{{\mathbb {R}}}^d}} \circ \textrm{ext}$</span>. Hence, the continuity of <span class="mathjax-tex">$\text {id}_\tau ^{{{{\mathbb {R}}}^d}}$</span> implies the continuity of <span class="mathjax-tex">$\text {id}_\tau ^\Omega $</span>, which, in turn, by Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a> implies <span class="mathjax-tex">$q_1\le q_2$</span>. <span class="mathjax-tex">$\square $</span></p> <p>Next we show that, in case of <span class="mathjax-tex">$\tau _1=\tau _2$</span>, the embedding <span class="mathjax-tex">$\text {id}_\tau ^{{{{\mathbb {R}}}^d}}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ90">4.48</a>) holds under weaker assumptions on the parameters than the ones stated in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar10">2.10</a> (iii)-(a). Namely, in this case, we do not need any condition on the parameters <span class="mathjax-tex">$q_1, q_2$</span>.</p> <h3 class="c-article__sub-heading" id="FPar60">Corollary 4.22</h3> <p>Let <span class="mathjax-tex">$0< p_1,p_2\le \infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$0\le \tau _i\le \frac{1}{p_i}$</span>, with <span class="mathjax-tex">$q_i<\infty $</span> if <span class="mathjax-tex">$\tau _i=\frac{1}{p_i}$</span>, <span class="mathjax-tex">$i=1,2$</span>. If</p><div id="Equ204" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \frac{s_1-s_2}{d}= \frac{1}{p_1}-\tau _1-\frac{1}{p_2}+ \tau _2>0 \quad \text{ and } \quad \tau _1=\tau _2, \end{aligned}$$</span></div></div><p>then the embedding <span class="mathjax-tex">$\text {id}_\tau ^{{{{\mathbb {R}}}^d}}$</span> in (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ90">4.48</a>) is continuous.</p> <h3 class="c-article__sub-heading" id="FPar61">Proof</h3> <p>This result can be proved in the same way as its counterpart for embeddings on domains, cf. Substep 1.2 of the proof of Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar41">4.9</a>. Therefore, we omit the argument here. <span class="mathjax-tex">$\square $</span></p> <p>Lastly, we turn to the Triebel-Lizorkin-type spaces and state a result which gives us sufficient and necessary conditions for the continuity of the embedding on <span class="mathjax-tex">${{{\mathbb {R}}}^d}$</span>, when <span class="mathjax-tex">$\tau _2$</span> is large and <span class="mathjax-tex">$\tau _1$</span> is small. Specifically, we assume that</p><div id="Equ91" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \tau _2 \ge \frac{1}{p_2}\,\,\,\, \text{ with } \,\,\,\, q_2=\infty \,\,\,\, \text{ if } \,\,\,\, \tau _2=\frac{1}{p_2} \qquad \text{ and } \qquad \tau _1\le \frac{1}{p_1} \,\,\,\, \text{ with } \,\,\,\, q_1<\infty \,\,\,\, \text{ if } \,\,\,\, \tau _1=\frac{1}{p_1},\nonumber \\ \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.49) </div></div><p>as this case that was not considered in Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar11">2.11</a> (i).</p> <h3 class="c-article__sub-heading" id="FPar62">Corollary 4.23</h3> <p>Let <span class="mathjax-tex">$0< p_1,p_2<\infty $</span>, <span class="mathjax-tex">$s_i\in {{\mathbb {R}}}$</span>, <span class="mathjax-tex">$0<q_i\le \infty $</span>, <span class="mathjax-tex">$\tau _i\ge 0$</span>, <span class="mathjax-tex">$i=1,2$</span>. Assume that condition (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ91">4.49</a>) holds. Then the embedding</p><div id="Equ92" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} \text {id}_{\tau }^{{{{\mathbb {R}}}^d}}: {F}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d}) \hookrightarrow {F}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}) \end{aligned}$$</span></div><div class="c-article-equation__number"> (4.50) </div></div><p>holds if, and only if, <span class="mathjax-tex">$\quad \displaystyle \frac{s_1-s_2}{d}\ge \frac{1}{p_1}- \tau _1 - \frac{1}{p_2}+\tau _2$</span>.</p> <h3 class="c-article__sub-heading" id="FPar63">Proof</h3> <p>The sufficiency part follows from the fact that</p><div id="Equ205" class="c-article-equation"><div class="c-article-equation__content"><span class="mathjax-tex">$$\begin{aligned} {F}_{p_1,q_1}^{s_1,\tau _1}({{{\mathbb {R}}}^d})\hookrightarrow B^{s_1+d(\tau _1-\frac{1}{p_1})}_{\infty , \infty }({{{\mathbb {R}}}^d}) \hookrightarrow B^{s_2+d(\tau _2-\frac{1}{p_2})}_{\infty , \infty }({{{\mathbb {R}}}^d}) = {F}_{p_2,q_2}^{s_2,\tau _2}({{{\mathbb {R}}}^d}), \end{aligned}$$</span></div></div><p>due to (<a data-track="click" data-track-label="link" data-track-action="equation anchor" href="/article/10.1007/s10231-023-01327-w#Equ6">2.6</a>) and Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar3">2.3</a>, and the corresponding result for the classical Besov spaces.</p> <p>For the necessity, we use a similar argument as in the proof of Corollary <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar58">4.21</a>, via the extension operator from Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar23">3.6</a>. In this case, Theorem <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar30">4.1</a> (i) and Proposition <a data-track="click" data-track-label="link" data-track-action="subsection anchor" href="/article/10.1007/s10231-023-01327-w#FPar33">4.3</a> will give us the complete result. <span class="mathjax-tex">$\square $</span></p> </div></div></section> </div> <div id="MagazineFulltextArticleBodySuffix"><section aria-labelledby="Bib1" data-title="References"><div class="c-article-section" id="Bib1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Bib1">References</h2><div class="c-article-section__content" id="Bib1-content"><div data-container-section="references"><ol class="c-article-references" data-track-component="outbound reference" data-track-context="references section"><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="1."><p class="c-article-references__text" id="ref-CR1">Bui, H.-Q., Paluszyński, M., Taibleson, M.H.: A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces. Studia Math. <b>119</b>, 219–246 (1996)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1397492" aria-label="MathSciNet reference 1">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0861.42009" aria-label="MATH reference 1">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 1" href="http://scholar.google.com/scholar_lookup?&title=A%20maximal%20function%20characterization%20of%20weighted%20Besov-Lipschitz%20and%20Triebel-Lizorkin%20spaces&journal=Studia%20Math.&volume=119&pages=219-246&publication_year=1996&author=Bui%2CH-Q&author=Paluszy%C5%84ski%2CM&author=Taibleson%2CMH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="2."><p class="c-article-references__text" id="ref-CR2">Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge Univ. Press, Cambridge (1996)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0865.46020" aria-label="MATH reference 2">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 2" href="http://scholar.google.com/scholar_lookup?&title=Function%20Spaces%2C%20Entropy%20Numbers%2C%20Differential%20Operators&publication_year=1996&author=Edmunds%2CDE&author=Triebel%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="3."><p class="c-article-references__text" id="ref-CR3">El Baraka, A.: An embedding theorem for Campanato spaces. Electron. J. Differential Equat. <b>66</b>, 1–17 (2002)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1921139" aria-label="MathSciNet reference 3">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1002.46024" aria-label="MATH reference 3">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 3" href="http://scholar.google.com/scholar_lookup?&title=An%20embedding%20theorem%20for%20Campanato%20spaces&journal=Electron.%20J.%20Differential%20Equat.&volume=66&pages=1-17&publication_year=2002&author=Baraka%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="4."><p class="c-article-references__text" id="ref-CR4">El Baraka, A.: Function spaces of BMO and Campanato type, pp.109-115, in: Proc. of the 2002 Fez Conference on Partial Differential Equations, Electron. J. Differ. Equ. Conf. 9, Southwest Texas State Univ., San Marcos, TX (2002)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="5."><p class="c-article-references__text" id="ref-CR5">El Baraka, A.: Littlewood-Paley characterization for Campanato spaces. J. Funct. Spaces Appl. <b>4</b>, 193–220 (2006)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2227045" aria-label="MathSciNet reference 5">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1133.42031" aria-label="MATH reference 5">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 5" href="http://scholar.google.com/scholar_lookup?&title=Littlewood-Paley%20characterization%20for%20Campanato%20spaces&journal=J.%20Funct.%20Spaces%20Appl.&volume=4&pages=193-220&publication_year=2006&author=Baraka%2CA"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="6."><p class="c-article-references__text" id="ref-CR6">Gonçalves, H. F.: Non-smooth atomic decomposition of variable 2-microlocal Besov-type and Triebel-Lizorkin-type spaces. Banach J. Math. Anal <b>15</b>(31), 50 (2021)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4261774" aria-label="MathSciNet reference 6">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1475.46035" aria-label="MATH reference 6">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 6" href="http://scholar.google.com/scholar_lookup?&title=Non-smooth%20atomic%20decomposition%20of%20variable%202-microlocal%20Besov-type%20and%20Triebel-Lizorkin-type%20spaces&journal=Banach%20J.%20Math.%20Anal&volume=15&issue=31&publication_year=2021&author=Gon%C3%A7alves%2CH%C2%A0F"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="7."><p class="c-article-references__text" id="ref-CR7">Gonçalves, H.F., Moura, S.D.: Characterization of Triebel-Lizorkin-type spaces with variable exponents via maximal functions, local means and non-smooth atomic decompositions. Math. Nachr. <b>291</b>, 2024–2044 (2018)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3858675" aria-label="MathSciNet reference 7">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1401.42019" aria-label="MATH reference 7">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 7" href="http://scholar.google.com/scholar_lookup?&title=Characterization%20of%20Triebel-Lizorkin-type%20spaces%20with%20variable%20exponents%20via%20maximal%20functions%2C%20local%20means%20and%20non-smooth%20atomic%20decompositions&journal=Math.%20Nachr.&volume=291&pages=2024-2044&publication_year=2018&author=Gon%C3%A7alves%2CHF&author=Moura%2CSD"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="8."><p class="c-article-references__text" id="ref-CR8">Gonçalves, H.F., Haroske, D.D., Skrzypczak, L.: Compact embeddings on Besov-type and Triebel-Lizorkin-type spaces on bounded domains. Rev. Mat. Complut. <b>34</b>, 761–795 (2021)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4302241" aria-label="MathSciNet reference 8">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1478.46033" aria-label="MATH reference 8">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 8" href="http://scholar.google.com/scholar_lookup?&title=Compact%20embeddings%20on%20Besov-type%20and%20Triebel-Lizorkin-type%20spaces%20on%20bounded%20domains&journal=Rev.%20Mat.%20Complut.&volume=34&pages=761-795&publication_year=2021&author=Gon%C3%A7alves%2CHF&author=Haroske%2CDD&author=Skrzypczak%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="9."><p class="c-article-references__text" id="ref-CR9">Haroske, D.D., Moura, S.D., Skrzypczak, L.: Smoothness Morrey Spaces of regular distributions, and some unboundedness properties. Nonlinear Anal Series A: Theory, Methods Appl <b>139</b>, 218–244 (2016)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1352.46032" aria-label="MATH reference 9">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 9" href="http://scholar.google.com/scholar_lookup?&title=Smoothness%20Morrey%20Spaces%20of%20regular%20distributions%2C%20and%20some%20unboundedness%20properties&journal=Nonlinear%20Anal%20Series%20A%3A%20Theory%2C%20Methods%20Appl&volume=139&pages=218-244&publication_year=2016&author=Haroske%2CDD&author=Moura%2CSD&author=Skrzypczak%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="10."><p class="c-article-references__text" id="ref-CR10">Haroske, D.. D., Skrzypczak, L.: Continuous embeddings of Besov-Morrey function spaces. Acta Math. Sin. (Engl. Ser.) <b>28</b>, 1307–1328 (2012)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2928480" aria-label="MathSciNet reference 10">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1272.46025" aria-label="MATH reference 10">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 10" href="http://scholar.google.com/scholar_lookup?&title=Continuous%20embeddings%20of%20Besov-Morrey%20function%20spaces&journal=Acta%20Math.%20Sin.%20%28Engl.%20Ser.%29&volume=28&pages=1307-1328&publication_year=2012&author=Haroske%2CD.%C2%A0D&author=Skrzypczak%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="11."><p class="c-article-references__text" id="ref-CR11">Haroske, D.D., Skrzypczak, L.: Embeddings of Besov-Morrey spaces on bounded domains. Studia Math. <b>218</b>, 119–144 (2013)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3125118" aria-label="MathSciNet reference 11">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1293.46021" aria-label="MATH reference 11">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 11" href="http://scholar.google.com/scholar_lookup?&title=Embeddings%20of%20Besov-Morrey%20spaces%20on%20bounded%20domains&journal=Studia%20Math.&volume=218&pages=119-144&publication_year=2013&author=Haroske%2CDD&author=Skrzypczak%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="12."><p class="c-article-references__text" id="ref-CR12">Haroske, D.D., Skrzypczak, L.: On Sobolev and Franke-Jawerth embeddings of smoothness Morrey spaces. Rev. Mat. Complut. <b>27</b>, 541–573 (2014)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3223579" aria-label="MathSciNet reference 12">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1311.46033" aria-label="MATH reference 12">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 12" href="http://scholar.google.com/scholar_lookup?&title=On%20Sobolev%20and%20Franke-Jawerth%20embeddings%20of%20smoothness%20Morrey%20spaces&journal=Rev.%20Mat.%20Complut.&volume=27&pages=541-573&publication_year=2014&author=Haroske%2CDD&author=Skrzypczak%2CL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="13."><p class="c-article-references__text" id="ref-CR13">Haroske, D. D., Skrzypczak, L.: Some quantitative result on compact embeddings in smoothness Morrey spaces on bounded domains; an approach via interpolation pp.181–191, in: Function Spaces XII, Banach Center Publ, vol. 119, Polish Acad. Sci., Warsaw (2019)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="14."><p class="c-article-references__text" id="ref-CR14">Haroske, D. D., Skrzypczak, L.: Entropy numbers of compact embeddings of smoothness Morrey spaces on bounded domains. J. Approx. Theory <b>256</b>, 24 pp. (2020)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="15."><p class="c-article-references__text" id="ref-CR15">Kozono, H., Yamazaki, M.: Semilinear heat equations and the Navier-Stokes equation with distributions in new function spaces as initial data. Comm. Partial Different. Equat. <b>19</b>, 959–1014 (1994)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1274547" aria-label="MathSciNet reference 15">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0803.35068" aria-label="MATH reference 15">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 15" href="http://scholar.google.com/scholar_lookup?&title=Semilinear%20heat%20equations%20and%20the%20Navier-Stokes%20equation%20with%20distributions%20in%20new%20function%20spaces%20as%20initial%20data&journal=Comm.%20Partial%20Different.%20Equat.&volume=19&pages=959-1014&publication_year=1994&author=Kozono%2CH&author=Yamazaki%2CM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="16."><p class="c-article-references__text" id="ref-CR16">Liang, Y., Yang, D., Yuan, W., Sawano, Y., Ullrich, T.: A new framework for generalized Besov-type and Triebel-Lizorkin-type spaces. Dissertationes Math. (Rozprawy Mat.) <b>489</b>, 1–114 (2013)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3099066" aria-label="MathSciNet reference 16">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1283.46027" aria-label="MATH reference 16">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 16" href="http://scholar.google.com/scholar_lookup?&title=A%20new%20framework%20for%20generalized%20Besov-type%20and%20Triebel-Lizorkin-type%20spaces&journal=Dissertationes%20Math.%20%28Rozprawy%20Mat.%29&volume=489&pages=1-114&publication_year=2013&author=Liang%2CY&author=Yang%2CD&author=Yuan%2CW&author=Sawano%2CY&author=Ullrich%2CT"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="17."><p class="c-article-references__text" id="ref-CR17">Mazzucato, A.L.: Besov-Morrey spaces: function space theory and applications to non-linear PDE. Trans. Amer. Math. Soc. <b>355</b>, 1297–1364 (2003)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1946395" aria-label="MathSciNet reference 17">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1022.35039" aria-label="MATH reference 17">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 17" href="http://scholar.google.com/scholar_lookup?&title=Besov-Morrey%20spaces%3A%20function%20space%20theory%20and%20applications%20to%20non-linear%20PDE&journal=Trans.%20Amer.%20Math.%20Soc.&volume=355&pages=1297-1364&publication_year=2003&author=Mazzucato%2CAL"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="18."><p class="c-article-references__text" id="ref-CR18">Meyer, Y.: Wavelets and Operators. Cambridge Univ. Press, Cambridge (1992)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0776.42019" aria-label="MATH reference 18">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 18" href="http://scholar.google.com/scholar_lookup?&title=Wavelets%20and%20Operators&publication_year=1992&author=Meyer%2CY"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="19."><p class="c-article-references__text" id="ref-CR19">Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. <b>43</b>, 126–166 (1938)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1501936" aria-label="MathSciNet reference 19">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0018.40501" aria-label="MATH reference 19">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 19" href="http://scholar.google.com/scholar_lookup?&title=On%20the%20solutions%20of%20quasi-linear%20elliptic%20partial%20differential%20equations&journal=Trans.%20Amer.%20Math.%20Soc.&volume=43&pages=126-166&publication_year=1938&author=Morrey%2CCB"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="20."><p class="c-article-references__text" id="ref-CR20">Moura, S.D., Neves, J.S., Schneider, C.: Traces and extensions of generalized smoothness Morrey spaces on domains. Nonlinear Anal. <b>181</b>, 311–339 (2019)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3902382" aria-label="MathSciNet reference 20">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1420.46030" aria-label="MATH reference 20">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 20" href="http://scholar.google.com/scholar_lookup?&title=Traces%20and%20extensions%20of%20generalized%20smoothness%20Morrey%20spaces%20on%20domains&journal=Nonlinear%20Anal.&volume=181&pages=311-339&publication_year=2019&author=Moura%2CSD&author=Neves%2CJS&author=Schneider%2CC"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="21."><p class="c-article-references__text" id="ref-CR21">Peetre, J.: On the theory of <span class="mathjax-tex">${{\cal{L} }}_{p,\lambda }$</span> spaces. J. Funct. Anal. <b>4</b>, 71–87 (1969)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 21" href="http://scholar.google.com/scholar_lookup?&title=On%20the%20theory%20of%20%24%24%7B%7B%5Ccal%7BL%7D%20%7D%7D_%7Bp%2C%5Clambda%20%7D%24%24%20L%20p%20%2C%20%CE%BB%20spaces&journal=J.%20Funct.%20Anal.&volume=4&pages=71-87&publication_year=1969&author=Peetre%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="22."><p class="c-article-references__text" id="ref-CR22">Rosenthal, M.: Local means, wavelet bases, representations, and isomorphisms in Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Math. Nachr. <b>286</b>, 59–87 (2013)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3019504" aria-label="MathSciNet reference 22">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1268.46027" aria-label="MATH reference 22">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 22" href="http://scholar.google.com/scholar_lookup?&title=Local%20means%2C%20wavelet%20bases%2C%20representations%2C%20and%20isomorphisms%20in%20Besov-Morrey%20and%20Triebel-Lizorkin-Morrey%20spaces&journal=Math.%20Nachr.&volume=286&pages=59-87&publication_year=2013&author=Rosenthal%2CM"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="23."><p class="c-article-references__text" id="ref-CR23">Rychkov, V.S.: On restrictions and extensions of the Besov and Triebel-Lizorkin spaces with respect to Lipschitz domains. J. London Math. Soc. <b>60</b>, 237–257 (1999)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=1721827" aria-label="MathSciNet reference 23">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0940.46017" aria-label="MATH reference 23">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 23" href="http://scholar.google.com/scholar_lookup?&title=On%20restrictions%20and%20extensions%20of%20the%20Besov%20and%20Triebel-Lizorkin%20spaces%20with%20respect%20to%20Lipschitz%20domains&journal=J.%20London%20Math.%20Soc.&volume=60&pages=237-257&publication_year=1999&author=Rychkov%2CVS"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="24."><p class="c-article-references__text" id="ref-CR24">Sawano, Y.: Wavelet characterizations of Besov-Morrey and Triebel-Lizorkin-Morrey spaces. Funct. Approx. Comment. Math. <b>38</b>, 93–107 (2008)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2433791" aria-label="MathSciNet reference 24">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1176.42020" aria-label="MATH reference 24">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 24" href="http://scholar.google.com/scholar_lookup?&title=Wavelet%20characterizations%20of%20Besov-Morrey%20and%20Triebel-Lizorkin-Morrey%20spaces&journal=Funct.%20Approx.%20Comment.%20Math.&volume=38&pages=93-107&publication_year=2008&author=Sawano%2CY"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="25."><p class="c-article-references__text" id="ref-CR25">Sawano, Y.: A note on Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces, Acta Math. Sin. (Engl. Ser.) <b>25</b>, 1223–1242 (2009)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2524944" aria-label="MathSciNet reference 25">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1172.42007" aria-label="MATH reference 25">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 25" href="http://scholar.google.com/scholar_lookup?&title=A%20note%20on%20Besov-Morrey%20spaces%20and%20Triebel-Lizorkin-Morrey%20spaces%2C%20Acta%20Math&journal=Sin.%20%28Engl.%20Ser.%29&volume=25&pages=1223-1242&publication_year=2009&author=Sawano%2CY"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="26."><p class="c-article-references__text" id="ref-CR26">Sawano, Y., Di Fazio, G., Hakim, D. I.: Morrey spaces. Introduction and Applications to Integral Operators and PDE’s. Vol. I, Monographs and Research Notes in Mathematics. Chapman & Hall CRC Press, Boca Raton, FL (2020)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="27."><p class="c-article-references__text" id="ref-CR27">Sawano, Y., Di Fazio, G., Hakim, D. I.: Morrey spaces. Introduction and Applications to Integral Operators and PDE’s. Vol. II, Monographs and Research Notes in Mathematics. Chapman & Hall CRC Press, Boca Raton, FL (2020)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="28."><p class="c-article-references__text" id="ref-CR28">Sawano, Y., Tanaka, H.: Decompositions of Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces. Math. Z. <b>257</b>, 871–905 (2007)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2342557" aria-label="MathSciNet reference 28">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1133.42041" aria-label="MATH reference 28">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 28" href="http://scholar.google.com/scholar_lookup?&title=Decompositions%20of%20Besov-Morrey%20spaces%20and%20Triebel-Lizorkin-Morrey%20spaces&journal=Math.%20Z.&volume=257&pages=871-905&publication_year=2007&author=Sawano%2CY&author=Tanaka%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="29."><p class="c-article-references__text" id="ref-CR29">Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces for non-doubling measures: Sawano, Y., Tanaka. H. Math. Nachr. <b>282</b>, 1788–1810 (2009)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2588836" aria-label="MathSciNet reference 29">MathSciNet</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 29" href="http://scholar.google.com/scholar_lookup?&title=Sawano%2C%20Y.%2C%20Tanaka.%20H&journal=Math.%20Nachr.&volume=282&pages=1788-1810&publication_year=2009"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="30."><p class="c-article-references__text" id="ref-CR30">Sawano, Y.: Besov-Morrey spaces and Triebel-Lizorkin-Morrey spaces on domains. Math. Nachr. <b>283</b>, 1456–1487 (2010)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2744140" aria-label="MathSciNet reference 30">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1211.46031" aria-label="MATH reference 30">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 30" href="http://scholar.google.com/scholar_lookup?&title=Besov-Morrey%20spaces%20and%20Triebel-Lizorkin-Morrey%20spaces%20on%20domains&journal=Math.%20Nachr.&volume=283&pages=1456-1487&publication_year=2010&author=Sawano%2CY"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="31."><p class="c-article-references__text" id="ref-CR31">Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. I. Eurasian Math. J. <b>3</b>, 110–149 (2012)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3024132" aria-label="MathSciNet reference 31">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1274.46074" aria-label="MATH reference 31">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 31" href="http://scholar.google.com/scholar_lookup?&title=Smoothness%20spaces%20related%20to%20Morrey%20spaces%20-%20a%20survey.%20I&journal=Eurasian%20Math.%20J.&volume=3&pages=110-149&publication_year=2012&author=Sickel%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="32."><p class="c-article-references__text" id="ref-CR32">Sickel, W.: Smoothness spaces related to Morrey spaces - a survey. II. Eurasian Math. J. <b>4</b>, 82–124 (2013)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3118894" aria-label="MathSciNet reference 32">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1290.46028" aria-label="MATH reference 32">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 32" href="http://scholar.google.com/scholar_lookup?&title=Smoothness%20spaces%20related%20to%20Morrey%20spaces%20-%20a%20survey.%20II&journal=Eurasian%20Math.%20J.&volume=4&pages=82-124&publication_year=2013&author=Sickel%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="33."><p class="c-article-references__text" id="ref-CR33">Tang, L., Xu, J.: Some properties of Morrey type Besov-Triebel spaces. Math. Nachr. <b>278</b>, 904–917 (2005)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2141966" aria-label="MathSciNet reference 33">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1074.42011" aria-label="MATH reference 33">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 33" href="http://scholar.google.com/scholar_lookup?&title=Some%20properties%20of%20Morrey%20type%20Besov-Triebel%20spaces&journal=Math.%20Nachr.&volume=278&pages=904-917&publication_year=2005&author=Tang%2CL&author=Xu%2CJ"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="34."><p class="c-article-references__text" id="ref-CR34">Triebel, H.: Theory of Function Spaces. Birkhäuser, Basel (1983)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0546.46027" aria-label="MATH reference 34">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 34" href="http://scholar.google.com/scholar_lookup?&title=Theory%20of%20Function%20Spaces&publication_year=1983&author=Triebel%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="35."><p class="c-article-references__text" id="ref-CR35">Triebel, H.: Theory of Function Spaces. II, Birkhäuser, Basel (1992)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?0763.46025" aria-label="MATH reference 35">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 35" href="http://scholar.google.com/scholar_lookup?&title=Theory%20of%20Function%20Spaces&publication_year=1992&author=Triebel%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="36."><p class="c-article-references__text" id="ref-CR36">Triebel, H.: Theory of Function Spaces. III, Birkhäuser, Basel (2006)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1104.46001" aria-label="MATH reference 36">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 36" href="http://scholar.google.com/scholar_lookup?&title=Theory%20of%20Function%20Spaces&publication_year=2006&author=Triebel%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="37."><p class="c-article-references__text" id="ref-CR37">Triebel, H.: Local Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics 20, European Mathematical Society. EMS), Zürich (2013)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 37" href="http://scholar.google.com/scholar_lookup?&title=Local%20Function%20Spaces%2C%20Heat%20and%20Navier-Stokes%20Equations%2C%20EMS%20Tracts%20in%20Mathematics%2020%2C%20European%20Mathematical%20Society&publication_year=2013&author=Triebel%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="38."><p class="c-article-references__text" id="ref-CR38">Triebel, H.: Hybrid Function Spaces, Heat and Navier-Stokes Equations, EMS Tracts in Mathematics 24, European Mathematical Society. EMS), Zürich (2015)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 38" href="http://scholar.google.com/scholar_lookup?&title=Hybrid%20Function%20Spaces%2C%20Heat%20and%20Navier-Stokes%20Equations%2C%20EMS%20Tracts%20in%20Mathematics%2024%2C%20European%20Mathematical%20Society&publication_year=2015&author=Triebel%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="39."><p class="c-article-references__text" id="ref-CR39">Triebel, H.: Theory of Function Spaces. IV, Birkhäuser, Basel (2020)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1445.46002" aria-label="MATH reference 39">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 39" href="http://scholar.google.com/scholar_lookup?&title=Theory%20of%20Function%20Spaces&publication_year=2020&author=Triebel%2CH"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="40."><p class="c-article-references__text" id="ref-CR40">Wojtaszczyk, P.: A Mathematical Introduction to Wavelets, London Mathematical Society Student Texts 37. Cambridge Univ. Press, Cambridge (1997)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 40" href="http://scholar.google.com/scholar_lookup?&title=A%20Mathematical%20Introduction%20to%20Wavelets%2C%20London%20Mathematical%20Society%20Student%20Texts%2037&publication_year=1997&author=Wojtaszczyk%2CP"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="41."><p class="c-article-references__text" id="ref-CR41">Yang, D., Yuan, W.: A new class of function spaces connecting Triebel-Lizorkin spaces and Q spaces. J. Funct. Anal. <b>255</b>, 2760–2809 (2008)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2464191" aria-label="MathSciNet reference 41">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1169.46016" aria-label="MATH reference 41">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 41" href="http://scholar.google.com/scholar_lookup?&title=A%20new%20class%20of%20function%20spaces%20connecting%20Triebel-Lizorkin%20spaces%20and%20Q%20spaces&journal=J.%20Funct.%20Anal.&volume=255&pages=2760-2809&publication_year=2008&author=Yang%2CD&author=Yuan%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="42."><p class="c-article-references__text" id="ref-CR42">Yang, D., Yuan, W.: New Besov-type spaces and Triebel-Lizorkin-type spaces including Q spaces. Math. Z. <b>265</b>, 451–480 (2010)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2609320" aria-label="MathSciNet reference 42">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1191.42011" aria-label="MATH reference 42">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 42" href="http://scholar.google.com/scholar_lookup?&title=New%20Besov-type%20spaces%20and%20Triebel-Lizorkin-type%20spaces%20including%20Q%20spaces&journal=Math.%20Z.&volume=265&pages=451-480&publication_year=2010&author=Yang%2CD&author=Yuan%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="43."><p class="c-article-references__text" id="ref-CR43">Yang, D., Yuan, W.: Characterizations of Besov-type and Triebel-Lizorkin-type spaces via maximal functions and local means. Nonlinear Anal. <b>73</b>, 3805–3820 (2010)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=2728556" aria-label="MathSciNet reference 43">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1225.46033" aria-label="MATH reference 43">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 43" href="http://scholar.google.com/scholar_lookup?&title=Characterizations%20of%20Besov-type%20and%20Triebel-Lizorkin-type%20spaces%20via%20maximal%20functions%20and%20local%20means&journal=Nonlinear%20Anal.&volume=73&pages=3805-3820&publication_year=2010&author=Yang%2CD&author=Yuan%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="44."><p class="c-article-references__text" id="ref-CR44">Yang, D., Yuan, W.: Relations among Besov-type spaces, Triebel-Lizorkin-type spaces and generalized Carleson measure spaces. Appl. Anal. <b>92</b>, 549–561 (2013)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3021276" aria-label="MathSciNet reference 44">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1278.42032" aria-label="MATH reference 44">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 44" href="http://scholar.google.com/scholar_lookup?&title=Relations%20among%20Besov-type%20spaces%2C%20Triebel-Lizorkin-type%20spaces%20and%20generalized%20Carleson%20measure%20spaces&journal=Appl.%20Anal.&volume=92&pages=549-561&publication_year=2013&author=Yang%2CD&author=Yuan%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="45."><p class="c-article-references__text" id="ref-CR45">Yuan, W., Haroske, D.D., Moura, S.D., Skrzypczak, L., Yang, D.: Limiting embeddings in smoothness Morrey spaces, continuity envelopes and applications. J. Approx. Theory <b>192</b>, 306–335 (2015)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3313487" aria-label="MathSciNet reference 45">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1334.46034" aria-label="MATH reference 45">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 45" href="http://scholar.google.com/scholar_lookup?&title=Limiting%20embeddings%20in%20smoothness%20Morrey%20spaces%2C%20continuity%20envelopes%20and%20applications&journal=J.%20Approx.%20Theory&volume=192&pages=306-335&publication_year=2015&author=Yuan%2CW&author=Haroske%2CDD&author=Moura%2CSD&author=Skrzypczak%2CL&author=Yang%2CD"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="46."><p class="c-article-references__text" id="ref-CR46">Yuan, W., Haroske, D.D., Skrzypczak, L., Yang, D.: Embedding properties of Besov-type spaces. Appl. Anal. <b>94</b>, 318–340 (2015)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3295700" aria-label="MathSciNet reference 46">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1320.46036" aria-label="MATH reference 46">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 46" href="http://scholar.google.com/scholar_lookup?&title=Embedding%20properties%20of%20Besov-type%20spaces&journal=Appl.%20Anal.&volume=94&pages=318-340&publication_year=2015&author=Yuan%2CW&author=Haroske%2CDD&author=Skrzypczak%2CL&author=Yang%2CD"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="47."><p class="c-article-references__text" id="ref-CR47">Yuan, W., Sickel, W., Yang, D.: Morrey and Campanato Meet Besov, Lizorkin and Triebel. Lecture Notes in Mathematics, vol. 2005. Springer-Verlag, Berlin (2010)</p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="48."><p class="c-article-references__text" id="ref-CR48">Yuan, W., Sickel, W., Yang, D.: Interpolation of Morrey-Campanato and related smoothness spaces. Sci. China Math. <b>58</b>, 1835–1908 (2015)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3383989" aria-label="MathSciNet reference 48">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1337.46030" aria-label="MATH reference 48">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 48" href="http://scholar.google.com/scholar_lookup?&title=Interpolation%20of%20Morrey-Campanato%20and%20related%20smoothness%20spaces&journal=Sci.%20China%20Math.&volume=58&pages=1835-1908&publication_year=2015&author=Yuan%2CW&author=Sickel%2CW&author=Yang%2CD"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="49."><p class="c-article-references__text" id="ref-CR49">Yang, D., Zhuo, C., Yuan, W.: Triebel-Lizorkin type spaces with variable exponent. Banach J. Math. Anal. <b>9</b>, 146–202 (2015)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3336888" aria-label="MathSciNet reference 49">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1339.46040" aria-label="MATH reference 49">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 49" href="http://scholar.google.com/scholar_lookup?&title=Triebel-Lizorkin%20type%20spaces%20with%20variable%20exponent&journal=Banach%20J.%20Math.%20Anal.&volume=9&pages=146-202&publication_year=2015&author=Yang%2CD&author=Zhuo%2CC&author=Yuan%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="50."><p class="c-article-references__text" id="ref-CR50">Yang, D., Zhuo, C., Yuan, W.: Besov-type spaces with variable smoothness and integrability. J. Funct. Anal. <b>269</b>, 1840–1898 (2015)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=3373435" aria-label="MathSciNet reference 50">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1342.46038" aria-label="MATH reference 50">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 50" href="http://scholar.google.com/scholar_lookup?&title=Besov-type%20spaces%20with%20variable%20smoothness%20and%20integrability&journal=J.%20Funct.%20Anal.&volume=269&pages=1840-1898&publication_year=2015&author=Yang%2CD&author=Zhuo%2CC&author=Yuan%2CW"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="51."><p class="c-article-references__text" id="ref-CR51">Zhuo, C.: Complex interpolation of Besov-type spaces on domains. Z. Anal. Anwend. <b>40</b>, 313–347 (2021)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4284347" aria-label="MathSciNet reference 51">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1480.46034" aria-label="MATH reference 51">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 51" href="http://scholar.google.com/scholar_lookup?&title=Complex%20interpolation%20of%20Besov-type%20spaces%20on%20domains&journal=Z.%20Anal.%20Anwend.&volume=40&pages=313-347&publication_year=2021&author=Zhuo%2CC"> Google Scholar</a> </p></li><li class="c-article-references__item js-c-reading-companion-references-item" data-counter="52."><p class="c-article-references__text" id="ref-CR52">Zhuo, C., Hovemann, M., Sickel, W.: Complex interpolation of Lizorkin-Triebel-Morrey Spaces on Domains. Anal. Geom. Metric Spaces <b>8</b>, 268–304 (2020)</p><p class="c-article-references__links u-hide-print"><a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="mathscinet reference" data-track-action="mathscinet reference" href="http://www.ams.org/mathscinet-getitem?mr=4178742" aria-label="MathSciNet reference 52">MathSciNet</a> <a data-track="click_references" rel="nofollow noopener" data-track-label="link" data-track-item_id="link" data-track-value="math reference" data-track-action="math reference" href="http://www.emis.de/MATH-item?1478.46017" aria-label="MATH reference 52">MATH</a> <a data-track="click_references" data-track-action="google scholar reference" data-track-value="google scholar reference" data-track-label="link" data-track-item_id="link" rel="nofollow noopener" aria-label="Google Scholar reference 52" href="http://scholar.google.com/scholar_lookup?&title=Complex%20interpolation%20of%20Lizorkin-Triebel-Morrey%20Spaces%20on%20Domains&journal=Anal.%20Geom.%20Metric%20Spaces&volume=8&pages=268-304&publication_year=2020&author=Zhuo%2CC&author=Hovemann%2CM&author=Sickel%2CW"> Google Scholar</a> </p></li></ol><p class="c-article-references__download u-hide-print"><a data-track="click" data-track-action="download citation references" data-track-label="link" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s10231-023-01327-w?format=refman&flavour=references">Download references<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p></div></div></div></section></div><section data-title="Acknowledgements"><div class="c-article-section" id="Ack1-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="Ack1">Acknowledgements</h2><div class="c-article-section__content" id="Ack1-content"><p> We are indebted to the referees of the first version of that paper for their valuable remarks which helped to improve the presentation.</p></div></div></section><section aria-labelledby="author-information" data-title="Author information"><div class="c-article-section" id="author-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="author-information">Author information</h2><div class="c-article-section__content" id="author-information-content"><h3 class="c-article__sub-heading" id="affiliations">Authors and Affiliations</h3><ol class="c-article-author-affiliation__list"><li id="Aff1"><p class="c-article-author-affiliation__address">Institute of Mathematics, Friedrich Schiller University Jena, 07737, Jena, Germany</p><p class="c-article-author-affiliation__authors-list">Helena F. Gonçalves & Dorothee D. Haroske</p></li><li id="Aff2"><p class="c-article-author-affiliation__address">Faculty of Mathematics and Computer Science, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 4, 61-614, Poznań, Poland</p><p class="c-article-author-affiliation__authors-list">Leszek Skrzypczak</p></li></ol><div class="u-js-hide u-hide-print" data-test="author-info"><span class="c-article__sub-heading">Authors</span><ol class="c-article-authors-search u-list-reset"><li id="auth-Helena_F_-Gon_alves-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Helena F. Gonçalves</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?sortBy=newestFirst&dc.creator=Helena%20F.%20Gon%C3%A7alves" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text"><span class="c-article-authors-search__links-text">You can also search for this author in</span><span class="c-article-identifiers"><a class="c-article-identifiers__item" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Helena%20F.%20Gon%C3%A7alves" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Helena%20F.%20Gon%C3%A7alves%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Dorothee_D_-Haroske-Aff1"><span class="c-article-authors-search__title u-h3 js-search-name">Dorothee D. Haroske</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?sortBy=newestFirst&dc.creator=Dorothee%20D.%20Haroske" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text"><span class="c-article-authors-search__links-text">You can also search for this author in</span><span class="c-article-identifiers"><a class="c-article-identifiers__item" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Dorothee%20D.%20Haroske" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Dorothee%20D.%20Haroske%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li><li id="auth-Leszek-Skrzypczak-Aff2"><span class="c-article-authors-search__title u-h3 js-search-name">Leszek Skrzypczak</span><div class="c-article-authors-search__list"><div class="c-article-authors-search__item c-article-authors-search__list-item--left"><a href="/search?sortBy=newestFirst&dc.creator=Leszek%20Skrzypczak" class="c-article-button" data-track="click" data-track-action="author link - publication" data-track-label="link" rel="nofollow">View author publications</a></div><div class="c-article-authors-search__item c-article-authors-search__list-item--right"><p class="search-in-title-js c-article-authors-search__text"><span class="c-article-authors-search__links-text">You can also search for this author in</span><span class="c-article-identifiers"><a class="c-article-identifiers__item" href="https://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=search&term=Leszek%20Skrzypczak" data-track="click" data-track-action="author link - pubmed" data-track-label="link" rel="nofollow">PubMed</a><span class="u-hide"> </span><a class="c-article-identifiers__item" href="https://scholar.google.co.uk/scholar?as_q=&num=10&btnG=Search+Scholar&as_epq=&as_oq=&as_eq=&as_occt=any&as_sauthors=%22Leszek%20Skrzypczak%22&as_publication=&as_ylo=&as_yhi=&as_allsubj=all&hl=en" data-track="click" data-track-action="author link - scholar" data-track-label="link" rel="nofollow">Google Scholar</a></span></p></div></div></li></ol></div><h3 class="c-article__sub-heading" id="corresponding-author">Corresponding author</h3><p id="corresponding-author-list">Correspondence to <a id="corresp-c1" href="mailto:leszek.skrzypczak@amu.edu.pl">Leszek Skrzypczak</a>.</p></div></div></section><section data-title="Additional information"><div class="c-article-section" id="additional-information-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="additional-information">Additional information</h2><div class="c-article-section__content" id="additional-information-content"><h3 class="c-article__sub-heading">Publisher's Note</h3><p>Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</p><p>All authors were partially supported by the German Research Foundation (DFG), Grant no. Ha 2794/8-1. The third author was also supported by National Science Center, Poland, Grant no. 2013/10/A/ST1/00091.</p></div></div></section><section data-title="Rights and permissions"><div class="c-article-section" id="rightslink-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="rightslink">Rights and permissions</h2><div class="c-article-section__content" id="rightslink-content"> <p><b>Open Access</b> This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit <a href="http://creativecommons.org/licenses/by/4.0/" rel="license">http://creativecommons.org/licenses/by/4.0/</a>.</p> <p class="c-article-rights"><a data-track="click" data-track-action="view rights and permissions" data-track-label="link" href="https://s100.copyright.com/AppDispatchServlet?title=Limiting%20embeddings%20of%20Besov-type%20and%20Triebel-Lizorkin-type%20spaces%20on%20domains%20and%20an%20extension%20operator&author=Helena%20F.%20Gon%C3%A7alves%20et%20al&contentID=10.1007%2Fs10231-023-01327-w&copyright=The%20Author%28s%29&publication=0373-3114&publicationDate=2023-03-27&publisherName=SpringerNature&orderBeanReset=true&oa=CC%20BY">Reprints and permissions</a></p></div></div></section><section aria-labelledby="article-info" data-title="About this article"><div class="c-article-section" id="article-info-section"><h2 class="c-article-section__title js-section-title js-c-reading-companion-sections-item" id="article-info">About this article</h2><div class="c-article-section__content" id="article-info-content"><div class="c-bibliographic-information"><div class="u-hide-print c-bibliographic-information__column c-bibliographic-information__column--border"><a data-crossmark="10.1007/s10231-023-01327-w" target="_blank" rel="noopener" href="https://crossmark.crossref.org/dialog/?doi=10.1007/s10231-023-01327-w" data-track="click" data-track-action="Click Crossmark" data-track-label="link" data-test="crossmark"><img loading="lazy" width="57" height="81" alt="Check for updates. Verify currency and authenticity via CrossMark" src="data:image/svg+xml;base64,<svg height="81" width="57" xmlns="http://www.w3.org/2000/svg"><g fill="none" fill-rule="evenodd"><path d="m17.35 35.45 21.3-14.2v-17.03h-21.3" fill="#989898"/><path d="m38.65 35.45-21.3-14.2v-17.03h21.3" fill="#747474"/><path d="m28 .5c-12.98 0-23.5 10.52-23.5 23.5s10.52 23.5 23.5 23.5 23.5-10.52 23.5-23.5c0-6.23-2.48-12.21-6.88-16.62-4.41-4.4-10.39-6.88-16.62-6.88zm0 41.25c-9.8 0-17.75-7.95-17.75-17.75s7.95-17.75 17.75-17.75 17.75 7.95 17.75 17.75c0 4.71-1.87 9.22-5.2 12.55s-7.84 5.2-12.55 5.2z" fill="#535353"/><path d="m41 36c-5.81 6.23-15.23 7.45-22.43 2.9-7.21-4.55-10.16-13.57-7.03-21.5l-4.92-3.11c-4.95 10.7-1.19 23.42 8.78 29.71 9.97 6.3 23.07 4.22 30.6-4.86z" fill="#9c9c9c"/><path d="m.2 58.45c0-.75.11-1.42.33-2.01s.52-1.09.91-1.5c.38-.41.83-.73 1.34-.94.51-.22 1.06-.32 1.65-.32.56 0 1.06.11 1.51.35.44.23.81.5 1.1.81l-.91 1.01c-.24-.24-.49-.42-.75-.56-.27-.13-.58-.2-.93-.2-.39 0-.73.08-1.05.23-.31.16-.58.37-.81.66-.23.28-.41.63-.53 1.04-.13.41-.19.88-.19 1.39 0 1.04.23 1.86.68 2.46.45.59 1.06.88 1.84.88.41 0 .77-.07 1.07-.23s.59-.39.85-.68l.91 1c-.38.43-.8.76-1.28.99-.47.22-1 .34-1.58.34-.59 0-1.13-.1-1.64-.31-.5-.2-.94-.51-1.31-.91-.38-.4-.67-.9-.88-1.48-.22-.59-.33-1.26-.33-2.02zm8.4-5.33h1.61v2.54l-.05 1.33c.29-.27.61-.51.96-.72s.76-.31 1.24-.31c.73 0 1.27.23 1.61.71.33.47.5 1.14.5 2.02v4.31h-1.61v-4.1c0-.57-.08-.97-.25-1.21-.17-.23-.45-.35-.83-.35-.3 0-.56.08-.79.22-.23.15-.49.36-.78.64v4.8h-1.61zm7.37 6.45c0-.56.09-1.06.26-1.51.18-.45.42-.83.71-1.14.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.36c.07.62.29 1.1.65 1.44.36.33.82.5 1.38.5.29 0 .57-.04.83-.13s.51-.21.76-.37l.55 1.01c-.33.21-.69.39-1.09.53-.41.14-.83.21-1.26.21-.48 0-.92-.08-1.34-.25-.41-.16-.76-.4-1.07-.7-.31-.31-.55-.69-.72-1.13-.18-.44-.26-.95-.26-1.52zm4.6-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.07.45-.31.29-.5.73-.58 1.3zm2.5.62c0-.57.09-1.08.28-1.53.18-.44.43-.82.75-1.13s.69-.54 1.1-.71c.42-.16.85-.24 1.31-.24.45 0 .84.08 1.17.23s.61.34.85.57l-.77 1.02c-.19-.16-.38-.28-.56-.37-.19-.09-.39-.14-.61-.14-.56 0-1.01.21-1.35.63-.35.41-.52.97-.52 1.67 0 .69.17 1.24.51 1.66.34.41.78.62 1.32.62.28 0 .54-.06.78-.17.24-.12.45-.26.64-.42l.67 1.03c-.33.29-.69.51-1.08.65-.39.15-.78.23-1.18.23-.46 0-.9-.08-1.31-.24-.4-.16-.75-.39-1.05-.7s-.53-.69-.7-1.13c-.17-.45-.25-.96-.25-1.53zm6.91-6.45h1.58v6.17h.05l2.54-3.16h1.77l-2.35 2.8 2.59 4.07h-1.75l-1.77-2.98-1.08 1.23v1.75h-1.58zm13.69 1.27c-.25-.11-.5-.17-.75-.17-.58 0-.87.39-.87 1.16v.75h1.34v1.27h-1.34v5.6h-1.61v-5.6h-.92v-1.2l.92-.07v-.72c0-.35.04-.68.13-.98.08-.31.21-.57.4-.79s.42-.39.71-.51c.28-.12.63-.18 1.04-.18.24 0 .48.02.69.07.22.05.41.1.57.17zm.48 5.18c0-.57.09-1.08.27-1.53.17-.44.41-.82.72-1.13.3-.31.65-.54 1.04-.71.39-.16.8-.24 1.23-.24s.84.08 1.24.24c.4.17.74.4 1.04.71s.54.69.72 1.13c.19.45.28.96.28 1.53s-.09 1.08-.28 1.53c-.18.44-.42.82-.72 1.13s-.64.54-1.04.7-.81.24-1.24.24-.84-.08-1.23-.24-.74-.39-1.04-.7c-.31-.31-.55-.69-.72-1.13-.18-.45-.27-.96-.27-1.53zm1.65 0c0 .69.14 1.24.43 1.66.28.41.68.62 1.18.62.51 0 .9-.21 1.19-.62.29-.42.44-.97.44-1.66 0-.7-.15-1.26-.44-1.67-.29-.42-.68-.63-1.19-.63-.5 0-.9.21-1.18.63-.29.41-.43.97-.43 1.67zm6.48-3.44h1.33l.12 1.21h.05c.24-.44.54-.79.88-1.02.35-.24.7-.36 1.07-.36.32 0 .59.05.78.14l-.28 1.4-.33-.09c-.11-.01-.23-.02-.38-.02-.27 0-.56.1-.86.31s-.55.58-.77 1.1v4.2h-1.61zm-47.87 15h1.61v4.1c0 .57.08.97.25 1.2.17.24.44.35.81.35.3 0 .57-.07.8-.22.22-.15.47-.39.73-.73v-4.7h1.61v6.87h-1.32l-.12-1.01h-.04c-.3.36-.63.64-.98.86-.35.21-.76.32-1.24.32-.73 0-1.27-.24-1.61-.71-.33-.47-.5-1.14-.5-2.02zm9.46 7.43v2.16h-1.61v-9.59h1.33l.12.72h.05c.29-.24.61-.45.97-.63.35-.17.72-.26 1.1-.26.43 0 .81.08 1.15.24.33.17.61.4.84.71.24.31.41.68.53 1.11.13.42.19.91.19 1.44 0 .59-.09 1.11-.25 1.57-.16.47-.38.85-.65 1.16-.27.32-.58.56-.94.73-.35.16-.72.25-1.1.25-.3 0-.6-.07-.9-.2s-.59-.31-.87-.56zm0-2.3c.26.22.5.37.73.45.24.09.46.13.66.13.46 0 .84-.2 1.15-.6.31-.39.46-.98.46-1.77 0-.69-.12-1.22-.35-1.61-.23-.38-.61-.57-1.13-.57-.49 0-.99.26-1.52.77zm5.87-1.69c0-.56.08-1.06.25-1.51.16-.45.37-.83.65-1.14.27-.3.58-.54.93-.71s.71-.25 1.08-.25c.39 0 .73.07 1 .2.27.14.54.32.81.55l-.06-1.1v-2.49h1.61v9.88h-1.33l-.11-.74h-.06c-.25.25-.54.46-.88.64-.33.18-.69.27-1.06.27-.87 0-1.56-.32-2.07-.95s-.76-1.51-.76-2.65zm1.67-.01c0 .74.13 1.31.4 1.7.26.38.65.58 1.15.58.51 0 .99-.26 1.44-.77v-3.21c-.24-.21-.48-.36-.7-.45-.23-.08-.46-.12-.7-.12-.45 0-.82.19-1.13.59-.31.39-.46.95-.46 1.68zm6.35 1.59c0-.73.32-1.3.97-1.71.64-.4 1.67-.68 3.08-.84 0-.17-.02-.34-.07-.51-.05-.16-.12-.3-.22-.43s-.22-.22-.38-.3c-.15-.06-.34-.1-.58-.1-.34 0-.68.07-1 .2s-.63.29-.93.47l-.59-1.08c.39-.24.81-.45 1.28-.63.47-.17.99-.26 1.54-.26.86 0 1.51.25 1.93.76s.63 1.25.63 2.21v4.07h-1.32l-.12-.76h-.05c-.3.27-.63.48-.98.66s-.73.27-1.14.27c-.61 0-1.1-.19-1.48-.56-.38-.36-.57-.85-.57-1.46zm1.57-.12c0 .3.09.53.27.67.19.14.42.21.71.21.28 0 .54-.07.77-.2s.48-.31.73-.56v-1.54c-.47.06-.86.13-1.18.23-.31.09-.57.19-.76.31s-.33.25-.41.4c-.09.15-.13.31-.13.48zm6.29-3.63h-.98v-1.2l1.06-.07.2-1.88h1.34v1.88h1.75v1.27h-1.75v3.28c0 .8.32 1.2.97 1.2.12 0 .24-.01.37-.04.12-.03.24-.07.34-.11l.28 1.19c-.19.06-.4.12-.64.17-.23.05-.49.08-.76.08-.4 0-.74-.06-1.02-.18-.27-.13-.49-.3-.67-.52-.17-.21-.3-.48-.37-.78-.08-.3-.12-.64-.12-1.01zm4.36 2.17c0-.56.09-1.06.27-1.51s.41-.83.71-1.14c.29-.3.63-.54 1.01-.71.39-.17.78-.25 1.18-.25.47 0 .88.08 1.23.24.36.16.65.38.89.67s.42.63.54 1.03c.12.41.18.84.18 1.32 0 .32-.02.57-.07.76h-4.37c.08.62.29 1.1.65 1.44.36.33.82.5 1.38.5.3 0 .58-.04.84-.13.25-.09.51-.21.76-.37l.54 1.01c-.32.21-.69.39-1.09.53s-.82.21-1.26.21c-.47 0-.92-.08-1.33-.25-.41-.16-.77-.4-1.08-.7-.3-.31-.54-.69-.72-1.13-.17-.44-.26-.95-.26-1.52zm4.61-.62c0-.55-.11-.98-.34-1.28-.23-.31-.58-.47-1.06-.47-.41 0-.77.15-1.08.45-.31.29-.5.73-.57 1.3zm3.01 2.23c.31.24.61.43.92.57.3.13.63.2.98.2.38 0 .65-.08.83-.23s.27-.35.27-.6c0-.14-.05-.26-.13-.37-.08-.1-.2-.2-.34-.28-.14-.09-.29-.16-.47-.23l-.53-.22c-.23-.09-.46-.18-.69-.3-.23-.11-.44-.24-.62-.4s-.33-.35-.45-.55c-.12-.21-.18-.46-.18-.75 0-.61.23-1.1.68-1.49.44-.38 1.06-.57 1.83-.57.48 0 .91.08 1.29.25s.71.36.99.57l-.74.98c-.24-.17-.49-.32-.73-.42-.25-.11-.51-.16-.78-.16-.35 0-.6.07-.76.21-.17.15-.25.33-.25.54 0 .14.04.26.12.36s.18.18.31.26c.14.07.29.14.46.21l.54.19c.23.09.47.18.7.29s.44.24.64.4c.19.16.34.35.46.58.11.23.17.5.17.82 0 .3-.06.58-.17.83-.12.26-.29.48-.51.68-.23.19-.51.34-.84.45-.34.11-.72.17-1.15.17-.48 0-.95-.09-1.41-.27-.46-.19-.86-.41-1.2-.68z" fill="#535353"/></g></svg>"></a></div><div class="c-bibliographic-information__column"><h3 class="c-article__sub-heading" id="citeas">Cite this article</h3><p class="c-bibliographic-information__citation">Gonçalves, H.F., Haroske, D.D. & Skrzypczak, L. Limiting embeddings of Besov-type and Triebel-Lizorkin-type spaces on domains and an extension operator. <i>Annali di Matematica</i> <b>202</b>, 2481–2516 (2023). https://doi.org/10.1007/s10231-023-01327-w</p><p class="c-bibliographic-information__download-citation u-hide-print"><a data-test="citation-link" data-track="click" data-track-action="download article citation" data-track-label="link" data-track-external="" rel="nofollow" href="https://citation-needed.springer.com/v2/references/10.1007/s10231-023-01327-w?format=refman&flavour=citation">Download citation<svg width="16" height="16" focusable="false" role="img" aria-hidden="true" class="u-icon"><use xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="#icon-eds-i-download-medium"></use></svg></a></p><ul class="c-bibliographic-information__list" data-test="publication-history"><li class="c-bibliographic-information__list-item"><p>Received<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2021-09-27">27 September 2021</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Accepted<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-03-10">10 March 2023</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Published<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-03-27">27 March 2023</time></span></p></li><li class="c-bibliographic-information__list-item"><p>Issue Date<span class="u-hide">: </span><span class="c-bibliographic-information__value"><time datetime="2023-10">October 2023</time></span></p></li><li class="c-bibliographic-information__list-item c-bibliographic-information__list-item--full-width"><p><abbr title="Digital Object Identifier">DOI</abbr><span class="u-hide">: </span><span class="c-bibliographic-information__value">https://doi.org/10.1007/s10231-023-01327-w</span></p></li></ul><div data-component="share-box"><div class="c-article-share-box u-display-none" hidden=""><h3 class="c-article__sub-heading">Share this article</h3><p class="c-article-share-box__description">Anyone you share the following link with will be able to read this content:</p><button class="js-get-share-url c-article-share-box__button" type="button" id="get-share-url" data-track="click" data-track-label="button" data-track-external="" data-track-action="get shareable link">Get shareable link</button><div class="js-no-share-url-container u-display-none" hidden=""><p class="js-c-article-share-box__no-sharelink-info c-article-share-box__no-sharelink-info">Sorry, a shareable link is not currently available for this article.</p></div><div class="js-share-url-container u-display-none" hidden=""><p class="js-share-url c-article-share-box__only-read-input" id="share-url" data-track="click" data-track-label="button" data-track-action="select share url"></p><button class="js-copy-share-url c-article-share-box__button--link-like" type="button" id="copy-share-url" data-track="click" data-track-label="button" data-track-action="copy share url" data-track-external="">Copy to clipboard</button></div><p class="js-c-article-share-box__additional-info c-article-share-box__additional-info"> Provided by the Springer Nature SharedIt content-sharing initiative </p></div></div><h3 class="c-article__sub-heading">Keywords</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=Besov-type%20space&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Besov-type space</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Triebel-Lizorkin-type%20spaces&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Triebel-Lizorkin-type spaces</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Smoothness%20Morrey%20spaces%20on%20domains&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Smoothness Morrey spaces on domains</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Limiting%20embeddings&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Limiting embeddings</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=Extension%20operator.&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">Extension operator.</a></span></li></ul><h3 class="c-article__sub-heading">Mathematics Subject Classification</h3><ul class="c-article-subject-list"><li class="c-article-subject-list__subject"><span><a href="/search?query=46E35&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">46E35</a></span></li><li class="c-article-subject-list__subject"><span><a href="/search?query=42B35&facet-discipline="Mathematics"" data-track="click" data-track-action="view keyword" data-track-label="link">42B35</a></span></li></ul><div data-component="article-info-list"></div></div></div></div></div></section> </div> </main> <div class="c-article-sidebar u-text-sm u-hide-print l-with-sidebar__sidebar" id="sidebar" data-container-type="reading-companion" data-track-component="reading companion"> <aside aria-label="reading companion"> <div class="app-card-service" data-test="article-checklist-banner"> <div> <a class="app-card-service__link" data-track="click_presubmission_checklist" data-track-context="article page top of reading companion" data-track-category="pre-submission-checklist" data-track-action="clicked article page checklist banner test 2 old version" data-track-label="link" href="https://beta.springernature.com/pre-submission?journalId=10231" data-test="article-checklist-banner-link"> <span class="app-card-service__link-text">Use our pre-submission checklist</span> <svg class="app-card-service__link-icon" aria-hidden="true" focusable="false"><use xlink:href="#icon-eds-i-arrow-right-small"></use></svg> </a> <p class="app-card-service__description">Avoid common mistakes on your manuscript.</p> </div> <div class="app-card-service__icon-container"> <svg class="app-card-service__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-clipboard-check-medium"></use> </svg> </div> </div> <div data-test="collections"> </div> <div data-test="editorial-summary"> </div> <div class="c-reading-companion"> <div class="c-reading-companion__sticky" data-component="reading-companion-sticky" data-test="reading-companion-sticky"> <div class="c-reading-companion__panel c-reading-companion__sections c-reading-companion__panel--active" id="tabpanel-sections"> <div class="u-lazy-ad-wrapper u-mt-16 u-hide" data-component-mpu><div class="c-ad c-ad--300x250"> <div class="c-ad__inner"> <p class="c-ad__label">Advertisement</p> <div id="div-gpt-ad-MPU1" class="div-gpt-ad grade-c-hide" data-pa11y-ignore data-gpt data-gpt-unitpath="/270604982/springerlink/10231/article" data-gpt-sizes="300x250" data-test="MPU1-ad" data-gpt-targeting="pos=MPU1;articleid=s10231-023-01327-w;"> </div> </div> </div> </div> </div> <div class="c-reading-companion__panel c-reading-companion__figures c-reading-companion__panel--full-width" id="tabpanel-figures"></div> <div class="c-reading-companion__panel c-reading-companion__references c-reading-companion__panel--full-width" id="tabpanel-references"></div> </div> </div> </aside> </div> </div> </article> <div class="app-elements"> <nav aria-label="expander navigation"> <div class="eds-c-header__expander eds-c-header__expander--search" id="eds-c-header-popup-search"> <h2 class="eds-c-header__heading">Search</h2> <div class="u-container"> <search class="eds-c-header__search" role="search" aria-label="Search from the header"> <form method="GET" action="//link.springer.com/search" data-test="header-search" data-track="search" data-track-context="search from header" data-track-action="submit search form" data-track-category="unified header" data-track-label="form" > <label for="eds-c-header-search" class="eds-c-header__search-label">Search by keyword or author</label> <div class="eds-c-header__search-container"> <input id="eds-c-header-search" class="eds-c-header__search-input" autocomplete="off" name="query" type="search" value="" required> <button class="eds-c-header__search-button" type="submit"> <svg class="eds-c-header__icon" aria-hidden="true" focusable="false"> <use xlink:href="#icon-eds-i-search-medium"></use> </svg> <span class="u-visually-hidden">Search</span> </button> </div> </form> </search> </div> </div> <div class="eds-c-header__expander eds-c-header__expander--menu" id="eds-c-header-nav"> <h2 class="eds-c-header__heading">Navigation</h2> <ul class="eds-c-header__list"> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springer.com/journals/" data-track="nav_find_a_journal" data-track-context="unified header" data-track-action="click find a journal" data-track-category="unified header" data-track-label="link" > Find a journal </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://www.springernature.com/gp/authors" data-track="nav_how_to_publish" data-track-context="unified header" data-track-action="click publish with us link" data-track-category="unified header" data-track-label="link" > Publish with us </a> </li> <li class="eds-c-header__list-item"> <a class="eds-c-header__link" href="https://link.springernature.com/home/" data-track="nav_track_your_research" data-track-context="unified header" data-track-action="click track your research" data-track-category="unified header" data-track-label="link" > Track your research </a> </li> </ul> </div> </nav> <footer > <div class="eds-c-footer" > <div class="eds-c-footer__container"> <div class="eds-c-footer__grid eds-c-footer__group--separator"> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Discover content</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals/a/1" data-track="nav_journals_a_z" data-track-action="journals a-z" data-track-context="unified footer" data-track-label="link">Journals A-Z</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/books/a/1" data-track="nav_books_a_z" data-track-action="books a-z" data-track-context="unified footer" data-track-label="link">Books A-Z</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Publish with us</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/journals" data-track="nav_journal_finder" data-track-action="journal finder" data-track-context="unified footer" data-track-label="link">Journal finder</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/authors" data-track="nav_publish_your_research" data-track-action="publish your research" data-track-context="unified footer" data-track-label="link">Publish your research</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/open-research/about/the-fundamentals-of-open-access-and-open-research" data-track="nav_open_access_publishing" data-track-action="open access publishing" data-track-context="unified footer" data-track-label="link">Open access publishing</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Products and services</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/products" data-track="nav_our_products" data-track-action="our products" data-track-context="unified footer" data-track-label="link">Our products</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/librarians" data-track="nav_librarians" data-track-action="librarians" data-track-context="unified footer" data-track-label="link">Librarians</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/societies" data-track="nav_societies" data-track-action="societies" data-track-context="unified footer" data-track-label="link">Societies</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springernature.com/gp/partners" data-track="nav_partners_and_advertisers" data-track-action="partners and advertisers" data-track-context="unified footer" data-track-label="link">Partners and advertisers</a></li> </ul> </div> <div class="eds-c-footer__group"> <h3 class="eds-c-footer__heading">Our brands</h3> <ul class="eds-c-footer__list"> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.springer.com/" data-track="nav_imprint_Springer" data-track-action="Springer" data-track-context="unified footer" data-track-label="link">Springer</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.nature.com/" data-track="nav_imprint_Nature_Portfolio" data-track-action="Nature Portfolio" data-track-context="unified footer" data-track-label="link">Nature Portfolio</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.biomedcentral.com/" data-track="nav_imprint_BMC" data-track-action="BMC" data-track-context="unified footer" data-track-label="link">BMC</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.palgrave.com/" data-track="nav_imprint_Palgrave_Macmillan" data-track-action="Palgrave Macmillan" data-track-context="unified footer" data-track-label="link">Palgrave Macmillan</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://www.apress.com/" data-track="nav_imprint_Apress" data-track-action="Apress" data-track-context="unified footer" data-track-label="link">Apress</a></li> <li class="eds-c-footer__item"><a class="eds-c-footer__link" href="https://link.springer.com/brands/discover" data-track="nav_imprint_Discover" data-track-action="Discover" data-track-context="unified footer" data-track-label="link">Discover</a></li> </ul> </div> </div> </div> <div class="eds-c-footer__container"> <nav aria-label="footer navigation"> <ul class="eds-c-footer__links"> <li class="eds-c-footer__item"> <button class="eds-c-footer__link" data-cc-action="preferences" data-track="dialog_manage_cookies" data-track-action="Manage cookies" data-track-context="unified footer" data-track-label="link"><span class="eds-c-footer__button-text">Your privacy choices/Manage cookies</span></button> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/legal/ccpa" data-track="nav_california_privacy_statement" data-track-action="california privacy statement" data-track-context="unified footer" data-track-label="link">Your US state privacy rights</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://www.springernature.com/gp/info/accessibility" data-track="nav_accessibility_statement" data-track-action="accessibility statement" data-track-context="unified footer" data-track-label="link">Accessibility statement</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/termsandconditions" data-track="nav_terms_and_conditions" data-track-action="terms and conditions" data-track-context="unified footer" data-track-label="link">Terms and conditions</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/privacystatement" data-track="nav_privacy_policy" data-track-action="privacy policy" data-track-context="unified footer" data-track-label="link">Privacy policy</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/home" data-track="nav_help_and_support" data-track-action="help and support" data-track-context="unified footer" data-track-label="link">Help and support</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://link.springer.com/legal-notice" data-track="nav_legal_notice" data-track-action="legal notice" data-track-context="unified footer" data-track-label="link">Legal notice</a> </li> <li class="eds-c-footer__item"> <a class="eds-c-footer__link" href="https://support.springernature.com/en/support/solutions/articles/6000255911-subscription-cancellations" data-track-action="cancel contracts here">Cancel contracts here</a> </li> </ul> </nav> <div class="eds-c-footer__user"> <p class="eds-c-footer__user-info"> <span data-test="footer-user-ip">8.222.208.146</span> </p> <p class="eds-c-footer__user-info" data-test="footer-business-partners">Not affiliated</p> </div> <a href="https://www.springernature.com/" class="eds-c-footer__link"> <img src="/oscar-static/images/logo-springernature-white-19dd4ba190.svg" alt="Springer Nature" loading="lazy" width="200" height="20"/> </a> <p class="eds-c-footer__legal" data-test="copyright">© 2025 Springer Nature</p> </div> </div> </footer> </div> </body> </html>