Document Zbl 0638.43007

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Mat. Pura Appl., IV. Ser.</a> <a href="/?q=in%3A153840" title="Articles in this Issue">148, 353-366 (1987)</a>. </div> <div class="abstract">The authors give a criterion for boundedness on the Hardy spaces $H^ p$ for functions M(${\mathcal L})$ of the sublaplacian ${\mathcal L}$ on a stratified Lie group, when M is bounded, and M and its dilates M(R$\lambda)$, $R>0$, are locally in the Lipschitz-Besov space $\Lambda$ (s;$\tau$,$\infty)$ uniformly with respect to R. The criterion requires that $r>0$ (not necessarily an integer), $s>r+(Q/2)+1$ and $Q(Q+r)<p<\infty$, where Q is the homogeneous dimension of the group. The proof is based on the atomic and molecular characterization of Hardy spaces.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A2843">T.Nakazi</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A4040531">Cited in <strong>38</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A43A80" title="MSC2020">43A80</a> </td> <td class="space"> Analysis on other specific Lie groups </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A22E30" title="MSC2020">22E30</a> </td> <td class="space"> Analysis on real and complex Lie groups </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A44A30" title="MSC2020">44A30</a> </td> <td class="space"> Multiple integral transforms </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A44A10" title="MSC2020">44A10</a> </td> <td class="space"> Laplace transform </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3ABesov+condition">Besov condition</a>; <a href="/?q=ut%3Amultiplier">multiplier</a>; <a href="/?q=ut%3AHardy+spaces">Hardy spaces</a>; <a href="/?q=ut%3Asublaplacian">sublaplacian</a>; <a href="/?q=ut%3Astratified+Lie+group">stratified Lie group</a>; <a href="/?q=ut%3ALipschitz-Besov+space">Lipschitz-Besov space</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0638.43007" data-ciurl="/ci/04040531" data-biburl="/bibtex/04040531.bib" data-amsurl="/amsrefs/04040531.bib" data-xmlurl="/xml/04040531.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/04040531.pdf" title="Zbl 0638.43007 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1007/BF01774295" aria-label="DOI for “$H^ p$-multipliers on stratified groups”" title="10.1007/BF01774295">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Baernstein, A.; Sawyer, E. T., Embedding and multiplier theorems for H^p(R^n), Memoirs of the Amer. Math. Soc., n. 318, 1-80 (1985) · <a href="/0566.42013" class="nowrap">Zbl 0566.42013</a></td> </tr><tr> <td>[2]</td> <td class="space">Coifman, R. R., Characterization of Fourier transforms of Hardy spaces, Proc. Nat. Acad. Sci. U.S.A., 71, 4133-4134 (1974) · <a href="/0287.42020" class="nowrap">Zbl 0287.42020</a></td> </tr><tr> <td>[3]</td> <td class="space">Coifman, R. R.; Rochberg, R., Representation theorems for holomorphic and harmonic functions in L^p, Astérisque, 77, 11-66 (1980) · <a href="/0472.46040" class="nowrap">Zbl 0472.46040</a></td> </tr><tr> <td>[4]</td> <td class="space">Calderón, A. P.; Torchinsky, A., Parabolic maximal functions associated with a distribution II, Advances in Math., 24, 101-171 (1977) · <a href="/0355.46021" class="nowrap">Zbl 0355.46021</a></td> </tr><tr> <td>[5]</td> <td class="space">Coifman, R. R.; Weiss, G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc., 83, 569-645 (1977) · <a href="/0358.30023" class="nowrap">Zbl 0358.30023</a></td> </tr><tr> <td>[6]</td> <td class="space">De Michele, L.; Mauceri, G., L^p-multipliers on the Heisenberg group, Michigan Math. J., 26, 361-373 (1979) · <a href="/0437.43005" class="nowrap">Zbl 0437.43005</a></td> </tr><tr> <td>[7]</td> <td class="space">Folland, G. B.; Stein, E. M., Hardy spaces on homogeneous groups, Mathematical Notes 28 (1982), Princeton, N.J.: Princeton University Press, Princeton, N.J. · <a href="/0508.42025" class="nowrap">Zbl 0508.42025</a></td> </tr><tr> <td>[8]</td> <td class="space">Guivarch, Y., Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France, 101, 333-379 (1973) · <a href="/0294.43003" class="nowrap">Zbl 0294.43003</a></td> </tr><tr> <td>[9]</td> <td class="space">Hemler, H. L., The molecular theory of H^p,q,s(H^n), Ph. D. Dissertation (1980), St. Louis: Washington University, St. Louis</td> </tr><tr> <td>[10]</td> <td class="space">Hulanicki, A., Subalgebra of L^1(G) associated with the Laplacian on a Lie group, Coll. Math., 31, 259-278 (1974) · <a href="/0316.43005" class="nowrap">Zbl 0316.43005</a></td> </tr><tr> <td>[11]</td> <td class="space">Hulanicki, A., Commutative subalgebra of L^1(G) associated with a subelliptic operator on a Lie group G, Bull. Amer. Math. Soc., 81, 121-124 (1975) · <a href="/0311.22010" class="nowrap">Zbl 0311.22010</a></td> </tr><tr> <td>[12]</td> <td class="space">Hulanicki, A.; Jenkins, J. W., Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc., 278, 703-715 (1983) · <a href="/0516.43010" class="nowrap">Zbl 0516.43010</a></td> </tr><tr> <td>[13]</td> <td class="space">Mauceri, G., Zonal multipliers on the Heisenberg group, Pacific J. of Math., 95, 143-159 (1981) · <a href="/0474.43009" class="nowrap">Zbl 0474.43009</a></td> </tr><tr> <td>[14]</td> <td class="space">G.Mauceri,Maximal operators and Riesz means on stratified groups, to appear in Symp. Math. · <a href="/0659.22009" class="nowrap">Zbl 0659.22009</a></td> </tr><tr> <td>[15]</td> <td class="space">Miyachi, A., On some Fourier multipliers for H^p(R^p, J. Fac. Sci. Univ. Tokyo, 27, 157-179 (1980) · <a href="/0433.42019" class="nowrap">Zbl 0433.42019</a></td> </tr><tr> <td>[16]</td> <td class="space">Taibleson, M. H.; Weiss, G., The molecular characterization of certain Hardy spaces, Astérique, 77, 67-149 (1980) · <a href="/0472.46041" class="nowrap">Zbl 0472.46041</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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