{"id":34,"date":"2015-10-05T12:44:35","date_gmt":"2015-10-05T10:44:35","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/mprats\/?p=34"},"modified":"2025-07-09T15:49:27","modified_gmt":"2025-07-09T13:49:27","slug":"marti-prats-and-xavier-tolsa-a-tp-theorem-for-sobolev-spaces-on-domains","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/mprats\/2015\/10\/05\/marti-prats-and-xavier-tolsa-a-tp-theorem-for-sobolev-spaces-on-domains\/","title":{"rendered":"Mart\u00ed Prats and Xavier Tolsa: A T(P) theorem for Sobolev spaces on domains"},"content":{"rendered":"\n<div class=\"wp-block-group is-nowrap is-layout-flex wp-container-core-group-is-layout-8f761849 wp-block-group-is-layout-flex\">\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link wp-element-button\" href=\"https:\/\/arxiv.org\/abs\/1406.4769\" target=\"_blank\" rel=\"noreferrer noopener\">arXiv<\/a><\/div>\n<\/div>\n\n\n\n<div class=\"wp-block-buttons is-layout-flex wp-block-buttons-is-layout-flex\">\n<div class=\"wp-block-button\"><a class=\"wp-block-button__link wp-element-button\" href=\"http:\/\/dx.doi.org\/10.1016\/j.jfa.2015.01.007\" target=\"_blank\" rel=\"noreferrer noopener\">J. Funct. Anal.<\/a><\/div>\n<\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given \\( 0&lt; s \\leq 1 \\), \\( 1 &lt; p &lt; \\infty \\) with \\(sp&gt;2\\) and a Lipschitz domain \\(\\Omega\\subset \\mathbb{C}\\), the Beurling transform \\(Bf=- {\\rm p.v.}\\frac1{\\pi z^2}*f\\) is bounded in the Sobolev space \\(W^{s,p}(\\Omega)\\) if and only if \\(B\\chi_\\Omega\\in W^{s,p}(\\Omega)\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In this paper we obtain a generalized version of the former result valid for any \\(s\\in \\mathbb{N}\\) and for a larger family of Calder\u00f3n-Zygmund operators in any ambient space \\(\\mathbb{R}^d\\) as long as \\(p&gt;d\\). In that case we need to check the boundedness not only over the characteristic function of the domain, but over a finite collection of polynomials restricted to the domain. Finally we find a sufficient condition in terms of Carleson measures for \\(p\\leq d\\). In the particular case \\(s=1\\), this condition is in fact necessary, which yields a complete characterization.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Recently, V. Cruz, J. Mateu and J. Orobitg have proved a T(1) theorem for the Beurling transform in the complex plane. It asserts that given \\( 0&lt; s \\leq 1 \\), \\( 1 &lt; p &lt; \\infty \\) with \\(sp&gt;2\\) and a Lipschitz domain \\(\\Omega\\subset \\mathbb{C}\\), the Beurling transform \\(Bf=- {\\rm p.v.}\\frac1{\\pi z^2}*f\\) is bounded &hellip; <a href=\"https:\/\/mat.uab.cat\/web\/mprats\/2015\/10\/05\/marti-prats-and-xavier-tolsa-a-tp-theorem-for-sobolev-spaces-on-domains\/\" class=\"more-link\">Continua la lectura de <span class=\"screen-reader-text\">Mart\u00ed Prats and Xavier Tolsa: A T(P) theorem for Sobolev spaces on domains<\/span> <span class=\"meta-nav\">&rarr;<\/span><\/a><\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[95],"tags":[39],"class_list":["post-34","post","type-post","status-publish","format-standard","hentry","category-papers-en","tag-papers-en"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/34","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/users\/53"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/comments?post=34"}],"version-history":[{"count":11,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/34\/revisions"}],"predecessor-version":[{"id":462,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/34\/revisions\/462"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/media?parent=34"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/categories?post=34"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/tags?post=34"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}