{"id":49,"date":"2017-10-05T13:04:12","date_gmt":"2017-10-05T11:04:12","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/mprats\/?p=49"},"modified":"2025-07-09T15:37:56","modified_gmt":"2025-07-09T13:37:56","slug":"marti-prats-sobolev-regularity-of-the-beurling-transform-on-planar-domains","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/mprats\/2017\/10\/05\/marti-prats-sobolev-regularity-of-the-beurling-transform-on-planar-domains\/","title":{"rendered":"Mart\u00ed Prats: Sobolev regularity of the Beurling transform on planar 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\/1507.04334\" 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:\/\/mat.uab.cat\/pubmat\/fitxers\/download\/FileType:pdf\/FolderName:v61(2)\/FileName:61217_01.pdf\" target=\"_blank\" rel=\"noreferrer noopener\">Publ. Mat.<\/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\">Consider a Lipschitz domain \\(\\Omega\\) and the Beurling transform of its characteristic function \\( \\mathcal{B} \\chi_\\Omega(z)= &#8211; {\\rm p.v.}\\frac1{\\pi z^2}*\\chi_\\Omega (z) \\). It is shown that if the outward unit normal vector \\( N \\) of the boundary of the domain is in the trace space of \\( W^{n,p}(\\Omega) \\) (i.e., the Besov space \\( B^{n-1\/p}_{p,p}(\\partial\\Omega) \\)) then \\(\\mathcal{B} \\chi_\\Omega \\in W^{n,p}(\\Omega) \\). Moreover, when \\( p&gt;2 \\) the boundedness of the Beurling transform on \\( W^{n,p}(\\Omega) \\) follows. This fact has far-reaching consequences in the study of the regularity of quasiconformal solutions of the Beltrami equation.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Consider a Lipschitz domain \\(\\Omega\\) and the Beurling transform of its characteristic function \\( \\mathcal{B} \\chi_\\Omega(z)= &#8211; {\\rm p.v.}\\frac1{\\pi z^2}*\\chi_\\Omega (z) \\). It is shown that if the outward unit normal vector \\( N \\) of the boundary of the domain is in the trace space of \\( W^{n,p}(\\Omega) \\) (i.e., the Besov space \\( &hellip; <a href=\"https:\/\/mat.uab.cat\/web\/mprats\/2017\/10\/05\/marti-prats-sobolev-regularity-of-the-beurling-transform-on-planar-domains\/\" class=\"more-link\">Continua la lectura de <span class=\"screen-reader-text\">Mart\u00ed Prats: Sobolev regularity of the Beurling transform on planar 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-49","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\/49","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=49"}],"version-history":[{"count":5,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/49\/revisions"}],"predecessor-version":[{"id":460,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/49\/revisions\/460"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/media?parent=49"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/categories?post=49"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/tags?post=49"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}