{"id":64,"date":"2015-10-15T13:13:19","date_gmt":"2015-10-15T11:13:19","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/mprats\/?p=64"},"modified":"2025-07-09T15:48:34","modified_gmt":"2025-07-09T13:48:34","slug":"marti-prats-singular-integral-operators-on-sobolev-spaces-on-domains-and-quasiconformal-mappings-phd-dissertation","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/mprats\/2015\/10\/15\/marti-prats-singular-integral-operators-on-sobolev-spaces-on-domains-and-quasiconformal-mappings-phd-dissertation\/","title":{"rendered":"Mart\u00ed Prats: Singular integral operators on sobolev spaces on domains and quasiconformal mappings (PhD dissertation)"},"content":{"rendered":"\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:\/\/www.tdx.cat\/bitstream\/handle\/10803\/314193\/mps1de1.pdf?sequence=1\" target=\"_blank\" rel=\"noreferrer noopener\">TDX<\/a><\/div>\n<\/div>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">In this dissertation some new results on the boundedness of Calder\u00f3n-Zygmund operators on Sobolev spaces on domains in \\(R^d\\). First a \\(T(P)\\)-theorem is obtained which is valid for \\(W^{n,p} (U)\\), where \\(U\\) is a bounded uniform domain of \\(R^d\\), \\(n\\) is a given natural number and \\(p&gt;d\\). Essentially, the result obtained states that a convolution Calder\u00f3n-Zygmund operator is bounded on this function space if and only if \\(T(P)\\) belongs to \\(W^{n,p} (U)\\) for every polynomial \\(P\\) of degree smaller than \\(n\\) restricted to the domain. For indices \\(p\\) less or equal than \\(d\\), a sufficient condition for the boundedness in terms of Carleson measures is obtained. In the particular case of \\( n=1 \\) and \\(p \\leq d\\), this Carleson condition is shown to be necessary in fact. The case where \\(n\\) is not integer and \\(0 &lt; n &lt; 1\\) is also studied, and analogous results to the former are obtained for a larger family of function spaces, the so-called Triebel-Lizorkin spaces. The thesis contains some optimal conditions to establish when the Beurling transform of a polynomial restricted to a domain is contained in a Sobolev space \\(W^{n,p}(U)\\), where \\(U\\) is a bounded planar lipschitz domain, in terms of the Besov regularity of the boundary of \\(U\\). This result, in combination with the one mentioned above, provides a condition to determine whether the Beurling transform is bounded on \\(W^{n,p}(U) \\) or not for \\(p&gt;2\\), which is optimal in case \\(n=1\\).<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Finally, an application of the aforementioned results is given for quasiconformal mappings in the complex plane. In particular, it is checked that the regularity \\(W^{n,p}(U)\\) of the Beltrami coefficient of a quasiconformal mapping for a bounded Lipschitz domain \\(U\\) with boundary parameterizations in a certain Besov space and \\(p&gt;2\\), implies that the mapping itself is in \\(W^{n+1,p}(U)\\).<\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this dissertation some new results on the boundedness of Calder\u00f3n-Zygmund operators on Sobolev spaces on domains in \\(R^d\\). First a \\(T(P)\\)-theorem is obtained which is valid for \\(W^{n,p} (U)\\), where \\(U\\) is a bounded uniform domain of \\(R^d\\), \\(n\\) is a given natural number and \\(p&gt;d\\). Essentially, the result obtained states that a convolution &hellip; <a href=\"https:\/\/mat.uab.cat\/web\/mprats\/2015\/10\/15\/marti-prats-singular-integral-operators-on-sobolev-spaces-on-domains-and-quasiconformal-mappings-phd-dissertation\/\" class=\"more-link\">Continua la lectura de <span class=\"screen-reader-text\">Mart\u00ed Prats: Singular integral operators on sobolev spaces on domains and quasiconformal mappings (PhD dissertation)<\/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":[28,95],"tags":[24,39],"class_list":["post-64","post","type-post","status-publish","format-standard","hentry","category-books-en","category-papers-en","tag-books","tag-papers-en"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/64","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=64"}],"version-history":[{"count":3,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/64\/revisions"}],"predecessor-version":[{"id":461,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/posts\/64\/revisions\/461"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/media?parent=64"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/categories?post=64"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/mprats\/wp-json\/wp\/v2\/tags?post=64"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}