{"id":352,"date":"2025-07-09T14:16:15","date_gmt":"2025-07-09T13:16:15","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=352"},"modified":"2026-05-28T19:12:40","modified_gmt":"2026-05-28T18:12:40","slug":"la-transformada-de-beurling-de-dominis-polinomials","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/la-transformada-de-beurling-de-dominis-polinomials\/","title":{"rendered":"La transformada de Beurling i aplicacions quasiconformes"},"content":{"rendered":"\n\nLa transformada de Beurling \\[Bf(z)=-\\frac1\\pi \\lim_{\\varepsilon\\to o}\\int_{|w-z|&gt;\\varepsilon} \\frac{f(w)}{(z-w)^2}\\, dm(w)\\] \u00e9s un operador important en an\u00e0lisi complexa. Aquesta import\u00e0ncia rau en el fet que si \\(f\\in W^{1,p}(\\mathbb C)\\), aleshores \\[B(\\bar \\partial f)=\\partial f.\\]\n\nAquest fet fa que sigui fonamental la seva comprensi\u00f3 per tal d&#8217;entendre certes EDP&#8217;s al pla, en particular \u00e9s cabdal en l&#8217;estudi de les aplicacions quasiconformes (homeomorfismes entre dominis del pla complex que deformen els angles de manera controlada). Possibles l\u00ednies de treball:\n<ul>\n \t<li>La transformada de Beurling \u00e9s un cas particular d&#8217;operador de Calder\u00f3n-Zygmund, i quan es treballa amb aquest tipus d&#8217;operadors sol ser fonamental saber com es comporten en actuar sobre constants. Si restringim l&#8217;estudi a dominis del pla \\(\\Omega\\), ens interessa con\u00e8ixer el comportament de \\(B\\chi_\\Omega\\). [2]\nEn dominis prou bons, usant les propietats esmentades, es pot calcular expl\u00edcitament la funci\u00f3 \\(B\\chi_\\Omega\\). Per exemple quan \\(\\Omega=\\mathbb D\\) la seva\u00a0 transformada \u00e9s \\(0\\) a l&#8217;interior del disc i es coneix el seu valor tamb\u00e9 sobre el complementari. [1]\nAplicarem aquests coneixements a l&#8217;estudi d&#8217;el\u00b7lipses, par\u00e0boles i dominis de vora polinomial, per la qual cosa ens convindr\u00e0 estendre les definicions a \\(BMO(\\mathbb C)\\). [3]<\/li>\n \t<li>La composici\u00f3 amb aplicacions quasiconformes que tenen certa regularitat preserva la integrabilitat de les derivades amb una certa p\u00e8rdua a l&#8217;exponent [4]. La soluci\u00f3 de l&#8217;equaci\u00f3 de Beltrami \\((I-\\mu B) (\\bar \\partial f)=\\mu\\) tamb\u00e9 permet inferir integrabilitat de les derivades de \\(\\partial f\\) si tenim integrabilitat de les derivades de \\(\\mu\\) [5]. Buscarem casos extremals de p\u00e8rdua d&#8217;integrabilitat en algun d&#8217;aquests dos contextos.<\/li>\n \t<li>Es poden desenvolupar temes d&#8217;an\u00e0lisi complexa en general, m\u00e9s enll\u00e0 de les propostes sobre transformada de Beurling i aplicacions quasiconformes, evidentment. En aquest cas, entrarem amb un enfocament prospectiu, a descobrir un camp en el que no soc expert i que tinc inter\u00e8s a entendre algun dia.\n\n \n<ul class=\"wp-block-list\">\n \t<li>M\u00e8todes d\u2019interpolaci\u00f3 complexa (Riesz-Thorin i Calder\u00f3n). Per exemple, llegir i entendre el resultat d&#8217;enguany <a href=\"https:\/\/arxiv.org\/pdf\/2605.27119\">https:\/\/arxiv.org\/pdf\/2605.27119<\/a><\/li>\n \n \t<li>Circle packing [Kenneth Stephenson]<\/li>\n \n \t<li>Teorema de l\u2019aplicaci\u00f3 de Riemann, amb aplicaci\u00f3 a dominis de frontera poligonal i arcs de disc: [Nehadi].<\/li>\n \n \t<li>Jensen formula, Blashke products. [Conway]<\/li>\n \n \t<li>Teorema dels nombres primers.<\/li>\n \n \t<li>Funci\u00f3 zeta de Riemann.<\/li>\n \n \t<li>Teorema de Picard.<\/li>\n \n \t<li>Funcions enteres (veure [Boas])<\/li>\n \n \t<li>Teorema de Muntz-Sz\u00e1sz.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n[1] Astala, Kari, Tadeusz Iwaniec, and Gaven Martin. <i>Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)<\/i>. Princeton University Press, 2008.\n\n[2] Prats, M. (2019). Sobolev regularity of quasiconformal mappings on domains. <i>Journal d\u2019Analyse Math\u00e9matique<\/i>, <i>138<\/i>(2), 513-562. https:\/\/doi.org\/10.1007\/s11854-019-0031-9\n\n[3] Prats, Mart\u00ed. &#8220;Sobolev regularity of the Beurling transform on planar domains.&#8221; <i>Publicacions Matem\u00e0tiques<\/i> 61.2 (2017): 291-336.\n\n[4] Oliva, M., &amp; Prats, M. (2017). Sharp bounds for composition with quasiconformal mappings in Sobolev spaces. <i>Journal of Mathematical Analysis and Applications<\/i>, <i>451<\/i>(2), 1026-1044. https:\/\/doi.org\/10.1016\/j.jmaa.2017.02.016\n\n[5] Prats, Marti. &#8220;Beltrami equations in the plane and Sobolev regularity.&#8221; <i>Communications on Pure and Applied Analysis<\/i> 17.2 (2018): 319-332.","protected":false},"excerpt":{"rendered":"<p>La transformada de Beurling \\[Bf(z)=-\\frac1\\pi \\lim_{\\varepsilon\\to o}\\int_{|w-z|&gt;\\varepsilon} \\frac{f(w)}{(z-w)^2}\\, dm(w)\\] \u00e9s un operador important en an\u00e0lisi complexa. Aquesta import\u00e0ncia rau en el fet que si \\(f\\in W^{1,p}(\\mathbb C)\\), aleshores \\[B(\\bar \\partial f)=\\partial f.\\] Aquest fet fa que sigui fonamental la seva comprensi\u00f3 per tal d&#8217;entendre certes EDP&#8217;s al pla, en particular \u00e9s cabdal en l&#8217;estudi de [&hellip;]<\/p>\n","protected":false},"author":53,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[40],"class_list":["post-352","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","tag-marti-prats"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/352","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/53"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=352"}],"version-history":[{"count":10,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/352\/revisions"}],"predecessor-version":[{"id":696,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/352\/revisions\/696"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=352"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=352"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=352"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}