{"paper":{"title":"Sobolev regularity of the Beurling transform on planar domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Mart\\'i Prats","submitted_at":"2015-07-15T19:17:57Z","abstract_excerpt":"Consider a Lipschitz domain $\\Omega$ and the Beurling transform of its characteristic function $\\mathcal{B} \\chi_\\Omega(z)= - {\\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 $p>2$ 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04334","kind":"arxiv","version":6},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}