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Using a priori estimates, in this paper we solve the following free boundary problem: if the Beurling transform of 1_D belongs to the Sobolev space W^{a,p}(D) for 0<a\\leq 1, 1<p<\\infty such that ap>1, then the outward unit normal N on bD, the boundary of D, is in the Besov space B_{p,p}^{a-1/p}(bD). The converse statement, proved previously by Cruz and Tolsa, also holds. So we have that B(1_D) is in W^{a,p}(D) if and only if N is in B_{p,p}^{a"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1201.5403","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-25T22:14:44Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"564b836d9f7a1db3d1f2d5876ab0ec34f0097889ad4fb3ce9e536537af926fb8","abstract_canon_sha256":"d74ac84d13633b3c977318dfa3562da7d1b553ebf499ee265ef2aa1a9f3c500c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:54.083274Z","signature_b64":"Ht8YWKxtGONoWLfTwEH1TOLnzYTq0WGR81IdDD3P07NRrbhoRVpmTfd+otoC1utpAh6hTbjXmOk7xFlZysVyAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5f3a5e2200f49b19f85f61fdd04cf7b2b0a89520ae2a2c3177c12d07fdb44f21","last_reissued_at":"2026-05-18T04:03:54.082709Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:54.082709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of C^1 and Lipschitz domains in terms of the Beurling transform","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Xavier Tolsa","submitted_at":"2012-01-25T22:14:44Z","abstract_excerpt":"Let D be a bounded planar C^1 domain, or a Lipschitz domain \"flat enough\", and consider the Beurling transform of 1_D, the characteristic function of D. 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