{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:L45F4IQA6SNRT6C7MH65ATHXWK","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"d74ac84d13633b3c977318dfa3562da7d1b553ebf499ee265ef2aa1a9f3c500c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-25T22:14:44Z","title_canon_sha256":"564b836d9f7a1db3d1f2d5876ab0ec34f0097889ad4fb3ce9e536537af926fb8"},"schema_version":"1.0","source":{"id":"1201.5403","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.5403","created_at":"2026-05-18T04:03:54Z"},{"alias_kind":"arxiv_version","alias_value":"1201.5403v1","created_at":"2026-05-18T04:03:54Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.5403","created_at":"2026-05-18T04:03:54Z"},{"alias_kind":"pith_short_12","alias_value":"L45F4IQA6SNR","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_16","alias_value":"L45F4IQA6SNRT6C7","created_at":"2026-05-18T12:27:14Z"},{"alias_kind":"pith_short_8","alias_value":"L45F4IQA","created_at":"2026-05-18T12:27:14Z"}],"graph_snapshots":[{"event_id":"sha256:b118f6969e17e803039db696bd4e3acc227ca2ada4d830d87a7c69e58a52da3f","target":"graph","created_at":"2026-05-18T04:03:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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. 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","authors_text":"Xavier Tolsa","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-25T22:14:44Z","title":"Regularity of C^1 and Lipschitz domains in terms of the Beurling transform"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5403","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:bca0d5e9573516b47e7a3474746cf8f92f8d5dd6b8da4fc2ea66a45313e77132","target":"record","created_at":"2026-05-18T04:03:54Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"d74ac84d13633b3c977318dfa3562da7d1b553ebf499ee265ef2aa1a9f3c500c","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2012-01-25T22:14:44Z","title_canon_sha256":"564b836d9f7a1db3d1f2d5876ab0ec34f0097889ad4fb3ce9e536537af926fb8"},"schema_version":"1.0","source":{"id":"1201.5403","kind":"arxiv","version":1}},"canonical_sha256":"5f3a5e2200f49b19f85f61fdd04cf7b2b0a89520ae2a2c3177c12d07fdb44f21","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f3a5e2200f49b19f85f61fdd04cf7b2b0a89520ae2a2c3177c12d07fdb44f21","first_computed_at":"2026-05-18T04:03:54.082709Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:54.082709Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Ht8YWKxtGONoWLfTwEH1TOLnzYTq0WGR81IdDD3P07NRrbhoRVpmTfd+otoC1utpAh6hTbjXmOk7xFlZysVyAw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:54.083274Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.5403","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bca0d5e9573516b47e7a3474746cf8f92f8d5dd6b8da4fc2ea66a45313e77132","sha256:b118f6969e17e803039db696bd4e3acc227ca2ada4d830d87a7c69e58a52da3f"],"state_sha256":"706328abbd4279a36703e5f88facb043f9afe894074bb0571872829805c72822"}