{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:L45F4IQA6SNRT6C7MH65ATHXWK","short_pith_number":"pith:L45F4IQA","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"},"canonical_sha256":"5f3a5e2200f49b19f85f61fdd04cf7b2b0a89520ae2a2c3177c12d07fdb44f21","source":{"kind":"arxiv","id":"1201.5403","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"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:L45F4IQA6SNRT6C7MH65ATHXWK","target":"record","payload":{"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"},"canonical_sha256":"5f3a5e2200f49b19f85f61fdd04cf7b2b0a89520ae2a2c3177c12d07fdb44f21","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"},"source_kind":"arxiv","source_id":"1201.5403","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:03:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4zPQe2hd9nPxDFppo/taNIAHMVz98PYYhdRiq6Wa7zVrhcY3Eu5STmp9U4tcsUXndmzVGA+AlvW8qfBSxoefBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T13:53:11.158362Z"},"content_sha256":"bca0d5e9573516b47e7a3474746cf8f92f8d5dd6b8da4fc2ea66a45313e77132","schema_version":"1.0","event_id":"sha256:bca0d5e9573516b47e7a3474746cf8f92f8d5dd6b8da4fc2ea66a45313e77132"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:L45F4IQA6SNRT6C7MH65ATHXWK","target":"graph","payload":{"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. 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5403","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:03:54Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tmij4PTs1f/RdEICYecp00g0RwTGghWOvODM45vIka/XU+xh2M5DYKJH4ARFG+SASLEzdYHb+XSMI/0YKJuJDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T13:53:11.158840Z"},"content_sha256":"b118f6969e17e803039db696bd4e3acc227ca2ada4d830d87a7c69e58a52da3f","schema_version":"1.0","event_id":"sha256:b118f6969e17e803039db696bd4e3acc227ca2ada4d830d87a7c69e58a52da3f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/L45F4IQA6SNRT6C7MH65ATHXWK/bundle.json","state_url":"https://pith.science/pith/L45F4IQA6SNRT6C7MH65ATHXWK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/L45F4IQA6SNRT6C7MH65ATHXWK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-05-26T13:53:11Z","links":{"resolver":"https://pith.science/pith/L45F4IQA6SNRT6C7MH65ATHXWK","bundle":"https://pith.science/pith/L45F4IQA6SNRT6C7MH65ATHXWK/bundle.json","state":"https://pith.science/pith/L45F4IQA6SNRT6C7MH65ATHXWK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/L45F4IQA6SNRT6C7MH65ATHXWK/bundle.json"},"state":{"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"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CvjWgjT82/AW1U93TLxvEgVM3WloYaqvNj2ZIEo4SitsWt3iApoXEayALV5zdHDT4uyindnKtb6E2iarD56tBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T13:53:11.161245Z","bundle_sha256":"e449b5d8699a653eb282244cee17e4a50cf920b68a8fc7cb1591541d44555649"}}