{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:FB5U6CI4NIT3ZQTYSDBM74P2ZX","short_pith_number":"pith:FB5U6CI4","schema_version":"1.0","canonical_sha256":"287b4f091c6a27bcc27890c2cff1facdedeb56506061cb0cc3f112a5b9ff3d99","source":{"kind":"arxiv","id":"1110.1302","version":3},"attestation_state":"computed","paper":{"title":"Calder\\'on-Zygmund kernels and rectifiability in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Joan Mateu, Laura Prat, Vasilis Chousionis, Xavier Tolsa","submitted_at":"2011-10-06T15:43:16Z","abstract_excerpt":"Let $E \\subset \\C$ be a Borel set with finite length, that is, $0<\\mathcal{H}^1 (E)<\\infty$. By a theorem of David and L\\'eger, the $L^2 (\\mathcal{H}^1 \\lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts $x / |z|^2,y / |z|^2,z=(x,y) \\in \\C$) implies that $E$ is rectifiable. We extend this result to any kernel of the form $x^{2n-1} /|z|^{2n}, z=(x,y) \\in \\C,n \\in \\mathbb{N}$. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose $L^2$-boundedness implies rectifiability"},"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":"1110.1302","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2011-10-06T15:43:16Z","cross_cats_sorted":[],"title_canon_sha256":"56297c6ee9e81c3b3049dabdbd3b27d25cd1e74e40b942bb37b1364406a4332a","abstract_canon_sha256":"3497e272546a42d3002b3be63a66a8ef67519e287571a670752a7e371c7392cf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:02:21.524387Z","signature_b64":"We3/nKJzyrG4gfBMALFKrjhq0qdjFW7JheR5zWUCXQUtIs4E7YZEJLDr5a/bI8O0oWH7NK9yfGWeWU1SuKuYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"287b4f091c6a27bcc27890c2cff1facdedeb56506061cb0cc3f112a5b9ff3d99","last_reissued_at":"2026-05-18T01:02:21.523708Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:02:21.523708Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Calder\\'on-Zygmund kernels and rectifiability in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Joan Mateu, Laura Prat, Vasilis Chousionis, Xavier Tolsa","submitted_at":"2011-10-06T15:43:16Z","abstract_excerpt":"Let $E \\subset \\C$ be a Borel set with finite length, that is, $0<\\mathcal{H}^1 (E)<\\infty$. By a theorem of David and L\\'eger, the $L^2 (\\mathcal{H}^1 \\lfloor E)$-boundedness of the singular integral associated to the Cauchy kernel (or even to one of its coordinate parts $x / |z|^2,y / |z|^2,z=(x,y) \\in \\C$) implies that $E$ is rectifiable. We extend this result to any kernel of the form $x^{2n-1} /|z|^{2n}, z=(x,y) \\in \\C,n \\in \\mathbb{N}$. We thus provide the first non-trivial examples of operators not directly related with the Cauchy transform whose $L^2$-boundedness implies rectifiability"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.1302","kind":"arxiv","version":3},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1110.1302","created_at":"2026-05-18T01:02:21.523825+00:00"},{"alias_kind":"arxiv_version","alias_value":"1110.1302v3","created_at":"2026-05-18T01:02:21.523825+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1110.1302","created_at":"2026-05-18T01:02:21.523825+00:00"},{"alias_kind":"pith_short_12","alias_value":"FB5U6CI4NIT3","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FB5U6CI4NIT3ZQTY","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FB5U6CI4","created_at":"2026-05-18T12:26:28.662955+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX","json":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX.json","graph_json":"https://pith.science/api/pith-number/FB5U6CI4NIT3ZQTYSDBM74P2ZX/graph.json","events_json":"https://pith.science/api/pith-number/FB5U6CI4NIT3ZQTYSDBM74P2ZX/events.json","paper":"https://pith.science/paper/FB5U6CI4"},"agent_actions":{"view_html":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX","download_json":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX.json","view_paper":"https://pith.science/paper/FB5U6CI4","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1110.1302&json=true","fetch_graph":"https://pith.science/api/pith-number/FB5U6CI4NIT3ZQTYSDBM74P2ZX/graph.json","fetch_events":"https://pith.science/api/pith-number/FB5U6CI4NIT3ZQTYSDBM74P2ZX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX/action/storage_attestation","attest_author":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX/action/author_attestation","sign_citation":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX/action/citation_signature","submit_replication":"https://pith.science/pith/FB5U6CI4NIT3ZQTYSDBM74P2ZX/action/replication_record"}},"created_at":"2026-05-18T01:02:21.523825+00:00","updated_at":"2026-05-18T01:02:21.523825+00:00"}