{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:LXLYLTZH5UJHFFYIZGUQD2MAHL","short_pith_number":"pith:LXLYLTZH","canonical_record":{"source":{"id":"1507.04334","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-15T19:17:57Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8e611481bce3b8cb8ba89d4d9e7163d48be77bb6fee0fb297a1cc90f97647e9a","abstract_canon_sha256":"856a5816a27bd8f9e3bbb188cd39ba730dd7315e05392ca7d05eababd3143034"},"schema_version":"1.0"},"canonical_sha256":"5dd785cf27ed12729708c9a901e9803ad9fdd4df05363dc4a373dae24c64c1cb","source":{"kind":"arxiv","id":"1507.04334","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04334","created_at":"2026-05-18T00:41:53Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04334v6","created_at":"2026-05-18T00:41:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04334","created_at":"2026-05-18T00:41:53Z"},{"alias_kind":"pith_short_12","alias_value":"LXLYLTZH5UJH","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LXLYLTZH5UJHFFYI","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LXLYLTZH","created_at":"2026-05-18T12:29:29Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:LXLYLTZH5UJHFFYIZGUQD2MAHL","target":"record","payload":{"canonical_record":{"source":{"id":"1507.04334","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-15T19:17:57Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"8e611481bce3b8cb8ba89d4d9e7163d48be77bb6fee0fb297a1cc90f97647e9a","abstract_canon_sha256":"856a5816a27bd8f9e3bbb188cd39ba730dd7315e05392ca7d05eababd3143034"},"schema_version":"1.0"},"canonical_sha256":"5dd785cf27ed12729708c9a901e9803ad9fdd4df05363dc4a373dae24c64c1cb","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:41:53.892137Z","signature_b64":"1dg3stK2fVYoH9zCxswnRk9av72fmc1CUGYvKVSFC3ezv/2uIRzHJarF0KrCvX9KUosKoeSDkeWCZlVp3md7DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5dd785cf27ed12729708c9a901e9803ad9fdd4df05363dc4a373dae24c64c1cb","last_reissued_at":"2026-05-18T00:41:53.891458Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:41:53.891458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.04334","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:41:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RWTgNELiKJkNViH/FM0KDMx9fKJ6M6fiBE0qslD5KV8zvwIEJyhhP9qGdvP5IFQ2BFRaMfv+rpcIgSJD6oUsDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:23:44.960067Z"},"content_sha256":"47523b1ab04a0853dcd5fd021f0a6d5875df745a1979bd181d872933a59de455","schema_version":"1.0","event_id":"sha256:47523b1ab04a0853dcd5fd021f0a6d5875df745a1979bd181d872933a59de455"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:LXLYLTZH5UJHFFYIZGUQD2MAHL","target":"graph","payload":{"graph_snapshot":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:41:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0PfTp1T0hj0dyaMg8rUQQLyLiRIpYKLpQZ0BSbjQGHMp0B3qsC4TWojJmhbf0WzfZw8RblED8v5UiNzvfuoxAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T17:23:44.960807Z"},"content_sha256":"ca47edf4affcd32c3226106118e9a6bc71f486a99f4a0a61541b4f2e11ee7e21","schema_version":"1.0","event_id":"sha256:ca47edf4affcd32c3226106118e9a6bc71f486a99f4a0a61541b4f2e11ee7e21"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LXLYLTZH5UJHFFYIZGUQD2MAHL/bundle.json","state_url":"https://pith.science/pith/LXLYLTZH5UJHFFYIZGUQD2MAHL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LXLYLTZH5UJHFFYIZGUQD2MAHL/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":"2026-06-08T17:23:44Z","links":{"resolver":"https://pith.science/pith/LXLYLTZH5UJHFFYIZGUQD2MAHL","bundle":"https://pith.science/pith/LXLYLTZH5UJHFFYIZGUQD2MAHL/bundle.json","state":"https://pith.science/pith/LXLYLTZH5UJHFFYIZGUQD2MAHL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LXLYLTZH5UJHFFYIZGUQD2MAHL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:LXLYLTZH5UJHFFYIZGUQD2MAHL","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":"856a5816a27bd8f9e3bbb188cd39ba730dd7315e05392ca7d05eababd3143034","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-15T19:17:57Z","title_canon_sha256":"8e611481bce3b8cb8ba89d4d9e7163d48be77bb6fee0fb297a1cc90f97647e9a"},"schema_version":"1.0","source":{"id":"1507.04334","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.04334","created_at":"2026-05-18T00:41:53Z"},{"alias_kind":"arxiv_version","alias_value":"1507.04334v6","created_at":"2026-05-18T00:41:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.04334","created_at":"2026-05-18T00:41:53Z"},{"alias_kind":"pith_short_12","alias_value":"LXLYLTZH5UJH","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_16","alias_value":"LXLYLTZH5UJHFFYI","created_at":"2026-05-18T12:29:29Z"},{"alias_kind":"pith_short_8","alias_value":"LXLYLTZH","created_at":"2026-05-18T12:29:29Z"}],"graph_snapshots":[{"event_id":"sha256:ca47edf4affcd32c3226106118e9a6bc71f486a99f4a0a61541b4f2e11ee7e21","target":"graph","created_at":"2026-05-18T00:41:53Z","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":"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","authors_text":"Mart\\'i Prats","cross_cats":["math.AP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-15T19:17:57Z","title":"Sobolev regularity of the Beurling transform on planar domains"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.04334","kind":"arxiv","version":6},"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":"sha256:47523b1ab04a0853dcd5fd021f0a6d5875df745a1979bd181d872933a59de455","target":"record","created_at":"2026-05-18T00:41:53Z","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":"856a5816a27bd8f9e3bbb188cd39ba730dd7315e05392ca7d05eababd3143034","cross_cats_sorted":["math.AP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-07-15T19:17:57Z","title_canon_sha256":"8e611481bce3b8cb8ba89d4d9e7163d48be77bb6fee0fb297a1cc90f97647e9a"},"schema_version":"1.0","source":{"id":"1507.04334","kind":"arxiv","version":6}},"canonical_sha256":"5dd785cf27ed12729708c9a901e9803ad9fdd4df05363dc4a373dae24c64c1cb","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5dd785cf27ed12729708c9a901e9803ad9fdd4df05363dc4a373dae24c64c1cb","first_computed_at":"2026-05-18T00:41:53.891458Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:41:53.891458Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1dg3stK2fVYoH9zCxswnRk9av72fmc1CUGYvKVSFC3ezv/2uIRzHJarF0KrCvX9KUosKoeSDkeWCZlVp3md7DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:41:53.892137Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.04334","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:47523b1ab04a0853dcd5fd021f0a6d5875df745a1979bd181d872933a59de455","sha256:ca47edf4affcd32c3226106118e9a6bc71f486a99f4a0a61541b4f2e11ee7e21"],"state_sha256":"1fd68e4335c13dc140139f6c843210c43ebbe60a7b13b6b48c681b14fa725262"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MUDk188uQ9COXD4F/g6+841wfcGwu2wWPDoJT18tATj7BXKCeyoV4/EwNLz7wlNHil8K/khnXzVEw34KoJzfBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T17:23:44.965221Z","bundle_sha256":"922e6a12e09da97a83244ccc1686fb92cb38a5522cc36467b1ed25db35a2cf94"}}