{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:V4NQQOHBYIV4BBAW4IGDCDPBGL","short_pith_number":"pith:V4NQQOHB","canonical_record":{"source":{"id":"1704.07560","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-25T06:55:46Z","cross_cats_sorted":[],"title_canon_sha256":"b97971089d4b819e336323afe703a4c0236083f61d4d88e79849a7c577dfaa83","abstract_canon_sha256":"ee4485862eaa7a70b290175aed760f0475076b52c966c56a125811bad115d09c"},"schema_version":"1.0"},"canonical_sha256":"af1b0838e1c22bc08416e20c310de132f0dfe30920f4c2e2462f1fcfb2993ce6","source":{"kind":"arxiv","id":"1704.07560","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.07560","created_at":"2026-05-18T00:43:50Z"},{"alias_kind":"arxiv_version","alias_value":"1704.07560v2","created_at":"2026-05-18T00:43:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.07560","created_at":"2026-05-18T00:43:50Z"},{"alias_kind":"pith_short_12","alias_value":"V4NQQOHBYIV4","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"V4NQQOHBYIV4BBAW","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"V4NQQOHB","created_at":"2026-05-18T12:31:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:V4NQQOHBYIV4BBAW4IGDCDPBGL","target":"record","payload":{"canonical_record":{"source":{"id":"1704.07560","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-25T06:55:46Z","cross_cats_sorted":[],"title_canon_sha256":"b97971089d4b819e336323afe703a4c0236083f61d4d88e79849a7c577dfaa83","abstract_canon_sha256":"ee4485862eaa7a70b290175aed760f0475076b52c966c56a125811bad115d09c"},"schema_version":"1.0"},"canonical_sha256":"af1b0838e1c22bc08416e20c310de132f0dfe30920f4c2e2462f1fcfb2993ce6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:50.246234Z","signature_b64":"P76b4pvS4f8XYcMakvI7CUAkPQfgIOCsmWpaIF6IluLh/DwtzLvL3ti3xGRvk1RTDgJN2MSPgYXDDTN4NqfVAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"af1b0838e1c22bc08416e20c310de132f0dfe30920f4c2e2462f1fcfb2993ce6","last_reissued_at":"2026-05-18T00:43:50.245539Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:50.245539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1704.07560","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:43:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CAgJq7Xc6evQW8PE8rGzSkT+eDWf54nkHcJgA7awMWaVpq7fZUnw8IS041fmv7nUHEkwak7ReGRR5lNbdGEoCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:22:10.062566Z"},"content_sha256":"0b90c24a1f151c661d6188922290017e4c7018670f963f00fbcbda64ce8ea302","schema_version":"1.0","event_id":"sha256:0b90c24a1f151c661d6188922290017e4c7018670f963f00fbcbda64ce8ea302"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:V4NQQOHBYIV4BBAW4IGDCDPBGL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Local elliptic regularity for the Dirichlet fractional Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrique Zuazua, Mahamadi Warma, Umberto Biccari","submitted_at":"2017-04-25T06:55:46Z","abstract_excerpt":"We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-\\Delta)^s$ on an arbitrary bounded open set $\\Omega\\subset\\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\\textrm{loc}}(\\Omega)$, while for $2\\leq p<\\infty$ we show that the solutions belong to $W^{2s,p}_{\\textrm{loc}}(\\Omega)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different method"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07560","kind":"arxiv","version":2},"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":"2026-05-18T00:43:50Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"22t19eylFf0vMebIAl8RkzZNohMhNn2JNGVkSWscWIvFG5RgT7CoE4G00oxTEUPBL+PxipYrbJiX9Mq5ubs/BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T10:22:10.062919Z"},"content_sha256":"9382a8990582079ee418cd5b59345ae9eb07334ea56ae53ccb9c01604cf155d3","schema_version":"1.0","event_id":"sha256:9382a8990582079ee418cd5b59345ae9eb07334ea56ae53ccb9c01604cf155d3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V4NQQOHBYIV4BBAW4IGDCDPBGL/bundle.json","state_url":"https://pith.science/pith/V4NQQOHBYIV4BBAW4IGDCDPBGL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V4NQQOHBYIV4BBAW4IGDCDPBGL/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":"2026-05-28T10:22:10Z","links":{"resolver":"https://pith.science/pith/V4NQQOHBYIV4BBAW4IGDCDPBGL","bundle":"https://pith.science/pith/V4NQQOHBYIV4BBAW4IGDCDPBGL/bundle.json","state":"https://pith.science/pith/V4NQQOHBYIV4BBAW4IGDCDPBGL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V4NQQOHBYIV4BBAW4IGDCDPBGL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:V4NQQOHBYIV4BBAW4IGDCDPBGL","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":"ee4485862eaa7a70b290175aed760f0475076b52c966c56a125811bad115d09c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-25T06:55:46Z","title_canon_sha256":"b97971089d4b819e336323afe703a4c0236083f61d4d88e79849a7c577dfaa83"},"schema_version":"1.0","source":{"id":"1704.07560","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1704.07560","created_at":"2026-05-18T00:43:50Z"},{"alias_kind":"arxiv_version","alias_value":"1704.07560v2","created_at":"2026-05-18T00:43:50Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1704.07560","created_at":"2026-05-18T00:43:50Z"},{"alias_kind":"pith_short_12","alias_value":"V4NQQOHBYIV4","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_16","alias_value":"V4NQQOHBYIV4BBAW","created_at":"2026-05-18T12:31:49Z"},{"alias_kind":"pith_short_8","alias_value":"V4NQQOHB","created_at":"2026-05-18T12:31:49Z"}],"graph_snapshots":[{"event_id":"sha256:9382a8990582079ee418cd5b59345ae9eb07334ea56ae53ccb9c01604cf155d3","target":"graph","created_at":"2026-05-18T00:43:50Z","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":"We analyze the local elliptic regularity of weak solutions to the Dirichlet problem associated with the fractional Laplacian $(-\\Delta)^s$ on an arbitrary bounded open set $\\Omega\\subset\\mathbb{R}^N$. For $1<p<2$, we obtain regularity in the Besov space $B^{2s}_{p,2,\\textrm{loc}}(\\Omega)$, while for $2\\leq p<\\infty$ we show that the solutions belong to $W^{2s,p}_{\\textrm{loc}}(\\Omega)$. The key tool consists in analyzing carefully the elliptic equation satisfied by the solution locally, after cut-off, to later employ sharp regularity results in the whole space. We do it by two different method","authors_text":"Enrique Zuazua, Mahamadi Warma, Umberto Biccari","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-25T06:55:46Z","title":"Local elliptic regularity for the Dirichlet fractional Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.07560","kind":"arxiv","version":2},"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":"sha256:0b90c24a1f151c661d6188922290017e4c7018670f963f00fbcbda64ce8ea302","target":"record","created_at":"2026-05-18T00:43:50Z","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":"ee4485862eaa7a70b290175aed760f0475076b52c966c56a125811bad115d09c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-04-25T06:55:46Z","title_canon_sha256":"b97971089d4b819e336323afe703a4c0236083f61d4d88e79849a7c577dfaa83"},"schema_version":"1.0","source":{"id":"1704.07560","kind":"arxiv","version":2}},"canonical_sha256":"af1b0838e1c22bc08416e20c310de132f0dfe30920f4c2e2462f1fcfb2993ce6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"af1b0838e1c22bc08416e20c310de132f0dfe30920f4c2e2462f1fcfb2993ce6","first_computed_at":"2026-05-18T00:43:50.245539Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:50.245539Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"P76b4pvS4f8XYcMakvI7CUAkPQfgIOCsmWpaIF6IluLh/DwtzLvL3ti3xGRvk1RTDgJN2MSPgYXDDTN4NqfVAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:50.246234Z","signed_message":"canonical_sha256_bytes"},"source_id":"1704.07560","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0b90c24a1f151c661d6188922290017e4c7018670f963f00fbcbda64ce8ea302","sha256:9382a8990582079ee418cd5b59345ae9eb07334ea56ae53ccb9c01604cf155d3"],"state_sha256":"604147307ae456021160ccfcd17e4bc046ecfd80f54d456a8406749fd16a23ee"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+UFBWPs9iAhSfdPV0F65/vaOUWak1m1cd5MNUbQ/KOyizbk5txOeAXhyWQcmXXK9PKkBwUzrPFTbN0VKbPTEBw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T10:22:10.064916Z","bundle_sha256":"4f58495bc4159bb2045c95249f5add38ec9b0904cfa1a14beeecb90a8bd3070f"}}