{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:DAAHRSVIGSURJ46BMGJWPI7RHU","short_pith_number":"pith:DAAHRSVI","schema_version":"1.0","canonical_sha256":"180078caa834a914f3c1619367a3f13d38056cff89d3ea210e64e66d7018fa1d","source":{"kind":"arxiv","id":"1202.5561","version":3},"attestation_state":"computed","paper":{"title":"Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optimal transport maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Guido De Philippis","submitted_at":"2012-02-24T21:25:22Z","abstract_excerpt":"The aim of this note is to show that Alexandrov solutions of the Monge-Ampere equation, with right hand side bounded away from zero and infinity, converge strongly in $W^{2,1}_{loc}$ if their right hand side converge strongly in $L^1_{loc}$. As a corollary we deduce strong $W^{1,1}_{loc}$ stability of optimal transport maps."},"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":"1202.5561","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-02-24T21:25:22Z","cross_cats_sorted":[],"title_canon_sha256":"7b452e62534eac4da02904fca3b4c21c6e17fb6d3912471d816d2a61d1438b46","abstract_canon_sha256":"4046375ea94b4c9873b384a01ec44f23d7bb281cb5df04d8650fb94e2b1869af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:40:38.621140Z","signature_b64":"me+rEFxzVK4437NpOMP87J1CweMNdvU+ivJSuIZVYecWBJ3qBxuWx7xgo4oDiPjyNlRVQOlyzYdPEiXJCUW0Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"180078caa834a914f3c1619367a3f13d38056cff89d3ea210e64e66d7018fa1d","last_reissued_at":"2026-05-18T03:40:38.620452Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:40:38.620452Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Second order stability for the Monge-Ampere equation and strong Sobolev convergence of optimal transport maps","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Guido De Philippis","submitted_at":"2012-02-24T21:25:22Z","abstract_excerpt":"The aim of this note is to show that Alexandrov solutions of the Monge-Ampere equation, with right hand side bounded away from zero and infinity, converge strongly in $W^{2,1}_{loc}$ if their right hand side converge strongly in $L^1_{loc}$. As a corollary we deduce strong $W^{1,1}_{loc}$ stability of optimal transport maps."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.5561","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":"1202.5561","created_at":"2026-05-18T03:40:38.620557+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.5561v3","created_at":"2026-05-18T03:40:38.620557+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.5561","created_at":"2026-05-18T03:40:38.620557+00:00"},{"alias_kind":"pith_short_12","alias_value":"DAAHRSVIGSUR","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"DAAHRSVIGSURJ46B","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"DAAHRSVI","created_at":"2026-05-18T12:27:01.376967+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/DAAHRSVIGSURJ46BMGJWPI7RHU","json":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU.json","graph_json":"https://pith.science/api/pith-number/DAAHRSVIGSURJ46BMGJWPI7RHU/graph.json","events_json":"https://pith.science/api/pith-number/DAAHRSVIGSURJ46BMGJWPI7RHU/events.json","paper":"https://pith.science/paper/DAAHRSVI"},"agent_actions":{"view_html":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU","download_json":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU.json","view_paper":"https://pith.science/paper/DAAHRSVI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.5561&json=true","fetch_graph":"https://pith.science/api/pith-number/DAAHRSVIGSURJ46BMGJWPI7RHU/graph.json","fetch_events":"https://pith.science/api/pith-number/DAAHRSVIGSURJ46BMGJWPI7RHU/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU/action/timestamp_anchor","attest_storage":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU/action/storage_attestation","attest_author":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU/action/author_attestation","sign_citation":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU/action/citation_signature","submit_replication":"https://pith.science/pith/DAAHRSVIGSURJ46BMGJWPI7RHU/action/replication_record"}},"created_at":"2026-05-18T03:40:38.620557+00:00","updated_at":"2026-05-18T03:40:38.620557+00:00"}