{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2024:TFJPVOCMEJOJTA7H4PHUQFS3I3","short_pith_number":"pith:TFJPVOCM","schema_version":"1.0","canonical_sha256":"9952fab84c225c9983e7e3cf48165b46da854f5564f1acee87379fd6cccb912f","source":{"kind":"arxiv","id":"2409.09437","version":2},"attestation_state":"computed","paper":{"title":"Harnack inequality for singular or degenerate parabolic equations in non-divergence form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Junyuan Fang, Sungwon Cho, Tuoc Phan","submitted_at":"2024-09-14T13:33:33Z","abstract_excerpt":"This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\\frac{1}{n}}$ class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. To prove the result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additio"},"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":"2409.09437","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-09-14T13:33:33Z","cross_cats_sorted":[],"title_canon_sha256":"52ee293e999c132d33619a3cae5b8f46828894f1cefa28166feae93da19bbeb9","abstract_canon_sha256":"36c6378cdf49f55145d1e82320fde4c224f75ab02fddbd9d4d3e34644dea4308"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:18:29.996164Z","signature_b64":"77ATZAvy/Sx7zPMH2dsuCV5LpekRyJ36aTj4++C4e4setOmuHkdX8qalDEtGjHg9sKUYlxQmq2hV7YJKZCgxAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9952fab84c225c9983e7e3cf48165b46da854f5564f1acee87379fd6cccb912f","last_reissued_at":"2026-07-05T09:18:29.995669Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:18:29.995669Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Harnack inequality for singular or degenerate parabolic equations in non-divergence form","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Junyuan Fang, Sungwon Cho, Tuoc Phan","submitted_at":"2024-09-14T13:33:33Z","abstract_excerpt":"This paper studies a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\\frac{1}{n}}$ class of Muckenhoupt weights. Krylov-Safonov Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. To prove the result, we introduce a class of generic weighted parabolic cylinders and the smallness condition on the weighted mean oscillation of the weight through which several growth lemmas are established. Additio"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.09437","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2409.09437/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2409.09437","created_at":"2026-07-05T09:18:29.995727+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.09437v2","created_at":"2026-07-05T09:18:29.995727+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.09437","created_at":"2026-07-05T09:18:29.995727+00:00"},{"alias_kind":"pith_short_12","alias_value":"TFJPVOCMEJOJ","created_at":"2026-07-05T09:18:29.995727+00:00"},{"alias_kind":"pith_short_16","alias_value":"TFJPVOCMEJOJTA7H","created_at":"2026-07-05T09:18:29.995727+00:00"},{"alias_kind":"pith_short_8","alias_value":"TFJPVOCM","created_at":"2026-07-05T09:18:29.995727+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/TFJPVOCMEJOJTA7H4PHUQFS3I3","json":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3.json","graph_json":"https://pith.science/api/pith-number/TFJPVOCMEJOJTA7H4PHUQFS3I3/graph.json","events_json":"https://pith.science/api/pith-number/TFJPVOCMEJOJTA7H4PHUQFS3I3/events.json","paper":"https://pith.science/paper/TFJPVOCM"},"agent_actions":{"view_html":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3","download_json":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3.json","view_paper":"https://pith.science/paper/TFJPVOCM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.09437&json=true","fetch_graph":"https://pith.science/api/pith-number/TFJPVOCMEJOJTA7H4PHUQFS3I3/graph.json","fetch_events":"https://pith.science/api/pith-number/TFJPVOCMEJOJTA7H4PHUQFS3I3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3/action/storage_attestation","attest_author":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3/action/author_attestation","sign_citation":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3/action/citation_signature","submit_replication":"https://pith.science/pith/TFJPVOCMEJOJTA7H4PHUQFS3I3/action/replication_record"}},"created_at":"2026-07-05T09:18:29.995727+00:00","updated_at":"2026-07-05T09:18:29.995727+00:00"}