{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:GQYP7T3DRUX5EGFACKWERW7O2T","short_pith_number":"pith:GQYP7T3D","schema_version":"1.0","canonical_sha256":"3430ffcf638d2fd218a012ac48dbeed4fcb14996463cc53dc3fab11464711f51","source":{"kind":"arxiv","id":"1010.1554","version":1},"attestation_state":"computed","paper":{"title":"A Harnack inequality and H\\\"older continuity for weak solutions to parabolic operators involving H\\\"ormander vector fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Garrett Rea","submitted_at":"2010-10-07T21:46:44Z","abstract_excerpt":"This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with H\\\"ormander vector fields. Adapting the iteration scheme of J\\\"urgen Moser for elliptic and parabolic equations in $\\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \\sum X_i (a_{ij} X_j u)$ we use this to show H\\\"older continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear"},"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":"1010.1554","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-10-07T21:46:44Z","cross_cats_sorted":[],"title_canon_sha256":"466d992f9f5638dda0f1c3d1e52db9aab9877a8ec5968df4cac15b7ff7eddf3b","abstract_canon_sha256":"df3403eea28c51e66459a2a52ce905f0ab03893f6de577079a11ae4952bf9dab"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:39:40.232279Z","signature_b64":"YI63igWY+EIRi//IDLq24/KElKSeZn4r1TY6ZN6uBIx+DtAl+acDkn1jvuRHuQIgfX52YFCk9G89J5zscFxADA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3430ffcf638d2fd218a012ac48dbeed4fcb14996463cc53dc3fab11464711f51","last_reissued_at":"2026-05-18T04:39:40.231613Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:39:40.231613Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Harnack inequality and H\\\"older continuity for weak solutions to parabolic operators involving H\\\"ormander vector fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Garrett Rea","submitted_at":"2010-10-07T21:46:44Z","abstract_excerpt":"This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with H\\\"ormander vector fields. Adapting the iteration scheme of J\\\"urgen Moser for elliptic and parabolic equations in $\\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \\sum X_i (a_{ij} X_j u)$ we use this to show H\\\"older continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1554","kind":"arxiv","version":1},"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":"1010.1554","created_at":"2026-05-18T04:39:40.231706+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.1554v1","created_at":"2026-05-18T04:39:40.231706+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1554","created_at":"2026-05-18T04:39:40.231706+00:00"},{"alias_kind":"pith_short_12","alias_value":"GQYP7T3DRUX5","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_16","alias_value":"GQYP7T3DRUX5EGFA","created_at":"2026-05-18T12:26:07.630475+00:00"},{"alias_kind":"pith_short_8","alias_value":"GQYP7T3D","created_at":"2026-05-18T12:26:07.630475+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/GQYP7T3DRUX5EGFACKWERW7O2T","json":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T.json","graph_json":"https://pith.science/api/pith-number/GQYP7T3DRUX5EGFACKWERW7O2T/graph.json","events_json":"https://pith.science/api/pith-number/GQYP7T3DRUX5EGFACKWERW7O2T/events.json","paper":"https://pith.science/paper/GQYP7T3D"},"agent_actions":{"view_html":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T","download_json":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T.json","view_paper":"https://pith.science/paper/GQYP7T3D","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.1554&json=true","fetch_graph":"https://pith.science/api/pith-number/GQYP7T3DRUX5EGFACKWERW7O2T/graph.json","fetch_events":"https://pith.science/api/pith-number/GQYP7T3DRUX5EGFACKWERW7O2T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T/action/storage_attestation","attest_author":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T/action/author_attestation","sign_citation":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T/action/citation_signature","submit_replication":"https://pith.science/pith/GQYP7T3DRUX5EGFACKWERW7O2T/action/replication_record"}},"created_at":"2026-05-18T04:39:40.231706+00:00","updated_at":"2026-05-18T04:39:40.231706+00:00"}