{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:A5ILK3IST4MDLNLCJ622D6XZA4","short_pith_number":"pith:A5ILK3IS","schema_version":"1.0","canonical_sha256":"0750b56d129f1835b5624fb5a1faf9071952b4838e604f82e4e13e73c5a11888","source":{"kind":"arxiv","id":"1004.1101","version":3},"attestation_state":"computed","paper":{"title":"Lipschitz continuity of solutions of Poisson equations in metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Renjin Jiang","submitted_at":"2010-04-07T14:41:41Z","abstract_excerpt":"Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $\\mu$, $Q\\in[1,\\infty)$. Suppose that $(X,d,\\mu)$ supports a 2-Poincar\\'e inequality and a Sobolev-Poincar\\'e type inequality for the corresponding \"Gaussian measure\". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation $\\Delta u=f$, where $f\\in L^p_\\loc$. When $p>Q$, the local Lipschitz continuity of $u$ is established."},"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":"1004.1101","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2010-04-07T14:41:41Z","cross_cats_sorted":[],"title_canon_sha256":"9b626031a354b3ea3a23eda104605cd44a097065b0d5e443593d32926ecf6bab","abstract_canon_sha256":"98670b5cfe43b711201ba4540381481b0c1faff2d9186546a9202e15cc6ad5d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:05.820792Z","signature_b64":"HbgNu6/53KUMsZdgtDCa3NchI/mHqHJyucPL+Nm3px6RlC+v1Dt5yMKbyhMoso29XYpdeeaaMY29bXllPjJRBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0750b56d129f1835b5624fb5a1faf9071952b4838e604f82e4e13e73c5a11888","last_reissued_at":"2026-05-18T04:13:05.820423Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:05.820423Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Lipschitz continuity of solutions of Poisson equations in metric measure spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Renjin Jiang","submitted_at":"2010-04-07T14:41:41Z","abstract_excerpt":"Let $(X,d)$ be a pathwise connected metric space equipped with an Ahlfors $Q$-regular measure $\\mu$, $Q\\in[1,\\infty)$. Suppose that $(X,d,\\mu)$ supports a 2-Poincar\\'e inequality and a Sobolev-Poincar\\'e type inequality for the corresponding \"Gaussian measure\". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation $\\Delta u=f$, where $f\\in L^p_\\loc$. When $p>Q$, the local Lipschitz continuity of $u$ is established."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1004.1101","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":"1004.1101","created_at":"2026-05-18T04:13:05.820478+00:00"},{"alias_kind":"arxiv_version","alias_value":"1004.1101v3","created_at":"2026-05-18T04:13:05.820478+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1004.1101","created_at":"2026-05-18T04:13:05.820478+00:00"},{"alias_kind":"pith_short_12","alias_value":"A5ILK3IST4MD","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_16","alias_value":"A5ILK3IST4MDLNLC","created_at":"2026-05-18T12:26:05.355336+00:00"},{"alias_kind":"pith_short_8","alias_value":"A5ILK3IS","created_at":"2026-05-18T12:26:05.355336+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/A5ILK3IST4MDLNLCJ622D6XZA4","json":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4.json","graph_json":"https://pith.science/api/pith-number/A5ILK3IST4MDLNLCJ622D6XZA4/graph.json","events_json":"https://pith.science/api/pith-number/A5ILK3IST4MDLNLCJ622D6XZA4/events.json","paper":"https://pith.science/paper/A5ILK3IS"},"agent_actions":{"view_html":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4","download_json":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4.json","view_paper":"https://pith.science/paper/A5ILK3IS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1004.1101&json=true","fetch_graph":"https://pith.science/api/pith-number/A5ILK3IST4MDLNLCJ622D6XZA4/graph.json","fetch_events":"https://pith.science/api/pith-number/A5ILK3IST4MDLNLCJ622D6XZA4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4/action/storage_attestation","attest_author":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4/action/author_attestation","sign_citation":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4/action/citation_signature","submit_replication":"https://pith.science/pith/A5ILK3IST4MDLNLCJ622D6XZA4/action/replication_record"}},"created_at":"2026-05-18T04:13:05.820478+00:00","updated_at":"2026-05-18T04:13:05.820478+00:00"}