{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:GR2RI4XKSABHOI4NGC2326ZM2U","short_pith_number":"pith:GR2RI4XK","schema_version":"1.0","canonical_sha256":"34751472ea900277238d30b5bd7b2cd52bcf5f7abc30079ef360e8bfb99bcaaf","source":{"kind":"arxiv","id":"1211.3171","version":2},"attestation_state":"computed","paper":{"title":"Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.DG","authors_text":"Alexandru Krist\\'aly, Shin-ichi Ohta","submitted_at":"2012-11-14T00:22:26Z","abstract_excerpt":"We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \\ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space."},"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":"1211.3171","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DG","submitted_at":"2012-11-14T00:22:26Z","cross_cats_sorted":["math.AP","math.MG"],"title_canon_sha256":"410676ffbe0c6adfc23f1d09996090de47ff897ce78b947d2383d477d2bd3d9c","abstract_canon_sha256":"f49218e5ccd6c5ab8794a172a2d81eb46e96b23f5a5a8d1231e93b934fe91c9f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:06.881181Z","signature_b64":"k02O3os3Zdn6Yt5Dg2SKW5XH2Y0P8e6EJyx5Rqr6I8ruMiL3EdM35/1GDzXb+uDIIukStax6FYYJnCZr/4f1Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"34751472ea900277238d30b5bd7b2cd52bcf5f7abc30079ef360e8bfb99bcaaf","last_reissued_at":"2026-05-18T02:57:06.880667Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:06.880667Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Caffarelli-Kohn-Nirenberg inequality on metric measure spaces with applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.MG"],"primary_cat":"math.DG","authors_text":"Alexandru Krist\\'aly, Shin-ichi Ohta","submitted_at":"2012-11-14T00:22:26Z","abstract_excerpt":"We prove that if a metric measure space satisfies the volume doubling condition and the Caffarelli-Kohn-Nirenberg inequality with the same exponent $n \\ge 3$, then it has exactly the $n$-dimensional volume growth. As an application, if an $n$-dimensional Finsler manifold of non-negative $n$-Ricci curvature satisfies the Caffarelli-Kohn-Nirenberg inequality with the sharp constant, then its flag curvature is identically zero. In the particular case of Berwald spaces, such a space is necessarily isometric to a Minkowski space."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.3171","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1211.3171","created_at":"2026-05-18T02:57:06.880741+00:00"},{"alias_kind":"arxiv_version","alias_value":"1211.3171v2","created_at":"2026-05-18T02:57:06.880741+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.3171","created_at":"2026-05-18T02:57:06.880741+00:00"},{"alias_kind":"pith_short_12","alias_value":"GR2RI4XKSABH","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_16","alias_value":"GR2RI4XKSABHOI4N","created_at":"2026-05-18T12:27:06.952714+00:00"},{"alias_kind":"pith_short_8","alias_value":"GR2RI4XK","created_at":"2026-05-18T12:27:06.952714+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/GR2RI4XKSABHOI4NGC2326ZM2U","json":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U.json","graph_json":"https://pith.science/api/pith-number/GR2RI4XKSABHOI4NGC2326ZM2U/graph.json","events_json":"https://pith.science/api/pith-number/GR2RI4XKSABHOI4NGC2326ZM2U/events.json","paper":"https://pith.science/paper/GR2RI4XK"},"agent_actions":{"view_html":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U","download_json":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U.json","view_paper":"https://pith.science/paper/GR2RI4XK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1211.3171&json=true","fetch_graph":"https://pith.science/api/pith-number/GR2RI4XKSABHOI4NGC2326ZM2U/graph.json","fetch_events":"https://pith.science/api/pith-number/GR2RI4XKSABHOI4NGC2326ZM2U/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U/action/storage_attestation","attest_author":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U/action/author_attestation","sign_citation":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U/action/citation_signature","submit_replication":"https://pith.science/pith/GR2RI4XKSABHOI4NGC2326ZM2U/action/replication_record"}},"created_at":"2026-05-18T02:57:06.880741+00:00","updated_at":"2026-05-18T02:57:06.880741+00:00"}