{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:R7PR2ITOUZ4IHPX4AO2SECR5WW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1dc89252fd14fc6fc534b2554a1dedfd34069118d18ea179ea419942d1fca4e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-04T02:31:27Z","title_canon_sha256":"71445677ded43161ff53afb58746381a84165240282c628e94be4054d25ab17d"},"schema_version":"1.0","source":{"id":"1504.00972","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.00972","created_at":"2026-05-18T01:27:05Z"},{"alias_kind":"arxiv_version","alias_value":"1504.00972v2","created_at":"2026-05-18T01:27:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.00972","created_at":"2026-05-18T01:27:05Z"},{"alias_kind":"pith_short_12","alias_value":"R7PR2ITOUZ4I","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_16","alias_value":"R7PR2ITOUZ4IHPX4","created_at":"2026-05-18T12:29:39Z"},{"alias_kind":"pith_short_8","alias_value":"R7PR2ITO","created_at":"2026-05-18T12:29:39Z"}],"graph_snapshots":[{"event_id":"sha256:5f8537c95dd4c50571e892c6ed720aff93a4d90a58136b1ddf8cc5dd77e5fc9c","target":"graph","created_at":"2026-05-18T01:27:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\\geq 3$ and we let $\\Sigma$ to be a closed submanifold of dimension $1 \\leq k \\leq N-2. $ In this paper we study existence and non-existence of minimizers of Hardy inequality with weight function singular on $\\Sigma$ within the framework of Brezis-Marcus-Shafrir. In particular we provide necessary and sufficient conditions for existence of minimizers.","authors_text":"El Hadji Abdoulaye Thiam","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-04T02:31:27Z","title":"Weighted Hardy inequality on Riemannian manifolds"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00972","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:4f6ff9badba1cdb90c7905d454c4757b0dd8b5b23f806414d66cf8438f46c6b0","target":"record","created_at":"2026-05-18T01:27:05Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1dc89252fd14fc6fc534b2554a1dedfd34069118d18ea179ea419942d1fca4e8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-04-04T02:31:27Z","title_canon_sha256":"71445677ded43161ff53afb58746381a84165240282c628e94be4054d25ab17d"},"schema_version":"1.0","source":{"id":"1504.00972","kind":"arxiv","version":2}},"canonical_sha256":"8fdf1d226ea67883befc03b5220a3db5b755e4e2fd0305c018185c65a11c6276","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8fdf1d226ea67883befc03b5220a3db5b755e4e2fd0305c018185c65a11c6276","first_computed_at":"2026-05-18T01:27:05.361203Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:27:05.361203Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"C20/BTK31vuM+YkuPRY2lewwLuE5ZnyaLkMH4a/bA85rCgOsoOjn8Ho3Wx9+mazhFcskOgOt3+HbR+/hc1SeDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:27:05.361938Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.00972","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4f6ff9badba1cdb90c7905d454c4757b0dd8b5b23f806414d66cf8438f46c6b0","sha256:5f8537c95dd4c50571e892c6ed720aff93a4d90a58136b1ddf8cc5dd77e5fc9c"],"state_sha256":"16fede8209ddc119c03caa205ef87f774cfded0adb37b6eb26ee488aa54c40c8"}