{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:JN6QI3UI7ZP3SX4ROGAPRMPLNN","short_pith_number":"pith:JN6QI3UI","canonical_record":{"source":{"id":"1202.0033","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-31T21:27:38Z","cross_cats_sorted":[],"title_canon_sha256":"3580b00e13694d8ec1425d71fa782ccaedc6a22b1eba0f158ea6538ad85f7fc9","abstract_canon_sha256":"23b3d955b332b290b1418307421fa30a6914b91251d4e0e9c7b884abc4840175"},"schema_version":"1.0"},"canonical_sha256":"4b7d046e88fe5fb95f917180f8b1eb6b485351cbb2153049d4184c213b891c1a","source":{"kind":"arxiv","id":"1202.0033","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.0033","created_at":"2026-05-18T03:44:33Z"},{"alias_kind":"arxiv_version","alias_value":"1202.0033v2","created_at":"2026-05-18T03:44:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0033","created_at":"2026-05-18T03:44:33Z"},{"alias_kind":"pith_short_12","alias_value":"JN6QI3UI7ZP3","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JN6QI3UI7ZP3SX4R","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JN6QI3UI","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:JN6QI3UI7ZP3SX4ROGAPRMPLNN","target":"record","payload":{"canonical_record":{"source":{"id":"1202.0033","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-31T21:27:38Z","cross_cats_sorted":[],"title_canon_sha256":"3580b00e13694d8ec1425d71fa782ccaedc6a22b1eba0f158ea6538ad85f7fc9","abstract_canon_sha256":"23b3d955b332b290b1418307421fa30a6914b91251d4e0e9c7b884abc4840175"},"schema_version":"1.0"},"canonical_sha256":"4b7d046e88fe5fb95f917180f8b1eb6b485351cbb2153049d4184c213b891c1a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:33.725429Z","signature_b64":"W/BXt+YRiv0W3b20/wO53K0/JdcKpP4VcfGngSdTgat2H1lUsNRuvSgddSFzYPD/h/s6jCAIxKPqd4UQFnCFCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b7d046e88fe5fb95f917180f8b1eb6b485351cbb2153049d4184c213b891c1a","last_reissued_at":"2026-05-18T03:44:33.724911Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:33.724911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1202.0033","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:44:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FhhPWy4Zz6GObVg2f8YaS4FB5JGO0NgHwEZwSrrrYNjO+sVTM5JIx2R4AGFvt9gyKS4Rs2bDc/y1EDd5szwIBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:06:41.756307Z"},"content_sha256":"fb36eb7000da9e30b93bb3fb2549162d1dee81bcf4af5f82692f738a67fddd26","schema_version":"1.0","event_id":"sha256:fb36eb7000da9e30b93bb3fb2549162d1dee81bcf4af5f82692f738a67fddd26"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:JN6QI3UI7ZP3SX4ROGAPRMPLNN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Weighted Hardy inequality with higher dimensional singularity on the boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Fethi Mahmoudi, Mouhamed Moustapha Fall","submitted_at":"2012-01-31T21:27:38Z","abstract_excerpt":"Let $\\Omega$ be a smooth bounded domain in $\\mahbb R^N$ with $N\\ge 3$ and let $\\Sigma_k$ be a closed smooth submanifold of $\\delta \\Omega$ of dimension $1\\le k\\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $\\Sigma_k$. In particular we provide sufficient and necessary conditions for existence of minimizers."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0033","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:44:33Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"oyfkOaOUn1Z8QxBT/4wxYUBijnuZJpPfWIMeXc7dtapV45WGncQkFqhMobmin4s/u9AVyo3BpEdnkfaTKNyeCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T17:06:41.756926Z"},"content_sha256":"b7d85dfc0417e79ac7166cef4533c8af92a534549015340dcf44d6a9dfd919ba","schema_version":"1.0","event_id":"sha256:b7d85dfc0417e79ac7166cef4533c8af92a534549015340dcf44d6a9dfd919ba"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JN6QI3UI7ZP3SX4ROGAPRMPLNN/bundle.json","state_url":"https://pith.science/pith/JN6QI3UI7ZP3SX4ROGAPRMPLNN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JN6QI3UI7ZP3SX4ROGAPRMPLNN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-09T17:06:41Z","links":{"resolver":"https://pith.science/pith/JN6QI3UI7ZP3SX4ROGAPRMPLNN","bundle":"https://pith.science/pith/JN6QI3UI7ZP3SX4ROGAPRMPLNN/bundle.json","state":"https://pith.science/pith/JN6QI3UI7ZP3SX4ROGAPRMPLNN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JN6QI3UI7ZP3SX4ROGAPRMPLNN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JN6QI3UI7ZP3SX4ROGAPRMPLNN","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":"23b3d955b332b290b1418307421fa30a6914b91251d4e0e9c7b884abc4840175","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-31T21:27:38Z","title_canon_sha256":"3580b00e13694d8ec1425d71fa782ccaedc6a22b1eba0f158ea6538ad85f7fc9"},"schema_version":"1.0","source":{"id":"1202.0033","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.0033","created_at":"2026-05-18T03:44:33Z"},{"alias_kind":"arxiv_version","alias_value":"1202.0033v2","created_at":"2026-05-18T03:44:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.0033","created_at":"2026-05-18T03:44:33Z"},{"alias_kind":"pith_short_12","alias_value":"JN6QI3UI7ZP3","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JN6QI3UI7ZP3SX4R","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JN6QI3UI","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:b7d85dfc0417e79ac7166cef4533c8af92a534549015340dcf44d6a9dfd919ba","target":"graph","created_at":"2026-05-18T03:44:33Z","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 $\\Omega$ be a smooth bounded domain in $\\mahbb R^N$ with $N\\ge 3$ and let $\\Sigma_k$ be a closed smooth submanifold of $\\delta \\Omega$ of dimension $1\\le k\\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $\\Sigma_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.","authors_text":"Fethi Mahmoudi, Mouhamed Moustapha Fall","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-31T21:27:38Z","title":"Weighted Hardy inequality with higher dimensional singularity on the boundary"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.0033","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:fb36eb7000da9e30b93bb3fb2549162d1dee81bcf4af5f82692f738a67fddd26","target":"record","created_at":"2026-05-18T03:44:33Z","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":"23b3d955b332b290b1418307421fa30a6914b91251d4e0e9c7b884abc4840175","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-31T21:27:38Z","title_canon_sha256":"3580b00e13694d8ec1425d71fa782ccaedc6a22b1eba0f158ea6538ad85f7fc9"},"schema_version":"1.0","source":{"id":"1202.0033","kind":"arxiv","version":2}},"canonical_sha256":"4b7d046e88fe5fb95f917180f8b1eb6b485351cbb2153049d4184c213b891c1a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b7d046e88fe5fb95f917180f8b1eb6b485351cbb2153049d4184c213b891c1a","first_computed_at":"2026-05-18T03:44:33.724911Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:33.724911Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W/BXt+YRiv0W3b20/wO53K0/JdcKpP4VcfGngSdTgat2H1lUsNRuvSgddSFzYPD/h/s6jCAIxKPqd4UQFnCFCA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:33.725429Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.0033","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb36eb7000da9e30b93bb3fb2549162d1dee81bcf4af5f82692f738a67fddd26","sha256:b7d85dfc0417e79ac7166cef4533c8af92a534549015340dcf44d6a9dfd919ba"],"state_sha256":"7a57d719c950e969907d8017987193509cab72ff62e54cd09ebaf635c5a5d2b4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AglUki2J3SwNb16Y+ob3mw6IJNs4+IKH6ohCtVR/qVzm1ZQveiDwyxvCkmmlk3wSmtINAHe8vNqnC4UIKJaGDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T17:06:41.760756Z","bundle_sha256":"87f0d878c9e9ce701799cc41b0c8364339149b1665a7d5e0d1c6859f860da961"}}