{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:TNZGQL3JVYXPZPNG3IEN36YGNG","short_pith_number":"pith:TNZGQL3J","canonical_record":{"source":{"id":"2404.05222","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-04-08T06:35:38Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"e953c1d8a96353a14e8bce8669ea46af5614171998c26353d93534010f1699e7","abstract_canon_sha256":"3a857652859cf2b35101a7ca881ba520c2a25ffe0ae1f2b19d212605c6330ee9"},"schema_version":"1.0"},"canonical_sha256":"9b72682f69ae2efcbda6da08ddfb0669945ba50628db188d2499fc7f54d2dfd3","source":{"kind":"arxiv","id":"2404.05222","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.05222","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"arxiv_version","alias_value":"2404.05222v1","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.05222","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"pith_short_12","alias_value":"TNZGQL3JVYXP","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"pith_short_16","alias_value":"TNZGQL3JVYXPZPNG","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"pith_short_8","alias_value":"TNZGQL3J","created_at":"2026-07-05T08:05:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:TNZGQL3JVYXPZPNG3IEN36YGNG","target":"record","payload":{"canonical_record":{"source":{"id":"2404.05222","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-04-08T06:35:38Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"e953c1d8a96353a14e8bce8669ea46af5614171998c26353d93534010f1699e7","abstract_canon_sha256":"3a857652859cf2b35101a7ca881ba520c2a25ffe0ae1f2b19d212605c6330ee9"},"schema_version":"1.0"},"canonical_sha256":"9b72682f69ae2efcbda6da08ddfb0669945ba50628db188d2499fc7f54d2dfd3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:05:27.650080Z","signature_b64":"U3o2ePVTSP47AO64i7Cp5L6MtiRxAGzXGiV2Iyc/4jKpYeHRQ+AKB5oiTMWF89X/LJ3yecyAp6GVuP1vbukFAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b72682f69ae2efcbda6da08ddfb0669945ba50628db188d2499fc7f54d2dfd3","last_reissued_at":"2026-07-05T08:05:27.649687Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:05:27.649687Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2404.05222","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T08:05:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u04ibskobpQ0zHjWczUB1QDjlWd5Lh1KJiaURBMmrtBpFUNEYjvH0s8q/9ktmdgOcjjGNR6o/kGSGvWIj4lcBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T17:11:16.130669Z"},"content_sha256":"337d8dbe0e1db068798e45153e8cb528b6885c601e7bf9616d0b5dbf45338afc","schema_version":"1.0","event_id":"sha256:337d8dbe0e1db068798e45153e8cb528b6885c601e7bf9616d0b5dbf45338afc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:TNZGQL3JVYXPZPNG3IEN36YGNG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fractional Hardy inequalities and capacity density","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Antti V. V\\\"ah\\\"akangas, Kaushik Mohanta, Lizaveta Ihnatsyeva","submitted_at":"2024-04-08T06:35:38Z","abstract_excerpt":"We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.05222","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2404.05222/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T08:05:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"PX7Gyxp7O0taGIS4BBLOqLZ0x4mC1UbQfGY07FINUI49Ae7Wo45WMMpFywoUBFyhrrX6bDzaooJQblplJMLADQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-22T17:11:16.131041Z"},"content_sha256":"44ded54f157d7f00406b627e31050c9e335fc8f8d2f2dabab233fd0d310a727f","schema_version":"1.0","event_id":"sha256:44ded54f157d7f00406b627e31050c9e335fc8f8d2f2dabab233fd0d310a727f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TNZGQL3JVYXPZPNG3IEN36YGNG/bundle.json","state_url":"https://pith.science/pith/TNZGQL3JVYXPZPNG3IEN36YGNG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TNZGQL3JVYXPZPNG3IEN36YGNG/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":"2026-08-22T17:11:16Z","links":{"resolver":"https://pith.science/pith/TNZGQL3JVYXPZPNG3IEN36YGNG","bundle":"https://pith.science/pith/TNZGQL3JVYXPZPNG3IEN36YGNG/bundle.json","state":"https://pith.science/pith/TNZGQL3JVYXPZPNG3IEN36YGNG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TNZGQL3JVYXPZPNG3IEN36YGNG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:TNZGQL3JVYXPZPNG3IEN36YGNG","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":"3a857652859cf2b35101a7ca881ba520c2a25ffe0ae1f2b19d212605c6330ee9","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-04-08T06:35:38Z","title_canon_sha256":"e953c1d8a96353a14e8bce8669ea46af5614171998c26353d93534010f1699e7"},"schema_version":"1.0","source":{"id":"2404.05222","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2404.05222","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"arxiv_version","alias_value":"2404.05222v1","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2404.05222","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"pith_short_12","alias_value":"TNZGQL3JVYXP","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"pith_short_16","alias_value":"TNZGQL3JVYXPZPNG","created_at":"2026-07-05T08:05:27Z"},{"alias_kind":"pith_short_8","alias_value":"TNZGQL3J","created_at":"2026-07-05T08:05:27Z"}],"graph_snapshots":[{"event_id":"sha256:44ded54f157d7f00406b627e31050c9e335fc8f8d2f2dabab233fd0d310a727f","target":"graph","created_at":"2026-07-05T08:05:27Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2404.05222/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove that a pointwise fractional Hardy inequality implies a fractional Hardy inequality, defined via a Gagliardo-type seminorm. The proof consists of two main parts. The first one is to characterize the pointwise fractional Hardy inequality in terms of a fractional capacity density condition. The second part is to show the deep open-endedness or self-improvement property of the fractional capacity density, which we accomplish in the setting of a complete geodesic space equipped with a doubling measure. These results are new already in the standard Euclidean setting.","authors_text":"Antti V. V\\\"ah\\\"akangas, Kaushik Mohanta, Lizaveta Ihnatsyeva","cross_cats":["math.AP"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-04-08T06:35:38Z","title":"Fractional Hardy inequalities and capacity density"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.05222","kind":"arxiv","version":1},"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":"sha256:337d8dbe0e1db068798e45153e8cb528b6885c601e7bf9616d0b5dbf45338afc","target":"record","created_at":"2026-07-05T08:05:27Z","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":"3a857652859cf2b35101a7ca881ba520c2a25ffe0ae1f2b19d212605c6330ee9","cross_cats_sorted":["math.AP"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-04-08T06:35:38Z","title_canon_sha256":"e953c1d8a96353a14e8bce8669ea46af5614171998c26353d93534010f1699e7"},"schema_version":"1.0","source":{"id":"2404.05222","kind":"arxiv","version":1}},"canonical_sha256":"9b72682f69ae2efcbda6da08ddfb0669945ba50628db188d2499fc7f54d2dfd3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b72682f69ae2efcbda6da08ddfb0669945ba50628db188d2499fc7f54d2dfd3","first_computed_at":"2026-07-05T08:05:27.649687Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:05:27.649687Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"U3o2ePVTSP47AO64i7Cp5L6MtiRxAGzXGiV2Iyc/4jKpYeHRQ+AKB5oiTMWF89X/LJ3yecyAp6GVuP1vbukFAQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:05:27.650080Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.05222","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:337d8dbe0e1db068798e45153e8cb528b6885c601e7bf9616d0b5dbf45338afc","sha256:44ded54f157d7f00406b627e31050c9e335fc8f8d2f2dabab233fd0d310a727f"],"state_sha256":"69451a0b14f3b3e5a2a08bb47b4b29303f31e1bf65d1945bc63b6dcea27d4149"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ARnb3/TsnAri+Wo9Ibsw9W/IuYwOXiQtIJqI9FL1sJwMG3c3DKNDNEqUXNkSDGAvFpBt5Rpw1amLPw51m267Ag==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-22T17:11:16.133659Z","bundle_sha256":"442b945bdb82821c2bed20698b68ee977e14dbfdcf3f4d3b5e7e2a2b1591ca8c"}}