{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:RSAWX2ZIJQCFGP3YW7J45TUU7R","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":"87aaf41349cefa79ae795e8a28269cfcd1a701780c7fc0261c79f1fb97fd93d3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-04T08:39:54Z","title_canon_sha256":"ff7484807adf184aa79ee2301a67991031cbfdfa61252e678dff06c02f607582"},"schema_version":"1.0","source":{"id":"2607.03769","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.03769","created_at":"2026-07-07T02:18:06Z"},{"alias_kind":"arxiv_version","alias_value":"2607.03769v1","created_at":"2026-07-07T02:18:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.03769","created_at":"2026-07-07T02:18:06Z"},{"alias_kind":"pith_short_12","alias_value":"RSAWX2ZIJQCF","created_at":"2026-07-07T02:18:06Z"},{"alias_kind":"pith_short_16","alias_value":"RSAWX2ZIJQCFGP3Y","created_at":"2026-07-07T02:18:06Z"},{"alias_kind":"pith_short_8","alias_value":"RSAWX2ZI","created_at":"2026-07-07T02:18:06Z"}],"graph_snapshots":[{"event_id":"sha256:ebd650f65188cd38988885a768b471a981e8e6428ee8a13f1cf9929c73a56f44","target":"graph","created_at":"2026-07-07T02:18:06Z","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/2607.03769/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Weighted estimates for the fractional integral operator $I_\\alpha$ are established and subsequently applied to derive corresponding Hardy--Sobolev inequalities. The weights are constructed from distance functions to bounded median porous sets and possess mixed homogeneity, which enables us to extend earlier results obtained for porous sets to a significantly broader class of geometries.","authors_text":"Diego Maldonado, Javier Soria","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-04T08:39:54Z","title":"Weighted estimates for fractional integrals with Distances to Bounded Median Porous Sets and applications to Hardy--Sobolev Inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03769","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:c78d16cbb98d79953b0e031293aa5e5f4e84a176e1f1533c9802b26e5f4388f2","target":"record","created_at":"2026-07-07T02:18:06Z","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":"87aaf41349cefa79ae795e8a28269cfcd1a701780c7fc0261c79f1fb97fd93d3","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-04T08:39:54Z","title_canon_sha256":"ff7484807adf184aa79ee2301a67991031cbfdfa61252e678dff06c02f607582"},"schema_version":"1.0","source":{"id":"2607.03769","kind":"arxiv","version":1}},"canonical_sha256":"8c816beb284c04533f78b7d3cece94fc6de8bf23c072454b72d861d9768e80a3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8c816beb284c04533f78b7d3cece94fc6de8bf23c072454b72d861d9768e80a3","first_computed_at":"2026-07-07T02:18:06.673629Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-07T02:18:06.673629Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fhPs03cmShwZDgJ5n6FU9haasgHwCaUV8AjZ2atG3+QkEfrWOT97AMQ+66CJyqclcTorP9I59LqTaTF3w80pAw==","signature_status":"signed_v1","signed_at":"2026-07-07T02:18:06.675519Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.03769","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c78d16cbb98d79953b0e031293aa5e5f4e84a176e1f1533c9802b26e5f4388f2","sha256:ebd650f65188cd38988885a768b471a981e8e6428ee8a13f1cf9929c73a56f44"],"state_sha256":"215e8e7c6059efaadb8855d2966e5f797fb8515a809f48ef1bb193fb34d50956"}