{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2025:A3MLWWDIUOWMRGGSKO42MTVS6X","short_pith_number":"pith:A3MLWWDI","schema_version":"1.0","canonical_sha256":"06d8bb5868a3acc898d253b9a64eb2f5ef38927509ea30a63ab89411b86540f9","source":{"kind":"arxiv","id":"2506.08115","version":1},"attestation_state":"computed","paper":{"title":"Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.AP","authors_text":"Konstantin Merz, Krzysztof Bogdan","submitted_at":"2025-06-09T18:14:52Z","abstract_excerpt":"Motivated by the study of relativistic atoms, we prove sharp heat kernel bounds for the Hardy operator $(-\\Delta)^{\\alpha/2}-\\kappa|x|^{-\\alpha}$ acting on functions of the form $u(|x|) |x|^{\\ell} Y_{\\ell,m}(x/|x|)$ in $L^2(\\R^d)$, when $\\alpha\\in(0,2]\\cap(0,d+2\\ell)$."},"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":"2506.08115","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.AP","submitted_at":"2025-06-09T18:14:52Z","cross_cats_sorted":["math.FA","math.PR"],"title_canon_sha256":"a76f8368b46d00b998344b9e392bfaf5abced612f44950ac9a2698c2604b4c1d","abstract_canon_sha256":"ae39accf8861a543db67be2cd606877d4ac96ed8b2c680b0750da6a79b055deb"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T11:18:44.957494Z","signature_b64":"P3tdDr/6Yx1jGEixLa6R9b61bdu+5JFumQ7J8C1BunyIgsH4IcwpRXpBkNIld8HAxueTwCJ+kVTNEqLouBoZCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"06d8bb5868a3acc898d253b9a64eb2f5ef38927509ea30a63ab89411b86540f9","last_reissued_at":"2026-07-05T11:18:44.957033Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T11:18:44.957033Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.AP","authors_text":"Konstantin Merz, Krzysztof Bogdan","submitted_at":"2025-06-09T18:14:52Z","abstract_excerpt":"Motivated by the study of relativistic atoms, we prove sharp heat kernel bounds for the Hardy operator $(-\\Delta)^{\\alpha/2}-\\kappa|x|^{-\\alpha}$ acting on functions of the form $u(|x|) |x|^{\\ell} Y_{\\ell,m}(x/|x|)$ in $L^2(\\R^d)$, when $\\alpha\\in(0,2]\\cap(0,d+2\\ell)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.08115","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/2506.08115/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2506.08115","created_at":"2026-07-05T11:18:44.957086+00:00"},{"alias_kind":"arxiv_version","alias_value":"2506.08115v1","created_at":"2026-07-05T11:18:44.957086+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2506.08115","created_at":"2026-07-05T11:18:44.957086+00:00"},{"alias_kind":"pith_short_12","alias_value":"A3MLWWDIUOWM","created_at":"2026-07-05T11:18:44.957086+00:00"},{"alias_kind":"pith_short_16","alias_value":"A3MLWWDIUOWMRGGS","created_at":"2026-07-05T11:18:44.957086+00:00"},{"alias_kind":"pith_short_8","alias_value":"A3MLWWDI","created_at":"2026-07-05T11:18:44.957086+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.04010","citing_title":"Sharp upper bounds for the density of relativistic atoms: Noninteracting case","ref_index":7,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X","json":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X.json","graph_json":"https://pith.science/api/pith-number/A3MLWWDIUOWMRGGSKO42MTVS6X/graph.json","events_json":"https://pith.science/api/pith-number/A3MLWWDIUOWMRGGSKO42MTVS6X/events.json","paper":"https://pith.science/paper/A3MLWWDI"},"agent_actions":{"view_html":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X","download_json":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X.json","view_paper":"https://pith.science/paper/A3MLWWDI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2506.08115&json=true","fetch_graph":"https://pith.science/api/pith-number/A3MLWWDIUOWMRGGSKO42MTVS6X/graph.json","fetch_events":"https://pith.science/api/pith-number/A3MLWWDIUOWMRGGSKO42MTVS6X/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X/action/storage_attestation","attest_author":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X/action/author_attestation","sign_citation":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X/action/citation_signature","submit_replication":"https://pith.science/pith/A3MLWWDIUOWMRGGSKO42MTVS6X/action/replication_record"}},"created_at":"2026-07-05T11:18:44.957086+00:00","updated_at":"2026-07-05T11:18:44.957086+00:00"}