{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:AAM6CJGUD7QWCAFDHN7QBK525P","short_pith_number":"pith:AAM6CJGU","schema_version":"1.0","canonical_sha256":"0019e124d41fe16100a33b7f00abbaebe211f9e51e1530619342f1d319fcac01","source":{"kind":"arxiv","id":"2606.09067","version":1},"attestation_state":"computed","paper":{"title":"On the Constant and Extremal Function for Weighted Hardy Inequality in $L_p$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ivan Gadjev","submitted_at":"2026-06-08T06:03:23Z","abstract_excerpt":"We study the behaviour of the smallest possible constant $d(a,b, p,\\epsilon)$ in Hardy inequality $$ \\int_a^b\\left(\\frac{1}{x}\\int_a^xf(t)dt\\right)^px^{\\epsilon}\\,dx\\leq d(a,b,p,\\epsilon)\\,\\int_a^b [f(x)]^px^{\\epsilon}\\, dx, \\quad 2\\le p<\\infty. $$ The exact rate of convergence of $d(a,b,p,\\epsilon)$ is established and the ``almost extremal'' function is found."},"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":"2606.09067","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-06-08T06:03:23Z","cross_cats_sorted":[],"title_canon_sha256":"1f9668ffda61b7d4beba3d05c0a84cb315f36b1f4d1ac1a4e65eb038dbf510cf","abstract_canon_sha256":"366b6f277547f81e2e871f0ff42cf9cba687395ac49fe68458cf12d7c6727285"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T02:07:56.647696Z","signature_b64":"2B9rrxF3WccdRN9WlwTZtVpb289X7WEoZrjmN8Zt9PFYwGUWovr7quQF32orWVjF8ur3NGuEgJh0vVYtbEzOBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0019e124d41fe16100a33b7f00abbaebe211f9e51e1530619342f1d319fcac01","last_reissued_at":"2026-06-09T02:07:56.646548Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T02:07:56.646548Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Constant and Extremal Function for Weighted Hardy Inequality in $L_p$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Ivan Gadjev","submitted_at":"2026-06-08T06:03:23Z","abstract_excerpt":"We study the behaviour of the smallest possible constant $d(a,b, p,\\epsilon)$ in Hardy inequality $$ \\int_a^b\\left(\\frac{1}{x}\\int_a^xf(t)dt\\right)^px^{\\epsilon}\\,dx\\leq d(a,b,p,\\epsilon)\\,\\int_a^b [f(x)]^px^{\\epsilon}\\, dx, \\quad 2\\le p<\\infty. $$ The exact rate of convergence of $d(a,b,p,\\epsilon)$ is established and the ``almost extremal'' function is found."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09067","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/2606.09067/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":"2606.09067","created_at":"2026-06-09T02:07:56.646705+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.09067v1","created_at":"2026-06-09T02:07:56.646705+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.09067","created_at":"2026-06-09T02:07:56.646705+00:00"},{"alias_kind":"pith_short_12","alias_value":"AAM6CJGUD7QW","created_at":"2026-06-09T02:07:56.646705+00:00"},{"alias_kind":"pith_short_16","alias_value":"AAM6CJGUD7QWCAFD","created_at":"2026-06-09T02:07:56.646705+00:00"},{"alias_kind":"pith_short_8","alias_value":"AAM6CJGU","created_at":"2026-06-09T02:07:56.646705+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P","json":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P.json","graph_json":"https://pith.science/api/pith-number/AAM6CJGUD7QWCAFDHN7QBK525P/graph.json","events_json":"https://pith.science/api/pith-number/AAM6CJGUD7QWCAFDHN7QBK525P/events.json","paper":"https://pith.science/paper/AAM6CJGU"},"agent_actions":{"view_html":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P","download_json":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P.json","view_paper":"https://pith.science/paper/AAM6CJGU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.09067&json=true","fetch_graph":"https://pith.science/api/pith-number/AAM6CJGUD7QWCAFDHN7QBK525P/graph.json","fetch_events":"https://pith.science/api/pith-number/AAM6CJGUD7QWCAFDHN7QBK525P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P/action/storage_attestation","attest_author":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P/action/author_attestation","sign_citation":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P/action/citation_signature","submit_replication":"https://pith.science/pith/AAM6CJGUD7QWCAFDHN7QBK525P/action/replication_record"}},"created_at":"2026-06-09T02:07:56.646705+00:00","updated_at":"2026-06-09T02:07:56.646705+00:00"}