{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:TWZ2LZDTJ3TYHYQ6ETCKRIIJCF","short_pith_number":"pith:TWZ2LZDT","schema_version":"1.0","canonical_sha256":"9db3a5e4734ee783e21e24c4a8a10911400b431ca5042e07cf24e4a3b3e58309","source":{"kind":"arxiv","id":"2605.25366","version":1},"attestation_state":"computed","paper":{"title":"A Median Version of Hardy's Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Gangsong Leng","submitted_at":"2026-05-25T02:44:53Z","abstract_excerpt":"Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\\in L^p(0,\\infty)$, $p>1$, let $A(t)$ be the average of $f$ over $(0,t)$, and let $M(t)$ be the lower median of $f$ over $(0,t)$. We show that \\[\n  \\int_0^\\infty |M(t)-A(t)|^p\\,dt\n  \\leq 2^{1-p}\\left(\\frac p{p-1}\\right)^p\n  \\int_0^\\infty f(t)^p\\,dt, \\] and that the constant is best possible. The proof is based on a pointwise rearrangement estimate coming from the half-measure property of the median, followed by"},"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":"2605.25366","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.MG","submitted_at":"2026-05-25T02:44:53Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"2889d862ab92e5567e7189b102686c4a293c5b473f041e2b3e8b5ea7c5f9367c","abstract_canon_sha256":"13321a0f5781a71a169a64a151e0c0aca7809b4d2c618c0086f55e67de390259"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-26T02:04:31.146914Z","signature_b64":"cwiEqidfnws+Pp7tONQ4TjMv/oF25okM/hy0zAQjA69EnsCF7xqddMT/x8AUZsR6+8ld1wNKcE2rOedl71rcAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9db3a5e4734ee783e21e24c4a8a10911400b431ca5042e07cf24e4a3b3e58309","last_reissued_at":"2026-05-26T02:04:31.146244Z","signature_status":"signed_v1","first_computed_at":"2026-05-26T02:04:31.146244Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A Median Version of Hardy's Inequality","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.MG","authors_text":"Gangsong Leng","submitted_at":"2026-05-25T02:44:53Z","abstract_excerpt":"Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\\in L^p(0,\\infty)$, $p>1$, let $A(t)$ be the average of $f$ over $(0,t)$, and let $M(t)$ be the lower median of $f$ over $(0,t)$. We show that \\[\n  \\int_0^\\infty |M(t)-A(t)|^p\\,dt\n  \\leq 2^{1-p}\\left(\\frac p{p-1}\\right)^p\n  \\int_0^\\infty f(t)^p\\,dt, \\] and that the constant is best possible. The proof is based on a pointwise rearrangement estimate coming from the half-measure property of the median, followed by"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25366","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/2605.25366/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":"2605.25366","created_at":"2026-05-26T02:04:31.146330+00:00"},{"alias_kind":"arxiv_version","alias_value":"2605.25366v1","created_at":"2026-05-26T02:04:31.146330+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2605.25366","created_at":"2026-05-26T02:04:31.146330+00:00"},{"alias_kind":"pith_short_12","alias_value":"TWZ2LZDTJ3TY","created_at":"2026-05-26T02:04:31.146330+00:00"},{"alias_kind":"pith_short_16","alias_value":"TWZ2LZDTJ3TYHYQ6","created_at":"2026-05-26T02:04:31.146330+00:00"},{"alias_kind":"pith_short_8","alias_value":"TWZ2LZDT","created_at":"2026-05-26T02:04:31.146330+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/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF","json":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF.json","graph_json":"https://pith.science/api/pith-number/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/graph.json","events_json":"https://pith.science/api/pith-number/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/events.json","paper":"https://pith.science/paper/TWZ2LZDT"},"agent_actions":{"view_html":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF","download_json":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF.json","view_paper":"https://pith.science/paper/TWZ2LZDT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2605.25366&json=true","fetch_graph":"https://pith.science/api/pith-number/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/graph.json","fetch_events":"https://pith.science/api/pith-number/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/action/storage_attestation","attest_author":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/action/author_attestation","sign_citation":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/action/citation_signature","submit_replication":"https://pith.science/pith/TWZ2LZDTJ3TYHYQ6ETCKRIIJCF/action/replication_record"}},"created_at":"2026-05-26T02:04:31.146330+00:00","updated_at":"2026-05-26T02:04:31.146330+00:00"}