{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:3QGK6RLPGYAJWRUSALN4ZEA7M7","short_pith_number":"pith:3QGK6RLP","schema_version":"1.0","canonical_sha256":"dc0caf456f36009b469202dbcc901f67ce60173c5385412848d111ee1c56e735","source":{"kind":"arxiv","id":"1501.03678","version":2},"attestation_state":"computed","paper":{"title":"An improved Hardy-Trudinger-Moser inequality","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaobao Zhu, Yunyan Yang","submitted_at":"2015-01-15T13:43:18Z","abstract_excerpt":"Let $\\mathbb{B}$ be the unit disc in $\\mathbb{R}^2$, $\\mathscr{H}$ be the completion of $C_0^\\infty(\\mathbb{B})$ under the norm $$\\|u\\|_{\\mathscr{H}}=\\left(\\int_\\mathbb{B}|\\nabla u|^2dx-\\int_\\mathbb{B}\\frac{u^2}{(1-|x|^2)^2}dx\\right)^{1/2},\\quad\\forall u\\in C_0^\\infty(\\mathbb{B}).$$ Denote $\\lambda_1(\\mathbb{B})=\\inf_{u\\in \\mathscr{H},\\,\\|u\\|_2=1}\\|u\\|_{\\mathscr{H}}^2$, where $\\|\\cdot\\|_2$ stands for the $L^2(\\mathbb{B})$-norm. Using blow-up analysis, we prove that for any $\\alpha$, $0\\leq \\alpha<\\lambda_1(\\mathbb{B})$, $$\\sup_{u\\in\\mathscr{H},\\,\\|u\\|_{\\mathscr{H}}^2-\\alpha\\|u\\|_2^2\\leq 1}\\int"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1501.03678","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","primary_cat":"math.AP","submitted_at":"2015-01-15T13:43:18Z","cross_cats_sorted":[],"title_canon_sha256":"b1d3410be345d8842803385f48b0895f2f573fff14cc9f03ee2a96b749784f6a","abstract_canon_sha256":"8c5c8a4dfa0e4108b2d204a113f75b93a07a4b394dfa36b6a35bcfd871c6e0d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:33:16.183334Z","signature_b64":"5uNbUQtHZixj8vhJJgG6kGYhemZxftGLqf2JAFrSOeX/kj+yG6sQBbOfp2/4CqPHP2gGFfZQe6GqeFi3HxBOCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dc0caf456f36009b469202dbcc901f67ce60173c5385412848d111ee1c56e735","last_reissued_at":"2026-05-18T01:33:16.182686Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:33:16.182686Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An improved Hardy-Trudinger-Moser inequality","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Xiaobao Zhu, Yunyan Yang","submitted_at":"2015-01-15T13:43:18Z","abstract_excerpt":"Let $\\mathbb{B}$ be the unit disc in $\\mathbb{R}^2$, $\\mathscr{H}$ be the completion of $C_0^\\infty(\\mathbb{B})$ under the norm $$\\|u\\|_{\\mathscr{H}}=\\left(\\int_\\mathbb{B}|\\nabla u|^2dx-\\int_\\mathbb{B}\\frac{u^2}{(1-|x|^2)^2}dx\\right)^{1/2},\\quad\\forall u\\in C_0^\\infty(\\mathbb{B}).$$ Denote $\\lambda_1(\\mathbb{B})=\\inf_{u\\in \\mathscr{H},\\,\\|u\\|_2=1}\\|u\\|_{\\mathscr{H}}^2$, where $\\|\\cdot\\|_2$ stands for the $L^2(\\mathbb{B})$-norm. Using blow-up analysis, we prove that for any $\\alpha$, $0\\leq \\alpha<\\lambda_1(\\mathbb{B})$, $$\\sup_{u\\in\\mathscr{H},\\,\\|u\\|_{\\mathscr{H}}^2-\\alpha\\|u\\|_2^2\\leq 1}\\int"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.03678","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1501.03678","created_at":"2026-05-18T01:33:16.182776+00:00"},{"alias_kind":"arxiv_version","alias_value":"1501.03678v2","created_at":"2026-05-18T01:33:16.182776+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.03678","created_at":"2026-05-18T01:33:16.182776+00:00"},{"alias_kind":"pith_short_12","alias_value":"3QGK6RLPGYAJ","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_16","alias_value":"3QGK6RLPGYAJWRUS","created_at":"2026-05-18T12:29:02.477457+00:00"},{"alias_kind":"pith_short_8","alias_value":"3QGK6RLP","created_at":"2026-05-18T12:29:02.477457+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/3QGK6RLPGYAJWRUSALN4ZEA7M7","json":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7.json","graph_json":"https://pith.science/api/pith-number/3QGK6RLPGYAJWRUSALN4ZEA7M7/graph.json","events_json":"https://pith.science/api/pith-number/3QGK6RLPGYAJWRUSALN4ZEA7M7/events.json","paper":"https://pith.science/paper/3QGK6RLP"},"agent_actions":{"view_html":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7","download_json":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7.json","view_paper":"https://pith.science/paper/3QGK6RLP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1501.03678&json=true","fetch_graph":"https://pith.science/api/pith-number/3QGK6RLPGYAJWRUSALN4ZEA7M7/graph.json","fetch_events":"https://pith.science/api/pith-number/3QGK6RLPGYAJWRUSALN4ZEA7M7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7/action/storage_attestation","attest_author":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7/action/author_attestation","sign_citation":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7/action/citation_signature","submit_replication":"https://pith.science/pith/3QGK6RLPGYAJWRUSALN4ZEA7M7/action/replication_record"}},"created_at":"2026-05-18T01:33:16.182776+00:00","updated_at":"2026-05-18T01:33:16.182776+00:00"}