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For $0 \\leq \\alpha < \\lambda(\\Omega)$, let us define $\\|u\\|_{1,\\alpha}^n =\\|\\nabla u\\|_n^n -\\alpha \\|u\\|_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $\\Omega$, \\[ \\sup_{u\\in W^{1,n}(\\Omega), \\int_\\Omega u dx =0, \\|u\\|_{1,\\alpha} =1} \\int_{\\O"},"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":"1708.03028","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2017-08-09T22:37:40Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"069a802b79ddff74f3e4d7fb4fe0ddbd4ad82b6c94dd424e50874245d2bbc15c","abstract_canon_sha256":"a6d5eca46a0ab02ed09891549a56dac17168ae09b4565ced21d1c7fe097ae690"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:15.168624Z","signature_b64":"/Iw2uEnk0sldTJemIb9Fuz1YKr1LVl6+bpq+Qvo9nPo95tqs9GbFxev/Ub8UNzSuX15IYBg7/vRGpV4hVE05AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"dfb66f5e3fdd5222b7983d6946868d19b960ce99750daf11c7dc76a2d389ecee","last_reissued_at":"2026-05-18T00:38:15.167930Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:15.167930Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Improved Moser--Trudinger inequality for functions with mean value zero in $\\mathbb R^n$ and its extremal functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.FA","authors_text":"Van Hoang Nguyen","submitted_at":"2017-08-09T22:37:40Z","abstract_excerpt":"Let $\\Omega$ be a bounded smooth domain in $\\mathbb R^n$, $W^{1,n}(\\Omega)$ be the Sobolev space on $\\Omega$, and $\\lambda(\\Omega) = \\inf\\{\\|\\nabla u\\|_n^n: \\int_\\Omega u dx =0, \\|u\\|_n =1\\}$ be the first nonzero Neumann eigenvalue of the $n-$Laplace operator $-\\Delta_n$ on $\\Omega$. For $0 \\leq \\alpha < \\lambda(\\Omega)$, let us define $\\|u\\|_{1,\\alpha}^n =\\|\\nabla u\\|_n^n -\\alpha \\|u\\|_n^n$. We prove, in this paper, the following improved Moser--Trudinger inequality on functions with mean value zero on $\\Omega$, \\[ \\sup_{u\\in W^{1,n}(\\Omega), \\int_\\Omega u dx =0, \\|u\\|_{1,\\alpha} =1} \\int_{\\O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.03028","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":""},"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":"1708.03028","created_at":"2026-05-18T00:38:15.168043+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.03028v1","created_at":"2026-05-18T00:38:15.168043+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.03028","created_at":"2026-05-18T00:38:15.168043+00:00"},{"alias_kind":"pith_short_12","alias_value":"363G6XR73VJC","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"363G6XR73VJCFN4Y","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"363G6XR7","created_at":"2026-05-18T12:30:58.224056+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/363G6XR73VJCFN4YHVUUNBUNDG","json":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG.json","graph_json":"https://pith.science/api/pith-number/363G6XR73VJCFN4YHVUUNBUNDG/graph.json","events_json":"https://pith.science/api/pith-number/363G6XR73VJCFN4YHVUUNBUNDG/events.json","paper":"https://pith.science/paper/363G6XR7"},"agent_actions":{"view_html":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG","download_json":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG.json","view_paper":"https://pith.science/paper/363G6XR7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.03028&json=true","fetch_graph":"https://pith.science/api/pith-number/363G6XR73VJCFN4YHVUUNBUNDG/graph.json","fetch_events":"https://pith.science/api/pith-number/363G6XR73VJCFN4YHVUUNBUNDG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG/action/storage_attestation","attest_author":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG/action/author_attestation","sign_citation":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG/action/citation_signature","submit_replication":"https://pith.science/pith/363G6XR73VJCFN4YHVUUNBUNDG/action/replication_record"}},"created_at":"2026-05-18T00:38:15.168043+00:00","updated_at":"2026-05-18T00:38:15.168043+00:00"}