{"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"}