{"paper":{"title":"A sharp Trudinger-Moser type inequality involving $L^{n}$ norm in the entire space $\\mathbb{R}^{n}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Guozhen Lu, Maochun Zhu","submitted_at":"2017-03-02T18:59:02Z","abstract_excerpt":"Let $W^{1,n} ( \\mathbb{R}^{n} $ be the standard Sobolev space and $\\left\\Vert \\cdot\\right\\Vert _{n}$ be the $L^{n}$ norm on $\\mathbb{R}^n$. We establish a sharp form of the following Trudinger-Moser inequality involving the $L^{n}$ norm \\[ \\underset{\\left\\Vert u\\right\\Vert _{W^{1,n}\\left(\\mathbb{R} ^{n}\\right) }=1}{\\sup}\\int_{ \\mathbb{R}^{n}}\\Phi\\left( \\alpha_{n}\\left\\vert u\\right\\vert ^{\\frac{n}{n-1}}\\left( 1+\\alpha\\left\\Vert u\\right\\Vert _{n}^{n}\\right) ^{\\frac{1}{n-1}}\\right) dx<+\\infty \\]in the entire space $\\mathbb{R}^n$ for any $0\\leq\\alpha<1$, where $\\Phi\\left( t\\right) =e^{t}-\\underset"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.00901","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"}