{"paper":{"title":"Extremal function for Moser-Trudinger type Inequality with Logarithmic weight","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Prosenjit Roy","submitted_at":"2016-02-15T08:04:26Z","abstract_excerpt":"On the space of weighted radial Sobolev space, the following generalization of Moser-Trudinger type inequality was established by Calanchi and Ruf in dimension 2 : If $\\beta \\in [0,1)$ and $w_0(x) = |\\log |x||^\\beta $ then $$ \\sup_{\\int_B |\\grad u|^2w_0 \\leq 1 , u \\in H_{0,rad}^1(w_0,B)} \\int_B e^{\\alpha u^{\\frac{2}{1-\\beta}}} dx < \\infty,$$ if and only if $\\alpha \\leq \\alpha_\\beta = 2\\left[2\\pi (1-\\beta) \\right]^{\\frac{1}{1-\\beta}}.$ We prove the existence of an extremal function for the above inequality for the critical case when $\\alpha = \\alpha_\\beta$ thereby generalizing the result of Car"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04585","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"}