{"paper":{"title":"Extremal functions of generalized critical Hardy inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Megumi Sano","submitted_at":"2018-08-02T07:21:27Z","abstract_excerpt":"In this paper, we show the existence and non-existence of minimizers of the following minimization problems which include an open problem mentioned by Horiuchi and Kumlin in 2012: \\begin{align*} G_a := \\inf_{u \\in W_0^{1,N}(\\Omega ) \\setminus \\{ 0\\} } \\dfrac{\\int_{\\Omega} |\\nabla u |^{N} \\,dx}{\\left( \\int_{\\Omega} |u|^{q} f_{a,\\beta}(x) dx \\right)^{\\frac{N}{q}}}, \\,\\text{where} \\,\\,f_{a, \\beta}(x):=|x|^{-N}\\left( \\log \\frac{aR}{|x|} \\right)^{-\\beta}. \\end{align*} First, we give an answer to the open problem when $\\Omega =B_R(0)$. Next, we investigate the minimization problems on general bounde"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.00695","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"}