{"paper":{"title":"Hardy's inequality in a limiting case on general bounded domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Futoshi Takahashi, Jaeyoung Byeon","submitted_at":"2017-07-13T08:08:52Z","abstract_excerpt":"In this paper, we study Hardy's inequality in a limiting case: $$\n  \\int_{\\Omega} |\\nabla u |^N dx \\ge C_N(\\Omega) \\int_{\\Omega} \\frac{|u(x)|^N}{|x|^N \\left(\\log \\frac{R}{|x|} \\right)^N} dx $$ for functions $u \\in W^{1,N}_0(\\Omega)$, where $\\Omega$ is a bounded domain in $\\mathbb{R}^N$ with $R = \\sup_{x \\in \\Omega} |x|$. We study the (non-)attainability of the best constant $C_N(\\Omega)$ in several cases. We provide sufficient conditions that assure $C_N(\\Omega) > C_N(B_R)$ and $C_N(\\Omega)$ is attained, here $B_R$ is the $N$-dimensional ball with center the origin and radius $R$. Also we prov"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04018","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"}