{"paper":{"title":"Hardy's inequality and curvature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.SP","authors_text":"A.A. Balinsky, R.T. Lewis, W.D. Evans","submitted_at":"2011-01-12T11:41:29Z","abstract_excerpt":"A Hardy inequality of the form \\[\\int_{\\tilde{\\Omega}} |\\nabla f({\\bf{x}})|^p d {\\bf{x}} \\ge (\\frac{p-1}{p})^p \\int_{\\tilde{\\Omega}} \\{1 + a(\\delta, \\partial \\tilde{\\Omega})(\\x)\\}\\frac{|f({\\bf{x}})|^p}{\\delta({\\bf{x}})^p} d{\\bf{x}}, \\] for all $f \\in C_0^{\\infty}({\\tilde{\\Omega}})$, is considered for $p\\in (1,\\infty)$, where ${\\tilde{\\Omega}}$ can be either $\\Omega$ or $\\mathbb{R}^n \\setminus \\Omega$ with $\\Omega$ a domain in $\\mathbb{R}^n$, $n \\ge 2$, and $\\delta({\\bf{x}})$ is the distance from ${\\bf{x}} \\in {\\tilde{\\Omega}} $ to the boundary $ \\partial {\\tilde{\\Omega}}.$ The main emphasis is"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.2331","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"}