{"paper":{"title":"A reverse Holder inequality for extremal Sobolev functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jesse Ratzkin, Tom Carroll","submitted_at":"2014-03-28T12:16:11Z","abstract_excerpt":"Let $n \\geq 2$, let $\\Omega \\subset \\mathbf{R}^n$ be a bounded domain with smooth boundary, and let $1 \\leq p \\leq 2$. We prove a reverse-Holder inequality for functions $u$ realizing the best constant in the Sobolev inequality, that is\n  $$\\mathcal{C}_p(\\Omega) = \\inf \\left \\{ \\frac{\\int_\\Omega |\\nabla v|^2}{\\left ( \\int_\\Omega |v|^p \\right )^{2/p}} \\right \\} = \\frac{\\int_\\Omega |\\nabla u|^2}{\\left ( \\int_\\Omega |u|^p \\right )^{2/p}}.$$\n  Our inequality has the form $\\| u \\|_{L^p} \\geq K \\| u \\|_{L^q}$ for any $q > p$, where $K$ depends only on $n$, $p$, $q$, and $\\mathcal{C}_p(\\Omega)$. This"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7355","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"}