{"paper":{"title":"Existence of extremal functions for a family of Caffarelli-Kohn-Nirenberg inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wenming Zou, Xuexiu Zhong","submitted_at":"2015-04-02T02:49:35Z","abstract_excerpt":"Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\\it (Compositio Mathematica,1984):} $$\\Big(\\int_\\Omega \\frac{|u|^r}{|x|^s}dx\\Big)^{\\frac{1}{r}}\\leq C(p,q,r,\\mu,\\sigma,s)\\Big(\\int_\\Omega \\frac{|\\nabla u|^p}{|x|^\\mu}dx\\Big)^{\\frac{a}{p}}\\Big(\\int_\\Omega \\frac{|u|^q}{|x|^\\sigma}dx\\Big)^{\\frac{1-a}{q}},$$ where $\\Omega \\subset \\R^N (N\\geq 2)$ is an open set; $p, q, r, \\mu, \\sigma, s, a$ are some parameters satisfying some balanced conditions. When $\\Omega$ is a cone in $\\R^N$ (for example, $\\Omega=\\R^N)$, we prove the sharp constant $C(p,q,r,\\mu,\\sigma,s)$ can be achieve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.00433","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"}