{"paper":{"title":"Global Sobolev inequalities and Degenerate P-Laplacian equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"David Cruz-Uribe, Emily Rosta, Scott Rodney","submitted_at":"2018-01-29T16:42:10Z","abstract_excerpt":"We prove that a local, weak Sobolev inequality implies a global Sobolev estimate using existence and regularity results for a family of $p$-Laplacian equations. Given $\\Omega\\subset\\mathbb{R}^n$, let $\\rho$ be a quasi-metric on $\\Omega$, and let $Q$ be an $n\\times n$ semi-definite matrix function defined on $\\Omega$. For an open set $\\Theta\\Subset\\Omega$, we give sufficient conditions to show that if the local weak Sobolev inequality %\n\\[ \\Big(\\fint_B\n  |f|^{p\\sigma}dx\\Big)^\\frac{1}{p\\sigma} \\leq C\\Big[ r(B)\\fint_B\n  |\\sqrt{Q}\\nabla f|^pdx + \\fint_B\n  |f|^pdx\\Big]^\\frac{1}{p} \\]\nholds for some"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.09610","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"}