Degenerate Sobolev inequalities from the classical Sobolev inequality

Abstract

We prove a degenerate Sobolev inequality of the form (integral Omega|u|pKdx)1p p K\, dx\bigg ) {\frac{1}{p}} \le C\Vert K\Vert _{L {n}(\Omega )}\bigg ( \int _\Omega \big |\sqrt{Q}\nabla u \big | p\, dx\bigg ) {\frac{1}{p}}, $$\end{document}where Q is a matrix function whose smallest eigenvalue is bounded below by a constant multiple of K-2p '\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K {-\frac{2}{p'}}$$\end{document}. As an application, we prove the exponential integrability of solutions of the Dirichlet problem for -K-1div(Q del u)=f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-K {-1}{{\,\textrm{div}\,}}(Q{{\,\mathrm{\nabla }\,}}u)=f$$\end{document}, f is an element of L infinity(K,Omega)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in L \infty (K,\Omega )$$\end{document}, building upon recent results in Cruz-Uribe, MacDonald, and Rodney (2024).