Degenerate Sobolev inequalities from the classical Sobolev inequality

Abstract

We prove a degenerate Sobolev inequality of the form

$$ \bigg (\int _\Omega |u|^p K\, dx\bigg )^{\frac{1}{p}} \le C\Vert K\Vert _{L^{n}(\Omega )}\bigg ( \int _\Omega \big |\sqrt{Q}\nabla u \big |^p\, dx\bigg )^{\frac{1}{p}}, $$

where Q is a matrix function whose smallest eigenvalue is bounded below by a constant multiple of \(K^{-\frac{2}{p'}}\). As an application, we prove the exponential integrability of solutions of the Dirichlet problem for \(-K^{-1}{{\,\textrm{div}\,}}(Q{{\,\mathrm{\nabla }\,}}u)=f\), \(f\in L^\infty (K,\Omega )\), building upon recent results in Cruz-Uribe, MacDonald, and Rodney (2024).