Sobolev inequality with non-uniformly degenerating gradient

Abstract

In this paper we prove the following weighted Sobolev inequality in a bounded domain $\Omega\subset \mathbb{R}^n$, $ n\geq 1$, of a homogeneous space $(\mathbb{R}^n, \rho, wdx)$, under suitable compatibility conditions on the positive weight functions $( v, w, \omega_1, \omega_2, \dots, \omega_n )$ and on the quasi-metric $\rho$, \begin{equation*} %\label {PS} \Big ( \int_\Omega \vert f \vert ^q v \, wdz \Big)^{\frac{1}{q}}\leq {C} \, \sum \limits_{i=1}^N \Big ( \int_\Omega \vert f_{z_i} \vert ^p \omega_i M_S w\, dz \Big)^{\frac{1}{p}}, \quad f\in \mathrm{Lip}_0(\overline \Omega), \end{equation*} where $q\geq p>1$ and $M_S$ denotes the strong maximal operator. Some corollaries on non-uniformly degenerating gradient inequalities are derived.

Keywords

• sobolev's inequality
• homogeneous space
• non-uniformly degenerating gradient