On a class of anisotropic Muckenhoupt weights and their applications to $p$-Laplace equations

TL;DR

This paper characterizes a class of anisotropic weights related to Muckenhoupt classes, explores their doubling properties, and applies these findings to establish weighted inequalities and analyze solutions to weighted p-Laplace equations.

Contribution

It identifies the optimal parameter range for anisotropic weights to belong to Muckenhoupt classes and studies their doubling properties, leading to new weighted inequalities for p-Laplace equations.

Findings

Identified the optimal parameter range for weights in Muckenhoupt class $A_p$.

Showed that these weights are doubling measures but not in $A_p$.

Derived anisotropic weighted Poincaré and Sobolev inequalities.

Abstract

In this paper, a class of anisotropic weights having the form of $|x'|^{\theta_{1}}|x|^{\theta_{2}}|x_{n}|^{\theta_{3}}$ in dimensions $n\geq2$ is considered, where $x=(x',x_{n})$ and $x'=(x_{1},...,x_{n-1})$. We first find the optimal range of $(\theta_{1},\theta_{2},\theta_{3})$ such that this type of weights belongs to the Muckenhoupt class $A_{p}$. Then we further study its doubling property, which shows that it provides an example of a doubling measure but is not in $A_{p}$. As a consequence, we obtain anisotropic weighted Poincar\'{e} and Sobolev inequalities, which are used to study the local behavior for solutions to non-homogeneous weighted $p$-Laplace equations.