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

Abstract

In this paper, a class of anisotropic weights having the form of $|x^{\prime }{|}^{{\theta }_1}|x{|}^{{\theta }_2}|x_n{|}^{{\theta }_3}$ in dimensions $n\ge 2$ is considered, where $x=(x^{\prime },x_n)$ and $x^{\prime }=(x_1,...,x_{n-1})$. 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 Poincaré and Sobolev inequalities, which are used to study the local behavior for solutions to non-homogeneous weighted $p$-Laplace equations.