Powers of distances to lower dimensional sets as Muckenhoupt weights

TL;DR

This paper establishes conditions under which the distance to a set raised to a power becomes a Muckenhoupt weight in Ahlfors spaces, with applications to PDE regularity and fractals.

Contribution

It provides new sufficient conditions for distance functions to be Muckenhoupt weights in metric measure spaces, linking geometric properties to harmonic analysis.

Findings

Distance to a set raised to a power can be a Muckenhoupt weight under specific conditions.

Applications to PDE regularity and fractal analysis are demonstrated.

The results extend the understanding of weights in metric measure spaces.

Abstract

Let $(X,d,\mu)$ be an Ahlfors metric measure space. We give sufficient conditions on a closed set $F\subseteq X$ and on a real number $\beta$ in such a way that $d(x,F)^\beta$ becomes a Muckenhoupt weight. We give also some illustrations to regularity of solutions of partial differential equations and regarding some classical fractals.