Powers of distances to lower dimensional sets as Muckenhoupt weights

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.