Powers of distances to lower dimensional sets as Muckenhoupt weights

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

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