Muckenhoupt $A_p$-properties of distance functions and applications to Hardy-Sobolev -type inequalities

Let $X$ be a metric space equipped with a doubling measure. We consider weights $w(x)=\operatorname{dist}(x,E)^{-\alpha}$, where $E$ is a closed set in $X$ and $\alpha\in\mathbb R$. We establish sharp conditions, based on the Assouad (co)dimension of $E$, for the inclusion of $w$ in Muckenhoupt's $A_p$ classes of weights, $1\le p<\infty$. With the help of general $A_p$-weighted embedding results, we then prove (global) Hardy-Sobolev inequalities and also fractional versions of such inequalities in the setting of metric spaces.