Weakly porous sets and Muckenhoupt Ap distance functions

We consider the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight $w\left(x\right)=\mathrm{dist}{(x,E)}^{-\alpha }$ belongs to the Muckenhoupt class $A_1$, for some $\alpha >0$, if and only if $E\subset R^n$ is weakly porous. We also give a precise quantitative version of this characterization in terms of the so-called Muckenhoupt exponent of E. When E is weakly porous, we obtain a similar quantitative characterization of $w\in A_p$, for $1<p<\infty$, as well. At the end of the paper, we give an example of a set $E\subset R$ which is not weakly porous but for which $w\in A_p\setminus A_1$ for every $0<\alpha <1$ and $1<p<\infty$.