Weakly porous sets and Muckenhoupt $A_p$ distance functions

TL;DR

This paper characterizes when distance-based weights related to weakly porous sets belong to Muckenhoupt classes, providing quantitative criteria and examples that distinguish different classes of these weights.

Contribution

It offers a new characterization of Muckenhoupt $A_p$ weights in terms of weak porosity of sets, including quantitative measures and specific examples.

Findings

Distance weights are in $A_1$ iff the set is weakly porous.

Quantitative criteria for $A_p$ membership based on Muckenhoupt exponent.

An example of a non-weakly porous set with weights in $A_p$ but not in $A_1$.

Abstract

We examine the class of weakly porous sets in Euclidean spaces. As our first main result we show that the distance weight $w(x)=\operatorname{dist}(x,E)^{-\alpha}$ belongs to the Muckenhoupt class $A_1$, for some $\alpha>0$, if and only if $E\subset\mathbb{R}^n$ 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 $E\subset\mathbb{R}$ which is not weakly porous but for which $w\in A_p\setminus A_1$ for every $0<\alpha<1$ and $1<p<\infty$.