Weak porosity on metric measure spaces

TL;DR

This work characterizes when distance-to-set weights on doubling metric measure spaces belong to the Muckenhoupt class $A_1$ by connecting geometric notions of weak porosity and the doubling behavior of maximal free holes. The authors develop dyadic weak porosity notions, prove a precise equivalence with the $A_1$-property for some $\alpha>0$, and provide sharp examples showing optimality and limitations. They also establish that annular decay spaces simplify the picture and introduce an explicit construction of doubling weights outside $A_\infty$, broadening the toolkit for weighted inequalities in metric spaces. Finally, the paper links the Muckenhoupt exponent $\mathrm{Mu}(E)$ to $A_1$ and $A_p$ membership, offering a quantitative framework for distance-function weights to sets.

Abstract

We characterize the subsets $E$ of a metric space $X$ with doubling measure whose distance function to some negative power $\textrm{dist}(\cdot,E)^{-α}$ belongs to the Muckenhoupt $A_1$ class of weights in $X$. To this end, we introduce the weakly porous sets in this setting, and show that, along with certain doubling-type conditions for the sizes of the largest $E$-free holes, these sets characterize the mentioned $A_1$-property. We exhibit examples showing the optimality of these conditions, and simplify them in the particular case where the underlying measure satisfies a qualitative annular decay property. In addition, we use some of these distance functions as a new and simple method to explicitly construct doubling weights in $\mathbb{R}^n$ that do not belong to $A_\infty.$

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Proposition 2.2
• Proposition 2.3
• Lemma 2.4
• proof
• Proposition 2.5
• Proposition 2.6