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arXiv · 2306.11419

Weak porosity on metric measure spaces

Carlos Mudarra
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Abstract

We characterize the subsets $E$ of a metric space $X$ with doubling measure whose distance function to some negative power $\textrm{dist}(\cdot,E)^{-\alpha}$ 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 $A_\infty.$

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Carlos Mudarra. 2023-06-20. Weak porosity on metric measure spaces. https://doi.org/10.1017/prm.2025.10087

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