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Weak porosity on metric measure spaces
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics , First View
- Published online by Cambridge University Press:
- 17 November 2025, pp. 1-48
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We characterize the subsets E of a metric space X with doubling measure whose distance function to some negative power
$\operatorname{dist}(\cdot,E)^{-\alpha}$ belongs to the Muckenhoupt A1 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 A1-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 
$A_\infty.$
