Weak porosity on metric measure spaces

2023/06/20 by Carlos Mudarra, Mudarra, Carlos · 3 citations
Decision Sciences · Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Equations Stability Results #Fuzzy and Soft Set Theory #Metric Geometry (math.MG)

paper · pdf · doi:10.48550/arxiv.2306.11419

openalex publication_date 2023/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30

Abstract

We characterize the subsets E of a metric space X with doubling measure whose distance function to some negative power \textrmdist(⋅,E)^-α 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 ℝⁿ that do not belong to A_∞.

Cited by

• Weak porosity in spaces of homogeneous type
• On the dimension distortion under fractionally smooth mappings
• New characterizations of Muckenhoupt A_p distance weights for p>1

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