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arXiv · 2411.19856

One-sided Muckenhoupt weights and one-sided weakly porous sets in $\mathbb{R}$

Hugo Aimar·Ivana Gómez·Ignacio Gómez Vargas·Francisco Javier Martín-Reyes
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Abstract

In this work, we introduce the geometric concept of one-sided weakly porous sets in the real line and show that a set $E\subset\mathbb{R}$ satisfies $d(\cdot,E)^{-\alpha}\in A_1^+(\mathbb{R})\cap L^1_\textrm{loc}(\mathbb{R})$ for some $\alpha>0$ if and only if $E$ is right-sided weakly porous. Furthermore, we find that the property of being both left-sided and right-sided weakly porous is equivalent to the recent weakly porous condition discussed in the bibliography, which, in turn, was previously found to be intimately related to the usual class of Muckenhoupt weights $A_1$.

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Hugo Aimar, Ivana Gómez, Ignacio Gómez Vargas, Francisco Javier Martín-Reyes. 2024-11-29. One-sided Muckenhoupt weights and one-sided weakly porous sets in $\mathbb{R}$. https://doi.org/10.1016/j.jfa.2025.111110

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