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

TL;DR

This paper introduces the concept of one-sided weakly porous sets in the real line and characterizes when certain Muckenhoupt weights belong to these sets, revealing new geometric and weight class relationships.

Contribution

It defines one-sided weakly porous sets and establishes their equivalence with conditions on Muckenhoupt weights, connecting geometric porosity with weight class properties.

Findings

Characterization of weights $d( ext{·},E)^{- ext{α}}$ as $A_1^+$ and locally integrable.

Equivalence between being both-sided weakly porous and recent weakly porous conditions.

Link between weak porosity and classical Muckenhoupt $A_1$ weights.

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 $A_1$.