Limiting conditions of Muckenhoupt and reverse H\"older classes on metric measure spaces

TL;DR

This paper characterizes the limiting classes $A_1$ and $RH_ty$ on metric measure spaces with doubling measures, revealing their symmetric behavior and providing simplified proofs of key properties and a boundedness result for the maximal function.

Contribution

It introduces a new characterization of $A_1$ and $RH_ty$ classes on metric measure spaces, extending understanding of their properties and relationships.

Findings

Characterization of $A_1$ and $RH_ty$ classes on metric measure spaces.

Symmetric behavior of limiting Muckenhoupt and reverse H"older classes.

Simplified proofs of properties including Jones factorization theorem.

Abstract

The natural maximal and minimal functions commute pointwise with the logarithm on $A_\infty$. We use this observation to characterize the spaces $A_1$ and $RH_\infty$ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt $A_p$ and reverse H\"older classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between $A_p$ and reverse H\"older descriptions. Finally, we apply the characterization to give simple proofs of several known properties of $A_1$ and $RH_\infty$, including a refined Jones factorization theorem. In addition, we show a boundedness result for the natural maximal function.