Characterizations of weak reverse H\"older inequalities on metric measure spaces

TL;DR

This paper provides ten different characterizations of functions satisfying a weak reverse H"older inequality on metric measure spaces, extending the concept of Muckenhoupt weights to nondoubling measures and exploring their properties.

Contribution

It introduces multiple new characterizations of weak reverse H"older functions and generalizes the concept of Muckenhoupt weights to nondoubling measures in metric spaces.

Findings

Ten characterizations of weak reverse H"older functions

Extension of Muckenhoupt $A_$ weights to nondoubling measures

Identification of conditions that fail for weak $A_$ weights

Abstract

We present ten different characterizations of functions satisfying a weak reverse H\"older inequality on an open subset of a metric measure space with a doubling measure. Among others, we describe these functions as a class of weak $A_\infty$ weights, which is a generalization of Muckenhoupt weights that allows for nondoubling weights. Although our main results are modeled after conditions that hold true for Muckenhoupt weights, we also discuss two conditions for Muckenhoupt $A_\infty$ weights that fail to hold for weak $A_\infty$ weights.