Self-improving properties of weighted norm inequalities on metric measure spaces

TL;DR

This paper proves self-improving properties of weighted norm inequalities for the Hardy-Littlewood maximal function on metric measure spaces with doubling measures, using a Whitney covering and Calderón-Zygmund inspired techniques.

Contribution

It provides direct proofs of self-improving properties without relying on reverse Hölder inequalities, advancing understanding of weighted inequalities in metric spaces.

Findings

Self-improving properties of Muckenhoupt condition established

New proof techniques avoiding reverse Hölder inequalities

Applicable to metric measure spaces with doubling measures

Abstract

This work discusses self-improving properties of the Muckenhoupt condition and weighted norm inequalities for the Hardy-Littlewood maximal function on metric measure spaces with a doubling measure. Our main result provides direct proofs of these properties by applying a Whitney covering argument and a technique inspired by the Calder\'on-Zygmund decomposition. In particular, this approach does not rely on reverse H\"older inequalities.