Improved Buckley's theorem on LCA groups

TL;DR

This paper advances the understanding of weighted inequalities for the Hardy-Littlewood maximal function on LCA groups by refining Buckley's theorem, establishing a reverse Hölder inequality, and confirming the open property for Muckenhoupt weights.

Contribution

It provides sharp quantitative bounds for maximal functions on LCA groups and introduces new results on Muckenhoupt weights, improving prior theoretical frameworks.

Findings

Sharp weighted norm inequalities for maximal functions on LCA groups

A precise reverse Hölder inequality for A_infinity weights

Validation of the open property for A_p weights

Abstract

We present sharp quantitative weighted norm inequalities for the Hardy-Littlewood maximal function in the context of Locally Compact Abelian Groups, obtaining an improved version of the so-called Buckley's Theorem. On the way, we prove a precise reverse H\"older inequality for Muckenhoupt $A_\infty$ weights and provide a valid version of the "open property" for Muckenhoupt $A_p$ weights.