A new quantitative two weight theorem for the Hardy-Littlewood maximal operator

TL;DR

This paper presents an improved quantitative two weight theorem for the Hardy-Littlewood maximal operator, providing a new proof that bypasses previous reliance on sharp reverse Hölder inequalities, applicable in spaces of homogeneous type.

Contribution

It introduces a novel quantitative two weight theorem for the Hardy-Littlewood maximal operator, extending results to spaces of homogeneous type without non-empty annuli condition.

Findings

Enhanced bounds for the Hardy-Littlewood maximal operator in weighted spaces.

New proof techniques avoiding reverse Hölder inequalities.

Applicability to broader spaces of homogeneous type.

Abstract

A quantitative two weight theorem for the Hardy-Littlewood maximal operator is proved improving the known ones. As a consequence a new proof of the main results in [HP] and in [HPR12] is obtained which avoids the use of the sharp quantitative reverse Holder inequality for $A_{\infty}$ proved in those papers. Our results are valid within the context of spaces of homogeneous type without imposing the non-empty annuli condition.