A note on two weight commutators of maximal functions on spaces of homogeneous type

TL;DR

This paper investigates two-weight estimates for commutators of maximal functions on spaces of homogeneous type, establishing control by sparse operators and providing bounds for maximal commutators.

Contribution

It introduces new quantitative bounds for two-weight commutators of maximal functions and relates them to sparse operators in spaces of homogeneous type.

Findings

Commutators are controlled by sparse operators in this setting.

Lower bounds for maximal commutators are established.

Provides quantitative estimates for weighted BMO space.

Abstract

We study the two weight quantitative estimates for the commutator of maximal functions and the maximal commutators with respect to the symbol in weighted BMO space on spaces of homogeneous type. These commutators turn out to be controlled by the sparse operators in the setting of space of homogeneous type. The lower bound of the maximal commutator is also obtained.