On the harmonic and geometric maximal operators

TL;DR

This paper investigates the boundedness properties of harmonic and geometric maximal operators in f, establishing new two-weight inequalities and testing conditions for these operators over general open set bases.

Contribution

It introduces novel two-weight norm inequalities and testing conditions for harmonic maximal operators in f, extending the understanding of their boundedness in a general setting.

Findings

Proved two weight norm inequalities for harmonic maximal operators.

Established testing conditions over unions of basis sets.

Identified a sufficient bump condition for two weight boundedness.

Abstract

We examine the harmonic and geometric maximal operators defined for a general basis of open sets in $\R^n$. We prove two weight norm inequalities for the harmonic maximal operator assuming testing conditions over characteristic functions of unions of sets from the basis. We also prove a that a bumped two weight $A_p$-like condition is sufficient for the two weight boundedness of the harmonic maximal operator.