Maximal inequalities for square functions and quantitative mean ergodic theorems associated to group metric measure spaces

TL;DR

This paper develops weighted inequalities for non-commutative square functions in group metric measure spaces and applies them to establish a quantitative mean ergodic theorem, extending harmonic analysis to non-commutative settings.

Contribution

It introduces new weighted inequalities for non-commutative square functions and uses these to prove a quantitative mean ergodic theorem in a non-commutative framework.

Findings

Weighted strong and weak type inequalities established

Extension of harmonic analysis techniques to non-commutative spaces

Quantitative mean ergodic theorem proved

Abstract

In this article, we establish weighted strong and weak type inequalities for non-commutative square functions that naturally arise in the analysis of differences between ball averages and martingale sequences within the framework of group metric measure spaces. Then we use these maximal inequalities to prove a quantitative mean ergodic theorem. Our study extends classical harmonic analysis techniques to the non-commutative setting, revealing intricate interactions between group structures, operator-valued functions, and associated filtration systems.