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

TL;DR

The paper develops weighted non-commutative square-function inequalities on group metric measure spaces, extending Muckenhoupt weight theory and Calderón–Zygmund techniques to operator-valued contexts. It introduces a non-commutative Calderón–Zygmund decomposition and proves weighted weak and strong type bounds for a square-function built from ball-averages and dyadic conditional expectations, enabling a quantitative mean ergodic theorem for group actions by power-bounded operators. A transference framework connects spatial averaging bounds to ergodic averages, yielding uniform bounds on the differences $(M_{r_i}-M_{r_{i+1}})$ in $L^p( cal)$-norms with row-column square summability. The results illuminate the interaction between group geometry, weights, and non-commutative martingale/quasi-probabilistic structures, with potential applications to non-commutative ergodic theory and harmonic analysis on groups.

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.

Theorems & Definitions (62)

• Theorem 1.1: hong2021noncommutative
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2
• Proposition 2.3: Mei07, Proposition 1.1
• Theorem 2.4
• Lemma 2.5
• Proposition 2.6
• Remark 2.7