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

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.