Operator-Valued Littlewood-Paley Theory and Atomic Hardy Spaces on Metric Measure Groups

This paper develops a comprehensive operator-valued Littlewood-Paley theory and investigates atomic Hardy spaces on general metric measure groups. Extending classical harmonic analysis, we define operator-valued Littlewood-Paley square functions for functions taking values in a Banach space $X$. We establish their boundedness properties and demonstrate their utility in characterizing atomic Hardy spaces $H^p_A(X)$ for a range of $p$. The underlying structure of metric measure groups, which encompasses Euclidean spaces, Lie groups, and spaces with local doubling conditions, presents significant challenges due to their non-commutative and non-Euclidean geometries. Our approach involves constructing suitable approximations of the identity adapted to these geometries and leveraging vector-valued singular integral theory. We prove a characterization theorem for operator-valued atomic Hardy spaces in terms of these Littlewood-Paley square functions. This work provides a robust framework for harmonic analysis of operator-valued functions on complex geometric settings, with potential applications in stochastic processes, partial differential equations, and signal processing on non-commutative domains.