Pseudo-Localization of Singular Integrals and Noncommutative Littlewood–Paley Inequalities

We prove weak type inequalities for a large class of noncommutative square functions. In conjunction with BMO type estimates, interpolation and duality, we obtain the corresponding equivalences in the whole Lp scale. Examples and applications include weak type (1, 1) boundedness for the noncommutative analogs of Stein's ergodic averages, Littlewood-Paley g-functions and Lusin-area integrals or optimal constants for the Littlewood-Paley inequality of noncommutative Lp valued functions.