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 will obtain the corresponding equivalences in the whole Lp scale. The main novelty of our approach relies on a row/column valued theory for noncommutative martingale transforms and operator-valued Calderon-Zygmund operators. This seems to be new in the noncommutative setting and might be regarded as a first step towards a vector-valued noncommutative theory. Some examples and applications are also explored.