  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.
