Noncommutative Davis type decompositions and applications

We prove the noncommutative Davis decomposition for the column Hardy space H-p(c) for every 0 < p <= 1. A new feature of our Davis decomposition is a simultaneous control of H-1(c) and H-q(c) norms for any noncommutative martingale in H-1(c) boolean AND H-q(c) when q >= 2. As applications, we show that the Burkholder/Rosenthal inequality holds for bounded martingales in a noncommutative symmetric space associated with a function space E that is either an interpolation of the couple (L-p, L-2) for some 1 < p < 2 or is an interpolation of the couple (L-2, L-q) for some 2 < q < infinity. We also obtain the corresponding Phi-moment Burkholder/Rosenthal inequality for Orlicz functions that are either p-convex and 2-concave for some 1 < p < 2 or are 2-convex and q-concave for some 2 < q < infinity.