Non - Commutative Martingale Transforms

We prove that non-commutative martingale transforms are of weak type (1, 1). More precisely, there is an absolute constant C such that if M is a semi-finite von Neumann algebra and (Mn)n=1 is an increasing filtration of von Neumann subalgebras of M then for any non-commutative martingale x = (xn) ∞ n=1 in L 1(M), adapted to (Mn)n=1, and any sequence of signs (εn) ∞ n=1, ∥∥∥∥ε1x1 + N ∑ n=2 εn(xn − xn−1) ∥∥∥∥ 1,∞ ≤ C ‖xN‖1 for every N ≥ 2. This generalizes a result of Burkholder from classical martingale theory to non-commutative setting and answers positively a question of Pisier and Xu. As applications, we get the optimal order of the UMD-constants of the Schatten class S when p → ∞. Similarly, we prove that the UMD-constant of the finite dimensional Schatten class S n is of order log(n+1). We also discuss the Pisier-Xu non-commutative Burkholder-Gundy inequalities.