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 L1(M), adapted to (Mn)n=1∞, and any sequence of signs (εn)n=1∞,∥ε1x1+∑n=2Nεn(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 unconditional Martingale differences (UMD)-constants of the Schatten class Sp when p→∞. Similarly, we prove that the UMD-constant of the finite-dimensional Schatten class Sn1 is of order log(n+1). We also discuss the Pisier–Xu non-commutative Burkholder–Gundy inequalities.