Inequalities for noncommutative differentially subordinate martingales

The classical differential subordination of martingales, introduced by Burkholder in the eighties, is generalized to the noncommutative setting. Working under this domination, we establish the strong-type inequalities with the constants of optimal order as p -> 1 and p -> infinity, and the corresponding endpoint weak-type (1, 1) estimate. In contrast to the classical case, we need to introduce two different versions of noncommutative differential subordination, depending on the range of the exponents. For the L-P-estimate, 2 <= p < infinity, a certain weaker version is sufficient; on the other hand, the strong-type (p,p) inequality, 1 < p < 2, and the weak-type (1, 1) estimate require a stronger version. As an application, we present a new proof of noncommutative Burkholder-Gundy inequalities. The main technical advance is a noncommutative version of the good lambda-inequality and a certain summation argument. We expect that these techniques will be useful in other situation as well. (C) 2018 Elsevier Inc. All rights reserved.