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→∞, 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 Lp-estimate, 2≤p<∞, a certain weaker version is sufficient; on the other hand, the strong-type (p,p) inequality, 1