Abstract
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<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 λ-inequality and a certain summation argument. We expect that these techniques will be useful in other situation as well.
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