Strong differential subordinates for noncommutative submartingales

Abstract

We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder's definition from the classical case (Ann. Probab. 22 (1994) 995-1025). Then we establish the maximal weak-type (1, 1) inequality under the additional assumption that the dominating process is a submartingale. The proof rests on a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale. We also show the corresponding strong-type (p, p) estimate for 1 < p < ∞ under the assumption that the dominating process is a nonnegative submartingale. This is accomplished by combining several techniques, including interpolationflavor method, Doob-Meyer decomposition and noncommutative analogue of good-λ inequalities.