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<\infty $ under the assumption that the dominating process is a nonnegative submartingale. This is accomplished by combining several techniques, including interpolation-flavor method, Doob–Meyer decomposition and noncommutative analogue of good-$\lambda$ inequalities.