Strong differential subordinates for noncommutative submartingales

We introduce a notion of strong differential subordination of noncommutative semimartingales, extending Burkholder’s definition from the classical case. 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 interpolation-flavor method, Doob–Meyer decomposition and noncommutative analogue of good-λ inequalities.