Weak type inequality for noncommutative differentially subordinated martingales

In the paper we focus on self-adjoint noncommutative martingales. We provide an extension of the notion of differential subordination, which is due to Burkholder in the commutative case. Then we show that there is a noncommutative analogue of the Burkholder method of proving martingale inequalities, which allows us to establish the weak type (1,1) inequality for differentially subordinated martingales. Moreover, a related sharp maximal weak type (1,1) inequality is proved. Research supported by MEN Grant 1 PO3A 012 29.