In this talk we will report some new advances regarding the theory of noncommutative differentially subordinate martingales. 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 $p\to 1$ and $p\to \infty$, 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 $L^p$-estimate, $2\leq p<\infty$, 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. We also introduce a notion of strong differential subordination of noncommutative semimartingales, establish the maximal weak-type (1,1) inequality under the additional assumption that the dominating process is a submartingale, and show the corresponding strong-type $(p,p)$ estimate for $1<p<\infty$ under the assumption that the dominating process is a nonnegative submartingale. Finally, we give some estimates of square functions for noncommutative differentially subordinate martingales. This is accomplished by combining several techniques, including interpolation flavor method, Doob-Meyer decomposition, a significant extension of the maximal weak-type estimate of Cuculescu and a Gundy-type decomposition of an arbitrary noncommutative submartingale.

This is several joint work with Adam Osekowski, Lian Wu, Narcisse R. and Dejian Zhou.