Square functions for noncommutative differentially subordinate martingales

Abstract:

We prove inequalities involving noncommutative differentially subordinate martingales.

More precisely, we prove that if xx is a self-adjoint noncommutative martingale and yy is weakly differentially subordinate to xx then yy admits a decomposition y=z+wy=z+w where zz and ww are two martingales such that:

∥Sc(z)∥1,∞+∥Sr(w)∥1,∞≤c∥x∥1.‖Sc(z)‖1,∞+‖Sr(w)‖1,∞≤c‖x‖1.

 We also prove strong-type (p,p)(p,p)  version of the above weak-type result for 1<p<21<p<2.  As a byproduct of our approach, we obtain new and constructive proof of the noncommutative Burkholder-Gundy inequalities for 1<p<21<p<2 with the optimal order of the constants as p→1p→1.