Weak differential subordination of martingales and its application in Harmonic Analysis

Differential subordination of real-valued martingales together with its basic properties have been discovered by Burkholder in 1984. In this talk we will discuss weak differential subordination of martingales, which is a generalization of differential subordination to the infinite dimensional setting, and provide extension of the corresponding L^p estimates for vector-valued martingales. As a corollary we extend the results of Bañuelos and Bogdan (2007) and Bañuelos, Bielaszewski, and Bogdan (2011) on sharp estimates for the norms of a broad class of even Fourier multipliers (including e.g. second order Riesz transforms) to infinite dimensions.