The Hilbert transform and orthogonal martingales in Banach spaces

Recently Banuelos and Kwaśnicki showed that the $L^p$-norms of the periodic Hilbert transform and the discrete Hilbert transform coincide for all $1<p<\infty$, which used to be an open problem for past 90 years. One of the key tools exploited by them was the $L^p$-estimate forreal-valued differentially subordinated orthogonal martingales. The goal of this talk is to generalize this type of estimates to vector-valued martingales. This will allow us to extend the result by Banuelos and Kwaśnicki both to the infinite-dimensional setting and more general norms. Moreover, as a corollary we will show new inequalities for decoupling constants in terms of the Hilbert transform norm and we will provide $L^p$-estimates for weakly differentially subordinated harmonic functions.