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Due to a classical argument of Doob it is well-known that the periodic Hilbert transform has a representation in terms of stochastic integrals with respect to a 2-dimensional Brownian motion. These stochastic integrals happen to be orthogonal martingales, so any estimates for orthogonal martingales lead to the same estimates for the periodic Hilbert transform. The goal of this talk is to show the converse dependence. Namely, we show that for any Banach space $X$ and any convex continuous functions $\Phi, \Psi:X \to \mathbb R_+$ one has that for any orthogonal martingales $M$ and $N$ such that $N$ is weakly differentially subordinate to $M$

\[

\mathbb E \Psi(N_t) \leq C_{\Phi, \Psi} \mathbb E \Phi(M_t),\;\;\; t\geq 0,

\]

where the sharp constant C_{\Phi, \Psi} coincides with the $\Phi, \Psi$-norm of the periodic Hilbert transform.

This estimate has a lot of applications. In particular, it will allow us to show that the $L^p$-norms of the periodic Hilbert transform and the discrete Hilbert transform coincide for all $1<p<\infty$ and for any Banach space $X$. This extends the result of Bañuelos and Kwaśnicki, who showed in 2017 that the $L^p$-norms of the periodic Hilbert transform and the discrete Hilbert transform are equal in the real-valued setting, which had been an open problem for past 90 years.

The talk is based on joint work with Adam Osękowski (University of Warsaw).