The Hilbert transform and orthogonal martingales in Banach spaces

Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$\mathbb E \Psi(N_t) \leq C_{\Phi,\Psi,X} \mathbb E \Phi(M_t),\;\;\; t\geq 0,$$ where $\Phi, \Psi:X \to \mathbb R_+$ are convex continuous functions and the least admissible constant $C_{\Phi,\Psi,X}$ coincides with the $\Phi,\Psi$-norm of the periodic Hilbert transform. As a corollary, it is shown that the $\Phi,\Psi$-norms of the periodic Hilbert transform, the Hilbert transform on the real line, and the discrete Hilbert transform are the same if $\Phi$ is symmetric. We also prove that under certain natural assumptions on $\Phi$ and $\Psi$, the condition $C_{\Phi,\Psi,X}<\infty$ yields the UMD property of the space $X$. As an application, we provide comparison of $L^p$-norms of the periodic Hilbert transform to Wiener and Paley-Walsh decoupling constants. We also study the norms of the periodic, nonperiodic and discrete Hilbert transforms, present the corresponding estimates in the context of differentially subordinate harmonic functions and more general singular integral operators.