The Hilbert transform and orthogonal martingales in Banach spaces

TL;DR

This paper establishes bounds for orthogonal martingales in Banach spaces using Hilbert transform norms, revealing connections between martingale inequalities, harmonic analysis, and the geometry of Banach spaces.

Contribution

It proves a key estimate linking orthogonal martingales and Hilbert transform norms, and shows that finiteness of these norms characterizes the UMD property of Banach spaces.

Findings

The $ ext{Phi}, ext{Psi}$-norms of various Hilbert transforms are equal under symmetry.

Finiteness of the Hilbert transform norms implies the UMD property of the space.

Provides comparisons of $L^p$-norms with decoupling constants.

Abstract

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.