# Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

## TL;DR

This paper characterizes UMD Banach spaces via weak differential subordination of martingales and derives sharp bounds for Fourier multipliers, including Riesz transforms, in these spaces.

## Contribution

It introduces weak differential subordination for martingales and characterizes UMD Banach spaces through associated inequalities, also providing sharp Fourier multiplier estimates.

## Key findings

- UMD Banach spaces characterized by martingale inequalities
- Sharp bounds established for Fourier multipliers in UMD spaces
- Includes estimates for second order Riesz transforms

## Abstract

In this paper we introduce the notion of weak differential subordination for martingales and show that a Banach space $X$ is a UMD Banach space if and only if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E \|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.