# Martingale decompositions and weak differential subordination in UMD Banach spaces

## TL;DR

This paper characterizes UMD Banach spaces via martingale decompositions and weak differential subordination, establishing equivalences with certain boundedness properties of martingale components and subordination relations.

## Contribution

It proves that UMD Banach spaces are characterized by the existence of specific martingale decompositions and subordination inequalities, extending classical results.

## Key findings

- UMD spaces characterized by Meyer-Yoeurp decompositions.
- Equivalence between UMD property and boundedness of martingale components.
- Weak differential subordination implies norm estimates in UMD spaces.

## Abstract

In this paper we consider Meyer-Yoeurp decompositions for UMD Banach space-valued martingales. Namely, we prove that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$, any $X$-valued $L^p$-martingale $M$ has a unique decomposition $M = M^d + M^c$ such that $M^d$ is a purely discontinuous martingale, $M^c$ is a continuous martingale, $M^c_0=0$ and \[ \mathbb E \|M^d_{\infty}\|^p + \mathbb E \|M^c_{\infty}\|^p\leq c_{p,X} \mathbb E \|M_{\infty}\|^p. \] An analogous assertion is shown for the Yoeurp decomposition of a purely discontinuous martingales into a sum of a quasi-left continuous martingale and a martingale with accessible jumps. As an application we show that $X$ is a UMD Banach space if and only if for any fixed $p\in (1,\infty)$ and for all $X$-valued martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $$ \mathbb E \|N_{\infty}\|^p \leq C_{p,X}\mathbb E \|M_{\infty}\|^p. $$