Martingale decompositions and weak differential subordination in UMD Banach 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 \[\mathbb{E}\big\|M^{d}_{\infty}\big\|^{p}+\mathbb{E}\big\|M^{c}_{\infty}\big\|^{p}\leq c_{p,X}\mathbb{E}\|M_{\infty}\|^{p}.\] 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 $\mathbb{E}\|N_{\infty}\|^{p}\leq C_{p,X}\mathbb{E}\|M_{\infty}\|^{p}$.