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, $M_0^c=0$ 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 $$\mathbb{E}\Vert N_{\infty }{\Vert }^p\le C_{p,X}\mathbb{E}\Vert M_{\infty }{\Vert }^p.$$

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