Martingale decompositions and weak differential subordination in UMD Banach spaces (1706.01731v2)

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 $$ \mathbb E |N_{\infty}|^p \leq C_{p,X}\mathbb E |M_{\infty}|^p. $$