Martingale decompositions and weak differential subordination in UMD Banach spaces

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