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

Probability;Functional Analysis