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∊ (1, ∞), any X-valued Lp -martingale M has a unique decomposition M = Md + Mc such that Md is a purely discontinuous martingale, Mc is a continuous martingale, M 0 c = 0 and E ∥ M ∞ d ∥ P + E ∥ M ∞ c ∥ p ≤ c p , X E ∥ M ∞ ∥ 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 ∊ (1, ∞) and for all X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate. E ∥ N ∞ ∥ P ≤ c p , X E ∥ M ∞ ∥ p .