  In this paper we introduce the notion of weak differential subordination for
martingales and show that a Banach space $X$ is a UMD Banach space if and only
if for all $p\in (1,\infty)$ and all purely discontinuous $X$-valued
martingales $M$ and $N$ such that $N$ is weakly differentially subordinated to
$M$, one has the estimate $\mathbb E \|N_{\infty}\|^p \leq C_p\mathbb E
\|M_{\infty}\|^p$. As a corollary we derive the sharp estimate for the norms of
a broad class of even Fourier multipliers, which includes e.g. the second order
Riesz transforms.
