Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces

We introduce the notion of weak differential subordination for martingales, and show that a Banach space X is UMD if and only if for all p epsilon (1, infinity) and all purely discontinuous X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate E parallel to N-infinity parallel to(p) <= CpE parallel to M-infinity parallel to(p). As a corollary we derive a sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.