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 ∈ (1, ∞) and all purely discontinuous X-valued martingales M and N such that N is weakly differentially subordinated to M, one has the estimate E || N_∞ ||^p ≤ C_pE|| M<sub>∞<...