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

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.