This is an announcement for the paper “Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces” by Ivan Yaroslavtsevhttps://arxiv.org/find/math/1/au:+Yaroslavtsev_I/0/1/0/all/0/1.

Abstract: 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 $M$ an $N$ such tha $N$ 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.

The paper may be downloaded from the archive by web browser from URL https://arxiv.org/abs/1703.07817