Fourier multipliers and weak differential subordination of martingales in UMD Banach spaces
Volume 243 / 2018
Studia Mathematica 243 (2018), 269-301
MSC: Primary 42B15, 60G46; Secondary 60B11, 60G42, 60G44, 60G51.
DOI: 10.4064/sm170329-25-8
Published online: 5 April 2018
Abstract
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 $p\in (1,\infty)$ 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 a sharp estimate for the norms of a broad class of even Fourier multipliers, which includes e.g. the second order Riesz transforms.
