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

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 $M$ and $N$ such that $N$ is weakly differentially subordinated to $M$, one has the estimate $\mathbb{E}\Vert N_{\infty }{\Vert }^p\le C_p\mathbb{E}\Vert M_{\infty }{\Vert }^p$. 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.

Minor revision

Cited by in corpus (2)

• Burkholder-Davis-Gundy inequalities in UMD Banach spaces
• $L^q$-valued Burkholder-Rosenthal inequalities and sharp estimates for stochastic integrals