Characterization of Banach function spaces that preserve the Burkholder square-function inequality

Abstract

Let $(X,\,\|\,\cdot\,\|_X)$ be a Banach function space over a nonatomic probability space. We give a necessary and sufficient condition on $X$ for the inequalities $c \|f_{\infty}\|_X \leq \|S(f)\|_X \leq C \|f_{\infty}\|_X$ to hold for all uniformly integrable martingales $f=(f_n)_{n \geq 0}$, where $f_{\infty}=\lim_n f_n$ a.s. and $S(f)=\left\{f_0^{\,2}+\sum_{n=1}^{\infty} (f_n - f_{n-1})^2\right\}^{1/2}$.