ACTA issues

On some inequalities for martingale transforms in Banach function spaces

Masato Kikuchi

Acta Sci. Math. (Szeged) 80:1-2(2014), 289-306

Abstract. Let $X$ be a Banach function space over a probability space. We consider the inequality $\norm{(v{\ast }f)_{\infty }}_X \le C\norm{f_{\infty }}_X$, where $f=(f_n)_{n \in\Z }$ is a uniformly integrable martingale, $v=(v_n)_{n \in\Z }$ is a predictable process such that $\sup_n \abs{v_n}\le1$ almost surely, and $v{\ast }f=((v{\ast }f)_n)_{n \in\Z }$ denotes the martingale transform of $f$ by $v$. The main result gives necessary and sufficient conditions on $X$ for this inequality to hold.

DOI: 10.14232/actasm-012-542-3

AMS Subject Classification (1991): 60G42, 46E30, 46N30

Keyword(s): martingale, martingale transform, Banach function space, rearrangement-invariant function space

Received May 30, 2012. (Registered under 42/2012.)