On a relation between norms of the maximal function and the square function of a martingale
Volume 132 / 2013
Colloquium Mathematicum 132 (2013), 13-26
MSC: Primary 60G42, 46E30; Secondary 60G46, 47B38, 46N30.
DOI: 10.4064/cm132-1-2
Abstract
Let $\varOmega $ be a nonatomic probability space, let $X$ be a Banach function space over $\varOmega $, and let $\mathcal {M}$ be the collection of all martingales on $\varOmega $. For $f=(f_n)_{n \in \mathbb {Z}_+}\in \mathcal M$, let $Mf$ and $Sf$ denote the maximal function and the square function of $f$, respectively. We give some necessary and sufficient conditions for $X$ to have the property that if $f, g \in \mathcal M$ and $\| Mg\| _X \le \| Mf\| _X$, then $\| Sg\| _X \le C\| Sf\| _X$, where $C$ is a constant independent of $f$ and $g$.
