Geodesic
On a relation between norms of the maximal function and the square function of a martingale
Masato Kikuchi1
1 Department of Mathematics University of Toyama 3190 Gofuku Toyama 930-8555, Japan
Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 13-26 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $g$.
DOI : 10.4064/cm132-1-2
Keywords: varomega nonatomic probability space banach function space varomega mathcal collection martingales nbsp varomega mathbb mathcal denote maximal function square function nbsp respectively necessary sufficient conditions have property mathcal where constant independent nbsp

Masato Kikuchi 1

1 Department of Mathematics University of Toyama 3190 Gofuku Toyama 930-8555, Japan 
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 the square function of a martingale
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Masato Kikuchi. On a relation between norms of
 the maximal function and
 the square function of a martingale. Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 13-26. doi: 10.4064/cm132-1-2

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