Geodesic
A CHARACTERIZATION OF BANACH FUNCTION SPACES ASSOCIATED WITH MARTINGALES
Glasgow mathematical journal, Tome 46 (2004) no. 1, pp. 143-153

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DOI

Let $(\Omega,\, \Sigma,\,\Prob)$ be a nonatomic probability space and let $\F=(\F_n)_{n{\in}\Z}$ be a filtration. If $f=(\,f_n)_{n{\in}\Z}$ is a uniformly integrable $\F$-martingale, let $\A_{\F}f=(\A_{\F}f_n)_{n{\in}\Z}$ denote the martingale defined by $\A_{\F}f_n =\E[|\,f_{\infty}||\F_n]\; (n \,{\in}\, \Z)$, where $f_{\infty}=\lim_n f_n$ a.s. Let $X$ be a Banach function space over $\Omega$. We give a necessary and sufficient condition for $X$ to have the property that $S(\,\hspace*{.2pt}f\hspace*{.3pt}) \,{\in}\, X$ if and only if $S(\A_{\F}f) \,{\in}\, X$, where $S(\,\hspace*{.2pt}f\hspace*{.3pt})$ stands for the square function of $f=(\,f_n)$.
DOI : 10.1017/S0017089503001617
Mots-clés : 46E30, 60G42 
KIKUCHI, MASATO. A CHARACTERIZATION OF BANACH FUNCTION SPACES ASSOCIATED WITH MARTINGALES. Glasgow mathematical journal, Tome 46 (2004) no. 1, pp. 143-153. doi: 10.1017/S0017089503001617
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     title = {A {CHARACTERIZATION} {OF} {BANACH} {FUNCTION} {SPACES} {ASSOCIATED} {WITH} {MARTINGALES}},
     journal = {Glasgow mathematical journal},
     pages = {143--153},
     year = {2004},
     volume = {46},
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     doi = {10.1017/S0017089503001617},
     url = {http://geodesic.mathdoc.fr/articles/10.1017/S0017089503001617/}
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