On certain martingale inequalities for maximal functions and mean oscillations

Masato Kikuchi, Yasuhiro Kinoshita

Abstract


Let $X$ be a Banach function space over a nonatomic probability space. For a uniformly integrable martingale $f=(f_n)$ with respect to a filtration ${\mathcal F}=({\mathcal F}_n)$, let $Mf =\sup_n |f_n|$ and $\theta_{\mathcal F}f=\sup_n E[|f_{\infty}- f_{n-1}| \mid{\mathcal F}_n]$. We give a necessary and sufficient condition on $X$ for the inequality $\parallel \theta_{\mathcal F}f \parallel_X \leq C\parallel Mf\parallel_X$ to hold.

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DOI: http://dx.doi.org/10.7146/math.scand.a-15191

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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