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On the maximal inequalities for martingales involving two functions


Authors: Mei Tao and Peide Liu
Journal: Proc. Amer. Math. Soc. 130 (2002), 883-892
MSC (2000): Primary 60G42, 43A17
DOI: https://doi.org/10.1090/S0002-9939-01-06095-6
Published electronically: August 28, 2001
MathSciNet review: 1866045
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Abstract: Let $\Phi (t)$ and $\Psi (t)$ be nonnegative convex functions, and let $\varphi $ and $\psi $ be the right continuous derivatives of $\Phi $ and $\Psi ,$ respectively. In this paper, we prove the equivalence of the following three conditions: (i) $\Vert f^{*}\Vert _\Phi \leq c\Vert f\Vert _\Psi ,$ (ii) $L^\Psi \subseteq $ $H^\Phi $and (iii) $\int_{s_0}^t\frac{\varphi (s)}sds\leq c\psi (ct), \forall t>s_0,$where $L^\Psi $ and $H^\Phi $ are the Orlicz martingale spaces. As a corollary, we get a sufficient and necessary condition under which the extension of Doob's inequality holds. We also discuss the converse inequalities.


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Additional Information

Mei Tao
Affiliation: College of Mathematics Sciences, Wuhan University, Hubei, 430072, People’s Republic of China
Address at time of publication: Department of Mathematiques, U.F.R. des Sciences et Techniques, 16, Route de Gray, -F-25030 Besancon Cedex, France
Email: meitaosuizhou@263.net

Peide Liu
Affiliation: College of Mathematics Sciences, Wuhan University, Hubei, 430072, People’s Republic of China
Email: pdliu@whu.edu.cn

DOI: https://doi.org/10.1090/S0002-9939-01-06095-6
Keywords: Martingale inequality, nonnegative submartingale, maximal function, Orlicz space
Received by editor(s): February 4, 2000
Received by editor(s) in revised form: August 25, 2000
Published electronically: August 28, 2001
Additional Notes: This research was supported by the National Science Foundation of the People’s Republic of China
Communicated by: Claudia M. Neuhauser
Article copyright: © Copyright 2001 American Mathematical Society

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