On the maximal inequalities for martingales involving two functions
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- by Mei Tao and Peide Liu PDF
- Proc. Amer. Math. Soc. 130 (2002), 883-892 Request permission
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) $\|f^{*}\| _\Phi \leq c\|f\|_\Psi ,$ (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.References
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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
- 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
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1866045