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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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.
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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