数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 531-538.doi: 10.1007/s10473-023-0204-6

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NEW DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES*

Zhiwei Hao, Libo Li   

  1. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China
  • 收稿日期:2022-02-08 修回日期:2022-05-21 出版日期:2023-03-25 发布日期:2023-04-12
  • 通讯作者: †Libo Li, lilibo@hnust.edu.cn.
  • 作者简介:Zhiwei Hao, E-mail: haozhiwei@hnust.edu.cn
  • 基金资助:
    This project was supported by the National Natural Science Foundation of China (11801001, 12101223), the Scientific Research Fund of Hunan Provincial Education Department (20C0780) and the Natural Science Foundation of Hunan Province (2022JJ40145, 2022JJ40146).

NEW DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES*

Zhiwei Hao, Libo Li   

  1. School of Mathematics and Computing Science, Hunan University of Science and Technology, Xiangtan 411201, China
  • Received:2022-02-08 Revised:2022-05-21 Online:2023-03-25 Published:2023-04-12
  • Contact: †Libo Li, lilibo@hnust.edu.cn.
  • About author:Zhiwei Hao, E-mail: haozhiwei@hnust.edu.cn
  • Supported by:
    This project was supported by the National Natural Science Foundation of China (11801001, 12101223), the Scientific Research Fund of Hunan Provincial Education Department (20C0780) and the Natural Science Foundation of Hunan Province (2022JJ40145, 2022JJ40146).

摘要: Let $1\leq q\leq \infty$, $b$ be a slowly varying function and let $ \Phi: [0,\infty ) \longrightarrow [0,\infty ) $ be an increasing convex function with $\Phi(0)=0$ and $\lim\limits_{r \rightarrow \infty}\Phi(r)=\infty$. In this paper, we present a new class of Doob's maximal inequality on Orlicz-Lorentz-Karamata spaces $L_{\Phi,q,b}$. The results are new, even for the Lorentz-Karamata spaces with $\Phi(t)=t^p$, the Orlicz-Lorentz spaces with $b\equiv1$, and weak Orlicz-Karamata spaces with $q=\infty$ in the framework of $L_{\Phi,q,b}$. Moreover, we obtain some even stronger qualitative results that can remove the $\vartriangle_2$-condition of Liu, Hou and Wang (Sci China Math, 2010, 53(4): 905--916).

关键词: martingales, Doob's inequality, Orlicz-Lorentz-Karamata spaces, convex functions

Abstract: Let $1\leq q\leq \infty$, $b$ be a slowly varying function and let $ \Phi: [0,\infty ) \longrightarrow [0,\infty ) $ be an increasing convex function with $\Phi(0)=0$ and $\lim\limits_{r \rightarrow \infty}\Phi(r)=\infty$. In this paper, we present a new class of Doob's maximal inequality on Orlicz-Lorentz-Karamata spaces $L_{\Phi,q,b}$. The results are new, even for the Lorentz-Karamata spaces with $\Phi(t)=t^p$, the Orlicz-Lorentz spaces with $b\equiv1$, and weak Orlicz-Karamata spaces with $q=\infty$ in the framework of $L_{\Phi,q,b}$. Moreover, we obtain some even stronger qualitative results that can remove the $\vartriangle_2$-condition of Liu, Hou and Wang (Sci China Math, 2010, 53(4): 905--916).

Key words: martingales, Doob's inequality, Orlicz-Lorentz-Karamata spaces, convex functions