DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

doi:10.1007/s10473-021-0116-2

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (1): 283-296.doi: 10.1007/s10473-021-0116-2

DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

刘培德

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

收稿日期:2019-09-15 修回日期:2020-09-14 出版日期:2021-02-25 发布日期:2021-04-06
作者简介:Peide LIU,E-mail:pdliu@whu.edu.cn
基金资助:
The project was supported by the NSFC (11471251).

DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

Peide LIU

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received:2019-09-15 Revised:2020-09-14 Online:2021-02-25 Published:2021-04-06
About author:Peide LIU,E-mail:pdliu@whu.edu.cn
Supported by:
The project was supported by the NSFC (11471251).

摘要/Abstract

摘要： In this paper we deal with the martingales in variable Lebesgue space over a probability space. We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space. The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob's maximal operators in martingale Lebesgue space with a variable exponent. In particular, we present two kinds of weak-type Doob's maximal inequalities and some necessary and sufficient conditions for strong-type Doob's maximal inequalities. Finally, we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p>1.

关键词: variable Lebesgue space, martingale inequality, norm convergence, Doob's maximal inequality

Abstract: In this paper we deal with the martingales in variable Lebesgue space over a probability space. We first prove several basic inequalities for conditional expectation operators and give several norm convergence conditions for martingales in variable Lebesgue space. The main aim of this paper is to investigate the boundedness of weak-type and strong-type Doob's maximal operators in martingale Lebesgue space with a variable exponent. In particular, we present two kinds of weak-type Doob's maximal inequalities and some necessary and sufficient conditions for strong-type Doob's maximal inequalities. Finally, we provide two counterexamples to show that the strong-type inequality does not hold in general variable Lebesgue spaces with p>1.

Key words: variable Lebesgue space, martingale inequality, norm convergence, Doob's maximal inequality

中图分类号:

46E30

刘培德. DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE[J]. 数学物理学报(英文版), 2021, 41(1): 283-296.

Peide LIU. DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE[J]. Acta mathematica scientia,Series B, 2021, 41(1): 283-296.

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DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

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