DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

doi:10.1007/s10473-021-0116-2

Acta mathematica scientia,Series B ›› 2021, Vol. 41 ›› Issue (1): 283-296.doi: 10.1007/s10473-021-0116-2

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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).

Abstract

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

CLC Number:

46E30

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

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