DOOB'S MAXIMAL INEQUALITIES FOR MARTINGALES IN VARIABLE LEBESGUE SPACE

• 来源期刊: 数学物理学报（英文版）
• 年，卷（期）: 2021,（1）
• 页码范围: 283-296
• 页数: 14页
• 语种: 英文
• DOI: 10.3969/j.issn.0252-9602.2021.01.016

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 opera-tors 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 partic-ular,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.