数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (4): 1151-1162.doi: 10.1016/S0252-9602(18)30805-1

• 论文 • 上一篇    下一篇

BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT

刘培德, 王茂发   

  1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • 收稿日期:2017-06-30 修回日期:2017-12-07 出版日期:2018-08-25 发布日期:2018-08-25
  • 通讯作者: Peide LIU,E-mail:pdliu@whu.edu.cn E-mail:pdliu@whu.edu.cn
  • 作者简介:Maofa WANG,E-mail:mfwang.math@whu.edu.cn
  • 基金资助:

    The first author was supported by NSFC (11471251). The second author was supported by NSFC (11271293).

BURKHOLDER-GUNDY-DAVIS INEQUALITY IN MARTINGALE HARDY SPACES WITH VARIABLE EXPONENT

Peide LIU, Maofa WANG   

  1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
  • Received:2017-06-30 Revised:2017-12-07 Online:2018-08-25 Published:2018-08-25
  • Contact: Peide LIU,E-mail:pdliu@whu.edu.cn E-mail:pdliu@whu.edu.cn
  • Supported by:

    The first author was supported by NSFC (11471251). The second author was supported by NSFC (11271293).

摘要:

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

关键词: variable exponent Lebesgue space, martingale inequality, Dellacherie theorem, Burkholder-Gundy-Davis inequality, Chevalier inequality

Abstract:

In this article, by extending classical Dellacherie's theorem on stochastic sequences to variable exponent spaces, we prove that the famous Burkholder-Gundy-Davis inequality holds for martingales in variable exponent Hardy spaces. We also obtain the variable exponent analogues of several martingale inequalities in classical theory, including convexity lemma, Chevalier's inequality and the equivalence of two kinds of martingale spaces with predictable control. Moreover, under the regular condition on σ-algebra sequence we prove the equivalence between five kinds of variable exponent martingale Hardy spaces.

Key words: variable exponent Lebesgue space, martingale inequality, Dellacherie theorem, Burkholder-Gundy-Davis inequality, Chevalier inequality