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New variable martingale Hardy spaces

Part of: Stochastic processes

Published online by Cambridge University Press:  23 April 2021

Yong Jiao ,
Dan Zeng  and
Dejian Zhou
Yong Jiao
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha410075, People's Republic of China (jiaoyong@csu.edu.cn; zengdan@csu.edu.cn and zhoudejian@csu.edu.cn)
Dan Zeng
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha410075, People's Republic of China (jiaoyong@csu.edu.cn; zengdan@csu.edu.cn and zhoudejian@csu.edu.cn)
Dejian Zhou
Affiliation:
School of Mathematics and Statistics, Central South University, Changsha410075, People's Republic of China (jiaoyong@csu.edu.cn; zengdan@csu.edu.cn and zhoudejian@csu.edu.cn)
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Abstract

We investigate various variable martingale Hardy spaces corresponding to variable Lebesgue spaces $\mathcal {L}_{p(\cdot )}$ defined by rearrangement functions. In particular, we show that the dual of martingale variable Hardy space $\mathcal {H}_{p(\cdot )}^{s}$ with $0<p_{-}\leq p_{+}\leq 1$ can be described as a BMO-type space and establish martingale inequalities among these martingale Hardy spaces. Furthermore, we give an application of martingale inequalities in stochastic integral with Brownian motion.

Keywords

variable exponentrearrangement functionmartingale inequalitiesatomic decompositionduality

MSC classification

Primary: 60G46: Martingales and classical analysis
Secondary: 60G42: Martingales with discrete parameter
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 152 , Issue 2 , April 2022 , pp. 450 - 478
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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