## Banach Journal of Mathematical Analysis

### Martingale Hardy spaces with variable exponents

#### Abstract

In this paper, we introduce Hardy spaces with variable exponents defined on a probability space and develop the martingale theory of variable Hardy spaces. We prove the weak-type and strong-type inequalities on Doob’s maximal operator, and we get a $(1,p(\cdot),\infty)$-atomic decomposition for Hardy martingale spaces associated with conditional square functions. As applications, we obtain a dual theorem and the John–Nirenberg inequalities in the frame of variable exponents. The key ingredient is that we find a condition with a probabilistic characterization of $p(\cdot)$ to replace the so-called log-Hölder continuity condition in $\mathbb{R}^{n}$.

#### Article information

Source
Banach J. Math. Anal. Volume 10, Number 4 (2016), 750-770.

Dates
Accepted: 11 January 2016
First available in Project Euclid: 20 September 2016

https://projecteuclid.org/euclid.bjma/1474373751

Digital Object Identifier
doi:10.1215/17358787-3649326

Mathematical Reviews number (MathSciNet)
MR3548624

Zentralblatt MATH identifier
06667677

Subjects
Primary: 60G46: Martingales and classical analysis
Secondary: 60G42: Martingales with discrete parameter

#### Citation

Jiao, Yong; Zhou, Dejian; Hao, Zhiwei; Chen, Wei. Martingale Hardy spaces with variable exponents. Banach J. Math. Anal. 10 (2016), no. 4, 750--770. doi:10.1215/17358787-3649326. https://projecteuclid.org/euclid.bjma/1474373751.

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