Banach Journal of Mathematical Analysis

Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications

Guangheng Xie and Dachun Yang

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Let (Ω,F,P) be a probability space, and let φ:Ω×[0,)[0,) be a Musielak–Orlicz function. In this article, we establish the atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces WHφs(Ω), WHφM(Ω), WHφS(Ω), WPφ(Ω), and WQφ(Ω). We then use these atomic characterizations to obtain the boundedness of σ-sublinear operators from weak martingale Musielak–Orlicz Hardy spaces to weak Musielak–Orlicz spaces, as well as some martingale inequalities which further clarify the relationships among these weak martingale Musielak–Orlicz Hardy spaces. All these results improve and generalize the corresponding results on weak martingale Orlicz–Hardy spaces. Moreover, we improve all the known results on weak martingale Musielak–Orlicz Hardy spaces. In particular, both the boundedness of σ-sublinear operators and the martingale inequalities, for weak weighted martingale Hardy spaces as well as for weak weighted martingale Orlicz–Hardy spaces, are new.

Article information

Banach J. Math. Anal., Volume 13, Number 4 (2019), 884-917.

Received: 14 November 2018
Accepted: 20 December 2018
First available in Project Euclid: 9 October 2019

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G42: Martingales with discrete parameter
Secondary: 60G46: Martingales and classical analysis 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis

probability space weak martingale Musielak–Orlicz Hardy space atom σ-sublinear operator martingale inequality


Xie, Guangheng; Yang, Dachun. Atomic characterizations of weak martingale Musielak–Orlicz Hardy spaces and their applications. Banach J. Math. Anal. 13 (2019), no. 4, 884--917. doi:10.1215/17358787-2018-0050.

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