Martingale Operators and Hardy Spaces with Continuous Time Generated by Them

Hao, Zhiwei; Yue, Jianlan; Weisz, Ferenc

doi:10.3390/math13162583

Open AccessArticle

Martingale Operators and Hardy Spaces with Continuous Time Generated by Them

by

Zhiwei Hao

¹,

Jianlan Yue

¹ and

Ferenc Weisz

^2,*

¹

School of Mathematics and Statistics, Hunan University of Science and Technology, Xiangtan 411201, China

²

Department of Numerical Analysis, Eötvös L. University, Pázmány P. Sétány 1/C, 1117 Budapest, Hungary

^*

Author to whom correspondence should be addressed.

Mathematics 2025, 13(16), 2583; https://doi.org/10.3390/math13162583

Submission received: 14 July 2025 / Revised: 9 August 2025 / Accepted: 10 August 2025 / Published: 12 August 2025

(This article belongs to the Special Issue New Aspects of Differentiable and Not Differentiable Function Theory)

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Abstract

In this paper, we introduce the martingale Hardy spaces and

B M O

spaces generated by an operator T in continuous time and establish the atomic decomposition theorem of the space

H_{p}^{T}

under the condition that T is predictable. We show that the

B M O_{q}

spaces generated by the operator T are all equivalent and consider the sharp operator. Using the real interpolation method, we identify the interpolation spaces between the Hardy spaces and the

B M O

spaces. With the aid of atomic decomposition, we establish some martingale inequalities between the Hardy spaces generated by two different operators.

Keywords: continuous martingales; Hardy spaces; atomic decompositions

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MDPI and ACS Style

Hao, Z.; Yue, J.; Weisz, F. Martingale Operators and Hardy Spaces with Continuous Time Generated by Them. Mathematics 2025, 13, 2583. https://doi.org/10.3390/math13162583

AMA Style

Hao Z, Yue J, Weisz F. Martingale Operators and Hardy Spaces with Continuous Time Generated by Them. Mathematics. 2025; 13(16):2583. https://doi.org/10.3390/math13162583

Chicago/Turabian Style

Hao, Zhiwei, Jianlan Yue, and Ferenc Weisz. 2025. "Martingale Operators and Hardy Spaces with Continuous Time Generated by Them" Mathematics 13, no. 16: 2583. https://doi.org/10.3390/math13162583

APA Style

Hao, Z., Yue, J., & Weisz, F. (2025). Martingale Operators and Hardy Spaces with Continuous Time Generated by Them. Mathematics, 13(16), 2583. https://doi.org/10.3390/math13162583

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Martingale Operators and Hardy Spaces with Continuous Time Generated by Them

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