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Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces

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Abstract

Given a probability space \((\Omega ,\mathcal {F},\mathbb P)\) and a rearrangement-invariant quasi-Banach function space X, the authors of this article first prove the \(\alpha \)-atomic (\(\alpha \in [1,\infty )\)) characterization of weak martingale Hardy spaces \(WH_X(\Omega )\) associated with X via simple atoms. The authors then introduce the generalized weak martingale \(\textrm{BMO}\) spaces which proves to be the dual spaces of \(WH_X(\Omega )\). Consequently, the authors derive a new John–Nirenberg theorem for these weak martingale \(\textrm{BMO}\) spaces. Finally, the authors apply these results to the generalized grand Lebesgue space and the weighted Lorentz space. Even in these special cases, the results obtained in this article are totally new.

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Acknowledgements

We would like to thank the referee for her/his very careful reading, very valuable comments and suggestions which indeed helped us to improve and clarify the presentation of this article.

Funding

This project is supported by the National Natural Science Foundation of China 12201647 and the Natural Science Foundation of Hunan Province of China 2021JJ40711.

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  1. School of Mathematics and Statistics, HNP-LAMA, Central South University, 410075, Changsha, People’s Republic of China

    Xingyan Quan, Niyonkuru Silas & Guangheng Xie

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  1. Xingyan Quan
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All authors participated in the writing and the research of this article, and have authorized the final version of the article submitted for publication. We conceive of no conflict of interest in the publication of this article. The work has not been published previously and it has not been submitted for publication elsewhere.

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Correspondence to Guangheng Xie.

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Quan, X., Silas, N. & Xie, G. Dual Spaces for Weak Martingale Hardy Spaces Associated with Rearrangement-Invariant Spaces. Potential Anal (2023). https://doi.org/10.1007/s11118-023-10104-6

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