Abstract
We introduce and investigate the weak weighted martingale Hardy spaces \(\Lambda_{p,\infty}^s(\omega)\), where \(0<p<\infty\), \(\omega\) is a weight and \(s\) is the conditional square function. This new family of spaces provides a framework which unifies various kinds of weak martingale Hardy spaces, including weak martingale Orlicz–Hardy spaces, weak martingale Karamata–Hardy spaces, weak martingale Orlicz–Karamata–Hardy spaces, and so on. We establish the atomic decompositions for \(\Lambda_{p,\infty}^s(\omega)\), and then apply the atomic decompositions to deduce some new martingale inequalities and duality theorems. We discuss similar results for the Hardy spaces \(\Lambda_{p,\infty}^*(\omega)\), \(\Lambda_{p,\infty}^S(\omega)\), \(\mathcal P_{p,\infty}(\omega)\) and \(\mathcal Q_{p,\infty}(\omega)\) as well. The results obtained here generalize the corresponding known results in various weak martingale Hardy spaces.
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The authors would like to express their gratitude to the referee for his/her careful reading and valuable comments which improved the presentation of this article.
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Wenfei Fan is supported by Hunan Provincial Innovation Foundation For Postgraduate (No. CX20210091).
Anming Yang is supported by the NSFC (No. 11801157), Hunan Provincial Natural Science Foundation (No. 2020JJ5030) and the Fundamental Research Funds for the Central Universities.
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Fan, W., Yang, A. Dual spaces and inequalities of new weak martingale Hardy spaces. Acta Math. Hungar. 169, 134–157 (2023). https://doi.org/10.1007/s10474-023-01293-y
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DOI: https://doi.org/10.1007/s10474-023-01293-y