Abstract
Let \(f\) be an integrable function which has integral \(0\) on \(\mathbb{R}^n \). What is the largest condition on \(|f|\) that guarantees that \(f\) is in the Hardy space \(\mathcal{H}^1(\mathbb{R}^n)\)? When \(f\) is compactly supported, it is well-known that the largest condition on \(|f|\) is the fact that \(|f|\in L \log L(\mathbb{R}^n) \). We consider the same kind of problem here, but without any condition on the support. We do so for \(\mathcal{H}^1(\mathbb{R}^n)\), as well as for the Hardy space \(\mathcal{H}_{\log}(\mathbb{R}^n)\) which appears in the study of pointwise products of functions in \(\mathcal{H}^1(\mathbb{R}^n)\) and in its dual BMO.
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Acknowledgements
The authors thank Fran¸cois Bouchut for having shared his results. They also thank the referees for valuable suggestions and comments.
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Bonami, A., Grellier, S. & Sehba, B.F. Global Stein theorem on Hardy spaces. Anal Math 50, 79–99 (2024). https://doi.org/10.1007/s10476-024-00003-2
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DOI: https://doi.org/10.1007/s10476-024-00003-2