Abstract
Given \(0<p<2\), we consider Mikhlin and Hörmander type multiplier theorems on \(H^p(\mathbb {R}^n)\) with restricted smoothness conditions. More precisely, we assume that \(m\in C^k(\mathbb {R}^n\setminus \{0\})\), where \(k=\big [n|\frac{1}{p}-\frac{1}{2}|\big ]+1\), meanwhile, we have restrictions on the order of differentiation with respect to each coordinate. In particular, if \(p>\frac{2}{3}\), we only need to differentiate at most once with respect to any single coordinate.
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Notes
In this paper, we don’t distinguish the notation |a| and \(|a|_1\) for multi-index \(\alpha \in \mathbb {N}^n\).
The authors would like to thank Yongfei Huang for suggesting this type of multipliers to us.
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Acknowledgements
We are grateful to the anonymous referee for the careful reading and the valuable comments that helped us to improve the manuscript. P. Chen was supported by the National Natural Science of Foundation of China (Grant No. 11501583) and the Guangdong Natural Science Foundation 2016A030313351. S. Huang, J. Qiang and Q. Zheng were supported by the National Natural Science of Foundation of China (Grant No. 11801188) and the Fundamental Research Funds for the Central Universities (Grant No. 2018KFYYXJJ041).
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Qiang, J., Chen, P., Huang, S. et al. On Some Fourier Multipliers for \(H^p(\mathbb {R}^n)\) with Restricted Smoothness Conditions. J Geom Anal 30, 3672–3697 (2020). https://doi.org/10.1007/s12220-019-00211-5
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DOI: https://doi.org/10.1007/s12220-019-00211-5