We study the high-dimensional Hausdorff operators on the Morrey space and on the Campanato space. We establish their sharp boundedness on these spaces. Particularly, our results solve an open question posted by E. Liflyand (2013).
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J. Chen, D. Fan, J. Li: Hausdorff operators on function spaces. Chin. Ann. Math., Ser. B 33 (2012), 537–556.
J. Chen, D. Fan, S. Wang: Hausdorff operators on Euclidean spaces. Appl. Math., Ser. B (Engl. Ed.) 28 (2013), 548–564.
J. Chen, X. Zhu: Boundedness of multidimensional Hausdorff operators on H1(Rn). J. Math. Anal. Appl. 409 (2014), 428–434.
C. Fefferman, E. M. Stein: Hp spaces of several variables. Acta Math. 129 (1972), 137–193.
Z. Fu, L. Grafakos, S. Lu, F. Zhao: Sharp bounds for m-linear Hardy and Hilbert operators. Houston J. Math. 38 (2012), 225–244.
A. K. Lerner, E. Liflyand: Multidimensional Hausdorff operators on the real Hardy space. J. Aust. Math. Soc. 83 (2007), 79–86.
E. Liflyand: Open problems on Hausdorff operators. Complex Analysis and Potential Theory. World Scientific, Hackensack, 2007, pp. 280–285.
E. Liflyand: Boundedness of multidimensional Hausdorff operators on H1(ℝn). Acta Sci. Math. 74 (2008), 845–851.
E. Liflyand: Hausdorff operators on Hardy spaces. Eurasian Math. J. 4 (2013), 101–141.
E. Liflyand, A. Miyachi: Boundedness of the Hausdorff operators in Hp spaces, 0 < p < 1. Stud. Math. 194 (2009), 279–292.
E. Liflyand, F. Móricz: The Hausdorff operator is bounded on the real Hardy space H1(ℝ). Proc. Am. Math. Soc. 128 (2000), 1391–1396.
E. Liflyand, F. Móricz: Commuting relations for Hausdorff operators and Hilbert transforms on real Hardy spaces. Acta Math. Hung. 97 (2002), 133–143.
F. Móricz: Multivariate Hausdorff operators on the spaces H1(ℝn) and BMO(ℝn). Anal. Math. 31 (2005), 31–41.
J. Ruan, D. Fan: Hausdorff operators on the power weighted Hardy spaces. J. Math. Anal. Appl. 433 (2016), 31–48.
J. Ruan, D. Fan, Q. Wu: Weighted Herz space estimates for Hausdorff operators on the Heisenberg group. Banach J. Math. Anal. 11 (2017), 513–535.
F. Weisz: The boundedness of the Hausdorff operator on multi-dimensional Hardy spaces. Analysis, München 24 (2004), 183–195.
J. Xiao: Lp and BMO bounds of weighted Hardy-Littlewood averages. J. Math. Anal. Appl. 262 (2001), 660–666.
The first author would like to express his deep thanks to Professor Silei Wang for his private communication and help. The authors cordially thank the anonymous referees for their valuable comments which led to the improvement of this paper.
The research was supported by the Zhejiang Provincial Natural Science Foundation of China (No. LY18A010015), the National Natural Science Foundation of China (Nos. 11771358, 11871436) and the “BoDa” Research Program of Zhejiang International Studies University (No. BD2019B5).
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Ruan, J., Fan, D. & Li, H. Hausdorff Operator on Morrey Spaces and Campanato Spaces. Czech Math J 70, 505–517 (2020). https://doi.org/10.21136/CMJ.2019.0386-18
- Hausdorff operator
- Morrey space
- Campanato space