Abstract
In this paper, we study the rough fractional Hausdorff operator on variable exponent Morrey–Herz spaces in the setting of the Heisenberg group. We define Morrey–Herz spaces with three variable exponents and then give sufficient and necessary conditions for the boundedness of the rough fractional Hausdorff operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
All data generated or analysed during this study are included in this published article.
References
Adamowicz, T., Harjulehto, P., Hästö, P.: Maximal operator in variable exponent Lebesgue spaces on unbounded quasimetric measure spaces. Math. Scand. 116(1), 5–22 (2015)
Almeida, A., Hästö, P.: Besov spaces with variable smoothness and integrability. J. Funct. Anal. 258(5), 1628–1655 (2010)
Brown, G., Móricz, F.: Multivariate Hausdorff operators on the spaces \(L^p({{\mathbb{R} }}^n)\). J. Math. Anal. Appl. 271(2), 443–454 (2002)
Chen, J., Fan, D., Li, J.: Hausdorff operators on function spaces. Chin. Ann. Math. Ser. B 33(4), 537–556 (2012)
Chuong, N.M., Duong, D.V., Dung, K.H.: Multilinear Hausdorff operator on variable exponent Morrey–Herz type spaces. Integral Transforms Spec. Funct. 31(1), 62–86 (2020)
Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces: Foundations and Harmonic Analysis. Springer-Basel, Berlin (2013)
Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Heidelberg (2011)
Drihem, D., Seghiri, F.: Notes on the Herz-type Hardy spaces of variable smoothness and integrability. Math. Ineqaul. Appl. 19(1), 145–165 (2016)
Dung, K.H., Duong, D.V., Chuong, N.M.: Rough Hausdorff operator and its commutators on the Heisenberg group. Adv. Oper. Theory 6(3), 1–19 (2021)
Duong, D.V.: Generalized multilinear Hausdorff operators on the Heisenberg group. Results Math. 76(2), 1–24 (2021)
Fang, J., Zhao, J.: Variable Hardy spaces on the Heisenberg group. Anal. Theory Appl. 32(3), 242–271 (2016)
Georgkis, C.: The Hausdorff mean of a Fourier–Stieltjes transform. Proc. Am. Math. Soc. 116(2), 465–471 (1992)
Hurwitz, W.A., Silverman, L.L.: On the consistency and equivalence of certain definitions of summability. Trans. Am. Math. Soc. 18(1), 1–20 (1917)
Izuki, M.: Boundedness of sublinear operators on Herz spaces with variable exponent and application to wavelet characterization. Anal. Math. 36(1), 33–50 (2010)
Lifyand, E., Móricz, F.: The Hausdorff operators is bounded on the real Hardy \(H^1({{\mathbb{R} }})\). Proc. Am. Math. Soc. 128(5), 1391–1396 (2000)
Lin, X., Sun, L.: Some estimates on the Hausdorff operator. Acta Sci. Math. 78(3), 669–681 (2012)
Liu, D., Zhao, J.: Multilinear Hausdorff operators on weighted Herz and Morrey–Herz spaces with variable exponent. J. Pseudo Differ. Oper. Appl. 13(1), 1–21 (2022)
Musielak, J.: Orlicz Spaces and Modular Spaces. Springer, Berlin (1983)
Siskakis, A.G.: The Cesàro operator is bounded on \(H^1\). Proc. Am. Math. Soc. 110(2), 461–462 (1990)
Thangavelu, S.: Harmonic Analysis on the Heisenberg Group. Progress in Mathematics, Bostan (1998)
Wang, S., Xu, J.: Weighted norm inequality for bilinear Calderón–Zygmund operators on Herz–Morrey spaces withvariable exponents. J. Inequal. Appl. 2019(1), 1–23 (2019)
Wu, Q., Fan, D.: Hardy space estimates of Hausdorff operators on the Heisenberg group. Nonlinear Anal. 164, 135–154 (2017)
Acknowledgements
The authors would like to thank the referees for their careful reading and useful suggestions, which have improved the paper.
Funding
The corresponding author, Jiman Zhao, is supported by Beijing Municipal Natural Science Foundation (Grant No. 1222008), National Natural Science Foundation of China (Grant no. 12271042) and the National Key Research and Development Program of China (Grant No. 2020YFA0712900).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have no relevant financial or non-financial interests to disclose.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, Z., Zhao, J. Rough Fractional Hausdorff Operators on Morrey–Herz Spaces with Variable Exponents. Results Math 79, 28 (2024). https://doi.org/10.1007/s00025-023-02039-6
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-023-02039-6