Abstract
We characterize the \(L^p\)–\(L^q\) compactness of Bergman type operators, which are singular integral operators induced by the modified Bergman kernel on the complex unit ball. Moreover, we characterize Schatten class and Macaev class Bergman type integral operators on the Lebesgue space and the Bergman space by the methods of spectral estimates and operator inequalities; we also give a relatively intrinsic characterization by introducing a concept of dimension of a compact operator. The Dixmier trace of Bergman type operators is also calculated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, A., Fournier, F.: Sobolev Spaces, Pure and Applied Mathematics. Elsevier/Academic Press, Amsterdam (2003)
Alpay, D., Kaptanoğlu, H.T.: Toeplitz operators on Arveson and Dirichlet spaces. Integr. Equ. Oper. Theory 58, 1–33 (2007)
Ando, T.: On the compactness of integral operators. Indag. Math. 24, 235–239 (1962)
Békollé, D., Bonami, A.: Estimates for the Bergman and Szegö projections in two symmetric domains of \(\mathbb{C} ^n\). Colloq. Math. 68, 81–100 (1995)
Bonami, A., Garrigós, G., Nana, C.: \(L^p\)-\(L^q\) estimates for Bergman projections in bounded symmetric domains of tube type. J. Geom. Anal. 24, 1737–1769 (2014)
Cheng, G., Fang, X., Wang, Z., Yu, J.: The hyper-singular cousin of the Bergman projection. Trans. Am. Math. Soc. 369, 8643–8662 (2017)
Cheng, G., Hou, X., Liu, C.: The singular integral operator induced by Drury–Arveson kernel. Complex Anal. Oper. Theory 12, 917–929 (2018)
Cobos, F., Peetre, J.: Interpolation of compactness using Aronszajn–Gagliardo functors. Israel J. Math. 68, 220–240 (1989)
Connes, A.: Noncommutative Geometry. Academic Press Inc., San Diego (1994)
Ding, L.: Schatten class Bergman-type and Szegö-type operators on bounded symmetric domains. Adv. Math. 401(108314), 1–27 (2022)
Ding, L., Wang, K.: The \(L^p\)-\(L^q\) boundedness and compactness of Bergman type operators. Taiwan. J. Math. 26(4), 713–740 (2022)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.G.: Higher Transcendental Functions, vol. I. McGraw-Hill, New York (1953)
Forelli, F., Rudin, W.: Projections on spaces of holomorphic functions in balls. Indiana Univ. Math. J. 24, 593–602 (1974)
Kantorovich, V., Akilov, P.: Functional Analysis. Pergamon, Oxford (1982)
Kaptanoğlu, H.T., Üreyen, A.: Singular integral operators with Bergman–Besov kernels on the ball. Integr. Equ. Oper. Theory 91, 1–30 (2019)
Krasnoselskiĭ, A.: On a theorem of M. Riesz. Soviet Math. Dokl. 1, 229–231 (1960)
Krasnoselskiĭ, A., Zabreĭko, P., Pustylnik, I., Sobolevskiĭ, E.: Integral Operators in Spaces of Summable Functions. Nordhoff, Groningen (1976)
Lin, F., Yang, X.: Geometric Measure Theory-An Introduction. Advanced Mathematics, Science Press, Beijing (2002)
Pau, J.: A remark on Schatten class Toeplitz operators on Bergman spaces. Proc. Am. Math. Soc. 142(8), 2763–2768 (2014)
Phong, H., Stein, E.: Estimates for the Bergman and Szegö projections on strongly pseudo-convex domains. Duke Math. J. 44, 695–704 (1977)
Rudin, W.: Real and Complex Analysis, 2nd edn. McGraw-Hill, New York (1974)
Rudin, W.: Function Theory in the Unit Ball of \(\mathbb{C} ^n\). Grundlehren der Math, Springer, New York (1980)
Ringrose, J.: Compact Non-self-Adjoint Operators. Van Nostrand Reinhold Co., London (1971)
Upmeier, H., Wang, K.: Dixmier trace for Toeplitz operators on symmetric domains. J. Funct. Anal. 271, 532–565 (2016)
Wong, B.: Characterization of the unit ball in \(\mathbb{C} ^n\) by its automorphism group. Invent. Math. 41, 253–257 (1977)
Zhu, K.: Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, vol. 226. Springer, New York (2005)
Zhu, K.: Operator Theory in Function Spaces. Mathematical Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence (2007)
Acknowledgements
The author would like to thank Professor Qi’an Guan and Professor Kai Wang for their helpful discussions. The author would also like to thank Professor H. Kaptanoğlu devoutly for sending us their recent work. The author was partially supported by NSFC (12201571) and Scientific Research Foundation of Zhengzhou University (32213181). The author thank the anonymous referees for constructive comments which helped to improve the paper.
Author information
Authors and Affiliations
Contributions
LD wrote the manuscript. All authors reviewed the manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Communicated by H. Turgay Kaptanoglu.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This article is part of the topical collection “Harmonic Analysis and Operator Theory” edited by H. Turgay Kaptanoglu, Aurelian Gheondea and Serap Oztop.
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
Ding, L. Compact Bergman Type Operators. Complex Anal. Oper. Theory 18, 5 (2024). https://doi.org/10.1007/s11785-023-01419-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11785-023-01419-8