Abstract
We give characterizations for Bergman-Orlicz spaces with standard weights via a Lipschitz type condition in the Euclidean, hyperbolic, and pseudo-hyperbolic metrics. As an application, we obtain the boundeness of the symmetric lifting operator from Bergman-Orlicz spaces on the unit disk into Bergman-Orlicz spaces on the bidisk.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bergmann S. Über die Kernfunktion eines Bereiches und ihr Verhalten am Rande I. J Reine Angew Math, 1933, 169: 1–42
Bergman S. The Kernel Function and Conformal Mapping. Providence, RI: American Mathematical Society, 1950
Bojarski B, Hajlasz P. Pointwise inequalities for Sobolev functions and some applications. Studia Math, 1993, 106: 77–92
Duren P, Schuster A. Bergman Spaces. Mathematical Surveys and Monographs 100. Providence, RI: American Mathematical Society, 2004
Hajłasz P. Sobolev spaces on an arbitrary metric space. Potential Anal, 1996, 5: 403–415
Hajłasz P, Koskela P. Sobolev met Poincare. Mem Amer Math Soc, 2000, 145(688): 1–101
Hedenmalm H, Korenblum B, Zhu K. Theory of Bergman Spaces. Graduate Texts in Mathematics 199. New York: Springer, 2000
Heinonen J. Lectures on Analysis on Metric Spaces. Berlin: Springer, 2001
Koskela P, Saksman E. Pointwise characterizations of Hardy-Sobolev functions. Math Res Lett, 2008, 15: 727–744
Nam K. Lipschitz type characterizations of harmonic Bergman spaces. Bull Korean Math Soc, 2013, 50: 1277–1288
Peng R, Xing X, Jiang L. Pointwise multiplication operators from Hardy spaces to weighted Bergman spaces in the unit ball of ℂn. Acta Math Sci, 2019, 39B(4): 1003–1016
Sehba B F. Derivatives characterization of Bergman-Orlicz spaces and applications. arXiv:1610.01954
Wulan H, Zhu K. Lipschitz type characterizations for Bergman spaces. Canad Math Bull, 2009, 52: 613–626
Yang D. New characterizations of Hajłasz-Sobolev spaces on metric spaces. Sci China Math, 2003, 46: 675–689
Zhu K. Spaces of Holomorphic Functions in the Unit Ball. Graduate Texts in Mathematics 226. New Nork: Springer, 2004
Author information
Authors and Affiliations
Corresponding author
Additional information
The work was supported by Hainan Province Natural Science Foundation of China (2018CXTD338), the National Natural Science Foundation of China (11761026 and 11761027), and Guangxi Natural Science Foundation (2020GXNSFAA159085).
Rights and permissions
About this article
Cite this article
Ma, R., Xu, J. Lipschitz Type Characterizations for Bergman-Orlicz Spaces and Their Applications. Acta Math Sci 40, 1445–1458 (2020). https://doi.org/10.1007/s10473-020-0516-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-020-0516-8