 

Shan Li and
Hansong Huang
Composition operators on distinct Bergman spaces over planar domains
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Published: 
October 17, 2020. 
Keywords: 
Bergman spaces, composition operators,
domains of C^{2}boundary. 
Subject: 
Primary: 47B33; Secondary: 30H05. 


Abstract
In this paper, we will consider composition operators defined between distinct Bergman spaces over planar domains. The smoothness on boundary of the domains plays an important role in our study. On one hand, an essential extension of Littlewood's Subordination Principle is obtained. Precisely, for each holomorphic map that is defined between bounded domains of smooth boundaries, the associated composition operator is always bounded. This essentially depends on a "standard decomposition" of holomorphic functions over a classical domain, bounded by finitely many disjoint circles.
On the other hand, the situation becomes complex if domains with cusp boundary points are concerned, and there exists a link between the boundary behavior of the function symbol and the boundedness of the associated composition operator, where a detailed discussion is presented. Finally, we give estimates of norms for some classes of such composition operators. A deep interplay of function theory, geometry, and operator theory is revealed.


Acknowledgements
This work is partially supported by NSFC (12071134, 11471113). We thank the referee for his/her suggestions to improve the paper.


Author information
Shan Li:
Department of Mathematics
East China University of Science and Technology
Shanghai 200237, China
lishan_math@163.com
Hansong Huang:
Department of Mathematics
East China University of Science and Technology
Shanghai 200237, China
hshuang@ecust.edu.cn

