Composition operator on the normal weight Zygmund space in high dimensions

ABSTRACT

Let μ be a normal function on $[0,1)$. Suppose ϕ is a holomorphic self-map on the unit ball B of ${\mathbf{C}}^{\mathbf{n}}$. In this paper, the authors characterize the conditions that the composition operator $C_{\varphi }$ is bounded or compact on the normal weight Zygmund space $Z_{\mu }\left(B\right)$ when n>1. At the same time, the authors give the sufficient and necessary conditions for some special cases.

KEYWORDS:

• Normal weight Zygmund space
• composition operator
• boundedness
• compactness
• high dimensions

AMS SUBJECT CLASSIFICATIONS:

• 32A36
• 47B38

Acknowledgments

The authors thank the referees for their useful suggestions!

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is supported by the National Natural Science Foundation of China (No. 11571104) and the Hunan Provincial Innovation Foundation For Postgraduate (No. CX2017B220).