Composition Ces`aro Operator on the Normal Weight Zygmund Space in High Dimensions

Citation:

Si XU,Xuejun ZHANG,Shenlian LI.Composition Ces`aro Operator on the Normal Weight Zygmund Space in High Dimensions[J].Chinese Annals of Mathematics B,2021,42(1):69~84
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Authors:

Si XU; Xuejun ZHANG;Shenlian LI

Foundation:

This work was supported by the National Natural Science Foundation of China (No.,11571104) and the Hunan Provincial Innovation Foundation for Postgraduate (No.,CX2018B286).
Abstract: Let n>1 and B be the unit ball in n dimensions complex space bf C^{n}. Suppose that \varphi is a holomorphic self-map of B and \psi\in H(B) with \psi(0)=0. A kind of integral operator, composition Ces\`{a}ro operator, is defined by T_{\varphi,\psi}(f)(z)=\int_{0}^{1}f[\varphi(tz)]R\psi(tz)\frac{\rmd t}{t}, \quad f\in H(B), \ z\in B.
In this paper, the authors characterize the conditions that the composition Ces\`{a}ro operator T_{\varphi, \psi} is bounded or compact on the normal weight Zygmund space \mathcal{Z}_{\mu}(B). At the same time, the sufficient and necessary conditions for all cases are given.

Keywords:

Normal weight Zygmund space, Composition Ces`aro operator,Boundedness and compactness

Classification:

32A36, 47B38
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