Composition Cesàro Operator on the Normal Weight Zygmund Space in High Dimensions

Abstract

Let n > 1 and B be the unit ball in n dimensions complex space C ⁿ. Suppose that φ is a holomorphic self-map of B and ψ ∈ H(B) with ψ(0) = 0. A kind of integral operator, composition Cesàro operator, is defined by ${T_{\varphi,\psi }}\left( f \right)\left( z \right) = \int_0^1 {f\left[ {\varphi \left( {tz} \right)} \right]R\psi \left( {tz} \right){{{\rm{d}}t} \over t}},\;\;\;\;f \in H\left( B \right),\;\;z \in B.$ In this paper, the authors characterize the conditions that the composition Cesàro operator T _φ,ψ is bounded or compact on the normal weight Zygmund space ${{\cal Z}_\mu }\left( B \right)$. At the same time, the sufficient and necessary conditions for all cases are given.

Keywords

Normal weight Zygmund space / Composition Cesàro operator / Boundedness and compactness