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

Let n>1 and B be the unit ball in n dimensions complex space Cⁿ.Suppose thatφis a holomorphic self-map of B andψ∈H(B)withψ(0)=0.A kind of integral operator,composition Cesàro operator,is defined by T_φψ(f)(z)=∫¹0fφ(tz)]Rψ(tz)dt/t,f∈(B)z∈B.In this paper,the authors characterize the conditions that the composition Cesàro operator T_φ,ψis bounded or compact on the normal weight Zygmund space Z_μ(B).At the same time,the sufficient and necessary conditions for all cases are given.

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

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.