Composition Cesàro Operator on the Normal Weight Zygmund Space in High Dimensions
Si Xu , Xuejun Zhang , Shenlian Li
Chinese Annals of Mathematics, Series B ›› 2021, Vol. 42 ›› Issue (1) : 69 -84.
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 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.
Normal weight Zygmund space / Composition Cesàro operator / Boundedness and compactness
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