Composition operators on the Lipschitz space in polydiscs期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

Composition operators on the Lipschitz space in polydiscs

Authors:

Email author" target="_blank">Zehua?Zhou Email author

Affiliation:

Department of Mathematics, Tianjin University, Tianjin 300072, China; LiuHui Center for Applied Mathematics, Nankai University and Tianjin University, Tianjin 300072, China

Abstract:

Let Uⁿ be the unit polydisc of Cⁿ and φ = (φ ₁,...,φ _n) a holomorphic self-map of Uⁿ. Let 0 ≤α 1. This paper shows that the composition operator C is bounded on the Lipschitz space Lip(Uⁿ) if and only if there exists M > 0 such that

$\sum\limits_{k,l = 1}^n {\left| {\frac{{\partial \phi _l }}{{\partial zk}}(z)} \right|\left( {\frac{{1 - \left| {z_k } \right|^2 }}{{1 - \left| {\phi _l (z)} \right|^2 }}} \right)^{1 - \alpha } } \leqslant M$

for z ∈ Uⁿ. Moreover C_φ is compact on Lipα(Uⁿ) if and only if C_φ is bounded on Lipα(Uⁿ) and for every ε>0, there exists a δ > 0 such that

whenever dist((z),∂Uⁿ).

Keywords:

Lipschitz space polydisc composition operator

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