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Composition operators on the Lipschitz space in polydiscs

Authors:

Zehua?Zhou author-information" > author-information__contact u-icon-before" > mailto:zehuazhou@hotmail.com" title=" zehuazhou@hotmail.com" itemprop=" email" data-track=" click" data-track-action=" Email author" data-track-label=" " >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

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

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

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

whenever dist((z),∂Uⁿ).

Keywords:

Lipschitz space polydisc composition operator.

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