Weighted Composition Operators from H ∞ to the Bloch Space on the Polydisc

Abstract

Let ${\mathbb{D}}^n$ be the unit polydisc of ${ℂ}^n$, $\varphi \left(z\right)=\left({\varphi }_1\left(z\right),\dots ,{\varphi }_n\left(z\right)\right)$ be a holomorphic self-map of ${\mathbb{D}}^n$, and $\psi \left(z\right)$ a holomorphic function on ${\mathbb{D}}^n$. Let $H\left({\mathbb{D}}^n\right)$ denote the space of all holomorphic functions with domain ${\mathbb{D}}^n$, $H^{\infty }\left({\mathbb{D}}^n\right)$ the space of all bounded holomorphic functions on ${\mathbb{D}}^n$, and $\mathfrak{B}\left({\mathbb{D}}^n\right)$ the Bloch space, that is, $\mathfrak{B}\left({\mathbb{D}}^n\right)=\left\{f\in H\left({\mathbb{D}}^n\right){|\Vert f\Vert }_{\mathfrak{B}}=\left|f\left(0\right)\right|+{sup}_{z\in {\mathbb{D}}^n}{\displaystyle \sum _{k=1}^n\left|(\partial f/\partial z_k)\left(z\right)\right|}\left(1-{\left|z_k\right|}^2\right)<+\infty \right\}$. We give necessary and sufficient conditions for the weighted composition operator $\psi C_{\varphi }$ induced by $\varphi \left(z\right)$ and $\psi \left(z\right)$ to be bounded and compact from $H^{\infty }\left({\mathbb{D}}^n\right)$ to the Bloch space $\mathfrak{B}\left({\mathbb{D}}^n\right)$.