Weighted Composition Operators from $H^{\infty}$ to the Bloch Space on the Polydisc

Abstract

Let $\mathbb{D}^n$ be the unit polydisc of $\mathbb{C}^n$, $\varphi (z) = (\varphi_1 (z), \ldots , \varphi_n (z))$ be a holomorphic self-map of $\mathbb{D}^n$, and $\epsi(z) $ a holomorphic function on $\matbb{D}^n$. Let $H (\mathbb{D}^n) $ denote the space of all holomorphic functions with domain $\matbb{D}^n$, $H^infty (\matbb{D}^n)$ the space of all bounded holomorphic functions on $\matbb{D}^n$, and $\mathfrak{B} (\matbb{D}^n)$ the Bloch space, that is, $\mathfrak{B} (\matbb{D}^n) = \{ h \in H (\matbb{D}^n) | \| f \|_{\mathfrak{B}} = |f (0) | +\sup_{z \in (\matbb{D}^n)} \sum_{k=1}^n |(\partial f / \partial z_k) (z) | (1- |z_k|^2 ) \lt + \infty \}$. We give necessary and sufficient conditions for the weighted composition operator $\psi C_\varphi$ induced by $\varphi (z)$ and $\psi (z)$ to be bounded and compact from $H^{\infty} (\mathbb{D}^n)$ to the Bloch space $\mathfrak{B}(\mathbb{D}^n)$.