Weighted Composition Operators between different Bloch-type Spaces in Polydisk

Abstract

Let $\varphi (z)=({\varphi }_1(z),...,{\varphi }_n(z))$ be a holomorphic self-map of $U^n$ and $\psi (z)$ a holomorphic function on $U^n,$ where $U^n$ is the unit polydisk of ${\mathbb{C}}^n.$ Let $p\ge 0,$ $q\ge 0$, this paper gives some necessary and sufficient conditions for the weighted composition operator $W_{\psi ,\varphi }$ induced by $\psi$ and $\varphi$ to be bounded and compact between $p$-Bloch space ${\mathcal{B}}^p(U^n)$ and $q$-Bloch space ${\mathcal{B}}^q(U^n).$

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