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Keywords:
weighted composition operator; Zygmund type space; Bloch type space; essential norm
weighted composition operator; Zygmund type space; Bloch type space; essential norm
Summary:
Let $u$ be a holomorphic function and $\varphi $ a holomorphic self-map of the open unit disk $\mathbb {D}$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $uC_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $\mathbb {D}$ in terms of $u$, $ \varphi $, their derivatives, and $\varphi ^n$, the $n$-th power of $\varphi $. Moreover, we obtain some similar estimates for the essential norms of the operators $uC_{\varphi }$, from which sufficient and necessary conditions of compactness of $uC_{\varphi }$ follows immediately.
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