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Weighted composition operators from \(H^{\infty }\) to the Bloch space on the polydisc. (English) Zbl 1152.47016
Let \({\mathbb D}^n\) be the unit polydisc of \({\mathbb C}^n\), \(\varphi=(\varphi_1,\dots,\varphi_n)\) be a self-map of \({\mathbb D}^n\), and let \(\psi\) be holomorphic on \({\mathbb D}^n\). The main result of the present paper establishes that the weighted composition operator \(\psi C_\varphi\) is bounded from \(H^\infty({\mathbb D}^n)\) to \({\mathcal B}({\mathbb D}^n)\), i.e., from bounded functions to functions in the Bloch space, if and only if \(\psi \in {\mathcal B}({\mathbb D}^n)\) and
\[ \sup_{z\in {\mathbb D}^n} | \psi(z)| \sum_{k,j=1}^n \biggl|\frac{\partial \varphi_j}{\partial z_k}(z)\biggr| \frac{1-| z_k| ^2}{1-| \varphi_j(z)| ^2}<\infty. \] The corresponding compactness result replacing the supremum by “little oh” conditions is also shown.

MSC:
47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
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