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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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##### References:
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