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Weighted composition operators from $$\alpha$$-Bloch space to $$H^{\infty}$$ on the polydisc. (English) Zbl 1130.47015
Let $$\mathbb D^n$$ be the polydisk in $$\mathbb C^n$$ and let $$\alpha>0$$. A holomorphic function $$f$$ in $$\mathbb D^n$$ is said to belong to the $$\alpha$$-Bloch class $${\mathcal B}^\alpha(\mathbb D)$$ if $\| f\|_{\mathcal B^\alpha}:=| f(0)| +\sup_{z\in\mathbb D^n}\sum_{k=1}^n \left| \frac{\partial f}{\partial z_k}(z)\right| (1-| z_k| ^2)^{\alpha}<\infty.$ Let $$\Psi$$ be a holomorphic function on $$\mathbb D^n$$ and $$\varphi=(\varphi_1,\dots,\varphi_n)$$ be a holomorphic selfmap of $$\mathbb D^n$$. The authors give necessary and sufficient conditions on $$\Psi$$ and $$\varphi$$ in order for the weighted composition operators $$\Psi C_\varphi$$ to be bounded and compact from $$\mathcal B^\alpha(\mathbb D^n)$$ to $$H^\infty(\mathbb D^n)$$. The results depend on whether $$0<\alpha<1$$, $$\alpha =1$$, or $$\alpha>1$$. In the case where $$\alpha>1$$, the authors $\sup_{z\in\mathbb D^n}| \Psi(z)| \sum_{j=1}^n\frac{1}{(1-|\varphi_j(z)|^2)^{\alpha-1}}<\infty.$ The authors also note that one of their results corrects a statement of S. Ohno in [Taiwanese J. Math. 5, No. 3, 555–563 (2001; Zbl 0997.47025)] on the compactness of weighted composition operators between the Bloch space and $$H^\infty(\mathbb D)$$.

##### MSC:
 47B33 Linear composition operators 32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA)) 47B38 Linear operators on function spaces (general)
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##### References:
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