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Composition operators on the Bloch space in polydiscs. (English) Zbl 1026.47018
Summary: Let $$\Omega$$ be a bounded Bergman domain in $$\mathbb{C}^n$$ (a domain $$\Omega\subset\mathbb{C}^n$$ is called a bounded Bergman domain if it is bounded and there exists a constant $$C$$ depending only on $$\Omega$$, such that $$H_{\psi(z)}(J\psi(z)u, J\psi(z)u)\leq CH_z(u,u)$$, for each $$z\in\Omega$$, $$u\in\mathbb{C}^n$$ and holomorphic self-map $$\psi$$ of $$\Omega$$, where $$H_z(u,u)$$ denotes the Bergman metric of $$\Omega$$ and $$J\psi$$ the Jacobian of $$\psi$$) and $$\phi$$ a holomorphic self-map of $$\Omega$$. This paper shows that the composition operator $$C_\phi$$ induced by $$\phi$$ is always bounded on the Bloch space $$\beta(\Omega)$$. For $$\Omega= U^n$$, the unit polydisc of $$\mathbb{C}^n$$, this paper gives a necessary and sufficient condition for $$C_\phi$$ to be compact on $$\beta(U^n)$$.
Reviewer: Reviewer (Berlin)

##### 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))
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