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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)

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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