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Bloch-to-BMOA compositions in several complex variables. (English) Zbl 1093.47025

For a positive integer \(n\), let \(B_n\) denote the open unit ball in the complex Euclidean space \(\mathbb C^n\). For a holomorphic function \(\varphi:B_n\to B_m\), the paper studies the boundedness and compactness of the composition operator \[ C_\varphi:{\mathcal B}(B_m)\to BMOA(B_n), \] where \(\mathcal B\) is the Bloch space. The main result of the paper states that, under a mild condition on the regularity of \(\varphi\), the boundedness of \(C_\varphi\) above is equivalent to the measure \[ d\mu(z)={(1-| z| ^2)| Rf(z)| \over (1-| \varphi(z)| ^2)^2}\,dv(z) \] being a Carleson measure, where \(dv\) is the volume measure and \(Rf(z)\) is the radial derivative of \(f\) at \(z\).
Reviewer: Kehe Zhu (Albany)

MSC:

47B33 Linear composition operators
32A18 Bloch functions, normal functions of several complex variables
32A70 Functional analysis techniques applied to functions of several complex variables
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