Weighted composition operators between different weighted Bergman spaces in polydiscs (Q879671)

Let \(D^n\) be the unit polydisc in \({\mathbb C}^n\), \(n\in{\mathbb N}\). For \(1<p<\infty\) and \(\alpha>-1\), let \(A_\alpha^p(D^n)\) be the usual weighted Bergman spaces, the weight being \(z\mapsto\prod_{k=1}^n(1-| z_k| ^2)^\alpha\). Let \(\phi:D^n\to D^n\) and \(\psi:D^n\to{\mathbb C}\) be holomorphic. The author characterizes when, for \(\alpha,\beta>-1\), \(\eta\geq 1\) and \(1<p<\infty\), the weighted composition operator \(f\mapsto\psi(\cdot)\cdot f(\phi(\cdot))\) defines a bounded (resp., compact) operator \(A_\alpha^p(D^n)\to A_\beta^{\eta p}(D^n)\). The characterizations are in terms of properties of suitable Carleson measures on \(\overline{D^n}\).