Composition operators on the polydisc induced by smooth symbols (Q927643)

Let \(\Phi\) be a holomorphic map from \(\mathbb{D}^n\) into itself, then it induces a composition operator \(C_{\Phi}\) on \(H(\mathbb{D}^n)\) defined by \(C_{\Phi}f=f\circ\Phi\). In this paper, the authors prove that for \(\Phi\in C^2(\mathbb{D}^n)\), \(C_{v}\) is bounded on weighted Bergman space \(A_\alpha^p(\mathbb{D}^n)\) if and only if the Jacobian of \(\Phi\) does not vanish on those points \(\zeta\) on the distinguished boundary \(\mathbb{T}^n\) such that \(\Phi(\zeta)\in\mathbb{T}^n\). Also, they show that there is a ``jump phenomenon'': if \(\Phi\) is smooth enough and if \(C_{\Phi}\) does not map \(A_\alpha^p(\mathbb{D}^n)\) into itself, then it does not map \(A_\alpha^p(\mathbb{D}^n)\) into the larger space \(A_\beta^p(\mathbb{D}^n)\), where \(\alpha<\beta<\alpha+\frac{m}{2n}\), and \(m\) is a positive integer such that \(n-m\) is the rank of the Jacobian matrix of \(\Phi\).