Global integral criteria for composition operators
Let \(\mathcal U\) and \(dm\) denote the unit disk and the two-dimensional Lebesgue measure on the complex plane \(\mathbb C\). The weighted Dirichlet space \(\mathcal D_{\alpha}\), \(\alpha\in(-1,\infty)\), consists of functions \(f\) analytic on \(\mathcal U\) for which \[ ||f||_{\mathcal D_{\alpha}}^2=|f(0)|^2+\int_{\mathcal U}|f'(z)|^2(1-|z|^2)^{\alpha} dm(z)<\infty. \] A function \(f\in\mathcal D_{\alpha}\) belongs to the conformal invariant version of the space \(\mathcal D_{\alpha}\), the space \(Q_{\alpha}\), \(\alpha\in(-1,\infty)\), if \[ ||f||_{Q_{\alpha}}^2=|f(0)|^2+\sup_{w\in \mathcal U}\int_{\mathcal U }|f'(z)|^2\left(1-\left|\frac{w-z}{1-\bar{w}z}\right|^2\right)^{\alpha} dm(z)<\infty. \] The \(Q_{\alpha}\) spaces were introduced in [\textit{R. Aulaskari} and \textit{P. Lappan}, Criteria for an analytic function to be Bloch and a harmonic or meromorphic function to be normal, Yang, Chung-Chun (ed.) et al, Complex analysis and its applications. Proceedings of the international conference, held in Hong Kong, January 15-17, 1993. Harlow: Longman Scientific \& Technical. Pitman Res. Notes Math. Ser. 305, 136-146 (1994; Zbl 0826.30027)] for \(p\in(1,\infty)\) and in [\textit{R. Aulaskari, J. Xiao} and \textit{R. Zhao}, Analysis 15, No.~2, 101-121 (1995; Zbl 0835.30027)] for \(p\in(0,1]\). For \(\alpha=0\) we have the classical Dirichlet space and for \(\alpha\in(-1,0)\) the \(Q_\alpha\) spaces include only constants. An analytic self map \(\varphi\) of \(\mathcal U\) induces a composition operator \(C_{\varphi}\) acting on \(\mathcal D_{\alpha}\), respectively, \(Q_{\alpha}\) by \(C_{\varphi}f=f\circ\varphi\), where \(f\in{\mathcal D_{\alpha}}\), respectively, \(f\in Q_{\alpha}\). In several theorems of the paper integral criteria for \(C_{\varphi}:\mathcal D_{\alpha}\to\mathcal D_{\alpha}\), \(C_{\varphi}:\mathcal D_{\alpha}\to Q_{\alpha}\) and \(C_{\varphi}:Q_{\alpha}\to\mathcal D_{\alpha}\), \(C_{\varphi}:Q_{\alpha}\to Q_{\alpha}\) to be bounded or compact are given.
scientific article; zbMATH DE number 1797504
scientific article