Biholomorphically invariant families amongst Carleson class (Q1866448)

For \(\alpha\in(0,1)\), the Carleson class \(T_\alpha\) consists of all meromorphic functions \(f\) defined in the unit disk \({\mathbb D}=\{z:|z|<1\}\) for which \[ \|f\|_{T_\alpha}=\Biggl(\int_{\mathbb D}(f^\#(z))^2(1-|z|^2)^{1-\alpha} dm(z)\Biggr)^{1/2}<\infty, \] where \(f^\#\) and \(dm\) mean the spherical derivative of \(f\) and Lebesgue area measure on \({\mathbb D}\) respectively. The author introduces the biholomorphically invariant families (amongst the Carleson class \(T_\alpha\)) \(\text{BIT}_\alpha\) and \(\text{BIT}_{\alpha,0}\) consisting of those \(f\in T_\alpha\) with \(\sup_{w\in {\mathbb D}}\|f\circ\varphi_w\|_{T_\alpha}<\infty\) and \(\lim_{|w|\to 1}\|f\circ\varphi_w\|_{T_\alpha}=0\) respectively, where \(\varphi_w(z)=(w-z)/(1-\overline wz)\), \(z,w\in {\mathbb D}\). In the paper under review the author gives criteria for \(f\) to belong to \(\text{BIT}_\alpha\) and \(\text{BIT}_{\alpha,0}\) by the Ahlfors-Shimizu characteristic of \(f\). Further, if \(f=IO/J\in T_\alpha\), where \(I,J\) are inner functions and \(O\) is an outer function, then conditions for \(O\) to belong to \(\text{BIT}_\alpha\) and \(\text{BIT}_{\alpha,0}\) are derived. In the main theorem it is shown that every function in \(\text{BIT}_\alpha\) (\(\text{BIT}_{\alpha,0}\) respectively) can be factored as the quotient of two functions in \(H^\infty\cap\text{BIT}_\alpha\) (\(H^\infty\cap\text{BIT}_{\alpha,0}\) respectively), where \(H^\infty\) is the class of bounded analytic functions in \({\mathbb D}\). For the classes \(T_0\), \(T_\alpha\) this question has been considered by \textit{R. Nevanlinna} [Analytic functions (1970; Zbl 0199.12501)], \textit{L. Carleson} [(Diss.) Uppsala: Appelbergs Boktryckeri AB. 79 S. (Zbl 0036.04701)] and \textit{A. Aleman} [Mich. Math. J. 39, 537-549 (1992; Zbl 0773.30025)].