Generalized weighted composition operators from area Nevanlinna spaces to Bloch-type spaces (Q439183)

Let \(H(D)\) be the class of all holomorphic functions on the unit disk \(D\) of the complex plane. By the generalized weighted composition operator in the title the authors mean the operator \(D^n_{\varphi, u} f: =uf^{(n)}\circ \varphi\), \(f\in H(D)\), where \(n\geq 0\) is an integer, \(\varphi\in H(D)\) is a self-map of \(D\), and \(u\in H(D)\).NEWLINENEWLINEGiven \(\alpha>-1\), let \(dA_\alpha(z)= (1-|z|^2)^\alpha dA(z)\), where \(dA\) is the area measure on \(D\). For \(1\leq p<\infty\) and \(\alpha>-1\), let \({\mathcal N}_\alpha^p(D)\) be the area Nevanlinna space consisting of all \(f\in H(D)\) such that \(\|f\|_{{\mathcal N}_\alpha^p}:= \left\{\int_{D}[\log(1+|f|)]^p\,dA_\alpha\right\}^{1/p}<\infty\). Given a positive continuous function \(\mu\) on \([0,1)\) satisfying certain growth conditions near \(1\), the authors introduce the Bloch-type space \({\mathcal B}_\mu(D)\) consisting of all \(f\in H(D)\) such that \(\|f\|_{{\mathcal B}_\mu}:=|f(0)|+\sup_{z\in D} \mu(|z|)|f'(z)|<\infty\).NEWLINENEWLINEIn this paper, the authors obtain characterizations of the boundedness and compactness of the operator \(D^n_{\varphi, u}\) acting from \({\mathcal N}_\alpha^p(D)\) into \({\mathcal B}_\mu(D)\).