Weighted composition operators on spaces of functions with derivative in Mixed-Norm and Hardy spaces

Abstract

Let \(\varphi\) be an analytic self-map of the open unit disk \(\mathbb{D}\) in the complex plane. For any function \(\psi\) analytic in the unit disk \(\mathbb{D}\), the weighted composition operator is given by \(W_{\varphi,\psi}f(z)=\psi(z) f(\varphi(z))\) for \(z\in \mathbb{D}\) and \(f\) analytic on \(\mathbb{D}\). For each \(0< p\leq\infty\), let \(S_p\) be the space of analytic functions on \(\mathbb{D}\) whose derivatives belong to the Hardy space \(H_p\). For \(0<p,q\leq\infty\) and \(\alpha>0\) let \(S(p,q,\alpha)\) be the space of functions with derivative in the mixed norm space \(H(p,q,\alpha)\). In this article we deal with boundedness of the weighted composition operators from \(S(p,q,\alpha)\) into \(S_p\) spaces for \(0<p,q\leq\infty\) and \(\alpha>0\).