The multiplication operator from mixed-norm to \(n\)-th weighted-type spaces on the unit disk (Q2885363)

The \(n\)th weighted-type spaces on the unit disk \(\mathbb D\) consist of all \(u \in H(\mathbb D)\) such that NEWLINE\[NEWLINEb(f)=\sup_{z\in \mathbb D}\mu(z)|f^{(n)}(z)|<\infty,NEWLINE\]NEWLINE where \(\mu(z)\) is a positive continuous function on \(\mathbb D\) such that \(\mu(z)=\mu(|z|)\). The multiplication operator \(M_{\Phi}\) is defined by \(M_{\Phi}f=\Phi f\). In the present paper, the authors obtain basic results on the boundedness and compactness of the multiplication operator from mixed-norm spaces to \(n\)th weighted-type spaces on the unit disk. The paper is written very carefully and nicely. The terms used in the paper have also been explained in detail.