Boundedness and compactness of the operator \(C_\varphi^{n,u}\) from mixed-norm spaces to weighted Bloch spaces (Q2912612)

let \(\mathbb{D}\) denote the unit disk in \(\mathbb{C}\). For \(u\in H(\mathbb{D})\) and a self-mapping \(\varphi\) of \(\mathbb{D}\), let NEWLINE\[NEWLINE \left( C_{\varphi}^{n, u}f\right)(z)=\int_0^zf^{(n)}\left(\varphi(\xi)\right)u(\xi)\,d\xi. NEWLINE\]NEWLINENEWLINENEWLINEIn this paper, by using the quantity NEWLINE\[NEWLINE \frac{\mu(z)|u(z)|}{\phi\left(|\varphi(z)|\right)\left(1-|\varphi(z)|^2\right)^{\frac1q+n}}, NEWLINE\]NEWLINE with \(\phi\) some normal function, the authors give some necessary and sufficient conditions for \(C_{\varphi}^{n, u}\) to be a bounded or a compact opertor from mixed-norm spaces to weighted Bloch spaces.