Weighted differentiation composition operators to Bloch-type spaces
The authors provide characterizations of both the boundedness and the compactness of the weighted (iterated) differentiation composition operator \(f\mapsto \psi\cdot( f^{(n)}\circ\varphi)\), viewed from BMOA and the (usual) Bloch space to Bloch spaces with standard weights \(\alpha\) (see Theorems 5 and 7 of the paper). The key condition is the boundedness of the functions \(z\in{\mathbb D}\mapsto \displaystyle\frac{(1-|z|)^\alpha}{(1-|\varphi(z)|)^n}|\psi'(z)|\) and \(z\in{\mathbb D}\mapsto \displaystyle\frac{(1-|z|)^\alpha}{(1-|\varphi(z)|)^{n+1}}|\psi(z)\varphi'(z)| \). As often happens, the compactness characterization is obtained by replacing the boundedness condition (near the circle) with the corresponding vanishing condition.
scientific article; zbMATH DE number 6306066
scientific article