Volterra composition operators on logarithmic Bloch spaces
Let \(\mathbb B\) be the unit ball in \(\mathbb C^n\) and \(H(\mathbb B)\) the set of all holomorphic functions on \(\mathbb B\). Let \({\mathfrak R} f(z)=\sum_{j=1}^n z_j \frac{\partial f}{\partial z_j} (z)\) be the radial derivative of \(f\in H(\mathbb B)\) and \[ A_f(z):=(1-| z| ^2)\left( \log\frac{e}{1-| z| ^2} \right) | {\mathfrak R} f(z)| . \] The logarithmic Bloch space \(LB\) is the set of all functions \(f\in H(\mathbb B)\) for which \( \beta(f):=\sup_{z\in \mathbb B}A_f(z)<\infty,\) and its little sibling is given by \[ \text{LB}_0= \Big\{f\in \text{LB}: \lim_{| z| \to1} A_f(z)=0\Big\}. \] Let \(T_{g,\varphi}\) be defined as follows: \[ T_{g,\varphi}f(z)=\int_0^1 f(\varphi(tz)){\mathfrak R}g(tz)\,\frac{dt}{t}. \] Here, \(g\in H(\mathbb B)\) and \(\varphi\) is a holomorphic selfmap of \(\mathbb B\). The author characterizes boundedness and compactness of these operators between the logarithmic Bloch spaces LB and \(\text{LB}_0\).
scientific article; zbMATH DE number 5379955