On a product-type operator from Bloch spaces to weighted-type spaces on the unit ball
For a holomorphic function \(f\) on the unit ball \(\mathbb B\) in \(\mathbb C^n\) let \(\mathcal R f\) denote the radial derivative of \(f\), i.e., \[ \mathcal R f(z) = \sum_{j=1}^n z_j \frac{\partial f}{\partial z_j}. \] Furthermore, for \(m\in \mathbb N\), set \(\mathcal R^m f =\mathcal R (\mathcal R^{m-1} f)\). For \(\mu\) a positive continuous function on \(\mathbb B\) (called a weight), \(H^\infty_\mu\) denotes the set of holomorphic functions \(f\) on \(\mathbb B\) for which \(\sup_{z\in\mathbb B} \mu(z)|f(z)| <\infty\). Finally, for \(m\in \mathbb N\), \(u\) holomorphic on \(\mathbb B\), and \(\varphi\) an automorphism of \(\mathbb B\), set \(\mathcal R^m_{u,\varphi}(f)(z) = u(z) \mathcal R^m f(\varphi(z))\). In the paper under review, the author investigates boundedness and compactness of the operator \(\mathcal R^m_{u,\varphi}\) from the Bloch space \(\mathcal B\) and the little Bloch space \(\mathcal B_0\) to the space \(H^{\infty}_\mu\). The main results of the paper are as follows: \(\mathcal R^m_{u,\varphi} : \mathcal B \text{ (or }\mathcal B_0\))
scientific article; zbMATH DE number 5862532
scientific article