On Stević type operator from \(H^\infty\) space to the logarithmic Bloch spaces
Let \(H(\mathbb D)\) be the space of analytic functions on the open unit disk in the complex plane. The logarithmic Bloch space \(\mathcal B_{\log}\) consists of the functions \(f\in H(\mathbb D)\) such that \[ \sup_{z\in\mathbb D} \left( 1-|z|^{2} \right) \left( \log \frac{2}{1-|z|} \right) |f'(z)|<\infty. \] Its subspace \(\mathcal B_{{\log},0}\) consists of such functions \(f\) that \[ \lim_{|z|\to 1} \left( 1-|z|^{2} \right) \left( \log \frac{2}{1-|z|} \right) |f'(z)| =0. \] For \(\psi_{1},\psi_{2}\in H(\mathbb D)\) and an analytic self-map \(\varphi\) of \(\mathbb D\), consider the linear operator on \(H(\mathbb D)\) of the form \[ f(z)\mapsto \psi_{1}(z)f(\varphi(z))+\psi_{2}(z)f'(\varphi(z)), \quad z\in\mathbb D. \] Necessary and sufficient conditions for the boundedness (compactness) of these operators acting from the Hardy space \(H^{\infty}(\mathbb D)\) into the space \(\mathcal B_{\log}\) (\(\mathcal B_{\log,0}\)) are obtained.
scientific article; zbMATH DE number 6506174
scientific article