On a Li--Stević integral-type operators between different weighted Bloch-type spaces
Let \(\mathbb{D}=\left\{ z\in \mathbb{C}:\left| z\right| <1\right\} \) be the unit disk of the complex plane \(\mathbb{C}\) and let \(H(\mathbb{D})\) denote the space of all holomorphic functions on \(\mathbb{D}\). Every holomorphic self-map \(\varphi \) of the unit disc \(\mathbb{D}\) induces through composition a linear composition operator \(C_{\varphi }\) from \(H(\mathbb{D})\) to itself. The boundedness and compactness of \(C_{\varphi }\) and also of the weighted composition operator \(uC_{\varphi }\) have been well studied in various papers. \textit{S.-X.\thinspace Li} and \textit{S.\,Stević} in [J.~Math.\ Anal.\ Appl.\ 338, No.\,2, 1282--1295 (2008; Zbl 1135.47021)], for \(f,g\in H(\mathbb{D})\), defined a linear operator by \(C_{\varphi }^{g}f(z)=\int_{0}^{z}f^{\prime }(\varphi(w))g(w)\,dw\). They studied the boundedness and compactness of the operator on the Zygmund space, the Bloch-type space \(B^{\alpha },\) and the little Bloch-type space \(B_{0}^{\alpha }\). The main goal of the present paper is to characterize the boundedness and compactness of the Li-Stević integral-type operator between different weighted Bloch-type spaces.
scientific article; zbMATH DE number 5534595
scientific article