Product-Type Operators from Logarithmic Bergman-Type Spaces to Zygmund–Orlicz Spaces

Abstract

Let \({\mathbb{D}}\) be the open unit disk in the complex plane \({\mathbb{C}}\) and \({H(\mathbb{D})}\) the class of all analytic functions on \({\mathbb{D}}\). Let \({\phi}\) be an analytic self-map of \({\mathbb{D}}\) and \({u \in H(\mathbb{D})}\). As a class of new space, we first study the properties of the logarithmic Bergman-type space. We also use Young’s function to define another new space as a generalization of Zygmund space, called the Zygmund–Orlicz space. We study some of its properties and show that the Zygmund–Orlicz space is isometrically equal to certain Zygmund-type space for a very special weight. This allows us to define the little Zygmund–Orlicz space which is a closed subspace of the Zygmund–Orlicz space. As some applications of properties obtained, we characterize the boundedness and compactness of the product-type operators \({D^{n} M_{u}C_{\phi}}\), \({D^{n}C_{\phi}M_{u}}\), \({C_\phi D^n M_u}\), \({M_uD^nC_{\phi}}\), \({M_{u}C_{\phi}D^{n}}\), and \({C_{\phi}M_{u}D^{n}}\) from the logarithmic Bergman-type space to the Zygmund–Orlicz space and the little Zygmund–Orlicz space.