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.