Generalized Volterra type integral operators on large Bergman spaces

Abstract

Let $\varphi$ be an analytic self-map of the open unit disk $\mathbb{D}$ and $g$ analytic in $\mathbb{D}$. We characterize boundedness and compactness of generalized Volterra type integral operators $$G{I}_{(\varphi ,g)}f(z)={\int }_0^z{f}^{\prime }(\varphi (\xi ))g(\xi )d\xi$$ and $$G{V}_{(\varphi ,g)}f(z)={\int }_0^{z}f(\varphi (\xi ))g(\xi )d\xi ,$$ acting between large Bergman spaces $A_{\omega }^p$ and $A_{\omega }^q$ for $0<p,q\le \infty$. To prove our characterizations, which involve Berezin type integral transforms, we use the Littlewood-Paley formula of Constantin and Peláez and establish corresponding embedding theorems, which are also of independent interest. When $\varphi (z)=z$, our results for $G{V}_{(\varphi ,g)}$ complement the descriptions of Pau and Peláez.