Composition and Volterra type operators on large Bergman spaces with rapidly decreasing weights

We characterize boundedness, compactness and Schatten class properties of generalized Volterra-type integral operators acting between large Bergman spaces $A_{\omega }^p$ and $A_{\omega }^q$ for $0<p,q\le \mathrm{\infty }$. To prove our characterizations, which involve Berezin-type integral transforms, we use the Littlewood-Paley formula of Constantin and Peláez and corresponding embedding theorems. Our results generalize the work on integration operators of Pau and Peláez in J. Funct. Anal. 259 (2010), 2727–2756.