Generalized Volterra type integral operators on large Bergman spaces

Let $\phi$ 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 $$GI_{(\phi,g)}f(z)= \int_{0}^{z}f'(\phi(\xi))\,g(\xi)\, d\xi$$ and $$GV_{ (\phi, g)}f(z)= \int_{0}^{z} f(\phi(\xi))\,g(\xi)\, d\xi,$$ acting between large Bergman spaces $A^p_\omega$ and $A^q_\omega$ 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 Pel\'aez and establish corresponding embedding theorems, which are also of independent interest. When $\phi(z) = z$, our results for $GV_{(\phi,g)}$ complement the descriptions of Pau and Pel\'aez.