Generalized Volterra type integral operators on large Bergman spaces

TL;DR

This paper characterizes the boundedness and compactness of generalized Volterra integral operators on large Bergman spaces, using Berezin transforms and Littlewood-Paley formulas, extending previous results and establishing new embedding theorems.

Contribution

It provides new characterizations of operator boundedness and compactness on large Bergman spaces, involving Berezin transforms and embedding theorems, for generalized Volterra operators.

Findings

Characterizations involve Berezin type integral transforms.

Established embedding theorems for large Bergman spaces.

Extended results for the case when (z) = z.

Abstract

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.