Higher-order Volterra-type integral operator on Hardy and Bergman spaces

TL;DR

This paper studies the properties of a higher-order Volterra-type integral operator on Hardy and Bergman spaces, providing sharp estimates and characterizations of boundedness and compactness based on Carleson measure conditions.

Contribution

It offers the first complete characterization of boundedness and compactness of $T_{g,n}$ on Hardy and Bergman spaces using Carleson measure criteria.

Findings

Established sharp norm and essential norm estimates for $T_{g,n}$.

Provided necessary and sufficient conditions for boundedness.

Characterized compactness via vanishing Carleson measures.

Abstract

We investigate the higher-order Volterra-type integral operator $T_{g,n}$ on the unit disk, defined for $n\in\mathbb N$ by \[ T_{g,n}[f](z) := \underbrace{\int_{0}^{z}\int_{0}^{t_1}\cdots\int_{0}^{t_{n-1}}}_{n\ \text{times}} f(t_n)g'(t_n)\,dt_n\cdots dt_1,\quad z\in\mathbb D, \] where $f$ and $g$ are analytic in the unit disk $\mathbb D$. We establish sharp norm and essential norm estimates, and give complete characterizations of boundedness and compactness of $T_{g,n}$ on Hardy spaces $H^p$ and weighted Bergman spaces $A_\alpha^p$, in terms of (vanishing) Carleson measure conditions determined by $|g'|$.