Volterra type integration operators from Bergman spaces to Hardy spaces

TL;DR

This paper characterizes the boundedness of Volterra type integration operators from weighted Bergman spaces to Hardy spaces in complex n-dimensional unit balls, extending previous one-dimensional results using advanced harmonic analysis techniques.

Contribution

It provides a complete characterization of boundedness for these operators in higher dimensions, solving previously open cases and generalizing earlier findings.

Findings

Complete boundedness criteria for $J_b$ operators in $ ext{C}^n$

Extension of one-dimensional results to higher dimensions

Application of harmonic analysis tools to operator theory

Abstract

We completely characterize the boundedness of the Volterra type integration operators $J_b$ acting from the weighted Bergman spaces $A^p_\alpha$ to the Hardy spaces $H^q$ of the unit ball of $\mathbb{C}^n$ for all $0<p,q<\infty$. A partial solution to the case $n=1$ was previously obtained by Z. Wu in \cite{Wu}. We solve the cases left open there and extend all the results to the setting of arbitrary complex dimension $n$. Our tools involve area methods from harmonic analysis, Carleson measures and Kahane-Khinchine type inequalities, factorization tricks for tent spaces of sequences, as well as techniques and integral estimates related to Hardy and Bergman spaces.