Recovering Hardy spaces from optimal domains of integration operators

TL;DR

This paper investigates the optimal domains of Volterra integration operators between Hardy spaces on the unit ball, revealing how these domains relate to classical Hardy and tent spaces depending on the parameters p and q.

Contribution

It characterizes the optimal domains for bounded Volterra operators between Hardy spaces, extending previous results and identifying when these domains are larger than Hardy spaces.

Findings

Optimal domains always strictly contain the Hardy space $H^p$.

Intersection of optimal domains equals $H^p$ if $p \,\geq\, q$.

For $p<q$, the intersection is a larger tent space of holomorphic functions.

Abstract

We study the optimal domains for bounded Volterra integration operators $T_g$ between Hardy spaces $H^p$ and $H^q$ of the unit ball. It is shown that the optimal domain of a bounded $T_g:H^p\to H^q$ always strictly contains $H^p$. Moreover, the intersection of the optimal domains is equal to $H^p$ if $p\geq q$, whereas if $p<q$, we show that this intersection is a genuinely larger tent space of holomorphic functions. In the unit disk, this problem was recently solved for $p=q$ by Bellavita, Daskalogiannis, Nikolaidis and Stylogiannis.