Recovering Hardy spaces from optimal domains of integration operators
Setareh Eskandari, Antti Perälä
TL;DR
The paper addresses whether Hardy spaces on the unit ball can be recovered from the optimal domains of bounded Volterra integration operators between Hardy spaces. It establishes a sharp boundedness criterion: for $1<q<p<\infty$ with $r=\frac{pq}{p-q}$, the operator $S_g$ is bounded from $H^p$ to $H^q$ if and only if $g\in H^r$, and, when $g(0)=0$, $\|T_g\|_{H^p\to H^q}\asymp \|g\|_{H^r}$, with proofs based on area-function descriptions and tent-space techniques (necessity via Kahane–Khintchine arguments). Consequently, $H^p=[H^r:H^q]$ for $p>q$, i.e., $H^p$ is recoverable from the optimal domains, while for $p<q$, $H^p\subsetneq [\mathcal B^\alpha:H^q]$ with $\alpha=1+n/q-n/p$, showing non-recoverability. The results are succinctly expressed via intersections of weighted tent spaces $AT^{q}_2(|R g|^2)$, unifying the $p>q$ and $p<q$ regimes and extending known disk results to the unit-ball setting. Open questions remain for the $p=q$ case in higher dimensions.
Abstract
We study the optimal domains for bounded Volterra integration operators $T_g$ between distinct Hardy spaces $H^p$ and $H^q$ of the unit ball. It is shown that the intersection of the optimal domains is equal to $H^p$ if $p> q$, whereas if $p<q$, we show that this intersection is genuinely larger. In the unit disk, this problem was recently solved for $p=q$ by Bellavita, Daskalogiannis, Nikolaidis and Stylogiannis.