Recovering Hardy spaces from optimal domains of integration operators

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 $p\ge q$, 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.