Boundedness and compactness of an integral operator between $H^\infty$ and a mixed norm space on the polydisk

This addendum to [1] completely characterizes the boundedness and compactness of a recently introduced integral type operator from the space of bounded holomorphic functions $H^\infty(\mathbb D^n)$ on the unit polydisk $\mathbb D^n$ to the mixed norm space $\mathscr A^{p,q}_\alpha(\mathbb D^n)$ with $p,q\in[1,+\infty)$ and $\alpha=(\alpha_1,\dots,\alpha_n)$ such that $\alpha_j>-1$ for every $j=1,\dots,n$. We show that the operator is bounded if and only if it is compact and if and only if $g\in\mathscr A^{p,q}_{\alpha+\vec q}$, where $\vec q=(q,\dots,q)$.