Metrically bounded Volterra-type operators on area Nevanlinna spaces

Abstract

In this paper, we study metrically bounded Volterra-type operators $J_g$ and $I_g$ on the area Nevanlinna spaces $N_{\alpha }^p$, where $1\le p<\infty$ and $\alpha >-1$. Choe, Koo and Smith \cite{CKS} obtained partial characterizations of the corresponding symbol classes, but the exact characterization was still open. We establish a Littlewood--Paley type characterization of $N_{\alpha }^p$ and use it to resolve this problem affirmatively. More precisely, $J_g$ is metrically bounded on $N_{\alpha }^p$ if and only if $g$ belongs to the Bloch space $\mathcal{B}$, while $I_g$ is metrically bounded on $N_{\alpha }^p$ if and only if $g\in H^{\infty }$.