Weighted spaces of holomorphic functions on the upper halfplane

We discuss weighted spaces $Hv(\mathbf{G})$ of holomorphic functions on the upper halfplane $\mathbf{G}$ where $v(w) = v(i \operatorname{Im} w)$, $w \in \mathbf{G}$, $\lim_{t\to 0} v(it)=0$ and $v(it)$ is increasing in $t$. We characterize those weights $v$ with moderate growth where $Hv(\mathbf{G})$ is isomorphic to $l_{\infty}$ and we show that this is never the case if $v$ is bounded.