We discuss weighted spaces Hv(G) of holomorphic functions on the upper halfplane G where v(w) = v(i Im w), w ∈ G, limt!0 v(it) = 0 and v(it) is increasing in t. We characterize those weights v with moderate growth where Hv(G) is isomorphic to l1 and we show that this is never the case if v is bounded.
