Bers-orlicz spaces on the product riemann surface

In this paper, the Bers-Orlicz spaces on the automorphic formA _α ^ϕ (G) (orEA _α ^ϕ (G)) andL _α ^ϕ (G) on the product Riemann surfaces are studied. We prove that eachf∈A _α ^ϕ (G) is a cusp form. Forf∈A _α ^ϕ (G), we give the reproducing formula. And, we give the projective operatorP _gga fromL _α ^ϕ (G) toA _α ^ϕ (G) toEA _α ^ϕ (G)). After giving some fundamental properties of the Poincaré series, we prove a dual theoremA _α ^ϕ (G)=(EA _α ^ϕ (G)). Supported by the National Nature Science Foundation of China