In ,  and , Forelli, Rudin and Schneider described the isometries of the Hp spaces over balls and polydiscs. Koranyi and Vagi  noted that their methods could be used to describe the isometries of the Hp spaces over bounded symmetric domains. Recently Kolaski  observed that the algebraic techniques used above and Rudin's theorem on equimeasurability extended to the Bergman spaces over bounded Runge domains. In this paper we use the same general argument to characterize the onto linear isometries of the weighted Bergman spaces over balls and polydiscs, (all isometries referred to are assumed to be linear).
2. Preliminaries. Horowitz  first defined the weighted Bergman space Ap,α (0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy