Isometries of Weighted Bergman Spaces

Source abstract

In [ 2 ], [ 8 ] and [ 10 ], Forelli, Rudin and Schneider described the isometries of the H p spaces over balls and polydiscs. Koranyi and Vagi [ 6 ] noted that their methods could be used to describe the isometries of the H p spaces over bounded symmetric domains. Recently Kolaski [ 4 ] 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 [ 3 ] first defined the weighted Bergman space A p,α (0 < p < ∞, 0 < α < ∞) to be the space of holomorphic functions f in the disc which satisfy (1)