The Fantappiè and Laplace transforms realize isomorphisms between analytic functionals supported on a convex compact set \(K\subset {\mathbb {C}^n}\) and certain spaces of holomorphic functions associated with K. Viewing the Bergman space of a bounded domain in \({\mathbb {C}^n}\) as a subspace of the space of analytic functionals supported on its closure, the images of the restrictions of these transforms have been studied in the planar setting. For the Fantappiè transform, this was done for simply connected domains (Napalkov Jr and Yulmukhametov, 1995), and for the Laplace transform, this was done for convex domains (Napalkov Jr and Yulmukhametov, 2004). In this paper, we study this problem in higher dimensions for strongly convex domains, and establish duality results analogous to the planar case. We also produce examples to show that the planar results cannot be generalized to all convex domains in higher dimensions.
