Paley–Wiener Theorems on the Siegel Upper Half-Space

In this paper we study spaces of holomorphic functions on the Siegel upper half-space U and prove Paley–Wiener type theorems for such spaces. The boundary of U can be identified with the Heisenberg group H n . Using the group Fourier transform on H n , Ogden and Vagi (Adv Math 33(1):31–92, 1979) proved a Paley–Wiener theorem for the Hardy space H 2 (U). We consider a scale of Hilbert spaces on U that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury–Arveson space, and the Dirichlet space D. For each of these spaces, we prove a Paley–Wiener theorem, some structure theorems, and provide some applications. In particular we prove that the norm of the Dirichlet space modulo constants D˙ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of U.