Paley–Wiener Theorems on the Siegel Upper Half-Space

Abstract

In this paper we study spaces of holomorphic functions on the Siegel upper half-space ${\mathcal U}$ and prove Paley–Wiener type theorems for such spaces. The boundary of ${\mathcal U}$ can be identified with the Heisenberg group ${\mathbb H}_n$. Using the group Fourier transform on ${\mathbb H}_n$, Ogden and Vagi (Adv Math 33(1):31–92, 1979) proved a Paley–Wiener theorem for the Hardy space $H^2({\mathcal U})$. We consider a scale of Hilbert spaces on ${\mathcal U}$ that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury–Arveson space, and the Dirichlet space ${\mathcal D}$. 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 $\dot{{\mathcal D}}$ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of ${\mathcal U}$.