Paley--Wiener theorems on the Siegel upper half-space

TL;DR

This paper establishes Paley-Wiener theorems for various holomorphic function spaces on the Siegel upper half-space, linking boundary group structures with function space properties.

Contribution

It extends Paley-Wiener theorems to a broad class of Hilbert spaces on the Siegel upper half-space, including Hardy, Bergman, Dirichlet, and Drury-Arveson spaces.

Findings

Proved Paley-Wiener theorems for multiple function spaces on $\\mathcal U$

Established structure theorems and invariance properties under automorphisms

Connected boundary group Fourier analysis with function space characterizations

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-Vagi 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 $\mathcal U$.