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

Nicola Arcozzi, Alessandro Monguzzi, Marco M. Peloso, Maura Salvatori

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 $\mathcal U$.

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 and prove Paley-Wiener type theorems for such spaces. The boundary of can be identified with the Heisenberg group . Using the group Fourier transform on , Ogden-Vagi proved a Paley-Wiener theorem for the Hardy space . We consider a scale of Hilbert spaces on that includes the Hardy space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in particular the Drury-Arveson space, and the Dirichlet space . 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 is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of .

Paper Structure

This paper contains 9 sections, 18 theorems, 127 equations.

Table of Contents

  1. Introduction and statement of the main results
  2. Preliminaries
  3. More on the Heisenberg group and the the Siegel upper half-space
  4. The Fock space and the Fourier transform on the Heisenberg group
  5. The Cauchy--Riemann equations and the Hardy space
  6. The weighted Bergman spaces
  7. The weighted Dirichlet spaces
  8. The Dirichlet space
  9. The Drury--Arveson norm on the unit ball and final remarks

Key Result

Theorem 1

Let $F\in H^2$. Then, there exists $\widetilde{F}_0\in L^2({\mathbb H}_n)$ such that $\widetilde{F}_h\to\widetilde{F}_0$ in $L^2({\mathbb H}_n)$, as $h\to 0^+$. Moreover, the function $\widetilde{F}_0$ is such that Conversely, if $f\in L^2({\mathbb H}_n)$ is such that (ii) and (iii) are satisfied, then setting then $F\in H^2$ is such that $\widetilde{F}_0=f$ and (i)-(iii) hold.

Theorems & Definitions (22)

  • Definition
  • Definition
  • Theorem : OV
  • Definition
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma 2.1
  • Lemma 3.1
  • Proposition 3.2
  • ...and 12 more