Paley–Wiener Theorems on the Siegel Upper Half-Space

Abstract

In this paper we study spaces of holomorphic functions on the Siegel upper halfspace U and prove Paley-Wiener type theorems for such spaces. The boundary of U can be identified with the Heisenberg group Hn. Using the group Fourier transform on Hn, Ogden-Vagi [OV79] 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 constantsḊ is the unique Hilbert space norm that is invariant under the action of the group of automorphisms of U.

Holomorphic function spaces on homogeneous Siegel domains

“…where Hol(C + ) denotes the space of holomorphic functions on C + , the Hardy space (modulo constant functions). [5] Following [72], for p, q ∈ ]0, ∞] and s > 0 one may also define the weighted Bergman space (modification if q = ∞) if and only if p, q ≥ 1 and 2s > s + 2. In addition, P s reproduces the elements of A p,q s (C + ), that is, P s f = f ∀f ∈ A p,q s (C + ) if and only if 2s > s + 1 + 1/ min(1, p) (p, q ∈ ]0, ∞]).…”

“…Arcozzi, Monguzzi, Peloso, and Salvatori [5] studied the spaces A 2,2 s (D) on Siegel upper half-spaces, and established Paley-Wiener theorems.…”

“…To the best of our knowledge, results of this kind were only known for the Hardy space, for weighted Bergman spaces on homogeneous tube domains, and for weighted Bergman spaces on Siegel upper half-spaces, both for relatively invariant weights (cf. [65,40,17,5]). Replacing Fourier analysis on N with Fourier analysis on the centre of N , analogous results were also known for the Hardy and the unweighted Bergman spaces on general Siegel domains, and on weighted Bergman spaces on homogeneous Siegel domains for relatively invariant weights (cf.…”

“…We shall then determine an explicit formula for the reproducing kernel of the Bergman space A 2,2 s (D) (Proposition 3.11) and determine its reproducing properties on other spaces (Proposition 3.13). We shall then develop Paley-Wiener theorems for A 2,2 s (D), and extend them below the critical index following [5] (Proposition 3.17). We mention again that, in a different form, such Paley-Wiener theorems were already known in the literature (cf.…”

Fractional Paley–Wiener and Bernstein spaces

“…We also point out that the investigation of these spaces has natural counterparts in the several variable setting. We refer the reader to [3] and the upcoming manuscript [17] where we present the development of this theory in the setting of several variables.…”