Paley--Wiener theorems on the Siegel upper half-space
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2017
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| _version_ | 1866917688878563328 |
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| author | Arcozzi, Nicola Monguzzi, Alessandro Peloso, Marco M. Salvatori, Maura |
| author_facet | Arcozzi, Nicola Monguzzi, Alessandro Peloso, Marco M. Salvatori, Maura |
| contents | 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 $\mathcal U$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1710_10079 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | Paley--Wiener theorems on the Siegel upper half-space Arcozzi, Nicola Monguzzi, Alessandro Peloso, Marco M. Salvatori, Maura Complex Variables 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 $\mathcal U$. |
| title | Paley--Wiener theorems on the Siegel upper half-space |
| topic | Complex Variables |
| url | https://arxiv.org/abs/1710.10079 |