Schwartz correspondence for real motion groups in low dimensions
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| Format: | Preprint |
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2024
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| _version_ | 1866916128586989568 |
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| author | Astengo, Francesca Di Blasio, Bianca Ricci, Fulvio |
| author_facet | Astengo, Francesca Di Blasio, Bianca Ricci, Fulvio |
| contents | For a Gelfand pair $(G,K)$ with $G$ a Lie group of polynomial growth and $K$ a compact subgroup, the "Schwartz correspondence" states that the spherical transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$ isomorphically onto the space ${\mathcal S}(Σ_{\mathcal D})$, where $Σ_{\mathcal D}$ is an embedded copy of the Gelfand spectrum in ${\mathbb R}^\ell$, canonically associated to a generating system ${\mathcal D}$ of $G$-invariant differential operators on $G/K$, and ${\mathcal S}(Σ_{\mathcal D})$ consists of restrictions to $Σ_{\mathcal D}$ of Schwartz functions on ${\mathbb R}^\ell$.
Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair $(M_n,SO_n)$ with $n=3,4$. The rather trivial case $n=2$ is included in previous work by the same authors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_10848 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Schwartz correspondence for real motion groups in low dimensions Astengo, Francesca Di Blasio, Bianca Ricci, Fulvio Functional Analysis 43A85, 43A90, 22E30 For a Gelfand pair $(G,K)$ with $G$ a Lie group of polynomial growth and $K$ a compact subgroup, the "Schwartz correspondence" states that the spherical transform maps the bi-$K$-invariant Schwartz space ${\mathcal S}(K\backslash G/K)$ isomorphically onto the space ${\mathcal S}(Σ_{\mathcal D})$, where $Σ_{\mathcal D}$ is an embedded copy of the Gelfand spectrum in ${\mathbb R}^\ell$, canonically associated to a generating system ${\mathcal D}$ of $G$-invariant differential operators on $G/K$, and ${\mathcal S}(Σ_{\mathcal D})$ consists of restrictions to $Σ_{\mathcal D}$ of Schwartz functions on ${\mathbb R}^\ell$. Schwartz correspondence is known to hold for a large variety of Gelfand pairs of polynomial growth. In this paper we prove that it holds for the strong Gelfand pair $(M_n,SO_n)$ with $n=3,4$. The rather trivial case $n=2$ is included in previous work by the same authors. |
| title | Schwartz correspondence for real motion groups in low dimensions |
| topic | Functional Analysis 43A85, 43A90, 22E30 |
| url | https://arxiv.org/abs/2402.10848 |