Schwartz correspondence for real motion groups in low dimensions

Abstract: 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 ${\mathcal S}(\Sigma_{\mathcal D})$, where $\Sigma_{\mathcal D}$ 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 ${\mathcal S}(\Sigma_{\mathcal D})$ consists of restrictions to $\Sigma_{\mathcal D}$ 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.