Schwartz correspondence for real motion groups in low dimensions

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.