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 S(K G/K) isomorphically onto the space S(ΣD), where ΣD is an embedded copy of the Gelfand spectrum in R^, canonically associated to a generating system D of G-invariant differential operators on $G/K$, and S(ΣD) consists of restrictions to ΣD of Schwartz functions on R^. 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 (Mₙ,SOₙ) with $n=3,4$. The rather trivial case $n=2$ is included in previous work by the same authors.