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 Rℓ, 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 Rℓ. 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.