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 (₃), where ₃ 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 (₃) consists of restrictions to ₃ 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.