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