Schwartz correspondence for real motion groups in low dimensions

Abstract 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}} ({₃}) S (Σ D), where {₃} 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}} ({₃}) S (Σ D) consists of restrictions to {₃} 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.