Schwartz correspondence for real motion groups in low dimensions

Abstract

AbstractFor 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 $${{\mathcal {S}}}(K\backslash G/K)$$ S ( K \ G / K ) isomorphically onto the space $${{\mathcal {S}}}(\Sigma _{{\mathcal {D}}})$$ S ( Σ D ) , where $$\Sigma _{{\mathcal {D}}}$$ Σ D is an embedded copy of the Gelfand spectrum in $${{\mathbb {R}}}^\ell $$ R ℓ , canonically associated to a generating system $${{\mathcal {D}}}$$ D of G-invariant differential operators on G/K, and $${{\mathcal {S}}}(\Sigma _{{\mathcal {D}}})$$ S ( Σ D ) consists of restrictions to $$\Sigma _{{\mathcal {D}}}$$ Σ D of Schwartz functions on $${{\mathbb {R}}}^\ell $$ 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 $$(M_n,SO_n)$$ ( 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.