On the Schwartz space isomorphism theorem for rank one symmetric space

Let X be a rank one Riemannian symmetric space of noncompact type. We recall that such a space can be realized as G/K, where G is a connected noncompact semisimple Lie group of real rank one with finite center and K is a maximal compact subgroup of G. Anker [2], in his paper gave a remarkably short and elegant proof of the Lp-Schwartz space isomorphism theorem for K bi-invariant functions on G under the spherical Fourier transform for (0 < p ≤ 2). The result for K bi-invariant functions was first proved by Harish-Chandra [6,7,8] (for p = 2) and Trombi and Varadarajan [12] (for 0 < p < 2). Eguchi and Kowata [4] addressed the isomorphism problem for the Lp-Schwartz spaces on X . In [2], Anker has successfully avoided the involved asymptotic expansion of the elementary spherical functions, which has a crucial role in all the earlier works. In this paper, we have exploited Anker’s technique to obtain the isomorphism of the Lp-Schwartz space (0 < p ≤ 2) under Fourier transform for functions on X of a fixed K-type. Let (δ ,Vδ ) be an unitary irreducible representation of K of dimension δ . Our basic Lp-Schwartz space S δ (X) is a space of Hom(Vδ ,Vδ )-valued C ∞ functions, the Eisenstein integral Φλ ,δ (x) is a Hom(Vδ ,Vδ )-valued entire function on C and Sδ (aC) consists of analytic functions on the strip a∗ ε = {λ ∈ C||Imλ | ≤ ε}. Anticipating these and other notations and definitions developed in §§2 and 3, we state the main result of the paper.