Image of the Schwartz space under spectral projection

Abstract

Let X = G/K be a Riemannian symmetric space of non compact type and rank-one. The spectral projection P λ f of a function f on X can be written P λ f = f * φ λ where φ λ is the elementary spherical function corresponding to the complex parameter λ. We characterize the image of the Schwartz space \({\mathcal S^p(X)}\) under the spectral projection for 0 < p ≤ 2.