Image of the Schwartz space under spectral projection

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.