Image of Schwartz Space Under Spectral Projection

Let $X=G\mathrm{/}K$ symmetric space of non compact type, where $G$ is a rank-one connected semisimple Lie group with finite center. We shall look at the transform $P_{\lambda }f(x)=f\ast {\phi }_{\lambda }(x)$, where, $\lambda \in \mathbb C$and$\varphi_\lambda$ is the elementary spherical function. We shall try to characterizes the image of the Schwartz spaces $S^p(X)$ where $0 < p \leq 2$ under the above transform.