Image of Schwartz Space Under Spectral Projection

Let $X= G/K$ 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 $P_\lambda f(x) = f \ast \varphi_\lambda(x)$, 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.