Image of Schwartz Space Under Spectral Projection

Abstract

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 $P_{\lambda }f(x)=f\ast {\phi }_{\lambda }(x)$, where, $\lambda \in \mathbb{C}$ and ${\phi }_{\lambda }$ is the elementary spherical function. We shall try to characterizes the image of the Schwartz spaces $S^p(X)$ where $0<p\le 2$ under the above transform.

21 pages,V2