On the Schwartz space isomorphism theorem for rank one symmetric space

In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism $(0<p\le 2)$ under the Fourier transform for the class of functions of left $\delta$-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker’s \cite{A} proof of the corresponding result in the case of left $K$-invariant functions on $X$. Thus we give a proof which relies only on the Paley–Wiener theorem.