Articles written in Proceedings – Mathematical Sciences
Volume 117 Issue 3 August 2007 pp 333-348
In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism $(0 < p\leq 2)$ under the Fourier transform for the class of functions of left 𝛿-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker’s  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.