On the Schwartz space isomorphism theorem for rank one symmetric space

TL;DR

This paper provides a simplified proof of the Schwartz space isomorphism theorem for rank one symmetric spaces, focusing on functions of left -type and utilizing only the Paley--Wiener theorem.

Contribution

It offers a more straightforward proof of the Schwartz space isomorphism theorem for rank one symmetric spaces, extending Anker's approach to a broader class of functions.

Findings

Simplified proof of the Schwartz space isomorphism theorem

Extension of Anker's proof to -type functions

Relies solely on the Paley--Wiener theorem

Abstract

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 $\delta$-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.