On the Schwartz space isomorphism theorem for the Riemannian symmetric spaces

TL;DR

This paper proves an isomorphism theorem for a subspace of Schwartz functions on Riemannian symmetric spaces using the Helgason Fourier transform, relying solely on the Paley-Wiener theorem without complex asymptotic analysis.

Contribution

It provides a new proof of the Schwartz space isomorphism theorem that avoids complex asymptotics, using only the Paley-Wiener theorem.

Findings

Established the isomorphism theorem for Schwartz subspaces on symmetric spaces.

Simplified proof avoiding higher asymptotics of spherical functions.

Applicable to $L^p$-Schwartz classes with finite spectral subsets.

Abstract

We deduce a proof of the isomorphism theorem for certain closed subspace $\mc S^p_\Gamma(X)$ of the $L^p$-Schwartz class functions $(0< p \leq 2) $ on a Riemannian symmetric space $X$ where $\Gamma$ is a finite subset of $\what{K}_M$. The Fourier transform considered is the Helgason Fourier transform. Our proof relies only on the Paley-Wiener theorem for the corresponding class of functions and hence it does not use the complicated higher asymptotics of the elementary spherical functions.