Image of Schwartz Space Under Spectral Projection

TL;DR

This paper investigates how the spectral projection transform maps Schwartz spaces on rank-one symmetric spaces, characterizing the image of these function spaces under the transform involving elementary spherical functions.

Contribution

It provides a characterization of the image of Schwartz spaces under spectral projection transforms on rank-one symmetric spaces of noncompact type.

Findings

Characterization of the spectral projection transform's image on Schwartz spaces.

Extension of classical Fourier analysis results to symmetric spaces.

Insights into the structure of elementary spherical functions.

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 \varphi_\lambda(x)$, where, $\lambda \in \mathbb C$ and $\varphi_\lambda$ is the elementary spherical function. We shall try to characterizes the image of the Schwartz spaces $S^p(X) $ where $0 < p \leq 2$ under the above transform.