$L^r$- Schwartz spaces on split rank one semisimple symmetric spaces

TL;DR

This paper investigates the structure of $L^r$-Schwartz spaces and their Fourier transforms on split rank one semisimple symmetric spaces, identifying the Fourier kernel via eigenfunctions linked to the discrete spectrum.

Contribution

It explicitly characterizes the Fourier transform kernel on these spaces and relates it to eigenfunctions of the Laplace--Beltrami operator.

Findings

The Fourier transform kernel is spanned by eigenfunctions of the discrete spectrum.

The study extends the understanding of Schwartz spaces on symmetric spaces.

Explicit description of the Fourier transform kernel for split rank one spaces.

Abstract

We study the left $K$-invariant $L^r$-Schwartz space and its Fourier transform on split rank one semisimple symmetric spaces $G/H$ for $0<r\leq 2$. We explicitly determine the kernel of the Fourier transform and show that it is spanned by eigenfunctions associated with the discrete spectrum of the Laplace--Beltrami operator on $G/H$.