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

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 $G/H$.