Abstract. We have in [1] proposed a definition of cusp forms on semisimple
symmetric spaces G/H, involving the notion of a Radon transform and a
related Abel transform. For the real non-Riemannian hyperbolic spaces, we
showed that there exists an infinite number of cuspidal discrete series, and
at most finitely many non-cuspidal discrete series, including in particular the
spherical discrete series. For the projective spaces, the spherical discrete series
are the only non-cuspidal discrete series. Below, we extend these results to
the other hyperbolic spaces, and we also study the question of when the Abel
transform of a Schwartz function is again a Schwartz function.