Asymptotics for the Radon transform on hyperbolic spaces

Let $G/H$ be a hyperbolic space over $\Bbb R,$ $\Bbb C$ or $\Bbb H,$ and let $K$ be a maximal compact subgroup of $G.$ Let $D$ denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of $D.$ For any $L^2$-Schwartz function $f$ on $G/H,$ we prove that the Abel transform ${\mathcal A}(Df)$ of $Df$ is a Schwartz function. This is an extension of a result established in [2] for $K$-finite and $K\cap H$-invariant functions.