  In this article we prove time-global smoothing effects of dispersive
pseudodifferential equations with constant coefficient radially symmetric
symbols on real rank one symmetric spaces of noncompact type. We also discuss
gain of regularities according to decay rates of initial values for the
Schroedinger evolution equation. We introduce some isometric operators and
reduce the arguments to the well-known Euclidean case. In our proof, Helgason's
Fourier transform and the Radon transform as an elliptic Fourier integral
operator play crucial roles.
