  We prove a local support theorem for the exponential Radon transform for
functions of exponential decay at infinity. We also show that our decay
condition is essentially sharp for the classical Radon transform for hyperbolic
type domains as holes, by showing streched exponential counterexamples. This
shows a difference of the support theorems for compact domains, where the decay
has to be just faster than any polynomial. Also, gives a refinement for
non-compact domains, where support theorem was proved only for functions with
compact support. Our method is a version of R. S. Strichartz and J. Boman.
