Local support theorem for the exponential Radon transform

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.