Local support theorem for the exponential Radon transform

TL;DR

This paper establishes a local support theorem for the exponential Radon transform for functions with exponential decay, highlighting the sharpness of decay conditions and refining support theorems for non-compact domains.

Contribution

It proves a new local support theorem for the exponential Radon transform and demonstrates the sharpness of decay conditions, extending previous results to non-compact domains.

Findings

Support theorem holds for exponential decay functions

Decay conditions are essentially sharp with counterexamples

Refines support theorems for non-compact domains

Abstract

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.