Support theorems for Radon transforms on real analytic line complexes in three-space

Abstract:

In this article we prove support theorems for Radon transforms with arbitrary nonzero real analytic measures on line complexes (three-dimensional sets of lines) in ${\mathbb {R}^3}$. Let $f$ be a distribution of compact support on ${\mathbb {R}^3}$. Assume $Y$ is a real analytic admissible line complex and ${Y_0}$ is an open connected subset of $Y$ with one line in ${Y_0}$ disjoint from $\text {supp}\;f$. Under weak geometric assumptions, if the Radon transform of $f$ is zero for all lines in ${Y_0}$, then $\text {supp}\;f$ intersects no line in ${Y_0}$. These theorems are more general than previous results, even for the classical transform. We also prove a support theorem for the Radon transform on a nonadmissible line complex. Our proofs use analytic microlocal analysis and information about the analytic wave front set of a distribution at the boundary of its support.