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

Authors:
Jan Boman and Eric Todd Quinto

Journal:
Trans. Amer. Math. Soc. **335** (1993), 877-890

MSC:
Primary 58G15; Secondary 44A12, 58G07

DOI:
https://doi.org/10.1090/S0002-9947-1993-1080733-8

MathSciNet review:
1080733

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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.

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Keywords:
Radon transform,
admissible line complex,
analytic microlocal analysis

Article copyright:
© Copyright 1993
American Mathematical Society