Abstract
We prove a support theorem for Pompeiu transforms integrating on geodesic spheres of fixed radiusr>0 on real analytic manifolds when the measures are real analytic and nowhere zero. To avoid pathologies, we assume thatr is less than the injectivity radius at the center of each sphere being integrated over. The proof of the main result is local and it involves the microlocal properties of the Pompeiu transform and a theorem of Hörmander, Kawai, and Kashiwara on microlocal singularities.
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Quinto, E.T. Pompeiu transforms on geodesic spheres in real analytic manifolds. Israel J. Math. 84, 353–363 (1993). https://doi.org/10.1007/BF02760947
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DOI: https://doi.org/10.1007/BF02760947