Translator Disclaimer
March 2018 On the microlocal analysis of the geodesic X-ray transform with conjugate points
Sean Holman, Gunther Uhlmann
Author Affiliations +
J. Differential Geom. 108(3): 459-494 (March 2018). DOI: 10.4310/jdg/1519959623

Abstract

We study the microlocal properties of the geodesic X-ray transform $\mathcal{X}$ on a manifold with boundary allowing the presence of conjugate points. Assuming that there are no self-intersecting geodesics and all conjugate pairs are nonsingular we show that the normal operator $\mathcal{N} = \mathcal{X}^t \circ \mathcal{X}$ can be decomposed as the sum of a pseudodifferential operator of order $-1$ and a sum of Fourier integral operators. We also apply this decomposition to prove inversion of $\mathcal{X}$ is only mildly ill-posed when all conjugate points are of order $1$, and a certain graph condition is satisfied, in dimension three or higher.

Funding Statement

This work was supported by the Engineering and Physical Sciences Research Council [grant number EP/M016773/1].

Citation

Download Citation

Sean Holman. Gunther Uhlmann. "On the microlocal analysis of the geodesic X-ray transform with conjugate points." J. Differential Geom. 108 (3) 459 - 494, March 2018. https://doi.org/10.4310/jdg/1519959623

Information

Received: 16 November 2015; Published: March 2018
First available in Project Euclid: 2 March 2018

zbMATH: 06846983
MathSciNet: MR3770848
Digital Object Identifier: 10.4310/jdg/1519959623

Rights: Copyright © 2018 Lehigh University

JOURNAL ARTICLE
36 PAGES

This article is only available to subscribers.
It is not available for individual sale.
+ SAVE TO MY LIBRARY

SHARE
Vol.108 • No. 3 • March 2018
Back to Top