ON THE MICROLOCAL ANALYSIS OF THE GEODESIC X-RAY TRANSFORM WITH CONJUGATE POINTS

We study the microlocal properties of the geodesic X-ray transform 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 N = X ^t ◦ 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 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.