We give a detailed microlocal study of X-ray transforms over geodesic-like families of curves with conjugate points of fold type. We show that the normal operator is the sum of a pseudodifferential operator and a Fourier integral operator. We compute the principal symbol of both operators and the canonical relation associated to the Fourier integral operator. In two dimensions, for the geodesic transform, we show that there is always a cancellation of singularities to some order, and we give an

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... example where that order is infinite; therefore the normal operator is not microlocally invertible in that case. In the case of three dimensions or higher if the canonical relation is a local canonical graph we show microlocal invertibility of the normal operator. Several examples are also studied. -3840 is published continuously online. Periodical rate postage paid at Berkeley, CA 94704, and additional mailing offices. APDE peer review and production are managed by EditFLOW ™ from Mathematical Sciences Publishers. PUBLISHED BY mathematical sciences publishers