CLASSICAL AND MICROLOCAL ANALYSIS OF THE X-RAY TRANSFORM ON ANOSOV MANIFOLDS

We complete the microlocal study of the geodesic x-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou (J. Differential Geom. 105:2 (2017), 177-208) and pursued by Guillarmou and Lefeuvre in (Ann. of Math. (2) 190:1 (2019), 321-344). We prove new stability estimates and clarify some properties of the operator Pi(m)-the generalized x-ray transform. These estimates rely on a refined version of the LivSic theorem for Anosov flows, especially on a new quantitative finite-time LivSic theorem.