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 and pursued by Guillarmou and the second author. We prove new stability estimates and clarify some properties of the operator ${\mathrm{\Pi }}_m$, 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.