Classical and microlocal analysis of the x-ray transform on Anosov manifolds

Abstract

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. $\left(2\right)$ 190:1 (2019), 321–344). 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 Livšic theorem for Anosov flows, especially on a new quantitative finite-time Livšic theorem.