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 ${\Pi }_m$ — 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.

中文翻译：

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Anosov流形上X射线变换的经典和微局部分析

我们完成与阿诺索夫黎曼流形的测地通过Guillarmou流启动测X射线的微局部研究变换（J. 差的Geom。 105：2（2017），177-208）和Guillarmou和Lefeuvre在（追求安的。数学。 $（2）$ 190：1（2019），321–344）。我们证明新的稳定性估计并阐明算子的一些性质${\Pi }_{米}$—广义X射线变换。这些估计依赖于Anosov流的Livšic定理的改进版本，尤其是新的定量有限时间Livšic定理。