Stability of a solution to an integral geometry problem in Sobolev norms (Q1587010)

The main result of the article is a stability estimate in Sobolev norms in the integral geometry problem for symmetric tensor fields on a compact Riemannian manifold with boundary. Previously, similar results were obtained for functions in a Euclidean space by F.~Natterer, an \(L_2\)-estimate for symmetric tensor fields on a Riemannian manifold was obtained by V.~A.~Sharafutdinov. In the article under review, the latter result is generalized to an estimate in Sobolev norms; namely, the \(H^k\)-norm of a symmetric tensor field on a compact dissipative Riemannian manifold is estimated by a weighted norm of its ray transform and the \(H^{k-1}\)-norm of its divergence. A particular type of a weighted Sobolev norm is introduced for functions defined on the boundary of the manifold of unit tangent vectors.