# Invariant distributions, Beurling transforms and tensor tomography in higher dimensions

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## Abstract

In the recent articles Paternain et al. (J. Differ Geom, 98:147–181, 2014, Invent Math 193:229–247, 2013), a number of tensor tomography results were proved on two-dimensional manifolds. The purpose of this paper is to extend some of these methods to manifolds of any dimension. A central concept is the surjectivity of the adjoint of the geodesic ray transform, or equivalently the existence of certain distributions that are invariant under the geodesic flow. We prove that on any Anosov manifold, one can find invariant distributions with controlled first Fourier coefficients. The proof is based on subelliptic type estimates and a Pestov identity. We present an alternative construction valid on manifolds with nonpositive curvature, based on the fact that a natural Beurling transform on such manifolds turns out to be essentially a contraction. Finally, we obtain uniqueness results in tensor tomography both on simple and Anosov manifolds that improve earlier results by assuming a condition on the terminator value for a modified Jacobi equation.

## Notes

### Acknowledgments

M.S. was supported in part by the Academy of Finland and an ERC Starting Grant (grant agreement no 307023), and G.U. was partly supported by NSF and a Simons Fellowship. The authors would like to express their gratitude to the Banff International Research Station (BIRS) for providing an excellent research environment via the Research in Pairs program and the workshop Geometry and Inverse Problems, where part of this work was carried out. We are also grateful to Hanming Zhou for several corrections to earlier drafts, and to Joonas Ilmavirta for helping with a numerical calculation. Finally we thank the referee for numerous suggestions that improved the presentation.

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