Invariant distributions, Beurling transforms and tensor tomography in higher dimensions
Paternain, G. P., Salo, M., & Uhlmann, G. (2015). Invariant distributions, Beurling transforms and tensor tomography in higher dimensions. Mathematische Annalen, 363(1-2), 305-362. https://doi.org/10.1007/s00208-015-1169-0
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Mathematische AnnalenDate
2015Copyright
© Springer-Verlag Berlin Heidelberg 2015. This is a final draft version of an article whose final and definitive form has been published by Springer. Published in this repository with the kind permission of the publisher.
In the recent articles [PSU13, PSU14c], 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.
...
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Springer Berlin HeidelbergISSN Search the Publication Forum
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