A1 Journal article (refereed)
Carleman estimates for geodesic X-ray transforms (2023)
Estimations de Carleman pour les transformées aux rayons X géodésiques
Paternain, G. P., & Salo, M. (2023). Carleman estimates for geodesic X-ray transforms. Annales Scientifiques de l’École Normale Supérieure, 56(5), 1339-1379. https://doi.org/10.24033/asens.2557
JYU authors or editors
Publication details
All authors or editors: Paternain, Gabriel P.; Salo, Mikko
Journal or series: Annales Scientifiques de l’École Normale Supérieure
ISSN: 0012-9593
eISSN: 1873-2151
Publication year: 2023
Publication date: 22/01/2024
Volume: 56
Issue number: 5
Pages range: 1339-1379
Publisher: Societe Mathematique de France
Publication country: France
Publication language: English
DOI: https://doi.org/10.24033/asens.2557
Publication open access: Not open
Publication channel open access:
Web address of parallel published publication (pre-print): https://arxiv.org/abs/1805.02163
Abstract
In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carleman estimate with logarithmic weights (resp. linear weights) on the frequency side. As a particular consequence, on negatively curved simple manifolds the geodesic X-ray transform with attenuation given by a general connection and Higgs field is invertible modulo natural obstructions. The proof is based on showing that the Pestov energy identity for the geodesic vector field completely localizes in frequency. Our approach works in all dimensions ≥2, on negatively curved manifolds with or without boundary, and for tensor fields of any order.
Keywords: differential geometry; partial differential equations; dynamical systems; inverse problems
Free keywords: Geodesic X-ray transform; Carleman estimate; Pestov identity; geodesic flow
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Ministry reporting: Yes
Preliminary JUFO rating: 3
