Abstract
Let \(I_{m}\) denote the Euclidean ray transform acting on compactly supported symmetric m-tensor field distributions f, and \(I_{m}^{*}\) be its formal \(L^2\) adjoint. We study a unique continuation result for the operator \(N_{m}=I_{m}^{*}I_{m}\). More precisely, we show that if \(N_{m}f\) vanishes to infinite order at a point \(x_0\) and if the Saint-Venant operator W acting on f vanishes on an open set containing \(x_0\), then f is a potential tensor field. This generalizes two recent works of Ilmavirta and Mönkkönen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher-order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher-order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.
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We thank the reviewer for bringing these three references to our attention.
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Acknowledgements
We express our gratitude to Prof. Vladimir A. Sharafutdinov for his inspiring monograph, “Integral Geometry of Tensor Fields” which has helped us immensely over the years and without which the results of this paper would not have been possible. S.K.S. was supported by Academy of Finland (Centre of Excellence in Inverse Modelling and Imaging, grant 284715) and European Research Council under Horizon 2020 (ERC CoG 770924).
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Agrawal, D., Krishnan, V.P. & Sahoo, S.K. Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields. J Geom Anal 32, 245 (2022). https://doi.org/10.1007/s12220-022-00981-5
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DOI: https://doi.org/10.1007/s12220-022-00981-5