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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Notes
We thank the reviewer for bringing these three references to our attention.
References
Abhishek, A., Mishra, R.K.: Support theorems and an injectivity result for integral moments of a symmetric \(m\)-tensor field. J. Fourier Anal. Appl. 25(4), 1487–1512 (2019). https://doi.org/10.1007/s00041-018-09649-7
Bhattacharyya, S., Krishnan, V.P., Sahoo, S.K.: Unique determination of anisotropic perturbations of a polyharmonic operator from partial boundary data. arxiv:2111.07610 (2021)
Courant, R., Hilbert, D.: Methods of mathematical physics. vol. II, Wiley Classics Library. Wiley, New York, Partial differential equations, Reprint of the 1962 original, A Wiley-Interscience Publication (1989)
Denisjuk, A.: Inversion of the X-ray transform for 3D symmetric tensor fields with sources on a curve. Inverse Probl. 22(2), 399–411 (2006). https://doi.org/10.1088/0266-5611/22/2/001
Desai, N.M., Lionheart, W.R.B.: An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data. Inverse Probl. 32(11), 115009 (2016). https://doi.org/10.1088/0266-5611/32/11/115009
Ghosh, T., Salo, M., Uhlmann, G.: The Calderón problem for the fractional Schrödinger equation. Anal. PDE 13(2), 455–475 (2020). https://doi.org/10.2140/apde.2020.13.455
Hamaker, C., Smith, K.T., Solmon, D.C., Wagner, S.L.: The divergent beam X-ray transform. Rocky Mt. J. Math. 10(1), 253–283 (1980). https://doi.org/10.1216/RMJ-1980-10-1-253
Ilmavirta, J., Mönkkönen, K.: Unique continuation of the normal operator of the X-ray transform and applications in geophysics. Inverse Probl. 36(4), 045014 (2020). https://doi.org/10.1088/1361-6420/ab6e75
Ilmavirta, J.: X-ray tomography of one-forms with partial data. SIAM J. Math. Anal. 53(3), 3002–3015 (2021). https://doi.org/10.1137/20M1344779
John, F.: The ultrahyperbolic differential equation with four independent variables. Duke Math. J. 4(2), 300–322 (1938). https://doi.org/10.1215/S0012-7094-38-00423-5
Krishnan, V.P., Mishra, R.K.: Microlocal analysis of a restricted ray transform on symmetric \(m\)-tensor fields in \(\mathbb{R}^n\). SIAM J. Math. Anal. 50(6), 6230–6254 (2018). https://doi.org/10.1137/18M1178530
Krishnan, V.P., Manna, R., Sahoo, S.K., Sharafutdinov, V.A.: Momentum ray transforms. Inverse Probl. Imaging 13(3), 679–701 (2019). https://doi.org/10.3934/ipi.2019031
Krishnan, V.P., Manna, R., Sahoo, S.K., Sharafutdinov, V.A.: Momentum ray transforms, II: range characterization in the Schwartz space. Inverse Probl. 36(4), 045009 (2020). https://doi.org/10.1088/1361-6420/ab6a65
Kotake, T., Narasimhan, M.S.: Regularity theorems for fractional powers of a linear elliptic operator. Bull. Soc. Math. France 90, 449–471 (1962)
Krishnan, V.P., Sharafutdinov, V.A.: Ray transform on Sobolev spaces of symmetric tensor fields, I: higher order Reshetnyak formulas (2021)
Lionheart, W., Sharafutdinov, V..: Reconstruction algorithm for the linearized polarization tomography problem with incomplete data. In: Imaging Microstructures, Contemp. Math., vol. 494, pp. 137–159. Amer. Math. Soc., Providence. https://doi.org/10.1090/conm/494/09648 (2009)
Lionheart, W.R.B., Withers, P.J.: Diffraction tomography of strain. Inverse Probl. 31(4), 045005 (2015). https://doi.org/10.1088/0266-5611/31/4/045005
Mishra, R.K., Sahoo, S.K.: Injectivity and range description of integral moment transforms over \(m\)-tensor fields in \(\mathbb{R}^n\). SIAM J. Math. Anal. 53(1), 253–278 (2021). https://doi.org/10.1137/20M1347589
Mishra, R.K., Sahoo, S.K.: The generalized Saint Venant operator and integral moment transforms, arXiv:2203.13669 (2022)
Novikov, R., Sharafutdinov, V.A.: On the problem of polarization tomography I. Inverse Probl. 23(3), 1229–1257 (2007). https://doi.org/10.1088/0266-5611/23/3/023
Rüland, A.: Unique continuation for fractional Schrödinger equations with rough potentials. Commun. Partial Differ. Equ. 40(1), 77–114 (2015). https://doi.org/10.1080/03605302.2014.905594
Riesz, M.: Intégrales de riemann-liouville et potentiels. Acta Sci. Math. 9(1–1), 1–42 (1938-40)
Sharafutdinov, V.A.: Uniqueness theorems for the exponential X-ray transform. J. Inverse Ill-Posed Probl. 1(4), 355–372 (1993). https://doi.org/10.1515/jiip.1993.1.4.355
Sharafutdinov, V.A.: Integral geometry of tensor fields. Inverse and Ill-Posed Problems Series, VSP, Utrecht (1994). https://doi.org/10.1515/9783110900095
Stroock, D.W.: Weyl’s Lemma, one of many. In: Groups and Analysis, London Math. Soc. Lecture Note Ser., vol. 354, pp. 164– 173. Cambridge Univ. Press, Cambridge. https://doi.org/10.1017/CBO9780511721410.009 (2008)
Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)
Uhlmann, G.: Travel time tomography. J. Korean Math. Soc. 38(4), 711– 722, Mathematics in the new millennium (Seoul, 2000) (2001)
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