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April 2018, 12(2): 293-314. doi: 10.3934/ipi.2018013

Support theorem for the Light-Ray transform of vector fields on Minkowski spaces

Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA

Received  October 2016 Revised  December 2017 Published  February 2018

Fund Project: Partly supported by NSF Grants DMS 1301646 and DMS 1600327.

We study the Light-Ray transform of integrating vector fields on the Minkowski time-space $\boldsymbol{{\rm R}}^{1+n}$, $n≥ 2$, with the Minkowski metric. We prove a support theorem for vector fields vanishing on an open set of light-like lines. We provide examples to illustrate the application of our results to the inverse problem for the hyperbolic Dirichlet-to-Neumann map.

Citation: Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems & Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013
References:
[1]

A. Begmatov, A certain inversion problem for the ray transform with incomplete data, Siberian Math. Journal, 42 (2001), 428-434.

[2]

M. Bellassoued and I. Ben Aicha, Stable determination outside a cloaking region of two timedependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, Journal of Mathematical Analysis and Applications, 449 (2017), 46-76, arXiv: 1605.03466. doi: 10.1016/j.jmaa.2016.11.082.

[3]

M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Problem. Imaging, 5 (2011), 745-773. doi: 10.3934/ipi.2011.5.745.

[4]

I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time dependent coefficient, Inverse Problems, 31 (2015), 125010, 21 pp, arXiv: 1506.01935.

[5]

J. Boman, Helgason's support theorem for Radon transforms-a new proof and a generalization, In Mathematical Methods in Tomography (Oberwolfach, 1990), volume 1497 of Lecture Notes in Math., pages 1-5. Springer, Berlin, 1991.

[6]

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: 10.1215/S0012-7094-87-05547-5.

[7]

J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic line complexes in three-space, Trans. Amer. Math. Soc., 335 (1993), 877-890. doi: 10.1090/S0002-9947-1993-1080733-8.

[8]

J. M. Bony, Equivalence des Diverses Notions de Spectre Singulier Analytique, Sèminaire Goulaouic-Schwartz, 1976/77, no. 3.

[9]

J. Bros and D. Iagolnitzer, Support Essentiel et Structure Analytique Des Distributions, Sèminaire Goulaouic-Lions-Schwartz, 1975/76, no. 18.

[10]

A. Denisiuk, Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve, Inverse Problems, 22 (2006), 399-411. doi: 10.1088/0266-5611/22/2/001.

[11]

B. FrigyikP. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108. doi: 10.1007/s12220-007-9007-6.

[12]

A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240. doi: 10.1215/S0012-7094-89-05811-0.

[13]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232. doi: 10.1016/0022-1236(90)90011-9.

[14]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, In Integral Geometry and Tomography (Arcata, CA, 1989), volume 113 of Contemp. Math., pages 121-135. Amer. Math. Soc., Providence, RI, 1990.

[15]

V. Guillemin, On some results of Gel'fand in integral geometry, In Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984), volume 43 of Proc. Sympos. Pure Math., pages 149-155. Amer. Math. Soc., Providence, RI, 1985.

[16]

V. Guillemin and S. Sternberg, Geometric Asymptotics, American Mathematical Society, Providence, R. I., 1977.

[17]

S. Helgason, The Radon Transform, volume 5 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, second edition, 1999.

[18]

S. Holman and P. Stefanov, The weighted Doppler transform, Inverse Problem. Imaging, 4 (2010), 111-130. doi: 10.3934/ipi.2010.4.111.

[19]

L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math, 24 (1971), 671-704. doi: 10.1002/cpa.3160240505.

[20]

C. E. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2), 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.

[21]

Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equation from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046, arXiv: 1603.09600. doi: 10.1137/16M1076708.

[22]

V. P. Krishnan, A support theorem for the geodesic ray transform on functions, J. Fourier Anal. Appl., 15 (2009), 515-520. doi: 10.1007/s00041-009-9061-5.

[23]

V. P. Krishnan and P. Stefanov, A support theorem for the geodesic ray transform of symmetric tensor fields, Inverse Probl. Imaging, 3 (2009), 453-464. doi: 10.3934/ipi.2009.3.453.

[24]

M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, Communications in Mathematical Physics, (2017), 1-27, arXiv: 1505.03123. doi: 10.1007/s00220-017-3029-0.

[25]

R. B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Inverse Probl. In Spectral and scattering theory (Sanda, 1992), volume 161 of Lecture Notes in Pure and Appl. Math., pages 85-30. Dekker, New York, 1994.

[26]

C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to Neumann map, Partial Differential Equations, 39 (2014), 120-145. doi: 10.1080/03605302.2013.843429.

[27]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on simple surfaces, arXiv: 1109.0505, Invent. Math., 193 (2013), 229-247. doi: 10.1007/s00222-012-0432-1.

[28]

L. Pestov and G. Uhlmann, Two dimensional simple compact manifolds with boundary are boundary rigid, Ann. Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[29]

A. Z. Petrov, Einstein Spaces. Translated from the Russian by R. F. Kelleher. Translation edited by J. Woodrow, Pergamon Press, Oxford-Edinburgh-New York, 1969.

[30]

E. T. Quinto, Real analytic Radon transforms on Rank one symmetric spaces, Proc. Math. Soc., 117 (1993), 179-186. doi: 10.1090/S0002-9939-1993-1135080-8.

[31]

A. G. Ramm, Property C and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219. doi: 10.1016/0022-247X(91)90391-C.

[32]

A. G. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330.

[33]

R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp.

[34]

R. Salazar, Stability estimate for the relativistic Schroedinger equation with time-dependent vector potentials, Inverse Problems, 30 (2014), 105005, 18pp.

[35]

M. Sato, Hyperfunctions and Partial Differential Equations, Proc. Int. Conf. Funct. Anal. Tokyo 1969, 91-4.

[36]

M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations. In Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., Springer, Berlin, 287 (1973), 265-529.

[37]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994.

[38]

J. Sjöstrand, Singularités analytiques microlocales, In Astérisque, 95, Soc. Math. France, Paris, volume 95 of Astérisque, (1982), 1-166.

[39]

P. Stefanov, Support theorems for the light ray transform on analytic lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259-1274.

[40]

P. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559. doi: 10.1007/BF01215158.

[41]

P. Stefanov and G. Uhlmann, Microlocal Analysis and Integral Geometry, Book in Progress.

[42]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2.

[43]

P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.

[44]

P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math., 130 (2008), 239-268. doi: 10.1353/ajm.2008.0003.

[45]

P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE, 5 (2012), 219-260. doi: 10.2140/apde.2012.5.219.

[46]

P. Stefanov and Y. Yang, The Inverse Problem for The Dirichket-to-Neumann Map on Lorentzian Manifolds, 2016.

[47]

F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1. Pseudodifferential Operators, The University Series in Mathematics. Plenum Press, New York, 1980.

[48]

G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120. doi: 10.1007/s00222-015-0631-7.

[49]

A. Waters, Stable determination of X-Ray transforms of time dependent potentials from partial boundary data, Comm. Partial Differential Equations, 39 (2014), 2169-2197. doi: 10.1080/03605302.2014.930486.

[50]

A. Waters and R. Salazar, Global stability for time dependent X-ray transforms on simple manifolds and applications, arXiv: 1311.1591, 2013.

show all references

References:
[1]

A. Begmatov, A certain inversion problem for the ray transform with incomplete data, Siberian Math. Journal, 42 (2001), 428-434.

[2]

M. Bellassoued and I. Ben Aicha, Stable determination outside a cloaking region of two timedependent coefficients in an hyperbolic equation from Dirichlet to Neumann map, Journal of Mathematical Analysis and Applications, 449 (2017), 46-76, arXiv: 1605.03466. doi: 10.1016/j.jmaa.2016.11.082.

[3]

M. Bellassoued and D. Dos Santos Ferreira, Stability estimates for the anisotropic wave equation from the Dirichlet-to-Neumann map, Inverse Problem. Imaging, 5 (2011), 745-773. doi: 10.3934/ipi.2011.5.745.

[4]

I. Ben Aicha, Stability estimate for hyperbolic inverse problem with time dependent coefficient, Inverse Problems, 31 (2015), 125010, 21 pp, arXiv: 1506.01935.

[5]

J. Boman, Helgason's support theorem for Radon transforms-a new proof and a generalization, In Mathematical Methods in Tomography (Oberwolfach, 1990), volume 1497 of Lecture Notes in Math., pages 1-5. Springer, Berlin, 1991.

[6]

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: 10.1215/S0012-7094-87-05547-5.

[7]

J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic line complexes in three-space, Trans. Amer. Math. Soc., 335 (1993), 877-890. doi: 10.1090/S0002-9947-1993-1080733-8.

[8]

J. M. Bony, Equivalence des Diverses Notions de Spectre Singulier Analytique, Sèminaire Goulaouic-Schwartz, 1976/77, no. 3.

[9]

J. Bros and D. Iagolnitzer, Support Essentiel et Structure Analytique Des Distributions, Sèminaire Goulaouic-Lions-Schwartz, 1975/76, no. 18.

[10]

A. Denisiuk, Inversion of the x-ray transform for 3D symmetric tensor fields with sources on a curve, Inverse Problems, 22 (2006), 399-411. doi: 10.1088/0266-5611/22/2/001.

[11]

B. FrigyikP. Stefanov and G. Uhlmann, The X-ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 89-108. doi: 10.1007/s12220-007-9007-6.

[12]

A. Greenleaf and G. Uhlmann, Nonlocal inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240. doi: 10.1215/S0012-7094-89-05811-0.

[13]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232. doi: 10.1016/0022-1236(90)90011-9.

[14]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, In Integral Geometry and Tomography (Arcata, CA, 1989), volume 113 of Contemp. Math., pages 121-135. Amer. Math. Soc., Providence, RI, 1990.

[15]

V. Guillemin, On some results of Gel'fand in integral geometry, In Pseudodifferential Operators and Applications (Notre Dame, Ind., 1984), volume 43 of Proc. Sympos. Pure Math., pages 149-155. Amer. Math. Soc., Providence, RI, 1985.

[16]

V. Guillemin and S. Sternberg, Geometric Asymptotics, American Mathematical Society, Providence, R. I., 1977.

[17]

S. Helgason, The Radon Transform, volume 5 of Progress in Mathematics, Birkhäuser Boston Inc., Boston, MA, second edition, 1999.

[18]

S. Holman and P. Stefanov, The weighted Doppler transform, Inverse Problem. Imaging, 4 (2010), 111-130. doi: 10.3934/ipi.2010.4.111.

[19]

L. Hörmander, Uniqueness theorems and wave front sets for solutions of linear differential equations with analytic coefficients, Comm. Pure Appl. Math, 24 (1971), 671-704. doi: 10.1002/cpa.3160240505.

[20]

C. E. KenigJ. Sjöstrand and G. Uhlmann, The Calderón problem with partial data, Ann. of Math. (2), 165 (2007), 567-591. doi: 10.4007/annals.2007.165.567.

[21]

Y. Kian, Recovery of time-dependent damping coefficients and potentials appearing in wave equation from partial data, SIAM J. Math. Anal., 48 (2016), 4021-4046, arXiv: 1603.09600. doi: 10.1137/16M1076708.

[22]

V. P. Krishnan, A support theorem for the geodesic ray transform on functions, J. Fourier Anal. Appl., 15 (2009), 515-520. doi: 10.1007/s00041-009-9061-5.

[23]

V. P. Krishnan and P. Stefanov, A support theorem for the geodesic ray transform of symmetric tensor fields, Inverse Probl. Imaging, 3 (2009), 453-464. doi: 10.3934/ipi.2009.3.453.

[24]

M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, Communications in Mathematical Physics, (2017), 1-27, arXiv: 1505.03123. doi: 10.1007/s00220-017-3029-0.

[25]

R. B. Melrose, Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces, Inverse Probl. In Spectral and scattering theory (Sanda, 1992), volume 161 of Lecture Notes in Pure and Appl. Math., pages 85-30. Dekker, New York, 1994.

[26]

C. Montalto, Stable determination of a simple metric, a covector field and a potential from the hyperbolic Dirichlet-to Neumann map, Partial Differential Equations, 39 (2014), 120-145. doi: 10.1080/03605302.2013.843429.

[27]

G. P. Paternain, M. Salo and G. Uhlmann, Tensor tomography on simple surfaces, arXiv: 1109.0505, Invent. Math., 193 (2013), 229-247. doi: 10.1007/s00222-012-0432-1.

[28]

L. Pestov and G. Uhlmann, Two dimensional simple compact manifolds with boundary are boundary rigid, Ann. Math., 161 (2005), 1093-1110. doi: 10.4007/annals.2005.161.1093.

[29]

A. Z. Petrov, Einstein Spaces. Translated from the Russian by R. F. Kelleher. Translation edited by J. Woodrow, Pergamon Press, Oxford-Edinburgh-New York, 1969.

[30]

E. T. Quinto, Real analytic Radon transforms on Rank one symmetric spaces, Proc. Math. Soc., 117 (1993), 179-186. doi: 10.1090/S0002-9939-1993-1135080-8.

[31]

A. G. Ramm, Property C and an inverse problem for a hyperbolic equation, J. Math. Anal. Appl., 156 (1991), 209-219. doi: 10.1016/0022-247X(91)90391-C.

[32]

A. G. Ramm and J. Sjöstrand, An inverse problem of the wave equation, Math. Z., 206 (1991), 119-130. doi: 10.1007/BF02571330.

[33]

R. Salazar, Determination of time-dependent coefficients for a hyperbolic inverse problem, Inverse Problems, 29 (2013), 095015, 17pp.

[34]

R. Salazar, Stability estimate for the relativistic Schroedinger equation with time-dependent vector potentials, Inverse Problems, 30 (2014), 105005, 18pp.

[35]

M. Sato, Hyperfunctions and Partial Differential Equations, Proc. Int. Conf. Funct. Anal. Tokyo 1969, 91-4.

[36]

M. Sato, T. Kawai and M. Kashiwara, Microfunctions and pseudo-differential equations. In Hyperfunctions and pseudo-differential equations (Proc. Conf., Katata, 1971; dedicated to the memory of André Martineau), Lecture Notes in Math., Springer, Berlin, 287 (1973), 265-529.

[37]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-Posed Problems Series, VSP, Utrecht, 1994.

[38]

J. Sjöstrand, Singularités analytiques microlocales, In Astérisque, 95, Soc. Math. France, Paris, volume 95 of Astérisque, (1982), 1-166.

[39]

P. Stefanov, Support theorems for the light ray transform on analytic lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259-1274.

[40]

P. Stefanov, Uniqueness of the multi-dimensional inverse scattering problem for time dependent potentials, Math. Z., 201 (1989), 541-559. doi: 10.1007/BF01215158.

[41]

P. Stefanov and G. Uhlmann, Microlocal Analysis and Integral Geometry, Book in Progress.

[42]

P. Stefanov and G. Uhlmann, Stability estimates for the X-ray transform of tensor fields and boundary rigidity, Duke Math. J., 123 (2004), 445-467. doi: 10.1215/S0012-7094-04-12332-2.

[43]

P. Stefanov and G. Uhlmann, Boundary rigidity and stability for generic simple metrics, J. Amer. Math. Soc., 18 (2005), 975-1003. doi: 10.1090/S0894-0347-05-00494-7.

[44]

P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math., 130 (2008), 239-268. doi: 10.1353/ajm.2008.0003.

[45]

P. Stefanov and G. Uhlmann, The geodesic X-ray transform with fold caustics, Anal. PDE, 5 (2012), 219-260. doi: 10.2140/apde.2012.5.219.

[46]

P. Stefanov and Y. Yang, The Inverse Problem for The Dirichket-to-Neumann Map on Lorentzian Manifolds, 2016.

[47]

F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. 1. Pseudodifferential Operators, The University Series in Mathematics. Plenum Press, New York, 1980.

[48]

G. Uhlmann and A. Vasy, The inverse problem for the local geodesic ray transform, Inventiones Mathematicae, 205 (2016), 83-120. doi: 10.1007/s00222-015-0631-7.

[49]

A. Waters, Stable determination of X-Ray transforms of time dependent potentials from partial boundary data, Comm. Partial Differential Equations, 39 (2014), 2169-2197. doi: 10.1080/03605302.2014.930486.

[50]

A. Waters and R. Salazar, Global stability for time dependent X-ray transforms on simple manifolds and applications, arXiv: 1311.1591, 2013.

Figure 1.  $\Gamma_{\rho_0}$ with $0 <c < 1$.
Figure 2.  $\Gamma_{0}$ with $0 <c <1$.
Figure 3.  $\Gamma_{0}$ with $0 <c < 1$.
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