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February  2018, 12(1): 229-237. doi: 10.3934/ipi.2018009

Parametrices for the light ray transform on Minkowski spacetime

Yiran Wang

Department of Mathematics, University of Washington, Box 354350, Seattle, WA 98195-4350, USA

Received  February 2017 Published  December 2017

We consider restricted light ray transforms arising from an inverse problem of finding cosmic strings. We construct a relative left parametrix for the transform on two tensors, which recovers the space-like and some light-like singularities of the two tensor.

Keywords: Light ray transforms, parametrix, space-times, Lagrangian distributions, inverse problems.
Mathematics Subject Classification: Primary: 53C65; Secondary: 35S30.
Citation: Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems & Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009
References:
[1]

J. Antoniano and G. Uhlmann, A functional calculus for a class of pseudodifferential operators with singular symbols, Proc. Symp. Pure Math., 43 (1985), 5-16.   Google Scholar

[2]

M. de Hoop, G. Uhlmann and A. Vasy, Diffraction from conormal singularities, Annales Scientifiques de l'École Normale Supérieure, 4e serie, 48 (2015), 351-408.  doi: 10.24033/asens.2247.  Google Scholar

[3]

A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities, J. reine angew. Math., 455 (1994), 35-56.   Google Scholar

[4]

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.  Google Scholar

[5]

A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Annales de l'institut Fourier, 40 (1990), 443-466.  doi: 10.5802/aif.1220.  Google Scholar

[6]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.  doi: 10.1016/0022-1236(90)90011-9.  Google Scholar

[7]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, Contemporary Mathematics, 113 (1990), 121-135.   Google Scholar

[8]

A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ⅱ, Duke Math. J., 64 (1991), 415-444.  doi: 10.1215/S0012-7094-91-06422-7.  Google Scholar

[9]

A. Greenleaf and G. Uhlmann, Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.  doi: 10.1007/BF02096882.  Google Scholar

[10]

V. Guillemin, Cosmology in $(2+1) $-Dimensions, Cyclic Models, and Deformations of $M_{2, 1} $ Annals of Mathematics Studies, No. 121, Princeton University Press, 1989.  Google Scholar

[11]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.  doi: 10.1215/S0012-7094-81-04814-6.  Google Scholar

[12]

L. Hörmander, Fourier integral operators. Ⅰ, Acta Mathematica, 127 (1971), 79-183.  doi: 10.1007/BF02392052.  Google Scholar

[13]

L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators Springer-Verlag, Berlin, Heidelberg, 2009.  Google Scholar

[14]

M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, preprint, arXiv: 1505.03123. Google Scholar

[15]

R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Communications on Pure and Applied Mathematics, 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.  Google Scholar

[16]

B. Palacios, G. Uhlmann and Y. Wang, Reducing streaking artifacts in quantitative susceptibility mapping, SIAM Journal of Imaging Sciences, 10 (2017), 1921-1934.   Google Scholar

[17]

P. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259–1274. arXiv: 1504.01184.  Google Scholar

show all references

References:
[1]

J. Antoniano and G. Uhlmann, A functional calculus for a class of pseudodifferential operators with singular symbols, Proc. Symp. Pure Math., 43 (1985), 5-16.   Google Scholar

[2]

M. de Hoop, G. Uhlmann and A. Vasy, Diffraction from conormal singularities, Annales Scientifiques de l'École Normale Supérieure, 4e serie, 48 (2015), 351-408.  doi: 10.24033/asens.2247.  Google Scholar

[3]

A. Greenleaf and A. Seeger, Fourier integral operators with fold singularities, J. reine angew. Math., 455 (1994), 35-56.   Google Scholar

[4]

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.  Google Scholar

[5]

A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Annales de l'institut Fourier, 40 (1990), 443-466.  doi: 10.5802/aif.1220.  Google Scholar

[6]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, Journal of Functional Analysis, 89 (1990), 202-232.  doi: 10.1016/0022-1236(90)90011-9.  Google Scholar

[7]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, Contemporary Mathematics, 113 (1990), 121-135.   Google Scholar

[8]

A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms. Ⅱ, Duke Math. J., 64 (1991), 415-444.  doi: 10.1215/S0012-7094-91-06422-7.  Google Scholar

[9]

A. Greenleaf and G. Uhlmann, Recovering singularities of a potential from singularities of scattering data, Communications in Mathematical Physics, 157 (1993), 549-572.  doi: 10.1007/BF02096882.  Google Scholar

[10]

V. Guillemin, Cosmology in $(2+1) $-Dimensions, Cyclic Models, and Deformations of $M_{2, 1} $ Annals of Mathematics Studies, No. 121, Princeton University Press, 1989.  Google Scholar

[11]

V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Math. J., 48 (1981), 251-267.  doi: 10.1215/S0012-7094-81-04814-6.  Google Scholar

[12]

L. Hörmander, Fourier integral operators. Ⅰ, Acta Mathematica, 127 (1971), 79-183.  doi: 10.1007/BF02392052.  Google Scholar

[13]

L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators Springer-Verlag, Berlin, Heidelberg, 2009.  Google Scholar

[14]

M. Lassas, L. Oksanen, P. Stefanov and G. Uhlmann, On the inverse problem of finding cosmic strings and other topological defects, preprint, arXiv: 1505.03123. Google Scholar

[15]

R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Communications on Pure and Applied Mathematics, 32 (1979), 483-519.  doi: 10.1002/cpa.3160320403.  Google Scholar

[16]

B. Palacios, G. Uhlmann and Y. Wang, Reducing streaking artifacts in quantitative susceptibility mapping, SIAM Journal of Imaging Sciences, 10 (2017), 1921-1934.   Google Scholar

[17]

P. Stefanov, Support theorems for the light ray transform on analytic Lorentzian manifolds, Proc. Amer. Math. Soc., 145 (2017), 1259–1274. arXiv: 1504.01184.  Google Scholar

Figure 1.  Illustration of complex $\mathscr{C}_0$
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