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August  2022, 16(4): 787-826. doi: 10.3934/ipi.2021076

Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

1. 

TIFR Centre for Applicable Mathematics, Sharada Nagar, Chikkabommasandra, Yelahanka New Town, Bangalore, India

2. 

Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk, 630090, Russia

Dedicated to the memory of our teacher Yuri Grigor'evich Reshetnyak

Received  June 2021 Revised  October 2021 Published  August 2022 Early access  December 2021

Fund Project: The first author was supported by India SERB Matrics Grant MTR/2017/000837, and the second author was supported by RFBR, Grant 20-51-15004 (joint French – Russian grant)

For an integer $ r\ge0 $, we prove the $ r^{\mathrm{th}} $ order Reshetnyak formula for the ray transform of rank $ m $ symmetric tensor fields on $ {{\mathbb R}}^n $. Roughly speaking, for a tensor field $ f $, the order $ r $ refers to $ L^2 $-integrability of higher order derivatives of the Fourier transform $ \widehat f $ over spheres centered at the origin. Certain differential operators $ A^{(m,r,l)}\ (0\le l\le r) $ on the sphere $ {{\mathbb S}}^{n-1} $ are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any $ r $ although the volume of calculations grows fast with $ r $. The algorithm is realized for small values of $ r $ and Reshetnyak formulas of orders $ 0,1,2 $ are presented in an explicit form.

Citation: Venkateswaran P. Krishnan, Vladimir A. Sharafutdinov. Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas. Inverse Problems and Imaging, 2022, 16 (4) : 787-826. doi: 10.3934/ipi.2021076
References:
[1]

I. M. Gel'fand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, Academic Press, New York-London, (1966).

[2]

S. Helgason, The Radon Transform, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4757-1463-0.

[3]

F. John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J., 2 (1938), 300-322.  doi: 10.1215/S0012-7094-38-00423-5.

[4]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York, London, Sydney, 1963.

[5]

V. P. KrishnanR. MannaS. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, Inverse Problems and Imaging, 13 (2019), 679-701.  doi: 10.3934/ipi.2019031.

[6]

R. Seeley, Complex powers of an elliptic operator, In Proceeding of Symposia in Pure Mathematics, Vol. X, Singular Integrals, American Mathematical Society, Providence, R.I., 1967,288–307.

[7]

V. A. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrecht (1994). doi: 10.1515/9783110900095.

[8]

V. A. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002, 20 pp. doi: 10.1088/1361-6420/33/2/025002.

[9]

V. A. Sharafutdinov, X-ray transform on Sobolev spaces, Inverse Problems, 37 (2021), 015007, 25 pp. doi: 10.1088/1361-6420/abb5e0.

[10]

V. A. Sharafutdinov, Radon transform on Sobolev spaces, Siberian Math. J., 62 (2021), 560-580. 

show all references

References:
[1]

I. M. Gel'fand, M. I. Graev, N. Ya. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, Academic Press, New York-London, (1966).

[2]

S. Helgason, The Radon Transform, Birkhäuser Boston, Inc., Boston, MA, 1999. doi: 10.1007/978-1-4757-1463-0.

[3]

F. John, The ultrahyperbolic differential equation with four independent variables, Duke Math. J., 2 (1938), 300-322.  doi: 10.1215/S0012-7094-38-00423-5.

[4]

S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Interscience Publishers, New York, London, Sydney, 1963.

[5]

V. P. KrishnanR. MannaS. K. Sahoo and V. A. Sharafutdinov, Momentum ray transforms, Inverse Problems and Imaging, 13 (2019), 679-701.  doi: 10.3934/ipi.2019031.

[6]

R. Seeley, Complex powers of an elliptic operator, In Proceeding of Symposia in Pure Mathematics, Vol. X, Singular Integrals, American Mathematical Society, Providence, R.I., 1967,288–307.

[7]

V. A. Sharafutdinov, Integral geometry of tensor fields, VSP, Utrecht (1994). doi: 10.1515/9783110900095.

[8]

V. A. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002, 20 pp. doi: 10.1088/1361-6420/33/2/025002.

[9]

V. A. Sharafutdinov, X-ray transform on Sobolev spaces, Inverse Problems, 37 (2021), 015007, 25 pp. doi: 10.1088/1361-6420/abb5e0.

[10]

V. A. Sharafutdinov, Radon transform on Sobolev spaces, Siberian Math. J., 62 (2021), 560-580. 

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