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

Krishnan Venkateswaran P.; Sharafutdinov Vladimir A.

doi:10.3934/ipi.2021076

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2022, Volume 16, Issue 4: 787-826. Doi: 10.3934/ipi.2021076

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

Krishnan Venkateswaran P.^1, and
Sharafutdinov Vladimir A.^2,

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: May 31, 2021

Revised: September 30, 2021

Early access: December 2021

Published: August 2022

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)

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Abstract

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.

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Mathematics Subject Classification: Primary: 44A12, 65R32; Secondary: 46F12.

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References

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