Range characterization of ray transform on Sobolev spaces of symmetric tensor fields

Venkateswaran P. Krishnan; Vladimir A. Sharafutdinov

doi:10.3934/ipi.2024014

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2024, Volume 18, Issue 6: 1272-1293. Doi: 10.3934/ipi.2024014

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Range characterization of ray transform on Sobolev spaces of symmetric tensor fields

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

^*Corresponding author: Venkateswaran P. Krishnan
^*Corresponding author: Venkateswaran P. Krishnan

Received: January 2024

Revised: March 2024

Early access: March 2024

Published: December 2024

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Abstract

The ray transform $ I $ integrates symmetric $ m $-tensor field in $ \mathbb{R}^n $ over lines. This transform on Sobolev spaces was studied in two earlier works of the second author where Reshetnyak formulas (isometry relations) and range characterization of ray transform of functions in dimensions $ n\geq 3 $ were established. The main focus of the current work is range characterization of ray transform of symmetric tensor fields on Sobolev spaces generalizing the earlier result proved for the case of functions. In dimensions $ n\geq 3 $, range characterization of the ray transform in Schwartz spaces is well-known; the main ingredient of the characterization is a system of linear differential equations of order $ 2(m+1) $ called John equations. Using zeroth order Reshetnyak formulas, the range of the ray transform on Sobolev spaces is characterized in dimensions $ n\geq 3 $ in this paper.

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

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