Range characterization of ray transform on Sobolev spaces of symmetric tensor fields
Venkateswaran P. Krishnan, V. A. Sharafutdinov · Inverse Problems and Imaging · 2024
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.