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

2021/06/22 by Krishnan, Venky P., Sharafutdinov, Vladimir A. · 1 citation
#44A12 #46F12 #65R32 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2106.11624

Abstract

For an integer r≥0, we prove the rth order Reshetnyak formula for the ray transform of rank m symmetric tensor fields on ℝⁿ. Certain differential operators A^(m,r,l) (0≤ l≤ r) on the sphere \mathbbS^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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• Range characterization of the ray transform on Sobolev spaces of symmetric tensor fields in two dimensions

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