Ray transform on Sobolev spaces of symmetric tensor fields, II: Range characterization

TL;DR

This paper extends the understanding of the ray transform on symmetric tensor fields by providing a Sobolev space range characterization in dimensions three and higher, building on previous higher order Reshetnyak formulas.

Contribution

It offers the first Sobolev space range characterization of the ray transform in dimensions n ≥ 3, utilizing higher order Reshetnyak formulas and John equations.

Findings

Range characterized in Sobolev spaces for n ≥ 3

Utilizes higher order Reshetnyak formulas

Connects to classical John equations

Abstract

The ray transform $I$ integrates symmetric $m$-tensor field in $\mathbb{R}^n$ over lines. This transform in Sobolev spaces was studied in our earlier work where higher order Reshetnyak formulas (isometry relations) were established. The main focus of the current work is the range characterization. In dimensions $n\geq 3$, the 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 $2(m+1)$ which are called John equations. Using the higher order Reshetnyak formulas, the range of the ray transform on Sobolev spaces is characterized in dimensions $n\geq 3$ in this paper.