Unique continuation results for certain generalized ray transforms of symmetric tensor fields

TL;DR

This paper establishes unique continuation properties for generalized ray transforms of symmetric tensor fields, extending previous results to higher order tensors and related transforms, with implications for tensor field analysis.

Contribution

It introduces the Saint-Venant operator as a generalization of the exterior derivative for higher order tensors, enabling new unique continuation results for tensor ray transforms.

Findings

Unique continuation results for the normal operator of tensor ray transforms.

Identification of the Saint-Venant operator as a key generalization.

Extension of results to momentum and transverse ray transforms.

Abstract

Let $I_{m}$ denote the Euclidean ray transform acting on compactly supported symmetric $m$-tensor field distributions $f$, and $I_{m}^{*}$ be its formal $L^2$ adjoint. We study a unique continuation result for the normal operator $N_{m}=I_{m}^{*}I_{m}$. More precisely, we show that if $N_{m}$ vanishes to infinite order at a point $x_0$ and if the Saint-Venant operator $W$ acting on $f$ vanishes on an open set containing $x_0$, then $f$ is a potential tensor field. This generalizes two recent works of Ilmavirta and M\"onkk\"onen who proved such unique continuation results for the ray transform of functions and vector fields/1-forms. One of the main contributions of this work is identifying the Saint-Venant operator acting on higher order tensor fields as the right generalization of the exterior derivative operator acting on 1-forms, which makes unique continuation results for ray transforms of higher order tensor fields possible. In the second half of the paper, we prove analogous unique continuation results for momentum ray and transverse ray transforms.