Unique Continuation Results for Certain Generalized Ray Transforms of Symmetric Tensor Fields

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 operator \(N_{m}=I_{m}^{*}I_{m}\). More precisely, we show that if \(N_{m}f\) 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 Mönkkönen 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.