Microlocal analysis of the X-ray transform in non-smooth geometry

TL;DR

This paper proves the injectivity of the geodesic X-ray transform on $L^2$ spaces for simple Riemannian metrics with limited smoothness, using microlocal analysis techniques adapted to non-smooth geometries.

Contribution

It extends microlocal analysis methods to establish injectivity of the X-ray transform for metrics with finite differentiability, including explicit derivative requirements based on dimension.

Findings

Injectivity holds for metrics with finite regularity.

Ellipticity and smoothing properties are established for the normal operator.

The results apply to metrics with limited smoothness, such as $C^{10}$ in dimension 2.

Abstract

We prove that the geodesic X-ray transform is injective on $L^2$ when the Riemannian metric is simple but the metric tensor is only finitely differentiable. The number of derivatives needed depends explicitly on dimension, and in dimension $2$ we assume $g\in C^{10}$. Our proof is based on microlocal analysis of the normal operator: we establish ellipticity and a smoothing property in a suitable sense and then use a recent injectivity result on Lipschitz functions. When the metric tensor is $C^k$, the Schwartz kernel is not smooth but $C^{k-2}$ off the diagonal, which makes standard smooth microlocal analysis inapplicable.