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arXiv · 2309.12702

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

Joonas Ilmavirta·Antti Kykkänen·Kelvin Lam
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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.

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Joonas Ilmavirta, Antti Kykkänen, Kelvin Lam. 2023-09-22. Microlocal analysis of the X-ray transform in non-smooth geometry. https://arxiv.org/abs/2309.12702

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