by Joonas Ilmavirta, Antti Kykkänen, Kelvin Lam
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.
Reference:
Microlocal analysis of the X-ray transform in non-smooth geometry (Joonas Ilmavirta, Antti Kykkänen, Kelvin Lam), 2023.
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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 $gin C^{10}$. 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.
[arXiv]
Bibtex Entry:
@unpublished{rough-microlocal-xrt,
author = {Joonas Ilmavirta and Antti Kykkänen and Kelvin Lam},
title = {{Microlocal analysis of the X-ray transform in non-smooth geometry}},
month = sep,
year = 2023,
arxiv = {2309.12702},
url={http://users.jyu.fi/~jojapeil/pub/rough-microlocal-xrt.pdf},
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.}
}