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

by Joonas Ilmavirta, Antti Kykkänen and 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.

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Reference:

Microlocal analysis of the X-ray transform in non-smooth geometry (Joonas Ilmavirta, Antti Kykkänen and Kelvin Lam), 2023. [show abstract] [hide 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}

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@unpublished{rough-microlocal-xrt,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 Kykk\"{a}nen 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.}
}