Pestov identities and X-ray tomography on manifolds of low regularity

Ilmavirta, Joonas; Kykkänen, Antti

by Joonas Ilmavirta, Antti Kykkänen

Abstract:

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.

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Reference:

Pestov identities and X-ray tomography on manifolds of low regularity (Joonas Ilmavirta, Antti Kykkänen), Inverse Problems and Imaging, 2021. (To appear.) [show abstract] [hide abstract] We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the Hölder scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure. [arXiv]

Bibtex Entry:

@article{rough-pestov,
	author = {Joonas Ilmavirta and Antti Kykk\"anen},
	title = {{Pestov identities and X-ray tomography on manifolds of low regularity}},
	month = dec,
	year = {2021},
	journal = {Inverse Problems and Imaging},
	abstract = {We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.},
	url={http://users.jyu.fi/~jojapeil/pub/rough-pestov.pdf},
        arxiv = {2112.05523},
        note = {To appear.}
}