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

IF 1.2 4区数学 Q2 MATHEMATICS, APPLIED

Inverse Problems and Imaging Pub Date : 2021-12-10 DOI:10.3934/ipi.2023017

Joonas Ilmavirta, Antti Kykkanen

引用次数: 4

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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低正则性流形上的Pestov恒等式和x射线层析成像

我们证明了测地X射线变换在标量函数上是内射的，在C^｛1，1｝$中具有$g\的简单黎曼流形$（M，g）$上的一个形式上是（螺线管）内射的。除了证明之外，我们还重新定义了与粗糙几何兼容的简单性。这种$C^｛1，｝$正则性在H\“老尺度上是最优的。本文的大部分内容致力于在这种非光滑结构上的单位球面丛上建立微分算子和曲率算子的微积分。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Inverse Problems and Imaging 数学-物理：数学物理

CiteScore

2.50

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Inverse Problems and Imaging publishes research articles of the highest quality that employ innovative mathematical and modeling techniques to study inverse and imaging problems arising in engineering and other sciences. Every published paper has a strong mathematical orientation employing methods from such areas as control theory, discrete mathematics, differential geometry, harmonic analysis, functional analysis, integral geometry, mathematical physics, numerical analysis, optimization, partial differential equations, and stochastic and statistical methods. The field of applications includes medical and other imaging, nondestructive testing, geophysical prospection and remote sensing as well as image analysis and image processing. This journal is committed to recording important new results in its field and will maintain the highest standards of innovation and quality. To be published in this journal, a paper must be correct, novel, nontrivial and of interest to a substantial number of researchers and readers.