$\ge 3$维非阿贝尔$X$射线变换的稳定性
Stability of the non-abelian $X$-ray transform in dimension $\ge 3$
摘要：
非阿贝尔$X$射线断层摄影试图从在$\partial M$的所谓散射数据$C_Φ$的测量恢复域$M$中的矩阵势$Φ:M\rightarrow \mathbb{C}^{m\times m}$。对于$\dim M\ge 3$（以及在适当的凸性和正则性条件下），前向映射$Φ\mapsto C_Φ$的内射性在[arXiv：1605.07894]中建立。在这篇文章中，我们通过证明Hölder型稳定性估计来扩展[arXiv：1605.07894]。作为应用，我们将$\dim M =2$ [arXiv：1905.00860]的统计一致性结果推广到更高维度。[arXiv：1605.07894]中的注入性证明依赖于Uhlmann-Vasy [arXiv：1210.2084]的一种新方法，该方法首先在$\partial M$下方的浅层中建立注入性，然后通过层剥离参数将其全局化。本文的主要技术贡献是这些参数的一个更定量的版本，特别是证明层深和稳定常数的统一界限。
英文摘要：
Non-abelian $X$-ray tomography seeks to recover a matrix potential $Φ:M\rightarrow \mathbb{C}^{m\times m}$ in a domain $M$ from measurements of its so called scattering data $C_Φ$ at $\partial M$. For $\dim M\ge 3$ (and under appropriate convexity and regularity conditions), injectivity of the forward map $Φ\mapsto C_Φ$ was established in [arXiv:1605.07894]. In this article we extend [arXiv:1605.07894] by proving a Hölder-type stability estimate. As an application we generalise a statistical consistency result for $\dim M =2$ [arXiv:1905.00860] to higher dimensions. The injectivity proof in [arXiv:1605.07894] relies on a novel method by Uhlmann-Vasy [arXiv:1210.2084], which first establishes injectivity in a shallow layer below $\partial M$ and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular proving uniform bounds on layer-depth and stability constants.
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