发表机构
Shandong University; University of Science and Technology of China（山东大学; 中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究次黎曼流形上平方Carnot--Carathéodory距离最优传输的稳定性，证明了映射的$W_1^{1/4}$估计和势的Hölder稳定性，并构造例子说明指数最优性。
AI 中文摘要
我们研究了平方Carnot--Carathéodory距离下最优传输的稳定性。我们证明了在具有有界源支撑和上密度界的完备黎曼流形与fat次黎曼流形上，最优传输映射满足尖锐的$W_1^{1/4}$稳定性估计。在具有双侧密度界的John域上，我们在源附近等正则的完备光滑次黎曼流形上，对任意紧致目标集，获得了Kantorovich势的对数$L^2$稳定性。对于势，我们在双生成结构上获得了所有Hölder指数$0<\alpha<1/2$，在黎曼和fat情形以及所有步长为二的Carnot群上获得了指数$1/2$。三原子例子表明，在任何水平秩至少为二的光滑次黎曼流形上，映射和势关于$W_1$的指数分别不能超过$1/4$和$1/2$。我们还构造了光滑传输族，其目标在$W_1$中以$t$阶变化，在步长为$s$的等正则区域中产生$t^{1/s}$阶的映射变化。映射估计由对偶缺陷的强制性得出，而势估计由Kantorovich泛函的定量凹性得出。
英文摘要
We study the stability of optimal transport for the squared Carnot--Carathéodory distance. We prove the sharp $W_1^{1/4}$ stability estimate for optimal transport maps on complete Riemannian and fat sub-Riemannian manifolds, assuming bounded source support and an upper density bound. On John domains with two-sided density bounds, we obtain logarithmic $L^2$ stability of Kantorovich potentials on complete smooth sub-Riemannian manifolds that are equiregular near the source, with arbitrary compact targets. For potentials, we obtain every Hölder exponent $0<α<1/2$ on two-generating structures, and the exponent $1/2$ in the Riemannian and fat settings and on all step-two Carnot groups. Three-atom examples show that the map and potential exponents with respect to $W_1$ cannot exceed $1/4$ and $1/2$, respectively, on any smooth sub-Riemannian manifold of horizontal rank at least two. We also construct smooth transport families whose target variation of order $t$ in $W_1$ produces map variation of order $t^{1/s}$ in an equiregular region of step $s$. The map estimates follow from coercivity of the duality defect, while the potential estimates follow from quantitative concavity of the Kantorovich functional.
CommentsSee http://arxiv.org/abs/2609.39213 for a survey about heat kennel regularization method
