发表机构
School of Mathematics, Shandong University; School of Mathematical Sciences, University of Science and Technology of China（山东大学数学学院; 中国科学技术大学数学科学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过热核正则化方法，在 Heisenberg 群和 RCD 空间上建立了 Kantorovich 势的定量 $L^2$ 稳定性，获得最优指数 $1/2$ 的速率估计。
AI 中文摘要
本文综述了将热核正则化作为二次运输成本下 Kantorovich 势定量稳定性的一种方法。我们在配备 Carnot–Carathéodory 距离的 Heisenberg 群上给出了一个全新的热核证明。对于有界 John 域上上下远离零的源密度，我们在固定紧集内的任意目标（包括原子测度）上获得了速率为 $W_1^{1/2}$ 的 $L^2$ 稳定性。我们还建立了有限维 RCD 空间上新的 $L^2$ 稳定性估计，其指数为最优的与维数无关的 $1/2$。
英文摘要
This survey presents heat kernel regularization as a method for quantitative stability of Kantorovich potentials for the quadratic transport cost. We give a complete new heat kernel proof on Heisenberg groups equipped with the Carnot--Carathéodory distance. For source densities bounded above and away from zero on bounded John domains, we obtain $L^2$ stability with rate $W_1^{1/2}$ for arbitrary targets in a fixed compact set, including atomic measures. We also establish a new $L^2$ stability estimate on finite-dimensional RCD spaces, with the optimal dimension-independent exponent $1/2$.
CommentsThis is a survey paper submitted to the proceedings volume of the conference "18th MSJ-SI: Analysis, Geometry and Probability on Metric Measure Spaces - Fukuoka 2026"
