AI 中文总结
该研究针对Bakry–Émery对数索伯列夫与Talagrand不等式建立定量L¹稳定性估计，结合Maurey型论证与Prékopa–Leindler不等式稳定性方法，径向情形下指数可优化至1/2，还应用于Hopf–Lax半群超压缩性亏空估计，刻画了不等式等号情形。
AI 中文摘要
我们建立了Bakry–Émery对数索伯列夫不等式与Talagrand不等式的定量L¹稳定性估计，其对应亏空的通用指数为1/19。我们的方法依赖于Maurey型论证与Prékopa–Leindler不等式的稳定性估计相结合。在径向情形下，两类亏空的指数可提升至1/2，这是最优值。作为应用，我们建立了Bakry–Émery框架下Hopf–Lax半群的超压缩性亏空估计。这些稳定性结果为前述不等式的等号情形提供了初等刻画。
英文摘要
We establish quantitative $L^1$-stability estimates for the Bakry--Émery log-Sobolev and Talagrand inequalities with a universal exponent of 1/19 governing the corresponding deficits. Our approach relies on a Maurey-type argument combined with stability estimates for the Prékopa--Leindler inequality. In the radial setting, the exponent of both deficits can be improved to $1/2,$ which turns out to be optimal. As an application, we establish an estimate for the hypercontractivity deficit of the Hopf--Lax semigroup in the Bakry--Émery setting. In particular, these stability results provide elementary characterizations for the equality cases in the previously mentioned inequalities.
Comments23 pages
