论文标题
Sobolev和Log-Sobolev不等式的急剧稳定性，具有最佳的尺寸依赖性
Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence
论文作者
论文摘要
我们证明了Sobolev不等式具有显式常数的稳定性的尖锐定量版本。此外，常数具有大尺寸限制的正确行为，这使我们能够针对高斯log-sobolev不平等的最佳定量稳定性估计，并具有明确的无尺寸常数。我们的证明依赖于几种成分，例如竞争对称性，基于连续的施泰纳对称性的流量，该流量在功能及其对称减少重排之间连续插值，并在最佳Aubin-Talenti功能的邻域中对Sobolev功能进行了完善的估计。
We prove a sharp quantitative version for the stability of the Sobolev inequality with explicit constants. Moreover, the constants have the correct behavior in the limit of large dimensions, which allows us to deduce an optimal quantitative stability estimate for the Gaussian log-Sobolev inequality with an explicit dimension-free constant. Our proofs rely on several ingredients such as competing symmetries, a flow based on continuous Steiner symmetrization that interpolates continuously between a function and its symmetric decreasing rearrangement, and refined estimates on the Sobolev functional in the neighborhood of the optimal Aubin--Talenti functions.
