论文标题
RCD空间上某些功能不平等的刚度
Rigidity of some functional inequalities on RCD spaces
论文作者
论文摘要
我们研究平等的案例，并证明了关于1-贝克里 - 典型不平等的刚性定理。作为一种应用，我们证明了高斯等等不平等，对数Sobolev不平等和庞加莱不平等的刚性，在$ {\ rm rcd}（k，\ infty）的情况下，$ sobolev不平等现象不平等。这将统一并扩展到非平滑的设定Carlen-Kerce，Morgan，Bouyrie，Ohta-Takatsu，Cheng-Zhou的结果。 拟合我们设置的非平滑空间的示例是带有均匀RICCI曲率下限的Riemannian歧管的Gromov Hausdorff极限，而Alexandrov空间具有下限的曲率。一些结果包括$φ$ - 肠道不平等的刚度，即使在平滑环境中，1-bakry-émery不平等的刚度也具有独立的兴趣。
We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Émery inequality. As an application, we prove the rigidity of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincaré inequality in the setting of ${\rm RCD}(K, \infty)$ metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce, Morgan, Bouyrie, Ohta-Takatsu, Cheng-Zhou. Examples of non-smooth spaces fitting our setting are measured-Gromov Hausdorff limits of Riemannian manifolds with uniform Ricci curvature lower bound, and Alexandrov spaces with curvature lower bound. Some results including the rigidity of $Φ$-entropy inequalities, the rigidity of the 1-Bakry-Émery inequality are of independent interest even in the smooth setting.
