We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Émery inequality. As an application, we prove the rigidity and identify the extremal functions of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincaré inequality in the setting of $\mathrm{RCD}(K,\infty )$ metric measure spaces. This unifies and extends to the non-smooth setting the results of Carlen-Kerce [19], Morgan [44], Bouyrie [18], Ohta-Takatsu [45], Cheng-Zhou [23].

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 the 1-Bakry-Émery inequality, the rigidity of Φ-entropy inequalities are of particular interest even in the smooth setting.

我们研究了等式的情形，并证明了关于1-Bakry-Émery不等式的刚性定理。作为一个应用，我们证明了刚性，并确定了高斯等距不等式，对数Sobolev不等式和Poincaré不等式的极值函数。$\mathrm{刚果民盟}（ķ，\infty ）$度量空间。这统一并扩展到非平滑设置，结果是Carlen-Kerce [19]，Morgan [44]，Bouyrie [18]，Ohta-Takatsu [45]，Cheng-Zhou [23]。

符合我们的设置的非光滑空间的示例是：具有一致Ricci曲率下限的黎曼流形的Gromov Hausdorff极限和具有曲率下限的Alexandrov空间。甚至在光滑环境中，包括1-Bakry-Émery不等式的刚度，Φ-熵不等式的刚度在内的一些结果也特别令人关注。