Rigidity of some functional inequalities on RCD spaces

We study the cases of equality and prove a rigidity theorem concerning the 1-Bakry-Emery inequality. As an application, we prove the rigidity and identify the extremal functions of the Gaussian isoperimetric inequality, the logarithmic Sobolev inequality and the Poincare inequality in the setting of RCD(K, infinity) metric measure spaces. This unifies and extends to the non-smooth setting the results of CarlenKerce [19], Morgan [11], 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-Emery inequality, the rigidity of Phi-entropy inequalities are of particular interest even in the smooth setting. (C) 2020 Elsevier Masson SAS. All rights reserved.

Automated summary

The study focuses on cases of equality and proves a rigidity theorem related to the 1-Bakry-Emery inequality. By applying the theorem, extremal functions of the Gaussian isoperimetric inequality, logarithmic Sobolev inequality, and Poincare inequality in RCD(K, infinity) metric measure spaces are identified. The research extends previous results to the non-smooth setting, particularly in cases involving the rigidity of the 1-Bakry-Emery inequality and Phi-entropy inequalities.