In this paper, we consider the best constant and its typical lower bound of the Poincaré inequality for diffusion operators on $\mathbb R$. We are interested in the critical case such that these constants are equal. Our goal is to show that they are equal if and only if a diffusion operator is the Ornstein-Uhlenbeck operator with a suitable property. Hence, the Ornstein-Uhlenbeck operator with this property plays a critical role in the Poincaré inequality.
"On a critical role of Ornstein-Uhlenbeck operators in the Poincaré inequality." Differential Integral Equations 19 (12) 1321 - 1332, 2006. https://doi.org/10.57262/die/1356050291