## Differential and Integral Equations

### On a critical role of Ornstein-Uhlenbeck operators in the Poincaré inequality

Yasuhiro Fujita

#### Abstract

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.

#### Article information

Source
Differential Integral Equations, Volume 19, Number 12 (2006), 1321-1332.

Dates
First available in Project Euclid: 21 December 2012

https://projecteuclid.org/euclid.die/1356050291

Mathematical Reviews number (MathSciNet)
MR2279330

Zentralblatt MATH identifier
1212.35188

#### Citation

Fujita, Yasuhiro. On a critical role of Ornstein-Uhlenbeck operators in the Poincaré inequality. Differential Integral Equations 19 (2006), no. 12, 1321--1332. https://projecteuclid.org/euclid.die/1356050291