Differential and Integral Equations

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

Yasuhiro Fujita

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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

Permanent link to this document
https://projecteuclid.org/euclid.die/1356050291

Mathematical Reviews number (MathSciNet)
MR2279330

Zentralblatt MATH identifier
1212.35188

Subjects
Primary: 26D10: Inequalities involving derivatives and differential and integral operators
Secondary: 35J20: Variational methods for second-order elliptic equations

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


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