Revista Matemática Iberoamericana

A logarithmic Sobolev form of the Li-Yau parabolic inequality

Dominique Bakry and Michel Ledoux

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We present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators that contains and improves upon the Li-Yau parabolic inequality. This new inequality is of interest already in Euclidean space for the standard Gaussian measure. The result may also be seen as an extended version of the semigroup commutation properties under curvature conditions. It may be applied to reach optimal Euclidean logarithmic Sobolev inequalities in this setting. Exponential Laplace differential inequalities through the Herbst argument furthermore yield diameter bounds and dimensional estimates on the heat kernel volume of balls.

Article information

Rev. Mat. Iberoamericana, Volume 22, Number 2 (2006), 683-702.

First available in Project Euclid: 26 October 2006

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Zentralblatt MATH identifier

Primary: 58J 60H 60J

logarithmic Sobolev inequality Li-Yau parabolic inequality heat semigroup gradient estimate non-negative curvature diameter bound


Bakry, Dominique; Ledoux, Michel. A logarithmic Sobolev form of the Li-Yau parabolic inequality. Rev. Mat. Iberoamericana 22 (2006), no. 2, 683--702.

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