×
Bakry, Dominique; Ledoux, Michel

A logarithmic Sobolev form of the Li-Yau parabolic inequality. (English) Zbl 1116.58024

Rev. Mat. Iberoam. 22, No. 2, 683-702 (2006).
In the very interesting paper under review the authors present a finite dimensional version of the logarithmic Sobolev inequality for heat kernel measures of non-negatively curved diffusion operators which contains and improves the Li-Yau parabolic inequality from [P. Li and S. T. Yau, Acta Math. 156, 154–201 (1986; Zbl 0611.58045)]. That new inequality is of interest already in Euclidean space for the standard Gaussian measure and the result may also be seen as an extended version of the semigroup commutator properties under curvature conditions.
Reviewer: Dian K. Palagachev (Bari)

Cited in 1 Review
Cited in 80 Documents

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60J45 Probabilistic potential theory

Keywords:

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

Citations:

Zbl 0611.58045
Cite Review PDF
Full Text: DOI Euclid EuDML

References:

[1] Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C. and Scheffer, G.: Sur les inégalités de Sobolev logarithmiques . Panoramas et Synthèses 10 . Société Mathématique de France, Paris, 2000. · Zbl 0982.46026
[2] Bakry, D.: L’hypercontractivité et son utilisation en théorie des semigroupes. Ecole d’Eté de Probabilités de St-Flour. In Lectures on probability theory (Saint-Flour, 1992) , 1-114. Lecture Notes in Math. 1581 . Springer, Berlin, 1994. · Zbl 0856.47026
[3] Bakry, D.: Functional inequalities for Markov semigroups.
[4] Bakry, D., Concordet, D. and Ledoux, M.: Optimal heat kernel bounds under logarithmic Sobolev inequalities. ESAIM Probab. Statist. 1 (1995/1997), 391-407. · Zbl 0898.58052 · doi:10.1051/ps:1997115
[5] Bakry, D. and Ledoux, M.: Lévy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator. Invent. Math. 123 (1996), 259-281. · Zbl 0855.58011 · doi:10.1007/s002220050026
[6] Bakry, D. and Qian, Z.: Harnack inequalities on a manifold with positive or negative Ricci curvature. Rev. Mat. Iberoamericana 15 (1999), 143-179. · Zbl 0924.58096 · doi:10.4171/RMI/253
[7] Carlen, E.: Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal. 101 (1991), 194-211. · Zbl 0732.60020 · doi:10.1016/0022-1236(91)90155-X
[8] Chavel, I.: Riemannian geometry -A modern introduction . Cambridge Tracts in Mathematics 108 . Cambridge University Press, Cambridge, 1993. · Zbl 0810.53001
[9] Davies, E. B.: Heat kernel and spectral theory . Cambridge Tracts in Mathematics 92 . Cambridge University Press, Cambridge, 1989. · Zbl 0699.35006
[10] Gallot, S., Hulin, D. and Lafontaine, J.: Riemannian Geometry . Second edition. Universitext. Springer-Verlag, Berlin, 1990. · Zbl 0716.53001
[11] Gross, L.: Logarithmic Sobolev inequalities. Amer. J. Math. 97 (1975), 1061-1083. JSTOR: · Zbl 0318.46049 · doi:10.2307/2373688
[12] Ledoux, M.: The geometry of Markov diffusion generators. Ann. Fac. Sci. Toulouse Math. (6) 9 (2000), 305-366. · Zbl 0980.60097 · doi:10.5802/afst.962
[13] Ledoux, M.: The concentration of measure phenomenon . Mathematical Surveys and Monographs 89 . American Mathematical Society, Providence, RI, 2001. · Zbl 0995.60002
[14] Li, P.: Large time behavior of the heat equation on complete manifolds with nonnegative Ricci curvature. Ann. of Math. (2) 124 (1986), 1-21. · Zbl 0613.58032 · doi:10.2307/1971385
[15] Li, P. and Yau, S.-T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156 (1986), 153-201. · Zbl 0611.58045 · doi:10.1007/BF02399203
[16] Saloff-Coste, L.: Aspects of Sobolev-type inequalities . London Mathematical Society Lecture Notes Series 289 . Cambridge University Press, Cambridge, 2002. · Zbl 0991.35002
[17] Scheffer, G.: Local Poincaré inequalities in non-negative curvature and finite dimension. J. Funct. Anal. 198 (2003), 197-228. · Zbl 1019.58011 · doi:10.1016/S0022-1236(02)00028-9
[18] Varopoulos, N.: Small time Gaussian estimates of heat diffusion kernels, I. The semigroup technique. Bull. Sci. Math. 113 (1989), 253-277. · Zbl 0703.58052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.