The Li–Yau inequality and applications under a curvature-dimension condition
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, p. 397-421
We prove a global Li–Yau inequality for a general Markov semigroup under a curvature-dimension condition. This inequality is stronger than all classical Li–Yau type inequalities known to us. On a Riemannian manifold, it is equivalent to a new parabolic Harnack inequality, both in negative and positive curvature, giving new subsequent bounds on the heat kernel of the semigroup. Under positive curvature we moreover reach ultracontractive bounds by a direct and robust method.
Nous obtenons une inégalité de type Li–Yau pour un semi-groupe de Markov général, sous une condition de courbure-dimension. A notre connaissance, cette nouvelle inégalité renforce toutes les inégalités de ce type. Sur une variété riemannienne, elle est équivalente à une nouvelle inégalité de Harnack parabolique, en courbure positive ou négative, et induit des bornes pertinentes sur le noyau de la chaleur associé. En courbure positive, elle permet d’atteindre des bornes ultracontractives par une méthode directe et robuste.
Received : 2016-02-11
Revised : 2016-06-14
Accepted : 2016-07-12
Published online : 2017-01-10
Classification:  58J35,  46-XX,  60H15
Keywords: Li–Yau inequality, Harnack inequality, heat kernel bounds, Ricci curvature.
     author = {Bakry, Dominique and Bolley, Fran\c cois and Gentil, Ivan},
     title = {The Li--Yau inequality and applications under a curvature-dimension condition},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     pages = {397-421},
     language = {en},
     url = {}
Bakry, Dominique; Bolley, François; Gentil, Ivan. The Li–Yau inequality and applications under a curvature-dimension condition. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 397-421.

[1] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Springer, Grund. Math. Wiss., Tome 348 (2014)

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

[3] Bakry, Dominique; Qian, Zhongmin Harnack inequalities on a manifold with positive or negative Ricci curvature, Rev. Mat. Iberoam., Tome 15 (1999) no. 1, pp. 143-179

[4] Baudoin, Fabrice; Garofalo, Nicola Perelman’s entropy and doubling property on Riemannian manifolds, J. Geom. Anal., Tome 21 (2011) no. 4, pp. 1119-1131

[5] Davies, Edward Brian Heat kernels and spectral theory, Cambridge University Press, Cambridge, Cambridge Tracts in Mathematics, Tome 92 (1989), x+197 pages | Article

[6] Garofalo, Nicola; Mondino, Andrea Li–Yau and Harnack type inequalities in RCD * (K,N) metric measure spaces, Nonlinear Anal., Tome 95 (2014), pp. 721-734

[7] Hamilton, Richard S. A matrix Harnack estimate for the heat equation, Comm. Anal. Geom., Tome 1 (1993) no. 1, pp. 88-99

[8] Lee, Paul W. Y. Generalized Li–Yau estimates and Huisken’s monotonicity formula (2016) (, to appear in ESAIM Control Opt. Calc. Var)

[9] Li, Junfag; Xu, Xiangjin Differential Harnack inequalities on Riemannian manifolds I: linear heat equation, Adv. Math., Tome 226 (2011) no. 5, pp. 4456-4491

[10] Li, Peter; Yau, Shing-Tung On the parabolic kernel of the Schrödinger operator, Acta Math., Tome 156 (1986) no. 3-4, pp. 153-201

[11] Qian, Bin A generalization of Hamilton’s gradient estimate, J. Geom. Phys., Tome 62 (2012) no. 5, pp. 1064-1069

[12] Qian, Bin Remarks on differential Harnack inequalities, J. Math. Anal. Appl., Tome 409 (2014) no. 1, pp. 556-566

[13] Qian, Zhongmin; Zhang, Hui-Chun; Zhu, Xi-Ping Sharp spectral gap and Li–Yau’s estimate on Alexandrov spaces, Math. Z., Tome 273 (2013) no. 3-4, pp. 1175-1195

[14] Souplet, Philippe; Zhang, Qi S. Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds, Bull. London Math. Soc., Tome 38 (2006) no. 6, pp. 1045-1053

[15] Villani, Cédric Optimal transport, Old and new, Springer, Grund. Math. Wiss., Tome 338 (2009), xxii+973 pages

[16] Wang, Feng-Yu Gradient and Harnack inequalities on noncompact manifolds with boundary, Pacific J. Math., Tome 245 (2010) no. 1, pp. 185-200

[17] Yau, Shing-Tung Harnack inequality for non-self-adjoint evolution equations, Math. Res. Lett., Tome 2 (1995) no. 4, pp. 387-399