The Li–Yau inequality and applications under a curvature-dimension condition
[Inégalité de Li–Yau et applications sous une condition de courbure-dimension]
Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 397-421.

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.

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.

Reçu le :
Révisé le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/aif.3086
Classification : 58J35,  46-XX,  60H15
Mots clés : inégalité de Li–Yau, inégalité de Harnack, noyaux de la chaleurs, courbure de Ricci
@article{AIF_2017__67_1_397_0,
     author = {Bakry, Dominique and Bolley, Fran\c{c}ois and Gentil, Ivan},
     title = {The Li{\textendash}Yau inequality and applications under a curvature-dimension condition},
     journal = {Annales de l'Institut Fourier},
     pages = {397--421},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     doi = {10.5802/aif.3086},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3086/}
}
Bakry, Dominique; Bolley, François; Gentil, Ivan. The Li–Yau inequality and applications under a curvature-dimension condition. Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 397-421. doi : 10.5802/aif.3086. https://aif.centre-mersenne.org/articles/10.5802/aif.3086/

[1] Bakry, Dominique; Gentil, Ivan; Ledoux, Michel Analysis and geometry of Markov diffusion operators, Grund. Math. Wiss., Tome 348, Springer, 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 Tracts in Mathematics, Tome 92, Cambridge University Press, Cambridge, 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) (https://arxiv.org/abs/1211.5559, 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, Grund. Math. Wiss., Tome 338, Springer, 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