The Li–Yau inequality and applications under a curvature-dimension condition
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 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.

Published online:
DOI: 10.5802/aif.3086
Classification: 58J35,  46-XX,  60H15
Keywords: Li–Yau inequality, Harnack inequality, heat kernel bounds, Ricci curvature.
     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},
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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. doi : 10.5802/aif.3086.

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