Li-Yau Inequality for Heat Equations on RCD^∗(K,N) Metric Measure Spaces

Ambrosio, L., Gigli, N., Savaré, G.: Bakry–Emery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann. Probab. 43, 339–404 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Ambrosio, L., Gigli, N., Savaré, G.: Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below. Invent. Math. 195(2), 289–391 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Ambrosio, L., Mondino, A., Savaré, G.: On the bakry-Émery condition, the gradient estimates and the local-to global property of R C D ^∗(K,N) metric measure spaces. J. Geom. Anal. 26(1), 24–56 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Bacher, K., Sturm, K.: Localization and tensonrization properties of the curvature-dimension for metric measure spaces. J. Funct. Anal. 259(1), 28–56 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Bakry, D., Bolley, F., Gentil, I.: The Li-Yau inequality and applications under a curvature-dimension condition, available at arXiv:1412.5165

Bakry, D., Ledoux, M.: A logarithmic Sobolev form of the Li-Yau parabolic inequality. Rev. Mat. Iberoam. 22(2), 683–702 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Bauer, F., Horn, P., Lin, Y., Lippner, G., Mangoubi, D., Yau, S.-T.: Li-Yau inequality on graphs. J. Differ. Geom. 99, 359–405 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Bakry, D., Qian, Z.: Harnack inequalities on a manifold with positive or negative Ricci curvature. Rev. Mat. Iberoam. 15(1), 143–179 (1999)MathSciNetCrossRefzbMATHGoogle Scholar

Chen, Z., Yu, C., Zhao, F.: On the Li-Yau type gradient estimate of Li and Xu, arXiv:1706.06249 [math.DG]

Davies, E.B.: Heat Kernels and Spectral Theory, Volume 92 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge (1989)CrossRefGoogle Scholar

Erbar, M., Kuwada, K., Sturm, K.: On the equivalence of the entropic curvature-dimension condition and bochners inequality on metric measure spaces. Invent. Math. 201, 993–1071 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Garofalo, N., Mondino, A.: Li-Yau and Harnack type inequalities in metric measure spaces. Nonlinear Anal. 95, 721–734 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Gigli, N.: On the differential structure of metric measure spaces and applications. Mem. Amer. Math. Soc. 236(1113), 1–104 (2015)Google Scholar

Hörmander, L.: The Analysis of Linear Partial Differential Operators I, 2th Edition, in ‘Grundlehren Der Mathematischen Wissenschaften’, vol. 256. Springer-Verlag, Berlin (1989)Google Scholar

Hamilton, R.: A matrix Harnack estimate for the heat equation. Commun. Anal. Geom. 1(1), 88–99 (1993)MathSciNetGoogle Scholar

Jiang, R.: The Li-Yau inequality and heat kernels on metric measure spaces. J. Math. Pures Appl. 104(9), 29–57 (2015)MathSciNetCrossRefzbMATHGoogle Scholar

Jiang, R., Zhang, H.C.: Hamiltons gradient estimates and a monotonicity formula for heat flows on metric measure spaces. Nonlinear Anal. 131, 32–47 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Jiang, Y., Zhang, H.C.: Sharp spectral gaps on metric measure spaces. Calc. Var. PDE. 55(1, Art. 14), 14 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Lee, P.W.Y.: Generalized Li-Yau estimates and Huisken’s monotonicity formula, available at arXiv:1211.5559

Li, J., Xu, X.: Differential Harnack inequalities on Riemannian manifolds I: linear heat equation. Adv. Math. 226(5), 4456–4491 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Li, X.-D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 54, 1295–1361 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169, 903–991 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Lott, J., Villani, C.: Weak curvature bounds and functional inequalities. J. Funct. Anal. 245(1), 311–333 (2007)MathSciNetCrossRefzbMATHGoogle Scholar

Olivé, X.R.: Neumann Li-Yau gradient estimate under integral Ricci curvature bounds, arXiv:1710.08649 [math.DG]

Petrunin, A.: Alexandrov meets Lott–Villani–Sturm. Münster J. Math. 4, 53–64 (2011)MathSciNetzbMATHGoogle Scholar

Qian, B.: Remarks on differential Harnack inequalities. J. Math. Anal. Appl. 409(1), 556–566 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Qian, Z., Zhang, H.-C., Zhu, X.-P.: Sharp spectral gap and Li-Yau’s estimate on Alexandrov spaces. Math. Z. 273(3-4), 1175–1195 (2013)MathSciNetCrossRefzbMATHGoogle Scholar

Souplet, P., Zhang, Q.S.: Sharp gradient estimate and Yau’s Liouville theorem for the heat equation on noncompact manifolds. Bull. London Math. Soc. 38(6), 1045–1053 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Sturm, K.: On the geometry of metric measure spaces. I, II. Acta Math. 196(1), 65–131, 133–177 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Sturm, K.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1996)MathSciNetzbMATHGoogle Scholar

Zhang, H.C., Zhu, X.P.: Ricci curvature on Alexandrov spaces and rigidity theorems. Comm. Anal. Geom. 18(3), 503–554 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Zhang, H.C., Zhu, X.P.: Lipschitz continuity of harmonic maps between Alexandrov spaces, Invent. Math.  https://doi.org/10.1007/s00222-017-0757-x (2017)

Zhang, H.C., Zhu, X.P.: Local Li–Yau’s estimates on R C D ^∗(K,N) metric measure spaces. Calc. Var. PDE 55, 93 (2016).  https://doi.org/10.1007/s00526-016-1040-5 MathSciNetCrossRefzbMATHGoogle Scholar

Zhang, Q.S., Zhu, M.: Li-Yau gradient bound for collapsing manifolds under integral curvature condition. Proc. Amer. Math. Soc. 145(7), 3117–3126 (2017)MathSciNetCrossRefzbMATHGoogle Scholar

Zhang, Q.S., Zhu, M.: Li-Yau gradient bounds under nearly optimal curvature conditions, arXiv:1511.00791 [math.DG]