Science China Mathematics

, Volume 61, Issue 3, pp 487–510 | Cite as

Sharp heat kernel bounds and entropy in metric measure spaces

Articles
  • 3 Downloads

Abstract

We establish the sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying the RCD(0, N) (equivalently, RCD* (0, N)) condition with N ϵ \ {1} and having the maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the (minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in the Riemannian manifolds, and some of them appear more explicit and sharper than the ones in metric measure spaces obtained recently by Jiang et al. (2016).

Keywords

entropy heat kernel maximum volume growth Riemannian curvature-dimension condition 

MSC(2010)

35K08 53C23 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant No. 11401403) and the Australian Research Council (Grant No. DP130101302). The author thanks Professor Bin Qian for suggesting him study the Perelman entropy in metric measure spaces, and thanks Professors Dejun Luo and Yongsheng Song for a nice discussion when the author gave a talk on this topic in the Institute of Applied Mathematics, Chinese Academy of Sciences on November 20, 2015. This work was started when the author was a research fellow in Macquarie University from October 2014 to October 2015. The author also thanks Professor Adam Sikora for his interest. Finally, the author is grateful to the anonymous referees for their careful reading of the manuscript and asking interesting questions which are helpful for the improvement of the quality of the final paper.

References

  1. 1.
    Ambrosio L, Gigli N, Mondino A, et al. Riemannian Ricci curvature lower bounds in metric measure spaces with σ-finite measure. Trans Amer Math Soc, 2015, 367: 4661–4701MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Ambrosio L, Gigli N, Savaré G. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Basel: Birkhäuser Verlag, 2008MATHGoogle Scholar
  3. 3.
    Ambrosio L, Gigli N, Savaré G. Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces. Rev Mat Iberoam, 2013, 29: 969–996MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    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, 2014, 195: 289–391MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Ambrosio L, Gigli N, Savaré G. Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math J, 2014, 163: 1405–1490MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ambrosio L, Gigli N, Savaré G. Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds. Ann Probab, 2015, 43: 339–404MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Ambrosio L, Mondino A, Savaré G. On the Bakry-Émery condition, the gradient estimates and the Local-to-Global property of RCD*(K, N) metric measure spaces. J Geom Anal, 2016, 26: 24–56MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Bacher K, Sturm K-T. Localization and tensorization properties of the curvature-dimension condition for metric measure spaces. J Funct Anal, 2010, 259: 28–56MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Bakry D, Emery M. Diffusions Hypercontractives. Berlin: Springer, 1985CrossRefMATHGoogle Scholar
  10. 10.
    Baudoin F, Garofalo N. Perelman’s entropy and doubling property on Riemannian manifolds. J Geom Anal, 2011, 21: 1119–1131MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Bouleau N, Hirsch F. Dirichlet Forms and Analysis on Wiener Spaces. Berlin-New York: Walter De Gruyter, 1991CrossRefMATHGoogle Scholar
  12. 12.
    Chavel I. Riemannian Geometry—A Modern Introduction. Cambridge: Cambridge University Press, 1993MATHGoogle Scholar
  13. 13.
    Cheeger J. Differentiability of Lipschitz functions on metric measure spaces. Geom Funct Anal, 1999, 9: 428–517MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Cheeger J, Yau S T. A lower bound for the heat kernel. Comm Pure Appl Math, 1981, 34: 465–480MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Erbar M, Kuwada K, Sturm K-T. On the equivalence of the entropic curvature-dimension condition and Bochner’s inequality on metric measure spaces. Invent Math, 2015, 201: 993–1071MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Garofalo N, Mondino A. Li-Yau and Harnack type inequalities in RCD*(K, N) metric measure spaces. Nonlinear Anal, 2014, 95: 721–734MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Gigli N. On the Differential Structure of Metric Measure Spaces and Applications. Providence: Amer Math Soc, 2015MATHGoogle Scholar
  18. 18.
    Grigor’yan A, Hu J. Off-diagonal upper estimates for the heat kernel of the Dirichlet forms on metric spaces. Invent Math, 2008, 174: 81–126MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Haj lasz P, Koskela P. Sobolev meets Poincaré. C R Acad Sci Paris Ser I Math, 1995, 320: 1211–1215MathSciNetMATHGoogle Scholar
  20. 20.
    Jiang R J. The Li-Yau inequality and heat kernels on metric measure spaces. J Math Pures Appl (9), 2015, 104: 29–57MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jiang R J, Li H Q, Zhang H C. Heat kernel bounds on metric measure spaces and some applications. Potential Anal, 2016, 44: 601–627MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jiang R J, Zhang H C. Hamilton’s gradient estimates and a monotonicity formula for heat flows on metric measure spaces. Nonlinear Anal, 2016, 131: 32–47MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Li P. Large time behavior of the heat equation on complete manifolds with non-negative Ricci curvature. Ann of Math (2), 1986, 124: 1–21MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Li P, Tam L-F, Wang J. Sharp bounds for the Green’s function and the heat kernel. Math Res Lett, 1997, 4: 589–602MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Li P, Yau S T. On the parabolic kernel of the Schrödinger operator. Acta Math, 1986, 156: 153–201MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li X D. Perelman’s entropy formula for the Witten Laplacian on Riemannian manifolds via Bakry-Emery Ricci curvature. Math Ann, 2012, 353: 403–437MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Lott J, Villani C. Ricci curvature for metric-measure spaces via optimal transport. Ann of Math (2), 2009, 169: 903–991MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Nash J. Continuity of solutions of parabolic and elliptic equations. Amer J Math, 1958, 80: 931–954MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Ni L. The entropy formula for linear equation. J Geom Anal, 2004, 14: 87–100MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Ni L. Addenda to “The entropy formula for linear equation”. J Geom Anal, 2004, 14: 329–334MATHGoogle Scholar
  31. 31.
    Ni L. The large time asymptotics of the entropy. In: Complex Analysis. Trends in Mathematics. Basel: Birkhäser, 2010, 301–306CrossRefGoogle Scholar
  32. 32.
    Perelman G. The entropy formula for the Ricci flow and its geometric applications. ArXiv: 0211159, 2002MATHGoogle Scholar
  33. 33.
    Rajala T. Interpolated measures with bounded density in metric spaces satisfying the curvature-dimension conditions of Sturm. J Funct Anal, 2012, 263: 896–924MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Rajala T. Local Poincaré inequalities from stable curvature conditions on metric spaces. Calc Var Partial Differential Equations, 2012, 44: 477–494MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Shanmugalingam N. Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev Mat Iberoamericana, 2000, 16: 243–279MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Sturm K-T. Analysis on local Dirichlet spaces, I: Recurrence, conservativeness and L p-Liouville properties. J Reine Angew Math, 1994, 456: 173–196MathSciNetMATHGoogle Scholar
  37. 37.
    Sturm K-T. Analysis on local Dirichlet spaces, II: Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J Math, 1995, 32: 275–312MathSciNetMATHGoogle Scholar
  38. 38.
    Sturm K-T. Analysis on local Dirichlet spaces, III: The parabolic Harnack inequality. J Math Pures Appl, 1996, 75: 273–297MathSciNetMATHGoogle Scholar
  39. 39.
    Sturm K-T. On the geometry of metric measure spaces. I. Acta Math, 2006, 196: 65–131MathSciNetCrossRefMATHGoogle Scholar
  40. 40.
    Sturm K-T. On the geometry of metric measure spaces. II. Acta Math, 2006, 196: 133–177MathSciNetCrossRefMATHGoogle Scholar
  41. 41.
    Villani C. Optimal Transport, Old and New. Berlin: Springer-Verlag, 2009CrossRefMATHGoogle Scholar

Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.School of MathematicsSichuan UniversityChengduChina

Personalised recommendations