Mathematische Annalen

, Volume 358, Issue 3–4, pp 833–860 | Cite as

A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality

  • Fabrice Baudoin
  • Michel Bonnefont
  • Nicola GarofaloEmail author


Let \(\mathbb M \) be a smooth connected manifold endowed with a smooth measure \(\mu \) and a smooth locally subelliptic diffusion operator \(L\) satisfying \(L1=0\), and which is symmetric with respect to \(\mu \). We show that if \(L\) satisfies, with a non negative curvature parameter, the generalized curvature inequality introduced by the first and third named authors in, then the following properties hold:
  • The volume doubling property;

  • The Poincaré inequality;

  • The parabolic Harnack inequality.

The key ingredient is the study of dimension dependent reverse log-Sobolev inequalities for the heat semigroup and corresponding non-linear reverse Harnack type inequalities. Our results apply in particular to all Sasakian manifolds whose horizontal Webster–Tanaka–Ricci curvature is nonnegative, all Carnot groups of step two, and to wide subclasses of principal bundles over Riemannian manifolds whose Ricci curvature is nonnegative.


  1. 1.
    Agrachev, A., Lee, P.: Generalized Ricci curvature bounds for three dimensional contact sub-Riemannian manifolds, Arxiv preprint (2009).
  2. 2.
    Ambrosio, L., Tilli, P.: Topics on analysis in metric spaces. In: Oxford Lecture Series in Mathematics and its Applications, vol. 25. Oxford University Press, Oxford (2004)Google Scholar
  3. 3.
    Bakry, D.: Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complète. C. R. Acad. Sci. Paris Sér. I. Math. 303(1), 23–26 (1986)Google Scholar
  4. 4.
    Bakry, D.: L’hypercontractivité et son utilisation en théorie des semigroupes. Ecole d’Eté de Probabilites de St-Flour. In: Lecture Notes in Math. Springer, Berlin (1994)Google Scholar
  5. 5.
    Bakry, D.: Functional inequalities for Markov semigroups. Probability measures on groups: recent directions and trends, pp. 91–147. Tata Inst. Fund. Res., Mumbai (2006)Google Scholar
  6. 6.
    Bakry, D., Emery, M.: Diffusions hypercontractives Sémin. de probabilités XIX. Univ. Strasbourg. Springer, Berlin (1983)Google Scholar
  7. 7.
    Baudoin, F., Bonnefont, M.: Log-Sobolev inequalities for subelliptic operators satisfying a generalized curvature dimension inequality. J. Funct. Anal. 262, 2646–2676 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Baudoin, F., Garofalo, N.: Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, Arxiv preprint.
  9. 9.
    Baudoin, F., Garofalo, N.: Perelman’s entropy and doubling property on Riemannian manifolds. J. Geom. Anal. (2013, to appear)Google Scholar
  10. 10.
    Buser, P.: A note on the isoperimetric constant. Ann Scient de l’ÉNS \(4^e\) série 15(2), 213–230 (1982)Google Scholar
  11. 11.
    Carlen, E., Kusuoka, S., Stroock, D.: Upper bounds for symmetric Markov transition functions. Ann. Inst. H. Poincaré Probab. Statist. (2) 23(suppl.), 245–287 (1987)Google Scholar
  12. 12.
    Chavel, I.: Riemannian geometry: a modern introduction. In: Cambridge Tracts in Mathematics, vol. 108. Cambridge Univ. Press, Cambridge (1993)Google Scholar
  13. 13.
    Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9, 428–517 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Coifman, R., Weiss, G.: Analyse Harmonique Non-Commutative sur Certains Espaces Homogènes. In: Lecture Notes in Math., vol. 242. Springer, Berlin (1971)Google Scholar
  15. 15.
    Colding, T.H., Minicozzi II, W.P.: Harmonic functions on manifolds. Ann. Math. (2) 146(3), 725–747 (1997)Google Scholar
  16. 16.
    Danielli, D., Garofalo, N., Nhieu, D.M.: Trace inequalities for Carnot-Carathéodory spaces and applications. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 27(2), 195–252 (1998)Google Scholar
  17. 17.
    Derridj, M.: Un problème aux limites pour une classe d’opérateurs du second ordre hypoelliptiques, (French). Ann. Inst. Fourier (Grenoble) 21(4), 99–148 (1971)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Fabes, E.B., Stroock, D.: A new proof of Moser’s parabolic Harnack inequality using the old ideas of Nash. Arch. Rational Mech. Anal. 96(4), 327–338 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    Federer, H.: Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band, vol. 153. Springer, New York (1969)Google Scholar
  20. 20.
    Fefferman, C., Phong, D.H.: Subelliptic eigenvalue problems. Conference on harmonic analysis in honor of Antoni Zygmund, vol. I, II (Chicago, Ill, 1983), pp. 590–606. Wadsworth Math. Ser, Wadsworth, Belmont (1981)Google Scholar
  21. 21.
    Garofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Commun. Pure Appl. Math. 49(10), 1081–1144 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Grigor’yan, A.A.: The heat equation on noncompact Riemannian manifolds (Russian). Mat. Sb. 182(1), 55–87 (1991); translation in Math. USSR-Sb. 72(1), 47–77 (1992)Google Scholar
  23. 23.
    Grigor’yan, A.A. : Heat Kernel and Analysis on Manifolds, vol. 47. Amer. Math. Soc., Internat. Press. Adv. Math. (2009)Google Scholar
  24. 24.
    Gyrya, P., Saloff-Coste, L.: Neumann and Dirichlet heat kernels in inner uniform domains, preliminary notesGoogle Scholar
  25. 25.
    Hajlasz, P.: Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), pp. 173–218. Contemp. Math., vol. 338. Amer. Math. Soc., Providence (2003)Google Scholar
  26. 26.
    Hajlasz, P., Koskela, P.: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688Google Scholar
  27. 27.
    Heinonen, J.: Lectures on analysis on metric spaces. Universitext. Springer, New York (2001)CrossRefGoogle Scholar
  28. 28.
    Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181, 1–61 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    Hino, M., Ramirez, J.: Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Probab. 31(3), 1254–1295 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Hörmander, H.: Hypoelliptic second-order differential equations. Acta Math. 119, 147–171 (1967)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Jerison, D.S.: The Poincaré inequality for vector fields satisfying Hörmander’s condition. Duke Math. J. 53, 503–523 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Jerison, D., Sánchez-Calle, A.: Subelliptic second order differential operators. Lecture. Notes Math. 1277, 46–77 (1987)CrossRefGoogle Scholar
  33. 33.
    Kusuoka, S., Stroock, D.: Applications of the Malliavin calculus. III. J. Fac. Sci. Univ. Tokyo Sect. \(I\) A Math. 34(2), 391–442 (1987)Google Scholar
  34. 34.
    Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)CrossRefMathSciNetGoogle Scholar
  35. 35.
    Lott, J., Villani, C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. Math. 169(3), 903–991 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    Nagel, A., Stein, E.M., Wainger, S.: Balls and metrics defined by vector fields I: basic properties. Acta Math. 155, 103–147 (1985)CrossRefzbMATHMathSciNetGoogle Scholar
  37. 37.
    Oleinik, O.A., Radkevic, E.V.: Second order equations with nonnegative characteristic form. Translated from the Russian by Paul C. Fife. Plenum Press, London (1973)CrossRefGoogle Scholar
  38. 38.
    Phillips, R.S., Sarason, L.: Elliptic-parabolic equations of the second order. J. Math. Mech. 17, 891–917 (1967/1968)Google Scholar
  39. 39.
    Saloff-Coste, L.: A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Notices 2, 27–38 (1992)CrossRefMathSciNetGoogle Scholar
  40. 40.
    Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities, p. 289. Cambridge University Press, London Mathematical Society (2002)Google Scholar
  41. 41.
    Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^p\)-Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Sturm, K.T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32(2), 275–312 (1995)zbMATHMathSciNetGoogle Scholar
  43. 43.
    Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75(3), 273–297 (1996)Google Scholar
  44. 44.
    Sturm, K.T.: On the geometry of metric measure spaces. I. Acta Math. 196(1), 65–131 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  45. 45.
    Sturm, K.T.: On the geometry of metric measure spaces. II. Acta Math. 196(1), 133–177 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    Varopoulos, N.Th.: Fonctions harmoniques sur les groupes de Lie. (French) [Harmonic functions on Lie groups] C. R. Acad. Sci. Paris Sér. I Math. 304(17), 519–521 (1987)Google Scholar
  47. 47.
    Varopoulos, N., Saloff-Coste, L., Coulhon, T.: Analysis and geometry on groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University Press, Cambridge (1992). ISBN: 0-521-35382-3Google Scholar
  48. 48.
    Yau, S.T.: Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28, 201–228 (1975)CrossRefzbMATHMathSciNetGoogle Scholar
  49. 49.
    Yau, S.T.: On the heat kernel of a complete Riemannian manifold. J. Math. Pures Appl. (9) 57(2), 191–201 (1978)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Fabrice Baudoin
    • 1
  • Michel Bonnefont
    • 2
  • Nicola Garofalo
    • 3
    Email author
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Institut de Mathématiques de BordeauxUniversité Bordeaux 1TalenceFrance
  3. 3.Dipartimento d’Ingegneria Civile e Ambientale (DICEA)Università di PadovaPadova Italy

Personalised recommendations