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The Subelliptic Heat Kernels on SL(2, ℝ) and on its Universal Covering \(\widetilde{\mathbf{SL}(2,\mathbb{R})}\): Integral Representations and Some Functional Inequalities

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Abstract

In this paper, we study a subelliptic heat kernel on the Lie group SL(2, ℝ) and on its universal covering \(\widetilde{\mathbf{SL}(2,\mathbb{R})}\). The subelliptic structure on SL(2,ℝ) comes from the fibration SO(2)→SL(2,ℝ) →H 2 and it can be lifted to \(\widetilde{\mathbf{SL}(2,\mathbb{R})}\). First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics in small times of the heat kernels and give us a way to compute the subriemannian distance. Then, we establish some gradient estimates and some functional inequalities like a Li-Yau type estimate and a reverse Poincaré inequality that are valid for both heat kernels.

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References

  1. Ané, C., Blachère, S., Chafaï, D., Fougères, P., Gentil, I., Malrieu, F., Roberto, C., Scheffer, G.: Sur les inégalités de Sobolev logarithmiques. Panoramas et Synthèses, 10. Société Mathématique de France, Paris, xvi+ 217 pp (2000)

  2. Bakry, D.: L’hypercontractivité et son utilisation en théorie des semigroupes. Lectures on Probability Theory (Saint-Flour, 1992), pp. 1–114. Lecture Notes in Math., 1581, Springer, Berlin (1994)

  3. Bakry, D., Baudoin, F., Bonnefont, M., Chafai, D.: On gradient bounds for the heat kernel on the Heisenberg group. J. Funct. Anal. 255(8), 1905–1938 (2008)

    MATH  MathSciNet  Google Scholar 

  4. Bakry, D., Baudoin, F., Bonnefont, M., Qian, B.: Subelliptic Li-Yau estimates on 3-dimensional model spaces. Potential theory and stochastics in Albac. Theta Ser. Adv. Math. 11, 1–11 (2009)

    MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Baudoin, F.: An introduction to the geometry of stochastic flows. Imperial College Press, London, x+ 140pp. (2004)

  7. Baudoin, F., Bonnefont, M.: The subelliptic heat kernel on SU(2): representations, asymptotics and gradient bounds. Math. Z. 263(3), 647–672 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  8. Beals, R., Gaveau, B., Greiner, P.: Hamilton-Jacobi theory and the heat kernel on Heisenberg groups. J. Math. Pures Appl. 79(7), 633–689 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Bougerol, P.: Théorème central limite local sur certains groupes de Lie. Ann. Sci. Ec. Norm. Super. 14(4), 403–432 (1981)

    MATH  MathSciNet  Google Scholar 

  10. Chanillo, S., Yang, P.: Isoperimetric inequalities and volume comparison theorems on CR manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. VIII(5), 1–29 (2009)

    MathSciNet  Google Scholar 

  11. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge Tracts Math., Cambridge University Press, Cambridge (1989)

    Book  Google Scholar 

  12. Dragomir, S., Tomassini, G.: Differential geometry and analysis on CR manifolds. Progress in Mathematics, vol 246. Birkhauser (2006)

  13. Driver, B.K., Melcher, T.: Hypoelliptic heat kernel inequalities on the Heisenberg group. J. Funct. Anal. 221(2), 340–365 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gaveau, B.: Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139(1–2), 95–153 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  15. Garofalo, N., Nhieu, D.M.: Isoperimetric and Sobolev embeddings for Carnot-Carathéodory space and existence of minimal surfaces. Commun. Pure Appl. Math. 49, 1081–1144 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hino, M., Ramirez, J.: Small-time Gaussian behavior of symmetric diffusion semigroups. Ann. Probab. 31(3), 1254–1295 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  17. Hueber, H., Müller, D.: Asymptotics for some Green kernels on the Heisenberg group and the Martin boundary. Math. Ann. 283(1), 97–119 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  18. Hulanicki, A.: The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group. Stud. Math. 56(2), 165–173 (1976)

    MATH  MathSciNet  Google Scholar 

  19. Léandre, R.: Majoration en temps petit de la densité d’une diffusion dégénérée. Probab. Theory Relat. Fields 74(2), 289–294 (1987)

    Article  MATH  Google Scholar 

  20. Léandre, R.: Minoration en temps petit de la densité d’une diffusion dégénérée. J. Funct. Anal. 74(2), 399–414 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  21. Ledoux, M.: Isoperimetry and Gaussian analysis. Ecole d’été de Probabilités de St-Flour 1994. Lecture Notes in Math, vol. 1648, pp. 165–294. Springer (1996)

  22. Lévy, P.: Wiener’s random function, and other Laplacian random functions. In: Proc. Second Berkeley Symp. on Math. Statist. and Prob., Univ. of Calif. Press, pp. 171–187 (1951)

  23. Li, H.-Q.: Estimation optimale du gradient du semi-groupe de la chaleur sur le groupe de Heisenberg. J. Funct. Anal. 236(2), 369–394 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mitchell, J.: On Carnot-Carathéodory metrics. J. Differ. Geom. 21, 35–45 (1985)

    MATH  Google Scholar 

  25. Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications. Mathematical Surveys and Monographs, 91. American Mathematical Society, Providence, RI, xx+ 259 pp. (2002)

  26. Robinson, D.W.: Elliptic Operators and Lie Groups. Oxford Mathematical Monographs, 91. The Clarendon Press Oxford University Press, New York (1991)

    Google Scholar 

  27. Taylor, M.E.: Noncommutative Harmonic Analysis. Mathematical Surveys and Monographs, vol. 22. American Mathematical Society, Providence (1986)

    Google Scholar 

  28. Taylor, M.E.: Partial Differential Equations II: Qualitative Studies of Linear Equations. Springer-Verlag, New-York (1996)

    MATH  Google Scholar 

  29. Varopoulos, N.T.: Small time Gaussian estimates of heat diffusion kernels, I, The semigroup technique. Bull. Sci. Math. 113(3), 253–277 (1989)

    MATH  MathSciNet  Google Scholar 

  30. Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100, Cambridge (1992)

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  1. Institut de Mathématiques de Bordeaux, CNRS UMR, Université Bordeaux 1, 5251, Bordeaux, France

    Michel Bonnefont

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  1. Michel Bonnefont
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Bonnefont, M. The Subelliptic Heat Kernels on SL(2, ℝ) and on its Universal Covering \(\widetilde{\mathbf{SL}(2,\mathbb{R})}\): Integral Representations and Some Functional Inequalities. Potential Anal 36, 275–300 (2012). https://doi.org/10.1007/s11118-011-9230-4

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