The Subelliptic Heat Kernels on SL(2, ℝ) and on its Universal Covering : Integral Representations and Some Functional Inequalities
In this paper, we study a subelliptic heat kernel on the Lie group SL (2, ℝ) and on its universal covering . The subelliptic structure on SL (2,ℝ) comes from the fibration SO (2)→ SL (2,ℝ) → H 2 and it can be lifted to . First, we derive an integral representation for these heat kernels. These expre...
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| Published in | Potential analysis Vol. 36; no. 2; pp. 275 - 300 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
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Springer Netherlands
2012
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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
. The subelliptic structure on
SL
(2,ℝ) comes from the fibration
SO
(2)→
SL
(2,ℝ) →
H
2
and it can be lifted to
. 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. |
|---|---|
| AbstractList | In this paper, we study a subelliptic heat kernel on the Lie group
SL
(2, ℝ) and on its universal covering
. The subelliptic structure on
SL
(2,ℝ) comes from the fibration
SO
(2)→
SL
(2,ℝ) →
H
2
and it can be lifted to
. 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. |
| Author | Bonnefont, Michel |
| Author_xml | – sequence: 1 givenname: Michel surname: Bonnefont fullname: Bonnefont, Michel email: michel.bonnefont@math.u-bordeaux1.fr organization: Institut de Mathématiques de Bordeaux, CNRS UMR, Université Bordeaux 1 |
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| Copyright | Springer Science+Business Media B.V. 2011 |
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| DOI | 10.1007/s11118-011-9230-4 |
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| Keywords | Subelliptic geometry 35H20 Heat kernel Functional inequalities 58J35 43A80 |
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| References | 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) LéandreRMinoration en temps petit de la densité d’une diffusion dégénéréeJ. Funct. Anal.198774239941410.1016/0022-1236(87)90031-00637.58034904825 BakryDLedouxMA logarithmic Sobolev form of the Li-Yau parabolic inequalityRev. Mat. Iberoam.20062268370210.4171/RMI/4701116.580242294794 LiH-QEstimation optimale du gradient du semi-groupe de la chaleur sur le groupe de HeisenbergJ. Funct. Anal.2006236236939410.1016/j.jfa.2006.02.0161106.220092240167 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) BougerolPThéorème central limite local sur certains groupes de LieAnn. Sci. Ec. Norm. Super.19811444034320488.60013654204 Varopoulos, N.T., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100, Cambridge (1992) HinoMRamirezJSmall-time Gaussian behavior of symmetric diffusion semigroupsAnn. Probab.20033131254129510.1214/aop/10554257791085.310081988472 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) Dragomir, S., Tomassini, G.: Differential geometry and analysis on CR manifolds. Progress in Mathematics, vol 246. Birkhauser (2006) 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) BaudoinFBonnefontMThe subelliptic heat kernel on SU(2): representations, asymptotics and gradient boundsMath. Z.2009263364767210.1007/s00209-008-0436-01189.580092545862 BealsRGaveauBGreinerPHamilton-Jacobi theory and the heat kernel on Heisenberg groupsJ. Math. Pures Appl.20007976336890959.350351776501 ChanilloSYangPIsoperimetric inequalities and volume comparison theorems on CR manifoldsAnn. Scuola Norm. Sup. Pisa Cl. Sci.2009VIII51292548248 Baudoin, F.: An introduction to the geometry of stochastic flows. Imperial College Press, London, x+ 140pp. (2004) LéandreRMajoration en temps petit de la densité d’une diffusion dégénéréeProbab. Theory Relat. Fields198774228929410.1007/BF005699940587.60073 HulanickiAThe distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg groupStud. Math.19765621651730336.22007418257 DriverBKMelcherTHypoelliptic heat kernel inequalities on the Heisenberg groupJ. Funct. Anal.2005221234036510.1016/j.jfa.2004.06.0121071.220052124868 GarofaloNNhieuDMIsoperimetric and Sobolev embeddings for Carnot-Carathéodory space and existence of minimal surfacesCommun. Pure Appl. Math.1996491081114410.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A0880.350321404326 TaylorMEPartial Differential Equations II: Qualitative Studies of Linear Equations1996New-YorkSpringer-Verlag0869.35003 RobinsonDWElliptic Operators and Lie Groups. Oxford Mathematical Monographs, 911991New YorkThe Clarendon Press Oxford University Press 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) GaveauBPrincipe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotentsActa Math.19771391–29515310.1007/BF023922350366.22010461589 DaviesEBHeat Kernels and Spectral Theory. Cambridge Tracts Math.1989CambridgeCambridge University Press10.1017/CBO9780511566158 BakryDBaudoinFBonnefontMChafaiDOn gradient bounds for the heat kernel on the Heisenberg groupJ. Funct. Anal.20082558190519381156.580092462581 MitchellJOn Carnot-Carathéodory metricsJ. Differ. Geom.19852135450554.53023 VaropoulosNTSmall time Gaussian estimates of heat diffusion kernels, I, The semigroup techniqueBull. Sci. Math.198911332532770703.580521016211 BakryDBaudoinFBonnefontMQianBSubelliptic Li-Yau estimates on 3-dimensional model spaces. Potential theory and stochastics in AlbacTheta Ser. Adv. Math2009111112681833 HueberHMüllerDAsymptotics for some Green kernels on the Heisenberg group and the Martin boundaryMath. Ann.198928319711910.1007/BF014575040639.31005973806 TaylorMENoncommutative Harmonic Analysis. Mathematical Surveys and Monographs, vol. 221986ProvidenceAmerican Mathematical Society |
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| Snippet | In this paper, we study a subelliptic heat kernel on the Lie group
SL
(2, ℝ) and on its universal covering
. The subelliptic structure on
SL
(2,ℝ) comes from... |
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| SourceType | Publisher |
| StartPage | 275 |
| SubjectTerms | Functional Analysis Geometry Mathematics Mathematics and Statistics Potential Theory Probability Theory and Stochastic Processes |
| Title | The Subelliptic Heat Kernels on SL(2, ℝ) and on its Universal Covering : Integral Representations and Some Functional Inequalities |
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