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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Bibliographic Details
Published inPotential analysis Vol. 36; no. 2; pp. 275 - 300
Main Author Bonnefont, Michel
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 2012
Subjects
Functional Analysis
Geometry
Mathematics
Mathematics and Statistics
Potential Theory
Probability Theory and Stochastic Processes
Subelliptic geometry
35H20
Heat kernel
Functional inequalities
58J35
43A80
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Summary: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.
ISSN:0926-2601
1572-929X
DOI:10.1007/s11118-011-9230-4