The subelliptic heat kernels on SL(2,R) and on its universal covering SL(2,R)˜: integral representations and some functional inequalities

Michel Bonnefont (1)

1 IMB - Institut de Mathématiques de Bordeaux

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 its universal covering. First, we derive an integral representation for these heat kernels. These expressions allow us to obtain some asymptotics in small time of the heat kernels and give us a way to compute the subriemannian distances. 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.

Domaines: Géométrie différentielle [math.DG]

Michel Bonnefont. The subelliptic heat kernels on SL(2,R) and on its universal covering SL(2,R)˜: integral representations and some functional inequalities. Potential Analysis, 2012, pp.36, no. 2, 275-300. ⟨hal-00779378⟩