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

摘要

In this paper, we study a subelliptic heat kernel on the Lie group SL(2, ℝ) and on its universal covering [(SL(2,mathbbR))tilde]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 [(SL(2,mathbbR))tilde]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.