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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 fromthe fibration $SO(2) -> SL(2,R) -> H^2$ and it can be lifted to its universalcovering. First, we derive an integral representation for these heat kernels.These expressions allow us to obtain some asymptotics in small time of the heatkernels and give us a way to compute the subriemannian distances. Then, weestablish some gradient estimates and some functional inequalities like aLi-Yau type estimate and a reverse Poincar\'e inequality that are valid forboth heat kernels.

The subelliptic heat kernels on SL(2,R) and on its universal covering  $\widetilde{SL(2,R)}$: integral representations and some functional  inequalities