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)−>H2 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 Poincar\'e inequality that are valid for
both heat kernels