  It is known that the couple formed by the two dimensional Brownian motion and
its L\'evy area leads to the heat kernel on the Heisenberg group, which is one
of the simplest sub-Riemannian space. The associated diffusion operator is
hypoelliptic but not elliptic, which makes difficult the derivation of
functional inequalities for the heat kernel. However, Driver and Melcher and
more recently H.-Q. Li have obtained useful gradient bounds for the heat kernel
on the Heisenberg group. We provide in this paper simple proofs of these
bounds, and explore their consequences in terms of functional inequalities,
including Cheeger and Bobkov type isoperimetric inequalities for the heat
kernel.
