Gradient bounds for the heat kernel on the Heisenberg group

Gradient bounds are a very powerful tool to study heat kernel measures and

regularisation properties for the heat kernel. In the elliptic case, it is easy

to derive them from bounds on the Ricci tensor of the generator. In recent

years, many efforts have been made to extend these bounds to some simple

examples in the hypoelliptic situation. The simplest case is the Heisenberg

group. In this talk, we shall discuss some recent developments (due to H.Q. Li)

on this question, and give some elementary proofs of these bounds.