Hypoelliptic heat kernel inequalities on the Heisenberg group

We study the existence of “Lp-type”gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian”on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p> 1. The gradient estimate for p = 2 implies a corresponding Poincare ́ inequality for the heat kernel. The gradient estimate for p = 1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel