Fundamental solution and sharp L-p estimates for Laplace operators in the contact complex of Heisenberg groups

In this paper we construct a fundamental solution for the Laplace operator on the contact complex in Heisenberg groups H-n (Rumin's complex) relying on the notion of currents in H-n given recently by Franchi, Serapioni and Serra Cassano. This operator is of order 2 on k intrinsic forms for k not equal n, but is of order 4 on n intrinsic forms. As an application, we prove sharp L-p a priori estimates for horizontal derivatives.