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

Abstract In this paper we construct a fundamental solution for the Laplace operator on the contact complex in Heisenberg groups (Rumin’s complex) relying on the notion of currents in given recently by Franchi, Serapioni and Serra Cassano. This operator is of order 2 on k intrinsic forms for k≠ n, but is of order 4 on n intrinsic forms. As an application, we prove sharp L^p a priori estimates for horizontal derivatives. Keywords: Heisenberg groups, Differential forms, Currents, Laplace operators, Fundamental solution Mathematics Subject Classification (2000): 43A80, 58A10, 58A25, 35A08