Fundamental solution and sharp Lp 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 ${\mathbb H}^{n}$ (Rumin’s complex) relying on the notion of currents in ${\mathbb H}^{n}$ 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