Analysis of the Hodge Laplacian on the Heisenberg group

We consider the Hodge Laplacian Δ on the Heisenberg group Hn, endowed with a left-invariant and U(n)-invariant Riemannian metric. For 0≤k≤2n+1, let Δk denote the Hodge Laplacian restricted to k-forms. In this paper we address three main, related questions: whether the L2 and Lp-Hodge decompositions, 1(2n+1)/2 and prove that this restriction on ρ is optimal. Finally, we extend this multiplier theorem to the Dirac operator.