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<p<∞, hold on Hn; whether the Riesz transforms dΔ−12k are Lp-bounded, for 1<p<∞; to prove a sharp Mihilin-Hörmander multiplier theorem for Δk, 0≤k≤2n+1. Our first main result shows that the L2-Hodge decomposition holds on Hn, for 0≤k≤2n+1. Moreover, we prove that L2Λk(Hn) further decomposes into finitely many mutually orthogonal subspaces ν with the properties: domΔk splits along the ν's as ∑ν(domΔk∩ν); Δk:(domΔk∩ν)→ν for every ν; for each ν, there is a Hilbert space ν of L2-sections of a U(n)-homogeneous vector bundle over Hn such that the restriction of Δk to ν is unitarily equivalent to an explicit scalar operator acting componentwise on ν. Next, we consider LpΛk, 1<p<∞. We prove that the Lp-Hodge decomposition holds on Hn, for the full range of p and 0≤k≤2n+1. Moreover, we prove that the same kind of finer decomposition as in the L2-case holds true. More precisely we show that: the Riesz transforms dΔ−12k are Lp-bounded; the orthogonal projection onto ν extends from (L2∩Lp)Λk to a bounded operator from LpΛk to the the Lp-closure pν of ν∩LpΛk. We then use this decomposition to prove a sharp Mihlin-Hörmander multiplier theorem for each Δk. We show that the operator m(Δk) is bounded on LpΛk(Hn) for all p∈(1,∞) and all k=0,…,2n+1, provided m satisfies a Mihlin-Hörmander condition of order ρ>(2n+1)/2 and prove that this restriction on ρ is optimal. Finally, we extend this multiplier theorem to the Dirac operator.