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: (1) whether the L2 and Lp-Hodge decompositions, 1 < p < ∞, hold on Hn; (2) whether the Riesz transforms dΔ -1 2 k are Lp-bounded, for 1 < p < ∞; (3) 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 Vν with the properties: • domΔk splits along the Vν's as ν(domΔk ∩ Vν); • Δk : (domΔk ∩ Vν) -→ Vν for every ν; • for each ν, there is a Hilbert space Hν of L2-sections of a U(n)-homogeneous vector bundle over Hn such that the restriction of Δk to Vν is unitarily equivalent to an explicit scalar operator acting componentwise on Hν. 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 -1 2 k are Lp-bounded; • the orthogonal projection onto Vν extends from (L2∩Lp)Λk to a bounded operator from LpΛk to the the Lp-closure Vp ν of Vν ∩ 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.