The authors consider the Hodge Laplacian Δ on the Heisenberg group H n , endowed with a left-invariant and U(n) -invariant Riemannian metric. For 0≤k≤2n 1 , let Δ k denote the Hodge Laplacian restricted to k -forms.

In this paper they address three main, related questions:

• (1) whether the L 2 and L p -Hodge decompositions, 1

• (2) whether the Riesz transforms dΔ −12 k are L p -bounded, for 1<<∞ ;
• (3) how to prove a sharp Mihilin-Hörmander multiplier theorem for Δ k , 0≤k≤2n 1 .