L^p-spectral multipliers for the Hodge Laplacian acting on 1-forms on the Heisenberg group

We prove that, if \Delta_1 is the Hodge Laplacian acting on differential 1-forms on the (2n+1)-dimensional Heisenberg group, and if m is a Mihlin-H\"ormander multiplier on the positive half-line, with L^2-order of smoothness greater than n+1/2, then m(\Delta_1) is L^p-bounded for 1