Spectral multipliers on Heisenberg-Reiter and related groups

Let L be a homogeneous sublaplacian on a 2-step stratified Lie group G of topological dimension d and homogeneous dimension Q. By a theorem due to Christ and to Mauceri and Meda, an operator of the form F(L) is bounded on L-p for 1 < p < infinity if F satisfies a scale-invariant smoothness condition of order s > Q/2. Under suitable assumptions on G and L, here we show that a smoothness condition of order s > d/2 is sufficient. This extends to a larger class of 2-step groups the results for the Heisenberg and related groups by Muller and Stein and by Hebisch and for the free group N-3,(2) by Muller and the author.