Spectral multiplier theorems of Euclidean type on new classes of two-step stratified groups

From a theorem of Christ and Mauceri and Meda, it follows that, for a homogeneous sublaplacian L on a two-step stratified group G with Lie algebra g, an operator of the form F(L) is of weak type (1, 1) and bounded on Lp(G) for 1 < p < ∞ if the spectral multiplier F satisfies a scale-invariant smoothness condition of order s > Q/2, where Q = dim g + dim[g, g] is the homogeneous dimension of G. Here we show that the condition can be pushed down to s > d/2, where d = dim g is the topological dimension of G, provided that d ≤ 7 or dim[g, g] ≤ 2.