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 $1<p<\mathrm{\infty }$ if $F$ satisfies a scale-invariant smoothness condition of order $s>Q\mathrm{/}2$. Under suitable assumptions on $G$ and $L$, here we show that a smoothness condition of order $s>d\mathrm{/}2$ is sufficient. This extends to a larger class of 2-step groups the results for the Heisenberg and related groups by M"uller and Stein and by Hebisch, and for the free group $N_{3,2}$ by M"uller and the author.