A sharp multiplier theorem for solvable extensions of Heisenberg and related groups

Let $G$ be the semidirect product $N \rtimes \mathbb{R}$, where $N$ is a stratified Lie group and $\mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous left-invariant sub-Laplacians on $N$ and $\mathbb{R}$ can be lifted to $G$, and their sum $\Delta$ is a left-invariant sub-Laplacian on $G$. In previous joint work of Ottazzi, Vallarino and the first-named author, a spectral multiplier theorem of Mihlin--H\"ormander type was proved for $\Delta$, showing that an operator of the form $F(\Delta)$ is of weak type $(1,1)$ and bounded on $L^p(G)$ for all $p \in (1,\infty)$ provided $F$ satisfies a scale-invariant smoothness condition of order $s > (Q+1)/2$, where $Q$ is the homogeneous dimension of $N$. Here we show that, if $N$ is a group of Heisenberg type, or more generally a direct product of M\'etivier and abelian groups, then the smoothness condition can be pushed down to the sharp threshold $s>(d+1)/2$, where $d$ is the topological dimension of $N$. The proof is based on lifting to $G$ weighted Plancherel estimates on $N$ and exploits a relation between the functional calculi for $\Delta$ and analogous operators on semidirect extensions of Bessel--Kingman hypergroups.