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–Hörmander 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 Métivier 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.