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

Let G be the semidirect product N R, where N is a stratified Lie group and R acts on N via automorphic dilations. Homogeneous left-invariant sub-Laplacians on N and R can be lifted to G, and their sum 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 , showing that an operator of the form F() is of weak type (1, 1) and bounded on L p(G) for all p ∈ (1,∞) 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 and analogous operators on semidirect extensions of Bessel–Kingman hypergroups.