Spectral multipliers for sub-Laplacians on solvable extensions of stratified groups

Let G = NA,where N is a stratified group and A =R acts on N via automorphic dilations. Homogeneous sub-Laplacians on N and A can be lifted to left-invariant operators on G, and their sum is a sub-Laplacian L on G. We prove a theorem of Mihlin–H¨ormander type for spectral multipliers of L.The proof of the theorem hinges on a Calder´on–Zygmund theory adapted to a sub-Riemannian structure of G and on L1-estimates of the gradient of the heat kernelassociated to the sub-Laplacian.