Spectral multipliers for maximally subelliptic operators

Consider a non-negative, self-adjoint, maximally subelliptic operator on a compact manifold. We show that the spectral multiplier is a singular integral operator under an appropriate Mihlin-Hörmander type condition. We establish the equivalence between non-isotropic Besov and Triebel-Lizorkin spaces adapted to the operator and those adapted to a Carnot-Carathéodory geometry on the manifold. We also give a Mihlin-Hörmander type condition for the boundedness of the spectral multiplier on non-isotropic $L^p$ Sobolev spaces.