Functional calculus on non-homogeneous operators on nilpotent groups

We study the functional calculus associated with a hypoelliptic left-invariant differential operator L on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator L-0 on the `local' contraction G(0) of G, as well as of the corresponding Rockland operator L-infinity on the `global' contraction G(infinity) of G. We provide asymptotic estimates of the Riesz potentials associated with L at 0 and at infinity, as well as of the kernels associated with functions of L satisfying Mihlin conditions of every order. We also prove some Mihlin-Hormandermultiplier theorems for L which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the `Plancherelmeasure' associated with L from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.