Functional calculus on non-homogeneous operators on nilpotent groups

Abstract

We study the functional calculus associated with a hypoelliptic left-invariant differential operator mathcal {L} on a connected and simply connected nilpotent Lie group G with the aid of the corresponding Rockland operator mathcal {L}_0 on the ‘local’ contraction G_0 of G, as well as of the corresponding Rockland operator mathcal {L}_infty on the ‘global’ contraction G_infty of G. We provide asymptotic estimates of the Riesz potentials associated with mathcal {L} at 0 and at infty , as well as of the kernels associated with functions of mathcal {L} satisfying Mihlin conditions of every order. We also prove some Mihlin–Hörmander multiplier theorems for mathcal {L} which generalize analogous results to the non-homogeneous case. Finally, we extend the asymptotic study of the density of the ‘Plancherel measure’ associated with mathcal {L} from the case of a quasi-homogeneous sub-Laplacian to the case of a quasi-homogeneous sum of even powers.