Functional Calculus on Non-Homogeneous Operators on Nilpotent Groups

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 \emph{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 $G_{\mathrm{\infty }}$ of $G$. We provide asymptotic estimates of the Riesz potentials associated with $\mathcal{L}$ at $0$ and at $\mathrm{\infty }$, as well as of the kernels associated with functions of $\mathcal{L}$ satisfying Mihlin conditions of every order. We also prove some Mihlin-H"ormander 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.