  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 $G_\infty$ 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-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.
