  We establish a Harnack inequality for a class of quasi-linear PDE modeled on
the prototype {equation*}
  \partial_tu= -\sum_{i=1}^{m}X_i^\ast (|\X u|^{p-2} X_i u){equation*} where
$p\ge 2$, $ \ \X = (X_1,..., X_m)$ is a system of Lipschitz vector fields
defined on a smooth manifold $\M$ endowed with a Borel measure $\mu$, and
$X_i^*$ denotes the adjoint of $X_i$ with respect to $\mu$. Our estimates are
derived assuming that (i) the control distance $d$ generated by $\X$ induces
the same topology on $\M$; (ii) a doubling condition for the $\mu$-measure of
$d-$metric balls and (iii) the validity of a Poincar\'e inequality involving
$\X$ and $\mu$. Our results extend the recent work in
\cite{DiBenedettoGianazzaVespri1}, \cite{K}, to a more general setting
including the model cases of (1) metrics generated by H\"ormander vector fields
and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci
curvature and Riemannian volume forms; and (3) metrics generated by non-smooth
Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the
Harnack inequality continues to hold when the Lebesgue measure is substituted
by any smooth volume form or by measures with densities corresponding to
Muckenhoupt type weights.
