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We establish a Harnack inequality for a class of quasi-linear PDE modeled onthe 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 fieldsdefined 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 arederived assuming that (i) the control distance $d$ generated by $\X$ inducesthe 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 settingincluding the model cases of (1) metrics generated by H\"ormander vector fieldsand Lebesgue measure; (2) Riemannian manifolds with non-negative Riccicurvature and Riemannian volume forms; and (3) metrics generated by non-smoothBaouendi-Grushin type vector fields and Lebesgue measure. In all cases theHarnack inequality continues to hold when the Lebesgue measure is substitutedby any smooth volume form or by measures with densities corresponding toMuckenhoupt type weights.

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian