We study the equations \[ \begin{aligned} \left( L - {\partial \over \partial t} \right) u(x,t) & = 0 \quad \text{ and }\tag{1.1} \\ Lu(x) & = 0 \tag{1.2}\end{aligned} \] associated to the operator \(L = \sum_ i X^ 2_ i - X_ 0\) on a compact manifold \(M\) with a positive measure \(\mu\), where \(X_ 1, X_ 2, \dots, X_ m\) are smooth vector fields on \(M\) and \(X_ 0 = \sum_ i c_ iX_ i\). Our main purpose is to prove (Theorem 3.1 and Theorem 3.2) Harnack inequalities for positive solutions of Eq. (1.1) and Eq. (1.2) and to derive (Theorem 4.1) an upper estimate for the fundamental solution of the operator \(L - {\partial \over \partial t}\).