We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE,(Formula presented.) in cylinders Ω × (0,T) where Ω ⊂ M is an open subset of a manifold M endowed with control metric d corresponding to a system of Lipschitz continuous vector fields X=(X_1,..., X_m) and a measure dσ. We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincaré inequality in the metric measure space (M,d,dσ). We also show that such hypothesis hold for a class of Riemannian metrics gε collapsing to a sub-Riemannian metric limε → 0 gε = g0 uniformly in the parameter ε ≥ 0.
Capogna, Luca; Citti, Giovanna; and Rea, Garrett, "A Subelliptic Analogue of Aronson-serrin's Harnack Inequality" (2013). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.