A subelliptic analogue of Aronson–Serrin’s Harnack inequality

We study the Harnack inequality for weak solutions of a class of degenerate parabolic quasilinear PDE in cylinders where is an open subset of a manifold endowed with control metric corresponding to a system of Lipschitz continuous vector fields and a measure . We show that the Harnack inequality follows from the basic hypothesis of doubling condition and a weak Poincaré inequality in the metric measure space . We also show that such hypothesis hold for a class of Riemannian metrics collapsing to a sub-Riemannian metric uniformly in the parameter .