Harnack inequality for a subelliptic PDE in nondivergence form

We consider subelliptic equations in non divergence form of the type Lu=∑i≤jaijXjXiu=0 where XjXj are the Grushin vector fields, and the matrix coefficient is uniformly elliptic. We obtain a scale invariant Harnack’s inequality on the XjXj’s CC balls for nonnegative solutions under the only assumption that the ratio between the maximum and minimum eigenvalues of the coefficient matrix is bounded. In the paper we first prove a weighted Aleksandrov–Bakelman–Pucci estimate, and then we show a critical density estimate, the double ball property and the power decay property. Once this is established, Harnack’s inequality follows directly from the axiomatic theory developed by Di Fazio, Gutierrez and Lanconelli in Di Fazio et al. (2008).