Harnack inequalities of Landis-type for Hörmander operators in non-divergence form

In this talk we will discuss the validity of Harnack inequalities for two classes of linear second order equations in nondivergence form. The first class is formed by degenerate-elliptic operators which are horizontally elliptic with respect to Heisenberg-type vector fields. The second one constitutes a class of evolution operators of Kolmogorov-Fokker-Planck type. The analogous of the Krylov-Safonov Harnack inequality for these classes of Hörmander operators with bounded measurable coefficients is still unknown, due to the absence of proper Aleksandrov-Bakelman-Pucci type estimates. We will show a perturbative approach to prove invariant Harnack inequalities for operators with coefficients satisfying either a Cordes-Landis assumption or a continuity hypothesis. This talk is based on joint works with F. Abedin and C.E. Gutiérrez.