Large sets at infinity and maximum principle on unbounded domains for a class of sub-elliptic operators

Maximum Principles on unbounded domains play a crucial rôle in several problems related to linear second-order PDEs of elliptic and parabolic type. In the present notes, based on a joint work with E. Lanconelli, we consider a class of sub-elliptic operators L in RN and we establish some criteria for an unbounded open set to be a Maximum Principle set for L. We extend some classical results related to the Laplacian (proved by Deny, Hayman and Kennedy) and to the sub-Laplacians on homogeneous Carnot groups (proved by Bonfiglioli and Lanconelli).