The main purpose is to carry out a “qualitative’’ study of some classes of sub-elliptic operators L modeled sub-Laplacians on real Lie groups.
As it is crystal clear from several celebrated results obtained in the ‘70s, the sub-elliptic operators posses a rich algebraic/geometrical structure; for this reason, by means of techniques borrowed from Geometrical Analysis (flows of vector fields, composition of flows, Campbell-Hausdorff Theorem ecc.), we aim to establish the following results: the validity of the Strong/Weak Maximum Principle on open sets (bounded or unbounded); Harnack’s inequalities and regularity estimates; Liouville-type tehorems; the validity of the Gibbons conjecture and related symmetry results.
Clearly, all the above topics can be adequately formulated for parabolic operators of the form L-∂_t (with L modeled on a sub-Laplacian).
Dott. Stefano Biagi email: email@example.com