Alessia Kogoj Università di Urbino

Liouville theorems in sub-Riemannian settings


We present some Liouville-type theorems, related to evolution operators and to their "time"-stationary counterpart. The operators we are dealing with are left translations invariant on a Lie group structure and, in some cases, are homogeneous with respect to a group of dilations. In all these cases the operators have smooth coefficients and are hypoelliptic at least in a neighborhood of the origin. We also present a polynomial Liouville-type theorem for X-elliptic operators with nonsmooth coefficients, by extending to this new setting a celebrated result by Colding and Minicozzi related to the Laplace-Beltrami operator on Riemannian manifolds. The results are contained in a series of papers in collaboration with A. Bonfiglioli, E. Lanconelli, Y.Pinchover, S. Polidoro and E. Priola.