Harnack Inequality and Fundamental Solution for Degenerate Hypoelliptic Operators

In this thesis we study subelliptic operators in divergence form on R^N, and we are interested in establishing Harnack inequalities related to these operators in various contexts. As a first result of the thesis, we prove a non-invariant Harnack inequality, passing through a Strong Maximum Principle; in doing so, we require the hypoellipticity of the operator to construct a Green function, that we have used (by means of techniques of Potential Theory) in order to obtain the Harnack inequality. In the second main result of this thesis, we prove a non-homogeneous invariant Harnack inequality for these subelliptic operators under low regularity assumption. Currently, it is known that the natural framework for Harnack-type theorems is the setting of doubling metric spaces; we suppose that the quadratic form of the operator can be naturally controlled by a family of locally-Lipschitz vector fields. Moreover, we assume that, with the associated Carnot-Carathà©odory metric d, N-dimensional Euclidean space is endowed by d with the structure of a doubling space (globally) and a Poincarà© inequality on any d-ball holds true. We use a Sobolev type inequality and the Moser iterative technique to prove a non-homegeneous invariant Harnack Inequality; as a consequence, we show the existence of the Green function using only the Harnack inequality.