Some Classes of Partial Differential Operators modelled on Sub-Laplacians

This thesis concerns with the Theory of Hormander operators and with some classes of hypoelliptic differential operators with non-negative characteristic form. The following are the main problems faced in the thesis. (1) Given a Hormander operator L on the whole of RN, is it possible to find a real Lie group on which L is left-invariant? (2) Given a homogeneous Hormander operator L on RN, there exists a ``well-behaved'' global fundamental solution for L? (3) Given a hypoelliptic partial differential operator L on RN with non-negative characteristic form (not necessarily of Hormander-type), is it possible to prove a Strong Maximum Principle and to develop a satisfactory Potential Theory? Problems (1)-to-(3) are faced with a unitary approach which crucially relies on the study of the geometry of the integral curves of suitable vector fields associated with the operator L and of their composition.