Potential theory results for a class of PDOs admitting a global fundamental solution

Abstract

We outline several results of Potential Theory for a class of linear par-tial differential operators L of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for L; under different geometrical assumptions on L (mainly, under global doubling/Poincar\ue9 assumptions), it is described how to obtainan invariant, non-homogeneous Harnack inequality. When L is equipped with a global fundamental solution \u393, further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on L ensuring that such a \u393 exists