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 partial differential operators of the second order in divergence form. Under essentially the sole assumption of hypoellipticity, we present a non-invariant homogeneous Harnack inequality for under different geometrical assumptions on (mainly, under global doubling/Poincaré assumptions), it is described how to obtain an invariant, non-homogeneous Harnack inequality. When is equipped with a global fundamental solution further Potential Theory results are available (such as the Strong Maximum Principle). We present some assumptions on ensuring that such a exists.