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