Erika Battaglia

In this seminar some recent results concerning Harnack inequalities will be presented
for several classes of sub-elliptic operators.
We will start by considering a class of sub-elliptic operators, in divergence form,
with low-regular coefficients under global doubling and Poincaré assumptions;
for these operators a non-homogeneous invariant Harnack inequality will be shown.
As a consequence, we will prove the solvability of the Dirichlet problem (in a suitable weak sense).
In the second part, we will consider a class of hypoelliptic non-Hormander operators
for which we have been able to construct a Green function;
with a completely different approach with respect to the case of doubling metric spaces,
we will conclude by showing (by means of techniques of Potential Theory)
how the solvability of the Dirichlet problem
has been a fundamental tool
in order to prove a homogeneous Harnack inequality in the framework of harmonic spaces.