On the De Giorgi - Nash - Moser theorem for hypoelliptic operators.
Jonas Hirsch (Leipzig)
Abstract:
I would like to present a relative simple approach to show uniform boundedness and
a weak Harnack inequality for general hypoelliptic operators i.e.
$$
\left(X_0-\sum_{i, j=1}^m X_i^t a^{i j} X_j\right) u=-\sum_{i=1}^m X_i^t f^i+g .
$$
where $\lambda \leq a^{i j} \leq \Lambda$ is uniformly elliptic but merely
measurable and the $X_i$ are given smooth vectorfields. Furthermore we assume
that they satisfy the H\"ormander condition i.e. their Lie-Algebra spans
$\mathbb{R}^{n+1}$.
The novelty is the avoidance of a "general" Sobolev embedding and a
"quantitative" Poincare inequality. Our approach shows that one
can somehow consider even the classical De Giorgi-Nash-Moser theorem as
a "perturbation" of the Poisson equation.
If time permits I would like to discuss as well how the geometry of
the hypoelliptic equations come into play to obtain as a consequence
the famous HÃ¶lder regularity.
This is joint work with H. Dietert.