## Notes of Andrea Bonfiglioli’s lecture

Maximal principles and Harnack inequalities for PDO’s in divergence form

1. Motivation

CR geometry (sub-Laplacians), stochastic PDE’s.

2. Introduction

2.1. Standing assumptions

1. Total nondegeneracy.
2. Smooth hypoellipticity.

Sometimes, we require that ${L-\epsilon}$ is hypoelliptic as well. Or even existence of a global, positive fundamental solution (unfortunately, this is known only for special classes, like homogeneous operators on nilpotent groups, Nagel Stein 1990.

2.2. Earlier work

Theorem 1 (Bony 1969) Maximum principle and Harnack inequality for a class of degenerate elliptic operators (sums of squares of Hörmander vectorfields).

Bony uses a Hopf-type lemma and maximum propagation to get maximum principle. Then is used to get Harnack inequality.

Huge litterature in the 1980’s : Fabes, Jerison, Serapioni Franchi, Lanconelli, Chanillo, Wheeden, Sanchez-Calle. All assume hypo-ellipticity.

Nowadays, the framework has been enlarged : doubling metric spaces satisfying Poincaré inequality.

2.3. Examples

Sub-Laplacians.

Fedii 1971 : sum of squares of non Hörmander vectorfields (a constant basis, whose vectors are multiplied with flat functions). This can be hypo-elliptic but not sub-elliptic (Fefferman-Phong 1981).

3. Results

3.1. Hopf Lemma

Let ${F}$ be the set where ${u}$ achieves its maximum. Let ${y\in F}$ and ${\nu}$ be orthogonal to ${F}$ (meaning that the interior of some ball centered at ${y+\epsilon \nu}$ and passing through ${y}$ is disjoint from ${F}$). Then…

3.2. From Hopf lemma to maximum principle

Theorem 2 Non total degeneracy and hypoellipticity imply strong maximum principle.

\proof

Principal vectorfields ${X}$ have to be tangent to ${F}$. This implies ${F}$ has to be invariant under ${X}$. How can one build them ? Use columns of the matrix defining the operator. Note that Hörmander’s condition need not hold for these vectorfields.

Amano 1979 observed that non total degeneracy and hypoellipticity imply connectivity of ${{\mathbb R}^n}$ with respect to such vectorfields plus a drift vectorfields. Thus maximum principle follows.

3.3. Harnack inequality

Theorem 3 Non total degeneracy and hypoellipticity of ${L-\epsilon}$ imply strong a Harnack inequality where, however, the constant depends on the shape of the domain and of the considered subdomain.

\proof

Follows Bony’s approach. Solve the Dirichlet for ${L-\epsilon}$ (based on maximal principle). Prove existence of the Green kernel of ${L-\epsilon}$. Get a weak Harnack inequality. Use potential theory to get Harnack from weak Harnack.

By maximum principle, the Green kernel ${k_\epsilon}$ of ${L-\epsilon}$ is positive. Then for ${u\geq 0}$ such that ${Lu=0}$, Bony proves that

$\displaystyle \begin{array}{rcl} u(x)\geq \epsilon\int u(y)k_\epsilon(x,y)\,d\nu(y). \end{array}$

Since ${k_\epsilon>0}$, this allows to locally bound ${u(x)}$ from below with the ${L^1_{loc}}$-norm of ${u}$. On the space of ${L}$-harmonic functions, the ${L^1_{loc}}$ and ${C^\infty}$ topologies coincide. This way, we get the weak Harnack inequality

$\displaystyle \begin{array}{rcl} \sup_K u \leq C(x_0)u(x_0). \end{array}$

3.4. Role of potential theory

Theorem 4 (Mokobodzki-Brelot 1964) Very abstract setting. Assume weak Harnack inequality holds and that Dirichlet problem on small open sets has a solution, then strong Harnack inequality holds.

4. More on potential theory

How can one characterize ${L}$-subharmonic functions ?

Use balls defined by Green’s function (${\Gamma}$-balls) to define inradius of a domain. Then a representation formula follows, based on the divergence theorem, with kernel expressible in terms of Green’s function. A mean value formula holds for ${L}$-harmonic functions on ${\Gamma}$-balls, with a correction term. The corresponding inequality characterizes sub-harmonicity. So does monotonicity of mean values on ${\Gamma}$-balls.