Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators

Abstract

We consider non-negative solutions $u:{\Omega }\longrightarrow \mathbb {R}$ of second order hypoelliptic equations

$ \mathcal {L} u(x) =\sum \limits _{i,j=1}^{n} \partial _{x_{i}} \left (a_{ij}(x)\partial _{x_{j}} u(x) \right ) + \sum \limits _{i=1}^{n} b_{i}(x) \partial _{x_{i}} u(x) =0 $

where Ω is a bounded open subset of $\mathbb {R}^{n}$ and x denotes the point of Ω. For any fixed x ₀ ∈ Ω, we prove a Harnack inequality of this type

$ \sup _{K} u \le C_{K} u(x_{0})\qquad \forall \ u \ \text { s.t. } \ \mathcal {L} u=0, u\geq 0, $

where K is any compact subset of the interior of the $\mathcal {L}$-propagation set of x ₀ and the constant C _K does not depend on u.