Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators

We consider non-negative solutions u:Ω→ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u:{\Omega }\longrightarrow \mathbb {R}$\end{document} of second order hypoelliptic equations ℒu(x)=∑i,j=1n∂xiaij(x)∂xju(x)+∑i=1nbi(x)∂xiu(x)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \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 $\end{document} where Ω is a bounded open subset of ℝn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb {R}^{n}$\end{document} and x denotes the point of Ω. For any fixed x0 ∈ Ω, we prove a Harnack inequality of this type supKu≤CKu(x0)∀us.t.ℒu=0,u≥0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \sup _{K} u \le C_{K} u(x_{0})\qquad \forall \ u \ \text { s.t. } \ \mathcal {L} u=0, u\geq 0, $\end{document} where K is any compact subset of the interior of the ℒ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathcal {L}$\end{document}-propagation set ofx0 and the constant CK does not depend on u.