Harnack inequality for hypoelliptic second order partial differential operators (Q334909)

The authors consider non-negative solutions of second order hypoelliptic equations \[ {\mathcal L}u(x)=\sum_{i,j=1}^n \partial_{x_i}(a_{ij}(x)\partial_{x_j}u(x) ) + \sum_{i=1}^n b_i(x)\partial_{x_i} u(x)=0 \quad \text{ in } \Omega \] where \(\Omega\) is a bounded open subset of \(\mathbb R^n\) and \(x\) denotes the point of \(\Omega.\) For any fixed \(x_0\in \Omega, \) the authors prove a Harnack inequality of the type \[ \sup_K u\leq C_K u(x_0)\qquad \forall \;u \quad \text{ s.t. } \;{\mathcal L} u=0, \;u\geq 0, \] 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 \(u.\)