Double Ball Property for non-divergence horizontally elliptic operators on step two Carnot groups

Abstract

Let $\mathcal{L}$ be a linear second order horizontally elliptic operator on a Carnot group of step two. We assume $\mathcal{L}$ in non-divergence form and with measurable coefficients. Then, we prove the Double Ball Property for the nonnegative sub-solutions of $\mathcal{L}$. With our result, in order to solve the Harnack inequality problem for this kind of operators, it becomes sufficient to prove the so called $ϵ$-Critical Density.