On the best possible character of the $L^Q$ norm in some a priori estimates for non-divergence form equations in Carnot groups

Let $\boldsymbol{G}$ be a group of Heisenberg type with homogeneous dimension $Q$. For every $0<\epsilon<Q$ we construct a non-divergence form operator $L^\epsilon$ and a non-trivial solution $u^\epsilon\in\mathcal{L}^{2,Q-\epsilon}(\Omega)\cap C(\overline{\Omega})$ to the Dirichlet problem: $Lu=0$ in $\Omega$, $u=0$ on $\partial\Omega$. This non-uniqueness result shows the impossibility of controlling the maximum of $u$ with an $L^p$ norm of $Lu$ when $p<Q$. Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as

where $m$ is the dimension of the horizontal layer of the Lie algebra and $(u_{,ij})$ is the symmetrized horizontal Hessian of $u$.