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

Let G \boldsymbol {G} be a group of Heisenberg type with homogeneous dimension Q Q . For every 0 > ϵ > Q 0>\epsilon >Q we construct a non-divergence form operator L ϵ L^\epsilon and a non-trivial solution u ϵ ∈ L 2 , Q − ϵ ( Ω ) ∩ C ( Ω ¯ ) u^\epsilon \in \mathcal {L}^{2,Q-\epsilon }(\Omega )\cap C(\overline {\Omega }) to the Dirichlet problem: L u = 0 Lu=0 in Ω \Omega , u = 0 u=0 on ∂ Ω \partial \Omega . This non-uniqueness result shows the impossibility of controlling the maximum of u u with an L p L^p norm of L u Lu when p > Q p>Q . Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as \[ sup Ω u ≤ C ( ∫ Ω | det ⁡ ( u , i j ) | d g ) 1 / m , \sup _\Omega u\le C\left (\int _{\Omega }|\operatorname {det}(u_{,ij})|\,dg\right ) ^{1/m}, \] where m m is the dimension of the horizontal layer of the Lie algebra and ( u , i j ) (u_{,ij}) is the symmetrized horizontal Hessian of u u