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

Let G be a group of Heisenberg type with homogeneous dimension Q. For every 0 < ε < Q we construct a non-divergence form operator L ^ε and a non-trivial solution u^ε ∈ ℒ^2,Q-ε (Ω) ∩ C (Ω̄) to the Dirichlet problem: Lu = 0 in Ω, u = 0 on ∂Ω. 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 sup_Ω u ≤ C (∫_Ω | det(u,ij)| dg)^1/m, where m is the dimension of the horizontal layer of the Lie algebra and (u,ij) is the symmetrized horizontal Hessian of u.