A Lewy-Stampacchia Estimate for quasilinear variational inequalities in the Heisenberg group

We consider an obstacle problem in the Heisenberg group framework, and we prove that the operator on the obstacle bounds pointwise the operator on the solution. More explicitly, if $\epsilon\ge0$ and $\bar u_\epsilon$ minimizes the functional $$ \int_\Omega(\epsilon+|\nabla_{\H^n}u|^2)^{p/2}$$ among the functions with prescribed Dirichlet boundary condition that stay below a smooth obstacle $\psi$, then 0 \le \div_{\H^n}\, \Big((\epsilon+|\nabla_{\H^n}\bar u_\epsilon|^2)^{(p/2)-1} \nabla_{\H^n}\bar u_\epsilon\Big) \qquad \le (\div_{\H^n}\, \Big((\epsilon+|\nabla_{\H^n}\psi|^2)^{(p/2)-1} \nabla_{\H^n}\psi\Big))^+.