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~$\bar u$ minimizes
the functional
$$ \int_\Omega |\nabla_{\H^n}u|^2$$
among the functions with prescribed Dirichlet boundary
condition that stay below a smooth obstacle~$\psi$, then
$$
0\leq \Delta_{\H^n} \bar u\leq \Big(\Delta_{\H^n}\psi\Big)^{+}.
$$
Moreover, we discuss how it could be possible to generalize the
previous
bound to a quasilinear setting once some regularity issues for the
equation
$$
\div_{\H^n}\Big(|\nabla_{\H^n}u|^{p-2}\nabla_{\H^n}u\Big)=f
$$
are satisfied.}
