Reverse Inequalities for Subelliptic Functions

We study a wedge $\mathscr{K}(A)$ of solutions of the inequality $A(u) \ge 0$, where $A$ is a linear elliptic operator of order $2m$. For the elements of the wedge, we establish an interior estimate of the form
$$ \|u;H_1^{2m}(\omega)\| \le C(\omega,\Omega)\|u;L(\Omega)\|, $$
where $\omega$ is a compact subset of $\Omega$, $H_1^{2 m}(\omega)$ is the Nikol'skii space, $L(\Omega)$ is the Lebesgue space of integrable functions, and the constant $C(\omega,\Omega)$ is independent of the function $u$. Similar estimates that hold up to the boundaries are proved for the functions from $\mathscr{K}(A)$ satisfying the boundary conditions.