Let $P(D)$ be a differential operator of order $m$. I would like to show that $P(D)$ is elliptic if for some open and bounded subset of $\mathbb{R}^n$ and for some real $s$ and $c>0$ and all $\varphi\in \mathscr{D}(\Omega)$: $$\|\varphi\|_{s+m}\leq c\|P(D)\varphi\|_s.$$
Many thanks in advance for your help.