Let $P(z)$ be a non-null complex polynomial in $n$ variables $z=(z_1,\dots,z_n)$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq N} c_{\alpha} z^{\alpha},
\end{equation}
where as usual for every $\alpha=(\alpha_1,\dots,\alpha_n) \in \mathbb{N}^{n}$ (here and in the following $\mathbb{N}$ denotes the set of all non-negative integers) we set $|\alpha|=\alpha_1+\dots+\alpha_n$, and $z^{\alpha}=z_1^{\alpha_1}\dots z_n^{\alpha_n}$. Consider $P$ as a polynomial function from $\mathbb{R}^n$ into $\mathbb{C}$:
\begin{equation}
P(x)=\sum_{|\alpha| \leq N} c_{\alpha} x^{\alpha} \quad (x \in \mathbb{R}^n).
\end{equation}
For any $m \in \mathbb{N}$, any $A \subseteq \mathbb{R}^n$, and any $\phi \in \mathcal{D}(\mathbb{R}^n)$ set:
\begin{equation}
||\phi||_{m,A} = \sup_{\substack{x \in A \\ |\alpha| \leq m}} |(D^{\alpha} \phi)(x)|.
\end{equation}
Let $M > L > 0$ and put $K=[-M,M]^n$ and $E=K \backslash (-L,L)^n$. I am trying to prove that for any $m \in \mathbb{N}$, there exist $C > 0$ and $m' \in \mathbb{N}$ such that we have
\begin{equation}
||\phi||_{m,E} \leq C ||P\phi||_{m',E} \quad \forall \phi \in \mathcal{D}_{K} \tag{I},
\end{equation}
where as usual $\mathcal{D}_{K}$ is the set of all complex-valued functions  $\phi \in C^{\infty}(\mathbb{R}^n)$ with support contained in $K$. See the notes below for an explanation of the origin and relevance of this question.

Thank you very much in advance for your attention.

NOTE (1). If we take $L=0$, so that $E=K$, then (I) is an immediate corollary of a remarkable result proved by Lars Hörmander in his wonderful work [On the Division of Distributions by Polynomials](http://link.springer.com/article/10.1007/BF02589517). Indeed, inequality (4.3) of this work (taken with $k=0$) implies that for any $m, p \in \mathbb{N}$, there exist $C > 0$ and $m', p' \in \mathbb{N}$ such that
\begin{equation}
\sup_{\substack{x \in \mathbb{R}^n \\ |\alpha| \leq p}} (1+|x|)^m |(D^{\alpha} \phi) (x)|  \leq C \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha| \leq p'}} (1+|x|)^{m'} |(D^{\alpha} (P\phi)) (x)| \quad \forall \phi \in \mathcal{S}(\mathbb{R}^n) \tag{II}.
\end{equation}
We can state (II) in another way. Define the linear subspace $\mathcal{M}_{P}$ of $\mathcal{S}(\mathbb{R}^n)$:
\begin{equation}
\mathcal{M}_{P}=\{\psi \in \mathcal{S}(\mathbb{R}^n): \psi=P \phi, \phi \in \mathcal{S}(\mathbb{R}^n) \},
\end{equation}
and consider the multiplication map $M_{P}:\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{M}_{P}$ defined by
\begin{equation}
M_{P}(\phi)=P\phi \quad (\phi \in \mathcal{S}(\mathbb{R}^n)),
\end{equation}
Then (II) is equivalent to say that $M_{P}$ has a continuous inverse (this statement is Theorem (1) in Hörmander's work).

NOTE (2). The relevance of (I) comes from the fact that it would allow to give a direct proof of Theorem (4) in Hörmander's paper, which states that every distribution can be divided by a non-null polynomial. See my post [Division of Distributions by Polynomials](http://mathoverflow.net/questions/260821/division-of-distributions-by-polynomials/261641#261641) for a careful explanation of the problem. The direct proof of Theorem (4) was sketched by ifw in his answer to that post. Unfortunately, ifw did not give a proof of (I) nor he suggested a possible line of attack, by simply saying that (I) could be proved by localizing (II) or by modifying properly Hörmander's original proof. Even though I studied very carefully Hörmander's original proof (which can also be found in Trèves, Linear Partial Differential Equations with Constant Coefficients, $\S$ 5.5), I could not modify it in order to obtain (I) nor I could get (I) by localizing (II).