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 $L > 0$ and put $E=\mathbb{R}^n \backslash (-L,L)^n$. I am trying to prove that for any $m \in \mathbb{N}$, there exist $K > 0$ and $m' \in \mathbb{N}$ such that we have \begin{equation} ||\phi||_{m,E} \leq K ||P\phi||_{m',E} \quad \forall \phi \in \mathcal{D}(\mathbb{R}^n) \tag{I}. \end{equation} Any help is welcome. Thank you very much in advance for your attention.

NOTE (1). If we take $E= \mathbb{R}^n$, 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. Indeed, inequality (4.3) of this work (taken with $k=0$) implies that for any $m, p \in \mathbb{N}$, there exist $K > 0$ 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 K \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 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 me 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 be also 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).