On an Inequality of Lars Hörmander

Let $P(z)$ be a non-null complex polynomial in $n$ variables $z=(z_1,\dots,z_n)$: $$P(z)=\sum_{|\alpha| \leq N} c_{\alpha} z^{\alpha},$$ 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 $\mathbb{C}$: $$P(x)=\sum_{|\alpha| \leq N} c_{\alpha} x^{\alpha} \quad (x \in \mathbb{R}^n).$$ For any $m \in \mathbb{N}$, any $A \subseteq \mathbb{R}^n$, and any $\phi \in \mathcal{D}(\mathbb{R}^n)$ set: $$||\phi||_{m,A} = \sup_{\substack{x \in A \\ |\alpha| \leq m}} |(D^{\alpha} \phi)(x)|.$$ 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 $$||\phi||_{m,E} \leq C ||P\phi||_{m',E} \quad \forall \phi \in \mathcal{D}_{K} \tag{I},$$ 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.

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. 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 $$\sup_{\substack{x \in \mathbb{R}^n \\ |\alpha| \leq p}} (1+|x|)^m |(D^{\alpha} \phi) (x)| \leq C \sup_{\substack{x \in \mathbb{R}^n \\ |\alpha| \leq p'}} (1+|x|)^{m'} |(D^{\alpha} (P\phi)) (x)| \quad \forall \phi \in \mathcal{S}(\mathbb{R}^n) \tag{II}.$$ We can state (II) in another way. Define the linear subspace $\mathcal{M}_{P}$ of $\mathcal{S}(\mathbb{R}^n)$: $$\mathcal{M}_{P}=\{\psi \in \mathcal{S}(\mathbb{R}^n): \psi=P \phi, \phi \in \mathcal{S}(\mathbb{R}^n) \},$$ and consider the multiplication map $M_{P}:\mathcal{S}(\mathbb{R}^n) \rightarrow \mathcal{M}_{P}$ defined by $$M_{P}(\phi)=P\phi \quad (\phi \in \mathcal{S}(\mathbb{R}^n)),$$ Then (II) is equivalent to say that $M_{P}$ has a continuous inverse (this statement is Theorem (1) in Hörmander's work).
NOTE (2). Inequality (I) was stated without proof by ifw in his answer to the post Division of Distributions by Polynomials (see also my own answer for a comment). If true, (I) 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. In one of his comments, ifw said 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).
• How this is related to the Hormander inequality namely for $P(D)$ being non zero linear differential operator such that for any open set in $R^n$ there exists a constqnt C such that the norm of F is less than or equal to C times the norm of P(D)(F) for every F . Where F is a test function ? Mar 25, 2017 at 15:10
• Yhe norm is$L^2$ Mar 25, 2017 at 15:10