Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z\_1,\dots,z\_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum\_{|\alpha| \leq \mu} c\_{\alpha} z^{\alpha},
\end{equation}
where as usual for every $\alpha=(\alpha\_1,\dots,\alpha\_\nu) \in \mathbb{N}^{\nu}$ (here and in the following $\mathbb{N}$ denotes the set of all non-negative integers) we set $|\alpha|=\alpha\_1+\dots+\alpha\_\nu$, and $z^{\alpha}=z\_1^{\alpha\_1}\dots z\_{\nu}^{\alpha\_\nu}$. Consider $P$ as a polynomial function from $\mathbb{R}^\nu$ into $\mathbb{C}$:
\begin{equation}
P(x)=\sum\_{|\alpha| \leq \mu} c\_{\alpha} x^{\alpha} \quad (x \in \mathbb{R}^{\nu}).
\end{equation}
For any $m \in \mathbb{N}$, any $S \subseteq \mathbb{R}^\nu$, and any $\phi \in \mathcal{D}(\mathbb{R}^\nu)$ set:
\begin{equation}
||\phi||\_{m,S} = \sup\_{\substack{x \in S \\ |\alpha| \leq m}} |(D^{\alpha} \phi)(x)|.
\end{equation}
Let $M > L > 0$ and put $Q=[-M,M]^\nu$ and $E=Q \backslash (-L,L)^\nu$. 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}\_{Q} \tag{I},
\end{equation}
where as usual $\mathcal{D}\_{Q}$ is the set of all complex-valued functions $\phi \in C^{\infty}(\mathbb{R}^\nu)$ with support contained in $Q$. 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 $E=Q$, 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 $n, m \in \mathbb{N}$, there exist $K > 0$ and $n', m' \in \mathbb{N}$ such that
\begin{equation}
\sup\_{\substack{x \in \mathbb{R}^\nu \\ |\alpha| \leq m}} (1+|x|)^n |(D^{\alpha} \phi) (x)| \leq K \sup\_{\substack{x \in \mathbb{R}^\nu \\ |\alpha| \leq m'}} (1+|x|)^{n'} |(D^{\alpha} (P\phi)) (x)| \quad \forall \phi \in \mathcal{S}(\mathbb{R}^\nu) \tag{II}.
\end{equation}
We can state (II) in another way. Define the linear subspace $\mathcal{M}\_{P}$ of $\mathcal{S}(\mathbb{R}^\nu)$:
\begin{equation}
\mathcal{M}\_{P}=\{\psi \in \mathcal{S}(\mathbb{R}^\nu): \psi=P \phi, \phi \in \mathcal{S}(\mathbb{R}^\nu) \},
\end{equation}
and consider the multiplication map $M\_{P}:\mathcal{S}(\mathbb{R}^\nu) \rightarrow \mathcal{M}\_{P}$ defined by
\begin{equation}
M\_{P}(\phi)=P\phi \quad (\phi \in \mathcal{S}(\mathbb{R}^\nu)),
\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). Inequality (I) was stated without proof by ifw in his answer to the post [Division of Distributions by Polynomials](https://mathoverflow.net/questions/260821/division-of-distributions-by-polynomials/261641#261641) (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).



On an Inequality of Lars Hörmander
Finally, I realized how to modify Hörmander's proof in order to prove (I).
I will describe here the necessary changes we have to do to Hörmander's proof. Clearly, all the notation is that of Hörmander's paper.


Let $C$ be a closed, convex set of $\mathbb{R}^\nu$. Then make the following changes in Section (4) of Hörmander's paper [On the Division of Distributions by Polynomials](http://link.springer.com/article/10.1007/BF02589517):


$\bullet$ take the suprema comparing in all the relations over $C$;


$\bullet$ after relation (4.2), replace everywhere $N^k$ with $N^k \cap C$ and $N^{k+1}$ with $N^{k+1} \cap C$;


$\bullet$ replace $\mathbb{R}^\nu$ with $C$ in (4.13);


$\bullet$ replace in every relation $S(\xi)$ with $S(\xi) \cap C$ and $S(\eta)$ with $S(\eta) \cap C$;


$\bullet$ replace in every relation $S(\xi,\eta)$ with $S(\xi,\eta) \cap C$ and $S\_{1}(\xi,\eta)$ with $S\_{1}(\xi,\eta) \cap C$.


With these changes, Hörmander's proof shows that to all $n, m \in \mathbb{N}$ and $k \leq \mu$ there are $n', m' \in \mathbb{N}$ and a constant $K > 0$ such that
\begin{equation}
\sup\_{\xi \in C} (1+|\xi|)^n |f|\_{m, (N^k \cap C)\_{\xi}} \leq K 
\sup\_{\xi \in C} (1+|\xi|)^{n'} |Pf|\_{m',\xi}, \quad (f \in C^{m'}(\mathbb{R}^\nu)).
\end{equation}
So, if $C$ is a compact convex set, the previous inequality, taken with $k=0$, implies that to all $m \in \mathbb{N}$ there are $ m' \in \mathbb{N}$ and a constant $K > 0$ such that
\begin{equation}
||f||\_{m,C} \leq K ||Pf||\_{m',C} \quad (f \in C^{m'}(\mathbb{R}^\nu)) \tag{III}.
\end{equation}
Now, assume that $E$ is a finite union of compact convex subsets $C\_1,\dots, C\_p$ of $\mathbb{R}^\nu$. By (III), for each $i=1,\dots,p$, there exist $m'\_{i} \in \mathbb{N}$ and $K\_i > 0$ such that 
\begin{equation}
||f||\_{m,C\_i} \leq K\_i ||Pf||\_{m'\_{i},C\_i} \quad (f \in C^{m'\_{i}}(\mathbb{R}^\nu)) \tag{III}.
\end{equation}
By taking $K=\max\{K\_1,\dots,K\_p\}$ and $m'=\max\{m'\_{1},\dots,m'\_{p}\}$, we then get
\begin{equation}
||f||\_{m,E} \leq K ||Pf||\_{m',E} \quad (f \in C^{m'}(\mathbb{R}^\nu)) \tag{IV}.
\end{equation}
Clearly (I) is a particular case of (IV), so we are done.
QED


REMARK. One could think that (IV) still holds if we take $E$ to be any compact set, but this is not the case, as the following counterexample shows. Let $\nu=1$, and set 
\begin{equation}
E=\{ 0 \} \cup \left\{ \frac{1}{n}: n=1,2,3,\dots \right\}.
\end{equation}
Take $P(x)=x$ and let $K > 0$. Choose $N > K$. By the celebrated Whitney's Extension Theorem (which is Theorem (I) in [Analytic Extensions of Differentiable Functions Defined in Closed Sets](http://www.ams.org/journals/tran/1934-036-01/S0002-9947-1934-1501735-3/)), there exists $f \in C^{\infty}(\mathbb{R})$, with support contained in $\left[ \frac{1}{N+1},2 \right]$, such that for $n=1,2,3,\dots,N$
\begin{equation}
f^{(m)}\left( \frac{1}{n} \right) = (-1)^m m! n^{m+1} \quad (m \in \mathbb{N}).
\end{equation}
We have $D^{m}(Pf)(x)=0$ for all $x \in E$ and all $m=1,2,\dots$, so that $||Pf||\_{m',E}=1$ for all $m' \in \mathbb{N}$. But $||f||\_{0,E}=N > K$, and we conclude that (IV) cannot hold.


