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 $\alpha=(\alpha\_1,\dots,\alpha\_n) \in \mathbb{N}^{n}$ 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}
Define the linear subspace $\mathcal{M}\_{\mathcal{S}}$ of the Schwartz space $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$:
\begin{equation}
\mathcal{M}\_{\mathcal{S}}= \{ \psi \in \mathcal{S}: \psi=P\phi, \phi \in \mathcal{S} \},
\end{equation}
and the linear continuous multiplication map $M\_{P}:\mathcal{S} \rightarrow \mathcal{M}\_{\mathcal{S}}$
\begin{equation}
M\_{P}(\phi)=P\phi \quad (\phi \in \mathcal{S}).
\end{equation}
In his work [On the Division of Distributions by Polynomials](http://link.springer.com/article/10.1007/BF02589517), Hörmander proved the following remarkable result (whose proof is unexpectedly very complicated).


Theorem (1). The map $M\_P$ has a linear continuous inverse $M\_{P}^{-1}:\mathcal{M}\_{\mathcal{S}} \rightarrow \mathcal{S}$.


From this result we can easily deduce the following 


Theorem (2). Let $T \in \mathcal{S}'$. Then there exists $S \in \mathcal{S}'$ such that $P \cdot S=T$.


Proof. The map $T \circ M\_{P}^{-1}: \mathcal{M}\_{\mathcal{S}} \rightarrow \mathbb{C}$ is a linear continuous functional, so by the Hahn-Banach Theorem it can be extened to a continuous linear functional $S$ on $\mathcal{S}$. $S$ satifies $S(P\phi)=T(\phi)$ for each $\phi \in \mathcal{S}$, so $P \cdot S = T$.
QED


Hörmander says that an exactly analogous argument proves the following result.


Theorem (3). Let $\Omega$ be an open set of $\mathbb{R}^n$, and $T \in \mathcal{D'}(\Omega)$. Then there exists $S \in \mathcal{D'}(\Omega)$ such that $P \cdot S=T$.


Could you see some way of proving this theorem by using Theorem (1)? The fact is that if we define the subpspace of $\mathcal{D}(\Omega)$
\begin{equation}
\mathcal{M}\_{\mathcal{D}}= \{ \psi \in \mathcal{D}(\Omega): \psi=P\phi, \phi \in \mathcal{D}(\Omega) \},
\end{equation}
and the linear continuous multiplication map $N\_{P}:\mathcal{D}(\Omega) \rightarrow \mathcal{M}\_{\mathcal{D}}$
\begin{equation}
N\_{P}(\phi)=P\phi \quad (\phi \in \mathcal{D}(\Omega)),
\end{equation}
I see no way of deducing from Theorem (1) that $N\_P$ has a linear continuous inverse. If we could do this, then of course we could prove Theorem (3) by using the same argument we used to prove Theorem (2). Any help is welcome.
Thank you very much in advance for your attention.


NOTE. Let me notice that there is instead a way of proving Theorem (3) by using Theorem (2) (but of course this was not what Hörmander had in mind). Let $\Gamma$ be the collection of all open rectangles $\omega$, such that the closure of $\omega$ is a compact set contained in $\Omega$. Clearly $\Gamma$ is an open covering of $\Omega$. Let $\omega \in \Gamma$ and choose $\xi \in \mathcal{D}(\Omega)$ such that $\xi=1$ on $\omega$. Since $\xi \cdot T$ is a distribution with compact support, it defines a tempered distribution, so that by Theorem (2) there exists $V \in \mathcal{S}'$ such that 
\begin{equation}
V(P \phi)= T(\xi \phi) \quad (\phi \in \mathcal{S}).
\end{equation}
In particular, we have
\begin{equation}
V(P\phi)=T(\xi \phi)=T(\phi) \quad (\phi \in \mathcal{D}(\omega)).
\end{equation}
Let us denote with $S\_{\omega}$ the restriction of $V$ to $\mathcal{D}(\omega)$. We have $S\_{\omega} \in \mathcal{D}(\omega)$. Moreover, if $T\_{\omega}$ is the restriction of $T$ to $\mathcal{D}(\omega)$, then we have $ H \cdot S\_{\omega} = T\_{\omega}$. In other terms, the equation $P \cdot S = T$ has a solution on $\omega$.


Now, we know that there exists a locally finite partition of unity $(\psi\_j)\_{j=1}^{\infty}$ in $\Omega$ subordinate to the open cover $\Gamma$ (see Rudin, Functional Analysis, Second Edition, Theorem (6.20)). This means that $(\psi\_j)\_{j=1}^{\infty}$ is a sequence in $\mathcal{D}(\Omega)$, with $\psi\_j \geq 0$, such that:


(i) each $\psi\_j$ has its support in some member of $\Gamma$,


(ii) $\sum\_{j=1}^{\infty} \psi\_j(x)=1$ for every $x \in \Omega$,


(iii) to every compact $K \subset \Omega$ correspond an integer $m$ and an open set $W \supset K$ such that
\begin{equation}
\psi\_1(x)+\dots+\psi\_m(x)=1,
\end{equation}
for all $x \in W$. 


Let $\omega\_j$ be the element of $\Gamma$ which contains the support of $\psi\_j$ according to (i). Then define
\begin{equation}
S(\phi)= \sum\_{j=1}^{\infty} S\_{\omega\_j}(\psi\_j \phi) \quad (\phi \in \mathcal{D}(\Omega)).
\end{equation}
Since for each $\phi \in \mathcal{D}(\Omega)$ only finitely many of the functions $\psi\_j \phi$ are different from zero, it is easy to see that $S$ is well defined, that $S \in \mathcal{D'}(\Omega)$ and that $P \cdot S = T$.
QED



Division of Distributions by Polynomials

> 
> Hörmander's result shows that ${\mathcal M}\_{\mathcal S}$ is an isomorphic copy of $\mathcal S$. Especially, this means that any linear-topological operation one performs on the side of $\mathcal S$ has its analogue on the side of ${\mathcal M}\_{\mathcal S}$. In particular, both are Fréchet spaces. Now, ${\mathcal D}\_K=\{u\in \mathcal S\mid \operatorname{supp}u\subseteq K\}$ for a compact $K\subset\Omega$ is a closed subspace of $\mathcal S$. In view of $M\_P{\mathcal D}\_K = {\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_K$ (note that $\operatorname{supp}\phi=\operatorname{supp}\left(P\phi\right)$ for $\phi\in{\mathcal S}$, since $P$ is a non-zero polynomial) and $\mathcal D(\Omega)= \varinjlim\_K {\mathcal D}\_K$, one has that ${\mathcal M}\_{\mathcal D} = \varinjlim\_K \left( {\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_K\right)$ is an LF space and the restriction $N\_P$ of $M\_P$ to $\mathcal D(\Omega)$ is a
>  linear-topological isomorphism between $\mathcal D(\Omega)$ and ${\mathcal M}\_{\mathcal D}$. Further, the injection of ${\mathcal M}\_{\mathcal D}$ into ${\mathcal D}(\Omega)$ is continuous and also open, the latter by an open mapping theorem due to Pták (see Theorem (4,4) [here](https://www.jstor.org/stable/24489403) and **Edit 1**). Hence, the inductive limit topology of ${\mathcal M}\_{\mathcal D}$ agrees with the topology induced by ${\mathcal D}(\Omega)$.
> 
> 
> 




---


**Edit 1.** Let $K,\,K'$ be compact subsets of $\Omega$ with $K\subset (K')^\circ$ and $\{\phi\_m\}\subset{\mathcal D}\_{K'}$ be a sequence with $P\phi\_m\to 0$ in ${\mathcal D}\_{K'}/{\mathcal D}\_K$. According to Pták's theorem, in order to show that the injection ${\mathcal M}\_{\mathcal D}\to {\mathcal D}(\Omega)$ is open it suffices to prove that $P\phi\_m\to 0$ in $\left({\mathcal M}\_D\cap{\mathcal D}\_{K'}\right)/\left({\mathcal M}\_D\cap{\mathcal D}\_K\right)$. This is the same as $\phi\_m\to 0$ in ${\mathcal D}\_{K'}/{\mathcal D}\_K$. 


We proceed as follows: As ${\mathcal M}\_{\mathcal S}$ is a closed subspace of ${\mathcal S}$, we can regard $({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'})/({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_K)$ as a closed subspace of ${\mathcal D}\_{K'}/{\mathcal D}\_K$ (see **Edit 2**). Under this identification, the convergence $P\phi\_m\to 0$ in ${\mathcal D}\_{K'}/{\mathcal D}\_K$ becomes $P\phi\_m\to 0$ in $({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'})/({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_K)$ (because $P\phi\_m\in {\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'}$). Now using the fact that $M\_P^{-1}$ descends to a linear-topologial isomorphism from $({\mathcal M}\_{\mathcal D}\cap{\mathcal D}\_{K'})/({\mathcal M}\_{\mathcal D}\cap{\mathcal D}\_K)$ onto ${\mathcal D}\_{K'}/{\mathcal D}\_K$, we arrive at the conclusion. $\Box$




---


**Edit 2.** The identification of $({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'})/({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_K)$ as a closed subspace of ${\mathcal D}\_{K'}/{\mathcal D}\_K$ makes use of the short exact sequence
$$
0 \longrightarrow {\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_K \longrightarrow {\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'} \longrightarrow (({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'}) + {\mathcal D}\_K)/{\mathcal D}\_K\longrightarrow 0
$$  

as well as the fact that $({\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'}) + {\mathcal D}\_K$ is a closed subspace of ${\mathcal D}\_{K'}$. To see this, let $\{\phi\_m\}$ be a sequence in ${\mathcal D}\_{K'}$ and $\{\psi\_m\}$ be a sequence in ${\mathcal D}\_K$ such that 
$$
P\phi\_m + \psi\_m\to \zeta \enspace \text{in ${\mathcal D}\_{K'}$.}
$$ 
Note that $P\phi\_m \to \zeta$ in $C^\infty(K'\setminus K^\circ)$, because the $\psi\_m$ vanish to infinite order on $\partial K$. The crux is to see that there exists a $\phi\in{\mathcal D}\_{K'}$ such that $P\phi = \zeta$ in $C^\infty(K'\setminus K^\circ)$. To prove this claim (see **Edit 3**), use Whitney's extension theorem, as in Hörmander's paper, to extend $\zeta|\_{K'\setminus K^\circ}$ to a function $\hat\zeta\in{\mathcal M}\_{\mathcal D}\cap {\mathcal D}\_{K'}$. Then $\phi =M\_P^{-1}\hat\zeta$ has the desired properties. Given that, one has 
$$
\zeta = P\phi + (\zeta-P\phi)
$$ 
with $\zeta-P\phi \in {\mathcal D}\_K$, and is done. $\Box$




---


**Edit 3.** Hörmander in his paper provided estimates which involve both local and global aspects. The global aspects are irrelevant in the present context. Locally, Hörmander obtained pointwise estimates of derivatives of $f$ in terms of derivatives of $Pf$. Especially, one has that, for each $r\in\mathbb N\_0$, there is an $s\in \mathbb N\_0$ such that
$$
 \|f\|\_{C^r(K'\setminus K^0)} \lesssim \|Pf\|\_{C^s(K'\setminus K^0)},
 \quad f\in \mathcal D\_{K'}. \tag{#}
$$
The **example** below will make this clear.


Now, apparently $\zeta/P\in C^\infty(K'\setminus K)$. Utilizing (#) with $f=\phi\_m$ shows that $\zeta/P$ extends smoothly from $K'\setminus K$ to the boundary of $K$. All what remains to be done is to extend $\zeta/P$ further from $K'\setminus K^\circ$ in an arbitrary fashion to a function $\phi\in{\mathcal D}\_{K'}$. $\Box$




---


**Example.** In 2-D, suppose one wants to estimate $f$, $f\_x$, $f\_y$ in terms of derivatives of $Pf$, where $P=x^2y\,Q$ and $Q(0)\neq0$. Then it holds 
$$
\|f\|\_{C^1(\overline U)} \lesssim \|Pf\|\_{C^4(\overline U)}
$$
for any sufficiently small $0$-neighborhood $U$ of $\mathbb R^2$. Indeed, replacing $f$ with $Qf$, one can assume that $P=x^2y$. Then
$$
f(0) = \frac12\,(Pf)\_{xxy}(0), \enspace f\_x(0) = \frac16\,(Pf)\_{xxxy}(0), \enspace f\_y(0) = \frac14\,(Pf)\_{xxyy}(0).
$$
As $P$ is less degenerate at other places near $0$, the estimate follows.


