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, 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