# Division of Distributions by Polynomials

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 $\alpha=(\alpha_1,\dots,\alpha_n) \in \mathbb{N}^{n}$ 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 $\mathbb{C}$: $$P(x)=\sum_{|\alpha| \leq N} c_{\alpha} x^{\alpha} \quad (x \in \mathbb{R}^n).$$ Define the linear subspace $\mathcal{M}_{\mathcal{S}}$ of the Schwartz space $\mathcal{S}=\mathcal{S}(\mathbb{R}^n)$: $$\mathcal{M}_{\mathcal{S}}= \{ \psi \in \mathcal{S}: \psi=P\phi, \phi \in \mathcal{S} \},$$ and the linear continuous multiplication map $M_{P}:\mathcal{S} \rightarrow \mathcal{M}_{\mathcal{S}}$ $$M_{P}(\phi)=P\phi \quad (\phi \in \mathcal{S}).$$ 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 $\mathcal{D}(\Omega)$ $$\mathcal{M}_{\mathcal{D}}= \{ \psi \in \mathcal{D}(\Omega): \psi=P\phi, \phi \in \mathcal{D}(\Omega) \},$$ and the linear continuous multiplication map $N_{P}:\mathcal{D}(\Omega) \rightarrow \mathcal{M}_{\mathcal{D}}$ $$N_{P}(\phi)=P\phi \quad (\phi \in \mathcal{D}(\Omega)),$$ 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 $$V(P \phi)= T(\xi \phi) \quad (\phi \in \mathcal{S}).$$ In particular, we have $$V(P\phi)=T(\xi \phi)=T(\phi) \quad (\phi \in \mathcal{D}(\omega)).$$ 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 $$\psi_1(x)+\dots+\psi_m(x)=1,$$ 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 $$S(\phi)= \sum_{j=1}^{\infty} S_{\omega_j}(\psi_j \phi) \quad (\phi \in \mathcal{D}(\Omega)).$$ 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

After a long discussion with other mathematicians, I arrived at the conclusion that there is no simple way to derive Theorem (3) from Theorem (1) and the Hahn-Banach Theorem, as supposed by Hörmander in his work.

Let us recall the argument Hörmander apparently had in mind when saying that Theorem (3)can be proved in the same way as Theorem (2).

Let $T\in{\mathcal D}'(\Omega)$. Consider the linear functional $T \circ N_{P}^{-1}:{\mathcal M}_{\mathcal D} \rightarrow \mathbb{C}$. This functional is continuous for the topology of ${\mathcal D}(\Omega)$. Indeed, let $\{P\phi_m\}\subset {\mathcal M}_{\mathcal D}\cap {\mathcal D}_K$ for some compact $K\subset \Omega$ be a sequence with $P\phi_m \to 0$ in ${\mathcal D}(\Omega)$. Then $\{\phi_m\}\subset {\mathcal D}_K$ and $P\phi_m\to 0$ in ${\mathcal S}$. From the latter and the continuity of $M_P^{-1}$ one concludes $\phi_m\to 0$ in ${\mathcal S}$, together with the former one then has $\phi_m\to 0$ in ${\mathcal D}_K\subset{\mathcal D}(\Omega)$. Eventually, the continuity of $T$ for the topology of ${\mathcal D}(\Omega)$ shows that $T(\phi_m)\to 0$. So $T \circ N_{P}^{-1}$ is sequentially continuous. This means that $T \circ N_{P}^{-1}$ is continuous if $\mathcal{M}_{D}$ is considered with the inductive limit topology $\tau'$ of the Fréchet sapces $\mathcal{M}_{D} \cap D_{K}$. If this topology would be clearly equal to the topology $\tau$ that $\mathcal{M}_{D}$ inherits from ${\mathcal D}(\Omega)$, we could conclude that $T \circ N_{P}^{-1}$ is continuous with respect $\tau$ and we could proceed by applying the Hahn-Banach Theorem to prove the Theorem (3), exactly as we did in the proof of Theorem (2).

Now note that $\mathcal{M}_{D}$ is a closed linear subspace of ${\mathcal D}(\Omega)$. Indeed, Theorem (1) in the post implies easily that $\mathcal{M}_{D}$ is closed in $\mathcal{S}(\mathbb{R}^n)$, and since the inclusion $\textit{i}: \mathcal{M}_{D} \rightarrow \mathcal{S}(\mathbb{R}^n)$ is continuous, $\mathcal{M}_{D}$ is also closed in ${\mathcal D}(\Omega)$.

We are so lead to consider the following general situation. Let $E$ be an LF-space wich is the strict inductive limti of the Fréchet spaces $E_n$, and $M$ a closed linear subspace of $E$. Let $\tau$ be the subspace topology of $M$ and let $\tau'$ be the strict inductive limit topology on $M$ defined by the sequence of spaces $M \cap E_n$. We may ask the following question (Q): does the set of all continuous linear functionals on $(M,\tau)$ coincide with set of of all continuous linear functionals on $(M,\tau')$? In other words, do the duals of $(M,\tau)$ and $(M,\tau')$ coincide?

Unfortunately, generally speaking the answer to (Q) is negative: for details and references see my post Continuos Linear Mappings.

Hörmander seems to have made the error of having tacitly assumed that the topologies $\tau$ and $\tau'$ coincide. If this was actually the case, he was not the only one: see Trèves, Topological Vector Spaces, Distributions, and Kernels (1967), pp.128-129, where the author humbly admits to have made this mistake a few times in his life.

Obviously, this considerations leave open the problem of establishing whether the map $N_{P}^{-1}$ is continuous. If we can prove this fact, then the same argument used in the proof of Theorem (2) can be applied to prove Theorem (3). A proof of the continuity of $N_{P}^{-1}$ has been finally given by ifw in his reply to my post Division of Distributions (see also my answer to that post for a detailed explanation of ifw's proof). Anyway, the proof is highly non-elementary: it makes use of Pták's Open Mapping Theorem discovered seven years after the publication of Hörmander's work. So for sure, had Hörmander been aware of the difficulty of proving the continuity of $N_{P}^{-1}$, he could not have skipped the proof of Theorem (3) in his paper.