Division of Distributions by Polynomials

Question

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

Answer 1

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 and Edit 1). Hence, the inductive limit topology of ${\mathcal M}_{\mathcal D}$ agrees with the topology induced by ${\mathcal D}(\Omega)$.

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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. $\Box$

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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. $\Box$

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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 $\phi\in{\mathcal D}_{K'}$. $\Box$

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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.

Answer 2

In this answer, I will fill the gaps in ifw's argument, and I will also add some final remarks about my original question. First of all, some notation: here as in ifw's answer, we set for any $\phi \in \mathcal{D}(\mathbb{R}^n)$, any $S \subset \mathbb{R}^n$ and any non-negative integer $m$: \begin{equation} ||\phi||_{C^{m}(S)} =\sup_{\substack{x \in S \\ |\alpha| \leq m}} |(D^{\alpha} \phi)(x)|. \end{equation}

Let us begin with a couple of elementary results which were used by ifw in his argument.

Proposition (1). Let $X$ be a Fréchet space and let $M, N$ be closed linear subspaces of $X$, with $N \subset M$. Consider the quotient map $\pi:X \rightarrow X/N$, which associate to each $x \in X$ the coset $x+N$. Then $\pi(M)$ is a closed subspace of $X/N$, and the identity map $id:M/N \rightarrow \pi(M)$ is an isomorphism of topological vector spaces.

Proof. The map $\rho:X/N \rightarrow X/M$, which associates to each coset $x+N$ the coset $x+M$ is a continuous linear map, with $\ker(\rho)=\pi(M)$, so that $\pi(M)$ is a closed linear subspace of $X/N$. We get that $\pi(M)$ is a Fréchet space. Since the identity map $id:M/N \rightarrow \pi(M)$ is linear, continuous and surjcetive, from the Open Mapping Theorem, we conclude that it is also oen, so it is an isomorphism of topological vector spaces (for a direct proof that $id:M/N \rightarrow \pi(M)$ is open, see Note (2) in the post Subspaces of Quotient Spaces. QED

Proposition (2). Let $E, F, G$ be Fréchet spaces, and let $f:E \rightarrow F$ and $g:F \rightarrow G$ be continuous linear maps such that $0 \rightarrow E \xrightarrow[]{f} F \xrightarrow[]{g} G \rightarrow 0$ is a short exact sequence (that is $f$ is injective, $g$ is surjective, and $\textsf{Im}(f)=\ker(g)$). Then the maps $f$ and $g$ induce the isomorphims of topological vector spaces $\tilde{f}:E \rightarrow \textsf{Im}(f)$ and $\tilde{g}:F/ \ker(g) \rightarrow G$, where $\tilde{f}(x)=f(x)$ for all $x \in E$, and $\tilde{g}(x+\ker(g))=g(x)$ for all $x + ker(g) \in F/ \ker(g)$.

Proof. From $\textsf{Im}(f)=\ker(g)$ we get that $\textsf{Im}(f)$ is a closed linear subspace of $F$, so it is a Fréchet space. Then, the Open Mapping Theorem implies that the linear continuous bijection $\tilde{f}:E \rightarrow \textsf{Im}(f)$ is open, so an isomorphims of topological vector spaces. Since $g$ is surjective, again an application of the Open Mapping Theorem implies that $g:F \rightarrow G$ is open. So the map $\tilde{g}:F/ \ker(g) \rightarrow G$ is continuous, bijective and open. QED

Now, note that, in order to apply Pták's Open Mapping Theorem, it enough to consider the case in which $K=[-L,L]^n$ and $K'=[-L',L']^n$, with $L' > L > 0$. Assume for the moment that the inequality (#) in Edit (3) of ifs's answer holds with these $K$ and $K'$. We see from (#) that for each multi-index $\alpha$ the sequence $D^{\alpha} \phi_m$ is Cauchy with respect to the norm $||.||_{C^{0}(\mathbb{R}^n \backslash K°)}$ (note that for any $r \in \mathbb{N}$ we have $||\phi_m||_{C^{r}(\mathbb{R}^n \backslash K°)}= ||\phi_m||_{C^{r}(K' \backslash K°)}$ since $\phi_m \in \mathcal{D}_{K'}$), so that $D^{\alpha} \phi_m$ converges to a continuous function $f_{\alpha}$ on $\mathbb{R}^n \backslash K°$, and by standard results we get that $D^{\alpha}f_{0}=f_{\alpha}$ on the interior of $\mathbb{R}^n \backslash K°$. Then by using Whitney extension theorem as given in Functions Differentiable on the Boundary, we can extend $f_{0}$ to a function $\phi \in D_{K'}$. Since $P \phi = P f_{0} = \zeta$ on $\mathbb{R}^n \backslash K°$, we get the desired result.

So, in order to complete ifw's proof, we should be able to prove that (#) holds in the case in which $K=[-L,L]^n$ and $K'=[-L',L']^n$. More explicitly, we have to prove that for these $K$ and $K'$, to all non-negative integer $r$ there exist a non-negative integer $s$ and $C > 0$ such that \begin{equation} ||f||_{C^{r}(K' \backslash K°)} \leq C ||Pf||_{C^{s}(K' \backslash K°)} \quad \forall \phi \in \mathcal{D}_{K'} \tag{#}. \end{equation}

I finally realized how to modify Hörmander's original proof in order to get this result: see my answer to the post On an Inequality of Lars Hörmander.

This fills all the gaps in ifw's argument, so that we can finally say ... QED!

Let us conclude by some final remarks about my original question. We record here the argument Hörmander apparently had in mind when saying that Theorem (3) is proven 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 ${\mathcal D}(\Omega)$.

So, we may ask the following general question (Q): let $E$ is an LF-space wich is the strict inductive limti of the Fréchet spaces $E_n$, and $M$ is a closed linear subspace of $E$, is it true that the strict inductive limit topology $\tau'$ on $M$ defined by the sequence of spaces $M \cap E_n$ is equal to the subspace topology $\tau$ of $M$?

Unfortunately, generally speaking the answer to (Q) is negative: see e.g. Trèves, Topological Vector Spaces, Distributions, and Kernels, pp.128-129, where the author also says to have made a few times in his life the error of assuming these topologies are equal. Did Hörmander make the same error when he wrote that Theorem (3) can be proved as Theorem (1)?

Obviously, now we know by ifw's argument that in our case the topologies $\tau$ and $\tau'$ actually coincide (to prove this fact was indeed the hard part of ifw's proof). Anyway, the proof is not elementary at all, so Hörmander could not have skipped it in his paper, if he had been aware of the problem. Note also that ifw's argument makes use of Ptak's Open Mapping Theorem, which was published only in 1965, seven years after the publication of Hormander's work.

The fact that (Q) has generally speaking a negative answer was known already to Grothendieck in 1954: see e.g. Hustad's work A Note on Inductive Limits of Linear Spaces. About explicit examples in which (Q) has a negative answer, see the works quoted in Mennicken and Moller, Well Located Subspaces of LF-spaces, in Barroso (ed.), Functional Analysis, Holomorphy, and Approximation Theory. In particular, see the work by Kascic and Roth A Closed Subspace. See also Jochen Wengenroth's answer to my post Continuous Linear Mappings.