# Convolution modulo smooth functions and compact singularity bounds

*Written by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026.  Public domain (CC0).*

A smooth part of a compact convolution kernel can be ignored when an equation is considered modulo smooth functions. This gives a larger domain of definition than ordinary convolution. We construct that operation and its composition law, prove the two necessary conditions for solving every distributional datum modulo smooth functions, and express the geometric condition through distances of singular supports.

Basic references are [Grubb's Fourier-distribution notes](https://web.math.ku.dk/~grubb/dist5.pdf), [Melrose's differential-analysis course](https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/), and Hörmander's *The Analysis of Linear Partial Differential Operators*, volumes I and II. The full isolated-singularity smoothing characterization is Theorem 1.1 of Frequency-selective singularities and smooth convolutions. Theorem 1.1 of Slow decrease and entire Fourier division identifies invertibility, and Corollary 6.1 of Recovering singularities from convolution profiles gives the exact compact singular-hull subtraction bound.

We use the Fourier norm, scalar duality and cutoff estimates in Measuring regularity with weighted Fourier spaces, and the all-order embeddings and localization in Local regularity, sharp embeddings, and compactness. Lemma 3.1 of Singular supports and arbitrary distribution data proves that membership of every derivative in one fixed local Sobolev order implies smoothness. Its increasing-order obstruction is the starting mechanism for Section 4; we prove here the additional kernel localization and two-domain argument for general compact convolution kernels. Section 13.10 of Metric and topological foundations supplies smooth cutoffs. All additional arguments are supplied below.

The seminorm Hahn–Banach extension used in the final local duality step is Proposition 1.1 of Continuous functionals, test families and compact limits.

Throughout, \(n\ge1\), all distributions may be complex, and their pairing with tests is bilinear. Write
\[
 \check a(\theta)=a(\theta(-\,\cdot)),\qquad
 S_a=\operatorname{sing\,supp}a,\qquad
 Q(X)=\mathcal D'(X)/C^\infty(X).
 \tag{1.1}
\]
Singular support is the complement of the largest open set on which the distribution agrees with a smooth function. Compact sets are allowed to be empty.

## 1. The local operation on singularity classes

We first record the required patching construction. Given an open cover of an open Euclidean \(X\), there is a locally finite smooth partition of unity subordinate to a refinement with compact supports in cover members. To see this, choose compact exhaustion sets \(K_j\subset\operatorname{int}K_{j+1}\), with their interiors covering \(X\), and put \(K_0=K_{-1}=\varnothing\). Cover each compact annulus \(K_j\setminus\operatorname{int}K_{j-1}\) by finitely many small balls whose closures lie in a member of the given cover and in \(\operatorname{int}K_{j+1}\setminus K_{j-2}\). At each point such a ball exists: the annulus misses \(K_{j-2}\) and lies in \(\operatorname{int}K_{j+1}\). Choose smaller balls still covering the annulus and nonnegative smooth bumps supported in the first balls and positive on the smaller ones. The full family is locally finite, since a compact subset of \(X\) is contained in some \(K_J\), and the supports for \(j>J+2\) miss \(K_J\). Its sum is positive everywhere; divide the bumps by that smooth positive sum. The quotient functions give the asserted partition. The compact exhaustion is the distance-and-radius exhaustion proved in the linked foundation.

**Lemma 1.1 (smooth errors and compact singular sets).** A compact smooth function convolved with a compact distribution is smooth. For compact distributions \(a,b\),
\[
 \operatorname{sing\,supp}(a*b)\subset S_a+S_b.
 \tag{1.2}
\]
If a compact distribution \(v\) has \(S_v\subset X\), then a cutoff \(\chi\in C_c^\infty(X)\) equal to one near \(S_v\) gives
\[
 S_{\chi v}=S_v,\qquad v-\chi v\in C_c^\infty(\mathbb R^n).
 \tag{1.3}
\]

*Proof.* Differentiate the smooth factor in the compact distributional pairing; its derivatives depend continuously and smoothly on the output variable on each compact set, and every derivative passes through the finite-order pairing. This proves the first assertion.

For (1.2), choose cutoffs supported in arbitrarily small neighborhoods of \(S_a,S_b\) and equal to one near those sets. Each removed part is smooth and compact. Every convolution term containing a removed part is smooth by the first assertion. The remaining term has ordinary support inside the sum of the two chosen support neighborhoods. Let their radii decrease to zero. Compactness shows that the intersection of these closed outer sums is \(S_a+S_b\): otherwise a sequence of approximating pairs would have a limit pair in the two compact singular sets representing a point outside their sum. If either singular set is empty, that factor is already smooth and the convolution is smooth. Finally (1.3) follows because the cutoff is one at every singular point and \(v\) is smooth everywhere else. The removed part is compactly supported. \(\square\)

**Theorem 1.2 (canonical convolution on the quotient).** Let \(a\) be compact and let \(X,Y\) be open with
\[
 Y-S_a\subset X.
 \tag{1.4}
\]
There is a well-defined linear map
\[
 a_*:Q(X)\longrightarrow Q(Y).
 \tag{1.5}
\]
It agrees with ordinary convolution wherever the latter is defined. Adding a compact smooth function to \(a\) does not change this map.

*Proof.* Let \(W\Subset Y\) be open. The compact difference \(\overline W-S_a\) lies in \(X\), hence has a positive margin from its closed complement. Choose a compact neighborhood \(L\) of \(S_a\) sufficiently small that \(\overline W-L\Subset X\). Choose \(\theta\in C_c^\infty\) equal to one near \(S_a\), supported in \(L\), and set \(a_W=\theta a\). Then
\[
 a-a_W\in C_c^\infty,\qquad
 \overline W-\operatorname{supp}a_W\Subset X.
 \tag{1.6}
\]
If \(S_a=\varnothing\), take \(a_W=0\).

For \(u\in\mathcal D'(X)\), multiply \(u\) by a fixed compact cutoff equal to one on a neighborhood of the second compact set in (1.6), extend this compact distribution by zero, and form \(a_W*u\) on \(W\). The local restriction is independent of that cutoff: the kernel only samples the displayed compact neighborhood.

Two choices of \(a_W\) differ by a compact smooth function. Their full supports both satisfy the second condition in (1.6). One input cutoff can serve their union of sampling compacts; convolution of the difference with that compact distribution is smooth. Thus their restrictions differ smoothly on \(W\). The same argument on overlaps of two output neighborhoods proves consistency of the local classes. Smooth input changes give smooth output changes.

Choose a locally finite partition of unity \(\eta_j\) on \(Y\), with compact supports inside neighborhoods \(W_j\Subset Y\). If \(g_j=a_{W_j}*u\) there, the locally finite sum of the distributions \(\eta_jg_j\), extended by zero, defines \(g\in\mathcal D'(Y)\). On \(W_i\), the difference between \(g\) and \(g_i\) is
\(\sum_j\eta_j(g_j-g_i)\), which is locally a finite sum of smooth functions. Hence \(g\) represents exactly the local classes just constructed. Any two global representatives with those local classes differ smoothly everywhere and therefore define the same member of \(Q(Y)\).

This proves (1.5), linearity and independence of choices. The argument also permits any compact representative \(a_W\) with \(a-a_W\) smooth and the proper sampling condition in (1.6); it need not be produced by the particular cutoff. This observation proves invariance under adding a compact smooth kernel. When ordinary convolution is defined, it is itself such a local representative, and has the same class. \(\square\)

**Lemma 1.3 (composition).** Suppose compact \(a,b\) and open \(X,Y,Z\) satisfy
\[
 Y-S_a\subset X,\qquad Z-S_b\subset Y.
 \tag{1.7}
\]
Then the convolution kernel \(b*a\) defines a map from \(Q(X)\) to \(Q(Z)\), and
\[
 b_*a_*=(b*a)_*.
 \tag{1.8}
\]
In particular a smooth compact product gives the zero composition.

*Proof.* Lemma 1.1 and (1.7) give
\(Z-S_{b*a}\subset Z-(S_b+S_a)\subset X\), so the product map is defined. On a fixed \(W\Subset Z\), first choose a compact representative \(b_W\) of \(b\), with support in a sufficiently small neighborhood of \(S_b\), so that \(\overline W-\operatorname{supp}b_W\Subset Y\). Next choose a compact representative \(a_W\) of \(a\), close enough to \(S_a\), so that
\[
 \overline W-\operatorname{supp}b_W-\operatorname{supp}a_W
                    \Subset X.
 \tag{1.9}
\]
An input cutoff equal to one near this compact makes all the convolutions ordinary compact convolutions on \(W\). Their associativity is the equality of the two iterated compact distributional pairings applied to the same smooth test. The factor cutoffs make every test compact on the required sampling neighborhoods, so the pairing can be taken in either order.

Furthermore \(b*a-b_W*a_W\) is smooth and compact: expand the difference into products containing \(a-a_W\) or \(b-b_W\) and apply Lemma 1.1. Thus \(b_W*a_W\) represents the product class on \(W\). The local composition agrees with it, and Theorem 1.2 patches these equalities. If either singular set is empty, the corresponding map is zero and the same conclusion follows directly. \(\square\)

## 2. Isolated singularities obstruct noninvertible kernels

Fix a compact kernel \(\mu\), nonempty open \(X_1,X_2\), and assume
\[
 X_2-S_\mu\subset X_1.
 \tag{2.1}
\]
The map \(\mu_*:Q(X_1)\to Q(X_2)\) is now defined. Invertibility always means the exact equivalent slow-decrease and entire-division conditions in the linked preceding theorem.

**Theorem 2.1 (invertibility is necessary).** If \(\mu_*\) is surjective, then \(\mu\) is invertible.

*Proof.* Suppose \(\mu\) is not invertible. The slow-decrease theorem gives a collapsed profile; the isolated-singularity smoothing theorem supplies a compact continuous \(v\), singular exactly at zero, with
\[
 \mu*v\in C_c^\infty.
 \tag{2.2}
\]
The factor \(v\) can have arbitrarily small support. Since \(S_v=\{0\}\), its quotient operation is an endomorphism of \(Q(X_2)\), regardless of its full support.

Choose \(x_0\in X_2\). Surjectivity gives \(\mu_*[u]=[\delta_{x_0}]\). Apply \(v_*\) and Lemma 1.3. The left side is zero because \(v*\mu\) is smooth. The right side is the class of the translate of \(v\) by \(x_0\), which has singular support \(\{x_0\}\) and is therefore nonzero in \(Q(X_2)\). To verify this local description, use any sufficiently small output neighborhood of \(x_0\) and the ordinary compact convolution \(v*\delta_{x_0}\); its restriction has precisely that singularity. This contradiction proves invertibility. Zero and smooth kernels are included: their quotient maps are zero, whereas a point mass defines a nonzero target class. \(\square\)

## 3. The compact singularity condition and its distance form

For each compact \(K_1\subset X_1\), consider the condition that there be compact \(K_2\subset X_2\) with
\[
 \begin{gathered}
 v\in\mathcal E'(X_2),\qquad
 \operatorname{sing\,supp}(\check\mu*v)\subset K_1\\
 \Longrightarrow\quad S_v\subset K_2.
 \end{gathered}
 \tag{3.1}
\]
This controls the singular support of the input, not its ordinary support.

**Proposition 3.1.** Condition (3.1) for every compact \(K_1\) forces invertibility of \(\mu\).

*Proof.* If \(\mu\) is not invertible, neither is \(\check\mu\): reflection replaces each real center and complex argument by its negative in the slow-decrease window bound, and reflection is its own inverse. For any \(x\in X_2\), apply the isolated-singularity theorem to \(\check\mu\), with a support ball compactly contained in \(X_2\), to find \(v_x\in\mathcal E'(X_2)\) singular exactly at \(x\) with \(\check\mu*v_x\) smooth. Apply (3.1) with \(K_1=\varnothing\). It would place every \(x\in X_2\) in one compact subset of \(X_2\), which is impossible for a nonempty open set in positive dimension. Such an open set cannot itself be a compact subset of itself: if it were compact, it would be closed as well, and the connected real space would make it either empty or all of \(\mathbb R^n\); the latter is not compact. \(\square\)

Write \(d_X(A)=\inf_{x\in A}\operatorname{dist}(x,\mathbb R^n\setminus X)\), with \(d_X(\varnothing)=\infty\), and with distance to an empty complement equal to infinity.

**Theorem 3.2 (singular-support distance criterion).** Assume \(\mu\) is invertible. Under (2.1), condition (3.1) for every compact \(K_1\) is equivalent to
\[
 d_{X_2}(S_v)
  =d_{X_1}\bigl(\operatorname{sing\,supp}(\check\mu*v)\bigr)
                  \qquad(v\in\mathcal E'(X_2)).
 \tag{3.2}
\]

*Proof of the forward distance inequality.* Put \(S=S_\mu\), \(T=S_v\), and \(W=\operatorname{sing\,supp}(\check\mu*v)\). Lemma 1.1 gives \(W\subset T-S\). If \(r<d_{X_2}(T)\), every closed \(r\)-ball about a point of \(T\) lies in \(X_2\). Subtract any point of \(S\) and use (2.1): its corresponding ball lies in \(X_1\). Therefore
\[
 d_{X_1}(W)\ge d_{X_2}(T).
 \tag{3.3}
\]
If the right side is infinite, use every finite \(r\). Empty \(T\) gives smooth \(v\), hence empty \(W\); invertibility conversely makes empty \(W\) imply empty \(T\), by the smoothing characterization.

*Proof that confinement gives equality.* Condition (3.1) also applies to every global compact distribution whose singular support lies inside \(X_2\), even if its ordinary support does not. Indeed apply the cutoff in (1.3) to that singular support. It preserves both singular sets in (3.2), because the removed smooth compact part has smooth convolution. The cutoff version lies in \(\mathcal E'(X_2)\).

If (3.3) is strict, its smaller side is finite and \(T\) is nonempty. Choose a nearest pair \(x_0\in T\), \(y_0\notin X_2\). It exists by compactness of \(T\) and closedness of the nonempty complement. Set
\[
 t=|y_0-x_0|=d_{X_2}(T)>0,\qquad
 e=(y_0-x_0)/t.
 \tag{3.4}
\]
For every \(0\le s<t\), \(T+se\subset X_2\). Every image singular set is \(W+se\), since translation commutes with convolution. They all lie in the fixed swept compact
\[
 K_1=W+[0,t]e\Subset X_1:
 \quad \operatorname{dist}(K_1,X_1^c)
                  \ge d_{X_1}(W)-t>0.
 \tag{3.5}
\]
The extended form of (3.1) confines every \(T+se\), \(s<t\), to one compact \(K_2\subset X_2\). But \(x_0+se\to y_0\notin X_2\), contradicting that fixed compact bound. Infinite \(d_{X_1}(W)\) satisfies the same swept-compact assertion by finite-ball containment. Thus equality holds.

*Proof that equality gives confinement.* An invertible compact kernel is nonsmooth, so \(S\) is nonempty. The exact singular-hull subtraction bound, applied to \(\check\mu\), gives
\[
 \operatorname{ch}T
   \subset \operatorname{ch}W+\operatorname{ch}S.
 \tag{3.6}
\]
The sign follows from \(\operatorname{ch}S_{\check\mu}=-\operatorname{ch}S\); the subtraction bound uses convex singular hulls. If \(W=\varnothing\), then \(T=\varnothing\), and confinement is immediate.

For a nonempty compact \(K_1\subset X_1\), put
\[
 B=\operatorname{ch}K_1+\operatorname{ch}S,\qquad
 \delta=\min\{1,d_{X_1}(K_1)\}>0,\qquad
 K_2=\{x\in B:\operatorname{dist}(x,X_2^c)\ge\delta\}.
 \tag{3.7}
\]
The hulls and their sum are compact. The final set is closed in \(B\), hence compact, and its positive distance condition places it in \(X_2\). For empty \(X_2^c\), the distance condition is automatic. Equations (3.2), (3.6) give \(T\subset K_2\) whenever \(W\subset K_1\). If \(K_1=\varnothing\), take \(K_2=\varnothing\). The hull of \(K_1\) need not lie in nonconvex \(X_1\); it is used only for boundedness. This proves (3.1) in all cases. \(\square\)

**Definition 3.3.** For an invertible compact kernel and a pair satisfying (2.1), call the pair **convex for singular supports** when it satisfies the equivalent confinement and distance conditions of Theorem 3.2. One can instead use confinement as the defining condition for arbitrary kernels; Proposition 3.1 then supplies invertibility.

The invertibility condition must be retained when using only the distance identity. For a smooth kernel with \(X_2=\mathbb R^n\), both distances in (3.2) are infinite for every compact input, but the image has no singularities and there is no uniform confinement of the input singularities. The empty-support convention does not make that kernel invertible.

## 4. Why solving every singularity class forces confinement

The proof must accommodate distributions of orders increasing along the equation domain. A single fixed negative Sobolev order for all input data would miss this obstruction.

Use \(D=-i\partial\) and \(H^s=\{v:\langle\xi\rangle^sF_v(\xi)\in L^2\}\), with the preceding Fourier normalization. Compact distributions belong to \(H^s\) for some sufficiently negative \(s\), because their transforms have polynomial growth. We need two elementary estimates. If \(v\in H^s\) is compact and \(\varphi\) is a compact smooth test, Fourier inversion and Cauchy–Schwarz give
\[
 \sup_x|D^\beta(v*\varphi)(x)|
   \le C_\beta\|v\|_{H^s}
                    \|\varphi\|_{H^{|\beta|-s}}.
 \tag{4.1}
\]
Indeed its Fourier integrand is \(\xi^\beta F_vF_\varphi\), and \(|\xi^\beta|\le\langle\xi\rangle^{|\beta|}\); the two weighted factors are square integrable.

If \(k\) is a compact smooth function, every derivative of \(k*\varphi\) has a bound by \(C_{k,\beta,r}\|\varphi\|_{H^r}\) for every real \(r\). Integration by parts makes \(F_k\) decay faster than every power, so the same Fourier Cauchy–Schwarz argument applies with arbitrary \(r\). Both statements remain true for reflected tests and translated evaluations.

**Theorem 4.1 (compact singularity confinement is necessary).** Surjectivity of \(\mu_*:Q(X_1)\to Q(X_2)\) implies (3.1) for every compact \(K_1\subset X_1\).

*Proof.* Suppose (3.1) fails for one compact \(K_1\). Take a compact exhaustion \(L_j\) of \(X_2\). Inductively choose \(v_j\in\mathcal E'(X_2)\) and \(x_j\in S_{v_j}\) such that
\[
 \operatorname{sing\,supp}(\check\mu*v_j)\subset K_1,
 \qquad
 x_j\notin L_j\cup\bigcup_{\ell<j}\operatorname{supp}v_\ell.
 \tag{4.2}
\]
At each stage the forbidden union is compact in \(X_2\), so failure of uniform confinement permits this choice. The points \(x_j\) are locally finite in \(X_2\).

Choose a fixed cutoff \(\chi\in C_c^\infty(X_1)\) equal to one near \(K_1\). Choose decreasing symmetric bounded neighborhoods \(Y_j\) of zero with
\[
 \begin{gathered}
 \operatorname{supp}\chi+\overline Y_1\Subset X_1,\qquad
 \operatorname{supp}v_j+\overline Y_j\Subset X_2,\\
 x_\ell\notin\operatorname{supp}v_j+\overline Y_j
                        \quad(\ell>j).
 \end{gathered}
 \tag{4.3}
\]
For empty \(K_1\) take \(\chi=0\). The last requirement is possible because a locally finite set of future points, disjoint from the compact support, has positive distance from that support; its intersection with a compact neighborhood is finite.

After shrinking \(Y_j\) further, choose a cutoff \(\theta_j\) equal to one near \(S_\mu\) and set \(\kappa_j=\theta_j\mu\), with its support close enough to \(S_\mu\) that
\[
 (\operatorname{supp}v_j+\overline Y_j)
                         -\operatorname{supp}\kappa_j
                              \Subset X_1.
 \tag{4.4}
\]
This uses the compact singular sampling set and (2.1), as in Theorem 1.2. When \(S_\mu=\varnothing\), take \(\kappa_j=0\). The kernel differences \(\mu-\kappa_j\) are smooth and compact. A single order \(r\ge0\) bounds every \(\kappa_j\) as a compact distribution: multiplying \(\mu\), of fixed finite order \(r\), by a smooth cutoff changes its constants but not that order. Consequently
\[
 |F_{\kappa_j}(\xi)|\le C_j\langle\xi\rangle^r
 \tag{4.5}
\]
with possibly different constants. Every compact kernel has such a global finite-order bound by the test topology on one compact neighborhood of its support.

Choose \(s_j\) with \(v_j\in H^{s_j}\), and set \(a_0=0\). The all-derivative Sobolev smoothness lemma supplies a multiindex \(\alpha_j\) for which
\[
 D^{\alpha_j}v_j\notin
       H^{s_j-a_{j-1}}_{\rm loc}(x_j+Y_j),
 \qquad a_j=|\alpha_j|>a_{j-1}.
 \tag{4.6}
\]
If every derivative belonged to that same local space, the lemma would make \(v_j\) smooth near its singular point. All derivatives of order at most \(a_{j-1}\) do belong there by the global Fourier bound; this proves the strict inequality.

The locally finite sum
\[
 f=\sum_{j\ge1}(-D)^{\alpha_j}\delta_{x_j}
                       \in\mathcal D'(X_2)
 \tag{4.7}
\]
is therefore well-defined: on each compact test support only finitely many summands act. Assume surjectivity for its class and choose \(u\in\mathcal D'(X_1)\). On a neighborhood of \(\operatorname{supp}v_j+\overline Y_j\), the actual local representative \(\kappa_j*u\) is defined and has
\[
 \kappa_j*u=f+g_j
 \tag{4.8}
\]
for a smooth \(g_j\) there, by Theorem 1.2. The smooth error may depend on \(j\); no uniform bound on these errors is used.

Put \(A_j=\check\kappa_j*v_j\). It is compactly supported in \(X_1\) by (4.4), and its singular support lies in \(K_1\), because its difference from \(\check\mu*v_j\) is smooth. Test (4.8) on \(v_j*\varphi\), for \(\varphi\in C_c^\infty(Y_j)\). This is a smooth compact test in \(X_2\). The bilinear convolution transpose and (4.3) give
\[
 \begin{aligned}
 (D^{\alpha_j}v_j*\varphi)(x_j)
 &=u(A_j*\varphi)-g_j(v_j*\varphi)\\
 &\quad-\sum_{\ell<j}
       (D^{\alpha_\ell}v_j*\varphi)(x_\ell).
 \end{aligned}
 \tag{4.9}
\]
Future terms vanish. All pairings here are ordinary local pairings, with their full supports inside the relevant domains by (4.3), (4.4). The truncated kernels justify this identity even when ordinary convolution with the full \(\mu\) would not be defined.

The finite sum in (4.9) is bounded by
\(C_j\|\varphi\|_{H^{a_{j-1}-s_j}}\), by (4.1). The smooth-error term has a bound by \(C_{j,t}\|\varphi\|_{H^t}\) for every real \(t\): insert a compact smooth cutoff of \(g_j\) equal to it near the sampling support, move the compact convolution onto that smooth test, and use Fourier duality. Its resulting test in the \(\varphi\) variable is smooth and compact.

Split the image as
\[
 A_j=A_j'+A_j'',\qquad A_j'=\chi A_j,\qquad
                 A_j''=(1-\chi)A_j.
 \tag{4.10}
\]
The second part is smooth and compact, since all image singularities lie where \(\chi=1\). Its support plus \(\overline Y_j\) lies in \(X_1\), as a subset of the full sampling compact in (4.4). Applying the finite order of \(u\) there and the smooth-kernel estimate following (4.1) bounds \(u(A_j''*\varphi)\) by \(C_{j,t}\|\varphi\|_{H^t}\) for any real \(t\).

Equations (4.5) and the exact Sobolev cutoff multiplier estimate give
\[
 A_j'\in H^{s_j-r}.
 \tag{4.11}
\]
Its convolution with \(\varphi\) is supported in the one fixed compact
\(\operatorname{supp}\chi+\overline Y_1\Subset X_1\). Let \(c\) be an order of \(u\) on this compact. Estimate (4.1) therefore gives
\[
 |u(A_j'*\varphi)|
                    \le C_j\|\varphi\|_{H^{c+r-s_j}}.
 \tag{4.12}
\]
If \(\chi=0\), this term vanishes and one may take \(c=0\). The key point is that \(c+r\) is fixed, although the cutoffs and smooth errors vary with \(j\).

Since the strictly increasing integers \(a_{j-1}\) tend to infinity, (4.9)–(4.12), with \(t=a_{j-1}-s_j\), give for all sufficiently large \(j\)
\[
 |(D^{\alpha_j}v_j*\varphi)(x_j)|
              \le C_j\|\varphi\|_{H^{a_{j-1}-s_j}}
                       \quad(\varphi\in C_c^\infty(Y_j)).
 \tag{4.13}
\]
Reflection about \(x_j\) converts the left side into the action of \(D^{\alpha_j}v_j\) on a test in \(x_j+Y_j\), and preserves the Sobolev norm. Multiply an arbitrary Schwartz test by a compact smooth cutoff in that neighborhood. The cutoff multiplier estimate makes the resulting action bounded in \(H^{a_{j-1}-s_j}\). Seminorm Hahn–Banach and the scalar \(L^2\) Fourier duality extend it to that Hilbert space and identify the compactly localized distribution with an element of \(H^{s_j-a_{j-1}}\). For precision, the bounded functional is represented by an \(L^2\) function after the isometry \(h\mapsto\langle\xi\rangle^{a_{j-1}-s_j}F_h\); the bilinear pairing reflects the frequency, which leaves this even weight unchanged. Fourier inversion identifies that representative with the original action on compact tests.

Varying the cutoff proves
\(D^{\alpha_j}v_j\in H^{s_j-a_{j-1}}_{\rm loc}(x_j+Y_j)\), contradicting (4.6). Thus failure of confinement prevents surjectivity, and (3.1) is necessary. \(\square\)

This argument proves necessity for every distributional datum, not just for a fixed-order subspace. It does not yet prove sufficiency of the geometric condition. That requires a separate construction of the quotient solutions.

## References

- Gerd Grubb, *Distributions and Operators*, Chapter 5, “Fourier transformation of distributions,” freely readable [lecture notes](https://web.math.ku.dk/~grubb/dist5.pdf).
- Richard B. Melrose, *Differential Analysis*, MIT OpenCourseWare, Fall 2004, freely readable [course materials](https://ocw.mit.edu/courses/18-155-differential-analysis-fall-2004/).
- Lars Hörmander, *The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis*, second edition, Springer, 1990.
- Lars Hörmander, *The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients*, Springer, 1983.
- The full isolated-singularity, slow-decrease and singular-hull bounds are the exact preceding lessons linked above. The increasing-order Sobolev obstruction in the linked polynomial chapter is extended here with the full general-kernel and two-domain localization argument.