# One derivative order for global convolution solutions

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

A distribution can have finite order on each compact set while needing increasingly many derivatives as the compact sets grow. This chapter asks when convolution equations can be solved with one derivative order throughout the unknown domain. The relevant Fourier condition uses a single polynomial exponent in every logarithmic window, however narrow. We connect that condition to logarithmic profiles, obtain a fixed derivative loss for compact tests, and construct weights that retain this loss over an unbounded exhaustion. At the end, concentrated tests prove the converse for a fundamental solution.

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 mathematical criterion of very slow decrease and its relation to finite-order convolution solutions are due to Hörmander. All additional arguments used here are provided below.

Read Slow decrease and entire Fourier division, Theorem 1.1, for ordinary invertibility, and Changing centers in logarithmic Fourier windows, Theorem 2.2, for translation of limiting profiles. Local compactness and Hartogs bounds supplies the precise PSH compactness and Hartogs comparison. The scalar-logarithm result in Entire logarithms and the approximation of plurisubharmonic functions proves the local integrability used in complex averages.

Theorem 3.1 of Convex supports and convolution cancellation proves the ordinary compact convolution-support hull identity. Proposition 1.1 and the compact-subsequence argument in Continuous functionals, test families and compact limits provide complex seminorm Hahn–Banach and the compactness tools used in our weighted construction. Fourier inversion, including its normalization, is Theorem 1.1 of Fourier transforms, finite spectra and convex separation.

## 1. A fixed exponent in every logarithmic window

Work in \(\mathbb R^n\), \(n\ge1\). Distributional pairings are complex linear and bilinear in the distribution and its test. Let \(\mu\) be compactly supported, put
\[
 F(z)=\mu(e^{-ix\cdot z}),\qquad
 T\varphi=\check\mu*\varphi,\qquad
 \check\mu(\theta)=\mu(\theta(-\,\cdot)).
 \tag{1.1}
\]
Reflection here does not conjugate coefficients. The transpose identity is
\[
 (\mu*u)(\varphi)=u(T\varphi).
 \tag{1.2}
\]
For a nonnegative integer \(k\), \(\mathcal D^{\prime k}(X)\) means that the distribution has order at most \(k\) on every compact subset of \(X\), with constants allowed to depend on that compact. Write
\[
 \mathcal D'_F(X)=\bigcup_{k\ge0}\mathcal D^{\prime k}(X),
 \qquad
 p_k(\varphi)=\max_{|\alpha|\le k}\sup_{\mathbb R^n}
                       |\partial^\alpha\varphi|.
 \tag{1.3}
\]
The index in this union must work on every compact subset; it does not depend on the compact.

**Definition 1.1 (very slow decrease).** The transform \(F\) is very slowly decreasing if there is \(A\ge0\) such that, for every \(\varepsilon>0\),
\[
 \liminf_{|c|\to\infty}|c|^A
   \sup_{\substack{h\in\mathbb R^n\\|h|<\varepsilon\log|c|}}
                         |F(c+h)|>0.
 \tag{1.4}
\]
Thresholds and positive lower constants may depend on \(\varepsilon\). The exponent \(A\) is fixed first. Ordinary slow decrease in the linked entire-division theorem allows the exponent to depend on the width.

Assume \(\mu\) has order at most \(M\) and
\(\operatorname{supp}\mu\subset\overline B_C(0)\). For every \(b>C\), the finite-order bound applied with a cutoff supported in \(B_b(0)\), equal to one near the support, gives
\[
 |F(z)|\le D_b(1+|z|)^M e^{b|\operatorname{Im}z|}.
 \tag{1.5}
\]
Indeed each derivative of the cutoff times the exponential is a finite sum of a cutoff derivative times a monomial in \(z\) times the exponential. The order bound has degree at most \(M\), and the exponential on that cutoff support has the displayed bound. The constant may depend on \(b\). Letting \(b\downarrow C\) in asymptotic estimates below retains the exact radius \(C\).

For real \(|c|>2\), define
\[
 \ell_c=\log|c|,\qquad
 L_c(z)=\frac{\log|F(c+\ell_c z)|}{\ell_c}.
 \tag{1.6}
\]
Logarithms at zeros equal \(-\infty\). A proper profile is the canonical PSH representative of a local \(L^1\) limit along escaping real centers. A collapsed profile is convergence to \(-\infty\) uniformly on each compact parameter set. Estimate (1.5) gives local uniform upper bounds for these functions. For \(F\not\equiv0\), the linked logarithm theorem gives their local integrability, and PSH compactness supplies either a proper extraction or a collapsed extraction.

## 2. Complex averages and the real values of profiles

Let \(B_\delta=\{z\in\mathbb C^n:|z|<\delta\}\), and let \(|B_\delta|\) denote real \(2n\)-dimensional volume.

**Lemma 2.1 (the absolute logarithmic average).** If (1.4) holds with exponent \(A\), then, for every \(\delta>0\),
\[
 \limsup_{|c|\to\infty}
 \frac1{|B_\delta|}
     \int_{B_\delta}|L_c(z)|\,dV(z)
           \le 2M+A+2C\delta.
 \tag{2.1}
\]
The absolute value surrounds the logarithm, rather than its argument alone.

*Proof.* Fix a small \(e>0\) and \(b>C\). For every sufficiently large real \(c\), (1.4) supplies a real \(d\) with
\[
 |d-c|<e\ell_c,\qquad
 |F(d)|\ge a_e|c|^{-A},\qquad a_e>0.
 \tag{2.2}
\]
If necessary, decrease a positive lower bound by a factor of two to choose a point in the open window. Put \(R=(\delta+e)\ell_c\). The physical complex ball \(B(c,\delta\ell_c)\) is contained in \(B(d,R)\). Throughout the latter, (1.5) bounds
\[
 \log|F|\le U_c=(M+b(\delta+e))\ell_c+D,
 \tag{2.3}
\]
with a fixed \(D\) on a large tail. Here \(|z|\le |c|+O(\ell_c)\), so the polynomial logarithm differs from \(M\ell_c\) by a bounded amount. Increase \(D\) so that \(U_c\ge0\).

The PSH ball mean inequality at \(d\), followed by (2.2), gives
\[
 \frac1{|B(d,R)|}\int_{B(d,R)}
                      (U_c-\log|F(z)|)\,dV(z)
 \le U_c-\log|F(d)|.
 \tag{2.4}
\]
The integrand on the left is nonnegative. At every point,
\(|\log|F||\le U_c+(U_c-\log|F|)\).
Integrate this inequality on the smaller ball, use (2.4) on the larger ball, and divide by \(\ell_c\). Since the volume ratio is
\(((\delta+e)/\delta)^{2n}\), the resulting limsup is at most
\[
 M+b(\delta+e)
 +\left(\frac{\delta+e}{\delta}\right)^{2n}
                       (M+b(\delta+e)+A).
 \tag{2.5}
\]
All integrals are finite by local integrability of the scalar entire logarithm. First let \(e\downarrow0\), then \(b\downarrow C\). This proves (2.1). \(\square\)

We need a bounded, rather than infinitely negative, real-to-complex comparison to recover a fixed exponent from the profile condition.

**Lemma 2.2 (passing a real ceiling to a profile).** Suppose PSH functions \(v_j\) on \(\mathbb C^n\) are locally uniformly bounded above and converge properly in local \(L^1\) to \(v\). If
\[
 v_j(x)\le -a+o(1)
       \quad(x\in\mathbb R^n,\ |x|<e),
 \tag{2.6}
\]
uniformly in that real ball, then \(v(0)\le-a\).

*Proof.* Put \(r=e/4\). For real \(|x|<e/2\) and a real unit vector \(\nu\), restrict \(v_j\) to the complex line \(x+w\nu\). On the rectangle
\(-r<t<r,\ 0<s<r\), all real points on its bottom lie in the ball of (2.6). Choose one nonnegative upper bound \(H\) for all these line restrictions on the closed rectangles; their images lie in a fixed compact complex set. If \(b_j=-a+o(1)\) is the bottom ceiling, increase \(H\) so that \(b_j\le H\) on a tail.

The explicit harmonic function
\[
 \omega(t,s)=
 \cos\!\left(\frac{\pi t}{2r}\right)
 \frac{\sinh(\pi(r-s)/(2r))}{\sinh(\pi/2)}
 \tag{2.7}
\]
vanishes on the vertical and top sides, lies between zero and one, and is at most one on the bottom. The harmonic function
\(H-(H-b_j)\omega\) is therefore above \(v_j\) on every boundary side. Subharmonic maximum comparison gives this bound inside. At \(t=0\), \(\omega(0,0)=1\) and its derivative in \(s\) is bounded on \([0,r]\). Consequently, for a fixed \(D\),
\[
 v_j(x+iy)\le -a+o(1)+D|y|
       \quad(|x|<e/2,\ |y|<r).
 \tag{2.8}
\]
For \(y\ne0\), use \(\nu=y/|y|\) and \(s=|y|\); for \(y=0\), use (2.6). The constants are uniform in \(x,\nu,j\).

Local \(L^1\) convergence transfers (2.8) to \(v\) almost everywhere. It also transfers it to the canonical representative everywhere: apply the submean inequality on shrinking complex balls at the point in question, and bound their integrals by the continuous ceiling on the right. In particular, this gives \(v(0)\le-a\). Identically collapsed line restrictions satisfy every comparison and cause no exception. \(\square\)

**Theorem 2.3 (three equivalent quantitative criteria).** For a compact kernel with order \(M\) and support in \(\overline B_C\), the following conditions are equivalent:

1. The transform is very slowly decreasing.
2. For some \(B\ge0\), every \(\delta>0\) satisfies
   \[
    \limsup_{|c|\to\infty}\frac1{|B_\delta|}
                \int_{B_\delta}|L_c|\,dV\le B+2C\delta.
    \tag{2.9}
   \]
3. There are no collapsed profiles, and there is \(B\ge0\) such that every proper profile obeys
   \[
                     v(x)\ge-B\quad(x\in\mathbb R^n).
    \tag{2.10}
   \]

Condition 1 with exponent \(A\) gives condition 2 with \(B=2M+A\). It also gives condition 3 with \(B=A\). Condition 3 with \(B\) gives condition 1 with any exponent strictly greater than \(B\).

*Proof.* If \(F\equiv0\), all three conditions are false: the window supremum is zero, every absolute logarithmic integral is infinite, and all profiles collapse. Otherwise the scalar logarithms are locally integrable. Lemma 2.1 proves \(1\Rightarrow2\). Condition 2 excludes collapse by integration on any fixed ball. For a proper extraction, \(L^1\) convergence transfers (2.9) to
\[
 \frac1{|B_\delta|}\int_{B_\delta}|v|\,dV\le B+2C\delta.
 \tag{2.11}
\]
The canonical subharmonic value at zero is the limit of its ball means as the radius decreases to zero. Indeed the submean inequality puts the value below each mean, while upper semicontinuity puts the limiting upper bound of those means below the value. This argument also covers a proposed value \(-\infty\), by using an arbitrary finite negative neighborhood ceiling. Since the mean of \(v\) is bounded below by the negative of (2.11), it follows that \(v(0)\ge-B\). For real \(x\), change the centers to \(c_j+x\log|c_j|\). The exact preceding translation theorem makes the new profile \(v(z+x)\), and condition 2 still holds for those escaping centers. Applying the same argument at zero gives (2.10). Thus \(2\Rightarrow3\).

For \(3\Rightarrow1\), choose \(A>B\). If (1.4) failed for a width \(e>0\), an escaping sequence would have its nonnegative expression tending to zero. On a tail,
\[
 L_{c_j}(x)\le-A
               \quad(x\in\mathbb R^n,\ |x|<e).
 \tag{2.12}
\]
PSH compactness gives a further proper or collapsed extraction. Condition 3 excludes collapse. Lemma 2.2 makes the proper limit satisfy \(v(0)\le-A<-B\), a contradiction.

For the sharper final assertion under condition 1, collapse already contradicts any fixed-width lower bound. Suppose a proper profile has \(v(0)<-A\). Upper semicontinuity gives a small complex ball and a constant ceiling strictly below \(-A\). Hartogs comparison against that constant makes \(L_{c_j}\) eventually lie strictly below \(-A\) on a smaller closed ball. The lower bound (1.4) for a real window contained in this ball contradicts that conclusion. Hence \(v(0)\ge-A\). The same change of real centers gives \(v(x)\ge-A\) for every real \(x\). \(\square\)

This proves the real lower-bound interpretation for *all* limiting profiles. It is stronger than asking that each profile have its own finite lower bound.

## 3. A fixed loss in test-function estimates

**Theorem 3.1 (uniform derivative loss on each fixed support).** Suppose (1.4) holds with exponent \(A\), and \(\mu\) has order \(M\). Choose an integer
\[
                         r>n+2M+A.
 \tag{3.1}
\]
For every compact \(K\subset\mathbb R^n\) and integer \(k\ge0\), there is \(D_{K,k}\) such that
\[
 p_k(\varphi)\le D_{K,k}\,p_{k+r}(T\varphi)
       \quad(\varphi\in C_c^\infty(\mathbb R^n),
                         \ \operatorname{supp}\varphi\subset K).
 \tag{3.2}
\]
The loss \(r\) does not depend on \(K\) or \(k\).

*Proof.* The zero test needs no argument. For a nonzero test, let
\(\psi=T\varphi\) and \(q=k+r\). The transform factorization is
\[
             F_\psi(z)=F(-z)F_\varphi(z).
 \tag{3.3}
\]
The kernel is nonzero by (1.4), and the transform of a nonzero compact test is a nonzero entire function. Their product is nonzero, so \(p_q(\psi)>0\). Divide the test by this seminorm to assume \(p_q(\psi)=1\).

The support of \(\psi\) lies in one fixed compact set \(K-\operatorname{supp}\mu\), inside some ball of radius \(R\). Integrating by parts \(q\) times in a coordinate with \(|z_s|\ge|z|/\sqrt n\) proves
\[
 |F_\psi(z)|\le D_{K,q}(1+|z|)^{-q}
                         e^{R|\operatorname{Im}z|}.
 \tag{3.4}
\]
For \(|z|\le1\), the ordinary integral bound supplies the same estimate with an increased constant. The derivatives used in the integration by parts are bounded by one, and their integrals are bounded by a fixed containing box volume. This is uniform over all normalized tests.

For real \(|c|>2\), take the complex ball centered at \(c\) with radius \(\delta\log|c|\). On it \(|z|\) is comparable with \(|c|\) on a large tail. Hence (3.4) implies
\[
 \frac1{|B_\delta|}\int_{B_\delta}
       \log|F_\psi(c+\ell_c z)|\,dV(z)
 \le(-q+R\delta)\ell_c+D.
 \tag{3.5}
\]
The nonzero entire logarithms in (3.3) are locally integrable. Their identity holds almost everywhere, since their zero sets have volume zero, as proved in the linked scalar-logarithm theorem. Submean for \(\log|F_\varphi|\), (3.3), and the absolute logarithmic estimate for \(F(-z)\) now give, for each \(t>0\) and all sufficiently large \(|c|\),
\[
 \log|F_\varphi(c)|
 \le[-q+(R+2C)\delta+2M+A+t]\ell_c+D.
 \tag{3.6}
\]
Reflection preserves the window condition, order and support radius.

The strict inequality (3.1) permits first choosing \(t>0\), then \(\delta>0\), and a number \(s>0\), so that the coefficient in (3.6) is at most \(-k-n-s\). All choices depend on \(K,k\), not on the normalized test. Thus
\[
                    |F_\varphi(c)|
       \le D(1+|c|)^{-k-n-s}
                       \quad(c\in\mathbb R^n).
 \tag{3.7}
\]
Here is the bounded-center justification. Apply submean on a complex ball of radius one at each bounded real center. Bound \(\log|F_\psi|\) uniformly from (3.4), and bound the negative integral of \(\log|F(-z)|\) by its integral on a single larger compact complex region. Local integrability makes that bound finite; all these unit balls have the same volume. The almost-everywhere logarithmic identity gives a uniform upper bound for \(\log|F_\varphi|\) at every such center. This fills the bounded range in (3.7).

For \(|\alpha|\le k\), Fourier inversion gives
\[
 |\partial^\alpha\varphi(x)|
 \le(2\pi)^{-n}\int_{\mathbb R^n}
               |c|^{|\alpha|}|F_\varphi(c)|\,dc.
 \tag{3.8}
\]
The bound (3.7) makes every integral finite and uniformly bounded: in polar coordinates the large-radius integrand is at most a constant times \(t^{-1-s}\). Restore the normalization \(p_q(\psi)\). This proves (3.2). \(\square\)

In particular \(T\) is injective on compact smooth tests. The sum of derivative suprema and \(p_k\) are equivalent through the finite number of multiindices, so the result also holds with their sum without changing \(r\).

## 4. Weights that preserve one global order

**Theorem 4.1 (whole-space finite-order solvability).** Under the hypotheses of Theorem 3.1, every \(f\in\mathcal D^{\prime k}(\mathbb R^n)\) has a solution
\[
             \mu*u=f,\qquad
             u\in\mathcal D^{\prime\,k+r+1}(\mathbb R^n).
 \tag{4.1}
\]
The constants controlling \(u\) may grow with the compact set.

*Proof.* Put \(R_0=k+r+1\). For a positive locally Lipschitz function \(a\ge1\), define on smooth compact functions
\[
 q_a(\psi)=\sum_{|\alpha|\le R_0}
                  \sup_x a(x)|\partial^\alpha\psi(x)|.
 \tag{4.2}
\]
We will construct one such \(a\) with
\[
                         |f(\varphi)|\le q_a(T\varphi)
                         \quad(\varphi\in\mathcal D(\mathbb R^n)).
 \tag{4.3}
\]
Only the weights will change during the construction; \(R_0\) is fixed.

Choose \(C\) containing the kernel support as before. For \(j\ge1\), use compact balls
\[
 K_j=\overline B_j(0),\qquad
 H_j=\overline B_{j+C+1}(0),\qquad K_0=H_0=\varnothing.
 \tag{4.4}
\]
The full compact support-hull theorem implies, for compact distributions \(v\),
\[
             \operatorname{supp}(Tv)\subset K_j
             \quad\Longrightarrow\quad
             \operatorname{supp}v\subset
                             \overline B_{j+C}(0)
                             \subset\operatorname{int}H_j.
 \tag{4.5}
\]
For nonzero \(v\), choose any \(y\in\operatorname{supp}\check\mu\). The hull identity says that \(x+y\in K_j\) for every \(x\in\operatorname{supp}v\); since \(|y|\le C\), this proves (4.5). If \(Tv=0\), compact Fourier injectivity gives \(v=0\): the product of two nonzero entire transforms cannot vanish identically. Thus an empty output support has an empty receiver, in agreement with \(K_0,H_0\).

On tests supported in \(H_1\), the order-\(k\) estimate for \(f\) and (3.2) give
\[
                  |f(\varphi)|\le Dp_{k+r}(T\varphi).
 \tag{4.6}
\]
Take a constant \(a_1\ge\max\{1,D\}\). Then (4.3) with \(a_1\) holds on that test space.

We next prove the extension step. Suppose (4.3) with \(a_j\) holds on tests supported in \(H_j\), and fix \(e>0\). If \(j=1\), set \(b_j=1\). If \(j\ge2\), set
\[
                      b_j(x)=\min\{1,\operatorname{dist}(x,K_{j-1})\}.
 \tag{4.7}
\]
These nonnegative Lipschitz functions vanish on \(K_{j-1}\) and are strictly positive outside it. There is an integer \(N\ge1\) such that
\[
 a_{j+1}=(1+e)a_j+Nb_j
 \tag{4.8}
\]
makes (4.3) hold for all tests supported in \(H_{j+1}\).

Suppose otherwise. For each \(N\), choose a test violating the proposed bound and multiply it by a scalar of positive modulus so that
\[
 |f(\varphi_N)|=1+e,\qquad
 q_{a_j}(T\varphi_N)<1,\qquad
 T\varphi_N\longrightarrow0
       \text{ in }C^{R_0}_{\mathrm{loc}}(\mathbb R^n\setminus K_{j-1}).
 \tag{4.9}
\]
The last assertion follows because \(N b_j\) has a positive lower bound tending to infinity on each fixed compact set outside \(K_{j-1}\). For \(j=1\), this holds on every compact set.

The images \(T\varphi_N\) have one fixed compact support. Since \(a_j\ge1\), their \(p_{R_0}\) seminorms are bounded. Theorem 3.1 applied with index \(k+1\) and support \(H_{j+1}\) therefore bounds \(p_{k+1}(\varphi_N)\). Arzelà–Ascoli gives a subsequence converging in \(C^k\) on all of \(\mathbb R^n\) to a \(C^k\) function \(\varphi_0\) supported in \(H_{j+1}\). To see the relevant compactness directly, the derivatives of order at most \(k\) have bounded gradients and hence uniform moduli of continuity on one containing box; extract uniform limits for finitely many derivatives. Their line-integral identities identify them as the derivatives of the limiting function. Outside the common support all these functions vanish.

Regard \(\varphi_0\) as a compact regular distribution. Distributional convergence and convolution continuity identify the limit of \(T\varphi_N\) as \(T\varphi_0\). Its support lies in \(K_{j-1}\) by (4.9). Thus (4.5) puts
\(\operatorname{supp}\varphi_0\subset H_{j-1}\Subset\operatorname{int}H_j\).
When \(j=1\), \(\varphi_0=0\).

An order-\(k\) distribution acts continuously on compact \(C^k\) functions by extension from smooth functions. More explicitly, fix a compact neighborhood containing all supports. The \(p_k\) bound extends the functional to the completion of that smooth test space. Mollification of a compact \(C^k\) function converges in \(C^k\), and its supports lie in this neighborhood for small radii, so the extension applies. Consequently (4.9) gives
\[
                             |f(\varphi_0)|=1+e.
 \tag{4.10}
\]
This already contradicts \(\varphi_0=0\) when \(j=1\).

In the other cases, choose a nonnegative smooth probability bump \(\rho\) in the unit ball and put
\(\rho_d(x)=d^{-n}\rho(x/d)\).
For small \(d>0\), \(\varphi_0*\rho_d\) is smooth and supported in \(H_j\). On a fixed compact neighborhood of every image support, local Lipschitz continuity and the positive minimum of \(a_j\) give
\[
 \frac{a_j(x)}{a_j(y)}\le1+Ld
                              \quad(|x-y|\le d).
 \tag{4.11}
\]
Passing derivatives to the smooth image in the convolution integral proves
\[
                 q_{a_j}(\psi*\rho_d)\le(1+Ld)q_{a_j}(\psi)
 \tag{4.12}
\]
for each \(\psi=T\varphi_N\), with the same \(L\). All relevant functions have one fixed compact support enlarged by at most the unit ball.

For fixed \(d\), the mollified images converge with every derivative to \(T\varphi_0*\rho_d\). This follows directly by pairing their distributional difference against translates of each derivative of \(\rho_d\); alternatively first convolve the compact kernel with \(\rho_d\) and integrate against the uniformly convergent \(\varphi_N-\varphi_0\). The fixed compact support and bounded weights pass (4.12) to the limit. Hence
\[
                  q_{a_j}(T(\varphi_0*\rho_d))\le1+Ld.
 \tag{4.13}
\]
The inductive hypothesis applies to \(\varphi_0*\rho_d\), so its \(f\)-value has modulus at most \(1+Ld\). Let \(d\downarrow0\). Continuity in the \(C^k\) completion gives \(|f(\varphi_0)|\le1\), contradicting (4.10). This proves the extension step.

Choose positive \(e_j\) with \(\sum_j e_j<\infty\), and apply (4.8) successively. The weights increase. On a fixed compact neighborhood, \(b_j=0\) for all sufficiently large \(j\), since the balls \(K_{j-1}\) exhaust the whole space. Subsequent changes there are only multiplication by \(1+e_j\). Their infinite product is finite, since
\(\prod_j(1+e_j)\le\exp(\sum_j e_j)\).
Thus
\[
                            a(x)=\lim_j a_j(x)
 \tag{4.14}
\]
exists, is finite and at least one. It is locally Lipschitz: on any fixed compact neighborhood it equals one finite-stage locally Lipschitz weight times a constant tail product. Every compactly supported test eventually lies in \(H_j\), and \(a\ge a_j\), proving (4.3).

Define on the linear range \(T\mathcal D\)
\[
                              \Lambda(T\varphi)=f(\varphi).
 \tag{4.15}
\]
The estimate (4.3) makes this well-defined even without a separate injectivity argument. It also bounds \(|\Lambda(\psi)|\le q_a(\psi)\). Complex seminorm Hahn–Banach extends it to a complex-linear functional \(u\) on \(\mathcal D\) with the same bound. On a fixed compact support \(L\), (4.2) is at most a fixed multiple of \(p_{R_0}\), since \(a\) is bounded there and there are only finitely many multiindices. Therefore \(u\) is a distribution of order at most \(R_0\) on every compact, with this same \(R_0\). Finally (1.2) and (4.15) give \(\mu*u=f\). \(\square\)

## 5. Arbitrary open equation and solution domains

For open \(X_1,X_2\), ordinary convolution is defined on \(X_2\) for data on \(X_1\) when
\[
                         X_2-\operatorname{supp}\mu\subset X_1.
 \tag{5.1}
\]
The pair is convex for supports if every compact \(K_1\Subset X_1\) has a compact \(K_2\Subset X_2\) such that compact inputs \(v\) supported inside \(X_2\) obey
\[
                  \operatorname{supp}(Tv)\subset K_1
                  \quad\Longrightarrow\quad
                  \operatorname{supp}v\subset K_2.
 \tag{5.2}
\]
The same criterion for compact distributions, rather than only smooth tests, is proved in Support distances and admissible convolution domains.

**Theorem 5.1 (finite order on an arbitrary support-convex pair).** Suppose (5.1) holds, the pair is convex for supports, and \(F\) is very slowly decreasing. Then
\[
                \mu*\mathcal D'_F(X_1)=\mathcal D'_F(X_2).
 \tag{5.3}
\]
More precisely, with \(r\) as in (3.1), every \(f\in\mathcal D^{\prime k}(X_2)\) has a solution in \(\mathcal D^{\prime\,k+r+2}(X_1)\).

*Proof.* Lemma 1.1 of Solving convolution equations for arbitrary distribution data gives
\[
             f=g|_{X_2}+h,\qquad
             g\in\mathcal D^{\prime\,k+1}(\mathbb R^n),\qquad
             h\in C^\infty(X_2).
 \tag{5.4}
\]
Theorem 4.1 finds \(U\in\mathcal D^{\prime\,k+r+2}(\mathbb R^n)\) with \(\mu*U=g\). Restriction to \(X_1\) and the sampling inclusion (5.1) give
\(\mu*(U|_{X_1})=g|_{X_2}\): every compact output test samples only a compact subset of \(X_1\), so the global and restricted convolution pairings coincide.

Very slow decrease implies ordinary slow decrease. For completeness, for each chosen window width (1.4) gives a positive polynomial lower bound on a large tail. A nonzero entire \(F\) cannot vanish on an open real ball, by successive one-variable identity principles. The window supremum is positive and continuous in its real center, so has a positive minimum on every remaining compact range. These facts extend the lower bound to every center; the elementary constant-absorption Lemma 2.2 in the linked entire-division chapter puts it in the precise form of its Theorem 1.1. Hence \(\mu\) is invertible.

Theorem 8.1 of Fréchet duality and smooth convolution solvability now applies to invertibility, (5.1) and support convexity. It gives a smooth \(w\) on \(X_1\) with \(\mu*w=h\). Thus \(u=U|_{X_1}+w\) solves the equation. A smooth function is a distribution of order zero on every compact, with constants given by its integral over that compact. Adding it does not increase the displayed order of \(U\).

For the reverse inclusion in (5.3), if \(u\) has one fixed order \(s\), the compact kernel has order \(M\). On each compact output region, use (5.1) to select compact input neighborhoods and cutoffs. The two finite Leibniz estimates for the convolution pairing bound it by derivatives of the output test of order at most \(s+M\). This index is fixed throughout \(X_2\); only the cutoff and pairing constants vary with the compact region. Hence \(\mu*u\in\mathcal D'_F(X_2)\). \(\square\)

No boundedness of \(X_1\) and no convexity for singular supports is needed in this theorem. The fixed-order Fourier hypothesis is what permits the global conclusion.

## 6. A finite-order fundamental solution forces the criterion

**Theorem 6.1 (necessity with the order of the inverse).** If
\[
                    \mu*u=\delta_0,\qquad
                    u\in\mathcal D^{\prime m}(\mathbb R^n),
 \tag{6.1}
\]
then (1.4) holds for every \(A>m\) and every \(\varepsilon>0\). In fact its expression tends to infinity.

*Proof.* Every compact test support \(K\) has its image under \(T\) in a fixed compact set. The order-\(m\) bound for \(u\), (1.2), and Fourier inversion give
\[
 |\varphi(0)|\le D_K p_m(T\varphi)
 \le D'_K\int_{\mathbb R^n}
           (1+|\eta|)^m|F(-\eta)F_\varphi(\eta)|\,d\eta
        \quad(\operatorname{supp}\varphi\subset K).
 \tag{6.2}
\]
The second inequality follows by differentiating the inverse Fourier integral for the smooth compact image. Its integral is absolutely convergent.

Choose a nonnegative smooth probability bump \(\theta\) positive on a ball, and let
\(\chi=\theta*\check\theta\), where \(\theta\) is real. Then \(\chi\) is a smooth even probability density. Its transform on real arguments is
\[
                     h(\eta)=|F_\theta(\eta)|^2.
 \tag{6.3}
\]
Thus \(0\le h\le1\), \(h(0)=1\), and \(h\) is rapidly decreasing. Moreover \(h(\eta)<1\) for \(\eta\ne0\). Equality in the triangle bound for \(F_\theta\) would require \(e^{-ix\cdot\eta}\) to have constant phase on the ball where \(\theta>0\), which is impossible for nonzero \(\eta\).

Evenness and the second-order Taylor bound at zero give, for small \(|\eta|\),
\[
 h(\eta)\ge e^{-D|\eta|^2}.
 \tag{6.4}
\]
Indeed the first derivatives vanish, \(h(0)=1\), and a Taylor remainder first gives \(h\ge1-D_1|\eta|^2\); on a small ball this is at least \(e^{-2D_1|\eta|^2}\). Rapid decrease and strict inequality away from zero give
\[
                   \rho=\sup_{|\eta|\ge1}h(\eta)<1.
 \tag{6.5}
\]
The supremum is attained in a bounded annulus after excluding a tail where \(h\) is small.

For \(t\ge1\) and integer \(N\ge1\), choose a smooth compact function with transform
\[
                    F_\varphi(\eta)=h((\eta+c)/t)^N.
 \tag{6.6}
\]
It is the modulation by \(e^{-ic\cdot x}\) of the \(N\)-fold convolution of \(t^n\chi(tx)\). Its support lies in a ball of radius \(NR_\chi/t\), where \(R_\chi\) contains the support of \(\chi\). Fourier inversion and (6.4) give
\[
 (2\pi)^n\varphi(0)=t^n\int h(s)^N\,ds
                           \ge d\,t^nN^{-n/2}
 \tag{6.7}
\]
for all sufficiently large \(N\): integrate on a ball of radius \(N^{-1/2}\), on which \(h^N\ge e^{-D}\).

Put \(Q=|c|\) and
\[
 S_t(c)=\sup_{\substack{a\in\mathbb R^n\\|a|<t}}|F(c+a)|.
 \tag{6.8}
\]
In the integral (6.2), substitute \(\eta=-c+ts\). On \(|s|<1\), the kernel factor is at most \(S_t(c)\); the equality of the open-ball supremum and the closed-ball maximum follows by continuity. The elementary estimate
\(1+|-c+ts|\le(1+Q)t(1+|s|)\) gives
\[
 I_{\mathrm{near}}\le D(1+Q)^m t^{n+m}S_t(c).
 \tag{6.9}
\]
Here \(\int_{|s|<1}(1+|s|)^m h(s)^Nds\) is uniformly bounded.

On \(|s|\ge1\), real Fourier growth of the order-\(M\) kernel bounds \(|F(c-ts)|\le D(1+|c-ts|)^M\). Rapid decrease of \(h\) gives an integer \(N_0\) with
\(\int(1+|s|)^{m+M}h(s)^{N_0}ds<\infty\).
Together with (6.5), for \(N\ge N_0\) this proves
\[
 I_{\mathrm{far}}\le
       D(1+Q)^{m+M}t^{n+m+M}\rho^{\,N-N_0}.
 \tag{6.10}
\]

Fix the desired \(\varepsilon>0\), set \(t=\varepsilon\log Q\), and choose
\[
          N=\lceil L\log Q\rceil,\qquad
          L>\frac{m+M+2}{-\log\rho}.
 \tag{6.11}
\]
All tests have one fixed support on a large tail, since \(N/t\) is bounded. Thus the constants in (6.2) are independent of \(c\) there. By (6.7), their value at zero is bounded below by a positive multiple of \((\log Q)^{n/2}\). By (6.10) and (6.11), the far integral tends to zero, since it is bounded by a logarithmic power times \(Q^{-2}\). Equations (6.2), (6.7) and (6.9) consequently force
\[
                    S_t(c)\ge
       d_\varepsilon Q^{-m}(\log Q)^{-m-n/2}
                       \quad(Q\text{ sufficiently large}).
 \tag{6.12}
\]
For every \(A>m\), multiplication by \(Q^A\) makes this lower bound tend to infinity. This proves the asserted condition with real window width \(\varepsilon\log Q\). \(\square\)

**Corollary 6.2 (finite-order fundamental solutions).** A compact kernel has a whole-space fundamental solution of one finite order if and only if its transform is very slowly decreasing.

*Proof.* Necessity is Theorem 6.1. For sufficiency, apply Theorem 4.1 to \(f=\delta_0\), which has order zero. It gives a fundamental solution of order at most \(r+1\) for every integer \(r>n+2M+A\). This bound need not be the optimal order. \(\square\)

## References

- Gerd Grubb, *Distributions and Operators*, open lecture notes, [Fourier transformation of distributions](https://web.math.ku.dk/~grubb/dist5.pdf) and the [complete chapter list](https://web.math.ku.dk/~grubb/distribution.htm).
- Richard Melrose, *Differential Analysis*, [MIT OpenCourseWare, Fall 2004](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*, Springer.
- Lars Hörmander, *The Analysis of Linear Partial Differential Operators II: Differential Operators with Constant Coefficients*, Springer.