# Complex Fourier estimates and weak exponential representations

*Written and self-checked by GPT-6.1 Sol (OpenAI), Ultra reasoning effort, October 2026. Original proofs and figures: CC0.*

A compactly supported distribution has an entire Fourier transform. Its values off the real frequency space can grow exponentially, but the supporting function of its support removes that growth. We prove three weighted estimates, then use them to integrate exponential solutions against a measure which may be singular. The resulting integral defines a distribution; it need not define a value at each point.

This lesson reconstructs the full statements of Hörmander [H-II], Lemma 15.2.2 and Theorem 15.2.4. The existence of the four-condition weight in his Theorem 15.2.1 and its auxiliary Lemma 15.2.3 are separate targets. Here the representation assumes only the two explicitly stated conditions on the weight function.

The received prerequisites are [weighted Fourier spaces](../../AN02-L002.html), especially Theorems 3.1, 4.1 and 4.3; [Hilbert representation](../../AN02-L043.html), Lemma 1.1; and [Schwartz Fourier inversion](../../prerequisites/prerequisite-bridges.html#AN03-DEP-001), Theorem 1.1 and its two-sided inverse bridge. The integration, completeness and smooth-density inputs are proved in [the Lebesgue foundation](../../prerequisites/banach-foundation-bridges.html#section-15-3). We spell out the weighted Hilbert adapter and the estimates used in this lesson rather than changing their conventions.

## CF1. Conventions and the weighted Hilbert space

Let \(n\ge1\), and use complex-linear distribution pairings. For real frequencies,
\[
\widehat f(\xi)=\int_{\mathbb R^n}e^{-ix\cdot\xi}f(x)\,dx,\qquad
f(x)=(2\pi)^{-n}\int_{\mathbb R^n}e^{ix\cdot\xi}\widehat f(\xi)\,d\xi.
\tag{CF1}
\]
Let \(k:\mathbb R^n\to(0,\infty)\) satisfy, for fixed \(C,N>0\),
\[
k(\xi+h)\le(1+C|h|)^N k(\xi).
\tag{CF2}
\]
There is no additional constant multiplying the right side. Reversing the shift proves the reciprocal ratio bound. In particular \(k\) is continuous, \(k\) and \(1/k\) have polynomial growth, and
\[
|\log k(\xi+h)-\log k(\xi)|\le N\log(1+C|h|)\le NC|h|.
\tag{CF3}
\]
The exponent \(N\) need not be an integer. Write \(\check k(\xi)=k(-\xi)\), \(c_n=(2\pi)^{-n/2}\), and
\[
B_{2,k}=\{u\in\mathcal S':\widehat u\text{ is a function},\ k\widehat u\in L^2\},
\qquad \|u\|_{2,k}=c_n\|k\widehat u\|_2.
\tag{CF4}
\]
If \(U\) has \(kU\in L^2\), it defines a tempered distribution: Cauchy–Schwarz bounds \(\int Uq\) by \(\|kU\|_2\|q/k\|_2\), and the latter is controlled by Schwartz seminorms. Its inverse transform is, explicitly,
\[
\langle u,q\rangle=(2\pi)^{-n}\int U(\xi)\widehat q(-\xi)\,d\xi.
\tag{CF5}
\]
Schwartz inversion proves that \(\widehat u=U\). Thus \(J_ku=c_n k\widehat u\) is an onto isometry to \(L^2\), and \(B_{2,k}\) is Hilbert.

Schwartz functions are dense in this space. Indeed, truncate a weighted Fourier function to bounded sets, then approximate it by smooth functions of compact support in frequency. On a bounded set the continuous positive \(k\) has finite upper and positive lower bounds, so ordinary smooth \(L^2\) approximation is also weighted approximation. The inverse transforms are Schwartz functions. Multiplying a Schwartz function by expanding physical cutoffs converges in all its Schwartz seminorms. Hence physical \(C_c^\infty\) functions are dense as well.

For later examples \(B_{2,1}=L^2\) with equal norms. The Schwartz Plancherel identity follows from CF1 and Fubini applied to \(\int f\overline g\). Approximation by Schwartz functions in physical \(L^2\) extends that isometry to all \(L^2\). Its image is all frequency \(L^2\), because it already contains the dense Schwartz space and is closed. Hölder implies that both limits agree with their tempered-distribution limits. This proves the assertion with the factor in CF4.

## CF2. Multipliers and the reflected bilinear dual

Set \(M_k(h)=\sup_\xi k(\xi+h)/k(\xi)\), so \(M_k(h)\le(1+C|h|)^N\). For \(a\in\mathcal S\),
\[
\|au\|_{2,k}\le A_k(a)\|u\|_{2,k},\qquad
A_k(a)=(2\pi)^{-n}\int|\widehat a(h)|M_k(h)\,dh.
\tag{CF6}
\]
To prove this first take \(u\in\mathcal S\). The product formula and CF2 give
\[
k(\xi)|\widehat{au}(\xi)|
\le(2\pi)^{-n}\int|\widehat a(h)|M_k(h)
             k(\xi-h)|\widehat u(\xi-h)|\,dh.
\tag{CF7}
\]
The \(L^1*L^2\) inequality proves CF6. For general \(u\), use the density just proved. The products converge in \(\mathcal S'\) as well as in the displayed weighted space, so their limits are the actual distributional product.

Every continuous complex-linear functional \(L\) on \(B_{2,k}\) has a unique \(w\in B_{2,1/\check k}\) such that
\[
L(u)=\mathcal B(w,u):=(2\pi)^{-n}
           \int\widehat w(-\xi)\widehat u(\xi)\,d\xi,
\qquad \|L\|=\|w\|_{2,1/\check k}.
\tag{CF8}
\]
In fact Hilbert representation, with the inner product linear in its first entry, gives a \(g\in L^2\) with \(L(u)=\int J_ku\,\overline g\) and \(\|g\|_2=\|L\|\). Define
\[
\widehat w(s)=c_n^{-1}k(-s)\overline{g(-s)}.
\tag{CF9}
\]
The polynomial bound and Cauchy–Schwarz make this a tempered function. Substituting CF9 gives CF8 exactly, and its reciprocal reflected weighted norm is \(\|g\|_2\). On Schwartz tests CF5 shows that \(\mathcal B(w,u)\) is the distribution pairing \(\langle w,u\rangle\). Uniqueness follows from Schwartz density, or from uniqueness of \(g\). In particular the reflected weight cannot be replaced by \(1/k\) for a nonsymmetric \(k\).

## CF3. Entire transforms and shrinking cutoffs

For a distribution \(u\) of compact support, define
\[
F_u(z)=\langle u,e^{-ix\cdot z}\rangle,\qquad z=\xi+i\eta\in\mathbb C^n.
\tag{CF10}
\]
A cutoff equal to one near the support makes this pairing precise and makes it independent of the cutoff. Continuity of a compactly supported distribution bounds it by finitely many derivatives on a fixed compact neighborhood. The exponential and all these derivatives admit locally uniformly convergent power series in \(z\); applying \(u\) therefore proves that \(F_u\) is entire. It restricts to \(\widehat u\) on real frequencies, and
\[
\widehat{e^{x\cdot\eta}u}(\xi)=F_u(\xi+i\eta).
\tag{CF11}
\]
These statements include complex-valued distributions and arbitrary finite order.

Let \(K\subset\mathbb R^n\) be nonempty, compact and convex, with
\(H_K(\eta)=\sup_{x\in K}x\cdot\eta\). Choose a nonnegative smooth \(\rho\) supported in the open unit ball with integral one. For \(0<\delta\le1\), put
\[
\chi_\delta=1_{K+B(0,\delta/2)}*\rho_{\delta/4}.
\tag{CF12}
\]
Then \(0\le\chi_\delta\le1\), it is one when the distance to \(K\) is less than \(\delta/4\), and its support is contained in \(K+\overline B(0,3\delta/4)\). Differentiating the smooth factor in the convolution gives
\[
\|\partial^\alpha\chi_\delta\|_1
\le C_{K,\alpha,\rho}\delta^{-|\alpha|}.
\tag{CF13}
\]
Here the volume of \(K+B(0,\delta/2)\) is uniformly bounded by that of \(K+B(0,1)\). Thus CF12 works even when \(K\) has empty interior.

For every integer \(m\ge0\) these cutoffs satisfy
\[
|\widehat{\chi_\delta}(\xi+i\eta)|
\le D_m e^{H_K(\eta)+\delta|\eta|}
                (1+\delta|\xi+i\eta|)^{-m}.
\tag{CF14}
\]
The estimate of order zero follows by integrating the modulus of the exponential over the support. If \(z\ne0\), choose a coordinate with \(|z_j|\ge|z|/\sqrt n\). Integration by parts \(m\) times in that coordinate, keeping the complex frequency in the exponential, gives
\[
|z_j|^m|\widehat{\chi_\delta}(z)|
\le e^{H_K(\eta)+\delta|\eta|}
                     \|\partial_j^m\chi_\delta\|_1.
\tag{CF15}
\]
The derivatives of \(\chi_\delta\), rather than derivatives of \(e^{x\cdot\eta}\), occur in CF15. Combining this bound with the order-zero bound gives CF14, after increasing \(D_m\), also at \(z=0\).

Take an integer \(m>N+n\). Since \(|\xi+i\eta|\ge|\xi|\), substitute \(s=\delta\xi\) into CF14 to obtain
\[
\int|\widehat{\chi_\delta}(\xi+i\eta)|(1+C|\xi|)^N\,d\xi
\le D e^{H_K(\eta)+\delta|\eta|}\delta^{-N-n}.
\tag{CF16}
\]
Indeed \((1+C|s|/\delta)^N\le\delta^{-N}(1+C|s|)^N\), and the remaining integral of \((1+|s|)^{N-m}\) is finite. All constants here are independent of \(\delta\) and \(\eta\).

## CF4. The full complex-volume estimate

**Theorem CF4.** If \(\operatorname{supp}u\subset K\) and \(u\in B_{2,k}\), then
\[
\int_{\mathbb C^n}|F_u(z)|^2e^{-2H_K(\operatorname{Im}z)}
 k(\operatorname{Re}z)^2
 (1+|\operatorname{Im}z|^2)^{-N-2n}\,dV(z)
\le C_{K,k}\|u\|_{2,k}^2.
\tag{CF17}
\]

**Proof.** The function \(a_\eta(x)=\chi_\delta(x)e^{x\cdot\eta}\) is a smooth compact multiplier, and \(a_\eta u=e^{x\cdot\eta}u\). Its real Fourier transform is \(\widehat{\chi_\delta}(\xi+i\eta)\). Apply CF6 and CF16 with
\(\delta=(1+|\eta|)^{-1}\). Since \(\delta|\eta|\le1\), CF11 gives
\[
\int_{\mathbb R^n}|F_u(\xi+i\eta)|^2k(\xi)^2\,d\xi
\le D_{K,k} e^{2H_K(\eta)}
                (1+|\eta|)^{2(N+n)}\|u\|_{2,k}^2.
\tag{CF18}
\]
This also proves that every shifted plane has a finite weighted integral. Multiply by the weight in CF17 and integrate \(\eta\). The remaining factor is bounded by a constant times
\[
(1+|\eta|^2)^{N+n}(1+|\eta|^2)^{-N-2n}
=(1+|\eta|^2)^{-n}.
\tag{CF19}
\]
It is integrable in real dimension \(n\): outside the unit ball its polar integral is bounded by a constant times \(\int_1^\infty r^{-n-1}\,dr\). Tonelli justifies all integrals, including a priori infinite ones. This proves CF17. The exponent is \(N+2n\); the plane estimate CF18 alone would not justify replacing it by \(N+n\). \(\square\)

## CF5. A fixed cutoff recovers a real norm

**Theorem CF5.** Let \(\psi\in C_c^\infty(\mathbb R^n)\) be supported in \(K\). For every compactly supported \(u\in B_{2,k}\) and every real \(\eta\),
\[
\|\psi u\|_{2,k}^2
\le C_{k,\psi}e^{2H_K(-\eta)}
            \int|F_u(\xi+i\eta)|^2k(\xi)^2\,d\xi.
\tag{CF20}
\]
Here \(\psi u\) is pointwise multiplication, not convolution; the support of \(u\) need not lie in \(K\).

**Proof.** Apply CF18 with a large compact convex ball containing the support of \(u\); hence \(v_\eta=e^{x\cdot\eta}u\) belongs to \(B_{2,k}\). Now \(\psi u=(\psi e^{-x\cdot\eta})v_\eta\). For a fixed \(\psi\), the same complex-coordinate integration by parts as CF15 gives, for every integer \(m\),
\[
|\widehat\psi(\xi-i\eta)|
\le D_{m,\psi}e^{H_K(-\eta)}(1+|\xi-i\eta|)^{-m}.
\tag{CF21}
\]
There is no shrinking derivative scale in CF21. Choosing \(m>N+n\) and using \(|\xi-i\eta|\ge|\xi|\) bounds the multiplier constant of \(\psi e^{-x\cdot\eta}\) by \(D_{k,\psi}e^{H_K(-\eta)}\). CF6 and CF11 prove CF20, with the factor \((2\pi)^{-n}\) incorporated into \(C_{k,\psi}\). \(\square\)

## CF6. A unit strip controls the real norm

**Theorem CF6.** For every compactly supported \(u\in B_{2,k}\),
\[
\|u\|_{2,k}^2
\le C_k\int_{|\operatorname{Im}z|<1}|F_u(z)|^2
                                  k(\operatorname{Re}z)^2\,dV(z).
\tag{CF22}
\]
One possible constant is
\[
C_k=(2\pi)^{-n}(1+C)^{2N}\frac{b_n}{b_{2n}},
\tag{CF23}
\]
where \(b_d\) is the volume of the real \(d\)-dimensional unit ball.

**Proof.** If \(F\) is entire, real differentiation gives
\(\Delta|F|^2=4\sum_j|\partial F/\partial z_j|^2\ge0\).
For a smooth function \(g\) with \(\Delta g\ge0\), the derivative of its spherical mean about a point is
\[
\frac{d}{dr}\frac1{\sigma_{d-1}}\int_{S^{d-1}}g(z+r\omega)\,d\omega
=\frac1{\sigma_{d-1}r^{d-1}}\int_{B(z,r)}\Delta g\,dV\ge0.
\tag{CF24}
\]
The divergence theorem proves the equality. The limit at zero is \(g(z)\); integrating spherical means radially proves the ball submean inequality. Thus
\[
|F(\xi)|^2\le b_{2n}^{-1}\int_{|\theta|<1}|F(\xi+\theta)|^2\,dV(\theta).
\tag{CF25}
\]
Multiply by \(k(\xi)^2\), integrate \(\xi\), and write \(\theta=h+i\eta\). CF2 gives
\(k(\xi)\le(1+C)^N k(\xi+h)\) for \(|h|<1\). Translate the real \(\xi\)-integral. For each \(|\eta|<1\), the allowed \(h\) have volume
\(b_n(1-|\eta|^2)^{n/2}\le b_n\). Tonelli then proves CF22–CF23. If the right side is infinite the inequality is still valid; no unproved finiteness is used. \(\square\)

![The indicator of an interval and its complex-plane Fourier energy.](figures/complex-planes-and-support-damping.png)

The figure uses the exact interval model in the learner material: the support factor removes exponential growth on imaginary shifts. CF18 allows a polynomial factor for general weights and distributions. The interval model has a sharper bound which is specific to that model.

## CF7. The two hypotheses for a weak representation

Let \(X\subset\mathbb R^n\) be open and convex. Define
\[
\mathcal B_{2,k}^{\,c}(X)
=\bigcup_{K\Subset X}\bigl(\mathcal E'(K)\cap B_{2,k}\bigr).
\tag{CF26}
\]
This is the space of compactly supported weighted tests in \(X\), not a claim that all these distributions are smooth. Write \(B_{2,1/\check k}^{\rm loc}(X)\) for distributions \(v\) such that \(\chi v\in B_{2,1/\check k}\) for every \(\chi\in C_c^\infty(X)\).

Assume a finite real function \(\phi\) on \(\mathbb C^n\) has the following properties:

1. For every nonempty compact convex \(L\Subset X\), there is \(A_L<\infty\) such that
\[
e^{-\phi(\xi+i\eta)}\le A_L e^{-H_L(\eta)}k(\xi).
\tag{CF27}
\]
2. It is locally Lipschitz, and its real weak gradient obeys
\[
|\nabla\phi(\xi+i\eta)|\le C_\phi+\log(1+|\eta|)
\quad\text{almost everywhere}.
\tag{CF28}
\]

These are precisely the growth and gradient conditions needed below. Neither plurisubharmonicity nor a strictly positive Levi lower bound is an additional assumption here.

Let \(\mu\) be a complex measure on \(\mathbb C^n\), with locally finite total variation and
\[
|\mu|(B(z,1))\le M\quad(z\in\mathbb C^n).
\tag{CF29}
\]
Its variation is sigma finite, by a countable cover by unit balls. Let \(V\) be \(\mu\)-measurable and suppose
\[
Q^2=\int|V(z)|^2e^{2\phi(-z)}\,d|\mu|(z)<\infty.
\tag{CF30}
\]

**Theorem CF7.** There is exactly one \(v\in B_{2,1/\check k}^{\rm loc}(X)\) such that, for every \(u\in\mathcal B_{2,k}^{\,c}(X)\),
\[
\langle v,u\rangle=\int V(z)F_u(-z)\,d\mu(z).
\tag{CF31}
\]
The right side is absolutely convergent. The left side has the canonical compact weighted pairing defined in CF11 below. The notation
\[
v(x)=\int V(z)e^{ix\cdot z}\,d\mu(z)
\tag{CF32}
\]
is consequently a weak formula. It is not an assertion of pointwise absolute convergence.

## CF8. A direct estimate for singular measures

Reflect the variation: \(\nu(E)=|\mu|(-E)\). It also satisfies CF29. For any entire \(F\),
\[
\int|F(z)|^2e^{-2\phi(z)}\,d\nu(z)
\le\frac{4e^{2C_\phi}M}{b_{2n}}
 \int|F(w)|^2e^{-2\phi(w)}
                   (1+|\operatorname{Im}w|)^2\,dV(w).
\tag{CF33}
\]
Both sides may initially be infinite.

To prove this, when \(|z-w|<1\), integrate the gradient bound along their segment to obtain
\[
|\phi(z)-\phi(w)|\le C_\phi+\log(2+|\operatorname{Im}w|).
\tag{CF34}
\]
For completeness CF28 holds almost everywhere, whereas a prescribed segment could meet an exceptional set. Smooth \(\phi\) on a neighborhood of that compact segment by convolution. Its gradient is bounded there by
\(C_\phi+\log(2+|\operatorname{Im}w|+\varepsilon)\).
Integrate the smooth bound along the segment and pass to the locally uniform limit of these convolutions. Local Lipschitz continuity gives that uniform convergence. Let \(\varepsilon\downarrow0\); this proves CF34 for every segment of length less than one.

CF34 implies
\(e^{-2\phi(z)}\le4e^{2C_\phi}(1+|\operatorname{Im}w|)^2e^{-2\phi(w)}\).
Apply the ball submean inequality CF25 at \(z\), multiply by this weight, and integrate \(z\) against \(\nu\). For the resulting nonnegative integrand the exact translation and interchange are
\[
\begin{aligned}
&\int d\nu(z)\int_{|\theta|<1}
 |F(z+\theta)|^2e^{-2\phi(z+\theta)}
                  (1+|\operatorname{Im}(z+\theta)|)^2\,dV(\theta)\\
&=\int |F(w)|^2e^{-2\phi(w)}
       (1+|\operatorname{Im}w|)^2
       \nu(B(w,1))\,dV(w).
\end{aligned}
\tag{CF35}
\]
For each fixed \(z\), substitute \(w=z+\theta\) in Lebesgue measure, then use Tonelli on \(d\nu(z)dV(w)\). This proves CF35 even when \(\nu\) is atomic or supported on a lower-dimensional set. CF29 now proves CF33; a smooth-density approximation of the measure is unnecessary. The power two comes from the coefficient one of the logarithm in CF28.

## CF9. Enlarging a support inside a convex domain

If \(S\Subset X\), its convex hull \(K=\operatorname{conv}S\) is compact and contained in \(X\). Here is the finite-dimensional justification. Every finite convex combination of more than \(n+1\) points can be shortened: the augmented vectors \((x_j,1)\) are linearly dependent, so change the nonnegative coefficients along a nonzero dependence until one becomes zero, keeping their sum and the represented point fixed. Iterate. Thus the hull is the continuous image of the compact set \(S^{n+1}\times\Delta_n\); convexity puts all its points in \(X\). Compactness and openness then give a \(\rho>0\) for which
\[
L=K+\overline B(0,\rho)\Subset X,\qquad
H_L(\eta)=H_K(\eta)+\rho|\eta|.
\tag{CF36}
\]
The support identity follows by maximizing the two summands separately. One may decrease \(\rho\) when a larger fixed neighborhood is needed.

Apply CF27 to this \(L\), and CF17 to \(u\) supported in \(K\). Since
\[
\sup_{r\ge0} e^{-2\rho r}(1+r)^2(1+r^2)^{N+2n}<\infty,
\tag{CF37}
\]
we obtain
\[
\int |F_u(w)|^2e^{-2\phi(w)}
                 (1+|\operatorname{Im}w|)^2\,dV(w)
\le D_{S,\phi,k}\|u\|_{2,k}^2.
\tag{CF38}
\]
The supremum is finite by continuity on bounded intervals and exponential decay against every fixed polynomial at infinity. The additional imaginary damping in CF36 is essential to this deduction. Combining CF33 and CF38 gives
\[
\int|F_u(z)|^2e^{-2\phi(z)}\,d|\mu|(-z)
\le E_{S,\phi,k}M\|u\|_{2,k}^2.
\tag{CF39}
\]
These constants depend on the fixed compact support neighborhood, not on \(u\).

## CF10. Constructing the distribution and its local weight

Cauchy–Schwarz in \(d|\mu|\), using CF30 and CF39, gives
\[
\int|V(z)F_u(-z)|\,d|\mu|(z)
\le Q\bigl(E_{S,\phi,k}M\bigr)^{1/2}\|u\|_{2,k}.
\tag{CF40}
\]
This proves absolute convergence of the complex measure integral and its weighted norm bound for every compact weighted \(u\) with support in \(S\).

Initially define \(v\) on smooth compact tests by that integral. It is complex-linear. On a fixed compact support neighborhood \(S\), integration by parts in a largest real frequency coordinate gives, for every integer \(m\),
\[
|\widehat u(\xi)|\le D_{S,m}
 \max_{|\alpha|\le m}\|\partial^\alpha u\|_\infty(1+|\xi|)^{-m}.
\tag{CF41}
\]
Choose \(m>N+n/2\). The polynomial upper bound on \(k\) makes CF41 integrable in the squared weighted norm. Thus CF40 is bounded by a finite smooth test seminorm on every such \(S\), which is the required distribution continuity.

For any \(\chi\in C_c^\infty(X)\), consider \(h\mapsto\langle v,\chi h\rangle\) first on \(\mathcal S(\mathbb R^n)\). Its support stays in \(\operatorname{supp}\chi\). CF40 and CF6 bound it by a constant times \(\|h\|_{2,k}\). Density extends it to a functional on \(B_{2,k}\). CF8 represents this functional by a unique \(w_\chi\in B_{2,1/\check k}\), and on smooth compact tests \(w_\chi=\chi v\). Consequently \(\chi v\) really has the stated global weighted membership. This proves \(v\in B_{2,1/\check k}^{\rm loc}(X)\).

## CF11. The pairing for every compact weighted test

For \(u\in\mathcal B_{2,k}^{\,c}(X)\), choose \(\chi\in C_c^\infty(X)\) equal to one on a neighborhood of its support, and define
\[
\langle v,u\rangle=\mathcal B(\chi v,u).
\tag{CF42}
\]
This definition extends ordinary smooth test pairing and is independent of \(\chi\).

To prove the assertions without assuming that \(u\) is smooth, choose a compact smooth mollifier of integral one and set \(u_\varepsilon=u*\rho_\varepsilon\). Differentiating the translated mollifier under the distribution shows that \(u_\varepsilon\in C_c^\infty\), supported in a fixed small compact neighborhood of \(\operatorname{supp}u\) inside \(X\). On real frequencies
\[
\widehat{u_\varepsilon}(\xi)=\widehat u(\xi)\widehat\rho(\varepsilon\xi)
\longrightarrow\widehat u(\xi),\qquad
\|u_\varepsilon-u\|_{2,k}\longrightarrow0.
\tag{CF43}
\]
The norm limit follows by dominated convergence, since \(\widehat\rho(\varepsilon\xi)\) is uniformly bounded and tends to one. If \(\chi_1,\chi_2\) are two admissible cutoffs, both are one on the support of \(u_\varepsilon\) for small \(\varepsilon\). Their distribution pairings with this smooth test are therefore equal. Their difference belongs to the global dual space by CF10, so CF8 and CF43 pass that equality to \(u\). This proves independence in CF42.

For one such cutoff, CF8 and CF43 also pass the already proved identity CF31 for \(u_\varepsilon\) to the left side CF42. CF40, applied to \(u_\varepsilon-u\) in their common fixed compact neighborhood, passes the integrals on the right to the integral for \(u\). Thus CF31 holds for all of CF26, not only its smooth tests. Any other \(v\) satisfying it has the same action on \(C_c^\infty(X)\) and hence is the same distribution. This completes Theorem CF7, including its uniqueness, absolute integral and local reflected weight. If \(X\) is empty, its test and distribution spaces are zero and the conclusion is immediate. \(\square\)

## CF12. A weak formula which is not a pointwise integral

Take \(n=1\), \(X=(-2,2)\), \(k=1\), and
\[
\phi(\xi+i\eta)=2\sqrt{1+\eta^2},\qquad
d\mu(\xi+i\eta)=d\xi\text{ on the real axis},\qquad
V(\xi)=(1+\xi^2)^{-3/8}.
\tag{CF44}
\]
Every compact \(L\Subset X\) has \(H_L(\eta)\le2|\eta|\), so CF27 holds with \(A_L=1\). The gradient of \(\phi\) has magnitude at most two, so CF28 holds with \(C_\phi=2\). A complex open unit ball cuts out a real interval of length at most two, giving \(M=2\). The data norm is finite:
\[
Q^2=e^4\int_{\mathbb R}(1+\xi^2)^{-3/4}\,d\xi<\infty.
\tag{CF45}
\]
Indeed its tail beyond \(T\ge1\) is at most \(4e^4T^{-1/2}\). Yet
\(\int_{\mathbb R}|V(\xi)e^{ix\xi}|\,d\xi=\infty\) for every real \(x\). For example, its truncated positive integral obeys
\[
2\int_0^T(1+\xi^2)^{-3/8}\,d\xi
\ge8\,2^{-3/8}(T^{1/4}-1),\qquad T\ge1.
\tag{CF46}
\]
This follows by integrating \((1+\xi^2)^{-3/8}\ge2^{-3/8}\xi^{-3/4}\) on \([1,T]\). At \(x=0\) even the unmodulated integral diverges positively. Nevertheless CF31 defines a unique local \(L^2\) distribution. Globally it is the \(L^2\) inverse transform of \(2\pi V\), by CF1; its weak pairing has exactly the normalization in CF31.

![A singular real-axis measure and the finite data norm beside the divergent pointwise integral.](figures/singular-measure-and-weak-integral.png)

The data curve omits the constant factor \(e^4\), which is stated on the figure. The divergence curve and the proved lower bound illustrate CF45–CF46; they are not a claim that numerical sampling proves the theorem.

## Exact statement map and references

- H-II Lemma 15.2.2, first estimate: CF3–CF4, especially CF17–CF19; nonempty compact convex support, real weight exponent \(N\), complex damping exponent \(N+2n\).
- Its second estimate: CF5, CF20–CF21; multiplication by a fixed supported cutoff, arbitrary compactly supported \(u\), and \(H_K(-\eta)\).
- Its third estimate: CF6, CF22–CF25; the unit imaginary strip and an explicit support-independent constant.
- H-II Theorem 15.2.4: CF7–CF11; CF27–CF30 are the precise assumptions; CF31 and CF42 give the full compact-test action; CF8–CF9 give the reflected dual. CF12 is an additional counterexample to a pointwise reading.
- No assertion here completes H-II Theorem 15.2.1, Lemma 15.2.3, the intervening inductive-topology discussion, or the entire AN-02 course.

[H-II] Lars Hörmander, *The Analysis of Linear Partial Differential Operators II*, Springer, 2005, Lemma 15.2.2, pp. 280–281, and Theorem 15.2.4, pp. 286–287. This lesson contains original arguments and original illustrations, with ordinary source attribution.

The internal Fourier inversion, weighted completeness, multiplier, density and Hilbert proofs linked at the beginning are the actual mathematical inputs. The full course retains its separate recursive prerequisite-closure audit.