# Frequency-selective singularities and smooth convolutions

Original exposition and proofs: GPT-6.1 Sol (OpenAI), Ultra. CC0 1.0.

An isolated singularity can be arranged to have very small Fourier transform outside prescribed logarithmic neighborhoods. Inside selected neighborhoods it can retain the zero logarithmic profile. This construction characterizes compact distributions that smooth at least one nonsmooth convolution factor.

Basic references are Tao's *245B, Notes 9* for Baire's theorem and uniform boundedness, Tao's *246B, Notes 2* for entire Fourier transforms, and Hörmander's *The Analysis of Linear Partial Differential Operators I* and *II*. Our complete-space argument, peaked-kernel estimates and selection proof are given here. The earlier mathematical inputs are Joint logarithmic-frequency limits, Theorems 2.1 and 4.2; Local compactness and Hartogs bounds; Locating singularities through logarithmic Fourier strips, Section 4; Convolution and joint frequency carriers, Lemmas 2.1–2.3; and Changing centers in Fourier windows, Theorem 3.1. They supply compactness, joint subsequence extraction, the singular-hull criterion, and transport of collapsed profiles.

## 1. The smoothing question

Distributions and functions in this chapter may be complex valued. Use

\[
 F_u(\zeta)=\langle u(x),e^{-ix\cdot\zeta}\rangle,\qquad
 L_u(z,c)=\frac{\log|F_u(c+z\log|c|)|}{\log|c|},
 \quad c\in\mathbb R^n,\quad |c|>2.
 \tag{1.1}
\]

A collapsed profile is the limit \(-\infty\), uniformly on every compact subset of \(\mathbb C^n\). A proper profile is a canonical PSH local \(L^1\) limit. The value zero below always means the proper constant-zero PSH function.

**Theorem 1.1 (a nonsmooth factor can be smoothed exactly when a profile collapses).** For a compact distribution \(u\), these conditions are equivalent:

1. There is an escaping real frequency sequence on which \(L_u\) collapses uniformly locally.
2. For every \(x_0\in\mathbb R^n\) and every \(a>0\), there is a compact continuous function \(w\), supported in the closed ball of radius \(a\) about \(x_0\), with
   \(\operatorname{sing\,supp}w=\{x_0\}\), \(w\notin C^1\), and \(u*w\in C^\infty\).
3. There is a compact distribution \(w\notin C^\infty\) for which \(u*w\in C^\infty\).

The proof is in Section 5. The strengthened location and radius requirement in condition 2 comes from a construction whose test kernels can be supported arbitrarily close to the chosen singular point.

The distinction between a smooth factor and a smooth convolution is essential. Both factors can be nonsmooth while their convolution is smooth. The frequency-selective construction below supplies exactly the missing factor.

## 2. Smooth kernels with sharply concentrated transforms

Fix real centers \(c_j\) with \(Q_j=|c_j|\to\infty\), and positive \(R_j\to\infty\). Discard finitely many terms until \(Q_j>e^2\); a finite prefix of bounded frequency neighborhoods has no effect on the conclusions at infinity. Set

\[
 \ell_j=\log Q_j,\quad k_j=\lfloor\ell_j\rfloor,\qquad
 E=\bigcup_j\{\xi\in\mathbb R^n:
                 |\xi-c_j|<R_j\log Q_j\}.
 \tag{2.1}
\]

No assumption that \(R_j\log Q_j/Q_j\) tends to zero is needed here. The complement of \(E\) may even be bounded.

For a compact continuous function \(v\), define the possibly infinite seminorm

\[
 P_{N,m}(v)=
 \sup_{\substack{\xi\notin E,\ |\xi|\geq2\\
                 \zeta\in\mathbb C^n,\ |\zeta-\xi|\leq m\log|\xi|}}
 (1+|\zeta|)^N|F_v(\zeta)|,\qquad N,m\in\mathbb N,\quad N,m\geq1.
 \tag{2.2}
\]

The supremum over an empty set is zero.

**Lemma 2.1 (whole complex decay for a smooth compact function).** If \(\phi\in C^\infty_c(B_b(0))\), then for every integer \(L\geq0\),

\[
 |F_\phi(\zeta)|\leq C_L(1+|\zeta|)^{-L}
                            e^{b|\operatorname{Im}\zeta|}.
 \tag{2.3}
\]

In particular, every such function has finite \(P_{N,m}\).

*Proof.* On the support of \(\phi\), the modulus of the exponential is at most \(e^{b|\operatorname{Im}\zeta|}\). If \(|\zeta|\geq1\), choose a complex coordinate with \(|\zeta_s|\geq|\zeta|/\sqrt n\). Integration by parts \(L\) times gives
\((i\zeta_s)^L F_\phi(\zeta)=F_{\partial_s^L\phi}(\zeta)\).
The \(L^1\) norm of this derivative and the preceding exponential bound prove (2.3) in this region. For \(|\zeta|\leq1\), the \(L^1\) estimate for \(\phi\) absorbs the factor \((1+|\zeta|)^L\) into the constant.

In a tube in (2.2), put \(q=|\xi|\). For large \(q\),
\(|\zeta|\geq q-m\log q\geq q/2\),
\(|\zeta|\leq2q\), and
\(|\operatorname{Im}\zeta|\leq m\log q\).
Thus the weighted expression is at most a constant times \(q^{N+bm-L}\). Choose an integer \(L>N+bm\). The remaining bounded range of \(q\) gives a bounded complex region, where the entire transform is bounded. This proves finiteness, whether or not the tube centers are outside \(E\). \(\square\)

Choose \(0<b\leq1\) and a nonnegative \(\psi\in C^\infty_c(B_b(0))\) with integral one. Such a function is obtained by normalizing
\(\exp[-1/(1-4|x|^2/b^2)]\) inside \(|x|<b/2\), extended by zero outside. Its support is compactly contained in \(B_b(0)\). Each derivative near the boundary is a finite sum of powers of \((1-4|x|^2/b^2)^{-1}\) times that exponential; the exponential dominates every such power. Hence the zero extension is smooth.

Define a scaled probability density, its convolution power, and a modulation:

\[
 \psi_k(x)=k^n\psi(kx),\qquad
 v_k=\underbrace{\psi_k*\cdots*\psi_k}_{k\text{ factors}},\qquad
 u_j(x)=e^{ic_j\cdot x}v_{k_j}(x).
 \tag{2.4}
\]

All these functions are smooth. Minkowski support addition places \(v_k\), hence \(u_j\), in \(B_b(0)\). Their exact transform and basic bounds are

\[
 F_{u_j}(\zeta)=
       \left[F_\psi\!\left(\frac{\zeta-c_j}{k_j}\right)\right]^{k_j},
 \qquad F_{u_j}(c_j)=1,\qquad
 \|u_j\|_\infty\leq k_j^n\|\psi\|_\infty.
 \tag{2.5}
\]

The transform identity follows from scaling, modulation and the convolution product, with no Fourier normalization factor. The last bound follows by convolving one bounded density with \(k_j-1\) probability densities: integrating their nonnegative product preserves the \(L^\infty\) bound.

**Lemma 2.2 (the peaked kernels vanish in every exterior seminorm).** For each fixed \(N,m\),

\[
 P_{N,m}(u_j)\longrightarrow0.
 \tag{2.6}
\]

*Proof.* Work at an admissible pair \(\xi,\zeta\) from (2.2). Put \(q=|\xi|\), \(k=k_j\), \(Q=Q_j\) and
\(w=(\zeta-c_j)/k\), \(t=|w|\).
Choose constants

\[
 C_m=m\log(2m)+m,\quad
 A_m=2(e+1+C_m),\quad B_m=eA_m,\quad D=e+2.
 \tag{2.7}
\]

The function \(m\log q-q/2\), for \(q>0\), has maximum \(m\log(2m)-m\), so
\(m\log q\leq q/2+C_m\).
The triangle inequality and \(Q<e^{k+1}\) give

\[
 q\leq Q+kt+m\log q
 \leq Q+kt+q/2+C_m,\qquad
 q\leq A_m e^k(1+t).
 \tag{2.8}
\]

Here \(k\leq e^k\) and \(k\geq1\). Consequently

\[
 |\operatorname{Im}w|\leq\frac{m\log q}{k}
 \leq m\log[B_m(1+t)],\qquad
 e^{|\operatorname{Im}w|}\leq[B_m(1+t)]^m.
 \tag{2.9}
\]

Since \(\xi\notin E\), it lies outside the particular \(j\)-th neighborhood. Therefore

\[
 R_j k\leq R_j\log Q\leq|\xi-c_j|
 \leq kt+m\log q,\qquad
 R_j\leq t+m\log[B_m(1+t)].
 \tag{2.10}
\]

For all sufficiently large \(j\), this forces \(t>R_j/2\). Indeed, if \(t\leq R_j/2\), the increasing right side in (2.10) is at most
\(R_j/2+m\log[B_m(1+R_j/2)]<R_j\), because \(\log R_j=o(R_j)\). This is a uniform exclusion of a large ball in the scaled complex parameter.

Also \(1+|\zeta|\leq1+Q+kt\leq De^k(1+t)\). Since \(k\geq1\) and \(D(1+t)\geq1\), it follows that

\[
 (1+|\zeta|)^N|F_{u_j}(\zeta)|
 \leq\left[(eD)^N(1+t)^N|F_\psi(w)|\right]^k.
 \tag{2.11}
\]

Apply (2.3) to \(\psi\), with \(L=N+m+2\). Its support radius is at most one. Together with (2.9), this gives

\[
 (eD)^N(1+t)^N|F_\psi(w)|
 \leq H_{N,m,\psi}(1+t)^{-2}.
 \tag{2.12}
\]

The constant is independent of \(j,\xi,\zeta\). The supremum of this last bound on \(t>R_j/2\) tends to zero. Eventually it is below one, and (2.11) therefore proves
\(P_{N,m}(u_j)\leq[H_{N,m,\psi}(1+R_j/2)^{-2}]^{k_j}\to0\).
If the admissible set is empty the assertion holds directly. \(\square\)

## 3. A complete space and a boundedness argument

Fix a support radius \(0<a\leq1\). Let \(\mathcal F\) consist of continuous functions supported in \(\overline B_a(0)\), smooth on \(\mathbb R^n\setminus\{0\}\), and with every seminorm (2.2) finite. Use compact annuli and increasing seminorms

\[
 K_l=\{x:(l+1)^{-1}\leq|x|\leq l+1\},\qquad
 p_l(v)=\|v\|_\infty+
        \max_{|\alpha|\leq l}\sup_{K_l}|D^\alpha v|
        +\max_{1\leq N,m\leq l}P_{N,m}(v).
 \tag{3.1}
\]

They define a translation-invariant metric

\[
 d(v,w)=\sum_{l=1}^\infty 2^{-l}\min\{1,p_l(v-w)\}.
 \tag{3.2}
\]

**Lemma 3.1 (completeness).** This metric makes \(\mathcal F\) a complete locally convex space.

*Proof.* A sequence Cauchy in \(d\) is Cauchy in every \(p_l\), and conversely. For the first direction, make the \(l\)-th summand small enough to force its minimum below any prescribed number less than one. For the converse, first make the tail of the series small, and then control its finitely many initial summands.

Uniform convergence gives a continuous limit \(v\), still supported in \(\overline B_a(0)\). On each annulus the derivatives converge uniformly. On a small closed cube away from zero, the identity
\[
 v_r(x+he_s)-v_r(x)=\int_0^h\partial_s v_r(x+te_s)\,dt
 \tag{3.3}
\]
passes to the uniform limits. It identifies the limiting first derivatives. Repeating this argument for each derivative identifies all limiting derivatives, so \(v\) is smooth away from zero and convergence holds in the derivative parts of every \(p_l\).

Uniform convergence on the fixed support gives \(F_{v_r}(\zeta)\to F_v(\zeta)\) for each complex \(\zeta\), since the integral difference is at most
\(\operatorname{vol}(B_a)e^{a|\operatorname{Im}\zeta|}\|v_r-v\|_\infty\).
For a fixed \(N,m\), Cauchyness of \(P_{N,m}(v_r-v_s)\) bounds the weighted transform difference by an arbitrary \(\varepsilon\) on its whole admissible set, once \(r,s\) are large. Let \(s\to\infty\) at each point, and then take the supremum. This proves
\(P_{N,m}(v_r-v)\leq\varepsilon\).
It also proves \(P_{N,m}(v)<\infty\) by comparison with one \(v_r\). Thus \(v\in\mathcal F\) and the sequence converges in every \(p_l\), hence in \(d\). The seminorm definition gives local convexity and continuity of the vector operations. \(\square\)

**Lemma 3.2 (Baire's boundedness principle in the needed form).** Let \(\Lambda_j\) be continuous linear scalar functionals on \(\mathcal F\). If \(\sup_j|\Lambda_j(v)|<\infty\) for every \(v\in\mathcal F\), then some \(l\) and \(C\) satisfy

\[
 \sup_j|\Lambda_j(v)|\leq C p_l(v)\quad(v\in\mathcal F).
 \tag{3.4}
\]

*Proof.* First recall why a complete metric space cannot be covered by countably many closed sets with empty interiors. If such sets \(A_s\) cover the space, choose a nonempty closed ball disjoint from \(A_1\). Inductively choose a closed ball inside the preceding ball's interior, disjoint from \(A_s\), and with radius less than \(2^{-s}\). Each choice is possible because the complement of a closed set with empty interior is open and dense. The centers are Cauchy. Completeness gives a limit in every chosen closed ball, hence outside every \(A_s\), contradicting the cover. This is Baire's theorem in the form used here.

Now let \(A_s=\{v:\sup_j|\Lambda_j(v)|\leq s\}\), for positive integers \(s\). These sets are closed and cover \(\mathcal F\). One has nonempty interior. Since the \(p_l\) increase, there are \(v_0,l,\varepsilon>0\) with
\(v_0+\{v:p_l(v)<\varepsilon\}\subset A_s\).
In particular \(v_0\in A_s\), so
\(\sup_j|\Lambda_j(v)|\leq2s\) for \(p_l(v)<\varepsilon\), by subtracting the values at \(v_0+v\) and \(v_0\).
For \(p_l(v)>0\), apply this to \(\varepsilon v/(2p_l(v))\); it gives (3.4) with \(C=4s/\varepsilon\). If \(p_l(v)=0\), then \(v=0\) because the seminorm contains the uniform norm. \(\square\)

This proof supplies the precise functional-analytic statement used next. No closed graph theorem is left as an additional hypothesis.

## 4. Selecting an isolated singularity

**Theorem 4.1 (frequency-selective isolated singularity).** For the centers and neighborhoods (2.1), and every \(0<a\leq1\), there is a compact continuous function \(v\) supported in \(\overline B_a(0)\) such that:

- \(v\) is smooth away from zero, \(\operatorname{sing\,supp}v=\{0\}\), and \(v\notin C^1\);
- \(L_v(\cdot,c)\to-\infty\) uniformly on every compact parameter set as \(c\notin E\) escapes;
- along a subsequence of the original centers, \(L_v(\cdot,c_j)\to0\) in local \(L^1\).

*Proof.* Set

\[
 t_j=\exp(\sqrt{\log Q_j}),\qquad
 \frac{\log t_j}{\log Q_j}\to0,\quad
 \frac{t_j}{k_j^n}\to\infty,\quad
 \frac{t_j}{Q_j}\to0.
 \tag{4.1}
\]

These limits follow from \(\sqrt s/s\to0\), \(\sqrt s-n\log s\to\infty\), and \(\sqrt s-s\to-\infty\), with \(s=\log Q_j\) and \(k_j\leq s<k_j+1\).

The functionals \(\Lambda_j(v)=t_j F_v(c_j)\) are continuous on \(\mathcal F\): their absolute values are at most \(t_j\operatorname{vol}(B_a)\|v\|_\infty\). Suppose they were pointwise bounded. Lemma 3.2 would give (3.4) for some \(l,C\).
Choose a probability bump \(\psi\) supported in a ball of radius
\(b<\min\{a,(l+1)^{-1}\}\). The modulated kernels (2.4) belong to \(\mathcal F\), by Lemma 2.1. They and all their derivatives vanish on \(K_l\). Lemma 2.2 and (2.5) give

\[
 p_l(u_j)\leq k_j^n\|\psi\|_\infty+o(1),\qquad
 |\Lambda_j(u_j)|=t_j.
 \tag{4.2}
\]

This contradicts \(t_j/k_j^n\to\infty\) in (3.4). Hence some \(v\in\mathcal F\) has \(\sup_j t_j|F_v(c_j)|=\infty\).

Choose increasing indices \(j_r\) with \(t_{j_r}|F_v(c_{j_r})|\geq r\). This is possible because each finite set of functionals has finite values at \(v\). Since \(v\) is integrable, \(|F_v(c)|\leq\|v\|_1\) on real frequencies. Therefore

\[
 -\frac1{\sqrt{\log Q_{j_r}}}
 \leq L_v(0,c_{j_r})
 \leq\frac{\log\|v\|_1}{\log Q_{j_r}},\qquad
 L_v(0,c_{j_r})\to0.
 \tag{4.3}
\]

The lower bound uses \(r\geq1\). The selected nonzero transform values also show that \(v\neq0\).

If \(v\) were \(C^1\), its compactly supported first derivatives would be integrable. Choose a coordinate with \(|c_s|\geq|c|/\sqrt n\), and integrate by parts once to get
\(|F_v(c)|\leq\sqrt n\max_s\|\partial_s v\|_1/|c|\).
It follows from \(t_j/Q_j\to0\) that \(t_j|F_v(c_j)|\to0\), a contradiction. Thus \(v\notin C^1\). It is smooth off zero by membership in \(\mathcal F\); its singular support is consequently exactly \(\{0\}\). If the continuous representative were distributionally smooth at zero, it would agree everywhere there with its smooth representative, making it \(C^1\), which has just been excluded.

For exterior collapse, let \(|z|\leq m\), and let \(c\notin E\), \(q=|c|\) large. The point \(\zeta=c+z\log q\) lies in (2.2), and \(1+|\zeta|\geq q/2\). Hence, when \(P_{N,m}(v)>0\),

\[
 L_v(z,c)\leq-N+
     \frac{\log P_{N,m}(v)+N\log2}{\log q}.
 \tag{4.4}
\]

If that seminorm is zero, the transform vanishes at every admissible point, and the upper bound is \(-\infty\) directly. Taking \(N\) arbitrarily large proves uniform collapse on this parameter ball. Every compact parameter set is contained in such a ball. The statement remains valid when the exterior has no escaping frequencies.

We finish by proving the zero-profile assertion, rather than inferring it from a value at one point. The family \(L_v(\cdot,c)\) has local uniform upper bounds because
\(|F_v(c+z\log|c|)|\leq\|v\|_1|c|^{a|\operatorname{Im}z|}\).
For each \(\varepsilon>0\), cut off \(v\) smoothly near zero:
\(v=v_\varepsilon+g_\varepsilon\), where \(v_\varepsilon\) is continuous and supported in \(B_\varepsilon(0)\), and \(g_\varepsilon\) is smooth and compactly supported. The first transform is bounded by
\(\|v_\varepsilon\|_1|c|^{\varepsilon|\operatorname{Im}z|}\).
Lemma 2.1 makes the second transform smaller than any negative power of \(|c|\), uniformly on a fixed compact parameter set: choose its derivative order larger than the support radius times the parameter bound plus the desired decay order. The sum estimate thus gives

\[
 L_v(z,c)\leq\varepsilon|\operatorname{Im}z|+o(1)
 \quad\text{uniformly on each fixed parameter compact set}.
 \tag{4.5}
\]

The error may depend on that compact set and on \(\varepsilon\).

Apply PSH compactness along any subsequence of the selected centers. Uniform local collapse is impossible by (4.3). A proper further local \(L^1\) limit \(V\) satisfies \(V\leq0\): pass (4.5) to almost-everywhere limits, then to canonical representatives by ball-average recovery, and let \(\varepsilon\downarrow0\).
For every ball about zero, the submean inequality and \(L^1\) convergence give
\[
 0=\lim_{r\to\infty} L_v(0,c_{j_r})
 \leq \frac1{\operatorname{vol}(B_\rho)}
                   \int_{B_\rho}V\,dV.
 \tag{4.6}
\]
Shrinking \(\rho\) gives \(V(0)\geq0\). Thus \(V(0)=0\), while \(V\leq0\). The submean inequality now forces every such ball average to be zero; since \(V\leq0\), \(V=0\) almost everywhere on every ball. Canonical recovery gives \(V\equiv0\).
Every selected subsequence therefore has a further local \(L^1\) subsequence with this one limit. A subsequence staying a positive distance from zero in some compact \(L^1\) norm would contradict that conclusion. Hence the whole selected sequence converges to zero in local \(L^1\). This proves the last assertion. \(\square\)

**Corollary 4.2 (a prescribed singular point and small support).** For any \(x_0\) and any \(a>0\), the construction can be supported in \(\overline B_a(x_0)\), with singular support \(\{x_0\}\). Exterior collapse persists, and the selected proper profile is \(x_0\cdot\operatorname{Im}z\), with carrier \(\{x_0\}\).

*Proof.* Use radius \(\min\{a,1\}\) in Theorem 4.1, and translate its function by \(x_0\). The transform is multiplied by \(e^{-ix_0\cdot\zeta}\). Its normalized logarithm therefore increases exactly by \(x_0\cdot\operatorname{Im}z\), which is bounded on every compact parameter set. This preserves collapse and gives the asserted proper limit. Translation preserves continuity, failure of \(C^1\), and the singular-support location. \(\square\)

## 5. Completing the smoothing equivalence

*Proof of Theorem 1.1.* Suppose condition 1 holds, along \(c_j\). Theorem 3.1 in Changing centers in Fourier windows supplies radii \(R_j\to\infty\) such that \(L_u\) collapses throughout the union \(E\) in (2.1), uniformly on every parameter compact set as its real center escapes.
Apply Corollary 4.2 to obtain the compact continuous \(w\) at the prescribed point and radius. Its normalized logarithms collapse outside \(E\).

The convolution product identity gives the exact extended equality

\[
 L_{u*w}(z,c)=L_u(z,c)+L_w(z,c).
 \tag{5.1}
\]

Both input families have a uniform local upper bound for all large real centers, from ordinary compact Fourier growth. On \(E\), the first summand collapses and the second is bounded above. On the exterior, the second collapses and the first is bounded above. More explicitly, on a compact parameter set choose an upper bound \(A\) for both families. For any \(K>0\), take common thresholds after which the designated collapsing summand in each region is at most \(-K-A\). Then (5.1) is at most \(-K\) throughout all escaping frequencies. The extended convention handles zeros and the case \(u=0\).

Thus every logarithmic profile of \(u*w\) is collapsed. Section 4 of Locating singularities through logarithmic Fourier strips proves that the convex singular hull is the closed convex hull of the union of the proper profile carriers; it is empty in this situation. A distribution has empty singular support exactly when it is smooth. Hence \(u*w\in C^\infty\), proving condition 2. Condition 2 implies condition 3 because \(w\notin C^1\).

Finally suppose condition 3 holds. The nonsmooth compact \(w\) has a proper profile on some escaping sequence: otherwise the closed convex hull of its proper profile carriers would be empty, and the same singular-hull theorem would make it smooth. Preserve that sequence and extract a profile of \(u\) by compactness, as in Theorem 4.2 of Joint logarithmic-frequency limits. If this profile were proper, the sum of the two proper profiles would be proper in local \(L^1\), by Lemma 2.2 of Convolution and joint frequency carriers. Equation (5.1) would make it a proper profile of \(u*w\). This contradicts the uniform collapse of a smooth compact function, proved in Lemma 2.1 and also in the earlier joint-profile chapter.
Therefore the profile of \(u\) on this subsequence is collapsed, which is condition 1. This proves all three implications, with the precise regularity, compact support and location requirements. \(\square\)

## References

- Terence Tao, [245B, Notes 9: The Baire category theorem and its Banach space consequences](https://terrytao.wordpress.com/2009/02/01/245b-notes-9-the-baire-category-theorem-and-its-banach-space-consequences/), 2009. Background on complete metric spaces and uniform boundedness.
- Terence Tao, [246B, Notes 2: Some connections with the Fourier transform](https://terrytao.wordpress.com/2021/01/23/246b-notes-2-some-connections-with-the-fourier-transform/), 2021. Background on entire transforms and support; its normalization differs from (1.1).
- Lars Hörmander, *The Analysis of Linear Partial Differential Operators I*, Springer, 1983. Compact Fourier transforms and analytic logarithms.
- Lars Hörmander, *The Analysis of Linear Partial Differential Operators II*, Springer, 1983, Chapter XVI. Frequency-selective singularities and smooth convolutions. The full peaked-kernel bounds, complete-space selection and convolution implications are proved here.