LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-14

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Compact distribution convolution preserves schwartz and tempered spaces

Statement

Let $v \in D^{'} (R^{n})$ have compact support. Then

$C_{v} : S \to S, (C_{v} φ) (x) = (v * φ) (x) = ⟨ v_{y}, φ (x - y) ⟩,$

is continuous and complex-linear. If $u \in S^{'}$ , define $u * v \in S^{'}$ by

$⟨ u * v, ψ ⟩ = ⟨ u_{x}, ⟨ v_{y}, ψ (x + y) ⟩ ⟩ .$

After restricting $u$ and $u * v$ to $D^{'}$ , this agrees with the ordinary distribution convolution in which one factor has compact support. All assertions hold in ZF.

Facts & Assumptions

Given: A compactly supported distribution $v$ , a Schwartz function $φ$ , and a tempered distribution $u$ (Tempered distribution).

[F1]

Compactly supported distributions act continuously on all smooth functions and obey a fixed compact finite-order estimate (Compactly supported distributions extend to smooth functions).

[F2]

The Schwartz seminorms/topology are those of Schwartz space and its seminorms and Schwartz topology and convergence.

[F3]

The smooth-parameter clause for compact distribution pairings holds in ZF (Distribution pairing with smooth parameter families).

[F4]

Distribution convolution with one compactly supported factor is well-defined by the addition-map pairing and is commutative (Convolution of distributions when one has compact support, Convolution of distributions is well defined under the support hypothesis).

[F5]

Restriction embeds $S^{'}$ into $D^{'}$ (Tempered distributions embed continuously in distributions).

Proof

technique · uniform compact-support estimates and transposition

1.1F1F3algebra

Choose a compact neighborhood $K$ of $supp v$ and an order $m$ for the estimate in [F1]. Differentiate by [F3] and expand $x^{α} = ((x - y) + y)^{α}$ .

$p_{α β} (C_{v} φ) \leq C_{v, K, α, β} \max_{∣ γ ∣ \leq m, ∣ δ ∣ \leq ∣ α ∣} p_{δ, β + γ} (φ) .$

Only finitely many terms occur because $y$ stays in $K$ . [F1, F2, F3, algebra]

2.1F2step 1.1

The estimates in step 1.1 show that $C_{v} φ$ is Schwartz and that $C_{v}$ is continuous. Replacing $y$ by $- y$ proves the same statement for $T_{v} ψ (x) = ⟨ v_{y}, ψ (x + y) ⟩$ ; equivalently $T_{v} = C_{\overset{ˇ}{v}}$ for the reflected compact distribution $\overset{ˇ}{v}$ .

3.1givenstep 2.1

The displayed candidate for $u * v$ is $u \circ T_{v}$ . By step 2.1 this is a continuous complex-linear functional on $S$ , hence tempered.

4.1F4step 3.1

For $ψ \in D$ , the inner function $T_{v} ψ$ is precisely the iterated-pairing test used by the addition-map definition in [F4].

$⟨ u * v, ψ ⟩ = ⟨ ρ (u) \otimes v, (x, y) \mapsto ψ (x + y) ⟩$

with the cutoff interpretation prescribed there. This proves agreement after the embedding [F5]. The cases $u = 0$ , $v = 0$ , or empty support give zero. Only the ZF smooth-parameter clause of [F3] was used; no integral-interchange clause or choice axiom entered. [F3, F4, F5, step 3.1] ∎

Depends on

Dependency tree · 2 levels

Sources

Semyon Dyatlov, Lecture notes for 18.155 (2022)