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

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Differentiation and polynomial multiplication preserve tempered distributions

Statement

For $u \in S^{'} (R^{n})$ , a multi-index $α$ , and a complex polynomial $P$ , define

$⟨ \partial^{α} u, φ ⟩ = (- 1)^{∣ α ∣} ⟨ u, \partial^{α} φ ⟩, ⟨ P u, φ ⟩ = ⟨ u, P φ ⟩ .$

Both results lie in $S^{'}$ . For fixed $α$ or $P$ , these operations are continuous in both the weak and strong dual topologies, and their restrictions to $D$ agree with the corresponding operations on $D^{'}$ .

Facts & Assumptions

Given: A tempered distribution $u$ , a multi-index $α$ , and a complex polynomial $P$ (Tempered distribution).

[F1]

Differentiation and polynomial multiplication are continuous linear endomorphisms of Schwartz space (Basic operations are continuous on Schwartz space, Smooth polynomially bounded multipliers on schwartz space).

[F2]

Weak and strong dual topologies use point tests and bounded test sets (Weak and strong topologies on tempered distributions).

[F3]

On $D^{'}$ , distributional differentiation has the same sign and smooth multiplication has the same transpose formula (Distributional derivative, Multiplication of a distribution by a smooth function).

Proof

technique · transposition and bounded-set transport

1.1F1given

Each displayed functional is the composition of $u$ with a continuous Schwartz endomorphism, followed in the derivative case by a scalar sign. It is therefore continuous and complex-linear on $S$ , hence belongs to $S^{'}$ .

2.1F2step 1.1

For a single test $φ$ , the absolute value after either operation is a source weak seminorm evaluated at the transformed test. Thus each operation is weakly continuous.

2.2F1F2step 1.1

A continuous linear Schwartz endomorphism sends bounded sets to bounded sets. For a bounded $B$ , the target strong seminorm is therefore the source strong seminorm on $\partial^{α} B$ or $P B$ (the derivative sign disappears under absolute values). This proves strong continuity.

3.1F3∎

If $ψ \in D$ , then $\partial^{α} ψ$ and $P ψ$ are again compactly supported tests. Evaluating the two displayed definitions on $ψ$ gives exactly the formulas in [F3]. Hence restriction to $D^{'}$ commutes with both operations. The cases $α = 0$ , constant $P$ , $P = 0$ , and $u = 0$ follow from the same formulas. No choice axiom is used.

Differentiation and polynomial multiplication preserve tempered distributions

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources