Manuscript XV-F@3 · XV-F-v3.tex · lemma · d4e74afa

Lemma: Multiplying a delta derivative by a function

The reconstruction-fixed ambiguity in the punctured normal-jet module › Disjoint classification and the exact role of jet-detection · lines 1952–1966 · proof included or linked

XV-F@3#lem:deltamult

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Let $z_{0} \in X$ , $g \in C^{\infty} (X)$ , and let $μ$ be of minimal length among the multi-indices with $\partial^{μ} g (z_{0}) \neq = 0$ . Then $g \partial^{μ} δ_{z_{0}} = (- 1)^{∣ μ ∣} \partial^{μ} g (z_{0}) δ_{z_{0}}$ .

Proof. For $ψ \in D (X)$ , $⟨ g \partial^{μ} δ_{z_{0}}, ψ ⟩ = (- 1)^{∣ μ ∣} \partial^{μ} (g ψ) (z_{0}) = (- 1)^{∣ μ ∣} \sum_{ν \leq μ} (ν μ) \partial^{ν} g (z_{0}) \partial^{μ - ν} ψ (z_{0})$ . For $ν \leq μ$ , $ν \neq = μ$ , $∣ ν ∣ < ∣ μ ∣$ and $\partial^{ν} g (z_{0}) = 0$ by minimality.

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