Bourbaki

Proposition 9 (Leibniz Formula)

Tag 030V, permanent. Cite it as [Bourbaki, Tag 030V].

Théories spectrales, chapter IV, § 3. Distributions et distributions tempérées

Let ff be a distribution on U and gg an indefinitely differentiable function on U. Let α∈Nn\alpha \in \mathbf{N}^n. One has the relation

α(α)βα−β \alpha (\alpha )\beta \alpha -\beta
∂(fg)=∑∂f∂g \partial (f g) =\sum\partial f \partial g
β \beta
β⩽α\beta \leqslant \alpha

Proceeding by induction on ∣α∣|\alpha | as in the proof of FVR, I, p. 28, prop. 2, it is enough to consider the case where ∣α∣=1|\alpha |= 1. The result then follows from the calculation

⟨∂α(fg),φ⟩=⟨fg,−∂αφ⟩=⟨f,−g∂αφ⟩ \langle \partial^{\alpha}(f g), \varphi \rangle =\langle f g,-\partial^{\alpha}\varphi \rangle =\langle f,-g\partial^{\alpha}\varphi \rangle
=⟨f,−∂α(gφ)+φ∂αg⟩=⟨g∂αf+f∂αg,φ⟩ =\langle f,-\partial^{\alpha}(g\varphi ) +\varphi \partial^{\alpha}g\rangle =\langle g\partial^{\alpha}f+f \partial^{\alpha}g, \varphi \rangle

valid for φ∈D(U)\varphi \in \mathscr{D}(U).

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

Cited by

Nothing.

Cites

Nothing.

Machine translation of the English, by gpt-5-mini, gpt-5.4, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.