Example

Tag 0EEL, permanent. Cite it as [Bourbaki, Tag 0EEL].

Variétés différentielles et analytiques, chapter 2, § 14. Opérateurs différentiels

Let $J$ be a finite set, and $X$ an open set of $K^J$ . Let $H$ be a scalar $n$ -differential operator of order $\leq k$ on $X$ . There exists a unique family of scalar functions

c(\alpha(1), \ldots, \alpha(n))_{(\alpha(1), \ldots, \alpha(n)) \in \mathbf{N}^J}, \quad \sum_{i=1}^n |\alpha(i)| \leq k

X

such that

H_X(f_1, \ldots, f_n) = \sum_{\alpha(1), \ldots, \alpha(n)} c(\alpha(1), \ldots, \alpha(n)) \Delta^{\alpha(1)}(f_1) \ldots \Delta^{\alpha(n)}(f_n)

for every family

(f_1, \ldots, f_n)

of functions of class

C^k

X

. In order that

H

be of class

C^h

, it is necessary and sufficient that the functions

c(\alpha(1), \ldots, \alpha(n))

be of class

C^h

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

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Cites

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