Global analysis, analysis on manifolds Basic notions and local calculus Theorem

Peetre theorem on local operators

Builds on Smooth manifold Vector bundle Section of a vector bundle Differential operator between vector bundles

Statement

Let $M$ be a smooth manifold and let $E$ and $F$ be smooth real vector bundles over $M$ . Let $\Gamma(E)$ and $\Gamma(F)$ be their spaces of smooth sections. Let $D \colon \Gamma(E) \to \Gamma(F)$ be an $\R$ -linear map that is local, that is, $\operatorname{supp}(Ds) \subseteq \operatorname{supp}(s) \quad \text{for all } s \in \Gamma(E).$ Then every point $p \in M$ has an open neighbourhood $U$ with an integer $k \ge 0$ such that the following holds. Over $U$ , $D$ is a linear differential operator of order at most $k$ with smooth coefficients. In local coordinates and trivialisations over $U$ , this means $(Ds)|_U = \sum_{\abs{\alpha} \le k} a_\alpha \, \partial^\alpha (s|_U)$ for every $s \in \Gamma(E)$ , with smooth matrix-valued functions $a_\alpha$ .

Remarks

No continuity of $D$ is assumed; it is a consequence. The idea of the proof is to show first that, near each point, $(Ds)(x)$ depends only on the $k$ -jet of $s$ at $x$ for one fixed $k$ . If this failed, one could build sections with small disjoint supports accumulating at a point and add them up, and locality would force a contradiction. Linearity then shows that $D$ is given by a linear map on $k$ -jets, which is exactly a differential operator of order $k$ .

Example: the exterior derivative $d \colon \Omega^j(M) \to \Omega^{j+1}(M)$ is local, and it is indeed a differential operator of order $1$ .

Locality is needed. On $M = \R$ the operator $(Df)(x) = \int_0^x f(t)\, dt$ is linear but not local, and it is not a differential operator.

The order need not be bounded on all of $M$ when $M$ is not compact. On $\R$ , let $\varphi_n$ be a smooth function with support in $(n, n + 1)$ and $\varphi_n \ne 0$ . Then $Df = \sum_{n \ge 1} \varphi_n f^{(n)}$ is a locally finite sum, so it is a local linear operator, but its order is unbounded. If $M$ is compact, finitely many sets $U$ cover $M$ , so $D$ is a differential operator of finite order.

Going deeper

For a student

The theorem gives an algebraic characterisation of differential operators by their support property. It is used to show that natural operators in differential geometry, such as those that commute with diffeomorphisms, are differential operators of finite order. Together with invariance arguments, it leads to classification results for natural operators on bundles of tensors. It also explains why locality is the key property of the exterior derivative, the Lie derivative and covariant derivatives.

Reading path

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Differential operator between vector bundles concept
Section of a vector bundle concept
Smooth manifold concept
Vector bundle concept
Peetre theorem on local operators statement

Tag	`Q3X1CNXASD`
URI	`m2://Q3X1CNXASD`
Kind	statement, theorem
Writing	W0 stub
MSC	58J99, 47F05
Status	This copy of the site has no build data. Run `m2 status Q3X1CNXASD`.
About	Differential operator between vector bundles
Uses	Smooth manifold, Vector bundle, Section of a vector bundle, Differential operator between vector bundles
Same as	Wikidata Q7160489 Wikipedia Peetre theorem