In mathematics, the Peetre theorem, named after Jaak Peetre, is a result of functional analysis that gives a characterisation of differential operators in terms of their effect on generalized function spaces, and without mentioning differentiation in explicit terms. The Peetre theorem is an example of a finite order theorem in which a function or a functor, defined in a very general way, can in fact be shown to be a polynomial because of some extraneous condition or symmetry imposed upon it. This article treats two forms of the Peetre theorem. The first is the original version which, although quite useful in its own right, is actually too general for most applications.
The original Peetre theorem
Let M be a smooth manifold and let E and F be two vector bundles on M. Let be the spaces of smooth sections of E and F. An operator is a morphism of sheaves which is linear on sections such that the support of D is non-increasing: supp Ds ⊆ supp s for every smooth sections of E. The original Peetre theorem asserts that, for every point p in M, there is a neighborhood U of p and an integer k such that D is a differential operator of order k over U. This means that D factors through a linear mappingiD from the k-jet of sections of E into the space of smooth sections of F: where is the k-jet operator and is a linear mapping of vector bundles.
Proof
The problem is invariant under local diffeomorphism, so it is sufficient to prove it when M is an open set in Rn and E and F are trivial bundles. At this point, it relies primarily on two lemmas:
Lemma 1. If the hypotheses of the theorem are satisfied, then for every x∈M and C > 0, there exists a neighborhood V of x and a positive integerk such that for any y∈V\ and for any section s of E whose k-jet vanishes at y, we have |Ds|
Lemma 2. The first lemma is sufficient to prove the theorem.
We begin with the proof of Lemma 1. We now prove Lemma 2.
A specialized application
Let M be a compact smooth manifold, and E and F be finite dimensional vector bundles on M. Let is a smooth function which is linear on the fibres and respects the base point on M: The Peetre theorem asserts that for each operator D, there exists an integer k such that D is a differential operator of order k. Specifically, we can decompose where is a mapping from the jets of sections of E to the bundle F. See alsointrinsic differential operators.
Example: Laplacian
Consider the following operator: where and is the sphere centered at with radius. This is in fact the Laplacian. We show will show is a differential operator by Peetre's theorem. The main idea is that since is defined only in terms of 's behavior near, it is local in nature; in particular, if is locally zero, so is, and hence the support cannot grow. The technical proof goes as follows. Let and and be the rank trivial bundles. Then and are simply the space of smooth functions on. As a sheaf, is the set of smooth functions on the open set and restrictionis function restriction. To see is indeed a morphism, we need to check for open sets and such that and. This is clear because for, both and are simply, as the eventually sits inside both and anyway. It is easy to check that is linear: Finally, we check that is local in the sense that. If, then such that in the ball of radius centered at. Thus, for, for, and hence. Therefore,. So by Peetre's theorem, is a differential operator.
