In mathematics, the (linear) 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 section s 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 (depending on U) such that D is a differential operator of order k over U. This means that D factors through a linear mapping iD 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: We begin with the proof of Lemma 1. We now prove Lemma 2.

A specialized application

Let M be a compact smooth manifold (possibly with boundary), and E and F be finite dimensional vector bundles on M. Let is a smooth function (of Fréchet manifolds) 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 i_D is a mapping from the jets of sections of E to the bundle F. See also intrinsic differential operators.

Example: Laplacian

Consider the following operator: where and S_r is the sphere centered at x_0 with radius r. This is in fact the Laplacian, as can be seen using Taylor's theorem. We show will show L is a differential operator by Peetre's theorem. The main idea is that since Lf(x_0) is defined only in terms of f's behavior near x_0, it is local in nature; in particular, if f is locally zero, so is Lf, and hence the support cannot grow. The technical proof goes as follows. Let and E and F be the rank 1 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 U and restriction is function restriction. To see L is indeed a morphism, we need to check for open sets U and V such that and. This is clear because for x \in V, both (Lu)|V and L(u|V) are simply, as the S_r eventually sits inside both U and V anyway. It is easy to check that L is linear: Finally, we check that L is local in the sense that. If, then such that f = 0 in the ball of radius r centered at x_0. Thus, for , for, and hence (Lf)(x) = 0. Therefore,. So by Peetre's theorem, L is a differential operator.