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.
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.
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.
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
is a mapping from the jets of sections of E to the bundle F. See also intrinsic differential operators.
is the sphere centered at
This is in fact the Laplacian, as can be seen using Taylor's theorem.
is a differential operator by Peetre's theorem.
The main idea is that since
is defined only in terms of
, it is local in nature; in particular, if
, and hence the support cannot grow.
The technical proof goes as follows.
trivial bundles.
are simply the space
is the set of smooth functions on the open set
is indeed a morphism, we need to check
for open sets
eventually sits inside both
is linear: Finally, we check that
is local in the sense that
in the ball of radius
So by Peetre's theorem,
is a differential operator.