Peetre theorem
Encyclopedia
In mathematics
, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis
that gives a characterisation of differential operator
s in terms of their effect on generalized function space
s, 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.
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.
We begin with the proof of Lemma 1.
We now prove Lemma 2.
), and E and F be finite dimensional vector bundle
s on M. Let
be the collection of smooth sections of E. An operator
is a smooth function (of Fréchet manifold
s) 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,
is a mapping from the jet
s of sections of E to the bundle F. See also intrinsic differential operators.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the (linear) Peetre theorem, named after Jaak Peetre, is a result of functional analysis
Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
that gives a characterisation of differential operator
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
s in terms of their effect on generalized function space
Function space
In mathematics, a function space is a set of functions of a given kind from a set X to a set Y. It is called a space because in many applications it is a topological space, a vector space, or both.-Examples:...
s, and without mentioning differentiation
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
in explicit terms. The Peetre theorem is an example of a finite order theorem in which a function or a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
, 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. Letbe the spaces of smooth sections of E and F. An operator
is a morphism of sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
which is linear on sections such that the support
Support (mathematics)
In mathematics, the support of a function is the set of points where the function is not zero, or the closure of that set . This concept is used very widely in mathematical analysis...
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
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
of order k over U. This means that D factors through a linear mapping iD from the k-jet of sections
Jet (mathematics)
In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
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 proof relies on choosing a smooth metric on M, and in E and F. 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 integer k such that for any y∈V\{x) and for any section s of E whose k-jet vanishes (jks(y)=0), we have |Ds(y)|
- Lemma 2. The first lemma is sufficient to prove the theorem.
We begin with the proof of Lemma 1.
- Suppose the lemma is false. Then there is a sequence xk tending to x, and a sequence of very disjoint balls Bk around the xk (meaning that the geodesic distance between any two such balls is non-zero), and sections sk of E over each Bk such that jksk(xk)=0 but Dsk(xk)≥C>0.
- Let ρ(x) is a standard bump function for the unit ball at the origin: a smooth real-valued function which is equal to 1 on B1/2(0), which vanishes to infinite order on the boundary of the unit ball.
- Consider every other section s2k. At x2k, these satisfy
- j2ks2k(x2k)=0.
- Suppose that 2k is given. Then, since these functions are smooth and each satisfy j2k(s2k)(x2k)=0, it is possible to specify a smaller ball B′δ(x2k) such that the higher order derivatives obey the following estimate:
- where
- Now
- is a standard bump function supported in B′δ(x2k), and the derivative of the product s2kρ2k is bounded in such a way that
- As a result, because the following series and all of the partial sums of its derivatives converge uniformly
- q(y) is a smooth function on all of V.
- We now observe that since s2k and
2ks2k are equal in a neighborhood of x2k,
- So by continuity |Dq(x)|≥ C>0. On the other hand,
- since Dq(x2k+1)=0 because q is identically zero in B2k+1 and D is support non-increasing. So Dq(x)=0. This is a contradiction.
We now prove Lemma 2.
- First, let us dispense with the constant C from the first lemma. We show that, under the same hypotheses as Lemma 1, |Ds(y)|=0. Pick a y in V\{x} so that jks(y)=0 but |Ds(y)|=g>0. Rescale s by a factor of 2C/g. Then if g is non-zero, by the linearity of D, |Ds(y)|=2C>C, which is impossible by Lemma 1. This proves the theorem in the punctured neighborhood V\{x}.
- Now, we must continue the differential operator to the central point x in the punctured neighborhood. D is a linear differential operator with smooth coefficients. Furthermore, it sends germs of smooth functions to germs of smooth functions at x as well. Thus the coefficients of D are also smooth at x.
A specialized application
Let M be a compact smooth manifold (possibly with boundaryManifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
), and E and F be finite dimensional vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s on M. Let
be the collection of smooth sections of E. An operator
is a smooth function (of Fréchet manifold
Fréchet manifold
In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space....
s) 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
Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
of order k. Specifically,
is a mapping from the jet
Jet (mathematics)
In mathematics, the jet is an operation which takes a differentiable function f and produces a polynomial, the truncated Taylor polynomial of f, at each point of its domain...
s of sections of E to the bundle F. See also intrinsic differential operators.