differential operator

Roughly speaking, a differential operator is a mapping, typically understood to be linear, that transforms a function into another function by means of partial derivativesMathworldPlanetmath and multiplicationPlanetmathPlanetmath by other functions.

On n, a differential operator is commonly understood to be a linear transformation of 𝒞(n) having the form


where the sum is taken over a finite number of multi-indices I=(i1,,in)n, where aI𝒞(n), and where fI denotes a partial derivative of f taken i1 times with respect to the first variable, i2 times with respect to the second variable, etc. The order of the operator is the maximum number of derivativesPlanetmathPlanetmath taken in the above formulaMathworldPlanetmathPlanetmath, i.e. the maximum of i1++in taken over all the I involved in the above summation.

On a 𝒞 manifold M, a differential operator is commonly understood to be a linear transformation of 𝒞(M) having the above form relative to some system of coordinates. Alternatively, one can equip 𝒞(M) with the limit-order topologyMathworldPlanetmathPlanetmath, and define a differential operator as a continuous transformation of 𝒞(M).

The order of a differential operator is a more subtle notion on a manifold than on n. There are two complications. First, one would like a definition that is independent of any particular system of coordinates. Furthermore, the order of an operator is at best a local concept: it can change from point to point, and indeed be unboundedPlanetmathPlanetmath if the manifold is non-compact. To address these issues, for a differential operator T and xM, we define ordx(T) the order of T at x, to be the smallest k such that


for all f𝒞(M) such that f(x)=0. For a fixed differential operator T, the function ord(T):M defined by


is lower semi-continuous, meaning that


for all yM sufficiently close to x.

The global order of T is defined to be the maximum of ordx(T) taken over all xM. This maximum may not exist if M is non-compact, in which case one says that the order of T is infiniteMathworldPlanetmath.

Let us conclude by making two remarks. The notion of a differential operator can be generalized even further by allowing the operator to act on sectionsPlanetmathPlanetmath of a bundle.

A differential operator T is a local operator, meaning that


if fg in some neighborhoodMathworldPlanetmathPlanetmath of x. A theoremMathworldPlanetmath, proved by Peetre states that the converseMathworldPlanetmath is also true, namely that every local operator is necessarily a differential operator.

  1. 1.

    Dieudonné, J.A., Foundations of modern analysisMathworldPlanetmath

  2. 2.

    Peetre, J. , “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand., v. 7, 1959, p. 211

Title differential operator
Canonical name DifferentialOperator
Date of creation 2013-03-22 12:20:29
Last modified on 2013-03-22 12:20:29
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Definition
Classification msc 53-00
Classification msc 35-00
Classification msc 47E05
Classification msc 47F05
Related topic Operator