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 derivatives and multiplication by other functions.

On ${ℝ}^n$, a differential operator is commonly understood to be a linear transformation of ${𝒞}^{\mathrm{\infty }}({ℝ}^n)$ having the form

$$f\mapsto \sum _I{a}^I{f}_I,f\in {𝒞}^{\mathrm{\infty }}({ℝ}^n),$$

where the sum is taken over a finite number of multi-indices $I=(i^1,\mathrm{\dots },i^n)\in {\mathbb{N}}^n$, where $a^I\in {𝒞}^{\mathrm{\infty }}({ℝ}^n)$, and where $f_I$ denotes a partial derivative of $f$ taken $i_1$ times with respect to the first variable, $i_2$ times with respect to the second variable, etc. The order of the operator is the maximum number of derivatives taken in the above formula, i.e. the maximum of $i_1+\mathrm{\dots }+i_n$ taken over all the $I$ involved in the above summation.

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

The order of a differential operator is a more subtle notion on a manifold than on ${ℝ}^n$. 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 unbounded if the manifold is non-compact. To address these issues, for a differential operator $T$ and $x\in M$, we define ${ord}_x(T)$ the order of $T$ at $x$, to be the smallest $k\in \mathbb{N}$ such that

$$T[f^{k+1}](x)=0$$

for all $f\in {𝒞}^{\mathrm{\infty }}(M)$ such that $f(x)=0$. For a fixed differential operator $T$, the function $ord(T):M\to \mathbb{N}$ defined by

$$x\mapsto {ord}_x(T)$$

is lower semi-continuous, meaning that

$${ord}_y(T)\ge {ord}_x(T)$$

for all $y\in M$ sufficiently close to $x$.

The global order of $T$ is defined to be the maximum of ${ord}_x(T)$ taken over all $x\in M$. This maximum may not exist if $M$ is non-compact, in which case one says that the order of $T$ is infinite.

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 sections of a bundle.

A differential operator $T$ is a local operator, meaning that

$$T[f](x)=T[g](x),f,g\in {𝒞}^{\mathrm{\infty }}(M),x\in M,$$

if $f\equiv g$ in some neighborhood of $x$. A theorem, proved by Peetre states that the converse is also true, namely that every local operator is necessarily a differential operator.

References

1. Dieudonné, J.A., Foundations of modern analysis

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