differential operator

Roughly speaking, a differential operator is a mapping,typically understood to be linear, that transforms a function intoanother function by means of partial derivatives and multiplication byother functions.

On ${ℝ}^n$, a differential operator is commonly understood to be alinear 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 partialderivative of $f$ taken $i_1$ times with respect to the firstvariable, $i_2$ times with respect to the second variable, etc.The order of the operator is the maximum number of derivativestaken 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 commonlyunderstood to be a linear transformation of ${𝒞}^{\mathrm{\infty }}(M)$ having theabove form relative to some system of coordinates. Alternatively, onecan equip ${𝒞}^{\mathrm{\infty }}(M)$ with the limit-order topology, and define adifferential operator as a continuous transformation of ${𝒞}^{\mathrm{\infty }}(M)$.

The order of a differential operator is a more subtle notion on amanifold than on ${ℝ}^n$. There are two complications. First, onewould like a definition that is independent of any particular systemof coordinates. Furthermore, the order of an operator is at best a localconcept: it can change frompoint to point, and indeed be unbounded if the manifold isnon-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 $f(x)=0$. Fora 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 $M$ isnon-compact, in which case one says that the order of $T$ is infinite.

Let us conclude by making two remarks. The notion of a differentialoperator can be generalized even further by allowing the operator toact 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 byPeetre states that the converse is also true, namely that every localoperator is necessarily a differential operator.

References

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

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