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

 $f\mapsto\sum_{I}a^{I}f_{I},\quad f\in\mathcal{C}^{\infty}(\mathbb{R}^{n}),$

where the sum is taken over a finite number of multi-indices $I=(i^{1},\ldots,i^{n})\in\mathbb{N}^{n}$, where $a^{I}\in\mathcal{C}^{\infty}(\mathbb{R}^{n})$, 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 $i_{1}+\ldots+i_{n}$ taken over all the $I$ involved in the above summation.

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

The order of a differential operator is a more subtle notion on a manifold than on $\mathbb{R}^{n}$. 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 $\mathop{\mathrm{ord}}\nolimits_{x}(T)$ the order of $T$ at $x$, to be the smallest $k\in\mathbb{N}$ such that

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

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

 $x\mapsto\mathop{\mathrm{ord}}\nolimits_{x}(T)$

is lower semi-continuous, meaning that

 $\mathop{\mathrm{ord}}\nolimits_{y}(T)\geq\mathop{\mathrm{ord}}\nolimits_{x}(T)$

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

The global order of $T$ is defined to be the maximum of $\mathop{\mathrm{ord}}\nolimits_{x}(T)$ 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

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

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.

1. 1.

Dieudonné, J.A., Foundations of modern analysis

2. 2.

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

Title differential operator DifferentialOperator 2013-03-22 12:20:29 2013-03-22 12:20:29 rmilson (146) rmilson (146) 10 rmilson (146) Definition msc 53-00 msc 35-00 msc 47E05 msc 47F05 Operator