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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 $\reals^n$ , a differential operator is commonly understood to be a linear transformation of $\cfty(\reals^n)$ having the form $$ f \mapsto \sum_{I} a^I f_I,\quad f\in \cfty(\reals^n), $$ where the sum is taken over a finite number of multi-indices $I=(i^1,\ldots,i^n)\in \natnums^n$ , where $a^I\in \cfty(\reals^n)$ , 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 $\cfty$ manifold $M$ , a differential operator is commonly understood to be a linear transformation of $\cfty(M)$ having the above form relative to some system of coordinates. Alternatively, one can equip $\cfty(M)$ with the limit-order topology, and define a differential operator as a continuous transformation of $\cfty(M)$ .
The order of a differential operator is a more subtle notion on a manifold than on $\reals^n$ . 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 $k\in\natnums$ such that $$T[f^{k+1}](x) = 0$$ for all $f\in\cfty(M)$ such that $f(x)=0$ . For a fixed differential operator $T$ , the function $\ord(T):M\rightarrow \natnums$ defined by $$x\mapsto \ord_x(T)$$ is lower semi-continuous, meaning that $$\ord_y(T)\geq \ord_x(T)$$ 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 $$T[f](x) = T[g](x),\quad f,g\in\cfty(M),\; x\in M,$$ 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
- Dieudonné, J.A., Foundations of modern analysis
- Peetre, J. , ``Une caractérisation abstraite des opérateurs différentiels'', Math. Scand., v. 7, 1959, p. 211
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