differential operator DifferentialOperator 2002-02-15 15:01:10 2008-02-10 21:10:15 Definition \usepackage{amsmath} \usepackage{amsfonts} \usepackage{amssymb} \newcommand{\reals}{\mathbb{R}} \newcommand{\natnums}{\mathbb{N}} \newcommand{\cnums}{\mathbb{C}} \newcommand{\cfty}{\mathcal{C}^\infty} \newcommand{\ord}{\mathop{\mathrm{ord}}\nolimits} \newtheorem{proposition}{Proposition} Roughly speaking, a {\em 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 {\em 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. \bigskip {\bf References} \begin{enumerate} \item Dieudonn\'e, J.A., {\em Foundations of modern analysis} \item Peetre, J. , ``Une caract\'erisation abstraite des op\'erateurs diff\'erentiels'', {\em Math. Scand.}, v. 7, 1959, p. 211 \end{enumerate}