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 , a differential operator is commonly understood to be a linear transformation of having the form
where the sum is taken over a finite number of multi-indices , where , and where denotes a partial derivative of taken times with respect to the first variable, 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 taken over all the involved in the above summation.
On a manifold , a differential operator is commonly understood to be a linear transformation of having the above form relative to some system of coordinates. Alternatively, one can equip with the limit-order topology, and define a differential operator as a continuous transformation of .
The order of a differential operator is a more subtle notion on a manifold than on . 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 and , we define the order of at , to be the smallest such that
for all such that . For a fixed differential operator , the function defined by
is lower semi-continuous, meaning that
for all sufficiently close to .
The global order of is defined to be the maximum of taken over all . This maximum may not exist if is non-compact, in which case one says that the order of 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 is a local operator, meaning that
Peetre, J. , “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand., v. 7, 1959, p. 211
|Date of creation||2013-03-22 12:20:29|
|Last modified on||2013-03-22 12:20:29|
|Last modified by||rmilson (146)|