differential operator (original) (raw)
On a 𝒞∞ manifold M, a differential operator is commonly understood to be a linear transformation of 𝒞∞(M) having the above form relative to some system of coordinates. Alternatively, one can equip 𝒞∞(M) with the limit-order topology
, and define a differential operator as a continuous transformation of 𝒞∞(M).
The order of a differential operator is a more subtle notion on a manifold than on ℝn. 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 localconcept: 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 Tand x∈M, we define ordx(T) the order of T at x, to be the smallestk∈ℕ such that
for all f∈𝒞∞(M) such that f(x)=0. For a fixed differential operator T, the function ord(T):M→ℕ defined by
is lower semi-continuous, meaning that
for all y∈M sufficiently close to x.
The global order of T is defined to be the maximum of ordx(T)taken over all x∈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.
- Peetre, J. , “Une caractérisation abstraite des opérateurs différentiels”, Math. Scand., v. 7, 1959, p. 211