What is a differential operator?

Question

Let $X$ be a real $C^{\infty}$-manifold.

Define $C_c^{\infty}(X)$ to be the $\mathbb{C}$-vector space of all morphisms of manifolds $X \rightarrow \mathbb{C} = \mathbb{R}^2$ which are compactly supported.

What is the meaning of a differential operator on $C_c^{\infty}(X)$?

Answer 1

Those functions having compact support or not is irrelevant, the definition is the same, so let us focus on the larger space $C ^\infty (X)$.

A linear map $P : C ^\infty (X) \to C ^\infty (X)$ is a differential operator if and only if in every patch of coordinates $U$ we can write, for all smooth $f$ (and in multi-index notation),

$$(Pf) (x) = \sum _{|\alpha| \le N_U} c_\alpha (x) (\partial _\alpha f) (x)$$

for some $N_U \in \Bbb N$ (that may depend on $U$) and $c_\alpha \in C ^\infty (U)$ for all $\alpha$.

If the $N_U$ (that correspond to all the coordinate patches) are bounded by some $N$, then we say that $P$ has order at most $N$. Notice that the maximum order $N_U$ may, in principle, vary from one coordinate patch to another.

Answer 2

Let's wait with the compact support. Roughly speaking, a differential operator on $C^\infty(X)$ is a map $D:C^\infty(X)\to C^\infty(X)$, such that for any $f\in C^\infty(X)$, the value of $D(f)$ at a point depends only on the values of $f$ and its derivatives at that point.

One way to explain the previous paragraph formally is as follows. Choose a connection $\nabla$ on $X$. Then $\nabla^k f$ is a tensor of type $(0,k)$ for every $f\in C^\infty(X)$ and $k\in\mathbb{N}$. A map $D:C^\infty(X)\to C^\infty(X)$ is a differential operator of order $l$, if it is given by$$f\mapsto\Phi\left(f,\nabla f,\ldots,\nabla^lf\right),$$for some $$\Phi:C^\infty(X)\times\Gamma(T^*M)\times\ldots\times\Gamma\left(T^*M^{\otimes l}\right)\to C^\infty(X)$$ which is local. Being local means that given tensors $T_0,\ldots,T_l$ and $T'_0,\ldots,T'_l$ and a point $p\in X$, if $$T_0(p)=T'_0(p),\ldots,T_l(p)=T'_l(p),$$then$$\Phi(T_0,\ldots,T_l)(p)=\Phi(T'_0,\ldots,T'_l)(p).$$

Finally, depending on our motivation, we may require $D(f)$ to be compactly supported whenever $f$ is.