Let $M$ be a smooth real manifold and let $D_{< \infty}(M)$ be the algebra of differential operators of finite degree on $M$; it is the union $\bigcup_n D_{\leq n}(M)$ of the sets of differential operators of degree at most $n \in \mathbb N$.
I have two closely related questions:
- Locally (on coordinate neighborhoods) $D_{< \infty}$ is generated as an $\mathbb R$-algebra by $D_{\leq 1}$. (I.e. by vector fields and smooth functions.) Is this true globally?
- (Stronger) Is each $D_{\leq n}$ globally generated as an $\mathbb R$-module by $(D_{\leq 1})^n$?
If $M$ has a finite partition of unity with each support contained in a coordinate neighborhood, this is true. In particular it is true for compact $M$.
I'm interested in this because, for example, the pullback would then be the unique extension as an algebra-isomorphism of the pullback on vector fields and functions.