# A differential operator analogy of certain fact in real analysis of smooth functions

Let $$E\to M$$ be a smooth vector bundle over a smooth manifold $$M$$.

Let $$D$$ be a differential operator defined on the space $$\Gamma(E)$$ of smooth sections of $$E$$. We fix a section $$s\in \Gamma(E)$$.

Assume that for every $$x\in M$$ there exist a natural number $$n$$ such that $$D^n(s)(x)=0$$.

Does this imply that $$D^k(s)=0$$, for some $$k\in \mathbb{N}$$?

If the answer is "No", what about if we assume that $$D$$ is an elliptic operator and $$M$$ is a compact manifold?

The motivation for this question is the following fact which I learned from the book of "R.P.Boas:A primer of real functions".

https://www.jstor.org/stable/10.4169/j.ctt5hh8x5

Fact: Assume that $$f:\mathbb{R} \to \mathbb{R}$$ is a smooth function with the property that for every $$x\in \mathbb{R}$$ there exist a natural number $$n$$ with $$f^{(n)}(x)=0$$.Then $$f$$ is a polynomial.

• (A version of) the real analytic result you mention is discussed in this MO question: mathoverflow.net/questions/34059/… – Sam Hopkins Apr 24 at 2:59
• @SamHopkins Thanks for your comment.That is about smooth finction. Among answer one can find a reference to Boas book. I learned this result from the book of Boas when i was a master student any way the link you mentioned is about that classical theorem not about sections of vector bundles. – Ali Taghavi Apr 24 at 5:23
• right, I just mention to link that in case other people were interested in learning more about the classic result you mentioned. – Sam Hopkins Apr 24 at 13:22
• @SamHopkins yes I see thanks for the link. – Ali Taghavi Apr 25 at 6:03