Let $E, F$ be two smooth vector bundles over a smooth manifold $M$. [Peetre's theorem](https://en.wikipedia.org/wiki/Peetre_theorem) states that any $\mathbb{R}$-linear morphism $D: \mathcal{E} \to \mathcal{F}$ of the sheaves of sections of $E$ and $F$ is a differential operator, i.e. there exist trivializing charts $U\_i$ such that on $D|\_{U\_i}$ is a differential operator in the sense of real analysis.


**Question 1:** Does this result also hold for complex manifolds?


By this I mean that $M$ is a complex manifold, $E$ and $F$ are holomorphic vector bundles and $D$ is $\mathbb{C}$-linear. We then also restrict to sheaves of holomorphic sections.
I don't think the proof of the real version can be adapted directly, since it relies on bump functions.


This is related to my [m.SE question](https://math.stackexchange.com/questions/3376025/example-of-an-additive-map-mathcalo-x-to-mathcalo-x-that-is-not-a-diffe) in which I showed that the result does not hold for the sheaf of rational functions on $\mathbb{C}$.


However, on any locally ringed space $(X, \mathcal{O})$ one can define differential operators of **order** $0$ as multiplication with a global section of $\mathcal{O}$ and differential operators of order $n$ inductively as $\mathbb{C}$-linear morphisms of sheaves $D: \mathcal{E} \to \mathcal{F}$ satisfying that $f D - D f$ is a differential operator of order $n-1$. General differential operators can then be defined as $\mathbb{C}$-linear morphisms of sheaves which locally are differential operators of finite order.  

This gives the expected result for the differential operators on a Scheme (EGA IV Prop 16.8.8) (i. e. on $n$-dimensional affine space they are polynomials in the $n$ partial derivatives). By Peetre's theorem this definition also coincides with the classical notion of differential operators on a smooth manifold.


**Question 2:** If $\mathcal{E}$ and $\mathcal{F}$ are sheaves of sections of holomorphic vector bundles over a complex manifold, does the above definition of differential operators produce the classical differential operators between holomorphic vector bundles?



Does Peetre's theorem hold in complex analysis?
Yes, for both the second question.


**Q2**: Since it's a local question, it suffices to consider $M$ open in $\mathbb{C}^n$ and $\mathcal{E},\mathcal{F}$ one dimensional trivial complex bundles. So let $\mathcal{O}$ denote the ring of holomorphic functions on $M$, let $z\_1,\ldots, z\_n \in \mathcal{O}$ denote the standard coordinates on $\mathbb{C}^n$ restricted to $M$ and use multi-index notation $\mu=(\mu\_1,\ldots,\mu\_n)\in\mathbb{N}^n$ with $z^\mu=z\_1^{\mu\_1}\cdots z\_n^{\mu\_n}$, $\mu!=\mu\_1!\cdots\mu\_n!$ and $|\mu|=\mu\_1+\ldots+\mu\_n$. It's easy to show that an operator of the form 
$$
\sum\limits\_{|\mu|\leq k}c\_\mu \frac{\partial^{|\mu|}}{\partial z^\mu}: \mathcal{O} \to \mathcal{O}$$
with $c\_\mu\in \mathcal{O}$, satisfies the algebraic definition of a $\mathbb{C}$-linear differential operator of rank at most $k$.


To show the converse, assume $D\colon \mathcal{O} \to \mathcal{O}$ is such an algebraic DO of rank at most $k$. We want to show $D = \sum\limits\_{|\mu|\leq k}c\_\mu \frac{\partial^{|\mu|}}{\partial z^\mu}$ for suitable coefficients $c\_\mu \in \mathcal{O}$. Define them recursively as


$$
c\_{\mu}=
\cases{
D (1) \quad \text{ when } \mu=0 \\
\left(D - \sum\limits\_{0\leq |\nu|< |\mu|} c\_\nu \frac{\partial^{|\nu|}}{\partial z^\nu}\right)\left(\frac{z^\mu}{\mu!}\right) \quad \text{ when } 1\leq |\mu|\leq k.
} 
$$
Its straightforward to checkt that $D$ and $\sum\limits\_{|\mu|\leq k}c\_\mu \frac{\partial^{|\mu|}}{\partial z^\mu}$ agree on polynomials in $z$ of degree at most $k$. 


To show that they also agree on any other $u\in \mathcal{O}$, let $x\in M$ and use Taylor expansion around $x$ to write $u=p+r$ where $p$ is a polynomial in $z$ of degree at most $k$ and $r$ is an element of $I^{k+1}\_x\subset \mathcal{O}$, the $k+1$st power of the vanishing ideal at $x$. Now use the fact that $D(I^{k+m})=I^m$, for any ideal $I\subset \mathcal{O}$ and any algebraic DO $D$ of rank at most $k$. So that
$$
(D u)(x)=D(p+r)(x)=(D p)(x)= \left(\sum\limits\_{|\mu|\leq k}c\_\mu \frac{\partial^{|\mu|}}{\partial z^\mu}p\right)(x)=\sum\limits\_{|\mu|\leq k}c\_\mu \frac{\partial^{|\mu|}}{\partial z^\mu}(p+r)(x)=\left(\sum\limits\_{|\mu|\leq k}c\_\mu \frac{\partial^{|\mu|}}{\partial z^\mu} u\right)(x).
$$
Since this holds for any $x$ in $M$ we conclude $D u =\sum\limits\_{|\mu|\leq k}c\_\mu \frac{\partial^{|\mu|}}{\partial z^\mu} u$.


**Edit:** After the clarification of the question what follows is not a correct answer.


**Q1** should now be easy: since the topology of $M$ as complex manifold agrees with its topology as real manifold, by Peetres theorem a $\mathbb{C}$-linear local operator $D : C^\infty(M,\mathbb{C}) \to C^\infty(M,\mathbb{C})$ satisfies the algebraic definition of a DO with smooth $f$ (in the sense of your post $fD - Df$ being of lower rank). Since holomorphic $f$ are also smooth, the algebraic definition holds also for holomorphic $f$. Now if $D$ moreover preserves the subring of holomorphic functions it is a complex differential operator by Q2.


