Let $U$ be any open subset of $\mathbb{R}^n$ (or, more general, of some smooth manifold). Define $\mathcal{D}_{-1}(U):=\{0\}$. For any two linear operators $A$ and $B$, the commutator operator $[A,B]$ is given by $A\circ B-B\circ A$. For any $f\in C^\infty(U)$ we define the operator $m_f\in \text{End}(C^\infty(U))$ such that $m_f\ g=f g$. Consider $P\in \text{End}(C^\infty(U))$. By definition we take that $P\in \mathcal{D}_k$ iff $[P,m_f]\in \mathcal{D}_{k-1}$.
My problem is that I want to show that the set, call it $\mathcal{D}_k'$, of all partial differential operators with smooth coefficient functions of order at most $k$ equals $\mathcal{D}_k$ for each $k\in\{0,1,2,\dots\}$.
It's easy to show that $\mathcal{D}'_0=\mathcal{D}_0$. The same goes for the statement that $\mathcal{D}'_k\subset \mathcal{D}_k$ for all $k$. Difficulty lies in the reverse inclusion. I have strong feeling that any $P\in \mathcal{D}_k$ can be represented by
$ P=\sum_{|\alpha|\leq k}\frac{C^\alpha_P(1)}{\alpha!}\partial^\alpha $.
Here $\alpha$ are multi-indices and $C^0_P:=P$ and $C^\alpha_P=[C^\beta_P,f_{\alpha-\beta}]$, where $\beta$ is any multi-index of an order lower than $\alpha$ such that $\alpha-\beta$ is well defined as a multi-index and its order is equal to one. Furthermore for any $f_{\alpha-\beta}$ gives the $(\alpha-\beta)^\text{th}$ coordinate of any $x\in U$: $f_{\alpha-\beta}(x)=x_{\alpha-\beta}$.
However I have not been able to complete the inductive argument. It should be possible to show this with just algebraic properties of differential calculus and the definition given above. But I don't see it.
Kindly appreciated,
Aris
I think I have found a solution. For anybody interested, here are some pointers in what I think to be the right direction.
Consider an operator $P\in \mathcal{D}^k$ and some smooth function $f$ and suppose we are interested in the value of $Pf$ at $x_0$. Then consider the $k^\text{th}$ Taylor expansion $f^k_{x_0}$ of $f$ around $x_0$. The error term in the expansion can be written in the form $E(x)=\sum_{|\beta|=k}g_\beta(x) (x-x_0)^k$, where $g_\beta$ are smooth functions. If you let $P$ work on f written in it's Taylor's expansion and use the inductive properties of class $\mathcal{D}^k$, and evaluate the function each time at $x=x_0$, you will finally get something that looks like what is stated in my problem description and the error term drops out.