Let $X=P\frac{\partial}{\partial x}+Q\frac{\partial}{\partial y}$ be  a  polynomial vector field on $\mathbb{R}^{2}$.  Consider the  following (Moyal) operator on $\mathbb{C}[x,y]$:

$\tilde{D}_{X}(f)=f_{x}*P+f_{y}*Q$   or $\tilde{D}_{X}(f)=P*f_{x}+Q*f_{y}$ where $*$ is the Moyal product. The later has the potential of being an elliptic  operator since it is  of even degree.


This is a noncommutative version of the standard derivation $D_{X}(f)=f_{x}P+f_{y}Q$:

I have two questions:

1) Motivated by  the Leibniz formula for $D_{X}$, I  ask that: what is a natural  formula  for $\tilde{D}_{X}(f*g)=?$ 


>2)Can the number of  limit  cycles of $X$ be  bounded by the  codimension of the range of $\tilde{D}_{X}$?  

In particular is the following statement, true?

>* For  a limit cycle $\gamma$ of  a vector field $X$ on the plane and for  an smooth function $f$ on $\mathbb{R}^{2}$, there is  a point $p$ on $\gamma$ such that $\tilde{D}_{X}(f)(p)=0$.



  A (commutative) motivation  for  the  second  question is the question in the following  [post](https://mathoverflow.net/questions/164059/codimension-of-the-range-of-certain-linear-operators)

**Note** The * statement is  a Moyal version of a  key lemma to  prove the  commutative version of the second question. By  commutative version I mean we replace the Moyal product by the usual product.