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) derivation on $\mathbb{C}[x,y]$:

$\tilde{D}_{X}(f)=f_{x}*P+f_{y}*Q$ where $*$ is the Moyal product. This is a noncommutativized version of the standard derivation $D_{X}(f)=f_{x}P+f_{y}Q$:

I have two questions:

1. Motivating 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;

Codimension of the range of certain linear operators

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