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 order.
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
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.
