Fredholm index vs. Limit cycle theory Let $A$  be  the  algebra  of all smooth  functions  $f: \mathbb{R}^2 \to  \mathbb{R}$  such that $f$ is  flat  at  the  origin and  is  real  analytic on $\mathbb{R}^2 \setminus \{0\}$.
Let  $B $ be the  subalgebra  of  $A$ consisting  of  all  Schwartz  functions in $A$.
For  every  polynomial  vector  field $X = P(x,y) \partial_{x}  + Q(x,y) \partial_{y}$  we define  the  differential operator  $D_{X}(U)=PU_{x}+QU_{y}$. Obviously  $A$ and  $B$  are invariant under this  differential operator.
Let  $X=(y-(x^3 -x))\partial_{x}-x \partial_{y}$  be  the  Van der  Pol  vector  field.

What  can  be  said  about the  codimension of  the  range  of  $D_{X}:A \to A$? Is it finite?
What  can  be  said  about the codimension of  the  range  of  $D_{X}: B \to B$? Is it  finite?

We  explain  about the  motivation for  consideration of  such  $A$  and  $B$:
We  require the  real  analyticity on the  punctured plane to avoid the obvious  infinite  codimension since  if  a  limit  cycle  surrounds  a  non resonance singularity, using  bump  functions,  one  can  show that the  codimension is  infinity, as  we  explained here. We  require  the  flatness at the unique  singularity at the  origin to  avoid some  obstruction for existence  of (even)  formal power  series  solutions to the  equation $D_{X}.g=f$ when   $X$ is  the  Van der Pol equation or  a  more  general  algebraic  vector  field  with  degenerate  singularity (vanishing some  first  Jets  at the  origin). The other reason for  this flat requirement is  that we would  like  to  not  engage with the problem of  global analytic  extension of  a local  real  analytic  solution (if it exists) to   $D_{X}.g=f$. For  definition of  $B$, we  require  the  Schwartz  condition in order to  apply  the  Fourier  transform to  convert  a first  order  PDE, associated  with  a  quadratic  system,   to  higher order  PDE to  have  a  possible  chance  to work with  an elliptic  PDE. This  is  explained in the  Remark 2  and  its  consecutive example  of  page  5  of the  following  note:
     https://arxiv.org/pdf/1302.0001.pdf

Finally, for  the Van der Pol vector  field $X$, what  can be  said  about  the  codimension of  the range  of  $D_{X}$  as  an operator  on the  space $C^{\omega}(\mathbb{R}^2)$, the  space of  real  analytic  functions  on the  plane?

What  about the codimension of the range of the operator $L_X$ when it acts on either $\Omega^1(\mathbb{R}^2)$ or $\Omega^1(\mathbb{R}^2)/Z^1(\mathbb{R}^2)$ where $Z^1(\mathbb{R}^2)$ is the space of closed $1$-forms?
Do we have a possible chance for "finite codimension" in the latter quotient operator $L_X:\Omega^1(\mathbb{R}^2)/(Z^1(\mathbb{R}^2) \to  \Omega^1(\mathbb{R}^2)/(Z^1(\mathbb{R}^2)$?
Note that the codimension of the range of $L_X$ is an upper bound for the number of closed orbits of $X$
The reason is written in the motivation part of the following post:
Integral Separation of disjoint submanifolds of $\mathbb{R}^n$
 A: The linearization of the vector field $X$ at the singular point zero is
$$DX|_0 = \begin{pmatrix} 1 & 1\\ -1 & 0\end{pmatrix},$$
the eigenvalues of which are
$$ \lambda_{1, 2} = \frac{1}{2} \pm \frac{\sqrt{3}}{2} i,$$
hence both have positive real part. This allows to apply Thm. 4.1 in this paper, to conclude the following: Let $U$ be an open neighborhood of zero such that for each $x \in U$, $\Phi_t(x)$ converges to zero as $t \rightarrow - \infty$. Then given $v$ on $U$ which is flat at zero, the equation
$$ D_X u = v$$
has a unique solution $u$ in the space of smooth functions on $U$ that are  flat at zero. I think that with some additional arguments, one could additionally show that $u$ is analytic, provided that $v$ is analyitic.
In any case, the trouble is that this only works locally around zero: One cannot choose $U = \mathbb{R}^n$, due to the existence of a limit cycle. The largest set one can choose for $U$ seems to be the interior of the limit cycle. 
Furthermore, it is not at all clear whether given a $v$ which is flat at zero extends continuously from the interior of the limit cycle to the limit cycle itself. 
In a similar vein, I could believe that one can with similar methods get existence of solutions on the outside of the limit cycle. The problem now really is to glue these solutions together to form a smooth function on the limit cycle. This seems to be a very hard but nevertheless very interesting problem. 
\Edit: I just noticed that the codimension of the range is infinite in any case: Let $x$ be in the limit cocycle. Then for any smooth function $u$ on $\mathbb{R}^2$, the function $t \mapsto u(\Phi_t(x))$ is periodic ($\Phi_t(x)$ being the flow of $X$). Hence 
$$D_X u\bigl(\Phi_t(x)\bigr) = \frac{\mathrm{d}}{\mathrm{d} t} u\bigl(\Phi_t(x)\bigr)$$
necessarily has a zero. Therefore, if $v$ is any function in the limit cocyle such that $v(x) \neq 0$ for all $x$ in the limit cycle, then it cannot be in the range. 
Examples for such functions $v$ that lie in the algebra $B$ are
$$ v(x) = e^{-1/x^2 - x^2} x^k$$
for any $k \in \mathbb{N}_0$. This gives infinitely many linearly independent elements that are not in the range of $D_X$.
\Edit2: Indeed as Ali points out, some linear combination of these could be in the range, so this does not necessarily mean that the cokernel is infinite-dimensional.
