TITLE: Green's function for *GJMS* operator
QUESTION [5 upvotes]: Consider a Riemannian manifold $(M^n, g)$ of dimension $n$ with a metric $g$. We assume $M$ to be closed (compact without boundary). Let's not assume any hypothesis on the Yamabe invariant of the manifold.
A GJMS operator on $M^n$ is a differential operator of order $n$ with principal symbol $(-\Delta_g)^{(n/2)}$. 
If the Yamabe invariant on $M^n$ is positive then the kernel of this operator is just the constants (for this statement cfr Gursky - "Conformal invariants and nonlinear elliptic equations", as for the Green's function in this case you can check Ndiaye - "Constant $Q$-curvature metrics in arbitrary dimension"). 
In general the constants on $M^n$ are just a subset of the kernel.

Does anyone know if there exist a Green function for such operator in this case? 

I cannot find any references for this case. Any help?

REPLY [2 votes]: I believe that the answer is Yes by way of the method of the parametrix. This equation is elliptic (the principal symbol is injective, being a power of the principal symbol of the Laplacian) so the parametrix is generated by a pseudo-differential operator.