Let $(M,g)$ be a closed Riemannian manifold of dimension $n>7$. In this setting I have been able to prove that the Green's function of a positive Paneitz-Branson operator is non-negative. Furthermore, we can also conclude that the Green's function can't vanish on open sets. I need the Green's function to be positive, though, so I was wondering if anyone had an idea as to how to establish positivity of the Green's function under the hypotheses used above.

The definition of the Paneitz-Branson operator $P_g$ of the metric is as follows:

$P_g:= (\Delta_g)^2 - div_g(a_n R_g g + b_n Ric_g)d + Q_g. $

Here $\Delta_g$ is the Laplace-Beltrami operator, $a_n = \frac{(n-2)^2 +4}{2(n-1)(n-2)}$, $b_n = \frac{-4}{n-2}$, and $Q$ is the $Q$-curvature of the metric $g$. Note that it is a fourth-order operator and hence classical maximum principles can't be used in general. Also, if the hypothesis that $g$ is locally conformally flat is used, that would be ok too. Of course, if a counter-example is found that would be of interest as well.