Given the Laplacian associated to a Riemannian manifold $(M^n, g)$, there is a Green's function $G(p,q): M \times M \to \mathbb{R}$ that satisfies an inequality of the form $$|G(p,q)| \leq Ad(p,q)^{2-n}$$ where $A$ depends on the metric $g$ in some way.

I was wondering if it is known how this constant $A$ depends on the Riemannian metric? In particular, it would be nice to know how the estimate changes under conformal change of the metric.

This is of course impossible in general since the new distance function $d'(p,q)$ could bear no resemblance to the former, but say the conformal change is "nice" enough so that $$C^{-1}d(p,q) \leq d'(p,q) \leq Cd(p,q)$$ where $C$ depends on the conformal function only. Then what can be said?