Let $M^n$ be a compact Riemannian manifold without boundary. Then the Green's function for the Laplacian, $G(p,q),$ satisfies $$G(p,q)\leq A \; d(p,q)^{2-n}$$ where $d(p,q)$ is the distance between the points $p,q\in M.$ We can assume that $\int_M G(p,y) dy=0.$
How can we estimate the constant $A$? Are there any simple examples, like real or complex projective spaces, where it is possible to estimate $A$?
Any suggestions or references would be extremely helpful.