TITLE: Estimates on the Green function of an elliptic second order differential operator.
QUESTION [8 upvotes]: Let $D$ be a linear differential elliptic operator of second order
with infinitely smooth coefficients acting on real valued functions
on a compact manifold $M$. Let us assume that $D$ has no free term, i.e. $D(1)=0$.
Let us fix a smooth positive measure
(density) $\mu$ on $M$. Does there exist a (integrable) Green
function $G\colon M\times M\to \mathbb{R} $ with the following properties:
(1) $\int_M G(x,y) \cdot D\phi(y) d\mu(y) =\int_M\phi(y) d\mu(y)
-\phi(x)$ for any function $\phi$ and $x\in M$ (this is the definition of Green function);
(2) $G$ is infinitely smooth outside of the diagonal;
(3) $G$ is bounded from below.
The last property can be asked in a stronger form:
(3') Does $G$ satisfy the asymptotic estimate near the diagonal:
$$c|x-y|^{2-n}\leq G(x,y)\leq C|x-y|^{2-n}$$
where $c,C>0$ and $n=\dim M>2$. If $n=2$ there should be a
logarithmic estimate.
I am pretty sure that this is true and should be well known. I would
need a reference. The special case when $D$ is the Laplacian for a
Riemannian metric on $M$ is contained explicitly in some textbooks I
am familiar with.

REPLY [2 votes]: Green's functions are constructed in Aubin's book for operators such as you mentioned, but with some sign condition on the lowest order term. I have not had a close look but my suspicion is that (3') is fine but for (3) you need a maximum principle.