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.