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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.

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    $\begingroup$ Semyon, (1) seems to assume that the kernel of $D$ consists only of constant functions but that's not necessarily the case. $\endgroup$
    – Deane Yang
    Jul 5, 2011 at 9:25
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    $\begingroup$ To add to what Deane said: lower order terms sometimes matter when you try to apply maximum principle. Another condition is needed. $\endgroup$
    – Willie Wong
    Jul 5, 2011 at 9:48
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    $\begingroup$ Thanks, Deane. You are right. In my situation $D$ has no free term. I have added this condition to the question. $\endgroup$
    – asv
    Jul 5, 2011 at 9:56
  • $\begingroup$ Semyon, I learned how to prove (1) and (2) from books on pseudodifferential operators, probably the one by Chazarain and Piriou. In particular, (2) just follows from local elliptic regularity estimates that are quite easy to prove using pseudodifferential operators. But (3) and (3') are deeper and, as Willie says, probably require more assumptions that you have stated. My guess (and that's all it is) is that your conclusion holds if $Du = \partial_i(a^{ij}\partial_ju)$. But this is essentially the same case as the Laplacian for a Riemannian metric. $\endgroup$
    – Deane Yang
    Jul 5, 2011 at 16:14
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    $\begingroup$ Deane, (1) and (2) are indeed standard. Regarding the operators of the form $Du=\partial_i(a^{ij}\partial_ju)$, they are not coordinate independent. Strictly speaking, even the Riemannian Laplacian does not have this form: $\partial_i$ should be replaced by covariant derivatives $\nabla_i$, and for this you need a metric. I would be surprised if there is no generalizations beyond Riemannian Laplacians. $\endgroup$
    – asv
    Jul 6, 2011 at 7:34

2 Answers 2

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I failed to find an explicit reference till now. However the result seems to be true. In the appendix to this paper we with a co-author have written it down (modulo some fact we could fid in the literature, mostly in Shimakura's book).

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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.

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  • $\begingroup$ Could you please give a more precise reference to this book and theorem, it would be helpful. In Aubin's book "Some Nonlinear Problems in Riemannian Geometry" I was able to find estimates on the Green function for a Riemannian Laplacian only. $\endgroup$
    – asv
    Dec 16, 2011 at 18:03
  • $\begingroup$ Yes, that is the book. I guess what I implicitly said was that there is not much difference between the Laplace-Beltrami operator and a general elliptic operator with smooth coefficients. It might help if you give some details on what part of the arguments you are having trouble with, when you extend the arguments to general elliptic operators. $\endgroup$
    – timur
    Dec 16, 2011 at 23:04

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