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Let $$ P:\Gamma(E)\rightarrow\Gamma(F) $$ be an elliptic partial differential operator, with index zero and closed image of codimension 1, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections of vector bundles $E\rightarrow M$ and $F\rightarrow M$ on a Riemannian manifold $(M,g)$ without boundary.

Question: What is the elliptic operator's associated Green's operator? What does the notion of a Green's operator of an elliptic operator refer to or mean?

Reference request: Can someone please point out a good reference on Green's operators for elliptic partial differential operators.

mdg
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1 Answers1

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Here is a set up analgous to the situation of Dirac operator, which in fact I do not really know whether applies to your case because I did not encounter nonlinear operatos before. Let $$ P:\Gamma(E)\rightarrow \Gamma(F) $$ be an elliptic differential operator. Then in particular $P$ is Fredholm and it has finite dimensional kernel and cokernel. So we can decompose a section $\phi \in \Gamma(E)$ by $$ \phi=\phi_1+\phi_2, \phi_1\in \ker (P), \pi(\phi)=\phi_2 $$ and similarly for $\phi\in \Gamma(F)$ we can define $\pi'$ to be the projection map.

Now the associated Green's operator $G:\Gamma(F)\rightarrow \Gamma(E)$ is defined by $$ G\circ P=Id-\pi, P\circ G=Id-\pi' $$

I think one way to think about this is via Hodge decomposition theorem: For sufficiently nice elliptic operators (like Dirac operators, Generalized Laplacian operators, etc) over a compact manifold we should have $$ \Gamma(E)=\ker(P)\oplus Im(P^{*}) $$ where $P^{*}$ is understood as the dual of $P$ in some sense. Then you can think of the Green's operator as a some kind of an inverse to $P$ by omitting elements from the kernel of $P^{*}$. For reference you can check Melrose's notes on index theorem, or the Mathoverflow link.

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Bombyx mori
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