# Green operator of elliptic differential operator and radius of convergence

Let $$E \to X$$ be a hermitian vector bundle over a compact Kähler manifold and let $$L$$ be a self-adjoint elliptic linear differential operator on $$E$$. Suppose that $$E \to X$$ and $$L$$ are real-analytic. Then, it is well-known that solutions to $$L u = v$$ are real-analytic if $$v$$ is real-analytic (e.g. [1] Theorem 6.6.1).

Now, let $$G : \Gamma(E) \to \Gamma(E)$$ be the corresponding Green operator, i.e. $$LG + H = \mathrm{Id}$$, where $$H$$ is the orthogonal projection to $$\ker L$$. It follows from the analyticity of solutions to $$Lu = v$$ that $$G$$ maps real-analytic sections to real-analytic sections.

My question is if we can say something about the regions of convergence of the power series of $$G(v)$$ from those of $$v$$. To make this more precise, I will state:

Question. If $$x \in X$$, is there a neighbourhood $$U$$ of $$x$$ in a coordinate chart such that for all $$v \in \Gamma(E)$$ whose power series converges in $$U$$, the power series of $$G(v)$$ also converges in $$U$$?

But any information about the convergence of $$G(v)$$ from that of $$v$$ would be useful.

Reference.

[1] C.B. Morrey, Multiple Integrals in the Calculus of Variations, Grundlehren Series Vol 130, Springer-Verlag, Berlin (1966).

• I think people use asymptotic estimates for Green's operator. Have you tried something standard like Duhamel's principle? – Bombyx mori Oct 15 at 15:13