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).
