Expansion of the fundamental solution of a second‐order elliptic operator with analytic coefficients

Abstract

Let L$L$ be a second‐order elliptic operator with analytic coefficients defined in B1⊆Rn$B_1\subseteq \mathbb {R}^n$ . We construct explicitly and canonically a fundamental solution for the operator, that is, a function u:Br0→R$u:B_{r_0}\rightarrow \mathbb {R}$ such that Lu=δ0$Lu=\delta _0$ . As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of |x|$\vert {x}\vert$ , plus homogeneous polynomials multiplied by log(|x|)$\log (\vert {x}\vert )$ if the dimension n$n$ is even) which improves the classical result of [6]. The control we have on the complexity of each homogeneous term is optimal and in particular, when L$L$ is the Laplace–Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to Kodaira [8]. The main ingredients of the proof are a harmonic decomposition for singular functions and the reduction of the convergence of our construction to a nontrivial estimate on weighted paths on a graph with vertices indexed by Z2$\mathbb {Z}^2$ .