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

Abstract

Let $L$ be a second-order elliptic operator with analytic coefficients defined in $B_1\subseteq\mathbb R^n$. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function $u:B_{r_0}\to\mathbb R$ such that $Lu=\delta_0$. As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of $|x|$, plus homogeneous polynomials multiplied by $\log(|x|)$ if the dimension $n$ is even) which improves the classical result of F. John (1950). The control we have on the "complexity" of each homogeneous term is optimal and in particular, when $L$ is the Laplace-Beltrami operator of an analytic Riemannian manifold, we recover the construction of the fundamental solution due to K. Kodaira (1949). 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 $\mathbb Z^2$.