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

Abstract

AbstractLet be a second‐order elliptic operator with analytic coefficients defined in . We construct explicitly and canonically a fundamental solution for the operator, that is, a function such that . As a consequence of our construction, we obtain an expansion of the fundamental solution in homogeneous terms (homogeneous polynomials divided by a power of , plus homogeneous polynomials multiplied by if the dimension 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 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 .