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Expansion of the fundamental solution of a second-order elliptic operator with analytic coefficients
Federico Franceschini, Federico Glaudo
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Let L be a second-order elliptic operator with analytic coefficients defined in B₁⊆ℝⁿ. We construct explicitly and canonically a fundamental solution for the operator, i.e., a function u:B_(r₀)→ℝ such that Lu=δ₀. 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 ℤ².
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