Instability Phenomena for the Fourier Coefficients

Let P be an elliptic differential operator on a non‐compact connected manifold X ; suppose that both X and the coefficients of P are real analytic. Given a pair of open sets D and σ in X with σ ⊂⊂ D ⊂⊂ X , we fix a sequence { e v } of solutions of Pu = 0 in D which are pairwise orthogonal under integration over both D and σ. By orthogonality is meant the orthogonality in the corresponding Sobolev spaces; we also assume a completeness of the system on σ. For a fixed y ε X/⊂, denote by k v (y) the Fourier coefficients of a fundamental solution ϕ(·, y ) of P with respect to the restriction of {e v } to σ. Suppose K is a compact set in D/σ , and let f be a distribution with support on K . In this paper we show, under appropriate conditions on K , that if the moments ( f, k v ) decrease sufficiently rapidly in a certain precise sense, then these moments vanish identically. In the most favorable cases, it is then possible to conclude that f = 0. This phenomenon was previously noticed by the first author and L. Zalcman for analytic and harmonic moments of f .