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 ∈ X\σ̄, denote by kv(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\σ̄, 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, kv 〉 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.