Fourier uniqueness sets and the Klein-Gordon equation

Abstract

Building on ideas from H. Hedenmalm, H., Montes-Rodríguez, A., Heisenberg uniquenes pairs and the Klein-Gordon equation. Ann. of Math. 173 (2011), 1507-1527, we introduce (local) Fourier uniqueness sets for spaces of measures supported on a given curve in the plane. For the classical conic sections, the Fourier transform of the measure solves a second order partial diffeential equation. We focus mainly on the one-dimensional Klein-Gordon equation, which is associated with the hyperbola. We define the Hilbert transform for the hyperbola, and use it to introduce a natural real Hardy space of absolutely continuous measures on the hyperbola. For that space of measures, we obtain several examples of (local) Fourier uniqueness sets. We also obtain examples of Fourier uniqueness sets in the context of all Borel measures on the curve. The proofs are based on the dynamics of Gauss-type maps combined with ideas from complex analysis. We also look at the Fourier uniqueness sets for one branch of the hyperbola, where the notion of defect becomes natural.