The null set of the Fourier transform for a surface carried measure

Abstract

Let d u du be a smooth positive measure carried by a smooth compact hypersurface S S that is strictly convex and without boundary in R n ( n ⩾ 2 ) {R^n}(n \geqslant 2) . Assume that both S S and d u du are symmetric about the origin. If d ^ u \hat du denotes the Fourier transform of d u du then we show that the null set of d ^ u \hat du is a disjoint union of a compact set and countably many hypersurfaces that are all diffeomorphic to the unit sphere S n − 1 {S^{n - 1}} .