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

Abstract:

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