$L^p$ integrability of functions with Fourier support on a smooth space curve

Abstract

We prove that if $f\in L^p({\mathbb{R}}^k)$ with $p<(k^2+k+2)/2$ satisfies that $\hat{f}$ is supported on a small perturbation of the moment curve in ${\mathbb{R}}^k$, then $f$ is identically zero. This improves the more general result of Agranovsky and Narayanan, and the exponents are sharp in all dimensions. In the process, we develop a mechanism that should lead to further progress on related problems.