Let $K$ be a compact subset in $\mathbb{R}^n$ with $m(K)=0$, Suppose $supp\hat{u}\subset K$ for some $u\in L^p$,where $2\leq p\leq \frac{2n}{n-1}$,can we get $u\equiv 0$ ?
Motivation: If $K$ is a compact non-degenerate hypersurface,then it's well known that $u(x)\leq C|x|^{-\frac{n-1}{2}}$,hence $u\in L^p$ for any $p>\frac{2n}{n-1}$,but if we restrict $p$ in $[2,\frac{2n}{n-1}]$,then is it possible that $u\equiv 0$ ?