$L^p$ integrability of functions with Fourier support on High codimension manifolds

Abstract

Let $n,k\ge 2$ and $N_{n,k}=\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)-1$. We prove that if $f\in L^p({\mathbb{R}}^{N_{n,k}})$ and $\hat{f}$ is supported on the $n$-dimensional Parsell-Vinogradov manifold, then $f\equiv 0$ whenever $1\le p<p_c(n,k)=n+\left(\genfrac{}{}{0ex}{}{n+k}{n+1}\right)$. This improves the bound by Agranovsky and Narayanan, and generalizes the moment-curve result of Guo-Iosevich-Zhang-Zorin-Kranich. The exponent $p_c(n,k)$ is sharp when $n=2$ or $k=2$, since it matches the convergence exponent $p_{n,k}^{\mathrm{Tar}}$ of Tarry's problem. Our result also yields a general lower bound $p_{n,k}^{\mathrm{Tar}}\ge p_c(n,k)$, improving Ikromov's bound by $n-1$. The proof combines the rotation symmetry of Parsell-Vinogradov manifold with the essential orthogonality of rotated covering boxes.

arXiv comment12 pages, 3 figures