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

Abstract

Let $n,k\ge 2$ and $N_{n,k} = \binom{n+k}{k}-1$. 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 $1\le p < p_{c}(n,k) = n+ \binom{n+k}{n+1}$. 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.