PreprintCC BY 4.0arXiv
Lp integrability of functions with Fourier support on High codimension manifolds
Abstract
Let n,k≥ 2 and Nn,k = binomn+kk−1. We prove that if f ∈ Lp(ℝNn,k) and f̂ is supported on the n-dimensional Parsell-Vinogradov manifold, then f ≡ 0 whenever 1≤ p < pc(n,k) = n+ binomn+kn+1. This improves the bound by Agranovsky and Narayanan, and generalizes the moment-curve result of Guo-Iosevich-Zhang-Zorin-Kranich. The exponent pc(n,k) is sharp when n=2 or k=2, since it matches the convergence exponent pn,kTar of Tarry's problem. Our result also yields a general lower bound pn,kTar ≥ pc(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.
Related
- An arithmetic integrability result for codimension-one foliations on complex projective spaces
- Lagrangian fibrations on hyperkahler manifolds have no multiple fibers in codimension one
- Semi-classical Szegő kernel asymptotics on CR manifolds of high codimension
- Integrability of a composition of functions
- Pettis integrability of functions with values in separable symmetrically-normed ideals and related norm estimates
BibTeX
@misc{bi2026integrability,
title = {{$L^p$ integrability of functions with Fourier support on High codimension manifolds}},
author = {Cheng Bi and Ruichen Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39739},
eprint = {2610.39739},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39739},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - L^p integrability of functions with Fourier support on High codimension manifolds
AU - Cheng Bi
AU - Ruichen Sun
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39739
UR - https://arxiv.org/abs/2610.39739
AB - Let n,k≥ 2 and N_n,k = binomn+kk−1. We prove that if f ∈ L^p(ℝ^N_n,k) and f̂ is supported on the n-dimensional Parsell-Vinogradov manifold, then f ≡ 0 whenever 1≤ p < p_c(n,k) = n+ binomn+kn+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^Tar of Tarry's problem. Our result also yields a general lower bound p_n,k^Tar ≥ 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.
ER -