Lower bounds for parameter-dependent oscillatory integrals and Fourier frames on hypersurfaces
Abstract
We prove lower bounds for Fourier transforms of surface measures |ψ dσ̂(ξ)| ≳ψ, K, σ, R0 σ(B(p, |ξ|−1)), given p∈ KSubset{ψ>0} , n⃗p∥ ξ, |ξ|≥ R0, on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of p∈ KSubset {ψ>0} and match the classical upper bound of Bruna-Nagel-Wainger. The key new analytic ingredient is a uniform lower bound for parameter-dependent oscillatory integrals on ℝ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which infp∈ K|ψ dσ̂(R n⃗p)|≲N R−N, for arbitrary fixed {0}Subset KSubset{ψ>0}. Though the estimate above has its own interest, our original motivation is to study Fourier frames for surface measures. In particular, we show that, if Γ is a closed convex smooth surface in ℝd, d≥ 2, of finite type and at most one principal curvature vanishes at each point, then the surface measure on Γ does not admit any Fourier frame. This weakens the non-vanishing curvature condition in a previous result of Iosevich, Lai, the second author, and Wyman. On centrally symmetric surfaces of non-vanishing Gaussian curvature, we obtain more domains without Fourier frames. For example, we show that {x∈ Sd−1:0<xd<δ}, d≥ 3, does not admit any Fourier frame for all δ>0, which generalizes the result of Chen and the second author on the hemisphere.
Related
- Weak type (1,1) bounds for rough oscillatory singular integrals
- Asymptotic evaluation of multidimensional oscillatory integrals
- Fourier Transform-Based Computation of Two-Electron Four-Center Coulomb Repulsion Integrals
- Categorical Torelli and Fourier-Mukai partners of hypersurfaces
- Approximation of solutions of parameter-dependent problems by residual neural networks
BibTeX
@misc{fan2026lower,
title = {{Lower bounds for parameter-dependent oscillatory integrals and Fourier frames on hypersurfaces}},
author = {Shilei Fan and Bochen Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04799},
eprint = {2610.04799},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04799},
}
RIS
TY - UNPB
TI - Lower bounds for parameter-dependent oscillatory integrals and Fourier frames on hypersurfaces
AU - Shilei Fan
AU - Bochen Liu
PY - 2026
DA - 2026/10/03
PB - arXiv
DO - 10.48550/arxiv.2610.04799
UR - https://arxiv.org/abs/2610.04799
AB - We prove lower bounds for Fourier transforms of surface measures |ψ dσ̂(ξ)| ≳_ψ, K, σ, R_0 σ(B(p, |ξ|^−1)), given p∈ KSubset{ψ>0} , n⃗_p∥ ξ, |ξ|≥ R_0, on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of p∈ KSubset {ψ>0} and match the classical upper bound of Bruna-Nagel-Wainger. The key new analytic ingredient is a uniform lower bound for parameter-dependent oscillatory integrals on ℝ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which inf_p∈ K|ψ dσ̂(R n⃗_p)|≲_N R^−N, for arbitrary fixed {0}Subset KSubset{ψ>0}. Though the estimate above has its own interest, our original motivation is to study Fourier frames for surface measures. In particular, we show that, if Γ is a closed convex smooth surface in ℝ^d, d≥ 2, of finite type and at most one principal curvature vanishes at each point, then the surface measure on Γ does not admit any Fourier frame. This weakens the non-vanishing curvature condition in a previous result of Iosevich, Lai, the second author, and Wyman. On centrally symmetric surfaces of non-vanishing Gaussian curvature, we obtain more domains without Fourier frames. For example, we show that {x∈ S^d−1:0<x_d<δ}, d≥ 3, does not admit any Fourier frame for all δ>0, which generalizes the result of Chen and the second author on the hemisphere.
ER -