发表机构
Central China Normal University; Southern University of Science and Technology（华中师范大学; 南方科技大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了有限型凸超曲面上曲面测度Fourier变换的一致下界，并利用参数依赖振荡积分的新工具，推广了Fourier框架不存在性的结果至更弱曲率条件。
AI 中文摘要
我们证明了凸超曲面上曲面测度的Fourier变换的下界：给定$p\in K\Subset\{\psi>0\}$，$\vec{n}_{p}\parallel \xi$，$|\xi|\ge R_0$，有$|\widehat{\psi\\,d\sigma}(\xi)| \gtrsim_{\psi, K, \sigma, R_0} \sigma(B(p, |\xi|^{-1}))$，其中超曲面为有限型且每点至多有一个消失的主曲率。这些下界在$p\in K\Subset \{\psi>0\}$的选择上是一致的，并且与Bruna-Nagel-Wainger的经典上界相匹配。关键的新分析工具是$\mathbb{R}$上具有凸相位的参数依赖振荡积分的一致下界。我们还给出了具有两个消失主曲率的有限型凸例子，对于这些例子，$\inf_{p\in K}|\widehat{\psi\\,d\sigma}(R \vec{n}_p)|\lesssim_N R^{-N}$，对任意固定的$\{0\}\Subset K\Subset\{\psi>0\}$成立。尽管上述估计本身具有意义，我们最初的动机是研究曲面测度的Fourier框架。特别地，我们证明：若$\Gamma$是$\mathbb{R}^d$（$d\geq 2$）中的闭凸光滑曲面，具有有限型且每点至多有一个主曲率消失，则$\Gamma$上的曲面测度不承认任何Fourier框架。这削弱了Iosevich、Lai、第二作者和Wyman先前结果中的非消失曲率条件。在具有非消失高斯曲率的中心对称曲面上，我们获得了更多没有Fourier框架的区域。例如，我们证明$\{x\in S^{d-1}:0<x_d<\delta\}$（$d\geq 3$）对任意$\delta>0$不承认任何Fourier框架，这推广了Chen和第二作者关于半球的结果。
英文摘要
We prove lower bounds for Fourier transforms of surface measures $$|\widehat{ψ\,dσ}(ξ)| \gtrsim_{ψ, K, σ, R_0} σ(B(p, |ξ|^{-1})), \text{ given }p\in K\Subset\{ψ>0\} ,\ \vec{n}_{p}\parallel ξ,\ |ξ|\ge R_0,$$ on convex hypersurfaces of finite type with at most one vanishing principal curvature at each point. These bounds are uniform in the choice of $p\in K\Subset \{ψ>0\}$ 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 $\mathbb{R}$ with convex phases. We also give convex examples of finite type with two vanishing principal curvatures for which $$\inf_{p\in K}|\widehat{ψ\,dσ}(R \vec{n}_p)|\lesssim_N R^{-N},\ \text{ for arbitrary fixed } \{0\}\Subset K\Subset\{ψ>0\}.$$ 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 $\mathbb{R}^d, d\geq 2$, 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 $\{x\in S^{d-1}:0<x_d<δ\}, d\geq 3$, does not admit any Fourier frame for all $δ>0$, which generalizes the result of Chen and the second author on the hemisphere.
Comments20 pages
