论文标题
有限场上的扁平磁盘的限制估计值
Restriction estimates for the flat disks over finite fields
论文作者
论文摘要
在本文中，我们研究了有限磁场上平面磁盘的限制估计。 Mockenhaupt和Tao最初研究了这个问题，但它们的结果仅针对尺寸$ n = 4,6 $解决。我们将其结果改进并扩展到所有维度$ n \ geq 6 $。更确切地说，我们获得了尖锐的$ l^2 \ to l^r $估计，即使在扁平磁盘上使用了最佳的傅立叶衰减估计，也无法通过在有限的字段上应用常规的stein-tomas参数来证明这一点。主要成分之一是发现和分析平盘上表面测量的傅立叶变换的明确形式。此外，基于对抛物面的限制估计的最新结果，我们解决了超出$ l^2 $限制估算的扁平磁盘的限制估计。
In this paper we study the restriction estimate for the flat disk over finite fields. Mockenhaupt and Tao initially studied this problem but their results were addressed only for dimensions $n=4,6$. We improve and extend their results to all dimensions $n\geq 6$. More precisely, we obtain the sharp $L^2 \to L^r$ estimates, which cannot be proven by applying the usual Stein-Tomas argument over a finite field even with the optimal Fourier decay estimate on the flat disk. One of main ingredients is to discover and analyze an explicit form of the Fourier transform of the surface measure on the flat disk. In addition, based on the recent results on the restriction estimates for the paraboloids, we address improved restriction estimates for the flat disk beyond the $L^2$ restriction estimates.
