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arXiv 1702.01231math.CA
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关于Fourier限制估计的几点注记

Some remarks on Fourier restriction estimates

Jongchon Kim

备注：
Remark was added after Theorem 1.2. After the submission of the previous version, we learned that Bassam Shayya proved the extension operator estimate in Theorem 1.2 up to the sharp line in arXiv:1512.03238. This note will remain unpublished
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摘要：

我们提供了$L^p \to L^q$改进的一些傅立叶限制估计使用多项式分割。设$S\subset \mathbb{R}^3$是具有严格正第二基本形式的紧致$C^\infty$曲面.我们从用于获得$q>3.25$的$L^\infty(S) \to L^q(\mathbb{R}^3)$界的Guth估计中推导出$q> 3.25$和$q\geq 2p'$的相关傅立叶扩展算子的尖锐$L^p(S) \to L^q(\mathbb{R}^3)$估计。基于Cho和Lee的工作，当$S$是$\mathbb{R}^3$中的双曲抛物面时，我们给出了一个稍弱的结果.最后，我们给出了高维截断抛物面的一些改进。

英文摘要：

We provide $L^p \to L^q$ refinements on some Fourier restriction estimates obtained using polynomial partitioning. Let $S\subset \mathbb{R}^3$ be a compact $C^\infty$ surface with strictly positive second fundamental form. We derive sharp $L^p(S) \to L^q(\mathbb{R}^3)$ estimates for the associated Fourier extension operator for $q> 3.25$ and $q\geq 2p'$ from an estimate of Guth that was used to obtain $L^\infty(S) \to L^q(\mathbb{R}^3)$ bounds for $q>3.25$. We present a slightly weaker result when $S$ is the hyperbolic paraboloid in $\mathbb{R}^3$ based on the work of Cho and Lee. Finally, we give some refinements for the truncated paraboloid in higher dimensions.

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