A Family of Fractal Fourier Restriction Estimates with Implications on the Kakeya Problem

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-01-10 DOI:10.1007/s00041-023-10065-9

Bassam Shayya

引用次数: 0

Abstract

In a recent paper, Du and Zhang (Ann Math 189:837–861, 2019) proved a fractal Fourier restriction estimate and used it to establish the sharp \(L^2\) estimate on the Schrödinger maximal function in \(\mathbb R^n\), \(n \ge 2\). In this paper, we show that the Du–Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in \(\mathbb R^n\) that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in \(\mathbb R^2\).

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分形傅里叶限制估计系列及其对 Kakeya 问题的影响

在最近的一篇论文中，杜和张（Ann Math 189:837-861，2019）证明了一个分形傅里叶限制估计，并用它建立了关于薛定谔最大函数在\(\mathbb R^n\),\(n \ge 2\)中的锐\(L^2\)估计。在本文中，我们证明了 Du-Zhang 估计是分形限制估计族的端点，该族中的每个成员（除原始估计外）都隐含着一个与多项式沃尔夫公理密切相关的 \(\mathbb R^n\) 中尖锐的 Kakeya 结果。我们还证明了我们家族的所有估计在（（mathbb R^2）中都是真的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.