在有规律性假设的Kakeya地图上

IF 0.6 3区数学 Q3 MATHEMATICS

Mathematical Research Letters Pub Date : 2023-01-01 DOI:10.4310/mrl.2023.v30.n1.a4

Yuqiu Fu, Shengwen Gan

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引用次数: 0

摘要

对于一个$n-$维的Kakeya集合$(n\geq 3)$，我们可以定义一个与之相关的Kakeya映射，它通过$[0,1]\times S^{n-1},$将Kakeya集合参数化，其中$S^{n-1}$被认为是单位方向的空间。我们证明了如果Kakeya映射是$\alpha-$ Hölder连续的$\alpha>\frac{(n-2)n}{(n-1)^2},$或连续的，并且在Sobolev空间$ H^{s} $对于某些$s>(n-1)/2,$，那么Kakeya集具有正的Lebesgue测度。

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On Kakeya maps with regularity assumptions

For a $n-$dimensional Kakeya set $(n\geq 3)$ we may define a Kakeya map associated to it which parametrizes the Kakeya set by $[0,1]\times S^{n-1},$ where $S^{n-1}$ is thought of as the space of unit directions. We show that if the Kakeya map is either $\alpha-$H\"{o}lder continuous with $\alpha>\frac{(n-2)n}{(n-1)^2},$ or continuous and in the Sobolev space $ H^{s} $ for some $s>(n-1)/2,$ then the Kakeya set has positive Lebesgue measure.

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来源期刊

Mathematical Research Letters 数学-数学

CiteScore

1.40

自引率

0.00%

发文量

审稿时长

6.0 months

期刊介绍： Dedicated to publication of complete and important papers of original research in all areas of mathematics. Expository papers and research announcements of exceptional interest are also occasionally published. High standards are applied in evaluating submissions; the entire editorial board must approve the acceptance of any paper.