一个弯曲的Kakeya集合的构造

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-05-30 DOI:10.1112/blms.70110

Tongou Yang, Yue Zhong

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引用次数: 0

摘要

我们在r2 $\mathbb {R}^2$中构造一个测度0的紧集，其中包含1到2之间每个孔径的抛物线。因此，我们改进了p$ L^p$ - L q$ L^q$范数在p$,q$ p,q$范围内对应的极大算子的下界。此外，我们的构造可以由抛物线推广到满足适当曲率条件的c2 $C^2$曲线族。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

Construction of a curved Kakeya set

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Construction of a curved Kakeya set

We construct a compact set in $R^{2}$ of measure 0 containing a piece of a parabola of every aperture between 1 and 2. As a consequence, we improve lower bounds for the $L^{p}$ - $L^{q}$ norm of the corresponding maximal operator for a range of $p, q$ . Moreover, our construction can be generalised from parabolas to a family of $C^{2}$ curves satisfying suitable curvature conditions.