发表机构
UCLA Department of Mathematics（加州大学洛杉矶分校数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明弯曲Kakeya问题可提升为高维线性Kakeya问题，若经典猜想成立，则全维曲线族在Hörmander型族中稠密，并深化了对相关问题的理解。
AI 中文摘要
我们证明许多弯曲Kakeya问题，最初由Hörmander振荡积分问题所激发，可以提升到更高维度的经典Kakeya问题。因此，如果经典Kakeya集猜想在所有维度上都成立，那么其弯曲Kakeya集具有全维度的曲线族在Hörmander型族中是稠密的。这与无处稠密的Bourgain条件形成对比，后者是弯曲Kakeya极大函数估计最佳情形的一个必要条件。此外，线性Kakeya集猜想将意味着对二次曲线Kakeya集的相当完整的理解，这一研究最早由Wisewell系统开展。这些结果增进了对Guo--Guth--Nadjimzadah--Shen--Zhang所提出问题的理解，并且更一般地表明，高维中的经典Kakeya猜想依赖于低维中广泛的弯曲Kakeya问题。
英文摘要
We prove that many curved Kakeya problems, originally motivated by Hörmander's oscillatory integral problem, lift to the classical Kakeya problem in higher dimensions. As a consequence, if the classical Kakeya set conjecture were true in all dimensions, the families of curves whose curved Kakeya sets have full dimension are dense among Hörmander-type families. This is in contrast to the nowhere dense Bourgain's condition, which is a necessary condition for best-case curved Kakeya maximal function estimates. Furthermore the linear Kakeya set conjecture would imply a fairly complete understanding of Kakeya sets of quadratic curves, as first studied systematically by Wisewell. These results give a better understanding of a question posed by Guo--Guth--Nadjimzadah--Shen--Zhang, and they more generally show that the classical Kakeya conjecture in high dimensions rests on a broad range of curved Kakeya problems in lower dimensions.
Comments15 pages, 1 figure
