AI 中文总结
研究在任意维度\(d\geq 2\)下构造卡克亚集，核心方法是推广经典构造，主要贡献是所构造的卡克亚集\(\delta\)邻域体积有改进，还得出了径向傅里叶乘子\(L^p\)有界性相关结论。
AI 中文摘要
我们在任意维度\(d\geq 2\)下构造卡克亚集，将经典的佩龙树构造推广到二维以上。我们构造的卡克亚集的\(\delta\)邻域体积至多为\(C|\log\delta|^{-(d - 1)}\)，改进了先前已知的构造，且推测是最优的。我们还根据具有对数光滑性的贝索夫空间得出了径向傅里叶乘子的\(L^p\)有界性的结论。
英文摘要
We construct Kakeya sets in arbitrary dimension $d\geq 2$, generalizing the classical Perron tree construction beyond dimension $2$. The Kakeya sets we construct have $δ$-neighbourhood of volume at most $C|\logδ|^{-(d-1)}$, which improves on previously known constructions, and which is conjecturally optimal. We further derive consequences for the $L^p$-boundedness of radial Fourier multipliers in terms of Besov spaces with logarithmic smoothness.
CommentsUpdated introduction and added references. Associated GitHub: https://github.com/rafaeljfernandezd/A-CONSTRUCTION-OF-KAKEYA-SETS-IN-ARBITRARY-DIMENSION
