On Kakeya maps with regularity assumptions

2020/09/18 by Yuqiu Fu, Fu, Yuqiu, Shengwen Gan +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Metric Geometry (math.MG) #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2009.09108

openalex publication_date 2020/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In \mathbb Rⁿ, we parametrize Kakeya sets using Kakeya maps. A Kakeya map is defined to be a map ϕ:B^n-1(0,1)× [0,1]→ ℝⁿ, (v,t)↦ (c(v)+tv,t), where c:B^n-1(0,1)→ ℝ^n-1. The associated Kakeya set is defined to be K:=Im (ϕ). We show that the Kakeya set K has positive measure if either one of the following conditions is true. (1) c is continuous and c|_S^n-2∈ C^α(S^n-2) for some α>((n-2)n)/((n-1)²), (2) c is continuous and c|_S^n-2∈ W^1,p(S^n-2) for some p>n-2.

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