Lacunarity, Kakeya-type sets and directional maximal operators

2014/04/24 by Kroc, Edward, Pramanik, Malabika
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1404.6241

Abstract

We develop a notion of finite order lacunarity for direction sets in \mathbb R^d+1. Given a direction set Ω that is sublacunary according to this definition, we construct random examples of Euclidean sets that contain unit line segments with directions from Ω and enjoy analytical features similar to those of traditional Kakeya sets of infinitesimal Lebesgue measure. This generalizes to higher dimensions a planar result due to Bateman. Combined with earlier work of Alfonseca, Bateman, Parcet and Rogers, this notion of lacunarity and Kakeya-type sets also yields a characterization in all dimensions for directional maximal operators to be L^p-bounded.

Related

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• Probabilistic Construction of Kakeya-Type Sets in ℝ² associated to separated sets of directions
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