Lacunarity, Kakeya-type sets and directional maximal operators

Edward Kroc; Malabika Pramanik

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arXiv · 1404.6241

Lacunarity, Kakeya-type sets and directional maximal operators

Edward Kroc·Malabika Pramanik

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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.

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BibTeX RIS

Edward Kroc, Malabika Pramanik. 2014-05-02. Lacunarity, Kakeya-type sets and directional maximal operators. https://arxiv.org/abs/1404.6241

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