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Kakeya Sets and Directional Maximal Operators in the Plane

Michael Bateman
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Abstract

We completely characterize the boundedness of planar directional maximal operators on L^p. More precisely, if Omega is a set of directions, we show that M_Omega, the maximal operator associated to line segments in the directions Omega, is unbounded on L^p, for all p < infinity, precisely when Omega admits Kakeya-type sets. In fact, we show that if Omega does not admit Kakeya sets, then Omega is a generalized lacunary set, and hence M_Omega is bounded on L^p, for p>1.

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Michael Bateman. 2007-03-19. Kakeya Sets and Directional Maximal Operators in the Plane. https://arxiv.org/abs/math/0703559

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