Kakeya Sets and Directional Maximal Operators in the Plane
Abstract
We completely characterize the boundedness of planar directional maximal operators on Lp. More precisely, if Omega is a set of directions, we show that MOmega, the maximal operator associated to line segments in the directions Omega, is unbounded on Lp, 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 MOmega is bounded on Lp, for p>1.
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