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 Ω is a set of directions, we show that MΩ, the maximal operator associated to line segments in the directions Ω, is unbounded on Lp for all p<∞ precisely when Ω admits Kakeya-type sets. In fact, we show that if Ω does not admit Kakeya sets, then Ω is a generalized lacunary set, and hence, MΩ is bounded on Lp for p>1