Kakeya sets and directional maximal operators in the plane

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 $p<\infty$ 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 $p>1$