2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125954We 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>1.20 pagesClassical Analysis and ODEsCombinatoricsProbability42B25, 60K35text