Abstract: This talk investigates the Lebesgue boundedness of planar directional maximal operators D. These are maximal averages of functions over line segments in the plane whose slopes lie in a specified set P. A large body of multi-authored work has identified a geometric property of P, called finite-order lacunarity, as a key factor in ensuring that D is Lebesgue bounded. While several variations of this notion exist in the literature, they all center on the distribution of gaps in the slope set P.

Building on earlier work, an article of Bateman (2009) asserted a dichotomy for such operators. Namely, D is bounded on Lp for all p>1, precisely when the slope set P is finite-order lacunary, or equivalently, when P does not admit Kakeya-type sets. Conversely, sublacunary direction sets P admit Kakeya-like phenomena, implying in turn that D is unbounded on Lp for all p>1. Recent work of Hagelstein, Radillo-Murguia, and Stokolos (2024) has identified a gap in the proof of this assertion and produced explicit counterexamples, revealing that the previously accepted characterization cannot hold as stated. We provide a rigorous resolution under an adjusted notion of admissible finite-order lacunarity.