Directional maximal operators in the plane

@misc{kroc2026directional,
  title = {{Directional maximal operators in the plane}},
  author = {Edward Kroc and Juyoung Lee and Malabika Pramanik},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.23871},
  eprint = {2608.23871},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.23871},
  note = {CC0},
}

TY  - UNPB
TI  - Directional maximal operators in the plane
AU  - Edward Kroc
AU  - Juyoung Lee
AU  - Malabika Pramanik
PY  - 2026
DA  - 2026/10/09
PB  - arXiv
DO  - 10.48550/arxiv.2608.23871
UR  - https://arxiv.org/abs/2608.23871
AB  - This monograph investigates the Lebesgue boundedness of planar directional maximal operators D_Ω. These are maximal averages of functions over line segments in ℝ^2 whose slopes lie in a specified set Ω⊆ℝ. A large body of work has identified a geometric property of Ω, called finite-order lacunarity, as a key factor in ensuring that D_Ω is Lebesgue bounded. While several variations of this notion exist, they all centre on the distribution of gaps in Ω. Building on earlier work, an article of Bateman(2009) asserted a dichotomy for such operators. Namely, D_Ω is bounded on L^p for all p∈ (1,∞) precisely when the slope set Ω is finite-order lacunary, or equivalently, when Ω does not admit Kakeya-type sets. Conversely, sublacunary direction sets Ω admit Kakeya-like phenomena, implying that D_Ω is unbounded on L^p for all p∈ [1,∞). Recent work of Hagelstein, Radillo-Murguia, and Stokolos(2024) identified a gap in the proof of this assertion and produced counterexamples for which the separation mechanism underlying that proof fails, demonstrating the need for a corrected framework. We establish the corrected characterization by introducing a new notion of admissible finite-order lacunarity that faithfully reflects the combinatorial structure of the direction set. This leads to a tree-theoretic characterization in terms of finite splitting number and provides the foundation for new geometric and probabilistic constructions establishing the equivalence between finite-order lacunarity, the absence of Kakeya-type sets, and the boundedness of directional maximal operators. The resulting framework not only resolves the gap in the earlier proof, but also identifies admissible finite-order lacunarity as the structural invariant governing these phenomena.
ER  -