{
  "abstract": "This monograph investigates the Lebesgue boundedness of planar directional maximal operators $D_Ω$. These are maximal averages of functions over line segments in $\\mathbb R^2$ whose slopes lie in a specified set $Ω\\subseteq\\mathbb R$. 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 $p\\in (1,\\infty)$ 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 $p\\in [1,\\infty)$. 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23871",
  "author_total": 3,
  "authors": [
    {
      "name": "Edward Kroc"
    },
    {
      "name": "Juyoung Lee"
    },
    {
      "name": "Malabika Pramanik"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9kzrjw8jksrpy.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23871",
  "fulltext_url": null,
  "id": "9kzrjw8jksrpy",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23871",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7204244421",
  "pdf_url": "https://arxiv.org/pdf/2608.23871",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9kzrjw8jksrpy.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2608.23871",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23871"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Directional maximal operators in the plane",
  "title_text": "Directional maximal operators in the plane",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/9kzrjw8jksrpy",
  "venue": "arXiv",
  "versions": []
}