{
  "abstract": "For the one-sided Hardy--Littlewood maximal operator $M_d^+$ on $\\mathbb R^d$, the natural two-weight Muckenhoupt condition was shown by Sawyer in 1986 to characterize the weak $(p,p)$ inequality in dimension one, and by Forzani, Martín-Reyes and Ombrosi in 2011 in dimension two. At the endpoint $p=1$ in dimension three, Ombrosi and Nazarov have recently given negative answers to both the Fefferman--Stein-type question and the related weighted weak-type $(1,1)$ question \\cite{OmbrosiNazarov} (personal communication). In this paper, we prove that the two-weight characterization fails for every $d\\geq 3$ and $p>1$. More precisely, for every $1<p<\\infty$ and every $d\\geq 3$, there exist weights $w$ and $v$ such that $$ A_{p,d}^+(w,v)<\\infty, \\qquad \\|M_d^+\\|_{L^p(v)\\to L^{p,\\infty}(w)}=\\infty. $$ The proof uses a finite two-dimensional Hardy operator whose weak operator norm is bounded below by $c_p(\\log N)^{1/p'}$. This operator is embedded into the three-dimensional lattice maximal operator and then transferred to the continuous setting. A tensor extension yields the same failure in every dimension $d>3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08975",
  "author_total": 1,
  "authors": [
    {
      "name": "Dinghuai Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3w0y8m03ftka4.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08975",
  "fulltext_url": null,
  "id": "3w0y8m03ftka4",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08975",
  "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": "W7212000297",
  "pdf_url": "https://arxiv.org/pdf/2609.08975",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3w0y8m03ftka4.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08975",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08975"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions",
  "title_text": "On the Two-Weight Problem for One-Sided Maximal Operators in Higher Dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/3w0y8m03ftka4",
  "venue": "arXiv",
  "versions": []
}