{
  "abstract": "We give a direct proof of the equivalence between the Muckenhoupt $\\mathcal{A}_2$ and the Helson--Szegő conditions on the unit circle $T$. A positive weight $w$ belongs to $\\mathcal{A}_2(T)$ if and only if $\\log w=u+Hv$, where $u,v$ are real bounded functions, $\\|v\\|_\\infty<\\fracπ{2}$, and $H$ denotes the periodic Hilbert transform. The classical proof combines the Helson--Szegő theorem with the Hunt--Muckenhoupt--Wheeden characterization of weighted boundedness of the Hilbert transform. Our proof derives the Helson--Szegő representation directly from the Muckenhoupt $\\mathcal{A}_2$ condition, without using boundedness of $H$ on $L^2(w)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32922",
  "author_total": 3,
  "authors": [
    {
      "name": "Christos Grigoriadis"
    },
    {
      "name": "Michail Paparizos"
    },
    {
      "name": "Georgios Psaromiligkos"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a803gjdst952s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32922",
  "fulltext_url": null,
  "id": "a803gjdst952s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32922",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223708930",
  "pdf_url": "https://arxiv.org/pdf/2610.32922",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a803gjdst952s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32922",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32922"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A direct proof of the equivalence of the A2 and the Helson--Szego conditions",
  "title_text": "A direct proof of the equivalence of the A2 and the Helson--Szego conditions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/a803gjdst952s",
  "venue": "arXiv",
  "versions": []
}