{
  "abstract": "For $β\\ge0$, let $A_1^{(β)}$ be the class of weights satisfying $$ [w]_{A_1^{(β)}}:= \\sup_{0<\\varepsilon\\le1} \\varepsilon^β[w]_{A_{1+\\varepsilon}}<\\infty. $$ These classes satisfy $A_1\\subsetneq A_1^{(β)}\\subsetneq\\bigcap_{p>1}A_p$ for $β>0$. For every $α\\ge0$, we prove that $w\\in A_1^{(β)}$ if and only if $$ w\\left(\\left\\{M_{L(\\log L)^α}f>λ\\right\\}\\right) \\le C\\int_{\\mathbb{R}^n} \\frac{|f(x)|}λ \\left[ \\log\\left(e+\\frac{|f(x)|}λ\\right) \\right]^{α+β} w(x)\\,\\mathrm{d}x. $$ The least constant $C$ is comparable to $[w]_{A_1^{(β)}}$. Also, we obtain corresponding characterizations for Calderón--Zygmund operators and their commutators.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32653",
  "author_total": 1,
  "authors": [
    {
      "name": "Dinghuai Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fh01zjebqjt7s.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32653",
  "fulltext_url": null,
  "id": "fh01zjebqjt7s",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32653",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223709449",
  "pdf_url": "https://arxiv.org/pdf/2610.32653",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fh01zjebqjt7s.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32653",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32653"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Properties and applications of Muckenhoupt $A_1$ growth weights",
  "title_text": "Properties and applications of Muckenhoupt A_1 growth weights",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/fh01zjebqjt7s",
  "venue": "arXiv",
  "versions": []
}