{
  "abstract": "Let $Ω\\subset\\mathbb C$ be a bounded John domain and set $δ(z)=\\text{dist}(z,\\partialΩ)$. For $1<p<\\infty$ and $α>\\text{dim}_A(\\partialΩ)-2$, we establish a norm equivalence between $\\int_Ω|g|^pδ^α\\,dA$ and $\\int_Ω|g'|^pδ^{α+p}\\,dA$ for analytic functions $g$ on $Ω$, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking $α=mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\\ge2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between $\\int_Ω|f'|^pδ^{p-2}\\,dA$ and $\\int_Ω|f''|^pδ^{2p-2}\\,dA$ when $1<p<2$. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every $s>1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.23940",
  "author_total": 2,
  "authors": [
    {
      "name": "Katsuhiko Matsuzaki"
    },
    {
      "name": "Huaying Wei"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4zfx74h6cfx7v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.23940",
  "fulltext_url": null,
  "id": "4zfx74h6cfx7v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.23940",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2608|Analytic and geometric function theory"
    ]
  ],
  "openalex": "W7204246024",
  "pdf_url": "https://arxiv.org/pdf/2608.23940",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-04",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4zfx74h6cfx7v.ris",
  "sources": [
    {
      "first_seen": "2026-11-05T15:21:00Z",
      "native_id": "2608.23940",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.23940"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Distance-Weighted Norm Equivalences for Analytic Functions on John Domains",
  "title_text": "Distance-Weighted Norm Equivalences for Analytic Functions on John Domains",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:25Z",
  "url": "https://research.draconitic.com/paper/4zfx74h6cfx7v",
  "venue": "arXiv",
  "versions": []
}