{
  "abstract": "We prove a sharp mean-width inequality for monotone zonal operators and derive global and differential bounds for Euclidean and hyperbolic-harmonic self-maps of the unit ball. Let $k:[0,π]\\to\\mathbb R$ be continuous and nonincreasing, and let $T_k$ be the associated zonal integral operator, \\[ (T_kF)(ξ)=\\int_{\\mathbb S^{n-1}} k\\!\\left(\\arccos\\langleξ,η\\rangle\\right) F(η)\\,dσ(η), \\] where $σ$ is normalized surface measure. For every measurable $F:\\mathbb S^{n-1}\\to\\overline{\\mathbb B^n}$, we prove \\[ w\\!\\left(\\operatorname{co}T_kF(\\mathbb S^{n-1})\\right) \\le2λ_1(k), \\] where $w$ denotes mean width, normalized by $w(\\overline{\\mathbb B^n})=2$, and $λ_1(k)$ is the eigenvalue of $T_k$ on the space of spherical harmonics of degree one. For strictly decreasing kernels, equality holds exactly for $F(η)=Qη$ almost everywhere, with $Q\\in O(n)$. For the ordinary Poisson kernel, the multiplier is $r$, yielding sharp mean-width, intrinsic-volume, and image-volume contraction. In particular, $|f(r\\mathbb B^n)|\\leω_n r^n$ without injectivity assumptions, answering the area and higher-dimensional volume question of Koh and Kovalev.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09316",
  "author_total": 2,
  "authors": [
    {
      "name": "Deguang Zhong"
    },
    {
      "name": "David Kalaj"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fcdg12bmm163t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09316",
  "fulltext_url": null,
  "id": "fcdg12bmm163t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09316",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.09316",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fcdg12bmm163t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09316",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09316"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps",
  "title_text": "Sharp mean-width and Jacobian bounds for Euclidean and hyperbolic harmonic maps",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/fcdg12bmm163t",
  "venue": "arXiv",
  "versions": []
}