{
  "abstract": "Let $C^1(I)$ denote the space of all continuously differentiable real-valued functions on the closed unit interval $I=[0,1]$. For $1\\le p\\le\\infty$, we equip $C^1(I)$ with the norm \\[ \\Vert{f}\\Vert_{(p)} := \\begin{cases} \\left(|f(0)|^p+\\Vert{f'}\\Vert_{I}^p\\right)^{1/p}, & 1\\le p<\\infty, \\\\ \\max\\{|f(0)|,\\Vert{f'}\\Vert_{I}\\}, & p=\\infty, \\end{cases} \\qquad (f\\in C^1(I)), \\] where $\\Vert{\\cdot}\\Vert_I$ denotes the supremum norm on $I$. Let $\\mathcal{S}_p^+:=\\{f\\in C^1(I):f(0)\\ge0,\\,f'\\ge0,\\,\\Vert{f}\\Vert_{(p)}=1\\}$ be the positive unit sphere of $C^1(I)$. A map $T:\\mathcal{S}_p^+\\to \\mathcal{S}_p^+$ is called norm-additive if \\[ \\|T(f)+T(g)\\|_{(p)}=\\|f+g\\|_{(p)} \\qquad (f,g\\in \\mathcal{S}_p^+). \\] For $1<p\\le\\infty$, we prove that every surjective norm-additive map $T\\colon\\mathcal{S}_p^+\\to\\mathcal{S}_p^+$ is of the form \\[ T(f)(t) = f(0)+\\int_0^t f'(τ(s))\\,ds \\] for all $f\\in\\mathcal{S}_p^+$ and $t\\in I$, where $τ\\colon I\\to I$ is a homeomorphism. Conversely, every map of this form is a surjective norm-additive map. Consequently, every such map extends to a surjective linear isometry from $(C^1(I),\\Vert{\\cdot}\\Vert_{(p)})$ onto itself. For $p=1$, however, we construct a surjective norm-additive map on $\\mathcal{S}_1^+$ which is not of the above form and admits no linear extension to $C^1(I)$, showing that the assumption $p>1$ is essential.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.02862",
  "author_total": 2,
  "authors": [
    {
      "name": "Kazuki Ezumi"
    },
    {
      "name": "Takeshi Miura"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5d9j90xbmd5a1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.02862",
  "fulltext_url": null,
  "id": "5d9j90xbmd5a1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.02862",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Banach Space Theory"
    ]
  ],
  "openalex": "W7220436759",
  "pdf_url": "https://arxiv.org/pdf/2610.02862",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5d9j90xbmd5a1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.02862",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.02862"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Norm-Additive Maps on the Positive Unit Sphere of $C^1([0,1])$",
  "title_text": "Norm-Additive Maps on the Positive Unit Sphere of C^1([0,1])",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/5d9j90xbmd5a1",
  "venue": "arXiv",
  "versions": []
}