{
  "abstract": "In this paper, for every $\\varepsilon>0$, we establish interior and global a priori $C^{1,α}$ estimates for some $α\\in(0,1)$ from $W^{2,1+\\varepsilon}$ bounds for bounded smooth $2$-admissible solutions of the $2$-Hessian equation, with the interior estimate requiring only an upper bound on the density. Under positive upper and lower density bounds, we prove global $W^{2,1+\\varepsilon}$ estimates for some $\\varepsilon>0$ on bounded strictly mean convex $C^{2,β}$ domains with $C^{2,β}$ boundary values. For densities bounded above and below by positive constants, the interior $C^{1,α}$ estimate follows by combining the first result with an interior $W^{2,1+\\varepsilon}$ estimate obtained by a slight modification of an existing argument. Combining the first result with the global $W^{2,1+\\varepsilon}$ estimate yields the global $C^{1,α}$ estimate. Known examples show that the $C^{2,β}$ regularity assumptions on the boundary and boundary values are optimal in the Hölder scale for both the global $W^{2,1+\\varepsilon}$ estimate and the global $C^{1,α}$ estimate.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35839",
  "author_total": 1,
  "authors": [
    {
      "name": "Yasheng Lyu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8sw98kne9f2qz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35839",
  "fulltext_url": null,
  "id": "8sw98kne9f2qz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35839",
  "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"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7224003497",
  "pdf_url": "https://arxiv.org/pdf/2610.35839",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8sw98kne9f2qz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:43Z",
      "native_id": "2610.35839",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35839"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "$C^{1,α}$ and $W^{2,1+\\varepsilon}$ regularity for the $2$-Hessian equation",
  "title_text": "C^1,α and W^2,1+ε regularity for the $2$-Hessian equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:43Z",
  "url": "https://research.draconitic.com/paper/8sw98kne9f2qz",
  "venue": "arXiv",
  "versions": []
}