{
  "abstract": "In this paper, we establish a doubling argument to obtain Hessian estimates for convex solutions to the Hessian quotient equation $\\frac{σ_n}{σ_k}(D^2u) = f(x,u,Du)$ for $k=n-1$ and $k=n-2$ under the condition that $\\log f$ is convex in the $Du$ variable. In particular, our approach is pointwise and does not make use of the Legendre transform or integral-based local maximum principles. We provide a counterexample demonstrating that interior estimates can fail if no structural assumption is imposed on $f$ in the $Du$ variable. Finally, we extend our doubling argument to general Hessian quotient equations $\\frac{σ_l}{σ_k}(D^2u) = f(x,u,Du)$ for $k \\in \\{l-1, l-2\\}$, under a similar structural condition imposed on $f$ in the $Du$ variable, alongside an additional structural concavity assumption on the operator introduced by Lu-Tsai 2026, which has very recently been established in independent works.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.21982",
  "author_total": 1,
  "authors": [
    {
      "name": "Cheuk Yan Fung"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c36acd131ap9e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.21982",
  "fulltext_url": null,
  "id": "c36acd131ap9e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.21982",
  "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|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7171468062",
  "pdf_url": "https://arxiv.org/pdf/2607.21982",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c36acd131ap9e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2607.21982",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.21982"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Doubling Argument of the Hessian Estimate for the Hessian Quotient Equations",
  "title_text": "Doubling Argument of the Hessian Estimate for the Hessian Quotient Equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/c36acd131ap9e",
  "venue": "arXiv",
  "versions": []
}