{
  "abstract": "We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying $$ λ\\,dx\\leqμ_u\\leqλ^{-1}\\,dx, \\qquad 0<λ<1. $$ The distributional Hessian satisfies the sharp endpoint estimate $$ D^2u\\in L_{\\mathrm{loc}}^{P,\\infty}, \\qquad P=\\frac{1+λ}{1-λ}. $$ This settles the weak estimate proposed by De Philippis and Figalli and identifies the larger optimal exponent $P$. We further construct strictly convex $C^1$ solutions for which the distribution function of the Laplacian is comparable to $T^{-P}$ for all sufficiently large $T$. Hence both the exponent and the weak Lorentz endpoint are optimal: $L^{P,\\infty}$ cannot be replaced by $L^{P,b}$ for any finite $b>0$. As applications, we obtain sharp estimates in Minty coordinates and optimal quantitative bi-Lipschitz restrictions for gradient maps, including an optimal exceptional-mass bound in quadratic-cost transport.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.21956",
  "author_total": 4,
  "authors": [
    {
      "name": "XiaoRui Lu"
    },
    {
      "name": "Jin Yan"
    },
    {
      "name": "Bao Yu"
    },
    {
      "name": "Yang Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dxrqe9eqa35jv.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.21956",
  "fulltext_url": null,
  "id": "dxrqe9eqa35jv",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.21956",
  "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": "W7222677737",
  "pdf_url": "https://arxiv.org/pdf/2610.21956",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dxrqe9eqa35jv.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.21956",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.21956"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Hessian integrability for the planar Monge--Ampère equation",
  "title_text": "Sharp Hessian integrability for the planar Monge--Ampère equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/dxrqe9eqa35jv",
  "venue": "arXiv",
  "versions": []
}