{
  "abstract": "Let $Ω\\subset \\mathbb C^n$ be a bounded $m$-hyperconvex domain and let $ψ\\in SH_m(Ω)$ be a fixed negative $m$-subharmonic function. In this paper we introduce a relative finite energy class $\\mathcal E_{m,ψ}(Ω),$ which may be viewed as a Hessian analogue of the relative energy classes appearing in the pluripotential theory of complex Monge--Ampère equations. We develop a systematic pluripotential theory in this setting. More precisely, we introduce a relative Hessian capacity associated with the prescribed singularity type $ψ$, construct relative mixed Hessian products, and establish their fundamental properties. We prove a monotone convergence theorem and a Bedford--Taylor type continuity theorem for Hessian measures in the class $\\mathcal E_{m,ψ}(Ω)$. A central result of the paper is a relative comparison principle, which yields uniqueness of solutions to complex Hessian equations with prescribed singularities. As an application, we establish an existence and uniqueness theorem for the equation $(dd^c u)^m\\wedgeβ^{n-m} =μ$ for a large class of positive Radon measures that do not charge $m$-polar sets. The results obtained here provide a relative finite energy framework for complex Hessian equations and extend several fundamental aspects of Cegrell's theory to the setting of prescribed singularity types.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12032",
  "author_total": 1,
  "authors": [
    {
      "name": "Truong Dinh Dat"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e9k77kw24ypbe.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12032",
  "fulltext_url": null,
  "id": "e9k77kw24ypbe",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12032",
  "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.CV"
    ],
    [
      "openalex",
      "2608|Geometry and complex manifolds"
    ]
  ],
  "openalex": "W7213137130",
  "pdf_url": "https://arxiv.org/pdf/2609.12032",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e9k77kw24ypbe.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:11Z",
      "native_id": "2609.12032",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12032"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Relative Finite Energy Classes for Complex Hessian Equations with Prescribed Singularities",
  "title_text": "Relative Finite Energy Classes for Complex Hessian Equations with Prescribed Singularities",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:11Z",
  "url": "https://research.draconitic.com/paper/e9k77kw24ypbe",
  "venue": "arXiv",
  "versions": []
}