{
  "abstract": "In his seminal paper on the $\\bar{\\partial}$-equation in infinite dimensions [Rab], P. Raboin establishes the existence of $C^\\infty$ solutions on a subspace $H_T$ of a Hilbert space $H$, given a closed $C^\\infty$ right-hand side $F$ and a self-adjoint Hilbert-Schmidt correlation operator $T$. His proof relies on a lemma stated without proof; we demonstrate that this lemma is false in general within infinite-dimensional spaces by constructing a counterexample based on a theorem by Borwein and Fabian [BF]. We show that the conclusion of [Rab] remains unaffected, as it follows from an independent argument. We then present, with a complete proof, a generalization obtained in 1985: the $C^\\infty$ regularity assumption on $F$ can be relaxed to $C^1$, and the assumption regarding the correlation operator can be extended to any injective nuclear operator.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 1,
  "authors": [
    {
      "name": "Alain Michel Didier Rapp"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5ba05pzp0hdq6.bib",
  "cited_by": 0,
  "doi": null,
  "fulltext_url": null,
  "id": "5ba05pzp0hdq6",
  "issns": [],
  "landing_url": "https://hal.science/hal-05799614v1",
  "language": "fr",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://hal.science/hal-05799614/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5ba05pzp0hdq6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:57Z",
      "native_id": "hal-05799614",
      "source": "hal",
      "url": "https://hal.science/hal-05799614v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "hal:0.math",
      "subfield": "general"
    }
  ],
  "title": "Sur l'équation ∂̄ dans un espace de Hilbert correction et généralisation d'un résultat de P. Raboin",
  "title_text": "Sur l'équation ∂̄ dans un espace de Hilbert correction et généralisation d'un résultat de P. Raboin",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:57Z",
  "url": "https://research.draconitic.com/paper/5ba05pzp0hdq6",
  "venue": "HAL",
  "versions": []
}