{
  "abstract": "We study the reciprocal problem for weighted Bergman spaces: if $f\\in A_α^p(\\Bn)$ and $\\inf_{\\Bn}|f|>0$, does it follow that $1/f\\in A_α^p(\\Bn)$? We determine the range of parameters for which the answer is affirmative. In particular, we prove that functions in $A_α^p(\\D)$ have the reciprocal property for all $α\\in \\R$ and $p\\geq 1,$ where $\\D$ denotes the unit disk in $\\C.$ Moreover, functions in $A_α^p(\\Bn)$ have the reciprocal property for all $α\\in \\R$ when $n=2$ and $p=2.$ In addition, we resolve the reciprocal problem in the three-dimensional Drury--Arveson space $H_3^2$ and obtain an equivalent condition in the four-dimensional space $H_4^2$. We also settle the reciprocal problem for general $A_α^p(\\Bn)$ under some additional conditions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.19896",
  "author_total": 3,
  "authors": [
    {
      "name": "Guangfu Cao"
    },
    {
      "name": "Li He"
    },
    {
      "name": "Shuqing Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dq27d9v4d6fee.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.19896",
  "fulltext_url": null,
  "id": "dq27d9v4d6fee",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.19896",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7213578812",
  "pdf_url": "https://arxiv.org/pdf/2609.19896",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dq27d9v4d6fee.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.19896",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.19896"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The Reciprocal Problem on Weighted Bergman Spaces",
  "title_text": "The Reciprocal Problem on Weighted Bergman Spaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/dq27d9v4d6fee",
  "venue": "arXiv",
  "versions": []
}