{
  "abstract": "We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $δ>1/2$ there is a constant $C_M(δ)$, independent of the group $G$, such that $\\|μ\\|_{M(G)}\\le1$ and $\\inf_{γ\\in\\widehat G}|\\widehatμ(γ)|\\geδ$ imply that $μ$ is invertible in $M(G)$ and $\\|μ^{-1}\\|_{M(G)}\\le C_M(δ)$. Together with Nikolski's negative results, this shows that $1/2$ is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras $A(K)$ of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at $δ=1/2$ in $A(K)$ for every infinite compact Abelian group $K$, and in the unitization of $L^1(G)$ for every nondiscrete locally compact Abelian group $G$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27714",
  "author_total": 1,
  "authors": [
    {
      "name": "Przemysław Ohrysko"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2vyys8qv49hcs.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27714",
  "fulltext_url": null,
  "id": "2vyys8qv49hcs",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27714",
  "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.FA"
    ],
    [
      "openalex",
      "2610|Advanced Operator Algebra Research"
    ]
  ],
  "openalex": "W7214207967",
  "pdf_url": "https://arxiv.org/pdf/2609.27714",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2vyys8qv49hcs.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27714",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27714"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Norm-Controlled Inversion in Measure Algebras",
  "title_text": "Norm-Controlled Inversion in Measure Algebras",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/2vyys8qv49hcs",
  "venue": "arXiv",
  "versions": []
}