{
  "abstract": "Berger and Coburn showed that boundedness of a Toeplitz operator $T_g$ on the Fock space controls the heat transform $g^{(t)}$ for $1/4<t<1$, and conjectured that $T_g$ is bounded if and only if $g^{(1/4)}$ is bounded. Looi recently disproved the forward implication by constructing a bounded $T_g$ with unbounded $g^{(1/4)}$. We prove that the reverse implication also fails, even for a real-valued admissible symbol $g$ with $g^{(1/4)}\\in C_0(\\mathbb{C}^n)$, and that the forward implication fails even when $T_g\\in S_p$ for every finite $p>1$. Our counterexamples come from Baire category and closed graph arguments, which yield no explicit symbols. Despite this two-sided failure, the forward implication has a sharp Schatten-class form: $\\|g^{(1/4)}\\|_\\infty\\le 2^n\\|T_g\\|_{S_1}$ for every admissible symbol $g$ with $T_g$ trace class. We show that the constant $2^n$ is optimal and that the same estimate holds with $S_p$ in place of $S_1$ for $T_g\\in S_p$ when $0<p\\le1$. For $p>1$, no corresponding uniform estimate holds, even for smooth compactly supported symbols. We also determine the optimal constants in the Berger--Coburn estimates for $1/4<t\\le1$. Finally, for $1\\le p<\\infty$, we show that the equivalence $T_g\\in S_p$ if and only if $g^{(1/4)}\\in L^p$ holds exactly when $p=2$, answering a question of Bauer, Coburn, and Isralowitz.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11853",
  "author_total": 1,
  "authors": [
    {
      "name": "Jani A. Virtanen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dy4me6qk0r35x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11853",
  "fulltext_url": null,
  "id": "dy4me6qk0r35x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11853",
  "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",
      "2604|Holomorphic and Operator Theory"
    ]
  ],
  "openalex": "W7212222549",
  "pdf_url": "https://arxiv.org/pdf/2609.11853",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dy4me6qk0r35x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:40Z",
      "native_id": "2609.11853",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11853"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical Schatten exponents for the Berger-Coburn endpoint problem",
  "title_text": "Critical Schatten exponents for the Berger-Coburn endpoint problem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:40Z",
  "url": "https://research.draconitic.com/paper/dy4me6qk0r35x",
  "venue": "arXiv",
  "versions": []
}