{
  "abstract": "We determine the approximation numbers of critical embeddings between isotropic Besov spaces on nonempty bounded Lipschitz domains in any dimension. For 0 < p_1 < 2 < p_2 <= infinity and s_1 - s_2 = d max{1/p_1, 1 - 1/p_2} > d(1/p_1 - 1/p_2), the approximation numbers have order N^{-1/2}(log(e+N))^Gamma for every 0 < q_1, q_2 <= infinity, with an explicit four-case formula for Gamma. This resolves the strictly compact critical Besov approximation-number problem identified in Vybiral's overview. For q_1 < q_2, the logarithmic exponent depends on the ordering of p_1' and p_2; it can vanish for one pair (p_1,p_2) and remain positive for another with the same fine indices. A row estimate for the approximation error gives lower bounds when the target fine index is below one. Gaussian finite-rank approximations acting on several levels give the sharp upper bounds when Gamma < 1/2. For q_1 < q_2, matching lower bounds for the two strict orderings of the inner indices follow from Malykhin's finite-window estimates and norm comparisons. The results include quasi-Banach indices and the full spaces of bounded coefficient sequences at infinite fine indices.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.25910",
  "author_total": 1,
  "authors": [
    {
      "name": "Feng Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/068bf537tgew0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.25910",
  "fulltext_url": null,
  "id": "068bf537tgew0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.25910",
  "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 Banach Space Theory"
    ]
  ],
  "openalex": "W7223016301",
  "pdf_url": "https://arxiv.org/pdf/2610.25910",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-19",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/068bf537tgew0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:57Z",
      "native_id": "2610.25910",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.25910"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Approximation numbers of critical Besov embeddings",
  "title_text": "Approximation numbers of critical Besov embeddings",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:44Z",
  "url": "https://research.draconitic.com/paper/068bf537tgew0",
  "venue": "arXiv",
  "versions": []
}