{
  "abstract": "Let $S_m$ be the Schur multiplier with symbol $m$. We characterize the boundedness of $S_m:\\mathcal S^1\\to\\mathcal S^{1,\\infty}$ by approximation of weighted columns in a common finite-dimensional subspace. If $W(m)$ is the weak $(1,1)$ bound and $\\fD(m)$ is the column approximation constant, then $\\tfrac12W(m)\\le\\fD(m)\\le2W(m)$. These estimates hold uniformly under amplification. We also prove the Hörmander--Mikhlin--Schur weak $(1,1)$ estimate in every Euclidean dimension under the usual integer-order condition. The proof combines Cuculescu's projections, adjacent dyadic grids and tensor polynomial interpolation on a binary block decomposition.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20339",
  "author_total": 1,
  "authors": [
    {
      "name": "Yongqiang Tian"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f1kdnx86abwxz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20339",
  "fulltext_url": null,
  "id": "f1kdnx86abwxz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20339",
  "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|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7222249105",
  "pdf_url": "https://arxiv.org/pdf/2610.20339",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f1kdnx86abwxz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2610.20339",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20339"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem",
  "title_text": "Weak (1,1) Shur multipliers and the Hörmander--Mikhlin--Schur theorem",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/f1kdnx86abwxz",
  "venue": "arXiv",
  "versions": []
}