{
  "abstract": "Let $\\mathfrak R=\\nabla(-Δ_D)^{-1/2}$ be the Riesz transform associated with the Dirichlet Laplacian on the upper half-space $\\mathbb R^n_+$, and let $H^1_r(\\mathbb R^n_+)$ denote the Hardy space of restrictions. We establish an endpoint theory for commutators with $\\mathfrak R$ on $H^1_r(\\mathbb R^n_+)$. For every $b\\in \\mathrm{BMO}(\\mathbb R^n_+)$, the commutator $[b,\\mathfrak R]$ extends boundedly from $H^1_r(\\mathbb R^n_+)$ to $L^{1,\\infty}(\\mathbb R^n_+)$. We then prove the sharp strong-endpoint characterization \\[ [b,\\mathfrak R]:H^1_r(\\mathbb R^n_+)\\to H^1_r(\\mathbb R^n_+) \\quad\\Longleftrightarrow\\quad b\\in \\mathrm{BMO}^{\\log}(\\mathbb R^n_+). \\] Moreover, the $\\mathrm{BMO}^{\\log}$ norm is quantitatively equivalent to the sum of the usual $\\mathrm{BMO}$ norm and the $H^1_r$ operator norms of the component commutators $[b,\\mathfrak R_j]$. The logarithmic condition records the distance to the boundary and is therefore intrinsic to the Dirichlet geometry.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.23672",
  "author_total": 4,
  "authors": [
    {
      "name": "The Anh Bui"
    },
    {
      "name": "Xuan Thinh Duong"
    },
    {
      "name": "Xuejing Huo"
    },
    {
      "name": "Luong Dang Ky"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7tx8sbfy2cc01.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.23672",
  "fulltext_url": null,
  "id": "7tx8sbfy2cc01",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.23672",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7213996181",
  "pdf_url": "https://arxiv.org/pdf/2609.23672",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7tx8sbfy2cc01.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.23672",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.23672"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space",
  "title_text": "Endpoint Estimates for Commutators of the Dirichlet Riesz Transform on the Half-Space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/7tx8sbfy2cc01",
  "venue": "arXiv",
  "versions": []
}