{
  "abstract": "In this paper we study singular integrals associated with the nonnegative kernels \\[ K_α(x,y,z)=\\frac{|z|^{α/2}}{\\|(x,y,z)\\|^{α+1}}, \\qquad α>0, \\] in the first Heisenberg group $\\mathbb H$, where $\\|\\cdot\\|$ denotes the Korányi norm. This family was introduced by Chousionis and Li. We first show that there exists a compact $1$-Ahlfors regular horizontal $C^1$ curve $E\\subset\\mathbb H$ such that the singular integrals associated with $K_α$ are unbounded on $L^2(E)$ for every $0<α<4$. This extends the counterexamples of Chousionis, Li and Zhang from $0<α<2$ to the full range $0<α<4$. Combined with the boundedness theorem of Fässler and Orponen for $α\\ge4$, our result determines the exact range of exponents for boundedness on every $1$-Ahlfors regular curve. We also consider the more general kernels \\[ K_Φ(p)=\\frac{1}{\\|p\\|} Φ\\!\\left(\\frac{|z|^{1/2}}{\\|p\\|}\\right), \\qquad p=(x,y,z)\\neq0, \\] where $Φ:[0,1]\\to[0,\\infty)$ is continuous and nondecreasing, with $Φ(0)=0$. We show that the associated singular integral is $L^2$-bounded on every $1$-Ahlfors regular curve if and only if $Φ(u)=O(u^4)$ as $u\\to0$. When this condition fails, we construct a compact $1$-Ahlfors regular horizontal $C^1$ curve on which the operator is unbounded. The curves in both constructions can be chosen as horizontal lifts of planar $C^1$ graphs with arbitrarily small slope.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.27730",
  "author_total": 1,
  "authors": [
    {
      "name": "Lingxiao Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/b5c9pzmn76esr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.27730",
  "fulltext_url": null,
  "id": "b5c9pzmn76esr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.27730",
  "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"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7223090542",
  "pdf_url": "https://arxiv.org/pdf/2610.27730",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/b5c9pzmn76esr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2610.27730",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.27730"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Sharp boundedness criteria for nonnegative kernels on regular curves in the Heisenberg group",
  "title_text": "Sharp boundedness criteria for nonnegative kernels on regular curves in the Heisenberg group",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/b5c9pzmn76esr",
  "venue": "arXiv",
  "versions": []
}