{
  "abstract": "In this paper, we focus on the positive reflectionless measures rigidity problem proposed by Xavier Tolsa in [16]. Let $1\\leq n<d$, we consider positive reflectionless measures for the $n$-dimensional Riesz transform in $\\mathbb R^d$. Under polynomial upper growth and a lower mass bound at large scales about one point, we obtain two flatness results for reflectionless measures. The first concerns globally defined Lipschitz graph measures with finite $\\dot H^{1/2}$ energy and bounded, uniformly positive parameter densities whose deviations from a constant belong to $L^2$. The second case concerns a reflectionless measure that coincides with a flat measure outside a compact set. In both cases, the measure is a constant multiple of the $n$-dimensional Hausdorff measure on an affine $n$-plane. For integers $s>n/2$, we further obtain $H^s$ estimates controlling the graph gradient and density perturbation by the normalized Riesz field under smallness of the slope and density perturbation in $L^\\infty$. The constants are independent of the supports.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.32179",
  "author_total": 3,
  "authors": [
    {
      "name": "Lu Chen"
    },
    {
      "name": "Haolin Liu"
    },
    {
      "name": "Hongyu Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5vrf5yc9qk02a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.32179",
  "fulltext_url": null,
  "id": "5vrf5yc9qk02a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.32179",
  "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": "W7214719205",
  "pdf_url": "https://arxiv.org/pdf/2609.32179",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5vrf5yc9qk02a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.32179",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.32179"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs",
  "title_text": "Rigidity of reflectionless Riesz measures and Sobolev estimates for Riesz transforms on Lipschitz graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/5vrf5yc9qk02a",
  "venue": "arXiv",
  "versions": []
}