{
  "abstract": "We determine the sharp $L^p$ ranges of the Riesz transform and the reverse Riesz inequality for a class of power-degenerate elliptic operators of Grushin type. When the degenerate variable has dimension at least two, the Riesz transform is bounded throughout the reflexive range. In dimension one, a capacity-driven transition occurs: weak degeneracy produces a sharp finite upper endpoint, whereas strong degeneracy separates the two half-lines and restores the full range. The reverse inequality displays a different sharp phenomenon. In the weakly degenerate one-dimensional regime it holds throughout the reflexive range except at a single Hardy-critical exponent, while in all remaining regimes it holds for every $1<p<\\infty$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.15599",
  "author_total": 1,
  "authors": [
    {
      "name": "Dangyang He"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ex42g4zd1k5ps.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.15599",
  "fulltext_url": null,
  "id": "ex42g4zd1k5ps",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.15599",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7169790938",
  "pdf_url": "https://arxiv.org/pdf/2607.15599",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-20",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ex42g4zd1k5ps.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:08Z",
      "native_id": "2607.15599",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.15599"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type",
  "title_text": "Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:08Z",
  "url": "https://research.draconitic.com/paper/ex42g4zd1k5ps",
  "venue": "arXiv",
  "versions": []
}