{
  "abstract": "We study the partially fractional Grushin-type operator \\begin{equation} \\mathcal L_{α,β}u=(-Δ_x)^αu-(1+β)^2|x|^{2β}Δ_y u, \\qquad (x,y)\\in\\mathbb{R}^n\\times\\mathbb{R}^m,\\quad 0<α<1,\\ β\\ge 0, \\end{equation} which couples fractional diffusion in $x$ with degenerate local diffusion in $y$. Working with the anisotropic dilations $(x,y)\\mapsto(ρx,ρ^{α+β}y)$ and an adapted quasi-distance, we prove a uniform fractional Grushin-Poincaré inequality on anisotropic cylinders and, via a Riesz-type potential built from a telescoping sum over dyadic cylinders, a global Sobolev inequality at the sharp critical exponent $2^*_{α,β}=2Q_{α,β}/(Q_{α,β}-2α)$, where $Q_{α,β}=n+(α+β)m$. We then establish a concentration-compactness principle for the associated energy space. For $β>0$ the weight $|x|^{2β}$ destroys translation invariance in $x$, and we show that concentration is confined to the degenerate hyperplane $ Σ= \\{(0,y):y\\in\\mathbb{R}^m\\}$; this localization yields the existence of extremals for the sharp constant $S_{α,β}(n,m)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.09001",
  "author_total": 3,
  "authors": [
    {
      "name": "S. Malhotra"
    },
    {
      "name": "L. Sharma"
    },
    {
      "name": "K. Sreenadh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dm76gphq1q09a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.09001",
  "fulltext_url": null,
  "id": "dm76gphq1q09a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.09001",
  "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"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.09001",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-06",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dm76gphq1q09a.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:42Z",
      "native_id": "2610.09001",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.09001"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator",
  "title_text": "Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:42Z",
  "url": "https://research.draconitic.com/paper/dm76gphq1q09a",
  "venue": "arXiv",
  "versions": []
}