{
  "abstract": "In this paper, we study a family of weighted Gagliardo--Nirenberg interpolation inequalities associated with Grushin--type operators on $\\mathbb{R}^n \\times \\mathbb{R}^m$. Specifically, we examine how to control the weighted norm $\\||x|^βu \\|_{L^B}$ by balancing fractional derivatives $D_x^s u$ with weighted derivatives $|x|^b D_y^r u$. Using two--parameter scaling, spatial translations, and Littlewood--Paley decompositions, we map out exact parameter conditions and demonstrate that the scaling exponents are both necessary and optimal. For multi-parameter weighted extensions, we leverage Khintchine-type random sums to show why $Q \\ge 2$ is strictly required. When $x$ and $y$ are radially symmetric, two key advantages emerge: the allowable parameter range for the weights expands, and compact embeddings of the energy space into $L^p$, $2 < p < B$, are completely restored. We establish this compactness through two complementary tools, geometric covering arguments and a generalized three--norm version of Strauss's lemma, that yields explicit pointwise weighted decay estimates. Finally, in the scale--invariant case $β=0$, a concentration--compactness analysis proves that the inequality's sharp constant is achieved by an optimal function, with minimizing sequences remaining precompact up to the natural three-parameter group of symmetries.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.11819",
  "author_total": 1,
  "authors": [
    {
      "name": "Amin Esfahani"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9n3yeaa4sp8zz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.11819",
  "fulltext_url": null,
  "id": "9n3yeaa4sp8zz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.11819",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7221096632",
  "pdf_url": "https://arxiv.org/pdf/2610.11819",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9n3yeaa4sp8zz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2610.11819",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.11819"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Weighted Gagliardo--Nirenberg Inequalities of Grushin Type: Symmetries, Compactness, and Optimizers",
  "title_text": "Weighted Gagliardo--Nirenberg Inequalities of Grushin Type: Symmetries, Compactness, and Optimizers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/9n3yeaa4sp8zz",
  "venue": "arXiv",
  "versions": []
}