{
  "abstract": "We create counterexamples to show that Pólya-Szegő type of inequality fails for the weighted nonlocal seminorm \\begin{equation*} [u]_{W^{s,p, α}(\\mathbb{R}^d)} := \\left( \\int_{\\mathbb{R}^d} \\int_{\\mathbb{R}^d} \\frac{|x|^{α_{1} p} |y|^{α_{2} p} |u(x)-u(y)|^{p}}{|x-y|^{d+sp}} \\, dx \\, dy \\right)^{\\frac{1}{p}} \\ , \\end{equation*} in all three regimes $α_1 +α_2 < 0, \\, α_1 +α_2 = 0$, and $α_1 +α_2 >0$. These counterexamples are such that it completely settles the first two cases for all parameters. We also provide necessary and sufficient conditions on the parameters for $ [u]_{W^{s,p, α}(\\mathbb{R}^d)} < \\infty$, thus extending an earlier known result of [Serena Dipierro and Enrico Valdinoci, A density property for fractional weighted Sobolev spaces]. We also provide another proof of the fractional C-K-N type inequality, for most of the parameters (Hardy and Sobolev are included), using a familiar technique which is involved in the proof of fractional Sobolev inequality of [Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des sciences mathematiques] together with a symmetrization technique.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08189",
  "author_total": 2,
  "authors": [
    {
      "name": "Prosenjit Roy"
    },
    {
      "name": "Vivek Sahu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4btzntgvv7a9x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08189",
  "fulltext_url": null,
  "id": "4btzntgvv7a9x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08189",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08189",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4btzntgvv7a9x.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08189",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08189"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Failure of Pólya-Szegő inequality for weighted nonlocal seminorm and on nonlocal C-K-N inequality",
  "title_text": "Failure of Pólya-Szegő inequality for weighted nonlocal seminorm and on nonlocal C-K-N inequality",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:49Z",
  "url": "https://research.draconitic.com/paper/4btzntgvv7a9x",
  "venue": "arXiv",
  "versions": []
}