{
  "abstract": "In this article, we first investigate the necessary and sufficient conditions on the ranges of $s,s_1,s_2\\in R$, $1\\leq p_1,p_2,q\\leq \\infty$, $θ_1,θ_2\\geq 0$ for the validity of the fractional Gagliardo--Nirenberg inequalities $$\\|D^su\\|_{L^q(R^d)}\\lesssim \\|D^{s_1}u\\|_{L^{p_1}(R^d)}^{θ_1}\\|D^{s_2}u\\|_{L^{p_2}(R^d)}^{θ_2}.$$ Secondly, we consider the anisotropic Gagliardo--Nirenberg inequalities $$\\|u\\|_{L^q(R^d)}\\lesssim \\|u\\|_{L^{p_0}(R^d)}^{θ_0} \\prod_{j=1}^n\\|D_{x_j}^{s_j}u\\|_{L^{p_j}(R^d)}^{θ_j}.$$ We show the sharp conditions of these inequalities except ``one\" case. Finally, we establish the profile decomposition associated with the above inequalities when $p_j = 2$, $0\\leq j\\leq n$. Using these results, we establish existence of extremizers and construct soliton solutions of relevant dispersive equations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.27841",
  "author_total": 1,
  "authors": [
    {
      "name": "Jie Chen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8v3a7k70c6x27.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.27841",
  "fulltext_url": null,
  "id": "8v3a7k70c6x27",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.27841",
  "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": "W7204833350",
  "pdf_url": "https://arxiv.org/pdf/2608.27841",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8v3a7k70c6x27.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.27841",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.27841"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications",
  "title_text": "Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/8v3a7k70c6x27",
  "venue": "arXiv",
  "versions": []
}