{
  "abstract": "We study the representation of bilinear forms associated with Riesz fractional gradients by symmetric integral kernels. Motivated by [SS18, Open Problem 1.10], we give a complete description of the constant coefficient case. We prove that any symmetric representing kernel is uniquely determined and we derive its explicit formula in terms of the constant matrix $A$. This yields a sharp characterization of the matrices for which the kernel is either nonnegative or uniformly positive. In particular, [SS18, Open Problem 1.10] has a negative answer (even for constant matrices). We also identify a class of nonconstant antisymmetric divergence free perturbations that are invisible to the fractional bilinear form and hence leave the representing kernel unchanged. The general case of nonconstant matrices exhibits a substantially different behavior and will be the main subject of our forthcoming paper [DSV26].",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.20110",
  "author_total": 3,
  "authors": [
    {
      "name": "Serena Dipierro"
    },
    {
      "name": "Caterina Sportelli"
    },
    {
      "name": "Enrico Valdinoci"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7m5khpgh4vg7k.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.20110",
  "fulltext_url": null,
  "id": "7m5khpgh4vg7k",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.20110",
  "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",
      "2604|Nonlinear Partial Differential Equations"
    ]
  ],
  "openalex": "W7222173255",
  "pdf_url": "https://arxiv.org/pdf/2610.20110",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7m5khpgh4vg7k.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.20110",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.20110"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients",
  "title_text": "Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/7m5khpgh4vg7k",
  "venue": "arXiv",
  "versions": []
}