{
  "abstract": "We extend the concept of microlocal defect functionals to test functions belonging to the space $\\mathrm{L}^\\infty\\cap\\mathrm{VMO}_c$. Following L.~Tartar's remark that an extension of such concepts to $\\mathrm{VMO}$ spaces should be possible, we establish a functional-analytic framework for this extension within the $\\mathrm{L}^p-\\mathrm{L}^q$ setting. Because the topological dual of $\\mathrm{VMO}$ is the Hardy space $\\mathcal{H}^1$, the resulting object takes the form of an H-distribution rather than a non-negative Radon measure. By assuming strict Hölder conjugate inequalities, we use the John-Nirenberg inequality over localized domains to construct these functionals. We show that the functional acts as a distribution on the inductive limit topology of the test spaces, giving geometric localisation principles for equations with rough $\\mathrm{VMO}$ coefficients, that is, coefficients admitting sharp transitions of vanishing mean oscillation. We illustrate the framework in three settings: stratified transport, zero-order non-local cross-phase energies, and sub-critical acoustic scattering in high-contrast media. These differ in the nonlinearity generating the companion sequence but share a common geometric conclusion, since in each case the macroscopic energy defect is confined to the characteristic variety of the underlying flow, being supported where $\\sum_j a_j({\\bf x})ξ_j=0$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.04294",
  "author_total": 1,
  "authors": [
    {
      "name": "Marin Mišur"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/45dtzyr2dq76f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.04294",
  "fulltext_url": null,
  "id": "45dtzyr2dq76f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.04294",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "1703|Advanced Mathematical Modeling in Engineering"
    ]
  ],
  "openalex": "W7167626916",
  "pdf_url": "https://arxiv.org/pdf/2607.04294",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/45dtzyr2dq76f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2607.04294",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.04294"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Microlocal defect functionals in VMO: Geometric localisation and applications to highly heterogeneous media",
  "title_text": "Microlocal defect functionals in VMO: Geometric localisation and applications to highly heterogeneous media",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/45dtzyr2dq76f",
  "venue": "arXiv",
  "versions": []
}