{
  "abstract": "A question of Bourgain and Brezis, later restated as Open Problem 7.1 by Brezis, asks whether the $H^{1/2}+W^{1,1}$ norm of a mean-zero scalar function is controlled by the ambient $H^{-1/2}+L^1$ sum norm of its gradient. We show that this estimate fails in every dimension $N\\geq 2$. More precisely, for every integer $K\\geq 16$, we construct a smooth real-valued mean-zero function $Φ_K$ on $\\mathbb{T}^N$ such that $|\\nablaΦ_K|{H^{-1/2}(\\mathbb{T}^N)+L^1(\\mathbb{T}^N)}\\leq C$ and $|Φ_K|{H^{1/2}(\\mathbb{T}^N)+W^{1,1}(\\mathbb{T}^N)}\\geq cK^{1/4}-C$, where the absolute constants $c,C>0$ are independent of $N$ and $K$. Equivalently, the ratio of the gradient-constrained decomposition cost to the ambient decomposition cost is unbounded. The construction applies the Hodge projection to a smooth approximation of a normal vector measure on a codimension-one sheet. Its potential records lacunary partial sums at geometrically separated distances from the sheet. A scale-path lower bound, combined with an odd critical Hardy inequality, excludes uniformly bounded $H^{1/2}+W^{1,1}$ decompositions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.35882",
  "author_total": 1,
  "authors": [
    {
      "name": "Chenye Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6jsh8htjthyd9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.35882",
  "fulltext_url": null,
  "id": "6jsh8htjthyd9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.35882",
  "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",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7224107381",
  "pdf_url": "https://arxiv.org/pdf/2610.35882",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6jsh8htjthyd9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.35882",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.35882"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A Counterexample to a Bourgain-Brezis Sum-Space Estimate",
  "title_text": "A Counterexample to a Bourgain-Brezis Sum-Space Estimate",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/6jsh8htjthyd9",
  "venue": "arXiv",
  "versions": []
}