{
  "abstract": "We completely characterize a two-depth version of Journé's covering lemma in the bi-parameter setting. Let $\\mathcal U$ be an incomparable family of bi-parameter dyadic rectangles contained in $Ω$, and write $e_i(I):=e_i(I;Ω)$ for the two embeddedness depths. For every coordinatewise nonincreasing $w\\colon\\mathbb N^2\\to[0,\\infty)$, the estimate \\[ \\sum_{I\\in\\mathcal U}w(e_1(I),e_2(I))|I|\\lesssim_w|Ω| \\] holds uniformly in $(\\mathcal U,Ω)$ if and only if the following fixed-total condition holds: \\[ \\|w\\|_{\\mathrm{ft}}:=\\sup_{N\\geq0}\\sum_{a=0}^Nw(a,N-a)<\\infty. \\] For instance, $\\|(a+1)^{-s}(b+1)^{-(1-s)}\\|_{\\mathrm{ft}}\\sim[s(1-s)]^{-1}$ for $0<s<1$, so this allows weights that are not summable in either variable, while one-depth theory holds precisely for summable weights. Exactly the same characterization holds for dyadic Zygmund rectangles $I=I^1\\times I^2\\times I^3$ with $\\ell(I^3)=\\ell(I^1)\\ell(I^2)$. We use probabilistic packing methods to prove the sufficiency, and these techniques also yield sparse refinements of the preceding estimates, which self-improve to various forms of exponential integrability.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.22628",
  "author_total": 1,
  "authors": [
    {
      "name": "Henri Martikainen"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/d146t2nqqpfzc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.22628",
  "fulltext_url": null,
  "id": "d146t2nqqpfzc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.22628",
  "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.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7204204431",
  "pdf_url": "https://arxiv.org/pdf/2608.22628",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/d146t2nqqpfzc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:36Z",
      "native_id": "2608.22628",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.22628"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "A characterization of two-depth Journé packing for bi-parameter and Zygmund rectangles",
  "title_text": "A characterization of two-depth Journé packing for bi-parameter and Zygmund rectangles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/d146t2nqqpfzc",
  "venue": "arXiv",
  "versions": []
}