{
  "abstract": "Let $I=(a,b)$ be an open interval with finite or infinite endpoints. For $1<p_1,p_2<\\infty$ and $0<q<\\infty$, we study the weighted bilinear Hardy operator $ \\mathcal H_2(f,g)(x)=(\\int_a^x f)(\\int_a^x g) $ from $L^{p_1}(v_1;I)\\times L^{p_2}(v_2;I)$ to $L^q(u;I)$, including the quasi-Banach target range $0<q<1$. With $\\vartheta=\\min\\{1,q\\}$, the target and bounded-bilinear-map spaces are treated through their complete powered metrics. Under explicit local integrability assumptions on the weights, compactness is equivalent to boundedness and vanishing endpoint-tail quasi-norms. Whenever the full operator is bounded, the $\\vartheta$-power of the directed combined-tail limit equals exactly the powered essential distance, finite-rank distance, and Hausdorff measure of noncompactness in the target metric. Equivalently, the unpowered quantities agree under the stated quasi-norm-radius normalisation. Piecewise-constant middle-window approximants give quantitative bounds for the bilinear approximation numbers. The finite-rank-distance identity then implies that these numbers converge to the essential (quasi-)norm. Applying published norm-equivalent boundedness characteristics to endpoint-truncated output weights yields explicit compactness criteria in the convex, mixed, moderately subcritical, and deeply subcritical regimes. Algebraic companion exponents are used to make the negative powers arising for $q<1$ explicit. Four worked examples include an asymmetric mixed-range threshold and a nonlocally convex $L^{1/2}(0,1)$ example for which $a_{m+1}^{(2)}(\\mathcal H_2)=O(m^{-1/2})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37998",
  "author_total": 1,
  "authors": [
    {
      "name": "Saikat Kanjilal"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ceqwwjnza7dh7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37998",
  "fulltext_url": null,
  "id": "ceqwwjnza7dh7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37998",
  "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"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214994986",
  "pdf_url": "https://arxiv.org/pdf/2609.37998",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ceqwwjnza7dh7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37998",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37998"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals",
  "title_text": "Compactness and Essential Norms of Weighted Bilinear Hardy Operators on General Intervals",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/ceqwwjnza7dh7",
  "venue": "arXiv",
  "versions": []
}