{
  "abstract": "Let $$ M_{r,p}^{(d)}(g)=\\int_{\\mathbb R^d}|x|^r|g(x)|^p\\,dx, \\qquad 1<p<\\infty. $$ Write $p'=p/(p-1)$ and $ω_d=|B(0,1)|$. A direct step in the Wigderson--Wigderson moment method asks for a single-function estimate $$ \\lVert g\\rVert_1\\le K\\lVert g\\rVert_q^{\\,1-pη} \\bigl(M_{r,p}^{(d)}(g)\\bigr)^η, \\qquad η=\\frac{d(1-1/q)}{r+d-dp/q}. $$ We give a self-contained proof that, in its scale-invariant regime, this estimate holds exactly when $r>d(p-1)$. We then quantify the transition: the best constant blows up like $[r-d(p-1)]^{-1/p'}$ at the critical line, while on the annulus $1\\le |x|\\le R$ the endpoint loss is exactly $(dω_d\\log R)^{1/p'}$. The critical line nevertheless has two sharp Lorentz endpoints. The strong moment controls $L^{1,p}$, $$ \\lVert g\\rVert_{L^{1,p}}\\le ω_d^{1/p'}M_{d(p-1),p}^{(d)}(g)^{1/p}, $$ and strengthening the weighted norm from $L^p$ to $L^{p,1}$ restores the strong target, $$ \\lVert g\\rVert_1\\le ω_d^{1/p'} \\lVert |x|^{d/p'}g\\rVert_{L^{p,1}}. $$ More generally, every power moment has a sharp target Lorentz space, and its secondary index is optimal. These results identify the endpoint lost by the direct same-$q$ reduction. They do not obstruct auxiliary-norm versions of the Wigderson framework, which reach the Cowling--Price range by a different route.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.22345",
  "author_total": 1,
  "authors": [
    {
      "name": "Dean Menezes"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fxj68x9g34xfp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.22345",
  "fulltext_url": null,
  "id": "fxj68x9g34xfp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.22345",
  "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.GM"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214056282",
  "pdf_url": "https://arxiv.org/pdf/2609.22345",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fxj68x9g34xfp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.22345",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.22345"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.GM",
      "subfield": "general"
    }
  ],
  "title": "A Lorentz Endpoint and Quantitative Obstruction for a Cowling-Price Moment Reduction",
  "title_text": "A Lorentz Endpoint and Quantitative Obstruction for a Cowling-Price Moment Reduction",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/fxj68x9g34xfp",
  "venue": "arXiv",
  "versions": []
}