{
  "abstract": "Harrop-Griffiths, Killip, Ntekoume and Vişan proved that the derivative nonlinear Schrödinger equation (DNLS) is globally well-posed in $L^2(\\mathbb{R})$. A key ingredient of their proof is a local smoothing estimate. It holds on $L^2$-equicontinuous sets of initial data, but it cannot hold on $L^2$-balls, because rescaled stationary solitons, all of mass $2π$, violate it. This raises a natural question: for which masses $m$ does local smoothing hold uniformly on the ball $\\{M(q_0)\\le m\\}$? We prove three results. First, we give a quantitative version of the local smoothing estimate in which the data enter only through the mass and an explicit frequency scale $κ(q_0)$. On time intervals of length $2T$ the bound grows like $1+κ(q_0)^2T$, and the stationary soliton shows that the power of $κ$ cannot be lowered below one. Second, we prove uniform local smoothing on balls of small mass, together with a scale-invariant Morrey-type form and a localized $L^4_{t,x}$ bound. Third, we analyse the soliton family from both sides. The uniform constant must blow up at least like $(2π-m)^{-1}$ as $m\\uparrow 2π$, while every soliton of mass $m<2π$ satisfies local smoothing with a constant $O((2π-m)^{-1})$. These facts single out the stationary-soliton mass $2π$, which is also the Hayashi--Ozawa coercivity threshold, as the natural candidate for the sharp threshold. We formulate this as a conjecture, give a spectral reformulation in terms of the Kaup--Newell scattering data, and discuss the obstacles to a proof.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.05330",
  "author_total": 1,
  "authors": [
    {
      "name": "Mohamed Majdoub"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/05tdf665hc10a.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.05330",
  "fulltext_url": null,
  "id": "05tdf665hc10a",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.05330",
  "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-ph"
    ],
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "math.MP"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.05330",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/05tdf665hc10a.ris",
  "sources": [
    {
      "first_seen": "2026-11-04T15:21:25Z",
      "native_id": "2611.05330",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.05330"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "Uniform local smoothing for the derivative NLS and the soliton mass threshold",
  "title_text": "Uniform local smoothing for the derivative NLS and the soliton mass threshold",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/05tdf665hc10a",
  "venue": "arXiv",
  "versions": []
}