{
  "abstract": "For $p\\geq2$ and $r\\geq 1$, we prove the sharp strong $r$-variation estimate for solutions to the free Schrödinger equation $$ \\bigl\\|V^r_{[0,1]}\\{e^{itΔ}f\\}\\bigr\\|_{L^p(\\mathbb R^2)} \\lesssim \\|f\\|_{H^s(\\mathbb R^2)}. $$ The estimate holds if $s> s(p,r)$ and fails if $s< s(p,r)$, where $$s(p,r)=\\max\\left\\{ \\frac{2}{r}+\\frac{1}{p}-\\frac12,\\, \\frac{2}{r}-\\frac{2}{p},\\, \\frac{1}{p}+\\frac{1}{r},\\, \\frac{2}{p}-\\frac13,\\, 1-\\frac{2}{p} \\right\\}.$$ In particular, $s(3,\\infty)=1/3$. Thus our result yields a sharp variational extension of the two-dimensional Schrödinger maximal estimate of Du, Guth, and Li [{Ann. Math.}, 186, 607--640 (2017)]. The proof employs fractal restriction estimates to eliminate the regularity loss originating from the summation in the variation norm.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31099",
  "author_total": 2,
  "authors": [
    {
      "name": "Yanping Chen"
    },
    {
      "name": "Yongrui Tang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1kq9netk445ht.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31099",
  "fulltext_url": null,
  "id": "1kq9netk445ht",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31099",
  "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.CA"
    ],
    [
      "openalex",
      "2610|Advanced Mathematical Physics Problems"
    ]
  ],
  "openalex": "W7223369446",
  "pdf_url": "https://arxiv.org/pdf/2610.31099",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1kq9netk445ht.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.31099",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31099"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Sharp Variational estimates for Schrödinger equation in $\\mathbb{R}^{2}$",
  "title_text": "Sharp Variational estimates for Schrödinger equation in ℝ^2",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:28Z",
  "url": "https://research.draconitic.com/paper/1kq9netk445ht",
  "venue": "arXiv",
  "versions": []
}