{
  "abstract": "One-parameter variational inequalities are well developed, whereas their multi-parameter counterparts remain much less understood. We explore two-parameter variational inequalities for averages over tori in $\\mathbb{R}^3$. To capture the underlying two-parameter structure, we introduce a local two-parameter $r$-variation norm that combines rectangular increments with variations along the boundary. The resulting variation operator pointwise dominates the corresponding two-parameter local maximal function and, unlike the maximal function, also captures oscillation across the two parameters. We establish sharp $L^p$--$L^q$ bounds for this variation operator up to endpoints. For comparison, we also obtain sharp $L^p$ bounds up to endpoints for the corresponding local one-parameter variation operator, revealing a genuine difference between the one- and two-parameter boundedness regions. The proof combines square function estimates for two-parameter propagators with local smoothing estimates through mixed-norm interpolation.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07422",
  "author_total": 4,
  "authors": [
    {
      "name": "Juyoung Lee"
    },
    {
      "name": "Sanghyuk Lee"
    },
    {
      "name": "Feng Zhang"
    },
    {
      "name": "Shuijiang Zhao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3f1dfcatfs4ks.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07422",
  "fulltext_url": null,
  "id": "3f1dfcatfs4ks",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07422",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7211999022",
  "pdf_url": "https://arxiv.org/pdf/2609.07422",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3f1dfcatfs4ks.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07422",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07422"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Two-parameter variational estimates for averages over tori",
  "title_text": "Two-parameter variational estimates for averages over tori",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/3f1dfcatfs4ks",
  "venue": "arXiv",
  "versions": []
}