{
  "abstract": "This paper establishes sharp variational estimates for families of averaging operators over smooth hypersurfaces in $\\mathbb{R}^{n+1}$. For a hypersurface with $k$ non-zero principal curvatures at a given point, we prove that the $r$-variation operator $V_r(A_t f : t>0)$ is bounded on $L^p$ for all exponents satisfying $(k+1)/k < p < r(k+1)$, with operator norm depending explicitly on a translation parameter $D$. In the special case $k=n$ (i.e., non-vanishing Gaussian curvature), we show that the range $p<r(n+1)$ is optimal, thereby recovering and extending classical results for the sphere. Furthermore, for convex hypersurfaces of finite type (in the sense of Iosevich-Sawyer), we obtain variational bounds in the range $p_0 < p < 2r$, where $p_0$ is the contact index of the associated hypersurface. The proofs combine long-short variational decompositions, stationary phase analysis, vector-valued singular integrals, and local smoothing estimates for Fourier integral operators. These results unify and generalize earlier work on maximal functions and variational inequalities for averages over hypersurfaces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.01816",
  "author_total": 2,
  "authors": [
    {
      "name": "Xudong Nie"
    },
    {
      "name": "Panwang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7yr2etm4wbtgt.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.01816",
  "fulltext_url": null,
  "id": "7yr2etm4wbtgt",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.01816",
  "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.CA"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.01816",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7yr2etm4wbtgt.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.01816",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.01816"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "On Variational Inequalities for Averages over Hypersurfaces",
  "title_text": "On Variational Inequalities for Averages over Hypersurfaces",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/7yr2etm4wbtgt",
  "venue": "arXiv",
  "versions": []
}