{
  "abstract": "We develop square-function estimates for Fourier multipliers whose local oscillation scale is \\[ ρ(R)=\\frac{R}{(\\log R)^{γ-1}}, \\qquad γ>1. \\] This scale lies strictly between the dyadic scale and every fixed power-subdyadic scale at high frequency. For high-frequency symbols satisfying a localized Sobolev condition on balls of radius comparable to $ρ(R)$, we prove a pointwise square-function estimate and a weighted $L^2$ multiplier inequality. After adjoining a smooth compactly supported low-frequency part, we derive unweighted $L^p$ bounds. The weighted estimate is governed by a logarithmic geometric maximal operator which is strongly bounded above the critical $L^r$ threshold, satisfies weak type at the critical equality, and fails even weak type below it. As a model application, consider \\[ L(ξ)=\\frac12\\log(e^2+|ξ|^2), \\qquad m_{γ,β}(ξ)=L(ξ)^{-β}e^{iL(ξ)^γ}. \\] For $p=2$, the associated multiplier is bounded on $L^2$ for every $β\\geq0$. For $1<p<\\infty$, $p\\neq2$, it is bounded on $L^p$ under the sufficient condition \\[ β> d(γ-1)\\left|\\frac12-\\frac1p\\right|. \\] At the critical equality we obtain the corresponding Lorentz endpoint estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.27746",
  "author_total": 1,
  "authors": [
    {
      "name": "Vicente Vergara"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8713a97zc524v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.27746",
  "fulltext_url": null,
  "id": "8713a97zc524v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.27746",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7162665364",
  "pdf_url": "https://arxiv.org/pdf/2605.27746",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8713a97zc524v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:43Z",
      "native_id": "2605.27746",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.27746"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Logarithmic oscillatory multipliers and log-subdyadic square functions",
  "title_text": "Logarithmic oscillatory multipliers and log-subdyadic square functions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/8713a97zc524v",
  "venue": "arXiv",
  "versions": []
}