{
  "abstract": "Let $L=-Δ+V$ be a Schrödinger operator on $\\mathbb{R}^n$, where $Δ$ is the Laplacian and $V$ satisfies the reverse Hölder inequality ${\\rm RH}_q$ for some $q>n/2$. In this paper, we study the behavior of the Littlewood--Paley operators $s_L$ and $S_L$, as well as the semigroup maximal operator $T^*_L$, on the space ${\\rm CMO}_L(\\mathbb{R}^n)$ associated with the Schrödinger operator $L$. It is known from previous work that these operators are bounded on ${\\rm BMO}_L(\\mathbb{R}^n)$. Our main result shows that they are, in fact, mappings from ${\\rm CMO}_L(\\mathbb{R}^n)$ into itself. To prove this, we develop several equivalent characterizations of ${\\rm CMO}_L(\\mathbb{R}^n)$ and employ a refined decomposition that partitions the parameter interval at $r_Bρ(x_B)$, instead of the customary $r_B^2$ or $ρ(x_B)^2$. The new strategy allows us to overcome a key technical obstacle that arises when applying existing methods to the ${\\rm CMO}_L$ setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.07593",
  "author_total": 3,
  "authors": [
    {
      "name": "Wanjun Li"
    },
    {
      "name": "Qingze Lin"
    },
    {
      "name": "Liang Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c5gyg8y2ff6r7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.07593",
  "fulltext_url": null,
  "id": "c5gyg8y2ff6r7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.07593",
  "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.AP"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7212001716",
  "pdf_url": "https://arxiv.org/pdf/2609.07593",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-07",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c5gyg8y2ff6r7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.07593",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.07593"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators",
  "title_text": "Littlewood--Paley operators and semigroup maximal operators on CMO spaces associated to Schödinger operators",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/c5gyg8y2ff6r7",
  "venue": "arXiv",
  "versions": []
}