{
  "abstract": "Let $(\\mathcal{H}_t)_{t \\geq 0}$ denote the standard Ornstein-Uhlenbeck semigroup over ambient space $\\mathbb R^d$ equipped with Gaussian measure $d\\gamma(x) = e^{-|x|^2} dx$. Further let $\\mathcal{H}^*(f)(x) = \\sup_{t > 0} \\mathcal{H}_t(|f|)(x)$ denote the associated maximal operator. Since $(\\mathcal{H}_t)_{t \\geq 0}$ is a Markovian semigroup over $(\\mathbb R^d,d\\gamma(x))$, it is well known that $\\mathcal{H}^*$ is bounded on $L^p(\\mathbb R^d,d\\gamma(x))$ for all $1 < p \\leq \\infty$. We show that $\\mathcal{H}^*$ is also bounded on the weighted space $L^p(\\mathbb{R}^d,\\omega(x)e^{-\\frac{p}{2}|x|^2}dx)$ for $1 < p <\\infty$, for weights $\\omega : \\mathbb R^d \\to (0,\\infty)$ belonging to a certain Muckenhoupt type class $A^\\alpha_p$, where $\\alpha \\in [0,1)$. Essentially, a weight $\\omega$ belongs to $A^\\alpha_p$ if and only if $\\omega(Q) \\omega^{-\\frac{1}{p-1}}(Q)^{p-1} \\leq C |Q|^p$ for all cubes $Q$ contained in any reference cube $N_\\alpha(R_x) \\subseteq \\mathbb R^d$, where the well chosen family $(N_\\alpha(R_x))_{x \\in \\mathbb R^d}$ covers $\\mathbb R^d$, $N_\\alpha(R_x)$ contains $x$ and the side length of $N_\\alpha(R_x)$ is comparable to $1/\\max(1,|x|)^\\alpha$, with constants depending only on $d$. The class $A^\\alpha_p$ contains the usual Muckenhoupt class. We also use the weight class $A_p^{loc}$ from \\cite[Definition 2.2]{B} defined in a similar way as $A^\\alpha_p$ but with dyadic subcubes contained in some $N(R)$ built over the Gaussian dyadic grid (see below), and we show that the local part $\\sup_{t >0}\\mathcal{H}_t(|f|\\chi_{N(R_x)})(x)$ is bounded on $L^p(\\mathbb{R}^d,\\omega(x)e^{-\\frac{p}{2}|x|^2}dx)$ if and only if $\\omega \\in A^{loc}_p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28571",
  "author_total": 2,
  "authors": [
    {
      "name": "Christoph Kriegler"
    },
    {
      "name": "Jérémie Moukambi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/frjzjegxn82qa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28571",
  "fulltext_url": null,
  "id": "frjzjegxn82qa",
  "issns": [],
  "landing_url": "https://hal.science/hal-05757041v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
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      "arxiv",
      "math.CA"
    ],
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      "arxiv",
      "math.FA"
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      "0.math"
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      "1.math.math-ap"
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      "1.math.math-ca"
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    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214273729",
  "pdf_url": "https://hal.science/hal-05757041/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/frjzjegxn82qa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:37Z",
      "native_id": "hal-05757041",
      "source": "hal",
      "url": "https://hal.science/hal-05757041v1"
    },
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28571",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28571"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Ornstein-Uhlenbeck semigroup maximal operator on weighted $L^p$ space",
  "title_text": "Ornstein-Uhlenbeck semigroup maximal operator on weighted L^p space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/frjzjegxn82qa",
  "venue": "HAL",
  "versions": []
}