{
  "abstract": "Let $G$ be the group $\\mathbb{R}_+\\ltimes \\mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\\mathrm{d} ρ$ which is of $ax+b$ type, and $L$ be the positive definite distinguished left-invariant Laplacian on $G$. Let $u=u(t,\\cdot)$ be the solution to $u_{tt}+Lu=0$ with initial conditions $u|_{t=0}=f$ and $u_t|_{t=0}=g$. In this article we show that for a fixed $t \\in\\mathbb{R}\\setminus\\{0\\}$, \\begin{align*} \\|u(t,\\cdot)\\|_{L^{1,\\infty}(G,\\mathrm{d}ρ)}\\leq C\\ (1+|t|)\\big(\\|(\\operatorname{Id}+L)^{α_0/2}f\\|_{L^1(G,\\mathrm{d}ρ)} +\\|(\\operatorname{Id}+L)^{α_1/2}g\\|_{L^1(G,\\mathrm{d}ρ)}\\big) \\end{align*} if and only if $α_0\\geq {n/2}$ and $α_1\\geq n/2-1$. Moreover, the polynomial-in-time growth $1+|t|$ in the above estimate is best possible.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.27730",
  "author_total": 2,
  "authors": [
    {
      "name": "Yunxiang Wang"
    },
    {
      "name": "Lixin Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c0zebhqf6d3mx.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.27730",
  "fulltext_url": null,
  "id": "c0zebhqf6d3mx",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.27730",
  "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"
    ],
    [
      "arxiv",
      "math.CA"
    ],
    [
      "openalex",
      "2604|Advanced Harmonic Analysis Research"
    ]
  ],
  "openalex": "W7214179527",
  "pdf_url": "https://arxiv.org/pdf/2609.27730",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c0zebhqf6d3mx.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:00Z",
      "native_id": "2609.27730",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.27730"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "The weak-type $(1,1)$ estimates for wave equation on $ax+b$ groups",
  "title_text": "The weak-type (1,1) estimates for wave equation on ax+b groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:00Z",
  "url": "https://research.draconitic.com/paper/c0zebhqf6d3mx",
  "venue": "arXiv",
  "versions": []
}