{
  "abstract": "Let $\\mathbb H^n$ be the $n$-dimensional hyperbolic space, and $Δ_{\\mathbb H^n}$ be the Laplace--Beltrami operator on $\\mathbb H^n$. Let $u=u(t,\\cdot)$ be the solution to the wave equation $\\partial^2_{t}u-\\big(Δ_{\\mathbb H^n}+\\frac{1}{4}(n-1)^2\\operatorname{Id}\\big)u=0$ with initial conditions $u(0,x)=f(x)$ and $\\partial_tu(0,x)=0$. In this article we show that for a radial function $f$, $2n/(n-1)\\leq p<\\infty$, $γ<1/p$ and $T\\geq 0$, there exists a constant $C_p$ independent of $f$ and $T$ such that $$ \\left(\\int_T^{T+1}\\|u(t,\\cdot)\\|^p_p\\mathrm d t\\right)^{\\frac1p}\\leq C_{p,γ}\\,\\mathrm e^{(n-1)\\left(\\frac12-\\frac1p\\right)T}\\left\\|(-Δ_{\\mathbb H^n})^{\\frac{n-1}2\\left(\\frac12-\\frac1p\\right)-\\frac\\gamma2}f\\right\\|_p. $$ We also show that the exponential part $\\mathrm e^{(n-1)(1/2-1/p)T}$ of the bound is optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.06886",
  "author_total": 4,
  "authors": [
    {
      "name": "Peng Chen"
    },
    {
      "name": "Xuan Thinh Duong"
    },
    {
      "name": "Yunxiang Wang"
    },
    {
      "name": "Lixin Yan"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8g57evyzdt9fp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.06886",
  "fulltext_url": null,
  "id": "8g57evyzdt9fp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.06886",
  "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": "W7220940585",
  "pdf_url": "https://arxiv.org/pdf/2610.06886",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8g57evyzdt9fp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:26Z",
      "native_id": "2610.06886",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.06886"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    }
  ],
  "title": "Inequalities for radial solutions to the wave equation on the hyperbolic space",
  "title_text": "Inequalities for radial solutions to the wave equation on the hyperbolic space",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:26Z",
  "url": "https://research.draconitic.com/paper/8g57evyzdt9fp",
  "venue": "arXiv",
  "versions": []
}