PreprintCC BY 4.0arXiv
Inequalities for radial solutions to the wave equation on the hyperbolic space
Abstract
Let ℍn be the n-dimensional hyperbolic space, and Δℍn be the Laplace--Beltrami operator on ℍn. Let u=u(t,·) be the solution to the wave equation ∂2tu−(Δℍn+1/4(n−1)2Id)u=0 with initial conditions u(0,x)=f(x) and ∂tu(0,x)=0. In this article we show that for a radial function f, 2n/(n−1)≤ p<∞, γ<1/p and T≥ 0, there exists a constant Cp independent of f and T such that (∫TT+1|u(t,·)|ppd t)1/p≤ Cp,γ e(n−1)(1/2−1/p)T|(−Δℍn)(n−1)/2(1/2−1/p)−γ/2f|p. We also show that the exponential part e(n−1)(1/2−1/p)T of the bound is optimal.
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BibTeX
@misc{chen2026inequalities,
title = {{Inequalities for radial solutions to the wave equation on the hyperbolic space}},
author = {Peng Chen and Xuan Thinh Duong and Yunxiang Wang and Lixin Yan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06886},
eprint = {2610.06886},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06886},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Inequalities for radial solutions to the wave equation on the hyperbolic space
AU - Peng Chen
AU - Xuan Thinh Duong
AU - Yunxiang Wang
AU - Lixin Yan
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.06886
UR - https://arxiv.org/abs/2610.06886
AB - Let ℍ^n be the n-dimensional hyperbolic space, and Δ_ℍ^n be the Laplace--Beltrami operator on ℍ^n. Let u=u(t,·) be the solution to the wave equation ∂^2_tu−(Δ_ℍ^n+1/4(n−1)^2Id)u=0 with initial conditions u(0,x)=f(x) and ∂_tu(0,x)=0. In this article we show that for a radial function f, 2n/(n−1)≤ p<∞, γ<1/p and T≥ 0, there exists a constant C_p independent of f and T such that (∫_T^T+1|u(t,·)|^p_pd t)^1/p≤ C_p,γ e^(n−1)(1/2−1/p)T|(−Δ_ℍ^n)^(n−1)/2(1/2−1/p)−γ/2f|_p. We also show that the exponential part e^(n−1)(1/2−1/p)T of the bound is optimal.
ER -