PreprintCC BY 4.0arXiv
The weak-type (1,1) estimates for wave equation on ax+b groups
Abstract
Let G be the group ℝ+ltimes ℝn endowed with Riemannian symmetric space metric d and the right Haar measure d ρ which is of ax+b type, and L be the positive definite distinguished left-invariant Laplacian on G. Let u=u(t,·) be the solution to utt+Lu=0 with initial conditions u|t=0=f and ut|t=0=g. In this article we show that for a fixed t ∈ℝ∖{0}, |u(t,·)|L1,∞(G,dρ)≤ C (1+|t|)(|(Id+L)α0/2f|L1(G,dρ) +|(Id+L)α1/2g|L1(G,dρ)) if and only if α0≥ n/2 and α1≥ n/2−1. Moreover, the polynomial-in-time growth 1+|t| in the above estimate is best possible.
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BibTeX
@misc{wang2026weaktype,
title = {{The weak-type $(1,1)$ estimates for wave equation on $ax+b$ groups}},
author = {Yunxiang Wang and Lixin Yan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27730},
eprint = {2609.27730},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27730},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The weak-type (1,1) estimates for wave equation on ax+b groups
AU - Yunxiang Wang
AU - Lixin Yan
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27730
UR - https://arxiv.org/abs/2609.27730
AB - Let G be the group ℝ_+ltimes ℝ^n endowed with Riemannian symmetric space metric d and the right Haar measure d ρ which is of ax+b type, and L be the positive definite distinguished left-invariant Laplacian on G. Let u=u(t,·) 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 ∈ℝ∖{0}, |u(t,·)|_L^1,∞(G,dρ)≤ C (1+|t|)(|(Id+L)^α_0/2f|_L^1(G,dρ) +|(Id+L)^α_1/2g|_L^1(G,dρ)) if and only if α_0≥ n/2 and α_1≥ n/2−1. Moreover, the polynomial-in-time growth 1+|t| in the above estimate is best possible.
ER -