Sharp $L^p$-estimates for wave equation on $ax+b$ groups

Let $G$ be the group $\mathbb{R}_+\ltimes \mathbb{R}^n$ endowed with Riemannian symmetric space metric $d$ and the right Haar measure $\mathrm{d} \rho$ 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 of $u_{tt}+Lu=0$ with initial conditions $u|_{t=0}=f$ and $u_t|_{t=0}=g$. In this article we show that for a fixed $t \in{\mathbb R}$ and every $1<p<\infty$, \begin{align*} \|u(t,\cdot)\|_{L^p(G)}\leq C_p\Big( (1+|t|)^{2|1/p-1/2|}\|f\|_{L^p_{\alpha_0}(G)}+(1+|t|)\,\|g\|_{L^p_{\alpha_1}(G)}\Big) \end{align*} if and only if \begin{align*} \alpha_0\geq n\left|{1\over p}- {1\over2}\right| \quad \mbox{and} \quad \alpha_1\geq n\left|{1\over p}- {1\over2}\right| -1. \end{align*} This gives an endpoint result for $\alpha_0=n|1/p-1/2|$ and $\alpha_1=n|1/p-1/2|-1$ with $1<p<\infty$ in Corollary 8.2, as pointed out in Remark 8.1 due to M\"{u}ller and Thiele [Studia Math. \textbf{179} (2007)].