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

Let $G$ be the group ${\mathbb{R}}_{+}⋉{\mathbb{R}}^n$ 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 $u{\mathrm{\mid }}_{t=0}=f$ and $u_t{\mathrm{\mid }}_{t=0}=g$. In this article we show that for a fixed $t\in \mathbb{R}$ and every $1<p<\mathrm{\infty }$, \begin{align*} |u(t,\cdot)|{L^p(G)}\leq C_p\Big( (1+|t|)^{2|1/p-1/2|}|f|{L^p_{α_0}(G)}+(1+|t|),|g|{L^p{α_1}(G)}\Big) \end{align*} if and only if \begin{align*} α_0\geq n\left|{1\over p}- {1\over2}\right| \quad \mbox{and} \quad α_1\geq n\left|{1\over p}- {1\over2}\right| -1. \end{align*} This gives an endpoint result for ${\alpha }_0=n\mathrm{\mid }1\mathrm{/}p-1\mathrm{/}2\mathrm{\mid }$ and ${\alpha }_1=n\mathrm{\mid }1\mathrm{/}p-1\mathrm{/}2\mathrm{\mid }-1$ with $1<p<\mathrm{\infty }$ in Corollary 8.2, as pointed out in Remark 8.1 due to Müller and Thiele [Studia Math. $\mathbf{\text{179}}$ (2007)].