The weak-type $(1,1)$ 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 to $u_{tt}+Lu=0$ 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 $t\in \mathbb{R}\setminus \{0\}$, \begin{align*} |u(t,\cdot)|{L^{1,\infty}(G,$\mathrm{d}$ρ)}\leq C\ (1+|t|)\big(|(\operatorname{Id}+L)^{α_0/2}f|{L^1(G,$\mathrm{d}$ρ)} +|(\operatorname{Id}+L)^{α_1/2}g|_{L^1(G,$\mathrm{d}$ρ)}\big) \end{align*} if and only if ${\alpha }_0\ge n\mathrm{/}2$ and ${\alpha }_1\ge n\mathrm{/}2-1$. Moreover, the polynomial-in-time growth $1+\mathrm{\mid }t\mathrm{\mid }$ in the above estimate is best possible.