  This note is to conclude $L^1-L^1$ estimates for solutions to the following
Cauchy problem for visco-elastic damped $\sigma$-evolution models:
\begin{equation} \begin{cases} u_{tt}+ (-\Delta)^\sigma u+ (-\Delta)^\sigma u_t
= 0, &\quad x\in \mathbb{R}^n,\, t \ge 0, \\ u(0,x)= u_0(x),\quad
u_t(0,x)=u_1(x), &\quad x\in \mathbb{R}^n, \label{pt1.1} \end{cases}
\end{equation} where $\sigma> 1$, in all space dimensions $n\ge 1$.
