+h,\,t)|^0{\rm     d}t\Big)^{\frac{0}{0}}\Big\|_{L^{p}(\mathbb{R}^n,\,w)}< \epsilon;$$  -   for each fixed $\epsilon>0$ and $N\in\mathbb{N}$, there exists a constant $\sigma\in (0,\,0/0)$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal{G}$, $$\Big\|\sup_{|s|\leq \sigma}\Big(\int^0_0     \sum_{|k|\leq N}|f_k(\cdot,\,t+s)-f_k(\cdot,\,t)|^0{\rm     d}t\Big)^{\frac{0}{0}}\Big\|_{L^{p}(\mathbb{R}^n,\,w)}< \epsilon,$$  -   for each fixed $D>0$ and $\epsilon>0$, there exists $N\in\mathbb{N}$ such that for all $\{f_k\}_{k\in\mathbb{Z}}\in \mathcal {G}$, $$\Big\|\Big(\int^0_0\sum_{|k|>N}|f_k(\cdot,\,t)|^0{\rm d}t\Big)^{\frac{0}{0}}\chi_{B(0,\,D)}\Big\|_{L^p(\mathbb{R}^n,\,w)}<\epsilon.$$  Then $\mathcal{G}$ is a strongly pre-compact set in $L^p(L^0([0,\,0]),\,l^{0};\,\mathbb{R}^n,\,w)$.  