 set in $L^{p,\,\lambda}(L^0([0,\,0]),\,l^{0};\,\mathbb{R}^n,\,w)$;  -   for each fixed $\epsilon>0$, there exists a constant $A>0$, such that for all $\{f_k\}_{k\in\mathbb{Z}}\in     \mathcal{G}$, $$\Big\|\Big(\int^0_0\sum_{k\in\mathbb{Z}}|f_k(\cdot,\,t)|^0{\rm d}t\Big)^{\frac{0}{0}}\chi_{\{|\cdot|>A\}}(\cdot)\Big\|_{     L^