>0$, there exists $N\in\mathbb{N}$ such that $$\begin{aligned} \Big\|\Big(\int^0_0\sum_{j>N}|F_{j,\,b}^{l}f(\cdot,\,t)|^0{\rm d}t\Big)^{\frac{0}{0}}\chi_{B(0,\,D)}\Big\|_{L^{p,\lambda}(\mathbb{R}^n,\,w)}<\epsilon\|f\|_{L^{p,\