}(L^q([0,\,0]),\,l^r;\,\mathbb{R}^n,\,w)=\big\{\vec{f}=\{f_k\}_{k\in \mathbb{Z}}:\, \|\vec{f}\|_ {L^{p}(L^q([0,\,0]),\,l^r;\,\mathbb{R}^n,\,w)}<\infty\big\},$$ with $$\|\vec{f}\|_{L^{p}(L^q([0,\,0]),\,l^r;\,\mathbb{R}^n,\,w)}=\Big\|\Big(\int^0_0\Big(\sum_{k\in \mathbb{Z}}|f_k(x,\,t)|^r\Big)^{\frac{q}{r}}{\rm d}t\Big)^{