 $$\begin{aligned} \|\Phi_{\epsilon}(\vec{f})\|_{L^{p_0}(L^{p_0}([0,\,0]),\,l^{p_0};\,\mathbb{R}^n,\,w)}\lesssim \|\vec{f}\|_{L^{p_0}(L^{p_0}([0,\,0]),\,l^{p_0};\,\mathbb{R}^n,\,w)}.\end{aligned}$$ By interpolation, we 