$$ which implies that for $p_0\in (0,\,\infty)$, $$\begin{aligned} \|\Phi_{\epsilon}(\vec{f})\|_{L^{p_0}(L^{\infty}([0,\,0]),\,l^{\infty};\,\mathbb{R}^n,\,w)}\lesssim \|\vec{f}\|_{L^{p_0}(L^{\infty}([0,\,0]),\,l^{\infty};\,\mathbb{R}^n,\,w)}