.\end{aligned}$$ It then follows from the assumption (b) that for all $\vec{f}\in \mathcal{G}$, $$\begin{aligned} \|\vec{f}-\Phi_{\epsilon}(\vec{f})\|_{L^p(L^0([0,\,0]),\,l^{0};\,\mathbb{R}^n,\,w)}&\leq& \big\|\vec{f}\chi_{\mathcal{D}}-\Phi_{\epsilon}(\vec{f}) \big\|_{L^p(L^0([0,\,0]),\,l^{0};\,\mathbb{R}^n,\,w)}\\ &