Let $J$ be a mollifier, e.g. a function in $J \in C^\infty(\mathbb R^n)$ with the properties $J\geq 0$ and $\int J(x) \mathrm dx=1$ and $J(x)=0$ for all $x$ with $|x|>1.$ Now define $J_\varepsilon (x):=\varepsilon ^{-n}J(\varepsilon ^{-1}x)$ and $(J_\varepsilon \star u)(x)=\int_{\mathbb R^n}J_\varepsilon (x-y)u(y) \mathrm dy$ (the symbol $\star$ denotes convolution). How does one prove, that for every function $u \in C^\infty_0(\mathbb R^n)$ (with compact support) the following inequality holds:
$\left \| u-J_\varepsilon \star u \right \|_1 \leq c \cdot \varepsilon \left \| u \right \|_{1,1}\quad ?$
Here, $||u||_{1,1}=||u||_{L^1}+\sum_{j=1}^n||\partial_j u||_{L^1}$.