The total variation of a function $v\in W^{1,1}(\mathbb{R}^d)$ is equal to the $L^1(\mathbb{R}^d)$ norm of its weak derivative, see Total variation of (weakly) differentiable functions (an actual reference might be nice too). My question is related to this.
Let $C>0$ be a fixed constant.
Let $\{u_k\}_{k=1}^\infty\subseteq L^1({\mathbb{R}^d})$ be a sequence such that : for all $k$
$\|u_k\|_{L^1(\mathbb{R}^d)} \leq C$, i.e uniformly bounded in $L^1$.
For all $ \phi\in C^\infty_c(\mathbb{R}^d;\mathbb{R}^d)$ we have
$$ \int_{\mathbb{R}^d} u_k \text{div}(\phi)~dx= \int_{\mathbb{R}^d} w_k \cdot \phi ~ dx$$
where we know two facts : $$ \|w_k\|_{L^1(\mathbb{R}^d)}\leq \frac{C}{k^2},\label{1}\tag{1} $$ and $$ \sup_{\|\phi\|_{\infty}\leq 1}\left\{\int_{\mathbb{R}^d} w_k \cdot \phi ~ dx \right\}\leq \frac{C}{k}.\label{2}\tag{2} $$ Note the powers in the bounds of \eqref{1} and \eqref{2}.
Question: from here can I claim the bound $$ \|\nabla u_k\|_{L^1{(\mathbb{R}^d)}} \leq \frac{C}{k} \;? $$
I believe so since fact \eqref{1} tells me for all $k$ that $u_k\in W^{1,1}(\mathbb{R}^d)$ and fact \eqref{2} along with the facts from "https://math.stackexchange.com/questions/118958/total-variation-of-weakly-differentiable-functions?rq=1" gives the desired bound.