Does uniform boundedness in $W^{1,1}$ implies strong convergence in $L^{1}$?

Question

Suppose $f_i$ is uniformly bounded in $W^{1,1}$. My question is, can we conclude that there exists a sub-sequence of $f_i$ convergent strongly to some $f$ in $L^{1}$? I am just reading a paper, this is one of the steps of the proof but without any reference. Maybe it's obvious to the author.

Answer 1

Yes. A corollary to the Rellich Kondrachov theorem says that for all $p\ge 1$, we have $W^{1,p}$ compactly contained in $L^p$. Evans PDE book has a nice treatment of this.

EDIT: Actually, if I recall correctly, for $\Omega \subseteq \mathbb R^n$, to prove that $W^{1,p}(\Omega)$ is compactly contained in $L^p(\Omega)$, we only need the Rellich Kondrachov theorem when $1 \le p \le n$. When $p > n$, Morrey's Inequality is strong enough.