Let $B$ be the unit ball in $\mathbb{R}^{12}$ and $f_n\in L^7(B)$ such that $\lVert f_n \rVert_{W^{1,4}(B)}$ is bounded. Is it true that there exists a subsequence weakly convergent in $W^{1,1}(B)$?
My guess is it doesn't exist. Since the domain is bounded then $f_n$ is also bounded in $W^{1,1}$ which is not reflexive. But i'm having trouble finding a counterexample. I tried $f_n(x)=|x|^{-12/7+1/n}$. It satisfies the assumptions but does it admit a weakly convergent subsequence in $W^{1,1}(B)$? Thank you
If $(f_n)$ is bounded in $W^{1,4}$, then it has a subsequence converging weakly in $W^{1,4}$. This sequence also converges weakly in $W^{1,1}$, due to the continuity of the embedding $W^{1,4}\hookrightarrow W^{1,1}$.