If $\{f_n\}_{n \in \mathbb{N}}$ is a sequence in $L^1(\Omega)$, we know that if the BV($f_n$) is uniformly bounded then there exists a subsequence which converges in $L^1$ norm(Because BV is embedding in $L^1$ is compact).
Is there any criteria other than uniform BV bound which ensures the existence of subsequence which converges in $L^1$