I am trying to prove this statement: given a sequence $\{f_n | f_n > 0, c_1 \leq \int_{\Omega} f_n \log{f_n} \leq c_2 \}$, here $\Omega$ is a bounded domain; can we prove the $L^1$ strong convergence of $f_n$?
With the given information, now I can only say that $f_n \rightharpoonup f$ in $L^1(\Omega)$ weakly, and $ \int_{\Omega} f \log f \leq \lim \inf \int_{\Omega} f_n \log f_n \leq c_2$ ... But this does not give me $L^1$ strong convergence.
The answer is no, not even for a subsequence. If this conclusion were correct, then the embedding $L^2(\Omega) \to L^1(\Omega)$ would be completely continuous for bounded $\Omega$. And that is not the case.
A subsequence of $f_n$ converges weakly in $L^1$, since the $L \log L$ bound implies equi-integrability, and by the Dunford-Pettis theorem, a bounded equi-integrable subset of $L^1$ is weakly relatively compact. Thank you to @MaoWao for clarifying this.
For a concrete counterexample, consider $$ \Omega = [0,2\pi], \; f_n(x) = 2 + \sin 2 n x \, . $$ Then all $f_n$ are bounded in $L^\infty$ and thus in $L \log L$, and their weak-$\ast$ limit in $L^1$ is $f(x) = 2$. But the convergence is not strong, since $\|f_n - f\|_{L^1} = 4 = const.$