Weak convergence problem Let $\Omega$ ∈ $\mathbb{R}^d$ be a bounded set and $d\ge$ 1. Consider function sequence $f_n ∈ L_3(\Omega)$ such that,
$$f_n \to f\mbox{ weakly in }  L_2(\Omega)\mbox{ and }  \|f_n\|_{L_3(\Omega)} \le M$$
for set constant $M$. We additionally know that,
$$ \|f_n\|_{L_2 (\Omega)} \to \|f\|_{L_2 (\Omega)} $$
Show that,
$$ \|f_n - f \|_{L_p(\Omega)} \to 0\mbox{ for  every } p \in [2,3)   $$
Whether limit function $f$ have to be an element of $ L_3(\Omega) $ ?
Thank for help.

It's well known that norm convergence in $L^2$ follows from weak convergence plus the convergence of the norms. This gives that $f_n\to f$ pointwise a.e. on a subsequence, and thus $f\in L^3$ by Fatou's Lemma.
Now
$$
\int_{|f_n-f|\le C} |f_n-f|^p \le C^{p-2}\|f_n-f\|_2^2
$$
and
$$
\int_{|f_n-f|>C} |f_n-f|^p \lesssim C^{p-3} ,
$$
since at least one of $f_n$, $f$ is $>C/2$ on this set and, as we observed, the $3$ norms are bounded. So if $2\le p<3$, then we can make $\|f_n-f\|_p$ arbitrarily small by taking both $C$ and $n$ large.
