Let $\{f_k\} \subset L^2(\Omega)$, where $\Omega \subset \mathbb{R}^n$ is a bounded domain and suppose that $f_k \to f$ in $L^2(\Omega)$. Now if $a \geq 1$ is some constant, is it possible to say that $|f_k|^a \to |f|^a$ in $L^p$ for some $p$ (depending on $a$ and also possibly depending on $n$)?
Showing the statement is true would probably require a smart way of bounding $\left| |f_k|^a - |f|^a \right|$ by a term including the factor $|f_k - f|^2$. However, I don't really know what to do with the fact that $a$ doesn't have to be an integer...