Relation between $L^p$ spaces.

Question

We say $f\in L_{p}(U)$ if $\int_{U}|f|^{p}<\infty$.

Can we say $u\in L_{4} \iff \ u^{2} \in L_{2}$, and $u \in L_{2} \iff u^{1/2} \in L_{4}$

Answer 1

Yes, that is correct, because of the identity $\lvert x^n\rvert=\lvert x\rvert^n$, which holds for all $x\in\Bbb C$ and $n\in\Bbb Z$. For $n\notin \Bbb Z$ there is the definition issue of what $x^n$ is when $x\notin[0,\infty)$, in which case $f^n$ is arguably not defined.