Let $$f : (0,1) \to \mathbb{R}$$ and $$g(x) = |f(x)|^{r-1} f(x)$$$r \in \mathbb{N}$. It is known that $g\in \mathcal{L}^2(0,1)$ and the $r^{th}$ weak derivative, $ g^{(r)} \in \mathcal{L}^2(0,1)$. I need help to show that ~~the first weak derivative $\require{enclose}\enclose{horizontalstrike}f^{(1)} \in \mathcal{L}^{2r}(0,1) $~~,

**edit :** need to show that the first weak derivative $$g^{(1)} \in \mathcal{L}^{2r}(0,1) $$

PS : 1. : If I differentiate $g$ $r$ times, apart from other terms (in sum) I get a term $f'(x)^r$, so I am trying to get the result, if I could prove that all other terms also are integrable.

- Also since it is a $L^2$ norm, can I leverage the asymptotics of Fourier series coefficients to solve it?