While studying uniqueness of a weak solution to KdV equation (on torus), I encountered a problem to bound
$$ \int_{\mathbb{T}}f(x)^{2} \partial_{xxxxx}(\mathbb{P}_{N}f)(x) dx $$
using $\|f\|_{H^{2}(\mathbb{T})}^{3}$.
Here, I'm only assuming $f \in H^{2}(\mathbb{T})$, and $\mathbb{P}_{N}$ is the Fourier truncation operator defined by
$$\mathbb{P}_{N}f(x) := \sum_{|k| \le N} \hat{f}(k)e^{2\pi i k x}$$
Using that $\|\partial_{x}u\|_{L^{\infty}} \lesssim \|u\|_{H^{2}}$, I'm left to bound
$$ \int_{\mathbb{T}} [ \partial_{xx}\mathbb{P}_{N}, u ](\partial_{x}u) \partial_{xx}\mathbb{P}_{N}u $$.
But I have no idea to get any estimate on this commutator. Any helps will be highly appreciated.