Let $0<\alpha<1$. Let $D_x^{\alpha}$ denote the Fourier multiplier given by $\xi\to |\xi|^{\alpha}$. Suppose $u:\mathbb{R}^d\to\mathbb{C}$ is Schwartz (or even just smooth with compact support). What kind of "regularity" does $D_x^{\alpha}|u|^{\alpha}$ have? Specifically, I'm interested in knowing if $|u|^{\alpha}$ belongs to the Sobolev space $W^{\alpha,p}$.
I found something slightly weaker in a paper on NLS due to Visan, https://arxiv.org/pdf/math/0508298.pdf . In appendix A, they have a proposition which proves the following:
Let $F$ be Holder continuous of order $0<\alpha<1.$ Then for every $0<\sigma<\alpha$, $1<p<\infty$, and $\frac{\sigma}{\alpha}<s<1$, we have
$$ \|D_x^{\sigma}F(u)\|_{p}\lesssim \||u|^{\alpha-\frac{\sigma}{s}}\|_{p_1}\|D_x^{s}u\|_{\frac{\sigma}{s}p_2}^{\frac{\sigma}{s}} $$ where $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $(1-\frac{\sigma}{\alpha s})p_1>1.$
Is anything known at the endpoint $\alpha=\sigma$ and $s=1$? In particular, do we have a bound of the form $$ \|D_x^{\alpha}F(u)\|_{p}\lesssim \|D_x^1u\|_{\alpha p}^{\alpha} $$ or something close to this?
