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?. Using the Littlewood-Paley characterization of Holder spaces, one can show that $|u|^{\alpha}$ lies in the Besov space $B_{p,\infty}^{\alpha}$ for $1\leq p\leq \infty$, which is defined by $$ \|u\|_{B_{p,\infty}^{\alpha}}=\|u\|_{L^p}+\sup_{j\in\mathbb{N}}2^{j\alpha}\|P_ju\|_{L^p} $$ where $P_j$ is the standard Littlewood-Paley projection onto frequency $\sim 2^j$. In fact, it's not too difficult to produce a quantitative bound essentially of the form $$ \||u|^{\alpha}\|_{B_{p,\infty}^{\alpha}}\lesssim \|u\|_{W^{1,\alpha p}}^{\alpha}. $$ This can be proven by using the Littlewood-Paley characterization of Holder spaces and then simply estimating the quantity $$ \sup_{h\in\mathbb{R}^d, 0<|h|<1}\frac{\||u(\cdot+h)|^{\alpha}-|u(\cdot)|^{\alpha}\|_{L^p}}{|h|^{\alpha}}. $$
What I'm wondering is if we can say something more. For instance, is it true that if $u$ is Schwartz, that $|u|^{\alpha}$ belongs to the slightly smaller Sobolev space $W^{\alpha,p}$? Do we also have a similar quantitative bound analogous to the above one? Any reference would be appreciated!