$H^s$ norm of non-integer power of functions

Question

Let $ \Omega = \mathbb{T}^d (1 \leq d \leq 3)$ be the $d$ dimensional torus and $ u \in H^2(\Omega) $ be a complex valued function. For some $ 0 < \alpha < 1 $, let $ g(u) = |u|^\alpha u $.

My question is: do we have some estimates for $ g(u) $ which can ensure that it belongs to some fractional Sobolev space $ H^s(\Omega) $ with $ 1<s<2 $, possibly depending on $\alpha$?

This question occurs in the analysis of nonlinear Schrödinger equation, where we have to handle some terms like $$ \| (e^{i \tau \Delta} - I) g(u) \|_{L^2} \leq C \tau^{\gamma} \| g(u) \|_{H^{2\gamma}} ,\quad 0 < \gamma \leq 1. $$

Of course, we can differentiate $ g(u) $ once, and get $$ \nabla g(u) = |u|^\alpha \nabla u + |u|^{2\alpha-2}u^2 \overline{\nabla u}. $$ Then it seems not clear how to obtain the fractional estimates of these terms.

It may be related that when $ d=1 $, the power function $ |x|^\alpha (0 < \alpha < 1) $ is in $ H^s $ for any $ s<\alpha+1/2 $. Are there similar results for general $ |u|^\alpha $ with $ u $ sufficiently smooth?

Update:

To be precise, the question is about how does the non-smoothness of the power nonlinearity, i.e. $|u|^\alpha (0<\alpha<1)$, affect the regularity of $g(u)$ provided that $ u $ is sufficiently smooth. I wonder what's the best one could expect for $ g(u) \in H^s $ with $ 1 <s<2 $ since $ g(u) $ is at least in $ H^1 $ by the Sobolev embedding of $ H^2 \hookrightarrow L^\infty $ when $ 1 \leq d \leq 3 $.

As one can see, for $0<\alpha<1$, if we differentiate $g(u)$ twice, there will be some term with negative power and, in general, $ g(u) $ is in $ H^1 $ but not in $ H^2 $ even though $ u \in H^2 $. While in the smooth case, saying the cubic case ($\alpha = 2$), one has $$ \| |u|^2 u \|_{H^s} \leq C \| u \|_{H^s}^3, \quad s > \frac{d}{2}, $$ which implies that we will not lose any regularity of $g(u)$ for sufficiently smooth $ u $ if the nonlinearity is smooth.

Some attempts in 1D:

Use the follwoing definition of the fractional Sobolev space $H^s$: $ u \in H^s $ for some $ 0 < s < 1 $ if and only if $$ u \in L^2(\Omega) \text{ and } \int_\Omega \int_\Omega \frac{|u(x) - u(y)|^2}{|x-y|^{2s+d}} dx dy < \infty. $$

By the inequality $$ |x^{\alpha} - y^{\alpha}| \leq |x - y|^\alpha, \quad x, y \in \mathbb{R}, \quad 0 < \alpha < 1, $$ and the Sobolev embedding into Holder space in 1D, one has $$ ||u(x)|^\alpha - |u(y)|^\alpha| \leq |u(x) - u(y)|^\alpha \leq C | x - y|^\alpha. $$

Then by the definition above, one has $ |u|^\alpha \in H^s $ for any $ 0\leq s<\alpha $. Then $ \nabla g(u) \in H^s $ since $ \nabla u \in H^1 $ and $|u|^\alpha \in H^s $ and thus $g(u) \in H^{1+s}$ for any $ 0 \leq s<\alpha $.

Answer 1

In Christ-Weinstein, JFA 100 (1991) 87-109 you can find the fractional chain rule (Proposition 3.1) $$ \|F(u)\|_{\dot H^s_r}\le C \|F'(u)\|_{L^p}\|u\|_{\dot H^s_q} $$ where $s\in(0,1)$, $p,q,r\in(1,\infty)$, $1/r=1/p+1/q$, $F\in C^1$, and $\dot H^s_r$ is the homogeneous Sobolev space with norm $\||D|^su\|_{L^r}$. You can use this inequality to estimate the first derivative of $g(u)$.

Answer 2

In general this depends on the value of $\alpha$, $d$ and $s$. For instance, when $d=1$, we have $$ \||u|^\alpha u\|_{\dot{H}^s}\lesssim\|u\|_{\infty}^\alpha \|u\|_{\dot{H}^s}.$$ For a proof, see for instance Lemma 4.1 in Tzvetkov and Visciglia (MR3593518). The usufulness of this inequality is when $s>1/2$, then $\|u\|_\infty\lesssim \|u\|_{H^s}$. Particularly, this has been used by Tzvetkov and Visciglia to show some well-posedness results for NLS on product spaces, which might be relevant to your research.