Let $u \in W^{\alpha,p}(0,T;X)$ for some reflexive Banach space $X$ (you can also take a Hilbert space if it helps) for $\alpha \in (0,1)$ and $p \geq 2$. I am fine with both the Sobolev-Slobodeckij norm or using the Riemann-Liouville fractional derivative (interpret the integral in Bochner sense)$$\partial_t^\alpha u(t):=\frac{d}{dt}(k*u)(t):=\frac{1}{\Gamma(1-\alpha)} \frac{d}{dt} \int_0^t \frac{u(s)}{(t-s)^\alpha} \,ds,$$ which might be more convenient. Further, let $f$ be nice enough. I state the following theorem which is a modification of Theorem 5.3.4/1 and 5.3.5/1 in "Sobolev Spaces of Fractional Order, Nemytskij Operator" by Runst&Sickel (I take Lipschitz instead of Hölder and remove the $L^\infty$ condition by making the $q$ smaller than $p$).
Let $\alpha \in (0,1)$, $p \geq 2$. If $f\in C^1_{loc}(\mathbb{R})$ with $f'\in \text{Lip}(\mathbb{R})$, $f(0)=0$, $f'(0)=0$, then it holds $f(u)\in W^{\alpha,q}(0,T)$ for some $q<p$ for all $u\in W^{\alpha,p}(0,T)$.
I want to have a vector-valued version of this Sobolev composition theorem, i.e., from $u \in W^{\alpha,p}(0,T;X)$ I want to have $f(u) \in W^{\alpha,p}(0,T;Y)$ with the same properties for $f:X \to Y$. I can't find this in literature.
In the case $\alpha=1$ the proof is pretty simple. There, we have the statement for $X=\mathbb{R}$: If $u \in W^{1,p}(0,T)$, then $f(u) \in W^{1,p}(0,T)$ if $f \in \text{Lip}(R)$ with Lipschitz constant $L$. Remember that $u \in W^{1,p}(0,T;X)$ iff there is a function $g \in L^p(0,T)$ such that $\|u(t)-u(s)\|_X \leq |\int_s^t g(y)dy|$. Now, take a function $u \in W^{1,p}(0,T;X)$. Then we have such a function $g$. By Lipschitz property the inequality is fulfilled for $f(u)$ with the function $Lg$, and thus, $f(u) \in W^{1,p}(0,T;Y)$.
I am not sure about the availability of such an inequality in the fractional-order case. One might be able to exploit that $k*u$ is in $W^{1,p}(0,T;X)$ if $u \in W^{\alpha,p}(0,T;X)$. Then we are in the classical setting and get regularity for $f(k*u)$, but we need $k*(f(u))$.
I am thankful for any input, references are also welcome.
