Consider $\alpha \in (0;1)$, $Q := (0;T)\times(0;L)$ and the Hölder spaces $C^\alpha(Q)$ and $C^{\alpha/2,\alpha}(Q)$ of functions $u\colon Q_T \to \mathbb R$ with the norms
$\Vert u\Vert_{C^\alpha(Q)} := \Vert u\Vert_\infty + \sup_{s\ne t,x} \frac{\vert u(s,x)-u(t,x)\vert}{\vert s-t\vert^\alpha} + \sup_{s,x\ne y} \frac{\vert u(s,x)-u(s,y)\vert}{\vert x-y\vert^\alpha}$ $\Vert u\Vert_{C^{\alpha/2,\alpha}(Q)} := \Vert u\Vert_\infty + \sup_{s\ne t,x} \frac{\vert u(s,x)-u(t,x)\vert}{\vert s-t\vert^{\alpha/2}} + \sup_{s,x\ne y} \frac{\vert u(s,x)-u(s,y)\vert}{\vert x-y\vert^\alpha}$
(The only difference is that once, we are taking $\alpha$-Hölder-continuity in $s$ direction and once, $\alpha/2$-Hölder-continuity. The space $C^{\alpha/2,\alpha}(Q)$ often appears in the theory of parabolic PDEs.)
Now consider a $C^1$-function $f\colon \mathbb R \to \mathbb R$. Theorem 2 of this paper states that the composition $u \mapsto f \circ u$ is a continuous function $C^\alpha(Q) \to C^\alpha(Q)$.
I am looking for a similar continuity result for the same function, but considered as $C^{\alpha/2,\alpha}(Q) \to C^{\alpha/2,\alpha}(Q)$. Probably it's possible to redo the whole proof of this theorem, but it seems it is very technical and would need a long time. I was wondering if there is already such kind of result or if there is an easier way to get this. Thank you for your help!