Let $\Omega$ be a regular open set in $\mathbb{R}^n$ and $T>0$. Let $C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$ be the space of functions $f$ which are $\frac{1+\alpha}{2}$-Hölder in time and with $\nabla f \in C^{\frac{\alpha}{2};\alpha}([0,T]\times \overline{\Omega})$.
I am wondering what can be said on the (weak) time derivative $\partial_t f$ of a function $f \in C^{\frac{1+\alpha}{2};1+\alpha}([0,T]\times \overline{\Omega})$. Is it true that it belongs to something like $C^{-1+\alpha}([0,T]\times \overline{\Omega})$ (being the space of distributions on $[0,T]\times \overline{\Omega}$ which equal the divergence of a $\alpha$-Hölder function in space)?
More precisely, does there exist a $\alpha$-Hölder function in space $g : [0,T]\times \overline{\Omega} \to \mathbb{R}^n$ such that $$ \partial_t f = \mathrm{div}\, g? $$
