On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative

We prove two maximal regularity results in spaces of continuous and H"older continuous functions, for a mixed linear Cauchy-Dirichlet problem with a fractional time derivative ${\mathbb{D}}_t^{\alpha }$. This derivative is intended in the sense of Caputo and $\alpha$ is taken in $(0,2)$. In case $\alpha =1$, we obtain maximal regularity results for mixed parabolic problems already known in mathematica literature.