  We show weighted non-autonomous $L^q(L^p)$ maximal regularity for families of
complex second-order systems in divergence form under a mixed regularity
condition in space and time. To be more precise, we let $p,q \in (1,\infty)$
and we consider coefficient functions in $C^{\beta + \varepsilon}$ with values
in $C^{\alpha + \varepsilon}$ subject to the parabolic relation $2\beta +
\alpha = 1$. If $p < \frac{d}{\alpha}$, we can likewise deal with spatial
$H^{\alpha + \varepsilon, \frac{d}{\alpha}}$ regularity. The starting point for
this result is a weak $(p,q)$-solution theory with uniform constants. Further
key ingredients are a commutator argument that allows us to establish higher a
priori spatial regularity, operator-valued pseudo differential operators in
weighted spaces, and a representation formula due to Acquistapace and Terreni.
Furthermore, we show $p$-bounds for semigroups and square roots generated by
complex elliptic systems under a minimal regularity assumption for the
coefficients.
