Weighted non-autonomous L^q(L^p) maximal regularity for complex systems under mixed regularity in space and time

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∈(1,∞) and we consider coefficient functions in C_t^β+ε with values in C_x^α+ε subject to the parabolic relation 2β+α=1. If [Formula presented], we can likewise deal with spatial [Formula presented] 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.