Non-Autonomous Maximal $L^p$-Regularity under Fractional Sobolev Regularity in Time

We prove non-autonomous maximal $L^p$-regularity results on UMD spaces replacing the common H"older assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q(\mathrm{\Omega })$ we obtain maximal $L^p$-regularity for $p\ge 2$ and elliptic operators in divergence form with uniform $VMO$-modulus in space and $W^{\alpha ,p}$-regularity for $\alpha >\frac{1}{2}$ in time.