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 $L^q(\Omega)$ 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.

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