Nonautonomous maximal $L^p$-regularity under fractional Sobolev regularity in time

Abstract

We prove nonautonomous maximal $L^p$-regularity results on UMD spaces, replacing the common Hölder assumption by a weaker fractional Sobolev regularity in time. This generalizes recent Hilbert space results by Dier and Zacher. In particular, on $L^q\left(\mathrm{\Omega }\right)$ 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.