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

TL;DR

This paper establishes non-autonomous maximal $L^p$-regularity results on UMD spaces using fractional Sobolev regularity in time, extending previous Hilbert space results to more general Banach spaces.

Contribution

It generalizes recent results by replacing H"older continuity with fractional Sobolev regularity in time, applicable to elliptic operators with $VMO$-modulus.

Findings

Maximal $L^p$-regularity achieved for $p \,\geq 2$ on $L^q(\Omega)$.

Applicable to divergence form elliptic operators with $VMO$-modulus.

Requires fractional Sobolev regularity $W^{\alpha,p}$ with $\alpha > 1/2$ in time.

Abstract

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.