Weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems

TL;DR

This paper establishes weighted non-autonomous maximal regularity results for complex second-order divergence form systems in $L^q(L^p)$ spaces, under mixed regularity conditions in space and time, with applications to semigroup bounds.

Contribution

It introduces a novel framework for weighted maximal regularity of complex systems with mixed space-time regularity, including new commutator and pseudo differential operator techniques.

Findings

Proves weighted non-autonomous $L^q(L^p)$ maximal regularity for complex systems.

Establishes $p$-bounds for semigroups and square roots of elliptic systems.

Develops a weak $(p,q)$-solution theory with uniform constants.

Abstract

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 $p,q \in (1,\infty)$ and we consider coefficient functions in $C^{\beta + \varepsilon}$ with values in $C^{\alpha + \varepsilon}$ subject to the parabolic relation $2\beta + \alpha = 1$. If $p < \frac{d}{\alpha}$, we can likewise deal with spatial $H^{\alpha + \varepsilon, \frac{d}{\alpha}}$ 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.