Implications of structured continuous maximal regularity

TL;DR

This paper investigates how maximal regularity estimates improve when the spatial norm differs from the supremum norm, leading to new proofs, extensions, and solutions to open problems in evolution equations and control theory.

Contribution

It introduces a novel approach to sharpen maximal regularity estimates using properties like weak compactness, with applications to existing theorems and an open problem in control theory.

Findings

Provides a new proof of Guerre-Delabriere's $ ext{L}^1$-maximal regularity

Extends Baillon's theorem in a new setting

Resolves an open problem on input-to-state stability for abstract systems

Abstract

We study how maximal regularity estimates with respect to the continuous functions improve automatically in cases where the spatial norm is fundamentally different from the supremum norm. More precisely, we invoke properties such as weak compactness of convolution-type operators related to the mild solutions of the underlying linear evolution equations to sharpen the a priori estimates. These results have several applications: such as a new proof of Guerre-Delabriere's result on $\mathrm{L}^1$-maximal regularity and an extension of Baillon's theorem; a simplification for well-known perturbation theorems for generation of $\mathrm{C}_0$-semigroups; and we resolve an open problem on input-to-state stability from control theory for a general abstract class of systems.