A refinement of Baillon's theorem on maximal regularity

TL;DR

This paper refines Baillon's theorem, demonstrating that the presence of $c_0$ in the space can be excluded under a new condition similar to maximal regularity with respect to L-infinity, thus advancing understanding of semigroup generators.

Contribution

It introduces a refined condition that excludes the $c_0$ alternative in Baillon's theorem, extending the scope of maximal regularity results.

Findings

Excluded the $c_0$ alternative under the new condition

Extended Baillon's theorem to a broader class of spaces

Provided a refined criterion for maximal regularity

Abstract

By Baillon's result, it is known that maximal regularity with respect to the space of continuous functions is rare; it implies that either the involved semigroup generator is a bounded operator or the considered space contains $c_{0}$. We show that the latter alternative can be excluded under a refined condition resembling maximal regularity with respect to $\mathrm{L}^{\infty}$.