Continuous maximal regularity in locally convex spaces

TL;DR

This paper extends the concept of maximal regularity for strongly continuous semigroups from Banach spaces to locally convex spaces, establishing new characterizations and equivalences under certain topological conditions.

Contribution

It generalizes classical results on maximal regularity to locally convex spaces and links it with admissible operators, broadening the theoretical framework.

Findings

Travis' characterization of C-maximal regularity applies to locally convex spaces.

Under certain topological assumptions, maximal regularity is equivalent to admissibility.

The work extends classical semigroup regularity results beyond Banach spaces.

Abstract

We study maximal regularity with respect to continuous functions for strongly continuous semigroups on locally convex spaces as well as its relation to the notion of admissible operators. This extends several results for classical strongly continuous semigroups on Banach spaces. In particular, we show that Travis' characterization of $\mathrm{C}$-maximal regularity using the notion of bounded semivariation carries over to the general case. Under some topological assumptions, we further show the equivalence between maximal regularity and admissibility in this context.