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Continuous maximal regularity in locally convex spaces

Karsten Kruse, Felix L. Schwenninger

TL;DR

This work extends maximal regularity with respect to continuous forcing ($C$-maximal regularity) from Banach spaces to broad classes of Hausdorff locally convex spaces by employing bounded semivariation and extrapolation spaces $X_{-1}$. It generalizes Travis' characterization by proving that $C$-maximal regularity is equivalent to $C$-admissibility under suitable topological assumptions, and it characterizes solvability of the abstract Cauchy problem through the continuity properties of $A(T\!*f)$. The results connect solvability of inhomogeneous problems to the semigroup's regularity structure, extend the theory to control-operator frameworks $B$ acting into extrapolation spaces, and provide a robust foundation for applications to Markov semigroups and interpolation spaces in non-normed settings. Collectively, the paper broadens semigroup regularity theory beyond Banach spaces and clarifies when spatial regularity transfers to time-regularity in locally convex contexts.

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.

Continuous maximal regularity in locally convex spaces

TL;DR

This work extends maximal regularity with respect to continuous forcing (-maximal regularity) from Banach spaces to broad classes of Hausdorff locally convex spaces by employing bounded semivariation and extrapolation spaces . It generalizes Travis' characterization by proving that -maximal regularity is equivalent to -admissibility under suitable topological assumptions, and it characterizes solvability of the abstract Cauchy problem through the continuity properties of . The results connect solvability of inhomogeneous problems to the semigroup's regularity structure, extend the theory to control-operator frameworks acting into extrapolation spaces, and provide a robust foundation for applications to Markov semigroups and interpolation spaces in non-normed settings. Collectively, the paper broadens semigroup regularity theory beyond Banach spaces and clarifies when spatial regularity transfers to time-regularity in locally convex contexts.

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 -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.
Paper Structure (6 sections, 29 theorems, 109 equations)

This paper contains 6 sections, 29 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Notions and preliminaries
  3. The abstract Cauchy problem and C-maximal regularity
  4. Families of bounded semivariation
  5. C-admissibility
  6. 3CU-spaces and 3C-spaces

Key Result

Proposition 2.2

Let $a,b\in\mathbb{R}$ with $a<b$, $X$ and $Y$ be Hausdorff locally convex spaces and $Y$ sequentially complete. If $f\in\mathrm{C}([a,b];X)$ and $\alpha\colon [a,b]\to \mathcal{L}(X;Y)$ is of bounded semivariation, then $f$ is Riemann--Stieltjes integrable w.r.t. $\alpha$ and the linear map is continuous.

Theorems & Definitions (74)

  • Remark 2.1
  • Proposition 2.2: hoenig1973
  • proof
  • Remark 2.3
  • Corollary 2.4
  • proof
  • Definition 3.1: choe1985, komura1968
  • Definition 3.2
  • Proposition 3.3
  • proof
  • ...and 64 more