Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups

TL;DR

This paper develops a smooth manifold structure on spaces of absolutely continuous functions into infinite-dimensional manifolds, studies smooth mappings between these spaces, and proves $L^p$-semiregularity of certain half-Lie groups of diffeomorphisms, including their evolution maps.

Contribution

It introduces a manifold structure on $AC_{L^p}$ spaces for infinite-dimensional manifolds and establishes $L^p$-semiregularity of specific half-Lie groups of diffeomorphisms, including continuity of evolution maps.

Findings

Defined smooth manifold structures on $AC_{L^p}$ spaces.

Proved smoothness of superposition operators between these spaces.

Established $L^p$-semiregularity and continuity of evolution maps for certain half-Lie groups.

Abstract

For $p\in [1,\infty]$, we define a smooth manifold structure on the set $AC_{L^p}([a,b],N)$ of absolutely continuous functions $\gamma\colon [a,b]\to N$ with $L^p$-derivatives for all real numbers $a<b$ and each smooth manifold $N$ modeled on a sequentially complete locally convex topological vector space, such that $N$ admits a local addition. Smoothness of natural mappings between spaces of absolutely continuous functions is discussed, like superposition operators $AC_{L^p}([a,b],N_1)\to AC_{L^p}([a,b],N_2)$, $\eta\mapsto f\circ \eta$, for a smooth map $f\colon N_1\to N_2$. For $1\leq p <\infty$ and $r\in \mathbb{N}$ we show that the right half-Lie groups $\text{Diff}_K^r(\mathbb{R})$ and $\text{Diff}^r(M)$ are $L^p$-semiregular. Here $K$ is a compact subset of $\mathbb{R}$ and $M$ is a compact smooth manifold. An $L^p$-semiregular half-Lie group $G$ admits an evolution map $\text{Evol}:L^p([0,1],T_e G)\to AC_{L^p}([0,1],G)$, where $e$ is the neutral element of $G$. For the preceding examples, the evolution map $\text{Evol}$ is continuous.