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

Abstract

For $p\in [1,\infty ]$, we define a smooth manifold structure on the set $A{C}_{L^p}([a,b],N)$ of absolutely continuous functions $\gamma :[a,b]\to N$ 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 $A{C}_{L^p}([a,b],N_1)\to A{C}_{L^p}([a,b],N_2)$, $\eta \mapsto f\circ \eta$, for a smooth map $f:N_1\to N_2$. For $1\le p<\infty$ 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 $\text{Evol}:L^p([0,1],T_{e}G)\to A{C}_{L^p}([0,1],G)$, where $e$ is the neutral element of $G$. For the preceding examples, the evolution map $\text{Evol}$ is continuous.

Minor errors, redaction and references corrected