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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.

Submission history

From: Matthieu Pinaud [view email]
[v1] Tue, 10 Sep 2024 13:46:00 UTC (31 KB)
[v2] Wed, 11 Sep 2024 16:38:38 UTC (31 KB)
[v3] Tue, 1 Apr 2025 12:49:21 UTC (36 KB)
[v4] Thu, 29 May 2025 03:22:04 UTC (37 KB)
[v5] Thu, 18 Jun 2026 15:43:54 UTC (39 KB)