取值于无限维流形的绝对连续函数流形与半李群的正则性性质
Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups
AI总结：
本文为取值于无限维流形的绝对连续函数定义了光滑流形结构，并证明了右半李群Diff_K^r(R)和Diff^r(M)是L^p-半正则的，其演化映射连续。
摘要：
对于$p\in [1,\infty]$，我们为所有实数$a<b$和每个具有局部加法的、建模在序列完备局部凸拓扑向量空间上的光滑流形$N$，在绝对连续函数$\gamma\colon [a,b]\to N$（具有$L^p$导数）的集合$AC_{L^p}([a,b],N)$上定义了一个光滑流形结构。讨论了绝对连续函数空间之间的自然映射的光滑性，例如对于光滑映射$f\colon N_1\to N_2$，叠加算子$AC_{L^p}([a,b],N_1)\to AC_{L^p}([a,b],N_2)$，$\eta\mapsto f\circ \eta$。对于$1\leq p <\infty$和$r\in \mathbb{N}$，我们证明了右半李群$\text{Diff}_K^r(\mathbb{R})$和$\text{Diff}^r(M)$是$L^p$-半正则的。这里$K$是$\mathbb{R}$的紧子集，$M$是紧致光滑流形。一个$L^p$-半正则半李群$G$允许一个演化映射$\text{Evol}:L^p([0,1],T_e G)\to AC_{L^p}([0,1],G)$，其中$e$是$G$的单位元。对于前面的例子，演化映射$\text{Evol}$是连续的。
英文摘要：
For $p\in [1,\infty]$, we define a smooth manifold structure on the set $AC_{L^p}([a,b],N)$ of absolutely continuous functions $γ\colon [a,b]\to N$ 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 $AC_{L^p}([a,b],N_1)\to AC_{L^p}([a,b],N_2)$, $η\mapsto f\circ η$, 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
