实解析微分同胚的李群是 $L^1$-正则的
Lie groups of real analytic diffeomorphisms are $L^1$-regular
AI总结：
本文证明紧致实解析流形的实解析微分同胚李群具有 $L^1$-正则性，并发展无限维李群及局部凸直极限非线性映射的相关工具结果。
摘要：
设 $M$ 为紧致实解析流形，$G$ 为 $M$ 的所有实解析微分同胚构成的李群，其建模空间是 $M$ 上实解析向量场的空间 $\mathfrak{g}$。我们研究 $M$ 上依赖时间的实解析向量场的流，这些向量场在时间上是可积函数，并研究流对这种依赖时间的向量场的依赖关系。值得注意的是，我们证明李群 $G$ 是 $L^1$-正则的，即 $L^1([0,1],\mathfrak{g})$ 中的每个 $[\gamma]$ 都有一个演化，该演化是 $[0,1]$ 上取值于 $G$ 的绝对连续函数，并光滑地依赖于 $[\gamma]$。作为证明工具，我们建立了关于无限维李群 $L^1$-正则性的新结果，以及关于局部凸直极限上非线性映射的连续性和复解析性的新结果。
英文摘要：
Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all real-analytic diffeomorphisms of $M$, which is modelled on the space ${\mathfrak g}$ of real-analytic vector fields on $M$. We study flows of time-dependent real-analytic vector fields on $M$ which are integrable functions in time, and their dependence on the time-dependent vector field. Notably, we show that the Lie group $G$ is $L^1$-regular in the sense that each $[γ]$ in $L^1([0,1],{\mathfrak g})$ has an evolution which is an absolutely continuous $G$-valued function on $[0,1]$ and depends smoothly on $[γ]$. As tools for the proof, we develop new results concerning $L^1$-regularity of infinite-dimensional Lie groups, and new results concerning the continuity and complex analyticity of non-linear mappings on locally convex direct limits.
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