论文标题
真实分析性差异的谎言群体为$ l^1 $ regular
Lie groups of real analytic diffeomorphisms are $L^1$-regular
论文作者
论文摘要
让$ m $成为紧凑的，真实的分析歧管，$ g $是$ m $的所有真实分析差异的谎言组，在$ m $上的真实分析矢量字段的空间$ {\ mathfrak g} $上进行了建模。我们研究了$ m $上的时间依赖性实用分析矢量场的流，这些介于$ m $，这是及时的集成函数及其对时间依赖性矢量场的依赖。值得注意的是，我们表明，从$ l^1中的每个$ [γ] $（[0,1]，{\ mathfrak g}）$中的每个$ [γ] $的意义上说，Lie Group $ g $是$ l^1 $ regular，{\ mathfrak g}）$具有一个演变，这是一个绝对连续的$ g $ g $ valuew函数在$ [0,1]上，并且在$ [0,1]上依赖于$ [γ] $。作为证明的工具，我们开发了有关无限维二比lie组的$ l^1 $定型性的新结果，以及有关非线性映射在局部凸出直接限制上的连续性和复杂分析性的新结果。
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.
