Lie groups of real analytic diffeomorphisms are L^1-regular

H. Glöckner, ArXiv:2007.15611 (2020).

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Abstract
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 (DFS)-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 $[\gamma]$ in $L^1([0,1],{\mathfrak g})$ has an evolution which is an absolutely continuous $G$-valued function on $[0,1]$ and smooth in $[\gamma]$. As tools for the proof, we develop several new results concerning $L^p$-regularity of infinite-dimensional Lie groups, for $1\leq p\leq \infty$, which will be useful also for the discussion of other classes of groups. Moreover, we obtain new results concerning the continuity and complex analyticity of non-linear mappings on open subsets of locally convex direct limits.
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arXiv:2007.15611
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Glöckner H. Lie groups of real analytic diffeomorphisms are L^1-regular. arXiv:200715611. Published online 2020.
Glöckner, H. (2020). Lie groups of real analytic diffeomorphisms are L^1-regular. In arXiv:2007.15611.
@article{Glöckner_2020, title={Lie groups of real analytic diffeomorphisms are L^1-regular}, journal={arXiv:2007.15611}, author={Glöckner, Helge}, year={2020} }
Glöckner, Helge. “Lie Groups of Real Analytic Diffeomorphisms Are L^1-Regular.” ArXiv:2007.15611, 2020.
H. Glöckner, “Lie groups of real analytic diffeomorphisms are L^1-regular,” arXiv:2007.15611. 2020.
Glöckner, Helge. “Lie Groups of Real Analytic Diffeomorphisms Are L^1-Regular.” ArXiv:2007.15611, 2020.

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