Lie groups of real analytic diffeomorphisms are $L^1$-regular
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 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 depends smoothly on $[\gamma]$. 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.
- Publication:
-
arXiv e-prints
- Pub Date:
- July 2020
- DOI:
- 10.48550/arXiv.2007.15611
- arXiv:
- arXiv:2007.15611
- Bibcode:
- 2020arXiv200715611G
- Keywords:
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- Mathematics - Functional Analysis;
- 22E65 (primary);
- 28B05;
- 34A12;
- 34H05;
- 46E20;
- 46E40 (secondary)
- E-Print:
- v5: 47 pages