The Regularity Problem for Lie Groups with Asymptotic Estimate Lie Algebras

Abstract

We solve the regularity problem for Milnor's infinite dimensional Lie groups in the asymptotic estimate context. Specifically, let $G$ be a Lie group with asymptotic estimate Lie algebra $\mathfrak{g}$, and denote its evolution map by $\mathrm{evol}:\mathrm{D}\equiv \mathrm{dom}[\mathrm{evol}]\to G$, i.e., $\mathrm{D}\subseteq C^0([0,1],\mathfrak{g})$. We show that $\mathrm{evol}$ is $C^{\infty }$-continuous on $\mathrm{D}\cap C^{\infty }([0,1],\mathfrak{g})$ if and only if $\mathrm{evol}$ is $C^0$-continuous on $\mathrm{D}\cap C^0([0,1],\mathfrak{g})$. We furthermore show that $G$ is k-confined for $k\in \mathbb{N}\bigsqcup \{\mathrm{lip},\infty \}$ if $G$ is constricted. (The latter condition is slightly less restrictive than to be asymptotic estimate.) Results obtained in a previous paper then imply that an asymptotic estimate Lie group $G$ is $C^{\infty }$-regular if and only if it is Mackey complete, locally $\mu$-convex, and has Mackey complete Lie algebra - In this case, $G$ is $C^k$-regular for each $k\in {\mathbb{N}}_{\ge 1}\bigsqcup \{\mathrm{lip},\infty \}$ (with ``smoothness restrictions'' for $k\equiv \mathrm{lip}$), as well as $C^0$-regular if $G$ is even sequentially complete with integral complete Lie algebra.

27 pages. Version as published at Indagationes Mathematicae (title refined; presentation improved; proof of Lemma 9 revised). arXiv admin note: text overlap with arXiv:1711.03508