L^1-regularity of strong ILB-Lie groups

TL;DR

This paper proves that strong ILB-Lie groups modeled on Fréchet spaces are L^1-regular, meaning certain integrable Lie algebra-valued functions correspond smoothly to curves in the group, with applications to weighted diffeomorphism groups.

Contribution

It establishes L^1-regularity for strong ILB-Lie groups and a broader class of Fréchet-Lie groups, including weighted diffeomorphism groups, extending previous results.

Findings

Proves L^1-regularity for strong ILB-Lie groups.

Extends L^1-regularity to a class of Fréchet-Lie groups.

Provides examples including weighted diffeomorphism groups.

Abstract

If G is a Lie group modeled on a Fr\'echet space, let e be its neutral element and g be its Lie algebra. We show that every strong ILB-Lie group G is L^1-regular in the sense that each f in L^1([0,1],g) is the right logarithmic derivative of some absolutely continuous curve c in G with c(0)=e and the map from L^1([0,1],g) to C([0,1],G) taking f to c is smooth. More generally, the conclusion holds for a class of Fr\'echet-Lie groups considered by Hermas and Bedida. Examples are given. Notably, we obtain L^1-regularity for certain weighted diffeomorphism groups.