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 g, and denote its evolution map by evol:D≡dom[evol]→G, i.e., D⊆C0([0,1],g). We show that evol is C∞-continuous on D∩C∞([0,1],g) if and only if evol is C0-continuous on D∩C0([0,1],g). We furthermore show that G is k-confined for k∈N⊔{lip,∞} 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∞-regular if and only if it is Mackey complete, locally μ-convex, and has Mackey complete Lie algebra – In this case, G is Ck-regular for each k∈N≥1⊔{lip,∞} (with “smoothness restrictions” for k≡lip); as well as C0-regular if G is even sequentially complete, with integral complete Lie algebra.