The Exponential Law for Exotic Spaces of Test Functions and Diffeomorphism Groups

We prove the exponential law A(E × F,G) ∼= A(E,A(F,G)) (bornological isomorphism) for the following classes A of test functions: B (globally bounded derivatives), W∞,p (globally p-integrable derivatives), S (Schwartz space), B[M ] (globally Denjoy–Carleman), W [M ],p (Sobolev– Denjoy–Carleman), and S [M ] [L] (Gelfand–Shilov). Here E,F,G are convenient vector spaces (finite dimensional in the cases of W∞,p and W [M ],p) and M = (Mk) is a weakly log-convex weight sequence of moderate growth. As application we give a new simple proof of the fact that the groups of diffeomorphisms DiffB, DiffW∞,p, and DiffS are C∞ Lie groups, and DiffB{M}, DiffW {M},p, and DiffS {L} , for non-quasianalytic M , are C {M} Lie groups, where DiffA = {Id +f : f ∈ A(Rn,Rn), infx∈Rn det(In + df(x)) > 0}. In this connection we also discuss stability under composition.