The Convenient Setting for Quasianalytic Denjoy–carleman Differentiable Mappings

For quasianalytic Denjoy–Carleman differentiable function classes C where the weight sequence Q = (Qk) is log-convex, stable under derivations, of moderate growth and also an L-intersection (see (1.6)), we prove the following: The category of C-mappings is cartesian closed in the sense that C(E,C(F,G)) ∼= C(E × F,G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C-diffeomorphisms is a regular C-Lie group but not better. Classes of Denjoy-Carleman differentiable functions are in general situated between real analytic functions and smooth functions. They are described by growth conditions on the derivatives. Quasianalytic classes are those where infinite Taylor expansion is an injective mapping. That a class of mappings S admits a convenient setting means essentially that we can extend the class to mappings between admissible infinite dimensional spaces E,F, . . . so that S(E,F ) is again admissible and we have S(E × F,G) canonically S-diffeomorphic to S(E,S(F,G)) (the exponential law). Usually this comes hand in hand with (partly nonlinear) uniform boundedness theorems which are easy Sdetection principles. For the C∞ convenient setting one can test smoothness along smooth curves. For the real analytic (C) convenient setting we have: A mapping is C if and only if it is C∞ and in addition C along C-curves (C along just affine lines suffices). We shall use convenient calculus of C∞ and C mappings in this paper; see the book [15], or the three appendices in [17] for a short overview. In [17] we succeeded to show that non-quasianalytic log-convex Denjoy-Carleman classes C of moderate growth (hence derivation closed) admit a convenient setting, where the underlying admissible locally convex vector spaces are the same as for smooth or for real analytic mappings. A mapping is C if and only if it is C along all C -curves. The method of proof there relies on the existence of C partitions of unity. In this paper we succeed to prove that quasianalytic log-convex Denjoy-Carleman classes C of moderate growth which are also L-intersections (see (1.6)), admit a convenient setting. The method consists of representing C as the intersection ⋂ {C : L ∈ L(Q)} of all larger non-quasianalytic log-convex classes C; this is the meaning of: Q is an L-intersection. In (1.9) we construct countably many classes Q which satisfy all these requirements. Taking intersections of derivation closed classes C only, or only of classes C of moderate growth, is not sufficient for yielding the intended results. Thus we have to strengthen many results from [17] before we are able to prove the exponential law. A mapping is C if and only if Date: April 24, 2012. 2000 Mathematics Subject Classification. 26E10, 46A17, 46E50, 58B10, 58B25, 58C25, 58D05, 58D15.