The convenient setting for quasianalytic Denjoy–Carleman differentiable mappings

Abstract

For quasianalytic Denjoy-Carleman differentiable function classes C Q where the weight sequence Q = (Q k ) 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 Q -mappings is cartesian closed in the sense that C Q (E, C Q (F, G)) ∼ = C Q (E × F, G) for convenient vector spaces. Applications to manifolds of mappings are given: The group of C Q -diffeomorphisms is a regular C Q -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.

Denjoy–Carleman Differentiable Perturbation of Polynomials and Unbounded Operators

“…(E) and (F) have been proved in [14] in 2003. (G) was proved in [19, 7.1]; it can be proved as (H) with some obvious changes, but it is not a special case since C ω does not correspond to a sequence which is an L-intersection (see 'definitions and remarks' below and [17]). (J) and (K) were proved in [19, 7.1].…”

“…Then C Q and C L are closed under composition and allow for the implicit function theorem. See [17] or [16] and references therein.…”

Iterates of differential operators and vector valued functions on non quasi analytic classes

“…Kriegl, Michor and Rainer [44] have studied recently non-quasianalytic curves f : R → E with values in a locally convex space E in the setting of non-quasianalytic classes of Denjoy-Carleman type.…”

“…We characterize when the three definitions of vector valued {ω}-ultradifferentiable function given in the previous section coincide if E is a Fréchet space or a complete LBspace. We start with the following example due to Kriegl, Michor and Rainer [44].…”

“…Let ω(t) = t 1/α , s := 1/α > 1, be a Gevrey weight. There are curves in E {ω} (R, E) which are not in E t {ω} (R, E) by [44,Example 3.2]. This fact is based on the existence of f ∈ E {ω} (R) such that…”