Exponential Laws for Classes of Denjoy-Carleman Differentiable Mappings

Roumieu classes of Denjoy-Carleman type are classically defined by posing growth conditions on the derivatives of smooth functions in terms of a (weight-)sequence of positive numbers. In two papers from 2008 and 2009 P. Michor, A. Rainer and myself were able to extend this notion first for non- quasianalytic classes and then for some quasi-analytic classes to locally convex spaces and to prove that under certain assumptions -- most important moderate growth -- on the weight sequence the corresponding function spaces satisfy exponential laws, in other words one obtains cartesian closed categories. The aim of this project is to extend these results to other types and classes of Denjoy-Carleman differentiable mappings. In particular one can think of: 1. Classes defined by growth conditions of the Fourier transform of compactly supported functions under consideration, as they have been proposed by Beurling 1961 and by Bjorck 1966. 2. Classes of quasianalytic mappings without the strong assumptions needed so far, where we assumed that the quasianalytic class can be written as an intersection of non-quasianalytic strongly log-convex classes. There is some hope that this is possible, since although the realanalytic class cannot be written as intersection in the required way cartesian closedness is nevertheless valid as has been proved by Michor and myself in 1990. 3. Intersections of classes as they have been studied by Chaumat and Chollet 1998, Beaugendre 2001, Schmets and Valdivia 2005 and 2006. 4. Classes of Beurling type, where instead of the existence and all-quantor is used in the definition.

Exponential laws for various classes of smooth mappings have been shown. An exponential law is understood to be an isomorphism between the space of mappings on a product and that of mappings from one factor to the function space on the other factor. The classes treated consist of Denjoy-Carleman differentiable mappings, i.e. mappings satisfying a growth condition on their derivatives. In particular, this has been shown for Denjoy-Carleman classes of Beurling and of Roumieu type, which are described by a weight matrix. This incisomorphismlasses described by a weight sequence or by a weight function.