Abstract
In this article, the notion of ultradifferentiable CR manifold is introduced and an ultradifferentiable regularity result for finitely nondegenerate CR mappings is proven. Here, ultradifferentiable means with respect to Denjoy–Carleman classes defined by weight sequences. Furthermore, the regularity of infinitesimal CR automorphisms on ultradifferentiable abstract CR manifolds is investigated.
Highlights
The primary focus of this article is the study of the regularity of CR mappings
Looking at the literature concerning this problem, one observes that most theorems about the regularity of CR mappings are of a similar form which can be summarised as follows: We consider a CR mapping H between two CR submanifolds M and M with some a priori regularity that extends to a holomorphic mapping defined on a wedge with edge M
If the mapping and/or the manifolds satisfy certain nondegeneracy conditions at some point it is proven that H is of optimal regularity near this point, that is smooth if M and M are smooth, or real analytic if the manifolds are real analytic
Summary
The primary focus of this article is the study of the regularity of CR mappings. Looking at the literature concerning this problem, one observes that most theorems about the regularity of CR mappings are of a similar form which can be summarised as follows: We consider a CR mapping H between two CR submanifolds M and M with some a priori regularity that extends to a holomorphic mapping defined on a wedge with edge M. The finitely nondegenerate condition on the mapping remains unchanged but the holomorphic extension obviously makes no sense in this situation It is replaced in the theorem of Berhanu–Xiao with the assumption that the fibres of the wavefront set of H do not include opposite directions. We are going to call weight sequences, that satisfy these conditions, normal With these results at hand and an M-almost-analytic version of the almost-analytic implicit function theorem used in Lamel [24] and Berhanu–Xiao [3], it is possible to prove the ultradifferentiable version of the regularity result of Lamel: Theorem 1.1 Let M be a normal weight sequence and M ⊆ CN , M ⊆ CN be two generic ultradifferentiable submanifolds of class {M}, p0 ∈ M, p0 ∈ M and H : (M, p0) → (M , p0) a Ck0 -CR mapping that is k0-nondegenerate at p0. It is possible to overcome this obstacle, but for that we have to work in a far more general setting, which will be done in a forthcoming paper
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