We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the ∂-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensions and of the FBI transform. This allows to extend its definition to ultradifferentiable manifolds. We also discuss ultradifferentiable versions of the elliptic regularity theorem and obtain a general quasianalytic Holmgren uniqueness theorem.

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Abstract. We study the regularity of infinitesimal CR automorphisms of abstract CR structures which possess a certain microlocal extension and show that there are smooth multipliers, completely determined by the CR structure, such that if X is such an infinitesimal CR automorphism, then λX is smooth for all multipliers λ. As an application, we study the regularity of infinitesimal automorphisms of certain infinite type hypersurfaces in C n .

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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.