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.

Abstract. We study the set D n,β (R d ) of orientation preserving diffeomorphisms of R d which differ from the identity by a Hölder C n,β 0 -mapping, where n ∈ N ≥1 and β ∈ (0, 1]. We show that with its natural Fréchet topology) andwith its natural inductive locally convex topology) however are C 0,ω Lie groups for any slowly vanishing modulus of continuity ω. In particular, D n,β− (R d ) is a topological group and a so-called half-Lie group (with smooth right translations). We prove that the Hölder spaces C n,β 0 are ODE closed, in the sense that pointwise time-dependent C n,βFor the latter and n ≥ 2, we show that the flow mapAs an application we prove that the corresponding Trouvé group G n,β (R d ) from image analysis coincides with the connected component of the identity of D n,β (R d ).

A remarkable theorem of Joris states that a function f is $$C^\infty $$ C ∞ if two relatively prime powers of f are $$C^\infty $$ C ∞ . Recently, Thilliez showed that an analogous theorem holds in Denjoy–Carleman classes of Roumieu type. We prove that a division property, equivalent to Joris’s result, is valid in a wide variety of ultradifferentiable classes. Generally speaking, it holds in all dimensions for non-quasianalytic classes. In the quasianalytic case we have general validity in dimension one, but we also get validity in all dimensions for certain quasianalytic classes.

The Borel problem for Denjoy-Carleman and Braun-Meise-Taylor classes has well-known optimal solutions. The unified treatment of these ultradifferentiable classes by means of one-parameter families of weight sequences allows to compare these optimal solutions. We determine the relations among them and give conditions for their equivalence in the Roumieu case.

The Trouvé group G A from image analysis consists of the flows at a fixed time of all time-dependent vectors fields of a given regularity A(R d , R d ). For a multitude of regularity classes A, we prove that the Trouvé group G A coincides with the connected component of the identity of the group of orientation preserving diffeomorphims of R d which differ from the identity by a mapping of class A. We thus conclude that G A has a natural regular Lie group structure. In many cases we show that the mapping which takes a timedependent vector field to its flow is continuous. As a consequence we obtain that the scale of Bergman spaces on the polystrip with variable width is stable under solving ordinary differential equations. Theorem 2.4 ([4, Theorem 2.2]). A function f : I → E is measurable by seminorm if and only if f is weakly measurable and for each p ∈ P there exists a nullset N p ⊆ I such that f (I \ N p ) is separable. For a simple function f : I → E, the integral over a measurable set J ⊆ I is clearly defined as J f (t) dt := y∈E λ(f −1 ({y}) ∩ J) · y, where λ is the Lebesgue measure on I. A function f : I → E which is measurable by seminorm is called integrable by seminorm if ∀p ∈ P, ∀n ∈ N : p • (f p n − f ) ∈ L 1 (I), ∀J ⊆ I measurable ∃F J ∈ E ∀p ∈ P : p F J − J f p n (t) dt n→∞ −→ 0;in this case F J =: J f (t) dt. For a complete locally convex space E, integrability of p • f for each p ∈ P already implies integrability by seminorm. If E is a Banach space, then clearly integrability by seminorm coincides with Bochner integrability and the respective integrals coincide. It is easily seen that for each p ∈ P,