# Importance of Denjoy-Carleman classes as a class.

Denjoy-Carleman classes of differentiable functions, say in Roumieu's form:

Given a log-convex sequence $M_n$ of positive number denote by $C_M=C_M(\mathbb{R}^n,0)$ the ring of germs of $C^\infty(\mathbb{R}^n)$ functions such that for every compact $K$ there are constants $A$ and $B$ such that $|D^{\alpha}f(x)|\leq AB^{|\alpha|}M_{|\alpha|}$ for every x in $K$ and every multi-index $\alpha$.

In what ways are the $C_M$ important as a classes? Especially in the case in which of $C_M$ is quasianalytic. These sets of smooth functions satisfy a number of properties (perhaps adding extra conditions on the sequence $M$): are local rings, closed under composition, differentiation, implicit function theorem, ... These properties make them important, and the definition is perhaps natural, so that makes them important in a sense. Now, is there some context in which $C_M$ are precisely the solution to some problem? This is what I mean by importance of $C_M$ as a class.

Suppose we allocate all quasianalytic functions (all functions belonging to some quasianalytic $C_M$ for some $M$) in classes in some other way that still satisfy all desirable properties above and some others. If there is some sense in which the $C_M$ is specially relevant as a class it is not to be expected that some new subdivision of the quasianalytic functions is also going to have that property.

[I put tags according to where I imagine there are experts with an answer to this question, instead of just the nature of the objects in the question. Is this a valid way of using tags?]

Denjoy-Carleman differentiable functions are also called ultradifferentiable functions of type $C^M$. The spaces dual to them are called ultra distributions; they are larger than spaces of distributions, and they contain linear functionals of infinite order. Dual spaces of real analytic function spaces are called hyper functions. They are of importance for PDE-theory. See in particular the Japanese school around Komatsu for that.
One more result: For a compact real analytic manifold the group of $C^M$-diffeomorphisms is an infinite dimensional regular $C^M$-Lie group, for an extension of the $C^M$-class to a convenient setting in infinite dimensions, but not better. Here $M$ must be log-convex and of moderate growth. See 1, 2, and 3 for these results.