On quasianalytic local rings

Abstract. This expository article is devoted to the local theory of ultradifferentiable classes of functions, with a special emphasis on the quasianalytic case. Although quasi-analytic classes are well-known in harmonic analysis since several decades, their study from the viewpoint of differential analysis and analytic geometry has begun much more recently and, to some extent, has earned them a new interest. Therefore, we focus on contemporary questions closely related to topics in local algebra. We study, in particular, Weierstrass division problems and the role of hyperbolicity, together with properties of ideals of quasianalytic germs. Incidentally, we also present a simplified proof of Carle-man’s theorem on the non-surjectivity of the Borel map in the quasianalytic case. 1. Ultradifferentiable function germs 1.1. Historical background. At the end of the nineteenth century [6, 7], Borel produced the first non-trivial examples of sets E of infinitely differentiable functions on the real line, containing nowhere analytic functions, and such that any element f in E satisfies the implication (1) (f (j)(0) = 0, j = 0, 1,...) = ⇒ (f = 0). Borel’s examples were typically given by restrictions to the real line of series of rational functions (2) f(z) = ν=