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On quasianalytic local rings. (English) Zbl 1139.32003
Summary: This expository article is devoted to the local theory of ultradifferentiable classes of functions, with a special emphasis on the quasianalytic case. Although quasianalytic classes are well-known in harmonic analysis for 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 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 Carleman’s theorem on the non-surjectivity of the Borel map in the quasianalytic case.

MSC:
32B05 Analytic algebras and generalizations, preparation theorems
26E10 \(C^\infty\)-functions, quasi-analytic functions
46E25 Rings and algebras of continuous, differentiable or analytic functions
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