The Journal of Geometric Analysis ( IF 1.2 ) Pub Date : 2021-06-19 , DOI: 10.1007/s12220-021-00718-w David Nicolas Nenning 1 , Armin Rainer 1 , Gerhard Schindl 1
A remarkable theorem of Joris states that a function f is \(C^\infty \) if two relatively prime powers of f are \(C^\infty \). 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.
中文翻译:
超微分的非线性条件
里斯的一个显着定理指出,函数˚F是\(C ^ \ infty \) ,如果两个比较主要的力量˚F是\(C ^ \ infty \) 。最近,Thilliez 证明了一个类似的定理在 Roumieu 类型的 Denjoy-Carleman 类中成立。我们证明了除法性质,相当于 Joris 的结果,在各种超微分类中都是有效的。一般来说,它适用于非拟分析类的所有维度。在准分析的情况下,我们在第一维上具有一般有效性,但我们也获得了某些准分析类在所有维度上的有效性。