If |$F$| is a |$C^{\infty }$| function whose composition |$F \circ \sigma $| with a blowing-up |${\sigma }$| belongs to a Denjoy–Carleman class |$C_M$|⁠, then |$F$|⁠, in general, belongs to a larger class |$C_{M^{(2)}}$|⁠; that is, there is a loss of regularity. We show that this loss of regularity is sharp. In particular, the loss of regularity of Denjoy–Carleman classes is intrinsic to arguments involving resolution of singularities.

中文翻译：

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Denjoy–Carleman类中的吹扫函数的清晰估计

如果| $ F $ | 是| $ C ^ {\ infty} $ | 函数的组成| $ F \ circ \ sigma $ | 用吹起| $ {\西格玛} $ | 属于Denjoy–Carleman类| $ C_M $ |⁠，然后| $ F $ |⁠通常属于较大的类| $ C_ {M ^ {（2）}} $ |⁠ ; 也就是说，失去了规律性。我们证明这种规律性的损失是明显的。尤其是，Denjoy-Carleman类的正则性损失是涉及奇点解析的论点所固有的。