Local boundary smoothness of an analytic function f in the unit ball of \({\mathbb {C}}^n\) is compared to the smoothness of its modulus. We prove that in dimensions 2 and higher two different (and natural) conditions imposed on the zeros of f imply two different drops of its smoothness compared to the smoothness of |f|. We also show that some of the drops are the best possible.

中文翻译：

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单位球中解析函数的局部Hölder边界光滑度与其模量的光滑度比较

将\（{\ mathbb {C}} ^ n \）的单位球中解析函数f的局部边界平滑度与其模量的平滑度进行比较。我们证明在2维和更高维上，施加于f的零的两个不同（自然）条件意味着与|的平滑度相比，其平滑度下降了两个不同。f |。我们还表明，有些滴是最好的。