On the local Hölder boundary smoothness of an analytic function in the unit ball compared with the smoothness of its modulus. (English) Zbl 1436.32023

Summary: 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.


32A40 Boundary behavior of holomorphic functions of several complex variables
26B35 Special properties of functions of several variables, Hölder conditions, etc.
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