Abstract
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.
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Acknowledgements
The author is kindly grateful to his scientific adviser academician Sergei V. Kislyakov for having posed the problem, for a number of helpful suggestions and for help in preparation of this article. We would also like to thank anonymous referees for their careful reading of the paper and for a number of helpful suggestions.
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Communicated by Mieczyslaw Mastylo.
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Vasilyev, I. On the Local Hölder Boundary Smoothness of an Analytic Function in the Unit Ball Compared with the Smoothness of Its Modulus. J Fourier Anal Appl 26, 28 (2020). https://doi.org/10.1007/s00041-020-09735-9
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DOI: https://doi.org/10.1007/s00041-020-09735-9