On the Sharpness of the Estimate in a Theorem Concerning the Half Smoothness of a Function Holomorphic in a Ball

Abstract

Let ????ⁿ be the unit ball and Sⁿ be the unit sphere in ℂⁿ, n ≥ 2. Let 0 < α < 1, and define a function f on as follows:

The main result of the paper is the following theorem: the function ζ ↦ |f(ζ)| on the unit sphere Sⁿ belongs to the Hölder class H^α(Sⁿ), while the function f does not belong to the Hölder class for any ε > 0.