On moduli of continuity of a conjugate function of several variables (Q1358412)

Let \(R^N\) be the \(N\)-dimensional Euclidean space and \(R^1= R\). The author investigates the moduli of continuity of a conjugate function of several variables. Main result of the paper is Theorem 3. In this theorem, the author generalizes the result reported by Lekishvili on the character of violation of invariance of the class \(\text{Lip}(\alpha, C(T^N))\) \((0<\alpha<1)\) with respect to the many-dimensional conjugation operator acting in this class [\textit{M. M. Lekishvili}, Mat. Zametki 23, 361-372 (1978; Zbl 0418.42009)]. He considers the general class \(\{f\in C(T^N):\omega(f, \delta)= O([\varphi(\delta)])\}\), where \(\varphi(\delta)\) is a modulus of continuity and \(\omega(f,\delta)\) is the total modulus of continuity of a function \(f(x)\). If \(N=1\) Theorem 3 coincides with a result of Bari and Stechkin.